optimisation of periodic flight trajectories · optimisation of periodic flight trajectories ....

TECHNISCHE UNIVERSITÄT MÜNCHEN

Lehrstuhl für Flugsystemdynamik

Optimisation of Periodic Flight Trajectories

Jakob Schoppe Maximilian Lenz

Vollständiger Abdruck der von der Fakultät für Maschinenwesen der Technischen Universität München zur Erlangung des akademischen Grades eines

Doktor-Ingenieurs

genehmigten Dissertation.

Vorsitzender: Univ.-Prof. Dr.-Ing. Mirko Hornung Prüfer der Dissertation: 1. Univ.-Prof. Dr.-Ing. Dr. h.c. Gottfried Sachs

2. Univ.-Prof. Dr. rer. nat. habil. Matthias Gerdts 3. Univ.-Prof. Dr.-Ing. Florian Holzapfel

Die Dissertation wurde am 23. 12. 2014 bei der Technischen Universität München einge-reicht und durch die Fakultät für Maschinenwesen am 29. 04. 2015 angenommen.

http://www.unibw.de/lrt1/gerdts/mitarbeiter/prof.-gerdts

Danksagung Die vorliegende Arbeit entstand wärend meiner Zeit an der Technischen Universität München als wissenschaftlicher Mitarbeiter am Lehrstuhl für Flugmechanik und Flugregelung, aus dem nach der Emeritierung von Professor Sachs der Lehrstuhl für Flugsystemdynamik unter dem Ordinariat von Professor Holzapfel hervorging. Ich danke Herrn Professor Sachs für sein Engagement, sein Interesse und seine Begeisterung, die er vielen der in dieser Arbeit behandelten Themen entgegengebracht hat und mir so eine sehr große Unterstützung war. Bei Herrn Professor Gerdts möchte ich mich für die vielfältige Hilfe, die er als Mathematiker, sowohl mir persönlich als auch unter anderem im Rahmen von Munich Aerospace der gesamten Optimierungsgruppe des Lehrstuhls, geleistet hat. Herrn Professor Holzapfel gilt mein Dank für die Beibehaltung der Optimierungsgruppe als elemtarem Bestandteil des Lehrstuhls sowie für die über all die Jahre geleistete Unterstützung. Herrn Professor Hornung möchte ich herzlich für die Führung des Prüfungsvorsitzes danken. München, im Dezember 2014 Jakob Lenz

Table of Contents

Nomenclature .....................................................................................................................................9

CHAPTER 1 INTRODUCTION ............................................................................... 19

1.1 Organisation of the Thesis ..................................................................................................... 19

1.2 Periodic Optimal Control ....................................................................................................... 23

1.3 Contributions of the Thesis .................................................................................................... 27

CHAPTER 2 FUNDAMENTALS OF FLIGHT MECHANICS .................................. 30

2.1 Equations of Motion .............................................................................................................. 30 2.1.1 Introductory Thoughts and Validity ....................................................................................... 30 2.1.2 Position .................................................................................................................................. 31 2.1.3 Translation ............................................................................................................................. 36 2.1.4 Attitude .................................................................................................................................. 43 2.1.5 Rotation ................................................................................................................................. 45 2.1.6 Application ............................................................................................................................. 50

2.2 External Forces and Moments ............................................................................................... 51 2.2.1 Gravity ................................................................................................................................... 51 2.2.2 Aerodynamics ........................................................................................................................ 68 2.2.3 Propulsion .............................................................................................................................. 69

2.3 Atmosphere........................................................................................................................... 72 2.3.1 Static Atmosphere ................................................................................................................. 72 2.3.2 Wind ....................................................................................................................................... 74

2.4 Modelling of the Receivable Solar Radiation within the Earth’s Atmosphere .......................... 79 2.4.1 Introduction to Celestial Mechanics ...................................................................................... 79 2.4.2 Radiation on the Earth’s Orbit ............................................................................................... 85 2.4.3 Determination of the Solar Zenith Angle ............................................................................... 93 2.4.4 Atmospheric Influences ....................................................................................................... 103 2.4.5 Results ................................................................................................................................. 111

CHAPTER 3 APPLIED OPTIMAL CONTROL ..................................................... 118

3.1 The Parameter Dependent Optimal Control Problem ........................................................... 119 3.1.1 General Problem Formulation ............................................................................................. 119 3.1.2 Shooting Methods ............................................................................................................... 122 3.1.3 Optimisation Framework ..................................................................................................... 126 3.1.4 Mesh Refinement................................................................................................................. 128

3.2 Function Evaluator .............................................................................................................. 129 3.2.1 Automatic Differentiation of Model / Constraints ................................................................ 129 3.2.2 Sensitivity Equations ........................................................................................................... 132 3.2.3 Scaling ................................................................................................................................. 137 3.2.4 Structure / Sparsity.............................................................................................................. 138

3.3 Post Optimal Sensitivity Analysis ......................................................................................... 140 3.3.1 First Order Sensitivity of Optimal Solution .......................................................................... 141 3.3.2 First and Second Order Sensitivity of Cost Function .......................................................... 143 3.3.3 Linear Change in Constraints or Cost Function .................................................................. 144

3.4 Bilevel Optimal Control Tasks .............................................................................................. 145 3.4.1 Analysis of Change of Structure of the Solution ................................................................. 147 3.4.2 Upper Level – Lower Level .................................................................................................. 148 3.4.3 Periodic Optimisation .......................................................................................................... 149

CHAPTER 4 PERIODIC OPTIMAL CONTROL FOR FLIGHT APPLICATIONS . 152

4.1 Unlimited Endurance Missions – Solar Aircraft .................................................................... 152 4.1.1 Modelling a Solar Aircraft .................................................................................................... 154 4.1.2 Optimal Periodic Trajectory ................................................................................................. 157 4.1.3 Results for Altitude Limitations............................................................................................ 160 4.1.4 No Upper Altitude Limit ....................................................................................................... 168

4.2 Configuration Changes – Powered Glider with Retractable Engine ...................................... 175 4.2.1 Modelling the Motor Gliders ................................................................................................ 177 4.2.2 Saw-Tooth Flight as Periodic Optimal Control Problem ..................................................... 184 4.2.3 Propeller – Piston Engine .................................................................................................... 187 4.2.4 Propeller – Electric Engine................................................................................................... 203 4.2.5 Jet Engine ............................................................................................................................ 207

4.3 Bounding Flight – Siskin ...................................................................................................... 212 4.3.1 Modelling a Generic Siskin .................................................................................................. 217 4.3.2 Bounding Flight as Periodic Optimal Control Problem ....................................................... 223 4.3.3 Initial Guess from Parabola ................................................................................................. 224 4.3.4 Reference Trajectories for One Cycle ................................................................................. 237 4.3.5 Multiple Cycles .................................................................................................................... 252 4.3.6 Bilevel Optimisation for Solution Structure Analysis ........................................................... 255

CHAPTER 5 CONCLUSION AND PERSPECTIVES ........................................... 259

APPENDIX ............................................................................................................. 266

A. Modelling of the Earth (flat – round – rotating) ......................................................................... 266

B. Quaternion Representation of Attitude .................................................................................... 270

C. Vector Field Classifications ..................................................................................................... 273

D. Matrix Differential Calculus ...................................................................................................... 276

E. Constraints and Optimisation Parameter Vector ...................................................................... 278

F. Jacobian and Hessian Matrix .................................................................................................. 280

G. Solar Cells and Maximum Power Point Trackers ..................................................................... 282

H. The Propulsion System of the Antares 20E Motor Glider ......................................................... 286

BIBLIOGRAPHY .................................................................................................... 288

9

Nomenclature Vector and Matrix Representation To allow a better distinction, vectors and matrices are given in bold type. Vectors are mostly represented by small letters and round brackets whilst for matrices capital letters and square brackets are used. Three-dimensional Cartesian vectors are additionally marked by an arrow above. The following naming convention is used (2nd derivative of position vector as example): ( ) ( ) 54

213: rr EO

KK

1 Point, the motion of which is described or at which the acting forces and moments are acting: The vector represents the 2nd derivative to time of the position of the centre of gravity, .

2 System, in which the vector is notated: The notation system is the intermediate kinematic frame, K .

3 Type of velocity or acceleration or origin of acting forces and moments: The vector represents the kinematic acceleration, K .

4 Reference system for the 1st derivative to time: The first derivative of the position of the centre of gravity (velocity) is with respect to the earth-centred earth-fixed frame, E .

5 Reference system for the 2nd derivative to time: The first derivative of the velocity of the centre of gravity (acceleration) is with respect to the north-east-down frame, O .

In case of two entries at position one, the relative position of the second to the first entry is represented. In case of angular velocities, at position one there are always two entries. System number two rotates around system number one.

Points – Abbreviations used Symbol Explanation

0 Origin – centre of the earth A Aerodynamic reference point G Centre of gravity P Arbitrary point R Reference point T Thrust reference point

Coordinate Systems – Abbreviations used Symbol Explanation

I Earth-centred inertial frame E Earth-centred earth-fixed frame O North-east-down frame

10 Nomenclature

Coordinate Systems – Abbreviations used Symbol Explanation

NED North-east-down frame K Kinematic frame K Intermediate (rotated) kinematic frame A Aerodynamic frame A Intermediate (rotated) aerodynamic frame S Stability axes frame B Body-fixed frame N Navigation frame

NAV Navigation frame T Thrust reference frame P Arbitrary frame fixed on aircraft

ROT Frame rotating around its x-axis in otherwise fixed position to B U North-west-up frame

NWU North-west-up frame

Types of Velocity/ Acceleration and Origin of Forces/ Moments Abbreviations used

Symbol Explanation K Kinematic A Aerodynamic W Wind P Propulsive G Gravity T Total

Other Coordinate Systems used

Coordinate Systems Symbol Explanation Unit

Spherical Coordinates ( )λϑ,,r r Radius vector [m] ϑ Polar distance [rad] λ Geocentric longitude [rad]

Geodetic Coordinates ( )h,,ϕλ

λ Geodetic longitude [rad] ϕ Geodetic latitude [rad] h Altitude [m]

Ellipsoidal Harmonic Coordinates ( )λβ ,,u u Semi-minor axis [m] β Reduced latitude [rad] λ Geocentric longitude [rad]

Nomenclature 11

Acronyms and Symbols used

Latin Small Letters Symbol Explanation Unit

a Acceleration [m/s2] a Speed of sound [m/s] a Semi-major axis [m] b Semi-minor axis [m] b Wing span [m] b Factor for induced lift coefficient [-] c Polar radius of curvature [m] c Mean aerodynamic chord [m] d Distance [m] d Layer thickness/ height [m] e Factor for zero lift coefficient [-] e First eccentricity [-] e Equator e Unit vector [-] f Flattening [-] f Factor [-]

sg Standard acceleration due to gravity at sea level [m/s2]

g Gravity vector [m/s2] h Altitude [m] h Altitude of geodetic coordinates ( )h,,ϕλ [m]

ek Coulomb's constant [V·m/(A·s)] l Distance [m] m Mass [kg]

Coordinate Systems Symbol Explanation Unit

Keplerian Orbital Elements a Semi-major axis of the orbit [m] e First eccentricity of the orbit [-] Ω Longitude of the ascending node of the ecliptic [rad] ψ Ecliptical longitude of the perihelion [rad] i Inclination of the ecliptic [rad]

Pω Argument of perihelion [rad] tP Time of perihelion passage [s] µ True anomaly [rad] Astronomic Coordinate System α Right ascension [rad] δ Declination [rad]

ee Obliquity of the ecliptic [rad]

Sunλ Ecliptic longitude [rad]

12 Nomenclature

Latin Small Letters Symbol Explanation Unit

n Coefficient [-] n Load factor [-] n Normal [-] p Distance to polar axis [m] p Pressure [N/m2] p Pole p ( )

BOB

xK ,ω= Roll rate [rad/s] p Parameter vector p Linear momentum [kg·m/s] q Point charge [A·s] q ( )

BOB

yK ,ω= Pitch rate [rad/s]

q, q0, q’ Abbreviations [-] q Dynamic pressure [N/m2] q0, q1, q2, q3 Quaternions [-] r Position [m] r Radius vector of spherical coordinates ( )λϑ,,r [m] r Distance between focal point and point on ellipse [m] r ( )

BOB

zK ,ω= Yaw rate [rad/s]

r Radius vector [m] r Vector of residuals/ interior point conditions s Variable s Semi span [m] s Distance [m] t Time [s]

u Semi-minor axis of ellipsoidal harmonic coordinates ( )λβ ,,u [m]

u Velocity in x-direction [m/s] u Vector of controls v Velocity in y-direction [m/s] w Abbreviation for normalisation to unit vector [-] w Velocity in z-direction [m/s] x Position [m] x State vector y Position [m] y Output vector z Position [m] z Optimisation parameter vector

Latin Capital Letters Symbol Explanation Unit

A Aphelion [-] C Force or moment coefficient [-] C Constraints vector D Aerodynamic drag [N] D Downward D Duplication matrix

Nomenclature 13


E Linear eccentricity [m] E Irradiance [W/m2] E Energy [J] F Focal point [-] F Function of ellipsoid of revolution [-] F Objective function vector F Vector field F Force [kg·m/s2]

G Newtonian gravitational constant [m3/(kg·s2)] GM Gravitational constant of earth [m3/s2] H Geopotential altitude [m] H Hessian matrix H

Angular momentum [kg·m2/s] I Unity matrix I Active set of constraints

RI Inertia Tensor [kg·m2] J Jacobian matrix J Cost functional K Commutation matrix L Aerodynamic lift [N] L Aerodynamic rolling moment [N·m] L Rolling moment [N·m] L Elimination matrix L Lagrange function M Aerodynamic pitching moment [N·m] M Mach number [-] M Mean anomaly [rad] M Pitching moment [N·m] M

Moment [kg·m2/s2]

ABM Transformation matrix from frame B to frame A [-]

ϕM Meridian radius of curvature [m] N Aerodynamic yawing moment [N·m] N Yawing moment [N·m]

ϕN Radius of curvature in the prime vertical [m]

( )nxO Of order x to the power of n P Power [W] P Perihelion [-] Q Aerodynamic sideforce [N] R Resultant force, total aerodynamic force [N] R Radius [m] R Gas constant [J/(kg·K)] Re Reynolds number [-] S Surface [m2] S Reference surface [m2] S Wing Reference Area [m2] S Sensitivity matrix

'S Second order sensitivity matrix

14 Nomenclature


T Thrust [N] T Temperature [K] T Period [s] U Potential of gravity [N·m/kg] V Velocity [m/s] V Potential of gravitation [N·m/kg] W Potential of gravity for geoid [N·m/kg] W Wind X Force into x-direction [N] Y Force into y-direction [N] Z Force into z-direction [N] Z Zenith angle, zenith distance [rad]

Latin Letters Acronym Explanation Unit

Abs Absolute AC Alternating current [A] act Active Ae Aerosols AE Autumnal equinox AM Air mass/optical mass (both normalised) [-] APH Aphelion passage ATM Air traffic management AU Astronomical unit [m] aux Auxiliary quantity [-] Bat Battery BFGS Broyden-Fletcher–Goldfarb-Shanno bound Bounding CF Cost function CMD Commanded con Configuration change CPV Concentrator photovoltaics CV Co-vertex passage cyc Cycle DC Direct current [A] DENMRA Density function-based mesh refinement algorithm diff Diffuse diff Difference disc Discretised DLST Day-light saving time [h] dof Degrees of freedom eE Equal Energy EGM Earth gravitational model el Electric EMB Earth moon barycentre EN Equinoctes Eng Engine EOT Equation of time [min]

Nomenclature 15


eq Equality ext Extended ext Extra-terrestrial EXT Extremum fin Final HALE High altitude long endurance hor Horizontal INC Incremental conductance ineq Inequality ini Initial ISA International standard atmosphere IVP Initial value problem JD Julian Date [d] KKT Karush–Kuhn–Tucker Lag Lagrange lin linear LST Local standard time [h] Mp Moving particle MPP Maximum power point MPPT Maximum power point tracker Mr Multiple reflections MS Multiple shooting MSL Mean sea level MST Mean solar time [h] Na Non-absorbed NAV Navigation NLP Non-linear program OCP Optimal control problem ODE Ordinary differential equation opt Optimum/ optimal P&O Perturb-and-observe per Periodic PER Perihelion passage PERL Passivated emitter rear locally diffused Ph Phase PIV Particle image velocimetry

PMOD Physikalisch-Meteorologisches Observatorium Davos (=WRC)

Prop Propeller PV Photovoltaics QP Quadratic programming Ra Rayleigh scattering ref Reference retr Retracted RPM Revolutions per minute [1/min] SC Solar constant [W/m2] SEM Standard error of mean SL Sea level SNOPT Sparse Nonlinear OPTimizer

16 Nomenclature


SQP Sequential quadratic programming SS Summer solstice ST Solstices trans Transition TSI Total solar irradiance [W/m2] UAS Unmanned aerial system UAV Unmanned aerial vehicle VE Vernal equinox ver Vertical VIS Visibility [km] VLBI Very long baseline interferometry Vol Volume [m3] WGS 84 World Geodetic System 1984 WRC World Radiation Centre (=PMOD) WS Winter solstice

Greek Small Letters Symbol Explanation Unit

α Angle of attack [rad]

solarα Solar azimuth angle [rad] α Right ascension [rad] β Angle of sideslip [rad] β Eccentric anomaly [rad]

β Reduced latitude of ellipsoidal harmonic coordinates ( )λβ ,,u [rad]

γ Temperature gradient [K/m] γ Climb angle [rad]

λ Geocentric longitude of ellipsoidal harmonic coordinates ( )λβ ,,u [rad]

Tδ Fraction of total thrust available [-] δ Declination [rad] e Glide number [-]

ee Obliquity of the ecliptic [rad]

ζ Rudder deflection [rad] η Elevator deflection [rad] η Efficiency factor [-] ϑ Polar distance of spherical coordinates ( )λϑ,,r [rad] θ Elevation angle (of sun) [rad] κ Isentropic exponent of air [-] λ Wavelength [m] λ Geodetic longitude of geodetic coordinates ( )h,,ϕλ [rad]

λ Geocentric longitude of spherical coordinates ( )λϑ,,r [rad]

Sunλ Ecliptic longitude [rad] µ Bank angle [rad]

Nomenclature 17

Greek Small Letters Symbol Explanation Unit

μ Lagrange multipliers µ Dynamic viscosity [kg/(s·m)] ν Kinematic viscosity [m2/s] ξ Aileron deflection [rad] ρ Density [kg/m3] τ Normalised time [-] τ Transmission factor/ radiation factor [-] ϕ Geodetic latitude of geodetic coordinates ( )h,,ϕλ [rad] φ Potential χ Course angle [rad] ψ Constraints vector ψ Mean longitude of perihelion [deg] ω Hour angle [rad]

Pω Argument of perihelion [deg] ABω Angular rate/ velocity between frames B and A [rad/s]

Greek Capital Letters Symbol Explanation Unit

Ψ Azimuth angle [rad] Θ Inclination/ pitch angle [rad] Φ Bank angle [rad]

ABΦ Matrix representation of vector cross product Φ Potential of the centrifugal force [N·m/kg] ( )BB

ABΩ Matrix representation of angular rate [rad/s] Ω Mean longitude of the ascending node [deg]

Numbers Number Explanation Unit

0 Reference/ on the reference surface 0 Lift independent 0 Initial/ nominal 1 Final

19

Chapter 1

Introduction

1.1 Organisation of the Thesis The thesis at hand starts with a detailed introduction followed by the chapter Fundamentals of Flight Mechanics. In this second chapter, first, the Equations of Motion are derived in state space representation. For a sophisticated analytical treatment of the equations of motion a rigid body is defined the density of which may vary. The position of the centre of gravity is to be seen as part of the rigid assumption and is thus fixed within the body of fixed shape. The body may thus accumulate or eject mass as long as the position of the centre of gravity remains unaffected. The influences of such mass flows are thoroughly investigated. The convective changes in linear and angular momentum due to the mass flows are derived. Within the translatory equations, the convective change in linear momentum due to a mass flow is appropriately correlated to the external forces acting on the body, enabling a clear decision on whether or not to include the mass flow terms in the equations of motion. Also the complete set of rotatory equations is derived and the mass flow terms which can influence the angular momentum are stated. Using the physical causal chain of an aircraft, the required modelling depth is explained. The higher the dynamics level of the trajectory, the higher is the required modelling depth. Concerning the External Forces and Moments, the focus is put on gravity. The aerodynamic and propulsion paragraphs are compact. The possibly important gyroscopic moments e.g. of the engines can be easily included within the propulsion modelling not contradicting the rigid body assumption. Though it is rather common to model the shape of the earth as an ellipsoid of revolution in accordance with the World Geodetic System 1984 (WGS 84) its consequence – namely the change in both magnitude and direction of the gravity vector with a change in altitude – is to be evaluated. This altitude dependent change of the direction of the gravity vector may, for example, influence the trajectory of high flying aircraft like solar powered unmanned aerial systems (UAS), and therefore needs to be investigated. In addition, the difference between the WGS 84 ellipsoid modelling and a much more complex modelling by the Earth Gravitational Model 2008 (EGM2008) is judged concerning its influence on common aircraft trajectories. For the static Atmosphere, the International Standard Atmosphere is used. The influence of temporal and spatial wind fields on the moving aircraft modelled as a point mass is determined. In case of respecting the spatial extension of the aircraft’s body, also the wind angular velocity and its influence on the roll, pitch, and yaw rate of the aircraft is calculated. The last part of the chapter is dedicated to the Modelling of the Receivable Solar Radiation within the Earth’s Atmosphere to investigate the possibilities of solar powered flight. The aim is to determine the receivable solar radiation – which is attenuated by the atmosphere – in dependence on date, local time, and position: geodetic longitude, geodetic latitude, and altitude. After a short introduction to celestial mechanics, the receivable radiation on the

20 Introduction

orbit of the earth moon barycentre (EMB) around the sun is determined. The averaged irradiance on this orbit results from high quality measurement campaigns, and the orbit itself is a Keplerian. The deviation from the true earth orbit to the EMB orbit is calculated and shown to have only very little influence on the radiation receivable. Then, the position of the sun, as seen from the earth, is determined very accurately (for the application in mind). For this task, the equation of time (EOT) is needed which describes the difference between the averaged apparent solar movement and the true apparent solar movement. The two dominating reasons for this difference are the eccentricity of the EMB orbit and the obliquity of the ecliptic. Knowing both, the receiver position as well as the apparent position of the sun, the atmospheric attenuation processes are investigated. The attenuation is dependent first, on the atmospheric conditions, and second on the length of the path a ray of light has to travel before reaching the receiver. It is shown how the total radiation is combined out of direct as well as of diffuse radiation. In addition, the spectra for both extra-terrestrial solar radiation as well as on ground receivable radiation are given. Finally, several results of the solar modelling are presented. The composition of the solar irradiance (direct, diffuse, total) for characteristic days of the year is demonstrated. The total daily integrated solar irradiance on a horizontal receiver plane is given in dependence on latitude and date, and it is shown by which factor it can be increased if the receiver is oriented towards the sun. For some applications, the then presented yearly insolation receivable is important where in addition the maximum and minimum (assuming a clear sky) daily insolation is given. The results of the solar modelling process are always validated against high quality measurement data. In chapter three Applied Optimal Control, the transcription of the infinite-dimensional parameter dependent optimal control problem into a finite-dimensional approximation (parameter dependent optimisation problem) is regarded. The aim is always to minimise a scalar cost function. The advantages and disadvantages of indirect and direct transcription methods are given. Using a direct transcription, the original optimal control problem is transformed into a non-linear programming problem (NLP) which is solved numerically. Afterwards, the solution accuracy has to be assessed and validated. For a converged optimisation, a post optimal sensitivity analysis may be conducted laying for example the basis for a bilevel optimisation. First, The Parameter Dependent Optimal Control Problem is regarded in its basic structure. The general problem formulation is stated and the classification as Bolza, Mayer, Lagrange or Tschebyscheff problem is explained. Then, it is shown that by applying the multiple shooting method for the discretisation the strengths of the single shooting method are maintained but its weaknesses are overcome. Subsequently, an optimisation framework and its functioning is described in general. It has two major components. The first component is the (commercially available) NLP solver, and the second is the function evaluator which is described in more detail later on and can be regarded as the engineering part. The paragraph closes with a physically motivated treatment of mesh refinement. The Function Evaluator should be able to handle all information provided by the model and inversely the model should contain all information possibly usable by the function evaluator. If the automatic differentiation routine described in the paragraph is implemented, the model contains not only the state dynamics and the outputs of the system but also its complete Jacobian as well as its complete Hessian. These matrices are required for the first and second order sensitivity equations which are derived. It is clearly shown which part of the sensitivity equations is to be delivered from the model and which part has to be calculated

Introduction 21

within the optimisation framework. The sensitivity equations enable, for example, the calculation of the Jacobian and the Hessian of the cost function (or the constraints vector) with respect to the optimisation parameter vector. This lays the basis for a post optimal sensitivity analysis. For numeric stability of the solution process, proper scaling is very important and all knowledge about the resulting sparse matrices and their structure (Jacobian, Hessian) has to be transferred to the NLP solver in order to be exploited. Using the Post Optimal Sensitivity Analysis, the consequences of a change in the parameter vector on the optimal solution are investigated up to the second order. It forms the basis for Bilevel Optimal Control Tasks. Here, a special class of bilevel problems is investigated where an upper level parameter optimisation problem delivers a set of parameters to one or more lower level optimal control problems. In case of two competing optimal control problems on the lower level, the set of parameters (conditions) under which the given task may be fulfilled equally well is of high interest. Concerning periodic optimisation, it enables for example a solution structure analysis by determining the switching conditions where one and two cycles are equally effective. For any deviation from these conditions it is thereby known whether one or two cycles are favourable. With the basis laid in chapters two and three, the fourth chapter Periodic Optimal Control for Flight Applications presents a number of optimised trajectories by applying the theoretical knowledge derived. Such different flying systems as solar aircraft, motor gliders featuring a retractable propulsive unit, and siskins performing a bounding flight are regarded. They all have in common that a periodic trajectory turns out to be suited best for the respective task. The first paragraph is named Unlimited Endurance Missions – Solar Aircraft. The aim is to investigate the trajectory requiring the least battery capacity and thus weight penalty for a solar aircraft to enable circling the earth in sustained flight. After the modelling description, the qualitative segments of the trajectory are presented and explained. First, trajectories with an upper altitude limit are optimised. The aircraft may either vary the lift coefficient or the lift coefficient is considered to be fixed but optimised to the respective value. If the restriction of the upper altitude limit is dropped, the aircraft can store more energy in terms of mechanical energy, and thus requires fewer batteries but as a trade-off has to be equipped with a pressure cabin for the pilot. Finally, unmanned solar aircraft are dealt with not facing such weight penalties. In the second paragraph Configuration Changes – Powered Glider with Retractable Engine, a motor glider featuring a completely retractable propulsive unit is investigated. It can either be equipped with a piston engine or an electric engine powering a propeller. As third alternative also a small jet engine may be installed. In any case the whole propulsive unit may be retracted into the fuselage leading to a strong change in the aerodynamic configuration of the glider. The aim is always to maximise range. In order to reach this aim, the motor glider has to fly a saw-tooth profile. This consists of an optimised climb followed by an engine stopping and cooling phase before the propulsive unit is retracted into the fuselage. Then, an optimised glide follows. Finally, the engine is extended again and started up. The durations for the configuration changes, the cooling and the starting up procedure are fixed and the durations for the climb and the glide have to be optimised. In case of the piston engine powering a propeller, also the influences of head-, tail-, and crosswinds are investigated. The resulting saw-tooth trajectories are always compared to steady state horizontal flights. In case of a propeller installed, these horizontal flights are to be performed

22 Introduction

at the minimum altitude allowed. In case of a jet engine, three altitudes for the comparative flights are regarded: minimum altitude, 3000m, optimised altitude in dependence on thrust available. Moreover, in case of the jet engine installed, different maximum thrust levels are investigated. Aim of the last paragraph Bounding Flight – Siskin is to analyse this instationary flight of many small birds as, for example, the siskin. The three different intermittent flight modes of birds namely flap bounding flight, undulating flight, and the chattering flight are presented. The thorough investigation is concerned with a rather fast bounding flight. The bird flaps its wings, then retracts them and flies a curved segment with the wings folded completely into the body before it extends the wings again to restart flapping. The cost function is to maximise range and therefore minimise the energy consumed per distance travelled. To derive results of generally valid nature, the underlying equations are normalised. Under the omission of the extending and the retraction phases, an analytical optimisation is conducted. This analysis shows the superiority of the bounding flight to stationary horizontal flight concerning energetic aspects, and is used as initial guess for the numeric optimisation. Since for the bird it can be of vital interest to make progress against a headwind, its influence on both, the stationary horizontal flight, as well as on the bounding flight is analytically investigated, too. By the normalisation, additionally the connection to small unmanned aerial systems is established. Then, the numerically optimised trajectories for different speed levels are presented and compared. After that, the velocity is held fixed and the cycle length is varied to show the possible variations within one cycle of fixed average speed. Afterwards, multiple connected periodic cycles are analysed with the respective cost function graphs presented. Finally, a bilevel optimisation is conducted for a solution structure analysis. Here, the post optimal sensitivity analysis delivers an initial guess for the distance at which one and two bounding flight cycles are equally effective. Starting from this initial guess, the bilevel optimisation then determines the true intersection of the cost functions quickly and accurately. The following appendices are associated with Chapter 2 Fundamentals of Flight Mechanics: In appendix A, a judgment on the modelling of the shape of the earth (flat – round - rotating) is developed based on the kinematic velocity flown. First, the translational dynamics are regarded with the earth modelled as a WGS 84 ellipsoid of revolution. While the centrifugal acceleration is only position dependent, both, the Coriolis, as well as the acceleration due to the transport rate increase with kinematic flight velocity. Threshold velocities where 1‰, 1%, 10%, and 100% of the standard acceleration of gravity is reached are given. Concerning the rotational dynamics, the rates of a standard turn are compared to the earth’s rotational rate and the transport rate which is again dependent on the kinematic flight velocity. In appendix B, the attitude equations of motion are given in quaternion representation. Thereby, the singularity within the classical representation at an inclination angle of 90° may be overcome. In appendix C, vector fields and potential functions are shortly regarded. It is explained that the scalar potential of every solenoidal, conservative vector field must be harmonic. The following appendices are associated with Chapter 3 Applied Optimal Control: In appendix D, a matrix vectorisation operator and the Kronecker product are introduced. Moreover, the first and second order derivatives of a matrix product to a matrix are given.

Introduction 23

In appendix E, the vector of the discretised constraints and the optimisation parameter vector as used in the optimisation framework are stated in detail. In appendix F, the structure of the automatically generated Jacobian and Hessian matrices is presented. The following appendices are associated with Chapter 4 Periodic Optimal Control for Flight Applications: In appendix G, a short introduction to first solar cells with reference to the solar spectrum usable, and second to maximum power point trackers is given. In appendix H, the propulsion system of the Antares 20E motor glider built by Lange Aviation is briefly presented.

1.2 Periodic Optimal Control Periodic optimal control problems arise in various fields. To mention only a few: depending on the time horizon, chemical reactors (like most production units) need to be or may be operated periodically as reviewed in Bailey (1974). For general production processes, see for example Maurer et al. (1998). For economic applications, one may further consider Feichtinger (1992) and Feichtinger et al. (1994). In spaceflight, for example, periodic orbits for interplanetary flight are investigated which are free-fall trajectories shuttling back and forth between the planets making a flyby at each terminal (Hollister, 1969). A publication, in which not a numerically optimised periodic trajectory is derived but the altitude is sinusoidally controlled, is given in Mehta (2013). Even by this rather simple strategy, maximum endurance is increased by nearly 13%. When dealing with periodic optimal control, two basic situations depending on the type of system to be optimised and the condition of its environment have to be distinguished. The first situation is given by the optimisation of a non-changing system in a non-changing environment with rather long time horizon. Here, the cyclic behaviour of the states and the controls is not obligatory. For many such systems, however, a non-steady operational mode is superior to a continuous operation. It is then sufficient to optimise one (short) cycle which is to be periodically repeated. If a quasi-steady operation is superior to a cyclic operation this would automatically result during the optimisation process. Then, the steady state operation can be seen as special form of periodic operation. Also conditions like shear wind fields which are not subject to changes but which are – in the optimal case – cyclically passed fall under this classification. The second situation is present at the optimisation of a.) a system undergoing active or non-active changes or b.) a system operated in a cyclically changing environment or c.) altogether. Active changes are controllable fundamental changes like extending or retraction of the engines. Non-active changes are changes which may not be directly controlled but are known (mathematically describable) like deteriorating processes enforcing maintenance. A cyclically changing environment would be the apparent movement of the sun or as second example a race track which has to be passed several times. In case of solar powered systems, even endoatmospheric trajectories of infinite time horizon can be investigated. Here, it is most likely that a periodic operation mode is superior to a steady state operation.

24 Introduction

In the beginning of periodic optimisation, rather simple control strategies were investigated. For autonomous problems with sinusoidal control, the −π test (Guardabassi, 1971) can prove the existence of a periodic control superior to a steady control – see also Bittanti et al. (1973), the survey of Guardabassi et al. (1974), Bernstein and Gilbert (1980), and also Breakwell (1987). Additional tests whether periodic control yields better process performance than optimal steady state control including tests based on relaxed steady state control are investigated in Gilbert (1976) and Gilbert (1977). A second variation analysis to explain the superiority of non-steady state flight is presented in Menon (1989). In Wang and Speyer (1990) necessary and sufficient conditions for local optimality of periodic processes are provided. Also in case of discrete systems, a cyclic operation may be superior to steady state operation. Two sufficient and a necessary condition are found by Bittanti et al. (1976). A chattering optimal solution is a solution where the controls alternate between distinctly different values at an infinite rate resembling to a high frequency bang-bang control. Such solutions might arise for problems where the physical systems are modelled with false simplifications assuming that certain dynamics are that fast that they could be neglected. By an asymptotic expansion about the chattering solution, these neglected dynamics can be re-included (Chuang et al., 1988). If such a solution occurs, it is, however, in the vast majority of cases appropriate to reconsider the modelling process as described in paragraph 2.1.6 and start the optimisation anew. Periodic flight path optimisation is strongly connected with the names Sachs and Speyer. Subsequently, a shortened chronologically ordered survey is given. Based on Speyer (1973), the non-optimality of steady state cruise for aircraft in a non-changing environment is shown in Speyer (1976), even though the truly optimal trajectory is not determined. Although not globally minimising the cost criterion of total fuel consumed per distance travelled (maximising range), possibly locally minimising periodic orbits are found for a hypersonic vehicle (3dof) in Speyer et al. (1980) by an indirect shooting method. Here, two reasons are given why cyclic trajectories are superior to steady state trajectories: if the minimum drag point (dependent on altitude and velocity) does not coincide with the engine design point, fuel efficiency is improved by cyclic motion between these points. But even if the points do not diverge, a potential and kinetic energy interchange of the aircraft will reduce specific fuel consumption by taking advantage of the decreased drag with an increase in altitude (decrease in atmospheric density). For additional information on the applied shooting method, see Speyer and Evans (1981). In Speyer et al. (1985), the point mass model of an atmospheric vehicle operating in the hypersonic region is used where a periodic orbit which involves the flat earth model is found to be locally fuel minimising. Both, the aerodynamic scramjet engine model and the shooting method are further developed in Chuang and Speyer (1987) giving more realistic trajectories and predictions on fuel saving possibilities. When so far maximum range has been considered, maximum endurance is investigated for propeller and turbojet aircraft in Sachs and Christodoulou (1986) by applying the minimum principle. It is shown that maximum endurance flights are to be composed of two phases, with the first phase flown at full thrust followed by a gliding phase without thrust. The physical reason for the superiority of cyclic flight is due to the high rate in specific energy added to the aircraft per fuel mass burnt when flying at a high speed level (phase one). This by far outweighs the increased drag work as compared to steady state flight. It is further explained that high improvements of endurance may only be reached for turbojet engines

Introduction 25

where maximum thrust is approximately independent of flight velocity. As concerning aircraft design, maximum endurance positively correlates to the installed thrust to weight ratio. But especially the installed maximum lift to drag ratio over-proportionally influences the benefits reached by periodic trajectories: an increase in this ratio is of greater positive effect in the cyclic case than in the steady state case. Due to an increase in necessary speed with an increase in altitude (drop in density), it is advantageous for the minimum altitude reached during the periodic trajectory to be at a level as low as possible. In Grimm et al. (1986), periodic trajectories (minimising fuel per distance) for an F-4-type aircraft are investigated including mass changes. Due to the relatively short periods, the practical effect of decreasing mass was found to be very small. In Sachs and Christodoulou (1987), maximum range and maximum endurance flights are investigated for aircraft with a high amount of thrust installed able to reach high subsonic Mach numbers. In case of maximum endurance, the savings by a cyclic flight are more significant than for maximum range while the velocities flown stay within the incompressible region. Drag, thrust, and fuel consumption models are included which show the important dependency on the Mach number flown and the density of the surrounding air. High fidelity data are provided up to Mach 0.9 in the compressible region where significant drag rise effects are present. Like in the endurance case, also for range maximising cyclic trajectories the two phase structure with a thrust bang-bang control is optimal. Within the incompressible region, it is shown that it is desirable to have a high maximum thrust installed which can in the second phase be completely cut off (gliding phase). A maximum altitude constraint is imposed and it is shown that with an increase in the admissible altitude region, the tendency away from a cyclic flight towards a steady state flight at very high altitude (far above 10km) increases. In the compressible region, the drag rise represents a key factor for cyclic cruise since it also acts as a kind of barrier. The altitude constraint can be removed and no velocities above Mach 0.79 are reached. The oscillatory behaviour in terms of the amplitudes of the Mach number and lift coefficient changes is reduced in the compressible region. For all numerical solutions an indirect multiple shooting approach has been chosen. In Sachs and Lesch (1990), it is shown that at the range optimisation for propeller aircraft singular arcs may occur in dependence on the fuel consumption modelling leading to a smooth connection of the full thrust phase and the idle phase. But in case of a propeller powered aircraft, the advantage of cyclic flight is very small. Singular controls in periodic trajectory optimisation are further analysed in Sachs and Lesch (1993) and in Sachs et al. (1993). An investigation of maximum endurance for a high performance aircraft equipped with variable camber control is performed in Sachs and Mehlhorn (1991). Beside throttle setting and lift coefficient, variable camber is added as a third control. The positive effect of an increase in the lift to drag ratios is especially present in the subsonic region where the gliding phase of the endurance flight is to be performed. Thereby, the superiority of cyclic flight can be further improved. An increase in the length of a period in order to reduce the total number of engine cycles in regard to alternately operating at high and low thrust settings shows a comparatively small reduction in endurance performance, even if the period length is doubled when compared with the optimal value (Sachs and Mehlhorn, 1993). The importance of reducing the energy subtracted from the aircraft due to drag (dissipated energy), is pointed out in Sachs (1991) when for a propeller-driven aircraft fuel consumption per distance travelled is to be optimised leading to a periodic trajectory. For the hypersonic vehicle investigated by Dewell and Speyer (1993), a maximum range cyclic flight consists of three distinctive flight regimes: a Keplerian arc above the

26 Introduction

atmosphere, a glide to minimum altitude and a powered climb out of the atmosphere. Fuel savings are quite substantial and lie within the region of 15% as compared to steady state flight. In Speyer (1995) and Speyer (1996), results in periodic optimal flight which have been reached over the past 25 years are described, and the maximum range trajectory for a generic hypersonic aircraft model is presented. Additionally, a feedback controller called the periodic regulator, is developed to mechanize optimal periodic processes in the presence of state deviations from the optimal periodic path and system parameter uncertainties. In Dewell and Speyer (1997), the acceleration of a hypersonic vehicle is more strictly constrained and an optimal periodic regulator is implemented which is able to accommodate the slow decay in vehicle mass. In hypersonic flight, extreme convective heating of both, the engine and the airframe necessitates active cooling. Therefore, the on-board hydrogen serves both, as coolant and as fuel. Engine cooling is considered in Sachs and Dinkelmann (1996). A simplified cooling model to couple thermal-optimal and fuel-optimal hypersonic flight is developed in Dewell et al. (1997) where the resultant range maximising trajectory is again periodic. In the optimal case, no hydrogen used for cooling may be ejected into the atmosphere but should be completely used to impart kinetic energy to the aircraft. For a mathematical model able to calculate conductive, convective, and radiative heating effects through the thermal protection system, where the instationary quasi-linear heat equation with non-linear boundary conditions may be included in the optimisation process, see Chudej et al. (2009). The resulting temperature of the heat shield can then be limited by an additional path constraint. In Chen et al. (2006), for a hypersonic waverider vehicle the optimal periodic trajectory for range maximisation is determined using highly sophisticated models of the vehicle and the environment. The concept has a door which may be placed over the engine inlet during the power off phase increasing drag but also lift. As additional benefit of the periodic cruise, the stagnation heat load is significantly reduced with only a small increase in maximum stagnation heat rate. Another task is the endurance maximisation of an aircraft circling above a target on ground (Chen and Speyer, 2007). Here, a subsonic aircraft is constraint to fly on the surface of a vertical cylinder. The periodic trajectory shows significant fuel savings. A different problem class where periodic trajectories result, is dynamic soaring both, of birds as well as of sailplanes making use of a shear wind field (e.g. modelled as a logarithmic wind profile). The energy neutral dynamic soaring cycles analysed, pose per definition a periodic optimisation problem (Sachs et al., 1989). Minimum shear wind strengths to enable dynamic soaring are derived in Sachs (1993). Two fundamentally different types of dynamic soaring trajectories possible (oval and bend type) are regarded in Sachs and Da Costa (2003). Though the oval type is inferior or at best equal to the bend type with regard to energy extraction from the moving air, it is of closed form and therefore – under the assumption of a non-changing environment – can be performed ad infinitum at the same position. Dynamic soaring for high performance sailplanes operating in high altitudes using wind regions associated with jet streams are investigated in Sachs and Da Costa (2006). That, by dynamic soaring, progress is possible even against the wind direction is shown in Sachs et al. (2011d). In this context, it is also advised to the dissertation Investigation on Manned Dynamic Soaring of Knoll (1995) and more recently the dissertations Boundary Layer Dynamic Soaring for Autonomous Aircraft: Design and Validation of Bower (2011) and Robust Trajectory Optimization and Control of a Dynamic Soaring Unmanned Aerial Vehicle of Flanzer (2012).

Introduction 27

1.3 Contributions of the Thesis The contributions of the thesis are to be found within the single chapters but also in the combination of the chapters and the application of the theoretical knowledge to real world periodic optimal control tasks. Within the derivation and the analysis of the optimised trajectories, the underlying physics is of paramount interest. Thereby, the knowledge gained within the optimisation process is valid for and may thus be applied to a whole class of aircraft trajectory optimisation problems rather than only a specific trajectory. A major advantage of the equations presented is the fact that in the vast majority of cases the complete derivation process is included. Therefore, all influences may be qualitatively evaluated and in case of assumptions or simplifications the validity can be easily checked. Moreover, it has been of high importance to assess the quality of derived equations but also of the results of the complete optimisation process by comparison with highly reliable measurement data. Thereby, the possibilities for errors either in the assumptions or in the derivation process can be minimised. Furthermore, it has to be mentioned that all the optimal trajectories presented are of very high quality both, with respect to numeric precision (very low primal and dual infeasibility requested) as well as stability (e.g. no oscillations, only physically meaningful control inputs). During the setting of the phases, not only the physical structure but also the dynamics level has been profoundly taken care of. For any trajectory presented, the mesh refinement process has been conducted several times. The clean definition of a rigid body to be of fixed external and internal shape, but at the same time allowing density changes within the body, is the basis for the derivation of the equations of motion. Due to the density changes allowed, the body may eject or accumulate mass, and the influences of such mass flows can be judged and integrated into the equations of motion. Concerning for example an ejected mass flow, the underlying physics are unveiled in (2-47), and it becomes evident that only the relative velocity between the body and the mass flow is of importance since the internal effects cancel out. Moreover, the influence of the change in angular momentum due to the mass flow can be qualitatively assessed. The gravity model presented in detail is valid for the WGS 84 ellipsoid of revolution and is given in ellipsoidal harmonic coordinates. In the derivation, the influences of gravitation and centrifugal force are clearly separated. It is then shown, how the gravity vector which changes both, magnitude and direction with altitude is to be transformed to the north-east-down system. This allows the direct implementation into the equations of motion. Furthermore, judging with respect to a given task of simpler gravity models is enabled which only include the downward component of the gravity vector but also of more complex ones like the earth geopotential model EGM96. A major advantage of the solar model presented, is that its derivation is started from very basic physics, and all possible influences are named and judged with respect to the application in mind. Thereby, every single term of the solar model can be set into connection to its physical origin. This allows for an easily scalable model but also for the testing of other models. The solar model can determine the radiation receivable at any time and any position within the atmosphere for any receiver orientation. It may therefore be used for flight applications in the broadest sense. It is for example suited to calculate the actual radiation receivable in flight and also the daily irradiance for any point on earth. Thus, it could for example also be used to asses two locations for a solar powered hydrogen

28 Introduction

plant, one featuring a horizontal receiver and one featuring a 2-axes orientation system (with or without partial shading). In the chapter on applied optimal control, the basic functioning of an optimisation framework is described which is suitable for direct discretisation. Within the framework, the function evaluator can be regarded as the engineering part while the NLP solver is associated with the discipline of mathematics. A clear distinction between the generation of information and the processing of information is made. The processing of information may to a very large extent be automated. The analytical knowledge about a system, for example, which is lost due to the discretisation process can be restored up to the second order by the automatically generated Jacobian and Hessian matrices. The first and second order sensitivity equations are derived which form the basis for the post optimal sensitivity analysis. In case of periodic trajectories, a solution structure analysis is of high interest if only a rather low number of cycles shall be flown. Using the bilevel optimisation, it is investigated under which conditions the switching from one periodic cycle to two periodic cycles should occur. The post optimal second order sensitivity analysis of the cost functions gives the initial guess for this bilevel optimisation task. The aim of all trajectories optimised for the solar aircraft is to minimise the required battery capacity for sustained unlimited flight. The period of the trajectory is one solar day. Thus, the true local solar time at the end of the cycle equals that at the very beginning. Due to the energy surplus during the day, the crucial part begins with the moment the aircraft does not receive sufficient solar power any more to be able to maintain horizontal flight. The aircraft can store the energy gained during the day either in the batteries or in terms of mechanical energy. If altitude limitations are imposed, the maximum of energy stored in potential energy is set. The trajectories are stabilised whereby the cost function increases by less than 1‰. First, manned systems are investigated. In case of an altitude limit set to 8500m, it is shown that the replacement of a variable lift coefficient by an optimised fixed lift coefficient value leads to no relevant increase in battery weight. If the upper altitude limit is dropped, a pressure cabin for the pilot is required, but since the amount of mechanically stored energy can be largely increased, the battery weight strongly diminishes. Finally, an unmanned solar aircraft is investigated which has no weight penalties imposed concerning the pilot. Two facts become evident. First, if the mechanical energy storing is limited, the energy storing in batteries (weight density) becomes even more important. Second, to avoid any unusable energy surplus, the aircraft should be equipped with strong enough engines. For a manned aircraft, thereby, the time for circling the world is reduced, and for an unmanned system the whole periodic trajectory can be shifted to higher altitudes increasing the level of safety. While the solar aircraft is operated in a cyclically changing environment, in the next paragraph, a motor glider which can actually change its own configuration is regarded. The motor glider is equipped with a propulsive unit (piston engine powering a propeller, electric engine powering a propeller or jet engine) which can be completely retracted into the fuselage. In the clean configuration, the gliding performance is strongly improved. In order to maximise range, the motor glider has to fly a saw-tooth manoeuvre consisting of six phases. The climb and the glide are freely optimised. Before the glide, the engine has first to be cooled down and stopped, and second is retracted into the fuselage. After the glide, the engine has to be extended again, and needs to be started-up. In contrast to the duration of the glide, the durations of the two phases before and the two phases afterwards are fixed. At the end of the period the motor glider has to be in the same configuration as at the very beginning, and as many states as possible as well as all physical controls are to be periodic,

Introduction 29

too. For the motor glider equipped with a piston engine the influences of a head-, tail-, and crosswind are investigated. For comparison, also the optimised saw-tooth profiles for a motor glider equipped with an electric engine as well as for a motor glider equipped with a jet engine are calculated. In case of a jet engine, in addition the amount of available thrust is varied. The fuel consumption in saw-tooth mode is always compared to the optimised steady state horizontal flight. The bounding flight manoeuvre investigated shall help a small bird like a siskin to save energy per distance travelled. It consists of four phases. First, the bird flaps its wings and flies a curved segment. Then, it retracts the wings and subsequently flies another curved segment resembling a parabola. In the fourth phase, the bird extends its wings again. While the duration of the configuration changes is fixed, the time both of flapping as well as of the ballistic phase is to be optimised. In order to obtain results of generally valid nature, and under the omission of the configuration changing phases, all underlying equations are normalised and analytically optimised. The superiority of the bounding flight either with a certain amount of lift in the bound phase or without lift during the bound phase is shown as compared to steady state horizontal flight. Thereby, also a very good initial guess for the numeric optimisation including all the four phases is given. Moreover, the effect of headwind is analysed analytically and the superiority of the bounding flight manoeuvre over steady state horizontal flight is shown concerning both, energy used per distance travelled and at the same time kinematic velocity, and thus progress against the wind. During the modelling of the siskin, the Jacobian as well as the Hessian are generated in a fully automated manner. The bounding flight manoeuvre is optimised for a large velocity spectrum and for a fixed velocity over a large interval in distance covered during the cycle. Thereby, the variability of the manoeuvre is analysed. Then multiple cycles are analysed. A rather large distance interval is regarded which may be covered by the bird by flying one to three consecutive cycles showing the required cost function values. In order to minimise cost, a specific distance may only be flown by a certain number of cycles. This, to some extent, may be regarded as an integer optimisation. The bilevel optimisation is used for a solution structure analysis. The aim is to determine the distance at which two cycles become equally effective as one cycle. In the first step, one cycle has to be optimised. Doubling this cycle gives a nearly perfect initial guess for two consecutive cycles. By using the post optimal sensitivity analysis, the cost functions in the respective optima are approximated by parabolas. The cost function intersection then gives the initial guess for the bilevel optimisation routine. The true intersection of the cost functions is then determined in very short time by conducting the bilevel optimisation with the NLP solver fully exploiting the knowledge provided by the Jacobian and the Hessian matrices.

30

Chapter 2

Fundamentals of Flight Mechanics

2.1 Equations of Motion

2.1.1 Introductory Thoughts and Validity In the following, the equations of motion for a rigid body are derived. A rigid body will maintain its external as well as its internal shape. The body has a fixed reference point, R . There is no relative motion between an arbitrary point, P , and the reference point, R , within the body.

( ) ( ) ( ) 0Vrr

===

BRPBRPRP

B

dtd

(2-1)

As consequence, the position of the centre of gravity, , of the body is also fixed with respect to its shape. The centre of gravity may therefore be chosen as the reference point.

( ) 0r

=BP (2-2)

The body’s density, however, is allowed to vary as long as the just mentioned assumptions are met. The body may thus, in principle, loose or accumulate mass.

0≠dt

d Pρ is possible (2-3)

Parts rotating at a high frequency are to be treated as external systems (e.g. rotating engine components). The resulting gyroscopic moments are treated in paragraph 2.2.3. The movement of the rigid body can be uniquely described by twelve independent states since it has six degrees of freedom in space. The states are grouped in sets of three: for the position, the translation, the attitude, and the rotation of the body. In addition, one differential equation is needed to describe the change in mass. For each element of the state vector, x , a first order differential equation is defined:

( ) ( )tdtd ,,uxfxx == (2-4)

with the inputs (controls and disturbances), u, and the time, t, if the system is not autonomous. Thus, the flying system is modelled by a state-space representation.

Fundamentals of Flight Mechanics 31

As second assumption beside (2-1), the earth is supposed to rotate at a constant angular velocity around its polar axis:

( ) 0ω

=IIE (2-5)

Moreover, effects of polar motion, nutation and precession (measurable by Very Long Baseline Interferometry, VLBI) are not regarded for the transformation between the inertial frame, I , and the ECEF-frame, E . The whole paragraph 2.1 is related to TUM-FSD (2014) to which this author contributed.

2.1.2 Position The aim of the position equations of motion is to describe the change in position of the aircraft in dependence on the kinematic velocity flown with respect to the earth and its attached atmosphere featuring no inner movement (ECEF-frame, E ) denoted in the north-

east-down frame (NED-frame, O ): ( )EO

RKV

. For the selection of elements of the state vector

for the position equations of motion, a determining factor is the type of modelling of the earth’s surface. Which type of modelling should be applied is discussed in the appendix A.

Flat Earth In case of the earth modelled as flat and non-rotating, the motion is described in a

Navigation frame, NavN . This Navigation frame results from a north-east-down frame locally

fixed on ground which is rotated positively around its z-axis by the angle Nχ .

The change in position is thus:

( )( ) ( )( ) ( )

E

O

E

N

ON

NN

NNEO

RKON

E

Nh

VV

zyx

NAV

NAV

NAV

−⋅

−=⋅=

1000 cos sin0 sin cos

χχχχ

VM (2-6)

If for the angle 0=Nχ , the above transformation matrix becomes the identity matrix and the

system is abbreviated as ON as the x-axis of this navigation frame points northward and its

y-axis points eastward. The kinematic velocity in the NED-frame may be derived from the kinematic velocity in the

kinematic frame by using the transformation matrix TKOM . KOM results from the rotations

about the course angle, RKχ , and the climb angle, R

Kγ .

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

( ) ( )OK

RK

RK

RK

RK

RK

RK

RK

RK

RK

RK

RK

RK

OK

−⋅⋅⋅−⋅

=γγ

γχχγχγχχγχ

cos0 sin sin sin cos cos sin sin cos sin cos cos

M (2-7)

32 Fundamentals of Flight Mechanics

For the transformation of the velocity vector between the kinematic frame and the north-

east-down frame only the first column of OKM is needed:

( )( ) ( )( ) ( )

( )

E

O

RK

RK

RK

RK

RK

RK

RK

RK

E

K

RK

OKEK

RKOK

E

O

E

N

VVVV

hVV

⋅−⋅⋅⋅⋅

=

⋅=⋅=

− γγχγχ

sin cos sin cos cos

00MVM

(2-8)

Round Earth: WGS 84 In case of a round earth, throughout this work, the earth is modelled as an ellipsoid of revolution in accordance with the World Geodetic System 1984 (WGS 84) definition as published by the National Imagery and Mapping Agency NIMA (2000). From there, Table 2.1-1 gives the four defining parameters:

Parameter Notation Value Unit Semi-major axis a 6.3781370 · 106 m Reciprocal of flattening f/1 298.257223563 -

Angular velocity of the earth IEω 72.921150 · 10-6 rad/s Earth’s Gravitational Constant (including mass of the earth’s atmosphere)

M 398.6004418 · 1012 m3/s2

Table 2.1-1: WGS 84 ellipsoid defining parameters - (National Imagery and Mapping Agency NIMA, 2000)

By using the WGS 84 defining parameters, numerous geometric and physical constants may be derived; see also Moritz (1984) as well as Hofmann-Wellenhof and Moritz (2006). The semi-minor axis, b , is found from:

abaf −

= thus ( )fab −⋅= 1 (2-9)

The polar radius of curvature, c , is calculated by:

fa

bac

−⋅==1

12

(2-10)

The first eccentricity, e , follows from the focal triangle – see also (2-99) and (2-100):

feab

−=−= 11 2 (2-11)

The first eccentricity squared may therefore be expressed in terms of the flattening:

22 2 ffe −= (2-12)


Table 2.1-2 gives the values of the above derived geometric constants.

Parameter Notation Value Unit Semi-minor axis b 6.3567523142 · 106 m Polar radius of curvature c 6.3995936258 · 106 m First eccentricity e 8.181919084262 · 10-2 - First eccentricity squared 2e 6.69437999014 · 10-3 -

Table 2.1-2: WGS 84 ellipsoid derived geometric constants

This enables the calculation of the radius of curvature in the prime vertical, ϕN , in

dependence on the geodetic latitude, φ. It measures the distance perpendicular (normal) to the surface of the ellipsoid until its intersection with the polar axis, zE – see also Figure 2.1-1:

( )ϕ

ϕϕ 22 sin1 ⋅−=

eaN (2-13)

The meridian radius of curvature, ϕM , is the radius of a locally tangent circle of the same

curvature as the ellipsoid in the respective meridian plane.

( )( ) ϕϕ

ϕ ϕϕ 22

2

23

22

2

sin11

sin1

1⋅−

−⋅=

⋅−

−⋅=

eeN

e

eaM (2-14)

The radius from the origin to a point on the surface of the ellipsoid is denoted by R – see also Figure 2.1-2:

( ) ( ) ϕϕϕ

ϕϕϕ

2222

22

2222

cossin1sin1

cossin1+⋅−⋅=

⋅−

+⋅−⋅= eN

e

eaR (2-15)

Figure 2.1-1 ECEF and NED frames - compare TUM-FSD (2014)

λ ϕ

x0

y0

z0

h),,P( ϕλ

xE

yE

zE

0 ϕ

h h),,P( ϕλ

xE

yE

zE

0

p

R

Nφ


On the left, Figure 2.1-1 shows the attitude of the ECEF-frame and of the NED-frame with the two rotations between the frames: about the geodetic longitude, λ, and about

( )2' πϕϕ +−= . On the right, the radius of curvature in the prime vertical, Nφ, the altitude

above the ellipsoid, h, for the point P and its distance from the polar axis, p, are depicted. A general presentation of the involved coordinate system is given in Figure 2.1-5.

The rotation matrix OEM and the angular rates, ( )EOω , (transport rate) between the ECEF

and the NED-frame are:

OE

OE

−⋅−⋅−−

⋅−⋅−=

ϕλϕλϕλλ

ϕλϕλϕ

sinsincoscoscos0cossin

cossinsincossinM ( )

E

EEO

⋅−

⋅=

λλϕ

λϕ

cos

sinω (2-16)

In (2-16) the strapdown equation (Stevens and Lewis, 2003) was used for the calculation of the transport rate; for the strapdown approach see also (2-60) and (2-61). Figure 2.1-2 shows the values of Nφ, Mφ, and R in dependence on the geodetic latitude, ϕ .

Figure 2.1-2 Radii for the WGS 84 ellipsoid of revolution

For all latitudes, φ, holds that:

( ) ( )baNM

ab 22

≤≤≤ ϕϕ ϕϕ (2-17)

The distance p between the point P and the rotational axis is found as:

( ) ϕϕ cos⋅+= hNp (2-18)

6,330

6,350b

6,370a

6,390

c

6,410

Latitude [rad]

Rad

ii [k

m]

−π/2 −π/4 0 π/4 π/2

Nφ

Mφ

R


The first derivative with respect to time of this distance is:

( ) ( ) ϕϕϕ ϕϕ sincos ⋅+⋅−⋅+= hNhNp (2-19)

Inserting the expression for the radius of curvature in the prime vertical (2-13) and its first order time derivative gives:

( )( )

ϕϕ

ϕϕϕ

ϕϕϕ sinsin1

cossin1

cossin22

23

22

2

⋅

+

⋅−⋅−⋅

+

⋅−

⋅⋅⋅⋅

=

he

ahe

ea

p

(2-20)

Grouping about the first order time derivative of the geodetic latitude, ϕ , and the change in

altitude, h , leads to:

( )( )

( )( )

ϕϕϕ

ϕ

ϕ

ϕϕ cossinsin1

sin1

sin1

cos

23

22

22

23

22

22

⋅+⋅

−

−

−⋅−

−

⋅⋅⋅

=

hhe

ea

e

ea

p

(2-21)

Thereby, the first order time derivative of the distance p is found to be:

( ) ϕϕϕ ϕ cossin ⋅+⋅+⋅−= hhMp (2-22)

Due to basic geometry, the first derivative to time of the distance between the reference point and the rotational axis has to be:

( ) ϕϕ cossin0 ⋅+⋅−= hup ERK

(2-23)

This gives the position equations of motion (see also TUM-FSD (2014)) in matrix-vector notation as:

( ) E

O

E

N

hVV

hM

hN

h

−⋅

−+

⋅+

=

100

001

0cos

10

ϕ

ϕ ϕ

ϕλ

(2-24)

The above equations are purely based on geometric considerations and are therefore valid for any modelling depth of the regarded flight system. The kinematic relations describe the motion of the flight system without regarding the underlying cause.


2.1.3 Translation The translation equations of motion deliver the derivative of the Cartesian velocity vector for

the centre of gravity, ( )EOKV

, by applying Newton’s second law of motion connecting the

change in linear momentum to the external forces acting on the body.

( )EO

K

K

K

EOK

wvu

=

V (2-25)

Moreover, the differential equation for the non-Cartesian translatory state vector composed

of the kinematic velocity, KV , the course angle,

Kχ , and the climb angle, Kγ , are derived.

( ) ( )EK

K

K

K

K

EK

KKTransTrans

V

dtd

===γχ

Vxx (2-26)

The centre of gravity, , of the body (see also Figure 2.1-3) is defined as:

0r =⋅∫

m

P dm or 0r =⋅⋅∫∫∫

Vol

PP dVolρ (2-27)

The centre of gravity may thus be found from an integral over all elements of mass of the rigid body or from an integral over all volume elements including their respective density.

Figure 2.1-3 Centre of gravity of the rigid body

The regarded rigid body is allowed to vary its mass by varying its density with the restriction that the centre of gravity of the body does not change its position within the body of fixed volume and shape. If the centre of gravity (2-27) is fixed within the volume of fixed shape, its first order time derivative with respect to the inertial frame, I , has to be zero:

0r =

⋅⋅

∫∫∫

Vol

PPI

dVoldtd ρ (2-28)

GPr

dVoldm P ⋅= ρP


The time derivative may be placed inside the volume integral and with the fixed centre of gravity (2-2) results:

( ) ( ) ( ) ( ) ( ) 0rrωr0

(

=⋅

⋅+⋅

⋅×+⋅ ∫∫∫∫∫∫

= Vol

PP

Vol

PPIBPBP dVoldtddVol ρρρ (2-29)

From the definition of the centre of gravity (2-27) it follows:

( ) ( )[ ] ( ) 0rrω

0

))) ()))

=⋅

⋅+⋅⋅× ∫∫∫∫∫∫

=

Vol

PP

Vol

PPIB dVoldt

ddVol ρρ (2-30)

What finally gives a relation for the allowable density changes within the rigid body with fixed centre of gravity:

( ) 0r =⋅

⋅∫∫∫

Vol

PP dVol

dtdρ

(2-31)

The above equation may be seen as a strict consequence of the rigid body assumption (2-1). The body may thus in principle vary its mass by varying its density. Any density changes, e.g. in the fuel tanks from fuel density to air density due to combustion, or vice versa, from fuel shifting between the fuel tanks, are allowed, as long as the overall centre of gravity remains unchanged. Equation (2-31) is of high importance since it allows the derivation of the translation equations of motion for a rigid body emitting or collecting mass in a very general form. Newton’s second law of motion is applied to the body allowing a mass flow over its boundaries: the sum of external forces acting on the body equals the change in linear momentum of the body itself plus the convective change in linear momentum due to accumulated or emitted masses. The mass flows over the boundaries are modelled as continua of moving particles ( Mp ) entering or leaving the body at a quasi-steady velocity.

( ) ( )IIconvBody

IBody

I

dtd

,ppF ∆+

=∑ (2-32)

For a derivation of the equations of motion including a convective term of the linear momentum see also Thomson (1966). The sum of the external forces thus equals the inertial change in velocity of all mass elements, plus the volume integral over the moving body where density changes occur, plus

the convective change in linear momentum due to j mass flows entering, inm , or i mass

flows leaving, outm , the rigid body.

( ) ( )

( ) ( )∑∑

∫∫∫∫

∑

⋅+

⋅−

⋅⋅+⋅

=

=

i

Iioutiout

j

Ijinjin

Vol

PIP

Km

IPK

I

MpMp mm

dVoldt

ddmdtd

,,,, VV

VV

F

ρ

(2-33)


More general equations of motion for variable mass systems based on Kane’s formalism are derived in Eke and Mao (2002). There, the system consists of a closed surface as well as its enclosed contents in solid and fluent phase. Reactions between the components, e.g. a combustion process, may occur and mass is allowed to cross the system boundary in any direction. The convective terms of (2-33) are regarded first. The inertial velocity of a moving particle (with centre of gravity, MP ) to be measured from the moving body (with centre of gravity,

) is:

( ) ( ) ( ) ( )III

I MpMpMp

dtd VVrV

+=

= (2-34)

A moving particle (MP) crossing the boundary surface of the rigid body is depicted in Figure 2.1-4.

Figure 2.1-4 Rigid body and exhausted mass flow

The inertial velocity of particles crossing the body boundary is by using (2-34):

( ) ( ) ( ) ( ) ( ) ( ) ( )MpMpMp outin

IBBoutin

IEEIoutin ,,, rωVrωVV

×++×+= (2-35)

In general the net mass flow through the surface, S , of a volume is:

∫∫ ⋅⋅=S

dm Mp SV

ρ (2-36)

Here, S

d denotes the surface normal and by definition points out of the volume (is directed outbound). A positive mass flow represents an outflow – see also Anderson (2011).

GRr Pr

RrzI

xIyI

Pr

MpG

Mpr

Mpr

OI

P


Looking at the second term of the right-hand side of (2-33) which describes the density changes within the body:

( )

( ) ( ) ( ) ( ) ( )[ ] dVoldt

d

dVoldt

d

Vol

PRPIBRIEER

Vol

PIP

∫∫∫

∫∫∫

⋅⋅×+×+

=⋅⋅

ρ

ρ

rωrωV

V

(2-37)

where in the above equation the rigid body assumption (2-1) is used and the reference point of the rigid body is explicitly included. Taking the constant terms out of the integral, explicitly including the centre of gravity of the rigid body, and using requirements defined in (2-31):

( ) ( ) ( ) ( ) ( )[ ]( ) ( ) ( )[ ]

( ) ( ) ( ) ( )))) ()))

0

rωrω

rωV

rωrωV

=

∫∫∫∫∫∫

∫∫∫

∫∫∫

⋅⋅×+⋅⋅×+

+⋅⋅×+

=⋅⋅×+×+

dVoldt

ddVoldt

d

dVoldt

d

dVoldt

d

Vol

PPIB

Vol

PRIB

Vol

PRIEER

Vol

PRPIBRIEER

ρρ

ρ

ρ

(2-38)

gives:

( ) ( ) ( ) ( )[ ] ( ) ( ) ∫∫∫∫∫∫ ⋅⋅×+⋅×+=⋅⋅Vol

PRIBRIEER

Vol

PIP dVol

dtdmdVol

dtd ρρ rωrωVV

(2-39)

Therefore the volume integral may be solved using the change in mass of the rigid body:

( ) ( ) ( ) ( )[ ] ( ) ( )RIBRIEER

Vol

PIP mmdVol

dtd rωrωVV

×⋅+×+⋅=⋅⋅∫∫∫ρ

(2-40)

Finally, the first term of the right-hand side of (2-33) is analysed. The absolute inertial acceleration of mass element, dm , at position, Pr , is:

( ) [ ] ( ) ( )[ ]IRPIRI

RPRIIII

P

dtd

dtd

dtd rVrrV

+

=

+

= (2-41)

For a rigid body (2-1) this gives:

( ) ( ) ( ) ( )[ ] ( ) ( ) ( )

×+

+×+

=

=

RPIBBRPI

RIEERIII

P

dtd

dtd rωrrωVV

0

(

(2-42)

and for a constant rotational rate of the earth (2-5) follows:


( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]

( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( )[ ]

××+×+××+×

+××+×+×

+×+×+=

==

=

RPIBIBBRPIBRPIBIBRPBIB

RIEIEERIERIIE

EREBERIEEB

RII

P

rωωrωrωωrω

rωωVωrω

VωVωVV

00

0

(

))())

(

(2-43)

The mass independent terms may be taken out of the integral of the first term of the right-hand side of (2-33) which thereby becomes:

( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]

( ) ( ) ( ) ( ) ( )

⋅××+⋅×

+⋅

××+×+×⋅+

=⋅

∫∫

∫

∫

m

RPIBIB

m

RPBIB

m

RIEIEEREBERIEEB

R

m

IIP

dmdm

dm

dm

rωωrω

rωωVωVωV

V

2 (2-44)

Decomposing the position vector RPr with the centre of gravity definition from (2-27):

( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]

( ) ( ) ( ) ( )

⋅+⋅××+⋅×+⋅×

+⋅

××+×+×⋅+

=⋅

==

∫∫∫∫

∫

)()

)()

00

rrωωrωrω

rωωVωVωV

V

m

P

m

RIBIB

m

PBIB

m

RBIB

RIEIEEREBERIEEB

R

m

IIP

dmdmdmdm

m

dm

2

(2-45)

gives for the first term of the right-hand side of (2-33):

( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]

( ) ( ) ( ) ( ) ( )[ ][ ]RIBIBRBIB

RIEIEEREBERIEEB

R

m

IIP

m

m

dm

rωωrω

rωωVωVωV

V

××+×⋅

+

××+×+×⋅+⋅

=⋅∫

2 (2-46)

If the system is:

- only emitting mass and

- the centre of gravity is chosen to be the reference point: ( ) 0r =R

then, when inserting (2-46), (2-40), and (2-35) into (2-33) it follows for the sum of external forces (change in the system’s linear momentum):


( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]( ) ( ) ( )[ ]( ) ( ) ( )[ ] ( ) ( ) ( )

×+⋅−×+⋅

−×+⋅

+

××+×+×⋅+⋅

=

⋅+⋅⋅+⋅=

=

∑∫∫∫∫

∑

MpMp

Mp

out

IBBout

IEEK

IEEK

IEIEEK

EBEK

IEEB

K

i

Iioutiout

Vol

PIP

Km

IIPK

T

mm

m

m

mdVoldt

ddm

rωVrωV

rωV

rωωVωVωV

VVV

F

2

,,ρ

(2-47)

In (2-47) it becomes evident that the internal changes as described in (2-40) have to be equivalent to the mass flow leaving the rigid body times its inertial velocity as given by the first two terms in the right-hand side of (2-35). The external effect of (2-40) is therefore cancelled out. As result, the sum of external forces acting on the rigid body only emitting mass is:

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]

( ) ( ) ( ) ×+⋅

−

××+×+×⋅+⋅

=

⋅+⋅⋅+⋅= ∑∫∫∫∫∑

MpMp

Mp

out

IBBout

IEIEEK

EBEK

IEEB

K

i

Iioutiout

Vol

PIP

Km

IIPK

T

m

m

mdVoldt

ddm

rωV

rωωVωVωV

VVVF

2

,,ρ

(2-48)

Equation (2-48) shows that the influence of the mass flow term on the system is independent of the system’s own velocity but is dependent on the exhaust velocity of the mass flow leaving the rigid body. Looking at the single elements of (2-48):

( )EBKV

Change of velocity of the centre of gravity in ECEF-frame with respect to the body-fixed frame

( ) ( )EK

IE Vω ×⋅2 Coriolis acceleration

( ) ( )[ ] ( )EK

OBEO Vωω ×+ Acceleration due to transport rate and body angular rates

( ) ( ) ( )[ ]IEIE rωω ×× Centrifugal acceleration due to rotation of the earth

( )Bout

MpV

Exhaust velocity of m in relation to the rigid body

( ) ( ) ( )[ ] ( )Mpout

OBEOIE rωωω ×++ Rotatory influence of exhaust position MPoutr

If the differentiation in equation (2-48) is denoted with respect to the NED-frame instead of the body-fixed frame, the result is:


( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]

( ) ( ) ( ) ×+⋅

−

××+×+×⋅+⋅

=∑

MpMp out

IOOout

IEIEEK

EOEK

IEEO

K

T

m

m

rωV

rωωVωVωV

F

2 (2-49)

The respective kinematic acceleration of the centre of gravity is found to be:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ][ ]( ) ( ) ( )

×+⋅

+××+×+×⋅−⋅= ∑MpMp

outIOO

out

IEIEEK

EOEK

IET

EOK

mm

m

rωV

rωωVωVωFV

21

(2-50)

The above equation is further analysed for the assumption of quasi-steady mass in appendix A. In case of a sufficiently small kinematic velocity (see appendix A), the earth may be modelled as non-rotating and flat whereby the NED-frame becomes an inertial frame:

( ) ( )Oout

T

EOK

Mp

mm

mVFV ⋅+⋅= ∑1

(2-51)

Equation (2-51) allows for a proper analysis whether or not the exhausted mass flow has to be included in the modelling process. To decide on the inclusion, the exhausted mass flow times its exhaust velocity is to be compared to the sum of external forces acting on the rigid body – in each dimension of (2-51). If:

( )1<<

⋅

∑ T

Oout

Mpm

F

V

(2-52)

the fraction in (2-52) is much smaller than one, the exhausted mass flow may be neglected in the modelling process. In most aircraft applications the acting forces (e.g. the thrust, drag, lift) are orders of magnitude higher than the product of mass flow times its exhaust velocity. In case of neglecting or non-existence (e.g. electric aircraft) of the exhausted mass, the translation equations of motion for a rigid body notated in the kinematic frame read:

( )K

zT

yT

xT

EO

K

K

K

K

EO

KK

FFF

mwvu

⋅=

=

,

,

,1

V (2-53)

The first order time derivative of the non-Cartesian state vector ( )EK

KV

composed of the

kinematic velocity, KV , the course angle,

Kχ , and the climb angle, Kγ , is derived from

(2-53):

( ) ( ) ( ) ( )EK

KK

OKEK

KK

EO

KK VωVV

×+= (2-54)


Inserting the single terms gives:

( )( )

( )

( )( ) ( ) ( )

( ) ( )

⋅−

⋅⋅=

×

⋅

⋅−+

=EK

KK

K

EK

KK

KK

K

EKK

K

E

K

K

K

K

K

K

K

K

EK

K

K

EO

KK

VV

VVV

γγχ

γχγ

γχ

cos00

cos

sin

00V (2-55)

which may be stated in matrix-vector notation as:

( ) ( )

( )

( )EO

KK

EK

K

EK

K

K

EK

K

K

K

K

V

V

VV

⋅

−

⋅=

100

0cos

10001

γγχ (2-56)

In case of a flat non-rotating earth (see appendix A) and neglecting or non-existence of the exhausted mass flow, the translation equations of motion then read:

( ) ( )

( ) K

EK

zT

EK

K

yT

xT

EK

K

K

K

K

V

FV

F

F

m

V

−

⋅⋅=

,

,

,

cos1

γγχ

(2-57)

2.1.4 Attitude The attitude equations are, like the position equations, purely kinematic equations. The attitude states of the rigid body are – in general – the Euler angles: the azimuth angle, Ψ , the inclination or pitch angle, Θ , and the bank angle, Φ . They relate the body-fixed frame to the NED-frame as depicted in Figure 2.1-5.

Figure 2.1-5 Coordinate systems, angular rates, and corresponding angles

KµKK αβ ,−KK γχ ,

ΦΘΨ ,,

AA γχ , AA αβ ,−

Aµ

IEω EOω',ϕλ

OBωσκ ,

KK

AA

O BEI P


For a general treatment of reference frames and associated (preferred) coordinate systems the reader is advised to the book of Zipfel (2007), chapter 3 Frames and Coordinate Systems and the therein cited literature. In Figure 2.1-5 the main coordinate systems are depicted. Starting from the only translating non-accelerated inertial frame, I , the rotating ECEF-frame, E , is developed. Two rotations are necessary to transform the ECEF-frame into the NED-frame: about the geodetic

longitude, λ, and about the angle ( )2' πϕϕ +−= .

The task to define the attitude equations of motion will – generally speaking – always involve the body-fixed frame and the NED-frame. Their relation may be directly established by using

the Euler angles Ψ , Θ , and Φ with the respective rotation matrix BOM as well as its

derivative. An equivalent representation is found using the rotated kinematic system OKM :

( ) ( ) ( )KKKOKKKKBBO µγχβα ,,,,,!

MMM ⋅=ΦΘΨ (2-58)

In case of the existence of wind, the aerodynamic frame, A , differs from the rotated

kinematic frame, K , opening a third branch to connect the body-fixed frame to the NED-frame. The system P results from the body-fixed frame, B , by the positive rotations about κ and σ around the z-axis of the body-fixed frame and subsequently the new y-axis. The rotation matrix from the NED-frame to the body-fixed frame results from the three sequentially conducted rotations about Ψ , Θ , and Φ .

BO

BO

Φ⋅ΘΦ⋅Ψ−Φ⋅Θ⋅ΨΦ⋅Ψ+Φ⋅Θ⋅ΨΦ⋅ΘΦ⋅Ψ+Φ⋅Θ⋅ΨΦ⋅Ψ−Φ⋅Θ⋅Ψ

Θ−Θ⋅ΨΘ⋅Ψ

=

coscossincoscossinsinsinsincossincossincoscoscossinsinsincossinsinsincos

sincossincoscosM

(2-59)

By the strapdown equation/ Poisson’s kinematical equations (Stevens and Lewis, 2003), the motion of the body-fixed frame relative to the NED-frame is described. The matrix components are the angular rates measured in the body-fixed frame.

( )BB

OBx

OBy

OBx

OBz

OBy

OBz

OBBOBBOB

−−

−=⋅=

00

0

ωωωω

ωωMMΩ (2-60)

This gives the kinematic roll, pitch, and yaw rate ( )TBOBK

OBK

OBK rqp ,, as:

( )

B

BB

OBK

OBK

OBK

BOB

rqp

ΨΘΦ

⋅

Θ⋅ΦΦ−Θ⋅ΦΦ

Θ−

=

Θ⋅Φ⋅Ψ+Φ⋅Θ−Θ⋅Φ⋅Ψ+Φ⋅Θ

Θ⋅Ψ−Φ=

=

coscossin0cossincos0

sin01

coscossincossincos

sinω

(2-61)


Inverting the matrix in (2-61) gives the Euler kinematical equations (Stevens and Lewis, 2003) which is the differential equation for the Euler angles:

B

OBK

OBK

OBK

Brqp

⋅

ΘΦ

ΘΦ

Φ−ΦΘ⋅ΦΘ⋅Φ

=

ΨΘΦ

coscos

cossin0

sincos0tancostansin1

(2-62)

Due to the uniqueness of the inversion, the singularity at 2/π±=Θ of the above matrix cannot be avoided in the representation of the attitude as in (2-62). For the classical quaternion representation of the attitude equations of motion see appendix B. The first order time derivative of the Euler angles may also be calculated by solely using quaternions and their derivatives as defined in appendix B (Lenz, 2006):

[ ] [ ] [ ] [ ]( )[ ]2

2031

110030213021120323

22

21

20

2182

qqqqqqqqqqqqqqqqqqqqqqqq

−⋅−+⋅+⋅−+++⋅−−+⋅

=Ψ (2-63)

[ ]( )[ ]2

2031

31201302

21

2

qqqq

qqqqqqqq

−⋅−

−+−⋅=Θ

(2-64)

[ ] [ ] [ ] [ ]( )[ ]2

2031

330010323223100123

22

21

20

2182


−⋅−+⋅+⋅−+++⋅+−−⋅

=Φ (2-65)

With the definition of the quaternions (B-3), the subsequently derived equations are valid for the Euler angles. The quaternions may, however, be defined for any other set of rotation

angles (e. g. ( )KKK µγχ ,, ) not changing the results shown if the rotational rates are

appropriately adapted. As for example in Fisch et al. (2012a): to avoid singularities within the translation equations of motion at aerodynamic climb angles of 2/π± , the quaternions are implemented to substitute the aerodynamic course angle, the aerodynamic climb angle, and

the aerodynamic bank angle ( )AAA µγχ ,, . Since the quaternions replace the aerodynamic

flight-path angles, they are termed aerodynamic quaternions.

2.1.5 Rotation The differential equation for the rotation shows the influence of external moments on the rotatory states ( )B

IBω .

The angular momentum IOdH

of a mass element, dm , at position Pr about the origin of an

inertial system, IO , is defined as:

( )[ ] dmd IPK

POI ⋅×= VrH

(2-66)


Integrating over all mass elements of a rigid body with reference point, R , gives the angular momentum as:

( ) ( ) ( ) ( )∫∫∫∫ ⋅×+⋅×+⋅×+⋅×

=

m

IRPK

RP

m

IRK

RP

m

IRPK

R

m

IRK

R

O

dmdmdmdm

I

rrVrrrVr

H

(2-67)

By taking the constant terms out of the integral and respecting the rigid body assumption (2-1), results:

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )∫∫

∫

⋅

×+×+×⋅+

+⋅

×+×+⋅×=

=

=

m

RPIBBRPK

RPIRK

m

RP

m

RPIBBRPK

RIRK

RO

dmdm

dmmI

rωrrVr

rωrrVrH

0

0

(

(

(2-68)

Decomposing the position vector RPr to show the static mass moment using the definition of the centre of gravity (2-27), leads to:

( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]∫∫∫

∫∫

⋅××+×⋅+×⋅+

+

⋅××+⋅××+⋅×=

=

=

m

RPIBRPIRK

m

PIRK

m

R

m

PIBR

m

RIBRIRK

RO

dmdmdm

dmdmmI

rωrVrVr

rωrrωrVrH

0

0

)()

)()

(2-69)

When transforming the last cross-product into a matrix-vector-product:

( ) ( ) ( ) ( ) ( )BIB

BBRP

BIB

BB

RPRP

RPRP

RPRP

BRP

BIB

xyxz

yzωΦωrω ⋅−=⋅

−−

−−=×

00

0

(2-70)

this gives for the angular momentum of the rigid body:

( ) ( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( ) ( )[ ]∫ ⋅⋅×−⋅×

+⋅××+⋅×=

m

IBRPRPIRK

R

RIBRIRK

RO

dmm

mmI

ωΦrVr

rωrVrH

(2-71)

Again transforming the last cross-product into a matrix-vector-product:

( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( )IBRIRK

RRIBRIRK

R

O

mmm

I

ωIVrrωrVr

H

⋅+⋅×+⋅××+⋅×

= (2-72)

wherein the inertia tensor, RI , in (2-72) is defined as:

( ) ( )∫ ⋅⋅−=m

RPRPR dmΦΦI (2-73)


The inertia tensor denoted in the body-fixed frame, B , reads:

( )( )[ ] ( )[ ] ( ) ( )[ ] ( ) ( )[ ]

( ) ( )[ ] ( )[ ] ( )[ ] ( ) ( )[ ]

( ) ( )[ ] ( ) ( )[ ] ( )[ ] ( )[ ]

⋅+⋅⋅−⋅⋅−

⋅⋅−⋅+⋅⋅−

⋅⋅−⋅⋅−⋅+

=

∫∫∫

∫∫∫

∫∫∫

22

22

22

mB

RPB

RP

mB

RPB

RP

mB

RPB

RP

mB

RPB

RP

mB

RPB

RP

mB

RPB

RP

mB

RPB

RP

mB

RPB

RP

mB

RPB

RP

BBR

dmyxdmzydmzx

dmzydmzxdmyx

dmzxdmyxdmzy

I

(2-74)

In case of the aircraft being symmetrical in its body-fixed x-z-plane, the cross-products of inertia

xyI and zyI vanish.

The linear momentum of the rigid body (2-1) is using additionally the centre of gravity definition (2-27):

( ) ( ) ( ) ( ) ( ) ( ) mmdmdm RIBIRK

m

IRPK

m

IRK

I ⋅×+⋅=⋅+⋅= ∫∫ rωVrVp

(2-75)

Inserted into the above angular momentum equation (2-72):

( ) ( ) ( ) ( ) ( ) mIRK

RIRIBROI ⋅×+×+⋅= VrprωIH

(2-76)

In the following, Newton’s second law of motion is applied to the angular momentum of the body. The sum of all external moments around the origin of the inertial system equals the change in the angular momentum of the body itself about the origin plus the convective change in angular momentum due to absorbed or emitted mass flows; see also Thomson (1966).

( ) ( )IOconvBody

OI

O III

dtd

,HHM ∆+

=∑ (2-77)

The convective term for the emitted mass flows reads:

( ) ( )∑ ⋅

×=∆

iiout

Iiout

IO

convBody mMpMpI,,,

VrH (2-78)

The velocity of the i-th mass flow leaving the rigid body may be calculated in accordance with (2-35). It is sensible to assume a rigid body of quasi-steady mass since in most aerospace configurations a rotatory influence of exhausted masses is not desired and thus avoided. Moreover, a general – due to mass flows – time-varying inertia tensor (in the body-fixed frame) is not accessible for an analytic analysis. Rocket (attitude) control thrusters can be modelled as external systems. Rotatory masses are regarded as additional systems and are exemplarily treated for the engine in the propulsion part: paragraph 2.2.3. The quasi-steady mass assumption reads:


( ) 0H0p

≡

∂∂

≡∂

∂mm

IOI

and (2-79)

The above equation should emphasise that there is no direct influence of a change in the

body’s mass on the body’s linear momentum, ( )Ip , or its angular momentum, IOH

. The

variation in mass is of negligible dynamic effect (see also Stengel (2004)). The change in the body’s mass has no effect as an external force (see also explanation of (2-47)) or an external moment, and the convective terms are omitted in (2-77) and (2-79). The body’s mass itself, however, may still be varying with time represented by the respective differential equation. The sum of external moments now equals the change in the angular momentum of the body without regarding influences due to mass changes. Equation (2-77) then reads:

( ) II OI

O

dtd HM

=∑ (2-80)

If the above assumptions are met, the inertia tensor is time-invariant:

( ) 0I =BR (2-81)

Performing the inertial derivation of equation (2-76) in (2-80):

( ) ( ) ( )[ ]( ) ( ) ( ) ( )( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( )[ ]mmm IR

KIBR

IBRK

RIRK

RIB

IIRIIRK

IBRIBBIBROI

⋅××+⋅×+⋅××

+×+×

+⋅×+⋅=∑

VωrVrVrω

prpV

ωIωωIM

(2-82)

The moments about the inertial origin, IO , can be decomposed into the moments about the

reference point, R , and the forces acting on the reference point times their lever arm, Rr ; see also Figure 2.1-6:

( ) ( ) ( ) ( )∑∑∑ ×+= RRROI FrMM

(2-83)

Figure 2.1-6 External forces and moments about reference point

R

Pr

( )∑ RF( )∑ RM

G

PRr

Rr Pr

zI

xIyIOI


With the above assumptions the derivative of the linear momentum equals the sum of external forces acting on the reference point:

( ) ( )∑= RII Fp

(2-84)

Inserting (2-84), (2-83), and (2-75) into (2-82) gives for the sum of external moments about the reference point:

( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( ) mIR

KIB

IBRK

R

IBRIBBIBRR

⋅

×+×+

+⋅×+⋅=∑VωVr

ωIωωIM

(2-85)

If the centre of gravity, , is chosen to be the reference point, R , and the change of rotational states is denoted in the body-fixed frame, B , their dynamic differential equation reads:

( ) ( ) ( ) ( ) ( ) ( )[ ] BIB

BB

BIB

B

BBB

BIB ωIωMIω

⋅×−⋅= ∑−1 (2-86)

Looking at the single elements: ( )B

M

Influence of external moments ( ) ( ) ( )[ ]B

IBBB

B

IB ωIω ⋅× Inertia coupling In Pamadi (2004) an investigation on the coupling between latitudinal and longitudinal motion is given in Chapter 7: Inertia Coupling and Spin. In case of a flat, non-rotating earth (for the decision criteria see appendix A), the rotational

states are the roll, pitch, and yaw rates, ( )BOBω :

( ) ( ) ( ) ( ) ( ) ( )[ ] BOB

BB

BOB

B

BBB

BOB ωIωMIω ⋅×−⋅= ∑−1

(2-87)

For a symmetric aircraft the inverse of the inertia matrix (2-74) is:

( )

⋅−⋅

⋅⋅−⋅

=−

x

xz

y

xz

xz

z

x

xz

z

xz

xz

z

x

BB

III

IIIIII

IIII0

00

011I (2-88)

As mentioned earlier, (2-87) makes use of the quasi-steady mass assumption (2-79). If the influence of a mass flow leaving the aircraft is of measurable effect on its attitude, in the modelling process equations (2-77) and (2-78) have to be used.


2.1.6 Application The above presented formulae provide a profound insight into the physical basis of flight mechanics and a deeper understanding to evaluate mass changes of the flying system. Of course, the derived equations have to be adapted for the special case of application. From (2-51), for example, the Tsiolkovsky rocket equation may directly be deduced by setting the external forces to zero. For a derivation of the ideal rocket equation see for example Forward (1995). The expected or ideally known dynamics level of the trajectory to be simulated or optimised, determines the modelling depth of the regarded flight system.

Figure 2.1-7 Causal chain – compare TUM-FSD (2014)

In Figure 2.1-7 the primal physical causal chain from a commanded elevator deflection,

CMDη∆ , to a change in altitude, h∆ , is exemplarily depicted. The true flight system is

therefore of order six. Here, the influence of different time scales becomes evident. From the point of view of the translational dynamics, the activities of the rotational dynamics have decayed enabling the lift coefficient to be used as a control input if only smooth variations in lift occur. The same holds for the actuator dynamics as seen from the rotational dynamics. If for example the pitch rate variations are smooth, an elevator deflection can be taken as control input. When a constant rate climb ( ).consth = from sea level to FL360 is regarded, a

point mass modelling is sufficient. For a high agility fighter aircraft in terrain following mode, however, the non-minimum phase behaviour of the altitude due to an elevator deflection may play a crucial role. Since in optimal control the control input may vary at an infinite rate, care has to be taken that the mathematical modelling matches the underlying physics. It may for a point mass modelling be necessary to take, for example, the first derivative of the lift coefficient as control input, or in case of a six-degree of freedom model the roll rate build-up may be appropriately modelled by using the second derivative of the roll rate as control input.

η∆ ηM q α L∆ γ hq γ h∆CMDη∆ η η

h hhh)4(h)6(h )5(h

Actuator Dynamics Rotational DynamicsMoments

Translational DynamicsForces

∫∫ ∫ α∆ ∫∫∫


2.2 External Forces and Moments The sum of external forces acting on the centre of gravity consists of the forces due to gravity, the sum of aerodynamic forces, and the sum of propulsive forces:

( ) ( ) ( ) ( )∑∑∑∑ ++= KPK

AK

K

T FFFF

(2-89)

The total of external moments acting on the centre of gravity is composed by the aerodynamic moments and the moments due to propulsion:

( ) ( ) ( )∑∑∑ += BPB

AB

T MMM

(2-90)

In the following, the emphasis is put on gravity, since a deep analysis within the aerodynamic and propulsive disciplines lies beyond the scope of this work. Real system aerodynamic and engine data sets may either be implemented using lookup-tables or may become accessible via curve fitting tools which then also allow the differentiation of the given data up to the requested degree to be used within the optimisation process.

2.2.1 Gravity Modelling gravity is closely related to the modelling of the figure of the earth. In the present paragraph, first it is shown, how gravity may be calculated when the WGS 84 ellipsoid modelling is applied. There, the earth is defined as an equipotential (= level) geocentric ellipsoid of revolution on which the gravity vector stands perpendicular and becomes slightly deflected with an increase in altitude. This non-trivial consequence is profoundly evaluated. Neglecting the deflection, an approximation for the reduction of gravity with altitude is shown. The WGS 84 ellipsoid is an approximation to the (true) equipotential surface of the earth called the geoid. At the end of the paragraph the Earth Gravitational Model 1996 (EGM96 – see also Lemoine et al. (1998)) and the further refined Earth Gravitational Model 2008 (EGM2008 – see also Pavlis et al. (2012)) are shortly presented. One of the first to calculate the attraction due to mass of homogenous ellipsoids has been Carl Friedrich Gauß in 1813 – see Gauß (1813) – beside Laplace (in 1782), Ivory (in 1809), Chasles (in 1838), and Dirichlet (in 1839). Of course, the density of the earth is strongly inhomogeneous, see Moritz (1990). But when a mean density is calculated from WGS 84 values, the deviation between the formulae given by Gauss and the WGS 84 modelling differ by less than 2‰, see Schatz (2010). The paragraph 2.2.1 is closely related to Moritz (1990) and especially Hofmann-Wellenhof and Moritz (2006) to which the reader is advised for a more detailed analysis. Here, it is important to present the basic ideas established there enabling the gravity modelling to allow for a proper evaluation of the results and the consequences for the simulation and optimisation of flight trajectories. The equipotential ellipsoid serves as a reference surface for both the geometry of the earth as well as for its external gravitational field. Gravity is the combination of gravitation and the


centrifugal force a body experiences when it is in constraint motion with the rotating earth or its static atmosphere. Newton’s law of gravitation describes the cumulative attraction of the point masses 1m and

2m , separated by the distance, l , with being the Newtonian gravitational constant.

Fravitation1,2 [N] 221

lmm ⋅

⋅= (2-91)

If the larger mass, 1m , is called the attracting mass and the smaller mass, 2m , is called the

attracted mass by which (2-91) is normalised, then the equation reads:

Fravitation

⋅=

kgN 2

1

lm (2-92)

which describes the force per unit mass, and may be abbreviated by 'g . For the earth, the

gravitational constant including the mass of its atmosphere (GM) may be found in Table 2.1-1. The potential of gravitation, V , of a point mass is a scalar function:

⋅=

kgNm 1

lmV (2-93)

The potential of gravity, U , is the sum of the potential of gravitation, V , and of the potential of the centrifugal force, Φ :

Φ+= VU (2-94)

In order to enable a full consistency with the equations of motion derived earlier, the potential of gravitation, V , and the potential of the centrifugal force, Φ , are treated separately. Only from (2-136) on, they are combined in the potential of gravity, U . The resultant scaled gravity vector (2-138) has the elements due to the centrifugal influence clearly marked. For the earth, the potential of gravitation, V , is found to be:

+= 3

1r

Or

MV (2-95)

where r is the magnitude of the radius vector to the attracted mass in spherical coordinates. Thereby it is shown, that at large distances every attracting mass acts like a point mass.


The gravitational force at a position on or outside the attracting body may directly be derived from the gradient of the potential (first derivative of the potential of gravitation, V , to the position):

( )

i

iinravitatio

zVyVxV

V

∂∂∂∂∂∂

=∇=F

(2-96)

The trace of the second order gravitational gradient tensor which includes the second derivatives of the potential of gravitation, V , has to satisfy Poisson’s equation:

ρπ ⋅⋅−=∂∂

+∂∂

+∂∂

=∆ zV

yV

xVV 42

2

2

2

2

2

(2-97)

Poisson’s equation has to be zero outside the attracting body where the density, ρ , is

supposed to be zero or of negligible influence. Harmonic functions are by definition the solutions of the resultant Laplace’s equation which reads:

02

2

2

2

2

2

=∂∂

+∂∂

+∂∂

=∆zV

yV

xVV (2-98)

Any solid body has a gravitational potential which is harmonic outside the body. For the definition of a conservative vector field and the definition of the harmonic potential as well as an example, see appendix C. For both, gravitation as well as centrifugal force the investigation is concerned with the influence of the rotating ellipsoid of revolution with an attached atmosphere of negligible mass on a point P outside the ellipsoid but within the atmosphere. For the calculation of gravity and gravitation, the reference ellipsoid and ellipsoidal harmonic coordinates are introduced. The linear eccentricity, E , results from the focal triangle:

222 Eba += (2-99)

The value of E for the WGS 84 modelling may be found in Table 2.2-1. For the WGS 84 ellipsoid defining parameters see Table 2.1-1. The linear eccentricity is the product of the first eccentricity, e , and the semi-major axis, a :

aeE ⋅= (2-100)


The elements of (2-100) are depicted in Figure 2.2-1.

Figure 2.2-1 Confocal ellipsoid through point P – compare Hofmann-Wellenhof and Moritz (2006)

As shown in Figure 2.2-1, the reference ellipsoid has the semi-major axis, a , the semi-minor axis, b , and the linear eccentricity, E . The two foci are 1F and 2F . With the same origin, foci

and axis of rotation, z , an ellipsoid is laid through the point P (= confocal ellipsoid). In Figure 2.2-2, the confocal ellipsoid through P is depicted in black. Moreover, the inscribed circle of radius u is given where u is the semi-minor axis of the new ellipsoid

through P . In addition the circumscribed circle of radius 22 Eu + is depicted which is the

semi-major axis of the new ellipsoid.

P

1F 2F

O

b

Ea 22u E+

u

z


Figure 2.2-2 Ellipsoidal harmonic coordinates for point P in front and top view – compare Hofmann-Wellenhof

and Moritz (2006)

The ellipsoidal harmonic coordinates of the point P are the semi-minor axis, u , the reduced latitude, β , and the geocentric longitude, λ . The three ellipsoidal harmonic coordinates are

orthogonal. The transformation to Cartesian coordinates is given by:

λβ coscos22 ⋅⋅+= Eux (2-101)

λβ sincos22 ⋅⋅+= Euy (2-102)

βsin⋅= uz (2-103)

x

y

P

P

1F 2FOβ

λ

planexy −

z

βcosu 22 ⋅+ E

u

22u E+ z⋅


To find the direction of the unit vectors for the ellipsoidal harmonic coordinates in point P ,

the derivative of the position vector, P

, with respect to each of the coordinates including a normalisation is to be performed. This gives:

⋅⋅+

⋅⋅+

⋅⋅+

+=

∂∂

⋅

∂∂

=

β

λβ

λβ

βsin

sincosu

coscosu

sinu1

22

22

222

22

Eu

Eu

EuE

uu

uP

Pe

(2-104)

⋅⋅⋅+−⋅⋅+−

⋅⋅+

=∂∂

⋅

∂∂

=β

λβλβ

βββ

β

cossinsinucossinu

sinu11 22

22

222

uEE

EP

Pe

(2-105)

−=

∂∂

⋅

∂∂

=0

cossin

1 λλ

λλ

λP

Pe

(2-106)

The unit vectors stand – and thus the ellipsoidal harmonic coordinates – perpendicular to each other and form a left-hand system. The abbreviation w of which the inverse is used for the normalisation of the first unit vector

in ellipsoidal harmonic coordinates, ue , in (2-104) is introduced as:

( ) 22

222

usin,E

Euuw+

⋅+=

ββ (2-107)

To show the normalisation, the unit vectors are written as:

⋅⋅+

⋅⋅+

⋅=

β

λβ

λβ

sin

sincosu

coscosu

122

22

Eu

Eu

wue (2-108)

⋅⋅⋅+−⋅⋅+−

⋅+⋅

=β

λβλβ

β

cossinsinucossinu

u1 22

22

22uEE

Ewe (2-109)


⋅⋅+⋅⋅+−

⋅⋅+

=0

coscosusincosu

cosu1 22

22

22λβλβ

βλ E

E

Ee (2-110)

The new ellipsoid laid through P is defined by:

012

2

22

22

=−+++

=uz

EuyxF (2-111)

Evaluating the gradient of (2-111) gives the normal to the surface of the ellipsoid:

uwuE

uE

u

u

uzEu

yEu

x

F e⋅⋅=

⋅⋅+

⋅⋅+

⋅=

+

+

=∇2

sin

sincosu

coscosu

2

2

2

2

22

22

2

22

22

β

λβ

λβ

(2-112)

The unit vector ue therefore stands perpendicular to the surface of the confocal ellipsoid laid

through P and points outwards.

Figure 2.2-3 Ellipsoidal unit vectors

ueβe

λe×

1F 2FO

z

P


In Figure 2.2-3 the ellipsoidal unit vectors in point P are shown. Additionally, the respective hyperboloid through P for const=β is sketched to show the variation of the ellipsoidal unit

vectors with u . The hyperboloid shows the direction of the normal gravity vector. Stating the Laplace equation (2-98) in ellipsoidal harmonic coordinates, reads:

( ) ( ) 0cossintan2

sin1

2

2

222

222

2

2

2

222

222

=

∂∂

⋅⋅+⋅+

+∂∂

⋅−∂∂

+∂∂

⋅+∂∂

⋅+

⋅⋅+

=∆

λββ

ββ

β

β

VEuEuVV

uVu

uVEu

EuV

(2-113)

Although in (2-113) the three ellipsoidal harmonic coordinates appear in mixed form in the single terms, the solution of (2-113) to determine the potential of gravitation, V , is obtained by separation of the variables. Here it is assumed that the potential of gravitation may be given as the product of three independent functions, each of them depending on one ellipsoidal harmonic coordinate only:

( ) ( ) ( ) ( )λβλβ 321,, ffufuV ⋅⋅= (2-114)

Due to the rotational symmetry of the ellipsoid, the non-zonal terms which depend on λ have to vanish, giving:

( ) ( ) ( ) ( ) ( )1sin361

tan

tan

31, 2

0

22

1

1

220 −⋅⋅⋅⋅+

⋅

⋅⋅−=

−

−

βωωβquqa

bEuE

aUuV IEIE (2-115)

where the subscript 0 denotes the respective value on the reference ellipsoid ( )bu = . 0U is

the potential of the normal gravity field on the ellipsoid and is for the moment unknown. The abbreviations in (2-115) are:

( )

⋅−

⋅

⋅+⋅= −

Eu

uE

Euuq 3tan31

21 1

2

2

(2-116)

⋅−

⋅

⋅+⋅= −

Eb

bE

Ebq 3tan31

21 1

2

2

0 (2-117)

With the series expansion of arctan(x):

( ) ...753

tan753

1 +−+−=− xxxxx (2-118)

the abbreviation ( )uq becomes:

( )

−

⋅+

⋅−

⋅⋅= ...

6312

358

154

21 753

uE

uE

uEuq (2-119)

and therefore ( )uq decays exponentially with an increase in u .


The gravitational potential, V , is therefore:

( ) ( )

+⋅

⋅

⋅⋅−=

−3

1

220

11

tan31,

uO

ubE

EaUuV IEωβ (2-120)

The aim is now to express 0U in terms of the mass, M , of the ellipsoid of revolution. For

large distances, (2-95) has to hold. Therefore the influence of ∞→u is investigated. The squared distance of the point P from the origin may be expressed in Cartesian, spherical, and ellipsoidal coordinates:

β2222222 cos⋅+==++ Eurzyx (2-121)

This gives for the inverse of u :

β222 cos11

⋅−=

Eru (2-122)

In order to evaluate (2-122) for large values of u , a power series expansion of the right-hand side is performed where s represents the reciprocal of r (2-123) and the limes is determined for small values of s (2-124):

rs 1

= (2-123)

( ) ( )sfrfsr 0limlim

→∞→= (2-124)

If the substitution described in (2-123) is done in (2-122), the result is:

β222 cos11

⋅⋅−=

Ess

u (2-125)

Performing a power series expansion for the right-hand side of (2-125), gives:

( ) ...2cos1 3

22

+⋅⋅

+⋅= sEssf β (2-126)

or re-substituting r :

( ) ...12cos1

3

22

+⋅⋅

+=r

Er

rf β (2-127)

Finally, (2-122) can be written as:

+= 3

111r

Oru

(2-128)


Inserting (2-128) into (2-120) gives for the gravitational potential:

( )

+⋅

⋅

⋅⋅−=

−3

1

220

11

tan31

rO

rbE

EaUV IEω (2-129)

This enables the comparison of (2-129) with (2-95). Thereby the desired relationship between the earth’s gravitational constant (see also Table 2.1-1) and the reference gravity

potential, 0U , is established:

( )

⋅

⋅⋅−=

−

bE

EaUM IE

1

220

tan31 ω

(2-130)

Thus, the reference gravity potential, 0U , on the reference surface, 0S , is:

( ) 2210 3

1tan abE

EMU IE ⋅⋅+

⋅= − ω (2-131)

It is important to mention, that the determination of the reference gravity potential, 0U , is

independent of the knowledge of the density distribution within the earth.

The reference gravity potential, 0U , is now inserted into (2-115) for the gravitational

potential on and above the reference ellipsoid:

( ) ( ) ( ) ( )1sin361tan, 2

0

221 −⋅⋅⋅⋅+

⋅= − βωβ

quqa

uE

EMuV IE

(2-132)

The potential describing the centrifugal force, Φ , is:

( )

⋅⋅=Φ 2

222

sm

21 pIEω (2-133)

with p being the distance from the rotational axis as in paragraph 2.1.2 in Figure 2.1-1.

The centrifugal force (more precisely: acceleration) is just as the gravitational force (2-96) derived from the gradient of the potential:

∂Φ∂

∂Φ∂

∂Φ∂

=Φ∇= 2sm

z

y

x

lCentrifugaF

(2-134)


For the ellipsoid through P , the potential describing the centrifugal force in ellipsoidal harmonic coordinates is (see Figure 2.2-2):

( ) ( ) ( ) βωβ 2222 cosu21, ⋅+⋅⋅=Φ Eu IE

(2-135)

Therefore, the normal gravity potential, U , which is the sum of the gravitational potential (2-132), V , and the potential associated with the centrifugal force (2-135), Φ , is:

( ) ( ) ( )

( ) ( ) ( ) ( ) βωβω

βββ

22222

0

221 cosu21

31sin

21tan

,,,

⋅+⋅⋅+

−⋅⋅⋅⋅+

⋅=

=Φ+=

− Equqa

uE

EM

uuVuU

IEIE (2-136)

(i) Acceleration due to Gravity In order to become directly accessible in Cartesian coordinates, when taking the gradient of U from (2-136), the same normalisation as for the ellipsoidal unit vectors (from equation (2-104) on) has to be included. The gravity vector in ellipsoidal harmonic coordinates is:

( )( )( )( )

( )

( )

( )

∂∂

⋅⋅+

∂∂

⋅+⋅

∂∂

⋅

=

=

λβ

β

ββ

β

βββ

β

λ

β

,cos

1

,1

,1

,,,

,

22

22

uUEu

uUEuw

uuU

w

ugugug

uu

g (2-137)

Due to the rotational symmetry, the term ( )βλ ,ug is always zero.

As result, the properly scaled gravity vector (with the centrifugal influence indicated) in the direction of the ellipsoidal harmonic coordinates is:

( )( )( )

( ) ( ) ( )

( ) ( ) ( )

⋅⋅⋅

⋅−+⋅

+⋅−

⋅⋅−

−⋅

′⋅

+⋅⋅

⋅++

⋅−

=

0

cossin1

cos31sin

211

,,,

2

0

222

22

222

022

22

22

ββω

βωβω

βββ

λ

β

IE

lcentrifuga

lcentrifuga

IEIE

u

quqaEu

Euw

uq

uqEu

EaEu

Mw

ugugug

)()

)) ()) (2-138)


Again, the subscript 0 denotes the respective value on the reference ellipsoid. The additional abbreviation in (2-138) is:

( ) 1tan113 12

2

−

⋅−⋅

+⋅=′ −

uE

Eu

Euuq (2-139)

The additional abbreviation (2-139) is the derivative of (2-116) to u times a factor:

( ) ( )uuq

EEuuq

∂∂

⋅

+−=′

22

(2-140)

With (2-138), the precise (in accordance with WGS 84 modelling) acceleration due to gravity can be determined on as well as above the ellipsoid. (2-138) is given in the above form, since on the northern part for all altitudes above the reference ellipsoid, the gravity vector

components ug and βg are always negative – see also Figure 2.2-3. ug therefore points

downwards, and on the northern hemisphere βg points into southern direction.

In the last section of this paragraph, the aim is to determine the acceleration due to gravity in the NED-frame at a position defined in the ECEF frame. If the position is given as shown

in Figure 2.1-1 as ( )hP ,,ϕλ using the WGS 84 modelling, first, the position has to be

calculated in Cartesian coordinates.

( ) ( ) ( ) ( ) ( )λβϕλ ,,0,,,, 00 bhzyx uEP

EP err ⋅+= (2-141)

For zero altitude, the geodetic latitude, 0ϕ , is transformed to the reduced latitude, 0β , via:

00 tantan ϕβ ⋅=ab

(2-142)

and the normalisation term, 0w , is:

20

222

0sin

aEbw β⋅+

= (2-143)

The semi-minor axis of the confocal ellipsoid through P is found from solving (2-111) to u :

( ) ( ) ( )[ ] ( ) ( ) ( )[ ] ( ) 22222222222 42

1 ErrrrEErrr

u

Pz

Pz

Py

Px

Pz

Py

Px ⋅⋅+−−−+−++⋅

=

(2-144)

This gives the confocal ellipsoid, which features, however, no longer an equipotential surface and thus the gravity vector has a tangential component.

The new reduced latitude, altβ , of the confocal ellipsoid is found from equations (2-101) to

(2-103) as:

( ) ( ) uEu

rr

rPy

Px

Pz

alt

22

22tan +

⋅+

=β (2-145)


Inserting (2-144) and (2-145) into (2-138) gives the gravity vector in coordinates of the confocal ellipsoid which then has to be transformed to the NED-frame. To find the direction of the new ellipsoidal unit vectors (index: alt), the angle between altu ,e

and the horizontal, altϕ , at the point P above the reference ellipsoid can be calculated using

the fact that altu ,e is a unit vector by:

( )( )2

,,

,,

1tan

zaltu

zaltualt

e

e

−=ϕ (2-146)

The difference between the geodetic latitude, 0ϕ , from which on the height, h , is measured,

and altϕ gives the tilting of the unit vectors:

altϕϕϕ −=∆ 0 (2-147)

A negative turn about ϕ∆ around the axis in the direction of alt,λe aligns altu ,e with ue and

thus the z-axis of the NED-frame. The transformation of the gravity vector derived in (2-138) to the NED-frame is therefore performed by:

( )( ) ( )

( ) ( )

( )

( )

−⋅

∆∆−

∆∆⋅=

⋅=

β

β

ϕϕ

ϕϕ β

,0

,

cos0sin010

sin0cos

ug

ugm

ggg

m

uD

E

N

OF

(2-148)

In the subsequent figures, always the geodetic latitude on the reference ellipsoid, 0ϕ , is

depicted which is simply termed latitude.

(ii) Gravity on the Ellipsoid On the reference ellipsoid the gravity vector is normal to the level surface (= equipotential

surface), 0S , which may also be seen from (2-138) recalling (2-99):

00, =βg (2-149)

but not above.

Table 2.2-1 is giving the values of the derived geometric and physical constants for the WGS 84 ellipsoid:

Parameter Notation Value Unit Geometric constant: Linear eccentricity E 5.2185400842 · 105 m Physical constants: Normal gravity potential on the ellipsoid 0U 62.6368517146 · 106 m2/s2

Normal gravity at the equator eg 9.78032533590 m/s2

Normal gravity at the pole pg 9.83218493786 m/s2

Auxiliary quantity auxm 3.44978650684 · 10-3 - Table 2.2-1: WGS 84 ellipsoid derived geometric and physical constants


The values for gravity and gravitation as represented in (2-138) are depicted in Figure 2.2-4. At the equator, both gravity and gravitation have their minimum. At the poles, the centrifugal influence vanishes.

Figure 2.2-4 Gravity and gravitation

The influence due to rotation reaches its maximum at the equator where the difference between gravity and gravitation is 3.456‰ – see also appendix (A-4).

The value of the standard acceleration of gravity at sea-level, sg , may be found in Table

2.3-1. It is, however, valid since the 3rd General Conference on Weights and Measures in 1901 (United States Department of Commerce, 2008) which explains that nowadays it corresponds to a geodetic latitude which deviates from 45° by 30’. The radius, R, from the origin to the circles on the surface of the ellipsoid where standard acceleration of gravity is present has the value of 6.367303 · 106 m. The formula of Somigliana which is valid on the reference ellipsoid may be directly derived from (2-138) and gives the value of normal gravity in dependence on geodetic latitude where the subscript D denotes the downward direction as measured in the NED-frame:

( )ϕϕ

ϕϕϕ

2222

22

sincos

sincos

⋅+⋅

⋅⋅+⋅⋅=

ba

gbgag pe

D (2-150)

In (2-150) eg is the normal gravity at the equator and pg is the normal gravity at the poles;

both values are given in Table 2.2-1. For all applications where the influence of latitude is important but the influence of altitude may be omitted, the formula of Somigliana is very easy to handle.

9.78

9.80

g

9.82

g

9.84

Latitude [rad]

Acc

eler

atio

n [m

/s2 ]

−π/2 −π/4 0 π/4 π/2

GravityGravitation

gs

gp

ge


(iii) Gravity above the Ellipsoid Above the ellipsoid, the normal gravity reduces with altitude and a southbound tangential component rises (on the northern hemisphere). In Figure 2.2-5 the gravity vector components resulting from (2-138) are depicted in the downward direction and the southern direction of the NED-frame. All transformations necessary in (2-148) were performed which attenuates the tangential effect. The results are normalised by the standard acceleration due

to gravity at sea level, sg , from Table 2.3-1. As shown in Figure 2.2-5, the downward

acceleration reduces with an increase in altitude. The downward acceleration is symmetric to the equator and is only shown for negative latitudes in Figure 2.2-5. In addition, the southbound component of the gravity vector (2-138) transformed into the NED-frame is shown for the northern hemisphere in Figure 2.2-5. This tangential component is more than 10,000 times smaller than the downward component. On the southern hemisphere the tangential gravity component is northbound (not shown in Figure 2.2-5). So also for high altitude missions, e.g. unmanned solar missions as investigated in this work, the tangential component may normally be neglected.

Figure 2.2-5 Influence of altitude on gravity in downward and southbound direction

(iv) Spherical Approximation Since the tangential component of the gravity vector may be omitted in most cases, a Taylor series expansion for the altitude above the reference ellipsoid may be applied:

( ) ( ) ...21, 2

2

2

+⋅∂

∂⋅+⋅

∂∂

+= hhgh

hgghg DD

DD ϕϕ (2-151)

which leads to:

0.98

0.99

1.00

1.01

Latitude [rad]

Dow

nwar

d A

ccel

erat

ion

[gs]

0

10

20

30

Sout

hbou

nd A

ccel

erat

ion

[ µg s]

0km10km20km30km

-π/4-π/2 π/4 π/20


( ) ( ) ( )

⋅+⋅⋅−++⋅−⋅=

22 3sin2121,

ah

ahfmfghg auxDD ϕϕϕ (2-152)

when the series (2-151) is truncated after the squared term and the dimensionless auxiliary quantity:

( )M

bamIE

aux⋅⋅

=22

ω (2-153)

is used and only terms linear in the flattening, f , are included in (2-152). For the value of

the auxiliary quantity, auxm , in case of the WGS 84 modelling applied see Table 2.2-1.

For the WGS 84 modelling and atmospheric altitudes, (2-152) is clearly dominated by the h/a term showing an approximately linear reduction of the gravity vector with altitude. The deviation between the precise values resulting from (2-138) and the values from (2-152)

using the formula of Somigliana (2-150) for ( )ϕDg stays below 2-6 m/s 10 5 ⋅ for altitudes up

to 30km. A strongly simplified gravity model uses the standard acceleration due to gravity at sea

level, sg , from Table 2.3-1 and assumes a reduction with altitude in accordance with

Newton’s law of gravitation (2-92) neglecting the rotational influence:

( )2

+

⋅=hr

rghgE

EsD (2-154)

In this model a mean radius of the earth, Er , of 6.368 · 106 m is used. By the model, the

gravity values for geodetic latitudes of around 4/π± are well approximated for altitudes below 30km.

(v) EGM96 and EGM2008 Model For a high fidelity representation of the earth’s gravitational field, it is advised to the WGS 84 Earth Gravitational Model 1996 (EGM96) which is a spherical harmonic expansion of the gravitational potential, V . The complete model is comprised of 130,317 coefficients but may be truncated appropriately. The defining and explaining document is: The Development of the Joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) Geopotential Model EGM96 (Lemoine et al., 1998). From EGM96 a refined WGS 84 geoid has been determined (gravity potential W0 = const.): the WGS 84 EGM96 Geoid features a 15 arcminute grid and is available from NIMA. The geoid with its constant gravity potential is commonly used as a reference surface for mean sea level (MSL) heights. “Heights above sea level” refer to this geoid (orthometric height). The geoid being an equipotential surface for the actual gravity potential deviates from the WGS 84 reference ellipsoid (equipotential surface for W = W0 = U0) by not more than -110m


to +85m (max. geoidal undulation = max. geoidal height). The geoidal undulation is depicted in Figure 2.2-6. One may remember the difference between the semi-major axis and the semi-minor axis of the ellipsoid to be 21,385m. Due to the small deviation of the geoid from the ellipsoid, the earth’s gravity field can be composed of the ellipsoidal gravity field plus a small linear term.

Figure 2.2-6 Geoidal undulation for EGM96

The refined Earth Gravitational Model 2008 (EGM2008) has been released by the U.S. National Geospatial-Intelligence Agency (NGA); see Pavlis et al. (2012). The gravitational model is complete to spherical harmonic degree and order 2159, and contains additional coefficients extending to degree 2190 and order 2159. Its maximum and minimum deviation from the WGS 84 ellipsoid is the same as for the EGM96 and the maximum difference between EGM2008 and EGM96 reaches values between -8.5m to +12m (mostly located in the Antarctica). A vertical deviation by 110m equals a difference in gravity values of less than 3.4 · 10-4 m/s2 when calculated using (2-152). Due to the relatively small values of undulation as depicted in Figure 2.2-6, for the simulation and optimisation of most aircraft trajectories gravity modelled in accordance with (2-152) is by far sufficient.


2.2.2 Aerodynamics The total aerodynamic force on the aircraft, R , can be split into the drag, D , the aerodynamic sideforce, Q , and the lift, L . All of the components of R result from the

motion between the aircraft and the surrounding air, and are due to only two basic sources (Anderson, 2011): 1.) the pressure distribution over the aircraft’s surface 2.) the shear stress distribution over the aircraft’s surface. If for a fixed flow direction, the resultant total aerodynamic force, R , on the aircraft of fixed shape is expressed as the product of a dimensionless force coefficient, RC , times the

dynamic pressure, q , as defined in (2-180) times the wing reference area, S , it reads:

RCSqR ⋅⋅= (2-155)

As for example shown in Anderson (2011), this dimensionless force coefficient, RC , is a

function of only the Reynolds number, Re , and the Mach number (2-179), M :

( )MRefCR ,= (2-156)

The Reynolds number is the ratio of the inertial forces to the viscous forces in a flow. In case of an aerofoil, the characteristic length is the mean aerodynamic chord, c , and Re is typically defined as:

air

A

air

A cVcVReνµ

ρ ⋅=

⋅⋅= (2-157)

with the density, ρ , the aerodynamic velocity, AV , the dynamic viscosity of the surrounding

air, airµ , and the kinematic viscosity of the surrounding air, airν , which is for the standard

atmosphere 14.607 · 10-6 m2/s (DIN ISO 2533, 1979). The aerodynamic force coefficients are modelled by analytic functions in dependence on the steady and unsteady flow angles. Within these functions numerical values for the respective factors occur; e.g. for the induced drag. These values are only valid within certain regions of the Reynolds number and the Mach number range and otherwise need to be adapted. The same holds for the aerodynamic moment coefficients. Lift, drag, and sideforce are defined in the aerodynamic frame, A , with respect to the centre of gravity, . Eventually the aerodynamic data set has to be transformed, e.g. to respect the distance between the aerodynamic reference point, A , and the centre of gravity, . The sum of aerodynamic forces is:

( ) ( )AL

Q

D

A

AAAA

A

CCC

SqL

QD

−

−⋅⋅=

−

−== ∑∑ FF

(2-158)

with the dynamic pressure, q , as defined in (2-180) and the wing reference area, S , as well

as the corresponding dimensionless aerodynamic force coefficients. The force coefficients are mainly functions of the steady flow angles which are namely the aerodynamic angle of


attack, Aα , and the aerodynamic angle of sideslip,

Aβ . To a certain extent the force

coefficients are also functions of the unsteady flow angles: the change in aerodynamic angle

of attack, Aα , and the change in aerodynamic angle of sideslip,

Aβ . High lift devices like

flaps or spoilers also play an important role. The aerodynamic moments about the centre of gravity, , are denoted in the body-fixed frame, B , and have to include the aerodynamic forces times their lever-arm:

( ) ( ) ( ) ( )[ ]∑∑∑ ⋅×+= AAABAB

AB

AAB

A FMrMM

(2-159)

The aerodynamic rolling moment, L , pitching moment, M , and yawing moment, N , about the aerodynamic reference point, A , are:

( )Bn

m

l

B

BAA

CsCcCs

SqNML

⋅⋅⋅

⋅⋅=

=∑ M

(2-160)

with the semi span, s , and the mean aerodynamic chord, c . The aerodynamic moment coefficients are mainly functions of the control surface deflections (aileron, elevator, and rudder, ξ , η , and ζ ) as well as of the normalised roll, pitch, and yaw rates, p*, q*, and r*:

A

ABA

Vbpp

⋅⋅

=2

* (2-161)

A

ABA

Vcq

q⋅

⋅=

2* (2-162)

A

ABA

Vbrr

⋅⋅

=2

* (2-163)

with the aerodynamic roll, pitch, and yaw rate ( )ABA

ABA

ABA rqp ,, as defined in (2-206) and

the wing span, b .

2.2.3 Propulsion The task of the propulsion system is to generate thrust, T . In a rather detailed analysis of the thrust, the inlet and the outlet impulse are treated separately – compare TUM-FSD

(2014). The inlet impulse is assumed to act at the thrust reference point, InletT , in the

negative x-direction of the aerodynamic frame, A . The x-axis of the propulsion system is assumed to correlate with the engine shaft; for the rotations between the propulsion system frame and the body-fixed frame see Figure 2.1-5 and its explanations. The outlet impulse is

assumed to act at the thrust reference point, OutletT , into the direction of the engine shaft.


( ) ( ) ( )

P

BPT

TP

A

BAT

TP

PT

TPBPAT

TPBABP

Outlet

Outlet

Inlet

Inlet

Outlet

Outlet

Inlet

Inlet

FF

⋅⋅+

−⋅⋅=

=⋅+⋅=

001

001

,,

,,

MM

FMFMF

(2-164)

The transformation matrices for (2-164) are BAM as given in (2-205) and the transformation

matrix between the propulsion frame and the body-fixed frame is (compare Figure 2.1-5 and its explanation):

BP

BP

−⋅⋅⋅−⋅

=σσ

σκκσκσκκσκ

cos0sinsinsincoscossinsincossincoscos

M (2-165)

The total moments about the centre of gravity of the aircraft due to the propulsion system respect the gyroscopic moments as well as the inlet and the outlet impulse times their lever-arms:

( ) ( ) ( ) ( ) ( ) ( )B

TTPB

TB

TTPB

TB

TyroscopicB

P

Outlet

Outlet

OutletInlet

Inlet

Inlet,, FrFrMM

×+×+= (2-166)

The gyroscopic moments are (compare (2-85)):

( ) ( ) ( ) ( ) ( ) ( )[ ]BPRotKBB

B

IBK

PB

PRotKBB

B

Tyroscopic

RotRot ωIωωIM

⋅×+⋅= (2-167)

where ( )BPRotKω describes the movement of the rotating parts of the engine around the shaft

and therefore only the x-component is non-zero. Since the main influence on the available thrust is exerted by the aerodynamic velocity and by the density of the surrounding air, a simplified thrust model can be established; see also

Brüning et al. (1993). The maximum thrust available, maxT , is a function of the reference

thrust, refT , the aerodynamic velocity, V , and the density of the surrounding air, ρ . The

reference thrust, refT , is valid at a certain velocity, refV , and is usually given at sea level

altitude to which the reference density, refρ , correlates – see also Table 2.3-1.

ρ

ρρ

n

ref

n

refref

V

VVTT

⋅

⋅=max (2-168)

The coefficients Vn and ρn in (2-168) depend on the type of engine installed and in case of

an air-breathing engine a distinction between a turbojet engine (subsonic) and a piston engine powering a propeller has to be made. The coefficients are stated in Table 2.2-2; in case of a range given for the coefficient values, the medium value is used in the calculations if not otherwise stated.


Coefficient air-breathing / propeller air-breathing / jet electric / propeller Vn -1 0 -1

ρn (h<11km) 0.7 – 0.8 0.7 – 0.8 0

ρn (isothermal) 1 1 0 Table 2.2-2: Thrust modelling coefficients

The fraction of actual thrust acting to maximum thrust available is Tδ :

[ ]1 0, max

∈=T

TTδ (2-169)

The pilot commands the thrust by setting the thrust lever to a certain position, CMDT ,δ , from

which the actual thrust results respecting the engine characteristics and performance what may be modelled by:

( )TCMDTTT

Tδδδ

δ

−⋅= ,1 (2-170)

Here, T

Tδ is the engine’s time constant which describes a reaction the faster, the smaller the

time constant’s value. For an air-breathing engine, the fuel consumption lies between an idle and a maximum value both scaling with density in accordance with the modelling as in (2-168) if a constant

efficiency of the engine is assumed. Assuming a linear dependency on Tδ , the fuel

consumption is:

( )[ ]ρ

ρρδ

n

refreffuelTreffuelreffuelfuel mmmm

⋅+⋅−= min,, min,, max,, (2-171)


2.3 Atmosphere

2.3.1 Static Atmosphere The atmosphere is assumed to behave as an ideal gas and has – in the static case – no relative motion to the surface of the earth. The atmospheric conditions are calculated in accordance with the International Standard Atmosphere (DIN ISO 2533, 1979). From there, Table 2.3-1 gives the defining parameters:

Parameter Notation Value Unit Acceleration of gravity at sea-level sg 9.80665 m/s2

Pressure at sea-level MSLrefp , 101.325 · 103 N/m2

Gas constant of air R 287.05287 J/(kg·K)

Temperature at sea-level MSLrefT , 288.15 K

Density at sea-level MSLref ,ρ 1.225 kg/m3

Nominal radius of the earth Er 6.356766 · 106 m Isentropic exponent of air κ 1.4 -

Table 2.3-1: ISO 2533 constants and nominal values

The geopotential altitude, H , assumes the reduction of the acceleration of gravity with

geometric altitude, h , under the mere influence of Newton’s law of mass attraction (2-92). It is calculated using the nominal radius of the earth, Er , from Table 2.3-1:

hrhrH

E

E +

⋅= (2-172)

The difference between the geopotential and the geometric altitudes increases for higher altitudes but remains small. In a geopotential altitude of 32km, the geometric altitude differs by 5‰ or 162m. The first three layers of the atmosphere (AL) are the polytropic troposphere (-2km to 11km) with a reference altitude of 0km (mean sea level - MSL), the isothermal lower stratosphere (11km to 20km), and the polytropic upper stratosphere (20km to 32km). Both of the last two, the lower limit is the respective reference geopotential altitude. The temperature gradients per change in geopotential altitude in the troposphere and the upper stratosphere are:

Notation Value Unit Tropref ,γ -6.5 · 10-3 K/m

Stratref ,γ 1.0 · 10-3 K/m Table 2.3-2: ISO 2533 temperature gradients


For the atmosphere layers where a linear change in temperature with geopotential altitude occurs, the pressure as a function of geopotential altitude is found to be:

( ) ( )

⋅−

−⋅+⋅=

ALref

sR

g

ALrefALref

ALrefALref HH

TpHp

,

,,

,, 1

γγ (2-173)

The density is calculated under the ideal gas assumption as:

TRp⋅

=ρ (2-174)

Therefore, with linearly changing temperature the density is:

( ) ( )

+

⋅−

−⋅+⋅=

1

,,

,,

,

1ALref

sR

g

ALrefALref

ALrefALref HH

TH

γγρρ (2-175)

For the isothermal lower stratosphere, the pressure in dependence on geopotential altitude is:

( ) ( )

−⋅−⋅=

ALref

ALrefsALref T

HHRgpHp

,

,, exp (2-176)

Due to (2-174), the same dependency for the density holds:

( ) ( )

−⋅−⋅=

ALref

ALrefsALref T

HHRgH

,

,, expρρ (2-177)

The speed of sound, a , is:

TRa ⋅⋅= κ (2-178)

With the Mach number, M , as the ratio of aerodynamic velocity and the speed of sound:

aVM A= (2-179)

the dynamic pressure, q , can be written as:

22

221 MpVq A ⋅⋅=⋅⋅=

κρ (2-180)

Deviations from the standard atmosphere may be implemented by changing the reference values for the temperature and the pressure at mean sea level.

ISAMSLrefTropref TTT ∆+= ,, (2-181)

ISAMSLrefTropref ppp ∆+= ,, (2-182)


2.3.2 Wind In the presence of wind, the kinematic and the aerodynamic translatory and rotatory states of the aircraft differ. First, the aircraft is regarded as a point mass. The wind velocity vector is averaged over the aircraft and acts on its centre of gravity, . The wind velocity vector is

given with respect to the ECEF-frame: ( )EWV

.

Figure 2.3-1 Influence of wind on the trajectory

In Figure 2.3-1 it is shown how the kinematic velocity differs from the aerodynamic velocity

due to the influence of wind. Now the track/ course, Kχ , differs from the heading, Ψ , by

more than the aerodynamic angle of sideslip, Aβ .

The wind triangle shows that the kinematic velocity is the vector sum of the aerodynamic velocity and the wind velocity:

( ) ( ) ( )EW

EA

EK VVV

+= (2-183)

Its first order time derivative is:

( ) ( ) ( )EO

W

EOA

EOK VVV += (2-184)

If static and convective wind fields appear, the wind velocity is a function of time and the

position which is exemplarily denoted in a local navigation system, ON :

( ) ( ) ( )( )OOO N

EN

W

EN

W t rVV

,= (2-185)

The wind vector-field consists of a time dependent reference velocity and a position

dependent gradient/ Jacobi matrix. If a local navigation system, ON , is used with its x-axis

pointing in the same direction as the x-axis of the NED-frame (compare (2-6) and its explanation) the wind velocity depicted in this navigation frame is:

KχΨ

BxOx

WV

Aβ

Ax

Kx

KV

AV


( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( )

O

OOO

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

OO N

NNN

EN

W

N

EN

W

N

EN

W

N

EN

W

N

EN

W

N

EN

W

N

EN

W

N

EN

W

N

EN

W

E

N

refWEN

W

zw

yw

xw

zv

yv

xv

zu

yu

xu

t rVV ⋅

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

+= , (2-186)

In short, the wind velocity reads:

( ) ( ) ( ) ( )[ ] ( )OOOO

O

OO N

NN

EN

W

NE

N

refWEN

W t rVVV

⋅∇+= , (2-187)

Differentiating (2-187) with respect to the NED-frame gives under the assumption of a flat earth the change in wind velocity the aircraft experiences as:

( ) ( ) ( )[ ] ( )[ ] ( )EN

KNN

EN

W

NE

N

refW

EO

N

W OOOO

O

OOt

tVVVV ⋅∇+

∂∂

= , (2-188)

Assuming a time-invariant wind field gives as change in wind velocity the aircraft experiences:

( ) ( )[ ] ( )EN

KNN

EN

W

NEO

N

W OOOO

O

OVVV ⋅∇= (2-189)

The assumption in (2-189) is that the changes in wind velocity evoked by the kinematic velocity of the aircraft passing through the field (last term in (2-188)) are far larger than the temporal changes in the wind field itself (first term of the right hand side in (2-188)). Therefore only the spatial changes of the wind field are regarded. The change in wind effect on the aircraft is then merely a function of the change in its position and thus a function of its kinematic velocity.

Note that since the x-axis of ON is directed northbound the transformation matrix between

the NED-frame and ON is the unity matrix:

IM =ONO (2-190)

Thus, for the transformation of the wind gradient holds:

( )[ ] ( )[ ] ( )[ ]OOO

O

OOOO

O

O NN

EN

W

NONNN

EN

W

NONOO

EO

W

O VMVMV

∇=⋅∇⋅=∇ (2-191)

Then, from (2-184), the change in aerodynamic velocity with respect to the NED-frame the aircraft experiences is the change in its kinematic velocity minus the change in wind velocity (2-189):

( ) ( ) ( )[ ] ( )EO

KOO

EO

W

OEO

OK

EO

OA VVVV

⋅∇−= (2-192)


The aerodynamic velocity vector may – just like its kinematic pendant – be either given as a Cartesian or as a non-Cartesian vector. Equivalent to the kinematic NED velocity stated in (2-8), the aerodynamic Cartesian velocity vector is:

( ) ( )( ) ( )( ) ( )

( )

E

O

A

A

A

A

A

A

A

A

E

A

A

OAEA

AOA

E

O

A

A

A

EO

A

VVVV

wvu

⋅−⋅⋅⋅⋅

=

⋅=⋅=

=γ

γχγχ

sincossincoscos

00MVMV

(2-193)

The non-Cartesian aerodynamic velocity vector results from the Cartesian by:

( ) ( ) ( )[ ] ( )[ ] ( )[ ]222 EA

EA

EA

EA

EA wvuV ++== V

(2-194)

( )( )

= E

A

EA

Auvarctanχ (2-195)

( )( )[ ] ( )[ ]

+−=

22arctan

EA

EA

EA

A

vu

wγ (2-196)

So far, rather large wind fields were regarded and it was sensible to average the wind influence over the whole aircraft. Especially for rather small but strong wind fields or vortices their position dependent influence on the aircraft’s body may be of interest. Just as in case of the wind triangle (2-183) also the kinematic angular velocity is the vector sum of the aerodynamic angular velocity and the wind angular velocity – see also Brockhaus et al. (2011) and Etkin (2005):

OAW

ABA

OBK ωωω += (2-197)

In the following, the elements of (2-197) are derived and explained. In Anderson (2011) it is proven, that the wind angular velocity is equal to one-half of the curl of the wind vector field (compare also appendix C), and thus the last term of (2-197) is:

( ) ( )[ ]

( ) ( )

( ) ( )

( ) ( )OO

EO

W

O

EO

W

O

EO

W

O

EO

W

O

EO

W

O

EO

W

EO

W

OO

OAW

yu

xv

xw

zu

zv

yw

∂∂

−∂

∂∂

∂−

∂∂

∂∂

−∂

∂

⋅=×∇⋅=21

21 Vω

(2-198)

The above formula shows nicely that the rotatory wind gradients are the non-diagonal

elements of the wind gradient matrix ( )[ ]OO

EO

W

O V

∇ (compare (2-186)) while the translatory

wind gradients are all located on the diagonal of the gradient matrix. The first line of (2-198)


gives the roll rate effect of wind on the aircraft, the second line the pitch rate effect, and the third line the yaw rate effect. Each of the elements of each line is equally effective.

( )O

OAW

OAW

OAW

OOAW

rqp

=ω (2-199)

with the wind roll, pitch, and yaw rates:

( )OOAWp is influenced by:

( )O

N

EN

W

O

O

yw

∂

∂and equally by: ( )

( )O

N

EN

W

O

O

zv

∂

∂⋅−1 (2-200)

( )OOAWq is influenced by:

( )

ON

EN

W

O

O

z

u

∂

∂ and equally by: ( )

( )O

N

EN

W

O

O

xw

∂

∂⋅−1 (2-201)

( )OOAWr is influenced by:

( )O

N

EN

W

O

O

xv

∂

∂ and equally by: ( )

( )O

N

EN

W

O

O

yu

∂

∂⋅−1 (2-202)

Pure vortices in the respective plane occur if:

( ) ( )

( ) ( )

( ) ( )O

N

EN

W

ON

EN

W

ON

EN

W

ON

EN

W

ON

EN

W

ON

EN

W

O

O

O

O

O

O

O

O

O

O

O

O

yu

xv

xw

zu

zv

yw

∂

∂−=

∂

∂

∂

∂−=

∂

∂

∂

∂−=

∂

∂

(2-203)

The aerodynamic effect of these wind vortices is exactly the same as the respective (negative) kinematic rotation of the aircraft (Brockhaus et al., 2011). If the kinematic angular velocities (2-61) are known and also the wind angular velocities (2-198), the aerodynamic angular velocity vector results from (2-197) as:

( ) ( ) ( )OOAWBOB

OBKB

ABA ωMωω ⋅−= (2-204)

In the last section of this paragraph an alternative to (2-204) is presented. In order to obtain

( )BABAω also the transformation matrix between the aerodynamic and the body-fixed frame,

BAM , may be used. It results from the rotations about the negative aerodynamic angle of

sideslip, Aβ− , and about the angle of attack, Aα – see also Figure 2.1-5:


BA

A

A

A

A

A

A

A

A

A

A

A

A

BA

⋅−⋅

−⋅−⋅=

αβαβαββ

αβαβα

cossinsincossin0cossin

sinsincoscoscosM (2-205)

This gives by using the strapdown equation (compare (2-60)) the exact aerodynamic roll,

pitch, and yaw rate ( )TBABA

ABA

ABA rqp , as:

( )B

ABA

ABA

ABA

B

A

A

A

A

A

BABA

rqp

=

⋅−

⋅=

αβα

αβ

cos

sin

ω (2-206)

which is the aerodynamic pendant to the kinematic rates in (2-61). When controlling the kinematic flight path of the aircraft, the above described wind influence (2-198) has to be balanced by the appropriate aerodynamic rates (2-206). Here, especially the change in aerodynamic angle of attack ( ) ( )B

ABAB

A q=α necessary to balance the wind

pitch rate (2-201) is to be mentioned since it has a significant unsteady effect (Stengel, 2004); e.g. when the aircraft climbs and meanwhile faces an increasing headwind (wind gradient with altitude).


2.4 Modelling of the Receivable Solar Radiation within the Earth’s Atmosphere

Aim of paragraph 2.4 is the mathematical modelling of the receivable solar radiation within the earth’s atmosphere as a function of time, t , position (geodetic longitude, λ , geodetic latitude, ϕ , and altitude, h ), and attitude of the receiver. The model shall be used for solar

flight applications in a broad sense. Be it the fixed or twistable solar cells mounted on an aircraft or solar cells mounted on ground (eventually equipped with a 2-axis tracking system) to produce energy for flight applications. For flight trajectory optimisations also pilot blinding may become important. Here, the exact position of the sun is needed as seen from the aircraft. First, the radiation on the earth’s orbit in space is calculated. The movement of the earth around the sun is modelled as a two-body-problem. Therefore, to be exact, the radiation on the orbit of the earth moon barycentre (EMB) is calculated as a function of date. For a receiver within the atmosphere the declination is developed as a function of date while the hour angle of the sun is a function of position and local time. The position of the sun as seen from a defined position on the earth is precisely developed and attenuation processes within the atmosphere are investigated. The possibilities for different types of solar powered flight missions are examined. Different levels of accuracy of the derived equations are developed and compared to best fulfil the given task. In this work an analytical derivation of all equations is presented. In case of the necessity of extremely precise equations, the developed equations may be used as a basis for a series representation of which the higher order terms are fitted to the data provided by the almanacs, e.g. The Astronomical Almanac and The Nautical Almanac for the respective year, both published by the US Department of Defense, Navy, Nautical Almanac Office.

2.4.1 Introduction to Celestial Mechanics In accordance with Beutler et al. (2005a) the three Kepler's laws of planetary motion read: I. The orbit of each planet around the sun is an ellipse with the sun at one of its foci. II. The planet revolves so that the line joining it to the sun sweeps out equal areas in equal

intervals of time. III. The periods of any two planets are proportional to the power of 3/2 of their mean

distances from the sun.


Figure 2.4-1 Keplerian orbital elements

To describe the motion of the celestial body, six orbital elements are needed (five geometric elements plus time – see for example Beutler et al. (2005a)) which are depicted in Figure 2.4-1 and Figure 2.4-2:

1.) the semi-major axis of the orbital ellipse, a , representing the energy of the orbit 2.) the first eccentricity of the orbital ellipse, e 3.) the longitude of the ascending node of the ecliptic, Ω 4.) the inclination of the ecliptic, i , with respect to the invariable plane of the solar

system 5.) the argument of perihelion, Pω 6.) the time of perihelion passage, Pt

The invariable plane of the solar system (Laplace's invariable plane) is the plane passing through its barycentre on which the vector of its total angular momentum stands perpendicular. The ecliptic is the plane of the orbital motion of the EMB around the sun. Kepler’s 2nd law is used to compute the true anomaly, µ , depicted in Figure 2.4-1 and also

in Figure 2.4-2. The first derivative to time of the true anomaly is not constant as in fulfilment of Kepler’s second law the planet travels fastest at its perihelion and slowest at its aphelion.

xSolySol

zSol

Ωi

µ

Planet

Perihelion

Sun

Aphelion

Pω


Figure 2.4-2 True and mean anomaly

The mean anomaly, M , describes the average movement of the planet and is defined as:

Ttt

M P−⋅= π2 (2-207)

with the period of a whole circulation, T . Its derivative to time is constant:

TM π2

= (2-208)

The mean anomaly can also be calculated by the eccentric anomaly, β , where the position

of the planet is projected on the circumscribed circle of the ellipse:

ββ sin⋅−= eM (2-209)

Figure 2.4-2 shows the true anomaly, µ , the mean anomaly, M , and the eccentric

anomaly, β , with the sun positioned at the second focal point, 2F .

1F 2F

βr

µM

ab

PerihelionAphelionae ⋅


The inverse calculation of β from M is only possible by a series expansion (Murison, 1998)

given in order of the first eccentricity, e :

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )76

5

4

32

2sin4814sin

1546sin

8027

sin192

13sin128275sin

384125

2sin614sin

31

sin813sin

832sin

21sin

eOMMMe

MMMe

MMe

MMeMeMeM

+

⋅+⋅−⋅⋅+

+

⋅+⋅−⋅⋅+

+

⋅−⋅⋅+

+

⋅−⋅⋅+⋅⋅+⋅+=β

(2-210)

Differentiating (2-209) with respect to time and using (2-208) gives for the first derivative of the eccentric anomaly:

βπβ

cos112⋅−

⋅=eT

(2-211)

Therefore, the change in eccentric anomaly is equivalent to the change in mean anomaly at the co-vertices, it is maximal at the perihelion and minimal at the aphelion. The velocity at the co-vertices equals the average velocity. To calculate the eccentric anomaly, β , from the true anomaly, µ , one may use Lenz (2005):

µµβ

cos1coscos⋅+

+=

ee

(2-212)

µµβ

cos11sinsin

2

⋅+−⋅

=e

e (2-213)

⋅

+

−=

2tan

11

2tan µβ

ee

(2-214)

The distance between the focal point and the respective point on the ellipse (position of the planet), r , is:

( )βcos1 ⋅−⋅= ear (2-215)

Differentiating the distance, r , with respect to time and inserting the first derivative of the

eccentric anomaly, β , from (2-211):

ββπββ

cos1sin2sin

⋅−⋅⋅⋅=⋅⋅⋅=

eea

Tea

dtdr (2-216)


The second derivative of r to time is:

( )( )e

eea

Tdtrd

−⋅⋅−

⋅⋅⋅

= β

βπ cos

cos112

3

2

2

2

(2-217)

Setting (2-217) to zero gives the extremal (maximum/ minimal; designated as EXT) increase

in distance at a true anomaly of 2πµ ±= (2-212):

2112

eea

TdtdrEXT

−⋅⋅⋅±=

π

(2-218)

whilst at the perihelion and the aphelion the distance does not change ( )0=β .

As stated in (2-233), the radiation on the orbit is inversely proportional to the square of the distance from the sun. In order to evaluate the change in radiation, the derivative to time of the inverse of the distance squared is calculated:

rrrdt

d⋅−=

32

21 (2-219)

and the second derivative to time is:

( )[ ]2422

2

321 rrrrrdt

d ⋅−⋅⋅−=

(2-220)

Setting (2-220) to zero with using (2-215) to (2-217) gives:

( )( )

[ ] 04coscos3cos1

23 22

222 =⋅−+⋅⋅

⋅−⋅

⋅

=⋅−⋅ ee

eea

Trrr ββ

βπ

(2-221)

from which the respective value of the eccentric anomaly, β , results at which the maximum

change in radiation occurs:

ee

64811cos

2++−=β (2-222)

For small values of the first eccentricity, e , the maximum change in radiation occurs close to the co-vertices where 0cos =β . The EMB orbit geometric elements are given in Table

2.4-1. Inserting this result into (2-219) gives the proportionality factor for the maximum change in radiation as:

( ) ( )42

223

2324817

4812122612221

e

eeaTr

rEXTrdt

dEXT+−

++−−⋅⋅⋅⋅=

−=

π

(2-223)


The second coordinate system, the astronomic coordinate system, is used to describe the apparent movement of the sun. Here, the centre lies within the earth while the sun is revolving. The equator as well as both poles are projected on the celestial sphere to give the celestial equator, the north celestial pole and the south celestial pole.

Figure 2.4-3 Astronomic coordinate system

As shown in Figure 2.4-3, the ecliptic is tilted to the equatorial plane by ee , the obliquity of

the ecliptic. The declination, δ , is measured from the celestial equator along the great circle which passes through the sun and the north celestial pole. It reaches its maximum in summer and becomes minimal in winter. The intersection between the plane of the celestial equator with the ecliptic gives the reference direction with the vernal point (occurrence of vernal equinox). From this reference

direction (vernal point) the ecliptic longitude, Sunλ , and the right ascension, α , are measured

which are related by:

Sune λeα tancostan ⋅= (2-224)

SunEarth

αδ

Sunλ

eeVernal

Equinox

AutumnalEquinox

SummerSolstice

WinterSolstice

CelestialEquator

NorthCelestial

Pole

SouthCelestial

Pole

CelestialSphere


The relation with the declination, δ , is given by:

Sunλδα coscoscos =⋅ (2-225)

This also shows that the right ascension, α , equals the ecliptic longitude, Sunλ , at the

equinoctes and the solstices where ππππλ 2 and 2/3 , ,2/ ,0=Sun .

Differentiating (2-224) with respect to time and inserting (2-225) gives:

Sune λδ

eα ⋅= 2coscos

(2-226)

At both equinoctes (EN) where the declination is zero, the change in right ascension, α , is

smaller than the change in ecliptic longitude, Sunλ :

ENSuneEN ,cos λeα ⋅= (2-227)

Whilst at both solstices (ST) where the declination equals the obliquity of the ecliptic, the

change in right ascension, α , is larger than the change in ecliptic longitude, Sunλ :

STSune

ST ,cos1 λ

eα ⋅= (2-228)

The obliquity of the ecliptic, ee , is one of the two reasons why the mean solar time differs

from the true solar time. The above equations are used in the explanation of the equation of time (EOT) describing this difference (2-270). In this context it is to be mentioned, that the

temporal change in ecliptic longitude, Sunλ , equals the change in true anomaly, µ , (Schaub,

1950):

µλ =Sun (2-229)

2.4.2 Radiation on the Earth’s Orbit When analysing the true earth moon barycentre (EMB) orbit, the inclination may be neglected. According to Cox (2000), the inclination of the invariable (Laplacian) plane of the solar system with respect to the mean ecliptic and equinox of J2000.0 is only 1.57869° and the inclination of the EMB orbit to the ecliptic is 0.00005°. The longitudes of the ascending node, Ω , and of the perihelion, ψ , are measured from the

intersection of the ecliptic and the equator – see Figure 2.4-4. Therefore:

Pωψ +Ω= (2-230)

where Pω is the argument of perihelion measured from the ascending node along the orbit.


Parameter Notation Value Unit Semi-major axis of EMB a 149.59789 · 106 km First eccentricity of EMB e 0.01671022 - Mean longitude of the ascending node Ω -11.26064 deg Mean longitude of perihelion ψ 102.94719 deg Astronomical unit AU 149.59787066 · 106 km Radius of sun Sunr 6.96 · 108 m

Table 2.4-1: Earth orbit geometric elements and sun data

The data in Table 2.4-1 is taken from Cox (2000) and also refers to the mean ecliptic and equinox of J2000.0 at the epoch J2000; only the radius of the sun is taken from Beutler et al. (2005b). To give an impression of the stability of the orbit, the true longitude of the perihelion, ψ , osculated with a peak-to-peak amplitude of 0.17° during 2013 – see The

Astronomical Almanac for the Year 2013 (Nautical Almanac Office, 2013).

Figure 2.4-4 Earth orbit around the sun

In Figure 2.4-4, the orbit of the earth around the sun is depicted with the linear eccentricity being strongly raised to visualise the ellipticity of the orbit. The day numbers are in accordance with Table 2.4-4; only the summer solstice was set to June 21. The true anomaly between the perihelion and the line of the equinoctes is:

( ) °= 05281.7777dµ (2-231)

Note, that since perihelion is passed on the 3rd day of the year, ( )d77µ is valid for the 80th

day of the year, the occurrence of vernal equinox, see also Table 2.4-4.

Winter Solstice~ December 22

Summer Solstice~ June 21

Aphelion~ July 4

Perihelion~ January 3

Vernal Equinox~ March 21

Autumnal Equinox~ September 23

Co-Vertex~ April 5

Co-Vertex~ October 5γ

Earth‘s rotational axisψ

( )d77µ


The whole earth receives only 4.5 · 10-8 % of the sun’s total radiation. The Physikalisch-Meteorologisches Observatorium Davos (PMOD) or World Radiation Centre (WRC) publishes the composite post-processed Total Solar Irradiance (TSI) time series from 1978 to present. From different measurement campaigns, the solar radiation on the mean earth orbit is given.

Figure 2.4-5 TSI at distance of 1AU

In Figure 2.4-5, the different measurement campaigns of the WRC are depicted in different colours and the black vertical bars indicate a variation of 0.1% in radiation. The average of minima for solar cycles 20 to 24 is 1365.461 Wm−2, with an average half amplitude of

0.489 Wm−2 to give an average radiation of 1365.950 Wm−2. The solar constant, SCE , is

defined as the receivable radiation at a distance of one astronomical unit, 1AU, and is set to:

2 1366mWESC = (2-232)

The receivable radiation on the EMB orbit, extE , outside the earth’s atmosphere (extra-

terrestrial) varies inversely with the square of the distance from the sun, it is therefore:

21

⋅=

rAUEE SCext (2-233)

Setting the semi-major axis of the EMB orbit and 1AU equal as in the original definition (for the precise values see Table 2.4-1) and using r from (2-215) gives:

( )2cos11

β⋅−⋅=

eEE SCext (2-234)

Sola

r Irra

dian

ce (W

m−2)

Mod

el

HF

AC

RIM

I

HF

AC

RIM

I

HF

AC

RIM

II

VIR

GO

Days (Epoch Jan 0, 1980) 0 2.0•103 4.0•103 6.0•103 8.0•103 1.0•104 1.2•104

1368 0.1%

1366

1364 Min20/21 Min21/22 Min22/23 Min23/24

1362

75 77 79 81 83 85 87 89 91 93 95 97 99 01 03 05 07 09 11 13

Year


This enables the calculation of the distance of the EMB in dependence on days of the year and the determination of the receivable solar radiation on this orbit.

Figure 2.4-6 TSI at distance of EMB orbit and distance to sun

As shown in Figure 2.4-6 and numerical data given in Table 2.4-3, the solar radiation (= irradiance) on the EMB orbit varies by more than 90W/m2 during the year since the distance to the sun changes by nearly up to 5 · 106 km.

Parameter Notation Value Unit Semi-major axis of earth-moon EarthMoona − 384.400 · 103 km

Mass earth Earthm 5.9737 · 1024 kg

Mass moon Moonm 7.3483 · 1022 kg

Obliquity of the ecliptic J2000.0 ee 23.43928108 deg Mean inclination of lunar orbit w.r.t. ecliptic Mooni 5.145396 deg

Mean sidereal revolution period MoonT 27.32166 d Table 2.4-2: Data for EMB and lunar orbit

The centre of the earth deviates from the EMB orbit by the maximum of the EMB distance,

EMBd , if the small inclination of the lunar orbit is set to zero:

kmmm

madEarthMoon

MoonEarthMoonEMB 4671=

+⋅= − (2-235)

which is calculated with the data given in Table 2.4-2 originating from Beutler et al. (2005b) and lies within the earth. Due to its small value, EMBd is of no influence to the receivable

1310

1340

1370

1400

1430

Days of Year [d]

Sola

r Irr

adia

nce

[W/m

2 ]

50 100 150 200 250 300 350 146

148

150

152

154

Dis

tanc

e to

Sun

[Mio

km

]


radiation. A deviation by the whole semi-major axis of the WGS 84 ellipsoid as defined in Table 2.1-1 would change the radiation by a maximum (at perihelion) of 0.1225W/m2 or 0.00867%. The largest change in distance per day of the earth to the sun occurs when the earth and the moon cross the orbit of the common barycentre. This maximum change is:

dkmdx EMBMaxEarthSun 1072

32166.272

21sin2, =

⋅⋅⋅=∆ −

π (2-236)

This value is by far too small to have any significant influence on the receivable radiation. The daily change in solar irradiance and the daily change in distance to the sun over the year is depicted in Figure 2.4-7 with numerical data given in Table 2.4-3. The extreme (maximum/ minimum) change in receivable extra-terrestrial solar radiation occurs in accordance with (2-222) close to the co-vertices and is by differentiating (2-233) and using (2-223):

( ) ( ) ( )42

223

4817

4812122622e

eeT

EEEXT SCext+−

++−−⋅⋅⋅⋅=

π (2-237)

Figure 2.4-7 Change in irradiance and in distance of EMB orbit position to sun per day

Between two days of the year, the radiation on the EMB orbit can vary by up to nearly 0.8W/m2 and the distance between the EMB and the sun can change by a little more than 43 · 103 km. This would equal the maximum deviation when a leap year has not been properly respected and thus the day-number was wrong by 1.

-0.8

-0.4

0

0.4

0.8

Days of Year [d]

∆ S

olar

Irra

dian

ce [W

/m2 /d

]

50 100 150 200 250 300 350 -50

-25

0

25

50

∆ D

ista

nce

to S

un [1

000

km/d

]


Parameter Notation Value Unit

Distance of perihelion Pr 147.098076 · 106 km

Distance of aphelion Ar 152.097704 · 106 km

Maximum difference in distance AP rr − 4.99962731 · 106 km

Relative difference in distance ( ) eaar AP =− // ∓1.01671022 % -

Radiation at perihelion ( )Pext rE 1412.82 W

Radiation at aphelion ( )Aext rE 1321.47 W

Maximum difference in radiation ( ) ( )AextPext rErE − 91.36 W

Relative radiation at perihelion ( ) 1/ −SCPext ErE +3.4277196 % -

Relative radiation at aphelion ( ) 1/ −SCAext ErE -3.2601028 % -

Max change in distance per day

dtdrEXT ±43.0089 · 103 km/d

Max change in radiation per day ( )extEEXT ∓0.787085 W/m2/d Table 2.4-3: EMB model derived geometric and physical values

Since the first eccentricity of the EMB orbit squared is only:

-32 10 0.27923145 ⋅=e (2-238)

the terms of ( )2eO and higher will be omitted in (2-210), when the eccentric anomaly, β , is

calculated from the mean anomaly, M , giving a sufficiently accurate radiation model (2-234) as:

( )[ ]MeMeEE SCext sin cos21

1⋅+⋅−

⋅= (2-239)

For the period, T , the Julian year is used:

dT 25.365= (2-240)

Without limiting the accuracy of the receivable solar radiation, the perihelion passage of the earth is set to the third day of the year:

dtP 3= (2-241)

The orbit of the EMB experiences only very small variations enabling this assumption (Calkin, 1988) and (Calkin, 1990). For the calendar use, the perihelion passage can be increased by 6 hours per year until the next leap year is reached. This constant perihelion passage gives the receivable solar radiation on the EMB orbit as:


−

⋅⋅+−

⋅⋅−=

ddt

ddt

mW

Eext

25.36532sin0.01671022

25.36532 cos0.033420441

1366 2

ππ (2-242)

where t has to be calculated as decimal days. From Winter et al. (1991), Table 2.4-4 gives the day numbers of the respective occurrence starting with a leap year: Year (number) 2012 (I) 2013 (II) 2014 (III) 2015 (IV) Vernal equinox 80 445 810 1,175 Summer solstice 173 538 903 1,268 Autumnal equinox 266 631 996 1,362 Winter solstice 356 721 1,086 1,451

Table 2.4-4: Day numbers of characteristic occurrences

Figure 2.4-8 Deviation in irradiance due to different EMB orbit models

As shown in Figure 2.4-8, the simplified model stated in equation (2-242) gives slightly too high values compared to the precise model stated in (2-234) when in the numerical calculations the small first eccentricity (see Table 2.4-1) is included up to its sixth power as stated in (2-210) and the denominator in the equation for the receivable radiation is ( )2cos1 β⋅− e as in (2-234) instead of ( )βcos21 ⋅− e as in (2-239). The maximum deviation

is at the perihelion where the result of the simplified model would need to be corrected by 0.40814W/m2. Such a small deviation justifies the use of the simplified model (2-242).

50 100 150 200 250 300 350-0.45

-0.3

-0.15

0

0.15

Days of Year [d]

Dev

iatio

n in

Irra

dian

ce [W

/m2 ]


Using the distance between the sun and the EMB (2-215) and correcting it by the movement of the earth and the moon around their common barycentre gives for the true distance of the earth from the sun:

⋅+⋅+=− t

TMdrx

MoonMoonEMBEarthSun

π2cos ,0 (2-243)

The value EMBd is given in (2-235) and the sidereal revolution period of the moon, MoonT , is

given in Table 2.4-2.

Figure 2.4-9 True earth perihelion passage range

In Figure 2.4-9, the earliest possible and the latest possible perihelion passage of the earth is depicted and set into relation with the nominal passage of the EMB on January 3. As shown in the figure, the perihelion passage of the earth may vary by up to ±1.3 days. This explains the strong variation in the earth’s perihelion passage date published in the almanacs – see also Calkin (1990). The perturbation of the earth’s path by other planets – mainly Jupiter and Venus – does not exceed 1% of the perturbation by the moon – see also Karanikolov (2010). The solar irradiance spectrum is quite well approximated by the radiosity of a black body of 5777K (at least for λ > 0.6µm). In Figure 2.4-10, the high fidelity ASTM E-490 Air Mass Zero solar spectral irradiance is compared to the spectrum of a black body emitting at 5777K and in addition the visible spectrum is shown. The American Society for Testing and Materials developed this extra-terrestrial (air mass = 0) reference spectrum (ASTM E-490) in 2000 – see also American Society for Testing and Materials (2006). The integrated spectral irradiance conforms to the value of the solar constant given in (2-232).

Dec 31 Jan 1 Jan 2 Jan 3 Jan 4 Jan 5 Jan 60

2

4

6

8

Date

∆ to

EM

B p

erih

elio

n [1

000

km]

EMB standard perihelion passageEarth earliest perihelion passageEarth latest perihelion passage


The factor by which the black body spectrum has to be reduced in value for direct

comparison is SCf :

5-2

102.1645 ⋅=

=a

rf Sun

SC (2-244)

with the numerical value resulting from data in Table 2.4-1. As for example shown in Baehr and Stephan (2004), 46.52% of the black body radiation lies within the visible spectrum between 0.38 and 0.78µm. The maximum emission of the black body (at 5777K) is at 0.50µm. If the true solar spectrum published by PMOD/ WRC is taken (Wehrli, 1985), 48% of the total energy lie within the visible range (Duffie and Beckman, 2006) and 95% lie within the range of 0.3 and 2.4µm (Iqbal, 1984).

Figure 2.4-10 Spectral irradiance outside atmosphere

2.4.3 Determination of the Solar Zenith Angle

The solar zenith angle (zenith distance), SunZ , is the angle between the normal of a

horizontal receiver plane, horn , and the vector pointing from the receiver plane to the sun,

Sunn . Its compliment is the elevation, Sunθ – see Figure 2.4-11.

SunSunZ θπ−=

2 (2-245)

0 0.4 0.8 1.2 1.6 20

500

1000

1500

2000

2500

Wavelength λ [µm]

Spec

tral

Irra

dian

ce [W

/(m2 µ

m)]

ASTM E-490 Spectrum5777 K Black BodyVisible Spectrum⋅


Figure 2.4-11 Solar zenith angle and elevation angle

In a geocentric reference system where the earth is modelled to be round (Figure 2.4-12), the position of the sun can be defined by the declination, δ , and the hour angle, ω . Due to the large distance of the sun, the geocentric and the geodetic latitude may be used equivalently when the receivable radiation is calculated. The hour angle is the angle measured on the celestial equator between the receiver meridian and the horary circle of the sun. Thus, at solar noon, the hour angle is zero and afterwards increases up to 360°.

To align the X1-axis with the Sunn -vector, a rotation around the Z1-axis about ω− has to be

performed followed by a rotation around the new y-axis about δ− . The transformation

matrix S1M is therefore:

S

S

1

1

cos0sinsinsincossincoscossinsincoscos

⋅⋅−⋅−⋅

=δδ

ωδωωδωδωωδ

M (2-246)

To align the X1-axis with the horn -vector, a rotation around the Y1-axis about ϕ− has to be

performed giving the transformation matrix h1M as:

h

h

1

1

cos0sin010

sin0cos

−=

ϕϕ

ϕϕM (2-247)

horn

SunnSunZ

Sunθ


Thereby the solar zenith angle, SunZ , can directly be calculated assuming horn and Sunn to

be unit vectors in the respective system where the y and z positions are zero:

( ) ( ) ( )δϕωδϕ sinsincoscoscos

cos 11

⋅+⋅⋅=

⋅= horSunSunZ nn

(2-248)

Figure 2.4-12 Geocentric frame with solar zenith angle ZSun

To transform the 1-system into a North-West-Up system (NWU or simply U) at the receiver place, the first rotation has to be performed around the Y1-axis about ( )2/πϕ +− and the

second rotation around the new x-axis about π giving the transformation matrix 1UM as:

1

1

sin0cos010

cos0sin

U

U

−

−=

ϕϕ

ϕϕM (2-249)

Thereby, the solar direction vector can be given in the U-system:

( )U

USun

⋅+⋅⋅⋅

⋅+⋅⋅−=

δϕωδϕωδ

δϕωδϕ

sin sin cos cos cos sin cos

sin cos cos cos sinn (2-250)

horn

Sunn 0)( >ϕ0=λ

[west] λω

δ

1X

1Z

1Y

SunZ

xE


The above equation can be used to calculate the time of sunrise and sunset by setting the last line of (2-250) and thus (2-248) to zero, it can be used to calculate the daily duration of sunshine and the position of the sun from sunrise to sunset.

If a solar azimuth angle, solarα , which is measured positively counter clockwise from the x-

axis (north direction) of the U-system, is introduced, its value may be calculated by using the y-component and the x-component of (2-250):

( )δϕωδϕ

ωδαsincoscoscossin

sincostan⋅+⋅⋅−

⋅=solar (2-251)

The solar azimuth angle, solarα , will therefore have a negative value at sunrise which

diminishes until -180° are reached at solar noon where it jumps to +180° and continuous to diminish until sunset occurs (north = 0°, east = -90°, south = ±180°, west = +90°). Since the position of the sun now is known for every instant, (2-250) is also used in calculations of pilot blinding. It should, however, be noted, that refraction processes in the atmosphere are not respected in (2-250). The refractive atmosphere changes the solar position maximally by 0.57 degrees when the sun is directly above the horizon (Iqbal, 1984). Note that with data from Table 2.4-1, the solar angular diameter is 0.53°. While in (2-250) the latitude, ϕ , is a component of the actual position, the declination, δ , changes with date,

and the hour angle, ω , changes continuously. If a receiver surface, of which the axes are aligned with the NWU system, is rotated around the z-axis about the negative surface azimuth angle, Surfaceα− , and subsequently tilted

around the new y-axis about the surface inclination angle, Surfaceβ , its surface normal is

found as:

( )U

UReceiver

⋅−

⋅=

Surface

SurfaceSurface

SurfaceSurface

cos sin sin

sin cos

ββα

βαn (2-252)

Thereby, the angle between the receiver normal and the vector pointing towards the sun,

unReceiver,SZ , can be calculated as:

( )( )

( )[ ]ωδαωδϕδϕαβδϕωδϕβ

sincossin cos cos sin sin coscossinsin sin cos cos coscos

cos

SurfaceSurfaceSurface

Surface

⋅⋅−⋅⋅−⋅⋅⋅+⋅+⋅⋅⋅

=unReceiver,SZ (2-253)

of which for a horizontal receiver (2-248) results when setting the surface azimuth angle and the surface inclination angle to zero. If the receiver is turned towards the south, πα =Surface , and tilted about the surface

inclination angle to match the geodetic latitude, ϕβ =Surface , (2-253) simplifies to:

( ) ωδ coscos cos ⋅=tilted,Sunouthbound_Receiver_sZ (2-254)


(i) Declination as a Function of Date

As stated in Table 2.4-2, the obliquity of the ecliptic, ee , is nearly 23.44°. Through the year,

the declination, δ , varies sinusoidal with the obliquity of the ecliptic as its maximum while the declination is to be zero at the equinoctes – see also Figure 2.4-3. To respect the eccentricity of the earth orbit, it is included in an eccentric term – compare (2-264):

( ) ( )

⋅+−⋅⋅= Mett

Te sin22sin δπeδ (2-255)

To become zero at vernal equinox, δt is set to 82d. The declination is therefore:

−

⋅⋅+−

⋅⋅°=d

dtd

dt25.36532sin0.03342044

25.365822sin44.23 ππδ (2-256)

The equation stated in Winter et al. (1991) which is presented without derivation and has an accuracy of ±0.3°, reads:

( )

−⋅⋅+−⋅⋅°= dt

ddt

d4.2

25.3652sin93.13.82

25.3652sin44.23 ππδ (2-257)

Figure 2.4-13 Declination modelling and comparison

In Figure 2.4-13, the declination over the year is depicted in accordance with (2-256). In addition the times of vernal equinox (VE), summer solstice (SS), autumnal equinox (AE), and winter solstice (WS) in accordance with Table 2.4-4 are indicated. Moreover, the difference in values for the declination between (2-256) and the formula given in (2-257) is shown. The difference stays below 0.125°. In Iqbal (1984), a high fidelity calculation from Spencer (1971)

-24

-12

0

12

24

Days of Year [d]

Dec

linat

ion

[deg

]

50 100 150 200 250 300 350 -0.24

-0.12

0

0.12

0.24

∆ D

eclin

atio

n [d

eg]

ΔD

eclin

atio

n[d

eg]

VE SS AE WS


is presented with an accuracy of 0.04°. The deviation between (2-256) and the formula of Spencer stays below 0.3°. The pure sinusoidal approach not respecting the eccentricity would give deviations up to 2°. For a more precise calculation of the declination with an accuracy of 0.01°, the reader is referred to Blanco-Muriel et al. (2001). As may be seen in Figure 2.4-13, the maximum change in declination (less than 0.5° per day) occurs at the equinoctes. The smallest change in declination per day is zero at the solstices. Therefore, in most applications it is sufficient to calculate with only one value of declination for each day.

(ii) Determination of the Hour Angle The hour angle, ω , measures the longitudinal component of the solar radiation. It represents the earth’s rotation in 24 hours equivalent to a mean solar day (synodic day):

[deg] /24/360dhdtSun

°⋅=ω (2-258)

with Sunt being the true local solar time to be given in decimal hours:

( ) [h] 60min/h

EOT3602412 0 +−⋅

°+−−= λλhhDLSTLSTtSun (2-259)

Here, LST is the local standard time in decimal hours and DLST is a correction of 1h in case of daylight saving time is present in LST (last Sunday in March until last Sunday in October for Europe). The correction about 12 hours sets the hour angle to zero at solar noon. The fourth term represents the sun’s movement by on average 15°/h and corrects between the local time zone which can only give the precise time at the reference meridian of the time

zone, 0λ , and the true meridian of the observer, λ , which both are to be measured in

degrees west as shown in Figure 2.4-12. These four terms give the mean solar time, MST:

( ) [h] 3602412 0 λλ −⋅

°+−−=

hhDLSTLSTMST (2-260)

The equation of time, EOT, gives the difference between mean solar time, MST, and the true

solar time, Sunt . Two reasons are represented in the equation of time, which calculates the

deviation from the sun’s true path from the averaged path represented by the definition that a day constantly takes 24 hours.

The equation of time determines how long it takes the earth by its own rotation, IEω , to correct for the angular difference between the averaged solar movement (2-207), M , and the true apparent solar movement measured by the right ascension (2-224), α , corrected by the perihelion offset:

IEPPSunt

t IEP

dtMω

αλω

α −+⋅

−= ∫ ,EOT

(2-261)


Respecting the homogeneous temporal change in true anomaly, µ , and in ecliptic

longitude, Sunλ , (2-229) – compare also (2-231) and Figure 2.4-4 – gives EOT as:

IESun

IEIEPPSunt

t IESun

IEMdtM

P ωαλ

ωµ

ωαλ

ωαλ

ωµ −

+−

=−

+⋅

−+

−= ∫ ,EOT

(2-262)

First Element of the Equation of Time: due to Eccentricity of the Earth Orbit While a synodic day lasts 24 hours, the sidereal day during which the earth performs a whole revolution takes only 23.934472 hours – see Beutler et al. (2005b) which is in accordance with the WGS 84 definition as stated in Table 2.1-1. In contrast to the synodic

day, the sidereal day is in fact constant. IEω is used as a measure of time and relates time to angular quantities. The first term of the EOT represents the difference between the mean anomaly, M , and the true anomaly, µ , divided by (measured by) the angular velocity of the earth:

IE

MEOTω

µ−=1 (2-263)

Thus it is calculated, how long it takes the earth to compensate the effect due to the difference between mean and true anomaly by its own rotation. In Schaub (1950), a series expression between M and µ is derived using the Lagrange

reversion theorem:

( ) ( ) ( ) ( )[ ] ( )432 sin33sin131212sin

45sin2 eOMMeMeMeM +⋅−⋅⋅+⋅+⋅+=µ (2-264)

Again, due to the small eccentricity of the EMB orbit, terms of ( )2eO or higher may be

neglected (2-238). Equation (2-263) then reads:

MeEOT IE sin21 ⋅−=

ω (2-265)

or in numeric values:

( ) 325.365

2sin6min7.638488051

−⋅⋅−= dt

dEOT π

(2-266)

Until the first co-vertex is reached ( )M >µ , a time difference of 7.6min builds up to vanish

at the aphelion, then the negative difference builds up until the second co-vertex is reached and the difference has vanished when the perihelion is reached again. The values of the first term of the EOT are depicted in Figure 2.4-14 over the year. The dashed blue lines represent the perihelion passage (PER), the passage of the first co-vertex (CV), the aphelion passage (APH) and the passage of the second co-vertex (CV). As shown in Figure 2.4-14, the deviation between the approximate sinusoidal solution from (2-266) and the precise solution using (2-264) is at a maximum of less than 1% or less than 5s whereby the exclusion of higher order terms in (2-265) is justified.


Figure 2.4-14 EOT 1st term

Second Element of the Equation of Time: due to Obliquity of the Ecliptic The second term of the EOT results from the obliquity of the ecliptic which also contributes to the difference between the average and the true solar position. Therefore, the difference

between the ecliptic longitude, Sunλ , and the right ascension, α , is again measured by the

angular velocity of the earth:

IESunEOTω

αλ −=2 (2-267)

Thus, it is again calculated, how long it takes the earth to compensate the effect due to the difference between ecliptic longitude and right ascension by its own rotation. As explained by (2-225), at both equinoctes and solstices, the ecliptic longitude equals the right ascension.

Schaub (1950) gives a Taylor series expansion (Maclaurin) for the difference between Sunλ

and α as:

( )

( ) ( ) ( )864

2

6sin2

tan314sin

2tan

21

2sin2

tan

eOMM

M

VEe

VEe

VEe

Sun

+⋅

⋅+⋅

⋅

−⋅

=−

ee

eαλ

(2-268)

-10

-5

0

5

10

Days of Year [d]

EOT

1st t

erm

[min

]

50 100 150 200 250 300 350 -6

-3

0

3

6

∆ to

pre

cise

sol

utio

n [s

]

PER CV APH CV


The second mean anomaly, VEM , assumes a mean sun which coincides with the true sun at

vernal equinox, VE, and is thus defined as:

TttM VE

VE−

⋅= π2 (2-269)

with the period, T , of a whole circulation (2-240).

Starting from vernal equinox where both the ecliptic longitude, Sunλ , and the right ascension,

α , are zero, first, as explained in (2-227), αλ >Sun until the summer solstice is reached

where both Sunλ and the right ascension, α , are 2/π , and already αλ <Sun as explained in

(2-228). Then at autumnal equinox, the difference between Sunλ and α has again vanished

and so on.

If the very small terms of ( )4eO or higher are neglected, (2-267) reads:

( )VEe

IE MEOT 2sin2

tan1 22 ⋅

⋅=

eω

(2-270)

or in numeric values:

( ) 8025.365

4sin08min9.835850412

−⋅⋅= dt

dEOT π

(2-271)

The second term of the EOT thus gives the consequences of equation (2-226). Between each equinox and each solstice a time difference of ±9.84 minutes builds up and vanishes again – see also equations (2-227) and (2-228). This is shown in Figure 2.4-15. Moreover, the figure shows, that the purely sinusoidal solution in (2-271) deviates from a precise

solution using (2-268) up to ( )6eO by less than 2.5% or less than 15s.

Figure 2.4-15 EOT 2nd term

-10

-5

0

5

10

Days of Year [d]

EOT

2nd

term

[min

]

50 100 150 200 250 300 350 -30

-15

0

15

30∆

to p

reci

se s

olut

ion

[s]

VE


Equations (2-266) and (2-271) give the equation of time as:

( )

( )

−⋅⋅+

+

−⋅⋅−=

dtd

dtd

EOT

8025.365

4sin08min9.83585041

325.365

2sin6min7.63848805

π

π

(2-272)

In Figure 2.4-16, the equation of time is depicted. The maximum value reaches 16.6 minutes whilst the minimum value is -14.3 minutes leading to an amplitude of 30.8 minutes which equals approximately 8° in hour angle.

Figure 2.4-16 Equation of time

Comparing (2-272) to the equation of time given in Winter et al. (1991) which should read:

( ) ( )

−⋅⋅⋅+

−⋅⋅⋅−⋅

=

dtdt

EOT

81180

9716.1sin1644.0 3180

9856.0sin1276.0min60 ππ (2-273)

and of which an accuracy of 10 seconds is claimed, shows a maximum difference of ±25 seconds. For the purpose to determine the receivable solar radiation, it is sufficient, to determine the extra-terrestrial radiation, the declination as well as the equation of time merely as a function of date due to the small changes per day. The maximum difference in EOT per day is less than 0.5 minutes. The only variables that may lead to a fast change in the hour angle are the LST (360°/d or 24h/d) and eventually a strong change in longitudinal component, λ , if the receiver moves very fast in east-western direction.

50 100 150 200 250 300 350-20

-10

0

10

20

Days of Year [d]

EOT

[min

]


2.4.4 Atmospheric Influences Within the atmosphere, the direct solar radiation can be scattered (Rayleigh, aerosols) leading to diffuse radiation of which one may assume 50% to reach the ground. Or, the direct radiation may be absorbed (water, CO2, ozone) which increases the inner energy of the atmosphere and leads to heat emissions at long wavelengths ( mµλ 3> ). While the

scattering reduces the strength of the radiation, the absorption (nearly) eliminates complete bands of the spectrum – see also chapter 6 of Iqbal (1984).

(i) Determination of Optical Mass The relative air mass (AM or optical mass) is the normalised mass of air (precisely speaking: atmospheric mass) a ray of light has to pass before reaching the receiver. The normalisation is performed by the mass of air perpendicular above the sea level receiver position.

Figure 2.4-17 Trajectory of solar ray for refractive atmosphere

In addition to the solar zenith angle, as shown in Figure 2.4-17, the path is elongated due to the density changes within the atmosphere leading to a refraction process. This also has to

be respected in the calculation of the normalised or reference air mass, refAM :

( )

( )∫

∫

⋅

⋅=

atm

atm

h

h

ref

dzz

dssAM

0

0

ρ

ρ (2-274)

dzds

O

Er

zs

atmh


Assuming a standard atmosphere as described in 2.3.1, the pressure drops to 5% of the reference value in an altitude of 20.6km. Thus 95% of the atmosphere’s mass is below that altitude. In general, the limit of the atmosphere is set at 100km (von Karman line). If this limit is assumed for the AM calculation (with data of the standard atmosphere), a ray of light can at maximum pass an atmospheric distance of 1132km if refraction processes are not

regarded. The smaller the elevation, Sunθ , the larger, however, the influence of refraction due

to density changes. This and more influences are respected in Kasten and Young (1989) to give the normalised

air mass, refAM , as a function of solar zenith angle, SunZ (for degrees and radians):

( ) ( ) 6364.1deg,deg, 07995.9650572.0cos

1−−°⋅+

=SunSun

ref ZZAM (2-275)

( ) ( ) 6364.14 1.6769115106.713025 cos1

−− −⋅⋅+=

SunSunref ZZ

AM (2-276)

The above formulae approximate the air mass above sea level as modelled in the international standard atmosphere ISO 2533 (1975) at reference conditions with a deviation of less than 0.125%. The maximum reference air mass is found to be 38 from the integration of the product of density times distance. For deviations from the reference values a pressure correction has to be performed – see Iqbal (1984). The same correction is used for altitudes of the receiver above sea level:

refref p

pAMAM ⋅= (2-277)

The minimum air mass may thus become zero.

Figure 2.4-18 Reference air mass in dependence on solar zenith angle

0 15 30 45 60 75 900

2

4

6

8

10

Solar zenith angle [deg]

Ref

eren

ce A

ir M

ass

[-]


In Figure 2.4-18, the dependence of the reference air mass on the zenith angle is shown

(2-275). An refAM of 1.5 corresponds to zenith angle of 48.3°, an refAM of 2 corresponds to

zenith angle of 60.1°, an refAM of 5 corresponds to zenith angle of 78.7°, and an refAM of

10 corresponds to zenith angle of 84.8°.

(ii) Receivable Direct Solar Radiation In the following, only the total, thus spectrally integrated radiation is modelled. The model gives the radiant energy for specified atmospheric conditions and altitudes. A parameterisation method is applied, where the overall spectrally integrated transmittance for every atmospheric constituent is calculated separately. The multiplication of the single transmittance factors gives the total transmittance of the atmosphere. Several such models are compared in Iqbal (1984). The model presented here is to a large extent based on the model of Bird and Hulstrom (1980) and Bird and Hulstrom (1981) which is named Model C in Iqbal (1984) and which he rates best since it is very accurate and yields results closest to the spectrally integrated approach. It is constructed using SOLTRAN for which the code LOWTRAN 3 and LOWTRAN 4 has been used: see Selby and McClatchey (1975) as well as Selby et al. (1978). In accordance with Baehr and Stephan (2004), the solar radiation on a receiver directed towards the sun is:

223975.0 COOHOAeRaextoriented EE τττττ ⋅⋅⋅⋅⋅⋅= (2-278)

Here, the attenuation of solar irradiance due to Rayleigh scattering (RA), Aerosols (Ae), Ozone (O3), Water (H2O), and carbon dioxide with other gases (CO2) is included. The factor of 97.5% accounts for the spectral interval considered by SOLTRAN which is 0.3µm to 3.0µm and has been further adapted by Iqbal (1984) to the formula given in Bird and Hulstrom (1981) to correlate with the actual solar constant as given in (2-232). In the following, the single terms are presented and analysed. In the subsequent figures, the altitude is always set to sea level and thus the air mass is always the reference air mass,

refAM .

The transmission factor due to Rayleigh (air molecular) scattering is an adapted formula from the one presented in Winter et al. (1991):

( )[ ]( ) ( ) 10AMfor 10 10

01AMfor 1 0903.0 exp

10

01.1 84.0

>−⋅+==

≤−+⋅⋅−=

=

AMdAMdAM

AMAMAM

RaAM

RaRaRa

Ra

ττττ

τ

(2-279)

The adaptation shown in the second line of (2-279) prevents an increase of the transmission factor for large values of AM as originally published as Bird Model in Bird and Hulstrom (1980) and analysed in Bird and Hulstrom (1981). (2-279) is in good accordance with the

graphical representation of Raτ in Bird and Hulstrom (1980) – see also Figure 2.4-19. Due to

the smallness of the air molecules, the scattering occurs at the lower end of the spectrum and is responsible for the blueness of the clear sky. Above 1.2µm, there is no Rayleigh scattering.


The transmission factor due to scattering by aerosol particles (mainly dust and small water drops, Mie scattering) is also presented in Winter et al. (1991) as:

−⋅= 66.0

9.0 265.197.0lnexpVIS

AMAeτ (2-280)

Here, VIS gives the horizontal visibility in km and may range between 5km to 180km. In

Figure 2.4-19 where the transmission factor Aeτ is depicted as a function of refAM , the

visibility is set to 100km.

Figure 2.4-19 Transmission factors due to Rayleigh scattering and aerosol particles

In contrast to scattering, absorption only takes place in narrow bandwidths of the spectrum. The dissociated (atomic) O and N above 100km absorb all radiation below 0.085µm while the molecular oxygen O2 and nitrogen N2 absorb all radiation up to 0.20µm – see also Baehr and Stephan (2004). Ozone is a nearly perfect absorber in the region 0.20µm to 0.30µm and has a weaker absorption band from 0.30µm to 0.35µm. Iqbal (1984) analyses the distribution of ozone with altitude and shows, that it is sensible to assume all ozone to be concentrated in a layer at an altitude of 22km. The transmission factor due to absorption by ozone is therefore only valid for altitudes below 22km but has no pressure correction in the AM term:

( ) 3035.03

3

3 48.13911611.0

1refO

refOO AMd

AMd⋅⋅+

⋅⋅−=τ (2-281)

The factor 3Od gives the layer thickness of ozone in cm, so 0.32 corresponds to a thickness

of 3.2mm. It varies depending on the season and the longitude of the receiver position

0 2 4 6 8 100.4

0.6

0.8

1

Reference Air Mass [-]

Tran

smis

sion

Fac

tors

[-]

τRaτAe


between 45.025.03

≤≤ Od . (2-281) is given in Winter et al. (1991) and is a slight

simplification of the original formula given in Bird and Hulstrom (1980). Water vapour and CO2 are the main absorbers in the infrared spectrum – see also Baehr and Stephan (2004). Water vapour is mainly concentrated in the lower layers of the atmosphere – see Iqbal (1984). Therefore, for altitudes below 11km the transmission factor for absorption by water vapour is in accordance with Winter et al. (1991):

( ) 683.022

2

2 03.791385.6496.2

1AMdAMd

AMd

OHOH

OHOH ⋅⋅++⋅⋅

⋅⋅−=τ (2-282)

The factor OHd2

in the above formula represents the height of a water column in cm at the

location of the receiver. A typical value is 2cm or 2g/cm2. (2-282) is again a slight simplification of the original formula given in Bird and Hulstrom (1980). Nowadays, the so called wet component of tropospheric refraction can be measured with very high accuracy using microwave techniques like Doppler systems, GPS or Very Long Baseline Interferometry (VLBI) giving the total precipitable water vapour content above the observer (Beutler et al., 2005a). The absorption due to CO2, O2, N2 and other gases – see Winter et al. (1991) or Bird and Hulstrom (1980) – is modelled as:

( )26.00127.0exp2

AMCO ⋅−=τ (2-283)

Figure 2.4-20 Transmission factors due to gases, ozone, and water

Both, Figure 2.4-19 and Figure 2.4-20 are valid for the receiver located at SL altitude and standard atmospheric conditions.

0 2 4 6 8 100.4

0.6

0.8

1


Tran

smis

sion

Fac

tors

[-]

τCO2

τO3

τH2O


In conclusion, a reduction in elevation angle, Sunθ , has a double negative effect since both

terms in the multiplication for the calculation for a horizontal receiver are reduced:

( ) orientedSunhorizontal EZE ⋅= cos (2-284)

In Figure 2.4-21, the receivable solar radiation on SL altitude is shown. The ASTM E-490 spectrum for extra-terrestrial radiation is shown as well as the ASTM G173-03 (valid since January 2003) for on ground receivable direct and circumsolar radiation (after the radiation has passed an AM of 1.5) is given – see also American Society for Testing and Materials (2012) which corresponds with the international standard ISO 9845-1:1992. In red, the direct plus the circumsolar radiation is given. This is the spectral irradiance within ±2.5 degrees (5 degree diameter) field of view centred on the (0.5 degree diameter) solar disk. It includes the radiation from the disk but excludes scattered sky and reflected ground radiation. The ASTM G173 spectra are an average for the 48 contiguous states of the United States of America over a period of one year. The tilt angle (see explanation of Figure 2.4-23) selected is 37° which is approximately the average latitude for the contiguous USA. The spectra are therefore valid for all northern latitudes between +25° and +49°. Most of the reduction in the visible range is due to Rayleigh and aerosol scattering – there is only a small and weak absorption band due to ozone around 0.6µm. The important absorption bands are indicated with the respective cause.

Figure 2.4-21 ASTM spectra with absorption bands

0 0.4 0.8 1.2 1.6 20

500

1000

1500

2000

2500

Wavelength λ [µm]

Spec

tral

Irra

dian

ce [W

/(m2 µ

m)]

ASTM E-490 SpectrumASTM G173-03 Dir & CircumsolarVisible Spectrum⋅

H2O, CO2

H2O, CO2

H2O

H2O

O3


(iii) Receivable Diffuse Radiation Diffuse radiation results from Rayleigh scattering, the scattering due to aerosols, and from multiple reflections (Mr) between the surface and the atmosphere.

MrdiffAediffRadiffdiffuse EEEE ,,, ++= (2-285)

Most of it lies within the visible spectrum – see Baehr and Stephan (2004). The orientation of the receiver is crucial for the calculation of receivable radiation originating from multiple reflections. Moreover, in case of multiple reflections, also the ground surface albedo is important - see Winter et al. (1991). For a deeper analysis, the reader is advised to Iqbal (1984) and Bird and Hulstrom (1981). The diffuse radiation due to multiple reflections is by far the smallest summand in (2-285) and at the same time the most difficult to calculate. Moreover, most solar aircraft have the cells installed on the top and oriented more or less horizontally. Thus the ground albedo and other aspects interesting for on-ground arranged solar cells are not investigated. So, in the following, only the diffuse radiation resulting from Rayleigh and aerosol scattering is treated – see Winter et al. (1991). The non-absorbed (Na) irradiance is given as:

( ) AeAbsCOOHOSunextNa ZEE ,223cos786.0 ττττ ⋅⋅⋅⋅⋅⋅= (2-286)

with the transmittance factor due to aerosol absorption only, AeAbs,τ , being (when assuming

continental aerosols):

( ) ( )06.1, 1112.01 AMAMAeAeAbs +−⋅−⋅−= ττ (2-287)

This gives the diffuse radiation due to Rayleigh scattering as:

( )( ) RadiffNa

RaNaRadiff E

AMAMEE ,02.1, 2

11

121 ττ

⋅⋅=+−

−⋅⋅= (2-288)

and the diffuse radiation due to aerosol scattering (Bird and Hulstrom, 1981) as:

( )( ) AediffNa

Sun

AeAbs

AeNaAediff E

AMAMZ

EE ,02.1,

, 21

1cos1

121 τ

ττ

⋅⋅=+−

+⋅

−⋅⋅= (2-289)

Combining these two equations leads to the diffuse radiation receivable:

( )AediffRadiffNaAediffRadiffdiffuse EEEE ,,,, 21 ττ +⋅⋅=+= (2-290)

In Figure 2.4-22, the diffuse radiation factors for aerosols and Rayleigh scattering are given. They correspond to their definition in (2-288) and (2-289) where the factor of a half assumes that 50% of the diffuse radiation reaches the ground. Moreover, it is shown, that at least 12% (at 1=refAM ) of the non-absorbed radiation is available as diffuse radiation for a

receiver mounted at SL altitude. The higher the air mass, the higher is the portion of non-

absorbed radiation which is available as diffuse radiation (for 10<refAM ).

The total radiation receivable is therefore the sum of the direct radiation as regarded in (2-284) plus the diffuse radiation (2-285).


Figure 2.4-22 Diffuse radiation factors and fraction of diffuse radiation

In ASTM G173-03 a Global Tilt is defined as the spectral radiation from the solar disk plus sky diffuse radiation and diffuse radiation reflected from ground on a south facing surface tilted 37 degree from the horizontal. If from this Global Tilt the direct and circumsolar radiation as shown in Figure 2.4-21 is subtracted, the diffuse radiation as shown in Figure 2.4-22 results. Most of the diffuse radiation lies within the visible spectrum and the absorption bands are not refilled.

Figure 2.4-23 Diffuse radiation

0 2 4 6 8 100

0.1

0.2

0.3

0.4


Fact

ors

[-]

τdiff, Aeτdiff, RaDiffuse part of ENa

0 0.4 0.8 1.2 1.6 20

100

200

300

400

Wavelength λ [µm]

Spec

tral

Irra

dian

ce [W

/(m2 µ

m)]

ASTM G173-03 diffuseVisible Spectrum⋅


In this work a cloudless sky is assumed. For the treatment of solar radiation under cloudy skies, the reader is advised to Chapter 8 of Iqbal (1984) which is solely dedicated to this matter and treats it in great detail.

2.4.5 Results First, the composition of the receivable radiation/ irradiance is investigated. All data plotted in this paragraph are valid for a receiver at sea level at [West] 11°−=λ and °= 40ϕ for the

173rd day of the year (summer solstice, SS). Due to DLST, the maximum is reached around 1 o’clock pm – see Figure 2.4-24. The effect represented by the EOT is small: EOT = -2.2 minutes. Larger is the effect of the deviation from the reference longitude by -4° equalling -16 minutes. The diffuse radiation is quite important (>25% of the value of direct irradiance) during the first hour of sunshine and during the last hour before sunset. During solar noon, its portion drops below 10% of total irradiance. For the calculation of Figure 2.4-24, the factor

3Od which gives the layer thickness of ozone

was set to 0.32 and the horizontal visibility was set to 180km. For comparison also the total radiation on the 356th day of the year (winter solstice, WS) is given. Since the effects described by the equation of time are very small (EOT = +0.9 minutes) but DLST is not present, the maximum in irradiance at the day of the winter solstice is reached around noon. The calculations of all subsequent figures are performed on the same basis. If variations occur, they are clearly indicated.

Figure 2.4-24 Components of radiation for 173rd day of the year and total irradiance at WS

5 9 13 17 210

300

600

900

1200

Local Standard Time [h]

Irrad

ianc

e [W

/m2 ]

Total at SSDirect at SSDiffuse at SSTotal at WS


For the determination of the solar elevation, Sunθ , one may use (2-248) together with (2-245)

and for the determination of the solar azimuth angle, solarα , (2-251) may be used. Thereby,

the angular position of the sun can be calculated as depicted in Figure 2.4-25. At the above described position on the 173rd day of the year, the sun rises at an angle of 58.7° east of north (thus 31.3° north of east). The maximum elevation is 73.44° (90° - 40° + 23.44°) since the declination is at its maximum on summer solstice. The sunset occurs at an angle of 58.7° west of north (thus 31.3° north of west).

Figure 2.4-25 Angular position of the sun - 173rd day of the year

Second, the influence of the altitude of the receiver is analysed. As shown in Figure 2.4-26, the total receivable radiation increases at solar noon by around 8%, when the altitude is increased from SL to 8.5km equalling a pressure reduction to 32.7% of reference pressure. The direct radiation increases by around 15% but the diffuse radiation is lowered by 60%.

5 9 13 17 21-200

-150

-100

-50

0

50

100

150

200


Ang

les

[deg

]

Elevation θSunSolar Azimuth αsolar


Figure 2.4-26 Influence of altitude on irradiance

In Figure 2.4-27, the effect of latitude on the receivable solar radiation is shown. For a position at the tropic of cancer, clearly the highest maximum is reached. But there, the total duration of sunshine is shorter than at 40° northern latitude and for most of the day, the receivable radiation is less than at 40°. This is a clear indication, that the integrated irradiance, namely the insolation, will not have its overall maximum at the tropic of cancer, even at summer solstice.

Figure 2.4-27 Influence of latitude on irradiance - 173rd day of the year

5 9 13 17 210

300

600

900

1200


Irrad

ianc

e [W

/m2 ]

Sea Level8.5km altitude

5 9 13 17 210

300

600

900

1200


Irrad

ianc

e [W

/m2 ]

Tropic of CancerLatitude of 40°Equator


In Figure 2.4-28, the integrated daily solar irradiance (= irradiation or radiant exposure or insolation) in dependence on latitude and date is shown. Two maxima can be identified at a latitude of °±≈ 40ϕ : one for summer on northern hemisphere and one for summer on

southern hemisphere. The maximum on northern hemisphere is weaker than the one on southern hemisphere since it corresponds to the aphelion whilst the southern maximum occurs at perihelion position of the earth. The net heat of combustion of one litre of Jet A-1 Kerosene lies between 33.17MJ and 35.95MJ (ExxonMobil Aviation, 2008). Thereby it may be judged what a huge amount of energy is provided by solar radiation on every square metre of the earth’s surface. Moreover, the Figure 2.4-28 shows that the variation in solar irradiance over the year increases, the farer the position of the receiver from the equator – see also Figure 2.4-32.

Figure 2.4-28 Daily integrated solar irradiance [MJ/m2] at SL

The received radiation may be increased, if the receiver is oriented towards the sun. The factor of receivable radiation for the oriented receiver to the non-oriented receiver is depicted in Figure 2.4-29. A 2-axis tracking system will at least increase the radiation by 36%. For latitudes around 48°, the radiation may be increased by more than 50% in summer and by more than 200% in winter. The calculations presented in Figure 2.4-29 match well with the measurement data provided in Kelly and Gibson (2009) for Phoenix, Arizona, USA. Phoenix is located at a northern latitude of 33.43° and the ratio varied from 1.39 in summer to 2.10 in winter. It is, however, important to mention that Figure 2.4-29 assuming a 2-axis tracking system is only valid for clear sky conditions. In Kelly and Gibson (2009) it is shown that during heavily overcast with almost only diffuse irradiance, a horizontal module orientation increases the solar energy capture by on average 47% compared to a 2-axis solar tracking system. Thus, in order to best capture the isotropically-distributed diffuse solar irradiance emanating from the clouds the receiver has to be placed parallel to the ground. See also chapter 11.5 Sky Diffuse Radiation Incident on an Inclined Plane in Iqbal (1984).

5

5 5

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30

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30

31.631.6

31.6

33.7

Days of Year [d]

Latit

ude

[deg

]

50 100 150 200 250 300 350-60

-40

-20

0

20

40

60


Figure 2.4-29 Compare sun-oriented to horizontal receiver at SL

When the atmospheric influences are neglected in the calculations for Figure 2.4-28, Figure 2.4-30 results. Thus, the extra-terrestrial radiation is given which is purely direct radiation. This is the theoretic upper limit of the receivable solar radiation for high altitude solar missions with horizontal receivers/ cells.

Figure 2.4-30 Daily integrated solar irradiance without atmosphere [MJ/m2]

1.3651.365

1.38

1.38

1.38

1.38

1.38

1.38

1.38

1.5

1.5

1.5

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1.5

1.5

1.5

2

2

2

2

2

2

3

33

3

5

5

5

7

7

Days of Year [d]

Latit

ude

[deg

]

50 100 150 200 250 300 350-60

-40

-20

0

20

40

60

5

5 5

5

10

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10

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41.8

41.8

41.8

41.8

41.8

44.5

Days of Year [d]

Latit

ude

[deg

]

50 100 150 200 250 300 350-60

-40

-20

0

20

40

60


If the integration of the receivable radiation on SL altitude with atmospheric influences respected (as shown in Figure 2.4-28) is performed for the whole year, the blue curve in Figure 2.4-31 results for a horizontal receiver. If a tracking algorithm keeps the receiver perpendicular to the incoming radiation, the red curve in Figure 2.4-31 results. In addition, for the radiation receivable throughout the year, the factor of the oriented to the horizontal receiver is given – compare also Figure 2.4-29. A 2-axis tracking system enables to increase the receivable amount of yearly insolation even at 60° latitude above the value available from a horizontal receiver at the equator. By the blue curve it is shown that for year-around operations of aircraft with horizontally mounted solar cells, if latitude is to be kept constant, the equator is favourable. The yearly amount of solar radiation at Munich (48°) is nearly 33% below the yearly radiation at the equator, even if the overall maximum of radiation per day in Munich is slightly higher than at the equator - see also Figure 2.4-32. When judging the yearly insolation, it should be remembered that 1MWh is equivalent to 3600MJ and thus more than the net heat of combustion of one hundred litres of Jet A-1 Kerosene (ExxonMobil Aviation, 2008). In Germany, the truly receivable on-ground insolation has been on average 1.134MWh/m2 in the year 2011 (Deutscher Wetterdienst, 2012). Germany has a size of 357,340km2 (Statistische Ämter des Bundes und der Länder, 2014) and in 2013 had a total energy consumption of 13.91Exajoule (Arbeitsgemeinschaft Energiebilanzen e. V., 2014) – with 1 Exajoule being 1018 Joule. Per year, the total solar insolation in Germany is thus more than one hundred times higher than the total energy consumed within the whole country.

Figure 2.4-31 Insolation per year at SL

-60 -40 -20 0 20 40 600

1

2

3

4

Latitude [deg]

Year

ly In

sola

tion

[MW

h/m

2 ]

-60 -40 -20 0 20 40 601

1.5

2

2.5

3

Fact

or [-

]

Oriented ReceiverHorizontal Receiver Factor


In Figure 2.4-32, the maximum and minimum of daily solar radiation on a horizontal plane during the year in dependence on latitude is shown. Note, that like for all previous figures, always a cloudless day with high visibility is assumed. Moreover, the relation between the maximum and the minimum insolation per day for a given latitude is depicted. Again, the two overall maxima for southern and northern hemisphere can be identified. The maximum of receivable solar radiation per day varies only a little with latitude. The bandwidth of the maxima is only 5.0MJ/m2. This means that a short term solar mission can be flown at nearly any latitude if the date is properly chosen. The variation in the values of the minima is, however, very large. While at the equator the influence of the seasons is little more than 3MJ/m2 (less than 15%), the factor of the maximum daily insolation to the minimum daily insolation for a given latitude reaches 5 at around +45°, 7 at around +50°, and 10 around +53° latitude.

Figure 2.4-32 Insolation per day at SL: extrema and factor

-60 -40 -20 0 20 40 600

10

20

30

40

50

Latitude [deg]

Dai

ly In

sola

tion

[MJ/

m2 ]

-60 -40 -20 0 20 40 600

5

10

15

20

25

Fact

or [-

]

Max InsolationMin Insolation

Factor Max to Min

118

Chapter 3

Applied Optimal Control

The present chapter briefly introduces the fundamentals of optimal control theory with the aim to use it in the optimisation of aircraft trajectories and the analysis of bird flight. Therefore, it has to be pointed out, that the flight path of a real flying system shall be optimised. All physical states and controls of the system investigated are continuous which has to be respected in the mathematical modelling and especially in the choice of the control variables. Moreover, the optimal control problem is in general a restricted optimal control problem, since constraints including the states and the controls exist. Due to the complexity of the optimal control problem, in the end, the optimal control task always has to be solved numerically. A fundamental difference lies in the arrangement of the discretisation technique and the optimisation method. The solution methods used are distinguished as: indirect methods and direct methods. The different procedures are described by Betts (2009a) as ‘optimise then discretise’ (indirect) and ‘discretise then optimise’ (direct). In general, the transcription method which converts the infinite-dimensional optimal control problem into a finite-dimensional approximation consists of three fundamental steps which are in case of direct transcription (Betts, 2009a): 1.) Transcribe the optimal control problem into a non-linear programing (NLP) problem by

discretisation. 2.) Solve the resulting NLP problem - e.g. by sequential quadratic programming (SQP). 3.) Assess the accuracy of the approximation (i.e. the finite-dimensional problem) and if

necessary refine the discretisation and then repeat the optimisation steps (mesh refinement).

The indirect methods use the necessary optimality conditions to transform the optimal control problem into a multi-point boundary value problem which is discretised and subsequently solved numerically. The general augmented minimum principle with the general augmented Hamiltonian including the adjoint variables (co-states) and the respective differential equations is used to calculate the optimal control history which is then inserted into the original optimal control problem. For the solution of the boundary value problem, for example, single or multiple shooting methods may be used or alternatively collocation methods or variational methods. For a further classification of indirect methods including a representative exemplary choice, see Büskens (1998) and Büskens (2002) as well as the therein cited literature. The advantage of the indirect methods is the high quality of the solution and the deep insight into the structure of the underlying problem as well as the possibility to easily verify the optimality of the solution. The main, and dominating, disadvantages of the indirect methods are that the user has to have a profound knowledge of optimal control theory and especially that the user has to know the structure of the optimal solution a priori. This includes the knowledge of the structure of the adjoint

Applied Optimal Control 119

differential equations which in the end may strongly differ from the initial guess to also be provided by the user. Moreover, the critical points (e.g. contact points, boundary points) including their number have to be known (Büskens, 1998). To overcome these problems, the direct methods discretise the infinite-dimensional optimal control problem into a finite-dimensional parameter optimisation problem. Constraints including states, controls or parameters can easily be included or added. An initial guess for the adjoint variables becomes obsolete and the radius of convergence increases (Büskens, 2002). For the solution of the resulting (constraint non-linear) optimisation problem, numerous methods are available of which the sequential quadratic programming (SQP) methods have proven to be among the most stable and effective routines (Büskens, 1998). The gradient based methods, however, require at least the gradient of the objective function and the Jacobian of the constraints – see also chapter 5.1 Direct Discretization Methods in Gerdts (2011). While shooting methods only discretise the controls (the states result from integration starting at initial values and are functions of the control variables), collocation methods discretise both the states and the controls and thus both the discretised states and controls are optimisation variables. If the collocation method uses global polynomials for the discretisation, it is named global or pseudospectral. In the standard collocation case a high dimensional NLP problem results featuring a sparse structure in the Hessian of the Lagrange function and the Jacobian of the constraints. In the shooting method case the NLP problem is of lower dimension and the structure is denser. In general, the dimension of the NLP problem is dominated by the number of discretisation points rather than by the dimension of the underlying system of differential equations (Büskens, 2002). For a more general comparison between indirect and direct methods the reader is advised to Betts (2009a) and for a further classification of methods to Ross and Fahroo (2002). The above mentioned indirect methods represent the classical indirect approach in the function space as classified by Gerdts (2011) while the above mentioned direct methods are to be applied on a finite-dimensional approximation of the original problem within this classification.

3.1 The Parameter Dependent Optimal Control Problem

3.1.1 General Problem Formulation

The system ( )puxfx ,,= of which the trajectory is to be optimised has n states, x , which

are influenced by m controls, u , and features pn optimisable real parameters, p . The

system has to obey its state dynamics (3-6) given as explicit first order differential

equations. The resulting optimal trajectory has to comply with the inin initial boundary

conditions, iniψ , with the finn final boundary conditions, finψ , as well as with intn additional

120 Applied Optimal Control

conditions set at the interior points, intr . Moreover, eqn equality conditions, eqC , are

imposed as well as ineqn inequality conditions, ineqC , which form the path constraints. In order to find the optimal trajectory for a system in the time interval:

finini ttt ≤≤ (3-1)

the subsequently stated optimal control problem (OCP) has to be solved. Determine the optimal control history:

( ) mopt t R∈u (3-2)

and the optimal values for the real parameters:

pnopt R∈p (3-3)

by which the optimal state trajectory results:

( ) nopt t R∈x (3-4)

that minimises the cost functional:

( ) ( )( ) ( ) ( )( ) ( ) ( )( ) dttttfttttetttJfin

ini

t

tLagfininifinini ⋅+= ∫ ,,,,,,,,,, puxpxxpux (3-5)

respecting the state dynamics:

( )[ ] ( ) ( ) ( )( )tttttdtd ,,, puxfxx == (3-6)

and the initial and final boundary constraints:

( )( )( )( ) 1 ,,,

1 ,,,

++≤∈=

++≤∈=

nmntt

nmntt

finn

finfinfinfin

inin

iniiniiniini

fin

ini

R

R

ψ0puxψ

ψ0puxψ (3-7)

and the interior point conditions:

( ) ( )( ) intint,int ..., ,1 ,,, with nittt iiii === 0puxr0r (3-8)

and the inequality path constraints:

( ) ( )( ) ineqnineqineq ttt R∈≤ C0puxC ,,, (3-9)

Equality constraints:

( ) ( )( ) eqneqeq ttt R∈= C0puxC ,,, (3-10)

may occur in single segments but not throughout the whole trajectory.


The above stated cost functional (3-5) comprises an algebraic term, e , and an integral Lagrange term, Lagf . Depending on the existence of the terms, the control problem is

differently named: Problem Name Cost Functional Elements

Bolza problem 0≠Lagf and 0≠e

Mayer problem 0=Lagf

Lagrange problem 0=e

Tschebyscheff problem ( ) ( )( )tttfJ ,,,max pux= Table 3.1-1 Problem classification

All problem types of Table 3.1-1 are equivalent and can be transformed into each other – see for example Büskens (1998). In the English literature, frequently the transcription Chebyshev is used instead of Tschebyscheff. The Tschebyscheff problem may also be solved by imposing an inequality constraint on the respective maximum which has to be fulfilled involving an optimisable real parameter. An example for a Tschebyscheff problem is the minimisation of the maximum battery capacity a solar aircraft has to carry in order to fulfil a defined task. As further shown in Büskens (1998), non-autonomous problems where the dynamics (3-6) explicitly depend on the time, t , can be transformed into autonomous problems and it is always possible to transform a free final time into a fixed final time and generally to normalise the time interval (3-1) to:

10 ≤≤τ (3-11)

Where the normalised time, τ, results from the true time, t , by:

( )inifin

ini

ttttt

−−

=τ (3-12)

The state equations (3-6) in normalised time, τ, thus read:

( ) ( )( )[ ] ( )[ ] ( )( ) ( )( ) ( )( ) ( )inifin tttttddtt

dtdt

dd

−⋅=⋅==′ ττττ

ττ

τ ,,, puxfxxx (3-13)

The Lagrange cost function (3-5) is exemplarily transformed for a quadratic change in a state variable:

( ) ττ

dttd

dxdtdtdxJ

inifin

t

tLag

fin

ini

⋅−

⋅

=⋅

= ∫∫

11

0

22

(3-14)

Moreover, it is shown in Büskens (1998), that a single phase optimal control problem can always be split up into a multi-phase optimal control problem and thus, the original interior point conditions may be used to form new phase boundaries. Then, transition conditions have to be fulfilled at the ( )1−i phase boundaries of the phn phases:

( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( )ph

iiniifiniiiiniiifiniiiniiifiniitrans

nitttttt

..., ,2

,,,,,, ,1,1,1,1,1,1,

=

=−−−−−−− 0ppuuxxC (3-15)


In general, multiphase OCPs are used if the number of states or controls or real parameters varies during the trajectory and if the system changes between an autonomous and a non-autonomous system. Then, (3-15) has to be appropriately adapted. In case of an aircraft with a retractable engine, the retraction and extending procedure is in general of fixed duration and also the time for the start-up procedure of the engine is fixed. Only after the engine has been started-up, the thrust is controllable by the thrust-lever command. Moreover, different constraints or cost functions may be activated during different phases. Throughout this work, in multi-phase OCPs only the first phase may feature initial boundary constraints while in all phases final boundary constraints may be set – the same holds for the Mayer part of the cost function.

3.1.2 Shooting Methods In case of a shooting method applied, as mentioned at the beginning of the chapter, the controls are discretised enabling the transformation from the infinite-dimensional optimal control problem to the finite-dimensional optimisation problem. The discretisation is performed on the – in case of aircraft optimisation problems – quite dense control grid. The complete state trajectory follows from the integration of (3-6) or (3-13) starting from the

initial state vector, inix , using the control history.

The controls are discretised on a mesh of ( )1+cn grid points:

( ) ( ) 1 .... .... 0 1121 ==<<<<<<<<= −+ finnniiini ccττττττττ (3-16)

For numerical reasons it should be avoided that a change at a control node influences more than the two neighbouring (previous and subsequent) segments/ intervals. Also the physical nature of the system to be optimally controlled does not allow a change in a control to affect the past. The discretised controls may therefore be chosen to be piecewise constant, piecewise linear or in general be represented by a linear combination of elementary basis-splines (B-splines). For definition of the B-splines, the reader is advised to De Boor (1972) and to Cox (1972). For a graphical representation of B-splines of order two, three, and four see for example chapter 5.1.3 Control Discretisation in Gerdts (2011). For the treatment of insertion of new knots in the control grid, the reader is advised to Lee (1982). If the control grid is chosen to be sufficiently dense, the underlying physical controls are mathematically well approximated in piecewise linear manner which equals a 2nd order B-spline representation. The control values between two grid points are:

( )( )

( )( ) ( )111

with +++

<≤−⋅−

−+= iiii

ii

ii τττ

ττττ

τ uuuu (3-17)

In Büskens (1998), different control approximations are compared and it is shown by citing Conte and De Boor (1980) that the error term of the approximated control reduces to

( )( )[ ]( )21max iiO ττ −+ if the above interpolation is chosen instead of the piecewise constant

interpolation – see chapter 6.7 Piecewise-Polynomial Approximation of Conte and De Boor (1980). Thus, the interpolation error can be reduced to a value as low as desired by densifying the control grid. A grid which is twice as dense as the original grid reduces the interpolation error by the factor four.


(i) Single Shooting In case of the single shooting method applied, the solver has to determine the initial state

vector, inix , and the control history to solve the OCP defined in paragraph 3.1.1. Two

problems arise: the user has to provide an initial guess for the solution, in which his knowledge of the state trajectory is excluded – at least for a direct implementation. If there is not taken an additional step for the generation of a control history from a guessed state trajectory, the user can only provide the initial guess for the initial state vector and the control history of which in most applications he has less knowledge than on the state trajectory. The second main problem is mentioned in Büskens (1998) when he cites Stoer

and Bulirsch (1978) where it is shown, that the influence of faulty initial data, inix , may

exponentially grow with time. Hereby, it is also shown, that when reducing the integration

interval initt − or equally the normalised integration interval iniττ − , the influence of badly

chosen initial state values can be over proportionally reduced (see also Lemma of Gronwall in Gerdts et al. (2014)). The idea is thus to split the integration interval into several sufficiently small subintervals on which the single shooting method is applied consecutively.

(ii) Multiple Shooting One of the first to describe the multiple shooting algorithm applied to flight path optimisation was Bulirsch (1971) who points out that by the multiple shooting approach the region of convergence is improved and the dependency upon the initial guess is reduced, without losing general advantages of the single shooting method. The total time interval (3-11) is divided into ( )1+MSn sub-intervals or segments:

( ) ( ) 1 .... .... 0 ,1,1,,2,1, =<<<<<<<<<= −+ finnMSnMSjMSjMSMSMSini MSMSττττττττ (3-18)

On each sub-interval the integration is restarted. In the end, the state trajectory has to be a continuous function which is piecewise composed of the respective trajectory parts of the sub-interval trajectories. In Figure 3.1-1, the normalised time interval is depicted on which exemplarily five multiple shooting nodes were inserted to give the state grid featuring segment I to VI. The dashed blue line shows the initial guess for the states. At every node, the integration is reset. In the second step, the numerical solver is allowed to vary the values at the nodes. In the end, the defects at node j which are the difference between the state values resulting from integration in segment j and the values of the states at the node j, have to vanish to give a continuous state trajectory (defect constraints). It is advantageous if the control grid (3-16) actually is a sub-grid of the state grid (3-18). This means that every multiple shooting node is also a control node. Thereby, it is avoided that a change at a control node within a certain state grid segment influences the state trajectory in another segment.


Figure 3.1-1 Multiple shooting method

The evaluation of the path constraints is performed on the respective grid which features

PCn grid points:

( ) ( ) 1 .... .... 0 ,1,1,,2,1, =≤<<<<<<<≤= −+ finnPCnPCkPCkPCPCPCini PCPCττττττττ

(3-19)

In most cases:

finnPCPCini PCττττ << ,1, and (3-20)

to avoid possible conflicts with initial and final boundary constraints. The resulting non-linear programming problem contains the optimisation parameter vector, z , which contains the initial states, inix , the states at the multiple shooting nodes, jMS ,x , the

control vectors at the control nodes, iu , and the real parameter vector, p .

( ) cMSijMSini ninj , ... ,0 and , ... ,1 ,,, , === puxxz (3-21)

In total, the number of the elements of the optimisation parameter vector is zn .

The state vector at the multiple shooting node j resulting from integration is thus:

( ) ( ) ( ) ( )( )( )

∫−

⋅⋅−+= −

jMS

jMS

dttt inifinjMSjMS

,

1,

,,, 1,,

τ

τ

ττ zzxfxzx (3-22)

As pointed out by Betts (2009a), the numerical methods to solve ODE initial value problems (IVP) – last term in (3-22) – are relatively mature. In chapter 3.5 Initial Value Problems, an overview about one step methods (Collocation: a.) Classical Runge-Kutta b.) Lobatto:

0x

defect

initial guess trajectory from integration

1,MSx

2,MSx

1,MSτ 2,MSτ 3,MSτ 4,MSτ 5,MSτ0 1

( )τx

τ

3,MSx

4,MSx5,MSx

III III IV V VI


trapezoidal, Hermite-Simpson c.) Gauss collocation d.) Radau collocation) and multi-step methods (Adams-Bashford, Adams-Moulton) is provided. For the solution of IVPs, see also chapter 4 Discretization Methods for ODEs and DAEs in Gerdts (2011). For a continuous trajectory of the states the defects depicted in Figure 3.1-1 have to vanish:

( ) 0xzxC0C =−== jMSjMSdefdef jMS ,,, with ,

τ (3-23)

Performing all the previous steps, the non-linear programming problem then reads: Minimise the cost functional:

( )( ) ( )( ) ( ) ( )( ) τdtftteJ Laginifin ⋅⋅−+= ∫1

0

,,,, zzxzzxzzx (3-24)

respecting the equality and inequality constraints vector:

( )( )( )( )( )( )( )( )

( )( )( )( )

===≤==

=

0zzxψ0zzxr0zzxC0zzxC0zzxC0zzxψ

C

,,,,,,

int

fin

def

ineq

eq

ini

(3-25)

of which the elements are described above and the integration in (3-22) is performed. The necessary optimality conditions, which are the Karush-Kuhn-Tucker conditions (KKT-

conditions), make statements on the scalar-valued Lagrange function, L , which is the cost function augmented by the product of the constraints times the Lagrange multipliers, μ :

( ) ( ) ( )zCμzμz ⋅+= TJL , (3-26)

Apart from the necessity of the gradient of the function to be zero, the multipliers of the actn

active constraints, actC , have to be non-negative to give a candidate for the optimal solution:

Iiiact ∈≥ for 0,µ (3-27)

where I is the set of active constraints (active set) – see Büskens (2002). For the non-active constraints of ineqC , holds:

0C <−actnon (3-28)

0μ =−actnon (3-29)

Thereby, it immediately follows, that non-active constraints are of no influence on the analysis of necessary and sufficient optimality conditions based on the KKT-conditions. For a proof of the Karush-Kuhn-Tucker conditions, see for example Gill et al. (1982). In Schittkowski (1985), the basic sequential quadratic programming (SQP) method is implemented as code and explained. In Schittkowski (1983), the active set strategy is introduced leading to equality constrained subproblems to avoid the recalculation of unnecessary gradients. A compact explanation of the SQP method may also be found in chapter 5.2.2 in Gerdts (2011).


The solver then only has to minimise the Lagrange function (3-26) by solving the KKT-conditions. The above stated problem may be numerically solved using commercially available high fidelity software like IPOPT (Interior Point Optimizer) as described by Wächter and Vigerske (2013) or SNOPT (Sparse Nonlinear Optimizer) as described by Gill et al. (2008). Other extensive software packages are for example SOCS (Sparse Optimal Control Software) from Boeing and GESOP (Graphical Environment for Simulation and Optimization) from ASTOS – see Betts (2009b) and Fischer et al. (2011). In GESOP, the user may solve multiphase multiple shooting and collocation problems using SNOPT or the optimizer SOCS. Moreover, WORHP (We Optimize Really Huge Problems) has to be mentioned which may be used as an optimisation environment independent NLP solver, has been additionally integrated within the ESA project ‘eNLP’ and ‘eNLPext’ in the ASTOS software and is currently further developed in a European Clean Sky project ‘AWACs’ (Adaption of WORHP to Avionics Constraints) involving the TUM-FSD. For a comparison and a demonstration of differences and especially similarities of direct (multiple) shooting, indirect multiple shooting, and direct transcription, the reader is advised to Betts (1998).

3.1.3 Optimisation Framework In the following, the main elements of an optimisation framework are described very briefly. As mentioned above, the aim is to minimise the Lagrange function (3-26), L . The objective function vector, F , handed to the solver is thus composed of the cost function (3-24) and the constraints vector for the discretised problem based on the original constraints vector (3-25):

=

disc

JC

F (3-30)

For the optimisation process, both the objective function vector, F , as well as the optimisation parameter vector (3-21), z , are arranged in chronological order in τ where at the same iτ the original order within F and z for the states, controls, and parameters is

retained – see also appendix E: (E-1) and (E-2). In Figure 3.1-2, the basic procedure within an optimisation framework exemplarily using SNOPT as NLP solver is depicted. The user has to provide the solver settings, e.g. when the solver regards the numerical solution found to be optimal: maximum primal and dual infeasibility. Moreover, the user has to provide an initial guess for the optimisation parameter vector, iniz . The NLP solver hands this initial optimisation parameter vector over

to the function evaluator which contains the system’s dynamics as well as their derivatives with respect to the optimisation parameter vector, z , and an integrator to calculate the trajectory as well as all output values. The function evaluator hands back the calculated cost function, J , the constraints vector, C , and the vector of the derivatives with respect to z :

( ) ( )Czz ∇∇ ,J . With these inputs, the NLP solver calculates the Lagrange function, L , and


the Jacobian, ( )Lz∇ , as well as the approximation of the Hessian of L , ( )Lzz∇ . Thereby,

the solver determines changes in the optimisation parameter vector, [ ]z∆ , which it

subsequently hands to the function evaluator. Solvers like SNOPT are able to determine the whole Jacobian via finite differences but are capable of using all user provided (analytical) gradients. The more gradients the user provides, the better for the convergence. Whilst some solvers require also the Hessian, SNOPT approximates it internally. An automatic differentiation routine in the model generation process can deliver the analytic Jacobian as well as the analytic Hessian. While all standard NLP solvers are able to exploit the knowledge provided by the precise Jacobian, some (e.g. IPOPT, WORHP) can also exploit the knowledge provided by the precise Hessian. The optimisation parameter vector, z , is updated until the solver stopping criteria are met. If the optimal solution has been found, [ ]optz and the vector of the Lagrange multipliers, optμ , is

handed to the function evaluator which calculates the optimal trajectory, the minimal cost function and the vector of the constraints. For further calculations, the Jacobian of the cost function, ( )minJz∇ , the Jacobian of the constraints vector, ( )optCz∇ , the vector of the

Lagrange multipliers in the optimised case, optμ , and if implemented the Hessian of the

Lagrange function, containing ( )minJzz∇ and ( )optCzz∇ , are available.

Figure 3.1-2 Optimisation framework structure

Optimisation Framework

Function Evaluator

( ) ( ) ( )[ ] ( )zfxyx =τττ ,,

( ) ( )Czz ∇∇ ,J

NLP Solver

( ) ( )LL zzz ∇∇ ,C,J

[ ]z

( ) ( )CC zz ∇∇ ,,, JJ

[ ] optopt μz ,

Cμ ⋅+= TJL

Solver settings

Initial guess

Optimised trajectory

( ) ( ) ( )[ ] ( )optzfxyx =τττ ,,

( ) ( ) optoptJ μCzz ,,min ∇∇

optJ C,min

[ ]z

[ ]z∆

( ) ( )optJ Czzzz ∇∇ ,min


3.1.4 Mesh Refinement The importance of a mesh refinement procedure is pointed out by Betts (2009a) where he describes the mesh refinement as a fundamental part for the transcription method of an optimal control problem: after the direct transcription, the resulting optimisation problem is solved as a sparse NLP problem and finally a mesh refinement follows before the optimisation steps are repeated. A thorough treatment of the matter is given in chapter 4.7 of Betts (2009a) and the therein cited literature. Especially when defining the control grid, the awareness of the physical nature of the underlying problem is of high importance. The causal chain represented in Figure 2.1-7 might be reconsidered. In case of linear control interpolation (3-17) and a three degrees of freedom modelling with the angle of attack, α , being a control variable, the maximum rate of change should be incorporated as:

( ) ii τττταα −=∆

∆∆

= +1max

max with (3-31)

Here, the time normalisation (3-11), of course, has to be included. For every control, the control grid thus has to be as dense, that its physical limit is represented. In case of a six degrees of freedom modelling with the elevator deflection, η , being a control variable, the

maximum rate of change should thus be incorporated as:

( ) ii ττττηη −=∆

∆∆

= +1max

max with (3-32)

If the above mentioned limits are reached, in most cases the modelling has to be reconsidered. If not, regions with lower control activity may be identified where the grid might be widened. Moreover, it should always be remembered, that a time normalisation cannot change the underlying physics. Trajectories of long durations in most cases have to be split up into different phases, sometimes of different dynamics levels. A 24 hours flight modelled with an equidistant grid of 250 control nodes is equivalent to allowing the pilot an input not earlier than nearly six minutes after the previous. For long trajectories of rather low dynamics level involving different segments sometimes the precise time of a control input determining this segment change is more important than the overall number of control nodes. If the control grid is implemented as a subgrid of the state grid, also optimisable phase boundaries may be suited to fulfil this task. An alternative is a bisectional algorithm for control node placing near the segment changes. For an automatic mesh refinement process, the reader is advised to chapter 6.4 Automatic Mesh Adaptation of Büskens (1998). Since for the constraints the evaluation is only performed at the nodes of the constraints grid, no information is available between the nodes. Here, bisectional methods may be implemented to adapt the grid where the constraints are likely to be violated. The DENMRA (density function-based mesh refinement algorithm) approach has turned out as very effective (Zhao and Tsiotras, 2011). Here, the mesh point distribution is based on density functions to increase the accuracy of the solution and improve numerical robustness. DENMRA adapts the grid by analysing points of discontinuity of the control


variables or on the non-smoothness of the state variables. An additional density function based algorithm is presented which also regards the controls and in addition the states of the dynamic system. DENMRA is discussed in Bittner et al. (2013) where an additional algorithm is presented which also regards both the controls as well as the states of the dynamic system.

3.2 Function Evaluator The function evaluator to a certain extent may be seen as the engineering part of an optimisation framework as depicted in Figure 3.1-2 while the NLP solver represents the mathematics part. In order to fulfil its tasks already shortly described in paragraph 3.1.3, the function evaluator has to have several abilities for which a number of implementation principles has to be regarded.

3.2.1 Automatic Differentiation of Model / Constraints To be used in an optimisation environment, the dynamics model should not only contain the dynamics (3-6) as well as the yn output values, y :

( ) ( ) ( )( )tttt ,,, puxgy = (3-33)

but also the Jacobian, J , and the Hessian, H , of the system.

Figure 3.2-1 System's model for optimisation

By the Jacobian and the Hessian, the analytical knowledge about the system which is lost due to the discretisation process is restored up to the second order. The Jacobian matrix describes a change in the outputs ( )yx, of the system in dependence

on a change in its inputs:

( ) ( )( )

( ) ( )[ ]yJxJpuxyxJ ,,,,

,

=

∂∂

= Ttt (3-34)

( ) ( ) ( )( )tttt ,,, puxfx =

( ) ( ) ( )( )tttt ,,, puxgy =

p

u

t

x x

J

y

H

( ) ( )( )t

t,,,

,puxyxJ

∂∂

=

( ) ( )tt

,,, puxJH

∂∂

=


where in normalised time (3-11) for an autonomous system holds for the Jacobian of the state derivatives:

( )( )( )( )[ ]

( )

( )[ ]

( )( )( ) ( )inifinTT ttt

ddt

tdtdt

dd

−⋅=⋅∂

∂

=∂

∂

=′ xJpux

x

pux

xxJ

τ

τττ

,,,, (3-35)

The Jacobian is given in detail in appendix F in equation (F-1). The Hessian matrix contains the second order derivatives of the outputs ( )yx, with respect

to the inputs. In case of a vectorised Jacobian – see appendix D for vec operator (D-1) – the Hessian is:

( ) ( )( )( )

( ) ( )[ ]TT

T

tvect yHxH

puxJH ,,,,

=

∂∂

= (3-36)

The Hessian is given in more detail in appendix F in equation (F-2). If Schwarz' theorem may be applied, which in this context is generally the case, the partial derivatives within the Hessian are commutative and the Hessian is thus symmetric within each layer – see appendix F. It is therefore sufficient to calculate the upper or lower triangular matrix and expand the full matrix via a duplication matrix, D , – see appendix F. Inversely, the elimination matrix, L , may be used to eliminate all supra-diagonal elements for the vectorised form of H - see (F-4). Alternatively, the symmetry may be used to check the correct calculation of the Hessian. In order to be automatically differentiable in Matlab, the model of the system, as depicted in Figure 3.2-1, has to contain solely analytic functions and differential equations of which the variables are to be defined as symbolic. All non-analytic data have to be fitted by analytic functions. All derivatives will be available as analytic symbolic functions which is a major difference to, for example, Mehlhorn and Sachs (1994). Automatic differentiation in this work is to be seen as symbolic differentiation automatically executed and not as a technique for automatically augmenting computer programs with statements for the computation of derivatives as described for example by Eberhard and Bischof (1999). The automatic differentiation process is described in full detail in Haslinger (2013). The whole model generation process is performed in Matlab using the Symbolic Math Toolbox. The result is a Simulink model of the whole system containing several subsystems. Within each subsystem, the respective equations are given as embedded Matlab functions. Moreover, the system’s Jacobian and Hessian are generated in vectorised form (D-1). For the Hessian the vectorisation is consecutively performed for each layer. The basic idea for the structure of the automatic differentiation process is the distinction between the generation of information and the processing of information. The user has to define the dynamic equations of the system (3-6) and the outputs of the system (3-33). Thereby, all knowledge about the system is used and all information is – in principle – defined and the information generation process is completed. Most parts of this information are, however, not directly accessible. Aim of the differentiation process is to make more parts of this information accessible: namely the Jacobian and the Hessian of the


system. Since no additional knowledge or information is added, the differentiation can be performed in a completely automatic way, to be seen as processing of information. In principle, the Matlab code for the automatic symbolic differentiation consists of three parts: the model definition, the automatic differentiation and the test simulation. In the model definition (first part), the user has to implement the first order differential equations for the dynamics and the equations for the calculations of the outputs. The user has to group all inputs to states, controls, parameters, the normalised time, and constants. The user may define whether the full Hessian shall be calculated or only certain elements. When implementing the equations for the subsystems the user only has to take care, that subsystems may only feature inputs originating from already defined subsystems or direct inputs to the complete system. Thereby, the proper application of the chain rule is guaranteed. The automatic symbolic differentiation part (second part) requires no user inputs. By using the Matlab Symbolic Math Toolbox, the first and – if chosen – second derivatives are calculated. Then, all outputs are vectorised and a Simulink model is built in which all calculations are grouped in the respective subsystem. Moreover, subsystems for the Jacobian and the Hessian are added – separated for the state derivatives and the outputs.

Figure 3.2-2 Automatically generated Simulink model

In the simulation part (third part), the automatically generated Simulink model is executed using an initial state vector and a given control history for the normalised time. The user may define a fixed or variable step integrator. The states integration block (lowest block in Figure 3.2-2) is only added for system simulation and is excluded from the simulation model which is then integrated in the function evaluator within the optimisation framework. In addition, the automatically generated analytic Jacobian and Hessian may be checked against their approximations generated by finite differences. As indicated in Figure 3.1-2, the function evaluator has to contain an own integrator to calculate the state trajectory and the sensitivity differential equation described in paragraph 3.2.2.

Hessian of outputs y

Jacobian of outputs y

Outputs y

Hessian of state derivatives x

Jacobian of state derivatives x

State derivatives x

States xMODEL

MATRICES

x

x

x


3.2.2 Sensitivity Equations The cost functional (3-24), J , and the constraints vector (3-25), C , as well as their first and (for bilevel optimisation) second derivatives with respect to the optimisation parameter vector, z : ( ) ( ) ( ) ( )CC zzzzzz ∇∇∇∇ ,,, JJ are also implemented within the function evaluator:

( ) TJJzz ∂

∂=∇ (3-37)

and:

( ) TzCCz ∂

∂=∇ (3-38)

The derivative with respect to z is dependent on the discretisation and the implementation. It therefore cannot be performed within the model generation process. However, the derivatives with respect to the states, x , the controls, u , the parameters, p , and eventually

(non-autonomous system) the time, t , may be automatically calculated. The dependency on the optimisation parameter vector of the discretised controls (3-17) and the constant parameters is rather straight forward. The dependency of the state vector, x , on the optimisation parameter vector, z , is however, non-obvious but described by the sensitivity equations.

First Order Within the function evaluator the sensitivity differential equations are evaluated by integration. The sensitivity matrix, S , describes a change in the state vector, x , with respect to a change in the optimisation parameter vector, z , – see also Gerdts (2006):

( ) ( )Ttt

zxS

∂∂

= (3-39)

The sensitivity matrix is thus of dimension [ ]zx nn × .

Using the state dynamics (3-13) and the optimisation parameter vector (3-21) gives the sensitivity differential equation:

TTTTTTTTT

tttt zf

zp

pf

zu

uf

zx

xf

zx

zxSS

S∂∂

⋅∂∂

⋅∂∂

+∂∂

⋅∂∂

+∂∂

⋅∂∂

+∂∂

⋅∂∂

=∂∂

=

∂∂

∂∂

=∂∂

=τ

τ

(3-40)

The initial value for S is:

Tini

iniini z

xS∂∂

= (3-41)

The elements of the Jacobian of the system:

Jfpf

uf

xf ,,, ∈

∂∂

∂∂

∂∂

∂∂

tTTT (3-42)

are all computed during the modelling process as described in paragraph 3.2.1. For a theoretical justification of (3-40), as mentioned in Gerdts et al. (2014), see section 3 in chapter 11.1 Parameter Dependency of the Solution in Demailly (1994).


The derivative of the controls is:

cMSTi

TTi

TjMS

Tini

T

ninj , ... ,0 and , ... ,1 , , ,

,,,,

==

∂∂

=

∂∂

∂∂

∂∂

∂∂

=∂∂

0uu00

pu

uu

xu

xu

zu

(3-43)

Equivalent to (3-21), MSn is the total number of multiple shooting nodes, and cn the total

number of control nodes. Using a piecewise linear control interpolation as described in (3-17), gives for node i :

( )( )

<≤

−−

−⊗=∂∂ +

+

lse

for 1 11

em

iiii

im

Ti 0

Iuu τττ

ττττ

(3-44)

and for control node )1( +i :

( )( )

( )

<≤

−−

⊗=∂

∂ ++

+ elsem

iiii

im

Ti 0

Iu

u 11

1

for τττττ

ττ (3-45)

For an explanation of the Kronecker product, ⊗ , see (D-5) in appendix D. The derivative of the constant (with respect to the states, controls or time) parameters is straight forward:

[ ] cMS

TTi

TjMS

Tini

T

ninj , ... ,0 and , ... ,1 , , ,

,,,,

===

∂∂

∂∂

∂∂

∂∂

=∂∂

I000

pp

up

xp

xp

zp

(3-46)

The last term in the sensitivity differential equation (3-40) can be omitted since:

0z

=∂∂

Tτ

(3-47)

It is advantageous to solve the sensitivity differential equation (3-40) for S in parallel with the same integrator and the same stepsize as the state dynamics (3-13) within the function evaluator. The gradient of the cost function (3-37) is then using the sensitivity, S (3-39):

( ) TTTTTT

JJJJJzp

pzu

uS

xzz ∂∂

⋅∂∂

+∂∂

⋅∂∂

+⋅∂∂

=∂∂

=∇ (3-48)


and the gradient of the constraints (3-38):

( ) TTTTTT zp

pC

zu

uCS

xC

zCCz ∂

∂⋅

∂∂

+∂∂

⋅∂∂

+⋅∂∂

=∂∂

=∇ (3-49)

As indicated in Figure 3.1-2, the function evaluator may now deliver the cost functional (3-24), J , the constraints vector (3-25), C , and the respective gradients (3-48) and (3-49) to the solver. In case of SNOPT this is sufficient, since the Hessian is always approximated internally, e.g. via BFGS updates after each major iteration – see Gill et al. (2008) and Broyden (1970a) as well as Broyden (1970b).

Second Order In case of bilevel optimal control tasks, also the second order sensitivity differential equation is needed. The matrix differential calculus presented in this section is based on Magnus and Neudecker (1985) and it is additionally advised to appendix D. The second order sensitivity matrix, 'S , describes a change in the (vectorised) sensitivity matrix (3-39), S , with respect to a change in the optimisation parameter vector, z :

( ) ( )( )( )tvect T Sz

S∂∂

=' (3-50)

The second order sensitivity matrix is thus of dimension ( )[ ]zzx nnn ×⋅ .

Differentiating the sensitivity differential equation (3-40) with respect to the optimisation parameter vector, z , gives:

∂∂

⋅∂∂

∂∂

+

∂∂

⋅∂∂

∂∂

+

⋅

∂∂

∂∂

=

∂∂

∂∂

T

TTT

T

TTT

T

TT

T

TT

vecvecvec

vec

zp

pf

zzu

uf

zS

xf

z

zx

z

(3-51)

For better readability in the following the transposed sign as well as the associated squared brackets are omitted. Using (D-6), the first term of the right hand side of (3-51) evaluates in case of piecewise linear controls to:

[ ] ( )( )TTnT

T

nT

TTvec

vecvec

zx zS

xfI

zxf

ISSxf

z ∂∂

⋅

∂∂

⊗+∂

∂∂

∂⋅⊗=

⋅

∂∂

∂∂

(3-52)


The second term of the right hand side of (3-51) evaluates to:

T

T

TnT

T

n

T

T

TTT

vecvec

vec

zx zzu

ufI

zuf

Izu

zu

uf

z

∂

∂∂

∂⋅

∂∂

⊗+∂

∂∂

∂⋅

⊗

∂∂

=

∂∂

⋅∂∂

∂∂

(3-53)

Since the first derivatives of the controls stated in (3-44) and (3-45) are constant with respect to u and thus z , the second derivatives of the controls in (3-53) are zero, giving:

T

T

n

T

TTTT

vecvec

x zuf

Izu

zu

uf

z ∂

∂∂

∂⋅

⊗

∂∂

=

∂∂

⋅∂∂

∂∂ (3-54)

The third term of the right hand side of (3-51) containing the constant (with respect to the states, controls or time) parameter vector, p , is:

T

T

TnT

T

n

T

T

TTT

vecvec

vec

zx zzp

pfI

zpf

Izp

zp

pf

z

∂

∂∂

∂⋅

∂∂

⊗+∂

∂∂

∂⋅

⊗

∂∂

=

∂∂

⋅∂∂

∂∂

(3-55)

Since (3-46) is constant, the second derivatives of the parameters are zero, and thus:

T

T

n

T

TTTT

vecvec

x zpf

Izp

zp

pf

z ∂

∂∂

∂

⋅

⊗

∂∂

=

∂∂

⋅∂∂

∂∂

(3-56)

When calculating the derivative with respect to z of the first part of the Jacobian matrix in (3-52), the result is:

TT

T

TT

T

T

T

T

T vecvecvecvec

zp

pxf

zu

uxf

Sx

xf

zxf

HHH

∂∂

⋅∂

∂∂

∂+

∂∂

⋅∂

∂∂

∂+⋅

∂

∂∂

∂=

∂

∂∂

∂

∈∈∈)) ()) )) ()) )) ())

(3-57)

The elements of (3-57) which may be directly taken from the automatically generated Hessian (3-36) of the model are indicated by H∈ . All other elements have to be provided by the function evaluator within the optimisation framework.


Calculating the derivative with respect to z of (3-54) for the second part of the Jacobian matrix of S (3-51), gives:

TT

T

TT

T

T

T

T

T vecvecvecvec

zp

puf

zu

uuf

Sx

uf

zuf

HHH

∂∂

⋅∂

∂∂

∂+

∂∂

⋅∂

∂∂

∂+⋅

∂

∂∂

∂=

∂

∂∂

∂

∈∈∈)) ()) )) ()) )) ())

(3-58)

And finally, the derivative with respect to z of (3-56) for the third part of the Jacobian matrix of S (3-51), is:

TT

T

TT

T

T

T

T

T vecvecvecvec

zp

ppf

zu

upf

Sx

pf

zpf

HHH

∂∂

⋅∂

∂∂

∂+

∂∂

⋅∂

∂∂

∂+⋅

∂

∂∂

∂=

∂

∂∂

∂

∈∈∈)) ()) )) ()) )) ())

(3-59)

Thereby the differential equation for the second derivative of the sensitivity equation (3-51) results as:

( )( )

[ ]

( )( )

∂∂

⋅∂

∂∂

∂+

∂∂

⋅∂

∂∂

∂+⋅

∂

∂∂

∂⋅

⊗

∂∂

+

∂∂

⋅∂

∂∂

∂+

∂∂

⋅∂

∂∂

∂+⋅

∂

∂∂

∂⋅

⊗

∂∂

+∂

∂⋅

∂∂

⊗

+

∂∂

⋅∂

∂∂

∂+

∂∂

⋅∂

∂∂

∂+⋅

∂

∂∂

∂⋅⊗

=

∂∂

∂∂

=∂∂

TT

T

TT

T

T

T

n

T

T

TT

T

TT

T

T

T

n

T

T

TTn

TT

T

TT

T

T

T

nT

TTT

vecvecvec

vecvecvec

vec

vecvecvec

vecvec

x

x

z

x

zp

ppf

zu

upf

Sx

pf

Izp

zp

puf

zu

uuf

Sx

uf

Izu

zS

xfI

zp

pxf

zu

uxf

Sx

xf

IS

zx

zS

z

(3-60)

Equivalent to the first derivative of the sensitivity equation (3-40), also the second derivative

(3-60) is integrated and solved for ( )( )T

vecz

SS∂

∂=' with initial value ( )( )

Tini

iniini

vecz

SS∂

∂=' .


This enables for example the calculation of the Hessian of the cost function (3-37) which is using the sensitivity S (3-39) and its derivative with respect to z (3-60):

( )

( )( )

∂∂

⋅∂

∂∂

∂+

∂∂

⋅∂

∂∂

∂+⋅

∂

∂∂

∂⋅

∂∂

+

∂∂

⋅∂

∂∂

∂+

∂∂

⋅∂

∂∂

∂+⋅

∂

∂∂

∂⋅

∂∂

+∂

∂⋅

∂∂

⊗

+

∂∂

⋅∂

∂∂

∂+

∂∂

⋅∂

∂∂

∂+⋅

∂

∂∂

∂⋅

=∂

∂∂

∂=∇

TT

T

TT

T

T

TT

T

TT

T

TT

T

T

TT

T

TTn

TT

T

TT

T

T

TT

T

T

JvecJvecJvec

JvecJvecJvec

vecJ

JvecJvecJvec

J

J

z

zp

pp

zu

up

Sx

pzp

zp

pu

zu

uuS

xu

zu

zS

xI

zp

px

zu

uxS

xxS

zz

zz

(3-61)

Of course, the Hessian of any other scalar function may be calculated equivalently. The Hessian of any vector function like the constraints is evaluated in accordance with (3-61) and only the dimensions of the unity matrices have to be adapted as described by Magnus and Neudecker (1985).

3.2.3 Scaling As pointed out by e.g. Betts (2009a), scaling the NLP problem is essential to robustly obtain the solution at a high convergence rate, see paragraph 4.8 which is dedicated to Scaling. He explains the necessity of scaling the states and the respective constraints at the same weights to a magnitude about one. He also makes statements on scaling of the variables, the ODE defect scaling, general constraint scaling, and objective scaling. In Büskens (1998), chapter 3.6 treats the scaling of non-linear optimisation problems and a literature overview is given. It is distinguished between scaling in advance to the solution, scaling during the solution process, and scaling afterwards. In the second case, an efficient implementation is of primary importance. The a posteriori scaling is especially suitable for parameter dependent problems which underlie small variations (e.g. homotopy). It is pointed out, that the closer the condition index of the Hessian of the scaled Lagrange function is to one, the faster the theoretic convergence rate. This is also achieved if in accordance with Betts (2009a) the condition number of the KKT matrix is close to one.


In SNOPT, by default, all linear constraints and variables are scaled by an iterative procedure that attempts to make the matrix coefficients as close as possible to one – see Gill et al. (2008). Also a scaling for non-linear constraints may be applied. The user should, however, try to avoid highly non-linear constraints. General scaling problems and solutions are provided in chapter 8 Practicalities of Gill et al. (1982).

3.2.4 Structure / Sparsity Since in most applications the number of non-zero elements in the Jacobian, J , and the Hessian, H , is far below 5%, the solver actively has to exploit this sparse structure. In order to do so, the matrices have to be constructed efficiently; see also chapter 4.6 NLP Considerations – Sparsity in Betts (2009a). The Jacobian of the discretised problem results from (3-30), as:

∇=∇=

disc

JC

FJ zz (3-62)

Exemplarily, the Jacobian for one cycle of the four-phased model for the siskin with a control discretisation and with a full discretisation (collocation) is shown – see also paragraph 4.3. In contrast to (3-21), the optimisation parameter vector, z , is ordered as:

( ) cMSijMSini ninj , ... ,0 and , ... ,1 ,,, , === uxxpz (3-63)

For a more detailed presentation of discC , see (E-1) and for details on z , see (E-2).

Figure 3.2-3 Jacobian sparsity pattern - multiple shooting

In Figure 3.2-3, the sparsity pattern of the Jacobian is depicted. For visibility reasons only five multiple shooting nodes and 20 control nodes were used per phase. Only 1268 out of

0 60 120 180

0

50

100

o

oF∂

z∂1

Phase 2Fixed Length

Phase 4Fixed Length

Phase 3Variable Length

Phase 1Variable Length


19928 possible elements (106 x 188) are non-zero equivalent to 6.36%. Due to the lack of path constraints, the typical triangular structure is absent. If, more realistically, optimisations are performed using 20 multiple shooting nodes and 80 control nodes, a matrix of 406 x 728 elements results containing 4793 non-zero entries out of 295568 possible elements equivalent to 1.62%.

Figure 3.2-4 Jacobian sparsity pattern - collocation

In Figure 3.2-4, the collocation sparsity pattern for the Jacobian is depicted for visibility reasons with a discretisation of 10 nodes per phase for the states and the controls. The matrix contains 2699 non-zero entries out of 60568 possible elements equalling 4.46% (dimension: 226 x 268). The optimisations presented in paragraph 4.3 were conducted with a discretisation of 200 nodes where thus the dimension is 4026 x 4828 of which there are 51149 non-zero elements out of a possible 19437528 which is 0.26%. The Hessian of the Lagrange function is the second derivative of the scalar Lagrange

function (3-26), , with respect to the optimisation parameter vector, , (compare also (3-75)):

(3-64)

The Hessian for the discretisation of 10 nodes (collocation) is depicted in Figure 3.2-5. Within an optimisation framework, the solver has to be explicitly told which elements of the Jacobian or Hessian may become non-zero. All other elements are definitely zero and the knowledge about the sparse structure can be exploited by the solvers implemented.


Figure 3.2-5 Sparsity pattern of Hessian - collocation

So far, classical optimal control theory is presented which, for example, is applied in the trajectory optimisation for air races as in Fisch et al. (2009b), Fisch et al. (2009a) or Fisch et al. (2010) where the highly realistically modelled aircraft finishes a given race track in minimum time. Another application is the optimisation of approach trajectories as in Fisch et al. (2012b) where in addition two methods to handle discrete controls as in case of the flaps setting or the gear position are presented and compared.

3.3 Post Optimal Sensitivity Analysis In the present paragraph, a post optimal sensitivity analysis is presented. Thereby, it is investigated, how the optimal solution reacts to changes. In order to be accessible, these changes have to be representable within the parameter vector, p .

As stated at the very beginning of the chapter, the whole optimal control problem may be dependent on the parameters (3-3), p . So far, an optimal control problem has been solved

and a nominal, constant parameter vector, 0p , resulted.

The present part analyses the influences of a change in the parameter vector:

0ppp −=∆ (3-65)

on the optimal control problem. This analysis will answer two questions: first, how will a change in the parameter vector alter the cost function, the constraints, and the optimisation variables. But also the inverse problem may be solved: How does the parameter vector have to be altered for a desired change in the cost function, the constraints, and the optimisation variables:

ppp ∆+= 0 (3-66)

The terms sensitivity analysis and post-optimality analysis may be used equivalently.

0 130 260

0

130

260

Sparsity Pattern of Hessian


The results presented in this paragraph are based on the work of Büskens (2002), and especially the results shown in his chapter 3 Parametric Sensitivity Analysis where Fiacco (1983) is stated as the main source.

3.3.1 First Order Sensitivity of Optimal Solution For a parameterised optimisation problem, the necessary optimality conditions (KKT-conditions) state that the gradient of the Lagrange function, L , with respect to the optimisation parameters, z , has to be zero as well as the active constraints, actC , have to

be zero:

( ) ( )( )

( ) ( )( )

0pzC

pzCμpzpzC

pμzpμzK zzz =

∇⋅+∇=

∇=

,,,

,,,

,,act

actT

act

act

actact

JL (3-67)

The Jacobian of (3-67) is the so called KKT matrix:

( ) ( ) ( ) ( ) ( )[ ]( )

( ) ( )[ ]( )

∇∇∇

=

∇∇∇⋅+∇

=∇

0pzCpzCpμz

0pzCpzCpzCμpzpμzK

z

zzz

z

zzzzzμz

,,,,

,,,,,,,

act

Tactact

act

Tactact

Tact

act

L

Jact

(3-68)

The basic idea is that implicitly defined functions for the optimisation parameters ( )pzz =

and the Lagrange multipliers ( )pμμ actact = exist which continue to fulfil (3-67) if 0p is

disturbed (altered) sufficiently small. Therefore (3-67) may be abbreviated as:

( ) [ ] 0pKpμzK ==,, act (3-69)

A Taylor series expansion of (3-69) is truncated after the linear term and evaluated in the nominal parameter, 0p , giving:

[ ] [ ] [ ] 0ppKpKpKpp =∆⋅∇+≈

00 (3-70)

With the KKT-conditions holding [ ] [ ]( )0pK0pK == 0 , , it requires:

[ ] ( ) [ ] 0pKp

pμzKpK p

ppp =∇=

∂∂

=∇ 0

00

,, act (3-71)


This gives with applying the chain rule:

[ ] ( )( ) ( ) ( ) 0

pμpz

pμzKpzC

pμzpK

p

pμz

p

zpp =

⋅∇+

∇

∇=∇

0

000,0,

00

00,00 ,,

,,,

dddd

L

actact

act

actact

(3-72)

with:

( ) ( ) ( )000,0000,0 ,,,, pzCμpzpμz zpzpzp actT

actact JL ∇⋅+∇=∇ (3-73)

As stated in Büskens (2002), the KKT matrix (3-68) is invertible if the strict second order optimality conditions based on the Hessian of the Lagrange function are met – sentence 2.8 in Büskens (2002) by which a strict local minimum results. If this is the case, the explicit representation of the sensitivity differentials is enabled:

( ) ( )[ ]( )

( )( )

∇

∇⋅

∇∇∇

−=

−

00

00,01

00

0000,0

,,,

,,,,

0

0

pzCpμz

0pzCpzCpμz

pμpz

p

zp

z

zzz

p

p

act

act

act

Tactact

act

LL

dddd

(3-74)

with:

( ) ( ) ( )000,0000,0 ,,,, pzCμpzpμz zzzzzz actT

actact JL ∇⋅+∇=∇ (3-75)

When calculating the sensitivity differentials, the elements of the KKT matrix either have to be derived analytically or by finite differences – one must not use the matrices resulting from update processes of SQP solvers. Due to the truncated Taylor series in (3-70), all calculations resulting from (3-74), of course, have to respect the limitations from this precondition. Therefore, (3-74) may only be applied, if the structure of the optimisation problem remains unchanged; especially the active set must not change. Then, a high quality suboptimal solution will result from pure matrix-vector multiplication. The new optimisation parameter vector and the new vector of the Lagrange multipliers result from a Taylor series expansion truncated after the linear term as:

( ) ppzzpz

p

∆⋅+≈0

0 dd

(3-76)

( ) ppμμpμ

p

∆⋅+≈0

0, dd act

actact (3-77)

In chapter 4.1.3 Error Analysis of Open-Loop-Real-Time-Optimisation of Büskens (2002), it is shown that for the second derivatives of the optimisation parameter vector, z , and the Lagrange multipliers associated to the active constraints, actμ , to the parameter vector, p ,


one has to evaluate:

( )( )( )( )

( )( )

∇∇

∇

∇

∇

∇

=

,,,,

,,,

,,,

,,,,,,

00

00

00,0

00,0

00,0

00,0

2

2

2

2

0

0

pzCpzC

pμzpμzpμzpμz

pμp

z

zp

zz

pzp

zzp

pzz

zzz

p

p

act

act

act

act

act

act

act LLLL

f

dddd

(3-78)

These second order derivatives thus require the calculations of the third order derivatives of the Lagrange function (including the derivative of the Hessian which is rarely available) and the second order derivatives of the active constraints. Therefore, in most cases, only the sensitivity differentials in (3-74) are accessible and only a linear approximation for z (3-76) and actμ (3-77) is possible.

3.3.2 First and Second Order Sensitivity of Cost Function In Büskens (1998) and Büskens (2002) it is shown, that the first order sensitivity of the cost function locally around 0p behaves accordingly to the first order sensitivity of the Lagrange

function:

( ) ( ) ( )00

000000 ,,,

pzp

p pzpzpz

ppz

ddJJ

ddJ

⋅∇+∇= (3-79)

And equally:

( ) ( ) ( )

( ) ( )0000

00,000,000

,,

,,,,,

00

pzCμpz

pμzp

pμzp

pz

pp

ppp

actT

act

actact

J

Ld

dLd

dJ

∇⋅+∇=

∇== (3-80)

The major advantage of the second representation in (3-80) is that the sensitivity differentials (3-74) do not have to be calculated and therefore the Hessian of the Lagrange function is not required. In many cases the gradients in (3-80) may be derived analytically. As stated in Büskens and Maurer (2000), the above formula (3-80) may be used in the first-order Taylor expansion of the objective functional:

( ) ( ) pp

pzpzp

∆⋅+≈0

00 ,,ddJJJ (3-81)


Moreover, it is shown, that also two equivalent representations of the second order sensitivity of the cost function exist. The first equation results directly from differentiating (3-80):

( ) [ ] [ ] [ ]000200

2

000

, pCpμp

pzp

ppz

pp

zpp

ppp

act

T

act

T

ddL

ddL

dJd

∇⋅

+∇⋅

+∇= (3-82)

An equal expression is:

( ) [ ] [ ] [ ]TT

ddL

ddL

ddL

dJd

⋅∇⋅+

⋅∇⋅

+∇=

0000

000200

2

2,

ppz

pzz

ppp

p pzp

pzp

pzp

ppz

(3-83)

Thus, in the second case the calculation of the sensitivity of the Lagrange multipliers may be avoided. But in any case the second order sensitivity of the cost function only requires the first order sensitivity differentials (3-74). Thereby, the second order Taylor expansion for the objective functional may be calculated – see also Büskens and Maurer (2000):

( ) ( ) ( ) pp

ppp

pzpzpp

∆⋅⋅∆⋅+∆⋅+≈00

2

2

00 21,,

dJd

ddJJJ T

(3-84)

3.3.3 Linear Change in Constraints or Cost Function The sensitivity differentials (3-74) strongly simplify, in case of linear changes, linp , in the

active constraints while the cost function remains unchanged:

( ) 0pzC =− linact (3-85)

Note that in (3-85) the new parameter (3-66) is subtracted, otherwise the sign in (3-87) and thus also in (3-90) and in (3-91) has to be altered. Due to the constant definition of the cost function ( ) ( )zpz JJ lin =, , follows using (3-73):

( ) 0pμzzp =∇ 0,0,0 ,, linactLlin

(3-86)

( )actlin nlinact IpzCp −=∇ 0,0 , (3-87)

Inserting the above equations into the sensitivity differentials (3-74), gives:

( ) ( )[ ]( )

( ) ( )

⋅

∇∇∇

=

+==

−

act

actnznjiji

n

l

act

Tactact

lin

act

lin L

dd

dd

I0

0pzCpzCpμz

pμ

pz

z

zzz

p

p

))))))) ())))))) ,,1,,

0

0

1

00

0000,0

,,,,

(3-88)


Thereby, the sensitivity differentials for linear changes in the constraints result as:

actactnjnilin

act

actznjilin

njnildd

njnildd

zz

z

,,1,,1,

,,1,,1,

,

,

0

0

===

===

++

+

p

p

pμpz

(3-89)

The first order sensitivity of the cost function (3-80) simplifies for the active set to the negative of the active Lagrange multipliers:

( )iact

ilinddJ

,0,,

00

0

, μp

pz

p

−= (3-90)

And the second order sensitivity of the cost function (3-82) simplifies to the negative of the sensitivity of the active Lagrange multipliers to the parameter vector:

( ) ( )0,

2,

002

0

, pp

μp

pz

pd

ddJd iact

ilin

−= (3-91)

With (3-90) and (3-91) under the above assumptions also an additional interpretation of the Lagrange multipliers is provided: the Lagrange multipliers describe the sensitivity of the cost function, J , to the parameter vector, p .

A linear change in the cost function gives only an additive term and is therefore of no influence on the optimal solution or the respective multipliers. The above mathematical results are given with more details in Büskens (2002).

3.4 Bilevel Optimal Control Tasks In this work, only a special class of bilevel optimal control problems is described where an upper level parameter optimisation problem delivers a set of parameters to one or more lower level optimal control problems (OCPs). In case of only one OCP on the lower level (or several non-competing OCPs on the lower level), the cost function of the upper level is in general competing with the cost function on the lower level. In this case, bilevel optimal control is an elegant way to avoid cost function combinations of the two levels (multi-objective optimization). A typical example would be an approach or a departure which on the upper level should minimise the noise impact on ground and on the lower level should minimise time or fuel consumption as it is the case in Richter et al. (2014). The method is also suitable to determine Pareto-optimal solutions – see for example Zhang et al. (2008). For the Pareto front determination of the cost of the participants in optimised ATM scenarios considering total as well as single costs, see Bittner et al. (2014).


In the second case, two or more optimal control problems are competing on the lower level. In most cases they have to fulfil the same task which is altered by the parameter optimisation level. The challenge at the upper level is to alter the parameter vector such that on the lower level the task is solved equally well. Thereby, the limit is defined at which one system becomes superior to the other system or the limit at which an operational mode should be preferred over an alternative mode. E.g.: an air race is considered to be fair, if (two) different types of aircraft can finish the race in the same time. The track layout must be adapted not to favour a certain aircraft – see Fisch et al. (2012c). If the demands on a periodic cycle are altered, it has to be decided whether the original cycle is adapted or split into two cycles – see also the next paragraph for switching decisions. In this context, the bilevel optimisation is closely related to the real time optimisation for parameterised NLP problems – chapter 4 in Büskens (2002). In this work, however, only the open loop approach will be presented since in most cases it is sufficient within the bilevel context. Here, the sensitivity analysis from the lower level problems is used for a direct computation of the gradient of the cost function of the upper level parameter optimisation problem. It is further advised to chapter 6.1 Parametric Sensitivity Analysis and Open-Loop Real-Time Control in Gerdts (2011). Falk and Liu (1995) point out that a sensitivity-based approach is rather promising for solving general non-linear bilevel programming problems. Also a short overview about other methods is provided. For a sophisticated general treatment of the matter the reader is advised to the book Practical Bilevel Optimization: Algorithms and Applications from Bard (1998) and to Colson et al. (2007) for an overview. In the aerospace sector, the number of publications on bilevel optimal control problems is rather limited. In Callies and Wimmer (2000), ascent trajectories for a hypersonic vehicle are optimised under the condition that in case of a complete engine failure an emergency landing site can be reached safely. This condition is transformed into a series of secondary problems while the optimisation of the ascent trajectory is the primary problem. Moreover to be mentioned are Ehtamo and Raivio (2001) where a missile aircraft encounter is analysed. In Raivio and Ehtamo (2000), a pursuit-evasion game is split up into two optimal control problems. At the lower level the pursuer’s minimisation problem is solved and at the upper level the evader’s trajectory is subsequently corrected by a feasible improving step. This process is iteratively repeated until the maximum distance for the evader is determined requiring him to get as close to a border line as possible. In each iteration, gradient information with respect to the cost function of the lower level OCP is used in the solution of the upper level OCP. For bilevel programming tasks in the context of air races, see Fisch (2011). Either hierarchical cost functions are implemented as to increase safety of the races under the assumption that the pilot flies the track in minimum time or the track layout is adapted to increase fairness between two competing aircraft. The safety criteria are, for example, minimum distance or time to crowd, minimum pilot blinding or minimum load factor fatigue index. For the second case of two competing OCPs on the lower level, the track layout is adapted for two different types of aircraft to finish the race in the same time and thus providing a maximum of fairness. Knowing the nominal solution ( )00 pzz = and the sensitivity differentials (3-74) enables an

approximation for the new solution, ( )pz , via a first order Taylor series approximation as

given in (3-76). Since all elements may be calculated off-line, only a matrix-vector


multiplication and a vector addition are required to give the sub-optimal new solution. Thus, the operation can be performed quasi instantly. In Büskens (2002), a method is presented by which the quality of the sub-optimal solution may be further improved using an additional feed-back correction step. Thereby, the elaborative calculation of the second order sensitivity differentials (3-78) is unnecessary – only the active constraints have to be evaluated once. In case of a linear change in the

constraints, the order for z improves to ( )3p∆O . This would, however, only be necessary if

the new sub-optimal initial solution had to be known extremely accurately.

3.4.1 Analysis of Change of Structure of the Solution For real time optimisation it is obligatory and in case of bilevel optimisation advantageous to assure that a change in the parameter vector, p , does not alter the active set. Either a

constraint might become active or an active constraint may leave the active set. The results presented in Büskens (2002) are based on Beltracchi and Gabriele (1988a) and Beltracchi and Gabriele (1988b). Using the sensitivity differentials (3-74), a first order Taylor series expansion for the Lagrange multipliers gives:

( ) ( ) ( ) pppμμpμpμ ∆⋅+=≈ 00,:~d

d actactactact (3-92)

A constraint leaves the active set, if the associated Lagrange multiplier becomes zero. For the approximated i-th multiplier to be zero:

( ) ( ) ( ) ( )000,

0,,,, , :~0 pzppp

pp Iid

di

iactiactiactiact ∈∆⋅+=≈=

µµµµ (3-93)

The respective j-th parameter is found by:

( ) ( )( )

( ) p

j

iact

iactjji nIi

dpd ,,1j , 00

0,

0,,0 ∈∈−≈ pz

ppp µ

µ

(3-94)

The denominator shall not become zero since otherwise the j-th parameter would have no influence on the i-th constraint. A constraint enters the active set, if it becomes zero. For the approximated i-th constraint holds:

( ) ( ) ( )00000,,

0,,, , ,,0 pzppzp

pz Iid

dCCC i

iactnoniactnoniact ∉∆⋅+≈= −

− (3-95)

with:


( )

( ) ( ) ( ) ( )00000,,0000,,

000,,

, ,,

,

pzpzppzpz

pzp

pz IiCddC

ddC

iactnoniactnon

iactnon

∉∇+⋅∇

=

−−

−

(3-96)

The respective j-th parameter which activated the constraint is found by:

( ) ( )( )

( ) p

j

iactnon

iactnonjji nIi

dpdC

C,,1j ,

,00

000,,

0,,0 ∈∉−≈

−

− pzpz

pp (3-97)

Again, the denominator of (3-97) should not become zero, since otherwise the j-th parameter would have no influence on the i-th constraint. With (3-94) and (3-97), for each parameter a range may be defined, in which, under the assumptions stated earlier, the active set will remain unchanged. If the parameter stays within the limits shown by (3-94) and (3-97), an optimisation based on a sub-optimal solution (starting solution/ initial guess) using the sensitivity analysis is very likely to converge. This may also be seen as a first step to define a parameter space in which a solution/ convergence of an optimal control problem can be expected. The next step would be (real-time) optimisation methods able to handle changes in the problem structure and to evaluate the necessary corrections. In Büskens (1998), an extended sensitivity analysis including directional derivatives is presented where the active set is allowed to change.

3.4.2 Upper Level – Lower Level In the following, the upper level features a parameter optimisation problem while the lower level features two competing OCPs. As mentioned at the beginning of the paragraph 3.4, the aim is to analyse when an operational mode for a system becomes superior to another operational mode. In Figure 3.4-1, the basic structure for a bilevel problem with two lower OCPs is depicted. The upper level parameter optimisation problem delivers the same parameter vector, p , to

both lower level problems. These may be solved simultaneously for the given p . In case of

convergence, the lower level problems deliver the complete information on the optimised trajectory as well as the valuable first and second order gradient information including:

pddJ

: the gradient of the cost function with respect to the parameter vector (3-80)

pμ

pz

dd

dd , : the sensitivity differentials (3-74)

pF

zF

dd

dd , : the gradients of the objective function vector, F

Lzz∇ , Lzp∇ : the Hessian and the partial derivatives of the Lagrange function


Thereby, the gradient of the upper level problem may be calculated. In general, the gradient of the upper level objective function to the parameter vector, p , is:

( ) TTupper

Tupper

upper ddJJ

Jpz

zpzpp ⋅

∂

∂+

∂

∂=∇ , (3-98)

In (3-98), the derivative of the upper level cost function, upperJ , to the lower level

optimisation parameter vector, z , has to be implemented analytically using the known dependencies on the states, controls, and parameters.

Figure 3.4-1 Bilevel problem containing two lower level OCPs

3.4.3 Periodic Optimisation The general parameter dependent OCP is described in paragraph 3.1.1. In order for a problem to be termed periodic, the values of several of its states as well as their derivatives at the end of a cycle have to correspond to the initial values. For the states this reads:

( ) ( ) nntt pern

finperiniperper ≤∈= with Rxx (3-99)

and for the state derivatives:

Parameter Optimisation Problem

OCP I

OCP II( ) ( )

pμ

pzpμpz

dd

dd ,,,

L of ,,,, ppzzzppz ∇∇∇∇∇


p


L of ,,,, ppzzzppz ∇∇∇∇∇

( ) ( )pμ

pzpμpz

dd

dd ,,,


( ) ( ) pernfinperiniper tt R∈= xx (3-100)

Due to the underlying system of first order explicit differential equations (3-6), with the piecewise linear controls implemented (3-17), equation (3-100) may be fulfilled by setting the respective initial and final controls equal:

( ) ( ) mmtt perm

finperiniperper ≤∈= with Ruu (3-101)

As mentioned above, the lower level problems are competing. Therefore it is interesting to determine the point (set of upper level parameters) at which the two lower level problems fulfil the given task equally well. This point can, for example, be used to obtain a race track layout which gives two different types of aircraft the same chance of winning (Fisch et al., 2012c). Another interesting application arises if a system is possibly operated in structurally different ways. If a system is periodically operated, the switching point can be determined at which the number of cycles has to be either increased or decreased. For example: from which distance on is it advantageous to fly two shorter cycles instead of one rather long cycle. In the following, such a switching decision is analysed where the cost functions of the lower level problems have to be equal. The cost function for the upper problem is then:

( ) ( )[ ]221 pp JJJupper −= (3-102)

The gradient of the upper level cost function with respect to the parameter vector, p , is:

( ) ( )[ ] ( ) ( )

−⋅−⋅= TTT

upper

ddJ

ddJJJ

ddJ

pp

pppp

p21

212 (3-103)

In case of periodic optimal control problems on the lower level, their cost function derivative with respect to the parameters defining the final boundary conditions is zero in many cases. The cost function, for example, increases if the optimal cycle is stretched but also increases if the optimal cycle is shrunken. The second derivative of the cost function may be calculated using (3-83). The starting point for the bilevel search then results from the intersection of the second order cost function approximations. The approximated lower level cost functions are therefore:

( ) ( ) ( ) 2,1 21

,,

2

2

0

,, =∆⋅⋅∆⋅+∆⋅+≈

=

kd

JdddJJJ k

kTkk

kkoptoptkk

koptkopt

pp

ppp

pppp

)()

(3-104)

with:

2,1 , =−=∆ kkoptk ppp (3-105)

Since in many periodic problems, in addition, the specific cost functions (e.g. per distance travelled) in the optimal case for one and for two cycles are equal, the change in the


parameter vector is determined solely by the quadratic term of the Taylor series approximation, and thus by:

( ) ( ) 222

2

2121

2

1

2,1,

0 pp

ppp

ppp

∆⋅⋅∆−∆⋅⋅∆=optopt

dJd

dJd TT

(3-106)

If a scalar parameter, p , which occurs linearly in the respective final boundary constraint to

determine the cycle length or duration is used, then in the optimal cases of the lower level problems the parameter value for two cycles, 2,optp , is double the value for one cycle, 1,optp :

1,2, 2 optopt pp ⋅= (3-107)

The value of the intersection of the cost functions for one and two cycles is then solely determined by the second order derivatives of the respective cost function with respect to the parameter defining the cycle length. Thus, the intersection is found from (3-106) as:

( ) ( )21,2

22

21,2

12

202,1,

optp

optp

ppdp

Jdppdp

Jd

optopt

⋅−⋅−−⋅= (3-108)

which is solved by the scalar parameter, p :

1,

22

2

21

2

22

2

22

2

21

2

2,1,

2,2,1,1 opt

pp

pppp

dpJd

dpJd

dpJd

dpJd

dpJd

p

optopt

optoptopt ⋅

−

−⋅

+= (3-109)

The result of (3-109) gives the switching point at which one cycle is equally efficient as two cycles. Inserting (3-109) into the first term of (3-108), gives the cost at the switching point:

( ) ( )

2

1,

22

2

21

2

22

2

22

2

21

2

1,,12,1

2,1,

2,2,1,

21

⋅−

−⋅

⋅+= opt

pp

pppoptoptswitch p

dpJd

dpJd

dpJd

dpJd

dpJd

pJpJ

optopt

optoptopt (3-110)

The parameter, p , determined by (3-109) gives the initial guess for the optimisation within

the upper level problem.

152

Chapter 4

Periodic Optimal Control for Flight Applications

In the current chapter, several optimisations performed are presented. First, unlimited endurance missions for solar aircraft are analysed where it is even possible to circle the world in sustained flight. The cost function is the minimisation of the maximum battery capacity required in order to reduce the imposed weight penalty from the batteries. Second, configuration changes of aircraft are regarded. A motor glider performs a saw-tooth flight in order to minimise fuel consumption per kilometre travelled. The motor glider features either a piston or an electric engine powering a propeller or may alternatively be equipped with a jet engine. In every case the whole propulsive unit may be retracted into the fuselage leading to a strong change in the aerodynamic configuration. Third, the flight of small birds as the siskin is analysed. From nature the bounding flight mode is known and it is calculated how much energy per distance may be saved if this intermittent flight style is applied. The bird alternates between flapping and non-flapping phases where with an increase in flight speed the time of the wings folded into the body also increases. In every case, if not explicitly mentioned otherwise, in this chapter standard values are used for the atmosphere model (paragraph 2.3.1) and the acceleration due to gravity at sea level,

sg , presented in the same paragraph in Table 2.3-1. In every optimisation performed, the

controls are always implemented as piecewise linear to give the respective values between two control nodes, e.g. for the integration of the states.

4.1 Unlimited Endurance Missions – Solar Aircraft Solar aircraft are the only aircraft principally able to perform unlimited endurance missions with the current state of the art solar cells and batteries. Due to the fact that solar aircraft do not burn any fuel, they are also the only long endurance aircraft of constant mass. In case of an unmanned solar aircraft, all limitations imposed by the pilot, like pressure restrictions, are dropped. This makes the advantages of the pressure independence of the electric engine fully accessible when flying in high altitude regions with in addition more solar power available. While the long endurance flight of the albatross requires permanent adaptation to the shear wind field (Sachs, 2005), solar aircraft have to store the solar energy received during the day in either potential energy or in batteries. Thus instead of a high frequency low amplitude altitude oscillation, the solar aircraft reaches the highest as well as the lowest altitude only once during a solar day. The results presented here are inspired by the Solar Impulse Project (Solar Impulse, 2014) where the aim is to fly around the world in a manned solar aircraft. As described in paragraph 2.4.5, in dependence on the date a suitable region on earth may be chosen for the flight. In contrast to a real mission, the aim of this work is the investigation of the undisturbed (neither in positive nor in a negative manner) optimal trajectory for such a solar

Periodic Optimal Control for Flight Applications 153

aircraft. Therefore, influences like wind or in general weather occurrences are omitted. As consequence, the aerodynamic velocity, AV , as well as the kinematic velocity, KV , match

and may be simply abbreviated by V . Numerous TUM-FSD publications and presentations at national and international conferences are to be mentioned: - The publication termed Periodic Optimal Flight of Solar Aircraft with Unlimited Endurance

Performance in the Journal of Applied Mathematical Sciences (Sachs et al., 2010b). - The presentation of this author at the CEAS (Council of European Aerospace Societies)

European Air and Space Conference during 10 to 13 September 2007 in Berlin, Germany termed Solar Aircraft Trajectory Optimization for Performance Improvement where the corresponding paper is published as Sachs et al. (2007). The presentation of this author at the CEAS 2009 European Air and Space Conference during 26 to 29 October in Manchester, United Kingdom termed Periodic Unlimited Endurance Flight of Solar Aircraft with Simplified Control Strategy where the corresponding paper is published as Sachs et al. (2009a).

- The presentation of this author at the XXIX OSTIV Congress of the International Scientific and Technical Organization for Gliding during 6 to 13 August 2008 in Luesse, Germany termed Unique Flight Performance Capabilities of Solar-Powered Airplanes (Sachs et al., 2008c).

- The presentation of this author at the 26th ICAS (International Council of the Aeronautical Sciences) Congress 2008 during 14 to 19 September in Anchorage, Alaska, USA termed Unique Performance Characteristics of Solar Aircraft where the corresponding paper is published as Sachs et al. (2008d).

- The presentation at the AIAA Atmospheric Flight Mechanics Conference and Exhibit during 18 to 21 August 2008 in Honolulu, Hawaii, USA termed Periodic Optimal Control for Solar Aircraft with Unlimited Endurance Capability where the corresponding paper is published as Sachs et al. (2008b). The presentation given by this author at the AIAA Guidance, Navigation, and Control Conference during 10 to 13 August 2009 in Chicago, USA termed Unlimited Endurance Performance of Solar UAVs with Minimal or Zero Electrical Energy Storage where the corresponding paper is published as Sachs et al. (2009e).

A short overview about solar aircraft is given in Ross and Frei (2006) while the general design and challenges for the design and operation of solar aircraft are described in Ross (2008) and the design process of high altitude solar UAVs is treated in de Mattos et al. (2013). The history of solar aircraft including an outlook is presented in Voit-Nitschmann (2001). The Icaré aircraft of the university of Stuttgart is described in Scholz et al. (1995) and the more general solar aircraft design in Rehmet et al. (1997). In Rizzo and Frediani (2008), an overview about the solar powered aircraft preliminary design is presented. Sustainable solar flight is investigated by Noth et al. (2007) in PART IV: Applications of Valavanis (2007) where in a broad range the advances in UAVs are described. Noth (2008) describes the whole design process for solar powered aircraft very well and demonstrates his achievements by a 27h test flight of the model he built. The history of solar powered flight is nicely described in the respective chapter number 1.2 of Noth (2008) starting with Sunrise I in 1974 (see also Boucher (1984) and Boucher (1985)), including the first manned flight in the Gossamer Penguin in 1980 (see also MacCready et al. (1983)) up to the HALE (High Altitude Long Endurance) aircraft Helios in 2003 concluding with the QinetiQ Zephyr aircraft which after completion of Noth’s thesis in 2010 reached a flight endurance of 14 days, 22 minutes, 8 seconds and altitudes in excess of 70,000 feet (QinetiQ, 2014).

154 Periodic Optimal Control for Flight Applications

Unlimited endurance missions for HALE aircraft between 55° northern to 55° southern latitude are investigated in Keidel (2000). The demonstrators Solitair I (designed for 48° northern latitude) and the project Solitair II, both featuring twistable solar panels are described well in design and mission. The HALE aircraft HELIPLAT is described in Romeo et al. (2004).

4.1.1 Modelling a Solar Aircraft All data to model a generic solar aircraft were delivered by Hannes Ross, design advisor of the Solar Impulse Project (Solar Impulse, 2014).

Figure 4.1-1Scheme of energy management and propulsion system

Figure 4.1-1 shows the scheme of energy management and the propulsion system. The solar cells cover 93% of the wing reference area and transform the solar radiation into electric power. The maximum power point tracker (MPPT) is needed to vary the ratio between voltage and current delivered from the solar cells in order to receive maximum power. The converter is a circuit which converts direct current from one voltage to another. Below, the respective efficiencies are listed in Table 4.1-1:

Element Efficiency Solar cells (monocristalline silicon) 20 % Maximum power point tracker, MPPT 95 % Electric lines 99.5 % Battery manager 99.5 % Batteries 96.5 % Converter 98.5 % Engine 93 % Propeller 77 %

Table 4.1-1 Efficiencies of elements of energy and propulsion system

For an introduction to solar cells and maximum power point trackers it is advised to appendix G.

MPPTη≈0.95

On-Board Consumption750W

Battery withBattery Manager

η≈0.96Converterη≈0.985

Engineη≈0.93

η≈0.2

Solar Cells

η≈0.77

Propeller


The batteries may be charged at full rate until 90% of their maximum capacity is reached. The remaining 10% may only be charged at a decreased rate. A complete discharging of the batteries is not possible, an unusable remainder of 10% of the nominal capacity is present. Factors causing capacity deterioration were not regarded due to the limited amount of charging and discharging cycles the batteries are subjected to. The charging process is modelled as:

( ) BatqBatBatBat QqQ ,δ⋅= (4-1)

Here, the possible battery charging rate, Batq , depends on the battery charging state, BatQ ,

and, of course, on the available electric power provided by the solar cells. A control Batq,δ

delivers a percentage between zero and full of the possible battery charging rate to the batteries. The propulsion system consists of the electric engines powering the propellers. The thrust, T , results from the maximum engine power, maxEng,P , which is scaled (controlled) by the

thrust lever setting, Tδ , divided by the flight velocity, V , with in addition respecting the

efficiencies of the propeller, Propη , and of the engine, Engη :

VP

T Tδηη

⋅⋅⋅= maxEng,

EngProp (4-2)

Within the regarded speed and altitude range, the maximum engine power is regarded as constant. The efficiency of the motor is assumed to be constantly at 93% and the propeller is assumed to have a constant efficiency factor of about 77%. The assumptions of constant efficiency factors for the electric engine and the propellers are supported by the data provided in appendix H. Note that the data provided there is valid for a constant pitch propeller. The efficiency of a variable pitch propeller will be higher. For the whole mission, a constant on-board consumption of 750W is assumed which may be taken from the solar cells if enough power is available or by discharging the batteries. The translation and position states of the aircraft are the velocity, V , the climb angle, γ ,

the altitude, h , and the distance travelled, s . For better readability the notation is simplified,

e.g. V instead of ( )EKK

KV . For an illustration of the forces acting, see exemplarily Figure

4.2-7.

The resulting equations of motion (from (2-57) and (2-8)) are:

γsin⋅−−

= gm

DTV (4-3)

γγ cos⋅−⋅

=Vg

VmL

(4-4)

γsin⋅= Vh (4-5)

γcos⋅= Vs (4-6)


In the aerodynamics model, the dependency of drag and lift from the corresponding coefficients is represented as well as from the density, ρ , the Velocity, V , and the wing

reference area, S . The lift coefficient, LC , is a permanently available control. The drag coefficient, DC , is a

function of the lift coefficient:

( ) SVCD D ⋅⋅⋅= 22/ρ (4-7)

SVCL L ⋅⋅⋅= 2)2/(ρ (4-8)

( )LDD CCC = (4-9)

The basic data to model the solar aircraft as well as derived values are presented in Table 4.1-2. The aircraft has a total mass of 1650kg of which the batteries are responsible for nearly one fourth. With 196m2, the overall wing area is rather large and its vast majority is covered by solar cells. The best lift to drag ratio of the aircraft is nearly 33 and the corresponding lift coefficient is around 0.8. At full throttle, the engines consume 25kW. During the whole flight, an on-board consumption of 750W has to be respected.

Parameter Notation Value Unit Mass of aircraft with batteries m 1650 kg

Mass of batteries Batm 400 kg

Wing reference area S 196 m2 Best glide number *e 1/32.91 - Lift coefficient for best lift/ drag ratio *

LC 0.8191 -

Maximum engine power (of all engines) maxEng,P 25 kW

Permanent on-board consumption permboardonP ,− 750 W Table 4.1-2 Model data for the solar aircraft

The solar radiation receivable is modelled in accordance with the equations given in paragraph 2.4. All the effects presented there are included, but always higher order terms which were shown to be of minimal influence are omitted. The period of the trajectory is one solar day. A solar day has passed, if the initial true local

solar time, ( )0Sunt , and the final true local solar time, ( )cycSun tt , calculated in accordance with

(2-259), match. A rather strong deviation from 24h can be expected, as the position change of the aircraft is – of course – included in the calculations.

In the results presented here, the standard acceleration due to gravity at sea level, sg , from

Table 2.3-1 is used. On the cost function investigated (4-10), a higher fidelity modelling as given in (2-138) and (2-148) had no measurable effect.


4.1.2 Optimal Periodic Trajectory The optimisation problem type posed to calculate unlimited endurance missions is to be seen as a Tschebyscheff problem with its general classification listed in Table 3.1-1. The aim is to minimise the maximum battery capacity needed:

( ) minmax!== BatQJ (4-10)

The value is given in integrated electric power originally delivered from the solar cells [kWh]. A minimisation of the required battery capacity is equivalent to keeping the battery weight penalty as small as possible. In addition, a Lagrange cost to stabilise the trajectory by reducing the control activity is defined as:

( ) ( ) ( )[ ]∫ ⋅⋅+⋅+⋅=cyct

BATqTLLag dtcbCaJ0

2,

22δδ (4-11)

It reaches values of around 1‰ of (4-10) as the three coefficients a, b, and c are chosen in the range between 1 and 10. In GESOP (Fischer et al., 2011), the first order time derivatives of the piecewise linearly interpolated controls are directly accessible. In order to fly an unlimited endurance mission with the energy provided by the sun, the

aircraft periodically has to be in a position of equal true local solar time, Sunt , as defined in

(2-259). Since for any solar aircraft it is impossible to reach a speed as fast as to maintain a constant solar time by a westbound flight, the critical segment is the night without any radiation receivable. In order to minimise the duration of the night, the optimal flight is directed eastbound. Therefore the geographic longitude, λ , may be used instead of the position x in which the distance travelled, s , is measured. The y-position is constant and thus the latitude, ϕ , is constant – both are thus not part of the state vector.

Figure 4.1-2 Qualitative segments of optimal solar aircraft trajectory

0 0.2 0.4 0.6 0.8 1

Fraction of Solar Day

Alti

tude

Segment 1 Segment 4Segment 3Seg. 2


In Figure 4.1-2, the four segments of an optimal trajectory for a solar aircraft with energy storage in batteries is depicted. The single segments are:

1.) climb to maximum altitude with fully charging the batteries 2.) continuity of solar power use to generate thrust during descent 3.) mere glide after sunset until reaching the minimum altitude allowed 4.) optimised horizontal flight at minimum altitude allowed using the energy stored in the

batteries until enough solar radiation is receivable to re-start with segment 1. In case of a solar UAV with no batteries available to power the engines, segment 4 has to be omitted and the optimal trajectory consists only of segments 1 to 3. Segment 1 is the energy storing segment. At the end of segment 1, the batteries are fully charged, the maximum altitude is reached (maximum of potential energy) and as to be seen in the optimisation results, even the velocity is at its maximum (maximum in kinetic energy). Thus, beside the energy stored in the batteries, also the mechanical energy is at its highest at the end of segment 1. In segments 2 and 3 no battery power is used for propulsion. In segment 2, the whole on-board consumption is satisfied by solar power and the remaining solar power is used for thrust generation. Segment 3 is a mere glide with the propellers in feathering position. Here, the on-board energy demand has to be met by discharging the batteries. Segment 4 is performed at the minimum altitude allowed since the minimum power, minP ,

necessary to propel the aircraft is inversely proportional to the root of the density, ρ , of the

surrounding air (Sachs et al., 2007):

ρ1~minP (4-12)

The lower the flight altitude, the higher is the density and thus the smaller is the minimum power required. In case of a constant minimum altitude limit set, due to (4-12) the vast majority of segment 4 is a quasi-steady flight. Several implementations were investigated and the respective optimisations converged. In the following, only single phase trajectories and trajectories comprising twelve phases are presented. The time interval always ranges from 0 to cyct , which is in case of twelve phases:

1211109876543210 tttttttttttttt cyc +++++++++++=≤≤ (4-13)

of which the durations of all phases are to be optimised. The controls are the thrust lever setting, Tδ , the lift coefficient, LC , and the battery control,

Batq,δ , all of which are available throughout the whole flight.


The optimisation parameters are the durations of the phases as well as a parameter imposing the maximum capacity on the batteries as path constraint. For twelve phases:

[ ]BatQptttttttttttt ,121110987654321 ,,,,,,,,,,,,=p (4-14)

All elements of (4-14) are to be non-negative and the last element, BatQp , , is after

appropriate scaling directly inserted in the cost function (4-10). For all states, lower and upper boundary constraints were implemented of which only the most important are to be mentioned:

( ) ( ) ( ) ( ) ( )( ) 0C ≤−−−−−= BatQBatBatBatineq ptQtQQtVVhththh ,min,minmaxmin ,,,, (4-15)

The solar aircraft has to respect the altitude constraints, must not fly below stall speed and must not charge or discharge the batteries out of their limit. For all controls, lower and upper boundary values were implemented:

10 ≤≤ Tδ (4-16)

max,min, LLL CCC ≤≤ (4-17)

10 , ≤≤ Batqδ (4-18)

Periodic boundary constraints were set on all states apart from the distance travelled, s , and in addition, the solar time has to be periodic:

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 0ψ =

−−

−−−=

0 ,0

,0 ,0 ,0

SunSunBatBat,, tttQtQ

hthtVtV

cyccyc

cyccyccycoutputsstatesper

γγ (4-19)

At the end of the cycle, the aircraft has to fly at the same velocity, V , with the same climb angle, γ , in the same altitude, h , as at the beginning of the cycle. Also the battery charging

state, BatQ , has to be periodic. The cycle has finished, when a whole solar day has passed.

Thus, the final true local solar time, ( )cycSun tt , has to be equal to the initial true local solar

time, ( )0Sunt . The true local solar time is calculated in accordance with (2-259) where the

longitudinal position changes of the aircraft are included. Moreover, also all controls have to be periodic:

( ) ( ) ( ) ( ) ( ) ( )[ ] 0ψ =−−−= 0,0,0 ,,, BatqcycBatqLcycLTcycTcontrolsper tCtCt δδδδ (4-20)

All optimisation results presented in paragraph 4.1.3 to paragraph 4.1.4 are obtained using the graphical user interface GESOP (Fischer et al., 2011) from ASTOS. Both, the single and the 12-phase model of the solar aircraft have been implemented in Simulink and afterwards were coded into C using the Real Time Workshop of Matlab. All other parts like the constraints, the limits of the states and the controls, the cost functions, the real parameters including their limits, and the control history for the initial guess were directly implemented in C. These files were then called by GESOP where SNOPT (Gill et al., 2008) has been used as numerical solver for the optimisation process with primal and dual infeasibility set to values equal or lower than 5.0 · 10-8. In all twelve phases, up to 50 additional multiple


shooting nodes were inserted as well as up to 75 additional control nodes. At every multiple shooting node also an additional control node was inserted. The path constraints were checked on a grid of up to 100 nodes per phase. The results were stored in Matlab-readable format. All plots are generated in Matlab. For each phase the duration was normalised to range from zero to one.

4.1.3 Results for Altitude Limitations In case of an upper boundary constraint imposed on the altitude, the energy storage to a certain amount has to be done by batteries, since the amount of kinetic energy the aircraft may reach is very small and the potential energy storage capability is limited by the altitude constraint.

(i) Variable Lift Coefficient First, the results for a classical aircraft configuration are presented where the lift coefficient is a control permanently available. The initial position is set at a geographic longitude,

2.0=iniλ (equal to 11.5°), and the whole flight takes place at a constant latitude, 7.0=iniϕ

(equal to 40.1°). The flight starts on the 155th day of the year at 7:45:00 local time. The minimum altitude is mh 1000min = due to a safety margin to the ground, and the maximum

altitude allowed is mh 8500max = which is a safety requirement for the pilot wearing a

pressure suit in the aircraft not featuring a pressure cabin.

Figure 4.1-3 Altitude profile for solar aircraft with altitude limits

In Figure 4.1-3, the optimised altitude profile for the solar aircraft is depicted. Segment 1, which includes the cycle boundary takes 38,971s [10h 49min 31s] during which the aircraft travels 488.33km. Segment 2 takes 6,774s [1h 52min 54s] where the aircraft passes 101.04km, segment 3 takes 14,122s [3h 55min 22s] where the aircraft passes 167.11km, and finally segment 4 takes 23,716s [6h 35min 16s] where the aircraft passes 244.82km. In

0 0.2 0.4 0.6 0.8 10

2000

4000

6000

8000

10000

Normalised Distance

Alti

tude

[m]

Segment 1 Segment 4Segment 3Seg. 2

Maximum Altitude


total, the cycle duration is therefore 83,583s [23h 13min 3s] or 2,817s [46min 57s] less than 24 hours which is explained by the eastward movement of in total 1,001.3km. The solar radiation receivable is shown in Figure 4.1-4. At the end of segment 1, just enough radiation is received to fly horizontally in 8500m. At the end of segment 4, enough radiation to initiate the climb is received.

Figure 4.1-4 Altitude profile and receivable solar power (altitude limits set)

Figure 4.1-5 Solar power and power consumption (altitude limits set)

In Figure 4.1-5, for a constant on-board consumption of 750W the solar power delivered from the solar cells as well as the power consumed on-board and by the engines is depicted. The green area represents the energy to be stored in the batteries in order to fly the trajectory. Its value is 2.076 · 108 Ws or 57.57kWh. For comparison, the maximum kinetic energy above stall speed is as little as 1.32 · 105 J and the potential energy stored by

0 0.2 0.4 0.6 0.8 10

2000

4000

6000

8000

10000

Alti

tude

[m]

Fraction of Solar Day starting at 7:45am0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

Sola

r Pow

er [k

W]

0 0.2 0.4 0.6 0.8 10

10

20

30

40

Fraction of Solar Day starting at 7:45am

Pow

er [k

W]

Solar Power Delivered [kW]Power Used (Engine + On-Board Cons.) [kW]Battery Capacity Required [kWh]Max Engine Power [kW]


the climb interval of 7500m is 1.214 · 108 J or 59% of the energy stored in the batteries. The respective altitude and velocity profiles are depicted in Figure 4.1-6.

Figure 4.1-6 Altitude and velocity profiles (altitude limits set)

With an energy density of the batteries of 240 Wh/kg, the required battery capacity is equivalent to a battery mass of 240kg being clearly less than the installed 400kg – also if the unusable remainder of the nominal capacity is included in the weight calculation. The gain in mechanical energy may be calculated as:

( ) ( ) J 10 · 1.2152

82min

2minmax, max

=−⋅+−⋅= VVmhhmgE hgainmech (4-21)

with the minimum speed being the lowest velocity reached at minimum altitude flight at the very end of segment 4:

( )min

minmin hVV = (4-22)

As may be seen in Figure 4.1-5, the area between the red solar power curve and the maximum engine power curve (black dashed line) is nearly as large as the green area showing the maximum battery capacity required. In fact, it is 96.4% as large but not constant as it reduces with flight speed but increases with altitude. If it is the task for the solar aircraft not only to minimise the maximum battery capacity (4-10) but also to maximise range, it should turn as much solar power delivered into thrust as possible. During the day, as soon as enough solar power is delivered for all engines to run – for example – at 90% of maximum power and also the on-board demand is satisfied, the batteries may be fully charged. With the batteries fully charged, thrust may be increased to maximum thrust. As soon as the solar power delivered drops below this value, the engine power has to be reduced accordingly. The night segments then remain unchanged. The energy stored in the batteries as well as in potential and kinetic energy is depicted in Figure 4.1-7. The figure is valid for the very end of segment 1 where the batteries are fully

0 0.2 0.4 0.6 0.8 10

2000

4000

6000

8000

10000

Alti

tude

[m]


12

14

16

18

20

Velo

city

[m/s

]


charged, the altitude is at its highest level allowed and the kinematic velocity reached is the fastest throughout the cycle.

Figure 4.1-7 Energy stored in battery, in potential energy, and in kinetic energy (altitude limits set)

The optimised trajectory shows only very little variation in climb angle – see Figure 4.1-8. The single segments are again marked by vertical lines finishing in open circles.

Figure 4.1-8 Climb angle history (altitude limits set)

in Battery in potential Energy in kinetic Energy0

50

100

150

200

250En

ergy

Sto

red

[MJ]

−π/60

−π/120

0

π/120

π/60


Gam

ma

[rad

]

0 0.2 0.4 0.6 0.8 1


Figure 4.1-9 Thrust lever setting and lift coefficient history (altitude limits set)

The four segments may also be identified by the control histories depicted in Figure 4.1-9. Only the thrust lever setting reaches one of its limits, namely the lower one as the aircraft performs the mere glide in segment 3. The variation in lift coefficient is rather small. In segments 3 and 4, the lift coefficient, LC , stays around 0.027 below that for flight at

minimum power, with *, 3

min LPL CC ⋅= (4-30). The night is slightly shortened, as due to the

mildly faster eastward movement the sun rises a bit earlier. The battery charging state shows the already mentioned maximum of 57.57kWh with its state history given in Figure 4.1-10. In segment 2, the continuity of solar power use during descent may be seen from the constant value of the battery charge followed by the constant discharge rate of 750W in segment 3 where no power is delivered to the engines. For visualisation purposes, the unusable amount of battery capacity is omitted in Figure 4.1-10.

Figure 4.1-10 Battery charging state history (altitude limits set)

0 0.2 0.4 0.6 0.8 10

20

40

60

80

Thru

st L

ever

Set

ting

δ T [%]

Fraction of Solar Day starting at 7:45am0 0.2 0.4 0.6 0.8 11.1

1.2

1.3

1.4

1.5

Lift

Coe

ffici

ent C

L [-]

CL for Minimum Power

0 0.2 0.4 0.6 0.8 10

15

30

45

60


Bat

tery

Cha

rge

[kW

h]


(ii) Constant Lift Coefficient The small variations in the lift coefficient (see Figure 4.1-9) enable a drastic control simplification where the lift coefficient is optimised to a constant value, fixLC , , which is

represented by the blue dashed line in Figure 4.1-11 and is found to be:

1.39132765, =fixLC (4-23)

A constant lift coefficient may be regarded as equivalent to a fixed elevator setting. For the longitudinal motion, the aerodynamic pitching moment about the centre of gravity as reference point (compare (2-160)) has to vanish:

( ) ( ) 0*,, =⋅⋅⋅= qCcSqM mBA ηα (4-24)

with the dynamic pressure (2-180), q , the wing reference area, S , the mean aerodynamic

chord, c , and the aerodynamic pitching moment coefficient, mC , which is in case of the

longitudinal motion and fulfilment of (4-24):

0*0 =⋅+⋅+⋅+= qCCCCC mqmmmm ηα ηα (4-25)

For a symmetric pull-up the pitch rate, q , equals the change in pitch angle which equals the

change climb angle:

γ =Θ=q (4-26)

From Figure 4.1-8, the maximum change in climb angle is found to be around:

max5

max 105 qs

rad=⋅= −γ (4-27)

Since the normalised pitch rate (2-162), *q , for the optimised trajectory is very small, it

may be omitted in (4-25). The pitching moment coefficient (4-25) then becomes only a function of the angle of attack, α , and the elevator deflection, η . The same holds for the lift

coefficient:

(0

0 *≈

⋅+⋅+⋅+= qCCCCC LqLLLL ηα ηα (4-28)

The fixed optimised lift coefficient, fixLC , , therefore determines the elevator deflection as:

α

ηαη

αα

η

L

Lmm

L

LfixLmm

CC

CC

CCC

CC

⋅−

−⋅+

−=

0,0

(4-29)


Figure 4.1-11 Comparison between variable and constant lift coefficient (altitude limits set)

Due to the energy surplus in segment 1, giving more degrees of freedom for the energy management during this segment, the constant lift coefficient lies very close to the values in the other segments (especially segments 3 and 4) where there are little degrees of freedom. This is also shown by the variation in the altitude profile depicted in Figure 4.1-12 which may be identified to be mainly in segment 1.

Figure 4.1-12 Altitude profiles for variable and constant lift coefficient (altitude limits set)

The corresponding thrust lever settings are given in Figure 4.1-13 where for the constant lift coefficient case a slightly higher maximum is reached during segment 1. In the subsequent segments there is practically no difference for the thrust lever settings between the variable and the constant lift coefficient case. The total distance travelled differs by less than 2.1km (the constant lift coefficient case is a bit shorter) and the total duration is practically the same since the constant lift coefficient trajectory numerically takes 6s longer than that for the variable lift coefficient.

0 0.2 0.4 0.6 0.8 11.2

1.3

1.4

1.5


Lift

Coe

ffici

ent C

L [-]

Variable CL optimisedConstant CL optimised

0 0.2 0.4 0.6 0.8 10

2000

4000

6000

8000

10000


Alti

tude

[m]



Figure 4.1-13 Thrust lever settings for variable and constant lift coefficient (altitude limits set)

By setting the lift coefficient to an optimised fixed value, which is very close to the value of the lift coefficient in the variable case of segments 3 and 4, the battery capacity in the constant lift coefficient case has to be increased by less than 1‰. Since the increase in lift coefficient at the end of segment 4 in the variable lift coefficient case is mainly due to a small velocity reduction (see also Figure 4.1-14), a constant lift coefficient is very suitable for segments 2, 3, and 4. Due to the energy surplus provided by the sun in segment 1, the possibly negative influence of a fixed lift coefficient on the cost function (4-10) may be neutralised by the thrust commanded requiring only little altering.

Figure 4.1-14 Velocity profiles for variable and constant lift coefficient (altitude limits set)

In Figure 4.1-14, the velocity profiles for a constant as well as for a variable lift coefficient are compared. In the constant lift coefficient case the profile is smoother. It may be noticed that the altitude profiles are virtually the same at the beginning and end of segment 1 – see

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8


Thru

st L

ever

Set

ting

δ T [%]

δT for variable CL optimisedδT for constant CL optimised

0 0.2 0.4 0.6 0.8 110

12

14

16

18


Velo

city

[m/s

]



also Figure 4.1-12. Since the kinetic energy is only around 1‰ of the potential energy or of the energy stored in the batteries, the velocity profile of the non-constant case may be used to demonstrate how well the optimisation procedure works. The difference in kinetic energy at the end of segment 1 is as little as 15.84kJ which equals 8·10-3 % of the energy stored in the batteries. Concerning the performance of the aircraft, the constant lift coefficient case is practically equivalent to the variable lift coefficient case but offers the advantage of a simpler control behaviour.

4.1.4 No Upper Altitude Limit In the following, the maximum altitude limit is dropped. First, optimisation results for a manned solar aircraft are presented. Second, the respective optimal trajectory for a solar UAV is given. Here, the limitations originating from the pilot are dropped which leads to a much lighter airplane.

(i) Manned Solar Aircraft Assuming that the maximum altitude limit is dropped, and assuming a constant overall mass of the aircraft, the weight saved through smaller batteries could be used for a pressure cabin. This is investigated in the following.

Figure 4.1-15 Altitude profile and receivable solar power (no upper altitude limit set, manned)

To minimise the cost function (4-10), the solar aircraft now has to climb up to an altitude of 15,392m – see Figure 4.1-15. Thereby, the power consuming segment 4 is shortened. It now starts only after 81% of the normalised cycle duration instead of already after 70% of the solar day having passed in case of the upper altitude limit set. In both cases, the time of sunset is after 53% of the cycle duration. Due to the higher flight altitude, also the receivable radiation is higher which may be seen by comparison between Figure 4.1-15 and Figure 4.1-4.

0 0.2 0.4 0.6 0.8 10

4000

8000

12000

16000

Alti

tude

[m]


10

20

30

40

Sola

r Pow

er [k

W]


In total, the new cycle duration is only 82,802s [23h 0min 2s] since the aircraft travels faster at the higher altitude and passes eastbound a distance of 1,278km during one cycle.

Figure 4.1-16 Solar power and power consumption (no upper altitude limit set, manned)

In case of no upper altitude limit set, the engines partially run at their maximum power of 25kW during segment 1 – see Figure 4.1-16. Since during the day the area between the red dashed solar power curve and the blue solid curve indicating the total power consumed is around 67% larger than the green area indicating the required battery capacity, a non-usable energy surplus exists. This surplus may only be accessed by stronger engines. The maximum altitude is reached at an engine power lower than maximum. After having the batteries fully charged, the discharging does not begin before segment 3. The solar aircraft ideally prolongs the flight at minimum altitude a bit, and thus segment 4, to store some energy in the batteries before initiating the climb – see also Figure 4.1-17. This enables a small stretching of the full thrust phase.

Figure 4.1-17 Battery charging and discharging (no upper altitude limit set, manned)

0 0.2 0.4 0.6 0.8 10

10

20

30

40


Pow

er [k

W]

Solar Power Delivered [kW]Power Used (Engine plus On-Board Cons.) [kW]Battery Capacity Required [kWh]

0 0.2 0.4 0.6 0.8 1-10

-5

0

5

10

15

Cha

rgin

g (+

) and

Dis

char

ging

(-) [

kW]



The required battery capacity now is 1.312 · 108 Ws or 36.45kWh. For comparison, the kinetic energy above stall speed stored at the end of segment 1 is as little as 4.41 · 105 J and the potential energy stored by the climb interval of 14,392m is now 2.329 · 108 J or 177% of the energy stored in the batteries – see also Figure 4.1-18 and compare to Figure 4.1-7.

Figure 4.1-18 Energy stored in battery, in potential energy, and in kinetic energy (no upper altitude limit set,

manned)

The respective altitude and velocity trajectories are depicted in Figure 4.1-19. From the

modelled thrust characteristics (2-168), the flight velocity for minimum power, minPV , and the

corresponding lift coefficient, min,PLC , are:

*,4

*

3 ;3 minmin LPLP CCVV ⋅== (4-30)

This velocity is given as reference velocity in Figure 4.1-19. The velocity profile thus directly

follows the altitude profile in its consequence for *V . As physical interpretation, the altitude and velocity profile strongly interact where energy storage is the main task for the altitude reached, and by the appropriately chosen velocity the power which is detracted from the airplane due to drag during the whole flight is minimised. With an energy density of the batteries of 240 Wh/kg the required energy stored in the batteries is equivalent to a battery mass of 152kg. Thus, up to 248kg could be used for a pressure cabin. This equals 15% of the aircraft’s total mass.

in Battery in potential Energy in kinetic Energy0

50

100

150

200

250

Ener

gy S

tore

d [M

J]


Figure 4.1-19 Altitude and velocity profile (no upper altitude limit set, manned)

The history of the climb angle is depicted in Figure 4.1-20. The maximum of the climb angle is now reached during the first third of segment 1 when the solar aircraft rather strongly increases altitude; afterwards, it continuously diminishes and at the end of segment 1 and for the whole of segment 2 it directly corresponds to the available solar power. In segment 3, the value for the optimal glide is reached.

Figure 4.1-20 Climb angle history (no upper altitude limit set, manned)

As already mentioned, the engines are partially run at full thrust whereby the complete range is used – see also Figure 4.1-21. The beginning and the end of this full-throttle phase can also be seen in the lift coefficient history. The variation in lift coefficient is even smaller than in case of an altitude limit set. Therefore a quite strong conformity between the speed behaviour and the behaviour of the density is present, since for a fixed lift coefficient the

0 0.2 0.4 0.6 0.8 10

4000

8000

12000

16000A

ltitu

de [m

]


15

20

25

30

0 0.2 0.4 0.6 0.8 110

15

20

25

30

Velo

city

[m/s

]

Ref. Velocity for Minimum Power

−π/40

−π/80

0

π/80

π/40


Gam

ma

[rad

]

0 0.2 0.4 0.6 0.8 1


velocity at the best lift to drag ratio is inversely proportional to the root of the density of the

surrounding air: ρ1~*V ; see (4-34).

Figure 4.1-21 Thrust lever setting and lift coefficient (no upper altitude limit set, manned)

A simplified thrust strategy might result from taking all solar power available during daytime and distracting the on-board consumption. All remaining power is then directly put into thrust. As soon as the available solar power is above 25.75kW, the batteries are charged. Only at segment 4 the engines are run at the respective power optimal for flight at minimum altitude.

(ii) Unmanned Solar Aircraft In the last section of this paragraph the trajectory for an unmanned solar aircraft is presented. It is assumed that due to the omission of the cabin and due to smaller batteries the mass of the aircraft is reduced to 1201.5kg. The lower altitude limit is set to be 500m. The available engine power as well as the aerodynamics remain unchanged. As mentioned at the beginning of the paragraph and shown by the optimal altitude profile in Figure 4.1-22, segment 4 is no longer existent. Segment 3 ends when the minimum altitude allowed is reached (coincides with the minimum velocity allowed which was set to be 8.5m/s). At the very end of segment 3, the engines are already running with power delivered from the solar cells. The optimal altitude profile and the receivable solar radiation are shown in Figure 4.1-22. The solar UAV climbs up to an altitude of 22,224m where also the maximum velocity of 38.16m/s is reached. Due to the high altitude, the receivable solar radiation is further increased. Within 81,750s [22h 42min 30s] the solar UAV travels 1,651km to the east to complete a cycle.

0 0.2 0.4 0.6 0.8 10

25

50

75

100

Thru

st L

ever

Set

ting

δ T [%]


1.2

1.3

1.4

1.5

Lift

Coe

ffici

ent C

L [-]

CL for Minimum Power


Figure 4.1-22 Altitude profile and receivable solar power for solar UAV

Figure 4.1-23 Solar power and power use for solar UAV

In Figure 4.1-23, the available solar power as well as the power use is depicted. The blue curve represents the power consumed by the engines as well as the on-board consumption and in addition the power delivered to the batteries (positive sign) and taken from the batteries (negative sign). This battery charging and discharging is represented by the black curve. During matching of the blue and the red curve no power is lost. The reason for the curves not to match above 25kW is the limitation of the engines – given by the black dashed line. Thus, the energy represented by the area between the red dashed solar power curve and the blue curve is the non-usable energy surplus which is lost. Since this energy surplus area is so large, there is no effect neither on the cost function (4-10) nor on the range if the battery charging within segment 1 is altered. Two possibilities were presented for energy storage: the batteries which represent additional weight and second, the storing in potential energy. In case of the solar UAV,

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

Alti

tude

[km

]


10

20

30

40

50

Sola

r Pow

er [k

W]

0 0.2 0.4 0.6 0.8 1-10

0

10

20

30

40

50


Pow

er [k

W]

Solar Power Delivered [kW]Used Electric Power [kW]Battery (Dis-)Charging [kW]Max Engine Power [kW]


stronger engines could reduce the amount of unused solar energy. By taking the aircraft to higher altitudes, the minimum altitude may be increased during the cycle representing more safety. The cost function would then be to fly with minimised energy storage and maximised minimum altitude. The trajectory presented here requires for the on-board consumption, which is permanently sustained, a usable battery capacity of 6.8kWh. The charging state of the batteries is given in Figure 4.1-24. As already mentioned above, due to the large energy surplus provided, the charging history of the battery may be freely altered as long as by the end of segment 1 the maximum capacity is reached.

Figure 4.1-24 Battery charging state history for solar UAV

The control histories for the thrust lever setting and the lift coefficient are given in Figure 4.1-25. The thrust lever takes values in the whole bandwidth between zero and one and also the bandwidth for the lift coefficient is broadened.

Figure 4.1-25 Thrust lever setting and lift coefficient for solar UAV

0 0.2 0.4 0.6 0.8 10

2

4

6

8


Bat

tery

Cha

rge

[kW

h]

0 0.2 0.4 0.6 0.8 10

25

50

75

100

Thru

st L

ever

Set

ting

δ T [%]


0.9

1.1

1.3

1.5

Lift

Coe

ffici

ent C

L [-]


4.2 Configuration Changes – Powered Glider with Retractable Engine In the following, different motor glider configurations featuring a retractable propulsive unit are investigated and compared. The aim is to maximise range. The motor glider is either equipped with a piston engine or an electric engine powering a propeller. Also the fuel minimal trajectory for a motor glider featuring a retractable jet engine is analysed. In order to maximise range, all types of motor gliders have to fly a saw-tooth manoeuvre depicted in principal in Figure 4.2-1.

Figure 4.2-1 Saw-tooth profile

The total cycle consists of six phases. In the first phase, the engine is already delivering thrust and the motor glider climbs from the lowest altitude allowed up to a certain altitude. This final altitude reached as well as the duration of the climb are results of the optimisation process. The motor glider has four controls: the thrust lever setting, CMDT ,δ , (only during the

first phase), the lift coefficient, LC , the commanded roll rate, CMDK ,µ , as well as the

commanded aerodynamic angle of sideslip, CMDA,β . In the second phase, the engine has to

be stopped and has to cool down before it is retracted into the fuselage leading to a strong change in the aerodynamic configuration. In contrast to the first phase, the time for cooling and retraction is fixed and therefore the duration of phases two and three is fixed. The fourth phase is the gliding phase, where nearly the starting altitude of the first phase is reached. Here, the potential energy is reduced as the glider tries to maximise range. The minimum altitude constraint must not be violated. Like for the first phase, the total time of the fourth phase is not fixed. Finally, in the fifth phase, the engine is extended again to be

t0

Altit

ude

h

Time cyc

Climb

Engine Stopping& Cooling

EngineRetraction

Glide

EngineExtension

EngineStart-up

OptimalClimb

Interval

Optimal Cycle Lengthhmin


started up and to build up thrust in the sixth phase. The durations of the last two phases are again fixed. In the optimisation process, an optimal cycle length is found where the duration of four of the six phases is fixed and the duration of the phases number one and four has to be optimised. In order to respect the dynamics of the system as well as the strongly different phase durations, both, the climb as well as the gliding phase are split into three phases each for the numeric optimisation. First, the respective manoeuvre is initiated, then conducted and finally levelled-off. In addition to a saw-tooth manoeuvre performed at calm, also head-, tail-, and crosswind influences on the optimal trajectory are investigated. Numerous TUM-FSD publications and presentations at national and international conferences are to be mentioned: - The presentation of this author at the XXIX OSTIV Congress of the International Scientific

and Technical Organization for Gliding during 6 to 13 August 2008 in Luesse, Germany termed Range Maximization by Sawtooth Mode Optimization for Motor Gliders with Retractable Engine where the corresponding paper is published in the Technical Soaring Magazine as Sachs et al. (2010c). The presentations at the XXX OSTIV Congress during 28 July to 6 August 2010 in Szeged, Hungary, termed Wind Effects on Maximum-Range Sawtooth Flight (Sachs et al., 2010d) and Maximum Range Performance of Electric Motor Gliders with Retractable Engine (Sachs et al., 2010a). The presentations of this author at the XXXI OSTIV Congress 6 to 15 August 2012 in Uvalde, Texas, USA termed Saw-Tooth Flight Performance of Motor Gliders with Retractable Jet Engine (Sachs et al., 2012f) and Unique Performance Characteristics of Electric Aircraft (Sachs, 2012b).

- The presentation of this author at the 11th VSDIA (Vehicle System Dynamics, Identification and Anomalies) conference during 10 to 12 November 2008 in Budapest, Hungary termed Periodic Optimal Control for Range Maximization of Motor Gliders where the corresponding paper is published as Sachs et al. (2008a), the presentation of this author at the 12th VSDIA conference during 8 to 10 November 2010 in Budapest, Hungary termed Wind Effects on Periodic Optimal Flight of Motor Gliders with Retractable Engine where the corresponding paper is published as Sachs et al. (2010e) and the presentation of this author at the 13th VSDIA conference during 5 to 7 November 2012 in Budapest, Hungary termed Periodic Optimal Flight of Motor Gliders with Retractable Jet Engine (Sachs et al., 2012e).

- The presentation of this author at the CEAS (Council of European Aerospace Societies) European Air and Space Conference during 26 to 29 October 2009 in Manchester, United Kingdom termed Range Maximization by Periodic Optimal Flight for Electric Motor Gliders with Retractable Engine where the corresponding paper is published as Sachs et al. (2009b). For actual developments in the electric aircraft sector, see also the references therein.

- The presentation of this author at the DGLR conference (Deutscher Luft- und Raumfahrtkongress) during 8 to 10 September 2009 in Aachen, Germany termed Sägezahnflug-Optimierung zur Maximierung der Reichweite von Motorseglern mit Klapptriebwerk (Sachs et al., 2009c) and the invited speech at the DGLR Bezirksgruppe Braunschweig on 26 April 2010 termed Trajektorien-Optimierung für Motorsegler mit Klapptriebwerk (Lenz, 2010).

- The presentation at the 9th International Conference on Mathematical Problems in Engineering, Aerospace, and Sciences (ICNPAA) during 10 to 14 July 2012 in Vienna, Austria termed Periodic Optimal Control for Range Maximization of Powered Sailplanes with Retractable Electric Motor where the corresponding paper is published as Sachs et al. (2012d).


- Several publications with presentations given by this author at the AIAA Guidance, Navigation, and Control (GNC) conference during 10 to 13 August 2009 in Chicago, USA termed Trajectory Optimization for Maximizing the Range of Powered Sailplanes with Retractable Propeller where the corresponding paper is published as Sachs et al. (2009d), at the AIAA GNC conference during 2 to 5 August 2010 in Toronto, Canada termed Wind Effects on the Maximum Range Performance of Motor Gliders with Retractable Engine where the corresponding paper is published as Sachs et al. (2010f), at the AIAA GNC conference during 8 to 11 August 2011 in Portland, Oregon, USA termed Head-, Tail- and Crosswind Effects on the Maximum Range of Powered Sailplanes with Retractable Engine where the corresponding paper is published as Sachs et al. (2011a), and at the AIAA GNC conference during 13 to 16 August 2012 in Minneapolis, Minnesota, USA termed Maximum Range Performance of Motor Gliders with Retractable Jet Engine where the corresponding paper is published as Sachs et al. (2012b) in the references of which also the actual developments in this sector are described.

4.2.1 Modelling the Motor Gliders The generic motor gliders are based on the Glaser-Dirks DG-808C Competition with data published in the flight handbook (DG Flugzeugbau, 2005). The DG-808C Competition is equipped with a SOLO 2625 piston engine powering a propeller.

Parameter Notation Value Unit Aircraft mass empty emptym 350 kg

Maximum take-off mass maxm 600 kg

Wing reference area S 11.8 m2 Wing span b 18 m Maximum power piston engine pistonPmax, 39 · 103 W

Maximum rate of climb at sea level SLhmax, 3.595 m/s

Maximum rate of climb in 3000m mh 3000max, 1.748 m/s

Maximum fuel consumption at sea level max,fuelm 3.209231 · 10-3 kg/s Table 4.2-1 Motor glider data from flight handbook (DG Flugzeugbau, 2005)

The maximum fuel consumption in the flight handbook (DG Flugzeugbau, 2005) is given in

litres per hour [l/h] and was transformed to [kg/s] using the fuel density lkgfuel /745.0=ρ .

As described in (2-52), the ejected (burnt) fuel mass times its exhaust velocity is to be related to the total forces acting on the aircraft in order to determine whether or not this mass flow must be included in the equations of motion. But even at an exhaust velocity of 250m/s, the resultant force would be as little as 0.8N. It is therefore neglected in the subsequent calculations. Both, in case of gliding as well as in case of climbing, the drag coefficient is modelled as a fourth degree polynomial of the lift coefficient with positive constants a, b, c, and d :

LLLLDLDD CdCcCbCaCCCC ⋅+⋅+⋅+⋅+== 234min,)( (4-31)


If the engine is extended (e.g. during climb) the subscript ext is used while in case of the engine being retracted the subscript retr is used (e.g. during glide). First, the gliding performance is regarded. In the flight handbook (DG Flugzeugbau, 2005) the measured velocity polar (the fact that measured values are published has to be remembered in the modelling process and its evaluation) which gives the vertical velocity,

verV , in dependence on the absolute velocity, absV , is depicted – see also Figure 4.2-3 where

the evaluated data points are shown in red. The lift coefficient results as:

2

2 12

−⋅

⋅⋅⋅

=abs

ver

absL V

VSV

gmCρ

(4-32)

The drag coefficient results from:

−⋅

⋅⋅⋅

=abs

ver

absD V

VSV

gmC 2

2ρ

(4-33)

For the modelling of the gliding polar, an altitude of 600m for the data provided is assumed with all atmospheric values resulting from the standard atmosphere as described in

paragraph 2.3.1. For the acceleration due to gravity, the standard value, sg , is used – see

Table 2.3-1. The coefficients calculated using (4-32) and (4-33) are shown as red crosses in Figure 4.2-2. Through the points a fourth degree polynomial as described in (4-31) is fitted. The resulting drag polar is depicted as blue curve in Figure 4.2-2.

Figure 4.2-2 Coefficient polar for motor glider in glide configuration

Moreover, the tangent on the polar is shown as black line and its touching point by an open circle in Figure 4.2-2. The respective values are given in Table 4.2-2:

Parameter Notation Value Unit Best glide number for gliding *

retre 1/51.4 - Lift coefficient for best lift/ drag ratio, gliding retroptLC ,, 0.8037 -

Velocity at best lift/ drag ratio, gliding *retrV 32.75 m/s

Table 4.2-2 Motor glider data from modelling for glide (engine retracted)

0 0.01 0.02 0.03 0.04 0.050

0.5

1

1.5

2

Drag Coefficient CD [-]

Lift

Coe

ffici

ent C

L [-]

Polar Modelled by Polynomial of Degree 4Data from Flight Handbook


The velocity at the best lift to drag ratio results from:

( ) optLCS

gmV,

2*

*

1

2

⋅⋅⋅+

⋅=

ρe (4-34)

All data in Table 4.2-2 are extremely close to the data stated in the flight handbook (DG Flugzeugbau, 2005) – the difference in velocity at the best lift to drag ratio, for example, is less than 0.03m/s.

To give the respective velocity polar, the vertical velocity, verV , as well as the absolute

velocity, absV , are calculated from:

( )[ ] 22

2

LLD

absCCCS

gmV+⋅⋅

⋅=

ρ (4-35)

( )[ ]( )[ ][ ]2

322

22

LLD

LDver

CCC

CCSgmV

+⋅

⋅⋅

−=ρ

(4-36)

The resulting velocity polar based on the fitted coefficient polar is depicted in Figure 4.2-3 as blue curve. The modelled velocity polar represents very well the evaluated data from the flight handbook (DG Flugzeugbau, 2005) which are shown by red crosses in Figure 4.2-3.

Figure 4.2-3 Velocity polar for motor glider in glide

In case of maximum rate climb, the engine is extended and delivers full thrust. In accordance with the flight handbook (DG Flugzeugbau, 2005), the maximum rate climb is performed at a velocity of 25m/s.

20 30 40 50 60

-2

-1.5

-1

-0.5

0

Absolute Velocity [m/s]

Vert

ical

Vel

ocity

[m/s

]

Polar Modelled by Polynomial of Degree 4Data from Flight Handbook


The rate of climb, h , results from the power correlation:

mgVD

mgVTh ⋅

−⋅

= (4-37)

The piston engine powering a propeller is modelled as a constant power propulsion system

(with 1−=Vn set in (2-168)). Using the coefficients from Table 2.2-2, gives for the maximum

thrust, maxT :

ρ

ρρ

n

ref

refref V

VTT

⋅⋅=max (4-38)

With the drag, D , modelled as described in (4-31), and with ρn set to 0.75, the following

implicit equation for the climb rate results:

( )

21

0

21

02

2

02

22

21

0

23

02

23

2

02

24

21

0min,

)()(

1

)(

1

)()(

1

)(

1

)()(1

⋅⋅

⋅−⋅⋅+

⋅−⋅⋅

−

⋅⋅

⋅−⋅⋅+

⋅−⋅⋅

−

⋅⋅−

⋅⋅⋅=

hCV

hV

hCd

hV

hCc

hCV

hV

hCb

hV

hCa

hCVChVT

mghh

L

IAS

IAS

L

IAS

L

L

IAS

IAS

L

IAS

L

L

IASD

n

refrefref

ρρ

ρρ

ρρ

ρρ

ρρ

ρρ

ρρ

ρρ

ρ

(4-39)

The atmospheric reference values for the thrust modelling are set to sea level and again for the atmosphere including the density calculation (2-175), the standard atmosphere as

described in paragraph 2.3.1 is assumed. The pilot controls the indicated airspeed, IASV .

The reference velocity, refV , is set to 25m/s.

In the flight handbook the measured rate of climb is linearly dependent on the altitude during the whole interval h∆ of 3000m:

( ) hhhh

hhh mSLSL ⋅

∆−

−= 3000max,max,max,

(4-40)

The reference thrust, refT , as well as the coefficients for the drag polar (4-31) are determined

numerically solving (4-39). The result is depicted in Figure 4.2-4 with numeric values given in Table 4.2-3. The maximum deviation in climb rate modelled to the measured data stated in the flight handbook (DG Flugzeugbau, 2005) is less than 0.07m/s in the altitude region of interest for the saw-tooth optimisation which is indicated by the vertical blue dashed lines in Figure 4.2-4 (600m to 2200m).


Figure 4.2-4 Climb performance of motor glider

Parameter Notation Value Unit

Best glide number for climbing *exte 1/10.6 -

Lift coefficient for best lift/ drag ratio, climbing extoptLC ,, 1.0304 -

Velocity at best lift/ drag ratio, climbing *extV 28.03 m/s

Reference thrust refT 1404 N Table 4.2-3 Motor glider data from modelling for climb (engine extended)

The resulting drag polar is depicted in Figure 4.2-5 including the tangent on the polar as well as its touching point for the best lift to drag ratio. In the climb configuration the best lift to drag ratio is only around one fifth of the best lift to drag ratio in gliding configuration. This may also be seen by the values of the drag coefficients depicted in Figure 4.2-2 (glide) and in Figure 4.2-5 (climb).

Figure 4.2-5 Coefficient polar for motor glider in climb

0 500 1000 1500 2000 2500 30001

2

3

4

Altitude [m]

Max

imum

Rat

e of

Clim

b [m

/s]

Modelled Climb PerformanceData from Flight Handbook

0 0.05 0.1 0.15 0.2 0.250

0.5

1

1.5

2

Drag Coefficient CD [-]

Lift

Coe

ffici

ent C

L [-]


The associated velocity polar for sea level altitude has been determined numerically and is given in Figure 4.2-6.

Figure 4.2-6 Velocity polar for motor glider in climb

In accordance with the flight handbook (DG Flugzeugbau, 2005), the maximum rate climb is performed at 25m/s or 90km/h. The maximum steady horizontal velocity is 41.83m/s. Respecting the underlying dynamics, the motor glider is modelled as a point mass. The main forces acting on the motor glider are depicted in Figure 4.2-7.

Figure 4.2-7 Motor glider with extended propeller and jet engine

With the point mass modelling, it is assumed that all forces act directly on the centre of gravity and it is further assumed that the thrust, T , acts opposite the drag, D . The translation states of the aircraft are the kinematic velocity, KV , the kinematic course angle,

Kχ , and the kinematic climb angle, Kγ . The equations of motion for the translatory states

result from (2-57), as:

( )( )

( ) ( )( ) ( )

( ) K

EK

K

K

EK

K

K

KEK

K

K

K

K

V

mgLV

L

mgDT

m

V

⋅−⋅⋅

⋅

⋅−−

⋅=

γµγ

µ

γ

γχ

coscoscos

sin

sin

1

(4-41)

20 25 30 35 40 45-2

-1

0

1

2

3

4

Absolute Velocity [m/s]

Vert

ical

Vel

ocity

[m/s

]

L

mgD

α VγT

T L

mgD

α Vγ


The position equations of motion are derived from (2-8), as:

( ) ( )( ) ( )

( )

E

O

K

K

K

K

K

K

K

K

E

OV

VV

hyx

⋅⋅⋅⋅⋅

=

γγχγχ

sincossincoscos

(4-42)

The positive x-direction of the navigation frame is the northern direction and the positive y-

direction is the eastern direction. Thus, as navigation frame the frame ON is used (compare

(2-6) and its explanation). Moreover, the normalised thrust, Tδ , is a state which is controlled by the respective

command, CMDT ,δ :

[ ] [ ]1 ,0 ,1 ,0 with ,, ∈∈

−= CMDTT

TCMDTT

TT

δδδδ

δδ

(4-43)

with the time constant, 5.0=T

Tδ . The actual thrust, T , results from:

TTT δ⋅= max (4-44)

with the maximum thrust available, maxT , modelled as described in (4-38).

In case of a combustion engine installed, also the mass of the aircraft, m , is a state which is

reduced by fuel consumption, fuelm :

fuelmm −= (4-45)

The fuel consumption is calculated as:

( )[ ] ( ) ρ

ρρδ

n

reffuelfuelfuelTfuel

hmmmm

⋅+−⋅= min,min,max, (4-46)

The minimum fuel consumption is the idle fuel consumption and is set to be 10% of the maximum fuel consumption. The ninth state of the aircraft is the bank angle, µ , with its first order time derivative being

controlled by the appropriate command:

CMDKK ,µµ = (4-47)

The lift coefficient, LC , is a control available in all phases. The drag coefficient, DC , is a

function of the lift coefficient as modelled in (4-31). Thereby, the actual lift and drag result as:

SVCL AL ⋅⋅⋅= 2)2/(ρ (4-48)

SVCD AD ⋅⋅⋅= 2)2/(ρ (4-49)


As mentioned at the beginning, the durations of climb (first phase) and glide (fourth phase) are to be optimised while the durations for stopping and cooling of the engine (second phase), the duration for retraction of the engine into the fuselage (third phase), the duration for extending the engine (fifth phase), as well as the duration for starting–up the engine (sixth phase) are fixed with values stated in Table 4.2-4.

Parameter Notation Value Unit Time for stopping and cooling, phase 2 coolingt 30 s

Time for engine retraction, phase 3 retractingt 15 s

Time for engine extending, phase 5 extendingt 15 s

Time for engine start-up, phase 6 upstartt − 5 s Table 4.2-4 Phases of fixed durations

The inclination angle of the engine and its supporting structure, the strut angle, engφ , is the

tenth state of the aircraft:

[ ]/2 ,0 πφ ∈eng (4-50)

During the extension and retraction phase, it changes at a constant rate of:

/30sπφ ±=eng (4-51)

which corresponds with the data of Table 4.2-4. During the other phases, it is held constant. The change between the drag polars for the retracted and the extended engine positions is assumed to comply with the projection of the propeller area into the flight direction. The drag coefficient, DC , is thus:

( ) ( )( ) ( )[ ]( ) ( )[ ]( ) ( )[ ]( ) ( )[ ] Lengretrextretr

Lengretrextretr

Lengretrextretr

Lengretrextretr

engretrDextDretrDLengDD

CdddCccc

Cbbb

Caaa

CCCCCC

⋅⋅−+

+⋅⋅−+

+⋅⋅−+

+⋅⋅−+

+⋅−+==

φ

φ

φ

φ

φφ

sin

sin

sin

sin

sin),(

2

3

4

min,,min,,min,,

(4-52)

The motor glider model is able to handle complex wind fields as all elements of equation (2-186) were implemented. Thereby, for example, shear wind fields may be included in the optimisation process.

4.2.2 Saw-Tooth Flight as Periodic Optimal Control Problem The aim for the motor glider is to minimise fuel consumption per kilometre travelled. The cost function, J , is:

( ) ( )min

)(0 !

=−

=cyc

cycfuelfuel

tstmm

J (4-53)

The value is given in kg of fuel per 100km of distance travelled, [kg/100km].


In case of an electric engine powering a propeller, the electric energy used per kilometre travelled is to be minimised:

( )min

)()(

!0 =

⋅==

∫cyc

t

Eng

cyc

cycel

ts

dtP

tstE

J

cyc

(4-54)

The inverse of the above equation, which is thus maximised is the distance which may be travelled per kWh of electric energy used by the engine [km/kWh]. In order to enable a direct comparison between (4-54) and (4-53), a fuel equivalent is additionally stated where it is assumed that the propeller electric engine combination has an efficiency of 75%.

In addition to the cost functions (4-53) or (4-54), a Lagrange cost, LagJ , is defined as:

( )[ ]∫ ⋅=cyct

EKK

KLag dtJ

0

2γ (4-55)

This Lagrange cost function mainly helps to stabilise the trajectory and keeps it rather easily flyable. It reaches values of around 1‰ of (4-53) or (4-54) which would be in terms of fuel consumption at maximum around 2g/100km.

The time interval ranges from 0 to cyct :

upstartextendingretractingcoolingcyc tttttttt −+++++=≤≤ 410 (4-56)

of which the durations of phases one and four are to be optimised, while the durations of all other phases are fixed – see also Table 4.2-4. The controls are the thrust lever setting, CMDT ,δ , which is only available during climb, the lift

coefficient, LC , the commanded roll rate, CMDK ,µ , as well as the commanded aerodynamic

angle of sideslip, CMDA,β . In case of range maximisation for calm as well as for all wind

cases presented, for the optimal trajectory the aerodynamic angle of sideslip is zero. The optimisation parameters are the duration of the climbing and gliding phase as well as for some optimisation runs the initial fuel mass.

[ ]inifuelmtt ,41 ,,=p (4-57)

The motor glider will always set the initial fuel mass to the lowest value allowed or to the minimum value required to fly a cycle. All elements of (4-57) have to be non-negative. For all states, lower and upper boundary constraints are implemented of which only the most important are to be mentioned:

( ) ( )( ) 0C ≤−−−= fuelTineq mtthh 0,1,min δ (4-58)

The glider has to respect the minimum altitude constraint, cannot have more than full thrust, and must have a non-negative fuel mass.


In the presence of wind, not the kinematic velocity but the aerodynamic pendant is limited:

( ) ( )( ) 0C ≤−−= maxmin , VtVtVV AAineq (4-59)

For all controls, lower and upper boundary values are implemented:

10 , ≤≤ CMDTδ (4-60)


max,,,min,, CMDKCMDKCMDK µµµ ≤≤ (4-62)

max,,,min,, CMDACMDACMDA βββ ≤≤ (4-63)

Periodic boundary constraints are set on all states apart from the x-position and the fuel mass:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0ψ =

−−−−

−−−−=

0 ,0 ,0 ,0

,0 ,0 ,0 ,0

engengTT, φφµµδδ

γγχχ

cyccyccyccyc

cyccyccyccycstatesper ttthth

ytyttVtV (4-64)

Moreover, also all controls have to be periodic:

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 0ψ =

−−

−−=

0 ,0

,0,0

,,

,,,

A,CMDcycA,CMDCMDKcycCMDK

LcycLCMDTcycCMDTcontrolsper tt

CtCtββµµ

δδ

(4-65)

All optimisation results presented in paragraph 4.2.3 to paragraph 4.2.5 are obtained using the graphical user interface GESOP (Fischer et al., 2011) from ASTOS. The model of the motor glider has been implemented in Simulink and afterwards was coded into C using the Real Time Workshop of Matlab. All other parts like the constraints, the limits of the states and the controls, the cost functions, the real parameters including their limits, and the control history for the initial guess were directly implemented in C. These files were then called by GESOP where SNOPT (Gill et al., 2008) has been used as numerical solver for the optimisation process with primal and dual infeasibility set to values equal or lower than 5.0 · 10-8. In order to handle the configuration changes as well as the different levels of control activity during the trajectory, in total ten phases were implemented to form a cycle. Climb and glide are split into three phases: they are initiated (50s), conducted (optimal time found by SNOPT), and levelled-off (50s). Therefore, for the optimiser eight phases are of fixed duration and two phases are of variable duration. In all ten phases, up to 150 additional multiple shooting nodes were inserted as well as up to 200 additional control nodes. At every multiple shooting node also an additional control node was inserted. The path constraints were checked on a grid of up to 100 nodes per phase. The results were stored in Matlab-readable format. All plots are generated in Matlab. For each phase, the duration was normalised to range from zero to one. In the following, the altogether four phases for initiation and levelling-off (each 50s) will not be mentioned but are included in climb and glide respectively.


4.2.3 Propeller – Piston Engine

(i) Calm In case of a piston engine installed, the optimal altitude profile for calm is given in Figure 4.2-8 where the minimum altitude constraint of 600m is shown as solid red line.

Figure 4.2-8 Altitude profile for saw-tooth flight with propeller piston engine combination

The motor glider climbs at full thrust from a little more than 600m to 2183.7m in 592.5s [9min 52s]. Then, the engine has to cool down for 30s before it is retracted into the fuselage which takes 15s. For the cooling and retracting a rather high loss of altitude has to be accepted. The fourth phase is the gliding phase where the glider maximises range – it takes 2216.6s [36min 57s]. Then, the engine is extended again which takes 15s and started-up during the sixth phase lasting 5s. The overall cycle duration is 2874.1s [47min 54s] during which the motor glider covers a distance of 93.199km using an altitude interval of 1583.7m. Most of the states are periodic (see above (4-64)) and for many states also their derivatives are periodic. To fly the complete cycle, the motor glider needs 1.7361kg of fuel. The cost function value is therefore 1.8628 kg/100km. As already mentioned, due to the combustion engine, the mass of the aircraft cannot be periodic. But the change in mass during the whole cycle is less than 0.29% of the aircraft’s total mass. And even in direct relation to thrust, a 17N fuel mass change can be considered as very mild.

0 10 20 30 40 50 60 70 80 90 100400

600

800

1000

1200

1400

1600

1800

2000

2200

Alti

tude

[m]

Distance [km]

Phase 1t1=592s

Phase 4t4=2217s


Figure 4.2-9 Velocity and climb velocity profiles (propeller - piston engine)

In Figure 4.2-9, the velocity and the climb velocity profiles for the saw-tooth flight are given. The initial and final values of the whole cycle, the climbing phase and the gliding phase are marked by open circles. The motor glider starts at a rather low velocity of 24.5m/s which is immediately increased to a level optimal for climb. The transition phases will be given in more detail later. After cooling and retracting – even though the engine is switched off – the velocity is slightly increased explaining the loss in altitude shown in Figure 4.2-8. For the optimal glide, the velocity has to be further increased. The constant change in velocity during glide is merely due to density changes of the atmosphere modelled in accordance with paragraph 2.3.1. The extending and starting procedure leads to a strong reduction in kinetic energy. The climb velocity linearly reduces during the climb. During the glide it stays quasi constant. In Figure 4.2-10, it is shown that the motor glider does not sustain the climb angle during optimised climb it reached shortly after the very beginning to account for the change in density. The exchange between potential and kinetic energy during phases two and three is shown by rather low climb angles. During glide, the climb angle is held completely constant at the optimal level.

20

25

30

35

40Ve

loci

ty [m

/s]

Distance [km]0 10 20 30 40 50 60 70 80 90 100-4

-2

0

2

4

Clim

b Ve

loci

ty [m

/s]


Figure 4.2-10 Climb angle profile (propeller - piston engine)

One of the controls of the motor glider is the lift coefficient shown in Figure 4.2-11. Since climb is performed at full thrust during the first phase, the lift coefficient corresponds directly with the velocity profile shown in Figure 4.2-9. For the optimal glide, it is held constant.

Figure 4.2-11 Lift coefficient history for saw-tooth flight (propeller - piston engine)

−π/20

−π/40

0

π/40

π/20

Distance [km]

Gam

ma

[rad

]

0 10 20 30 40 50 60 70 80 90 100

0 10 20 30 40 50 60 70 80 90 1000.4

0.8

1.2

1.6

2

Distance [km]

Lift

Coe

ffici

ent C

L [-]


Figure 4.2-12 IAS to fly saw-tooth manoeuvre (propeller - piston engine)

The pilot can relatively easy fly the manoeuvre. As depicted in Figure 4.2-12, during climb and glide he simply has to keep a constant indicated velocity, IAS. The strut angle of the engine and its supporting structure (4-50) is depicted in Figure 4.2-13 together with the commanded thrust. While the extending and retracting procedure linearly changes the strut angle by 2/π within 15s, the commanded thrust shows a bang-bang behaviour.

Figure 4.2-13 Strut angle and commanded thrust (propeller - piston engine)

0 10 20 30 40 50 60 70 80 90 10020

25

30

35

40

Distance [km]

Indi

cate

d A

irspe

ed [m

/s]

0

π/8

π/4

3π/4

π/2

Stru

t Ang

le [r

ad]

Distance [km]0 10 20 30 40 50 60 70 80 90 1000

25

50

75

100C

omm

ande

d Th

rust

[%]


1) Transition from Climb to Glide Now, the transition phases, namely cooling (phase 2) and engine retraction (phase 3) are regarded in more detail. In Figure 4.2-14, the altitude profile as well as the IAS for the transition from climb to glide are shown. At the end of the climb flown at full thrust, the motor glider reaches its peak altitude and also increases its kinetic energy. Then, the engine is switched off and cooled down for 30s (phase 2) before it is retracted into the fuselage (phase 3). During the cooling phase, the drag due to the extended engine is rather high and reduces with the projection of the propeller area into the flight direction during the retracing phase (4-52) – see also Figure 4.2-16. To maximise range, the speed is reduced at the high drag during cooling and the beginning of the retraction process and only after the retraction process is started, the velocity is significantly increased again. At the beginning of the glide, the altitude has to be further reduced to build up the optimal gliding speed. From around kilometre 19 on, the optimal gliding speed is reached.

Figure 4.2-14 Altitude and IAS profile during transition to glide (propeller - piston engine)

In order to optimally perform the transition from climb to glide, the climb angle has to be reduced to values far lower than that for optimal glide which is reached from kilometre 19 on as may be seen in Figure 4.2-15.

15 16 17 18 19 202000

2050

2100

2150

2200

Alti

tude

[m]

Distance [km]15 16 17 18 19 2020

25

30

35

40

Indi

cate

d A

irspe

ed [m

/s]

Phase 2t2=30s

Ph. 3t3=15s


Figure 4.2-15 Altitude and climb angle history during transition to glide (propeller - piston engine)

The optimal lift coefficient to be commanded by the pilot is depicted in Figure 4.2-16. For glide it is reduced to the optimal value given in Table 4.2-2. During climb and cooling, the drag coefficient is around one tenth of the lift coefficient. In the engine retracting phase, the polar changes in accordance with (4-52). In the gliding phase, the drag coefficient is around one fiftieth of the lift coefficient corresponding to a glide number of 0.02.

Figure 4.2-16 Lift and drag coefficient and glide number during transition to glide (propeller - piston engine)

15 16 17 18 19 202000

2050

2100

2150

2200A

ltitu

de [m

]

Distance [km]

−π/20

−π/40

0

π/40

π/20

Gam

ma

[rad

]G

amm

a [r

ad]

15 16 17 18 19 200

0.4

0.8

1.2

1.6

Lift

Coe

ffici

ent C

L [-]

Distance [km]15 16 17 18 19 200

0.04

0.08

0.12

0.16D

rag

Coe

ffici

ent C

D [-

], e [

-]

Drag Coefficient CD [-]Glide Number e [-]

Dra

g C

oeffi

cien

t CD

[-], ε

[-]


2) Transition from Glide to Climb At the end of the cycle, the engine and its supporting structure is extended again (phase 5) and started-up (phase 6). In Figure 4.2-17, the optimal altitude together with the IAS is shown. The drag increase (see also Figure 4.2-19) when extending the engine forces the motor glider to reduce its velocity since the loss in altitude is rather small. During the starting procedure, the minimum altitude allowed is touched and the thrust is built up – see also Figure 4.2-20. Then, the climb of the second cycle follows. Advantageously – as shown in Figure 4.2-12 – the required speed level for climb is lower than that for glide.

Figure 4.2-17 Altitude and IAS profile during transition to climb (propeller - piston engine)

During the extending procedure, the climb angle is first reduced in value and then increases to grow rather strongly during the engine start-up phase as shown in Figure 4.2-18.

90 91 92 93 94580

600

620

640

660

Alti

tude

[m]

Distance [km]90 91 92 93 9415

20

25

30

35

Indi

cate

d A

irspe

ed [m

/s]

65s

Phase 1of 2nd Cycle

Phase 5t5=15s


Figure 4.2-18 Altitude and climb angle history during transition to climb (propeller - piston engine)

In Figure 4.2-19, the drag increase when extending the engine and its supporting structure may well be seen. The drag increase is described by (4-52). During the engine start-up phase the lift coefficient reaches its maximum value as the minimum altitude is touched. Then, during climb, the drag is around one tenth of the lift corresponding to a glide number of 0.1.

Figure 4.2-19 Lift and drag coefficient and glide number during transition to climb (propeller - piston engine)

92 92.5 93 93.5 94580

600

620

640

660A

ltitu

de [m

]

Distance [km]

−π/20

−π/40

0

π/40

π/20

Gam

ma

[rad

]G

amm

a [r

ad]

Phase 5t5=15s

65s

Phase 1 of 2nd Cycle

92 92.5 93 93.5 940

0.6

1.2

1.8

2.4

Lift

Coe

ffici

ent C

L [-]

Distance [km]92 92.5 93 93.5 940

0.06

0.12

0.18

0.24

Dra

g C

oeffi

cien

t CD

[-],

e [-]

Drag Coefficient CD [-]Glide Number e [-]D

rag

Coe

ffici

ent C

D[-]

, ε [-

]


Figure 4.2-20 Thrust during start-up (phase 6) and initial climb of 2nd cycle (propeller - piston engine)

In Figure 4.2-20, the thrust build-up during the engine starting phase is depicted. At the beginning, the derivative of the thrust curve is zero and later increases nearly linearly in a slightly bowed manner until the maximum thrust for the respective velocity and altitude combination is reached. The above presented transition from glide to climb (see for example the altitude profile in Figure 4.2-17) fits well with the data provided by Lange Aviation for the Antares 18T (wing span 18m and maximum mass of 600kg equipped with a turbo piston engine) shown in Figure 4.2-21 which is taken from Lange (2013a). In Figure 4.2-21, the optimal extending and starting procedure is shown. In case of a malfunction of the electric starter, the wind milling of the propeller may be used to start-up the engine. While in case of a working starter the altitude loss is less than 20m, a starter malfunction requires at least 140m safety margin to the terrain.

93 93.1 93.2 93.3 93.4580

600

620

640

660A

ltitu

de [m

]

Distance [km]93 93.1 93.2 93.3 93.40

400

800

1200

1600

Thru

st [N

]Th

rust

[N]


Figure 4.2-21 Altitude loss during engine extending & start-up Antares 18T (Lange, 2013a)

In order to assess the superiority of the saw-tooth flight over stationary horizontal flight, the respective results are briefly described and summarised in Table 4.2-5. The motor glider can fly at full thrust horizontally at an altitude of 600m. In this mode, it reaches a flight velocity of nearly 42m/s but consumes 7.346kg/100km of fuel. Alternatively, the motor glider may perform an optimised horizontal flight in 600m. Therefore, the velocity is reduced to 29.72m/s and fuel consumption drops to 5.602kg/100km. Compared to this optimised horizontal flight the saw-tooth mode as described above saves more than two thirds of fuel per distance. At the same time, the average horizontal velocity for the saw-tooth flight is around 10% higher than that for optimised horizontal flight. The portion of powered flight in the saw-tooth mode is 20.62%. Due to the saw-tooth mode, the range of the motor glider is more than tripled.

Savings due to Saw-Tooth Mode

Altitude [m] 600 (hor.) 600 (hor.) Δ 1583.7

Average horizontal Velocity [m/s] 41.83 29.72 32.43

Consumption [kg/ 100km] 7.346 5.602 1.863

Savings [%] 66.74

Portion of Powered Flight [%] 20.62 Table 4.2-5 Savings due to saw-tooth mode piston engine calm

Time [s]0 10 20 30 40 50

-120

-60

-140

-100

-80

-20

-40

Req

uire

d Al

titud

e [m

]

20


(ii) Headwind and Tailwind In the following, the influence of a headwind or a tailwind on the optimal saw-tooth profile is investigated. The figures show the situation for an optimised manoeuvre at 10m/s headwind and tailwind respectively. For comparison, the trajectory for calm is also given.

Figure 4.2-22 Altitude profiles for saw-tooth flight in the presence of head- and tailwind

In case of a headwind, the pilot has to increase the climb interval (see Figure 4.2-22) and also the optimal cycle duration increases to 2,966s [49min 26s]. In case of tailwind, the opposite holds as the climb interval is reduced, the duration is lowered to 2,793s [46min 33s]. The ground distance travelled, however, is only 73.3km in the 10m/s headwind case while in the presence of a 10m/s tailwind, 115.0km are passed during an optimised cycle. The different levels of kinematic speed are also shown in Figure 4.2-23. Again, the initial and final values for the cycle are marked by open circles as well as the initial and final values of climb and glide.

Figure 4.2-23 Kinematic velocity profiles for saw-tooth flight in the presence of wind

0 10 20 30 40 50 60 70 80 90 100 110 120500

1000

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Alti

tude

[m]

10m/s HeadwindCalm10m/s Tailwind

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30

40

50

Normalised Cycle Duration

Kin

emat

ic V

eloc

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/s]



The wind influence can also be seen in the kinematic climb angle (Figure 4.2-24). The amplitude of the climb angle values increases with headwind and decreases with tailwind. At the beginning of the climb and the glide, the optimal values differ rather strongly, whilst at the end of the climb and the glide the values are very close.

Figure 4.2-24 Kinematic climb angles histories for saw-tooth flight in the presence of wind

The bang-bang structure of the thrust commanded is maintained. The whole climb is performed at full thrust.

Figure 4.2-25 Lift coefficients histories for saw-tooth flight in the presence of wind

In Figure 4.2-25 and in Figure 4.2-26, it is shown that in case of headwind and tailwind the pilot mainly has to adapt the gliding phase. In case of headwind, the optimal lift coefficient is lower than at calm and the optimal IAS is higher than at calm. Moreover, during the glide at headwind, the optimal lift coefficient and the optimal IAS are no longer constant but have to be slightly reduced (lift-coefficient) and slightly increased (IAS). For tailwind the opposite holds but the optimal variations are even smaller.

−π/16

−π/32

0

π/32

π/16


Gam

ma

[rad

]

0 0.2 0.4 0.6 0.8 1


0 0.2 0.4 0.6 0.8 10.4

0.8

1.2

1.6

2


Lift

Coe

ffici

ent C

L [-]



Figure 4.2-26 IAS level for saw-tooth flight in the presence of wind

The necessary adaptations of the saw-tooth manoeuvre in dependence on wind in terms of climb interval and (average) IAS during glide are shown in Figure 4.2-27 and in Figure 4.2-28. The stronger the headwind, the higher the climb interval and the faster the airspeed during glide to make progress against the wind. In case of tailwind always the opposite holds.

Figure 4.2-27 Optimal climb interval in dependence on wind

0 0.2 0.4 0.6 0.8 120

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/s]


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Cha

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in C

limb

Inte

rval

[m]

Head- and Tailwind [m/s]


Figure 4.2-28 Optimal average IAS during glide in dependence on wind

When the glider performs optimal saw-tooth manoeuvres at head- or tailwind, the minimum cost functions achievable in relation to calm are depicted in Figure 4.2-29. The glider may profit to a large extent from tailwinds and can handle moderate headwinds well. At strong headwinds, power limitations restrict the glider in making progress against the wind and the cost function rises rather strongly.

Figure 4.2-29 Change in cost function in dependence on wind

-20 -15 -10 -5 0 5 10 15 20-2

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in IA

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r Glid

e [m

/s]


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Rel

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ost [

%]



(iii) Crosswind In Figure 4.2-30, the optimal altitude profile for a 10m/s crosswind is depicted – again with the profile for calm as reference. The glider has to maximise the northbound distance per fuel mass burnt while facing a continuous wind from the west. In the 10m/s crosswind case, the glider slightly has to increase the climb interval to 2223m. The total cycle duration is 2904s [48min 24s] with an optimal cycle distance of 91.123km.

Figure 4.2-30 Altitude profile for saw-tooth flight in the presence of crosswind

In order to maximise range, in the presence of a crosswind, the glider has to avoid any aerodynamic angle of sideslip and must fly in a perfectly straight line perpendicular to the crosswind. It therefore has to adapt the heading as shown in Figure 4.2-31. The faster the kinematic velocity (resulting from the varying aerodynamic velocity and the constant wind), the smaller the heading – for the general wind situation, see also Figure 2.3-1. The bang-bang structure of the thrust is maintained.

Figure 4.2-31 Kinematic velocity and heading for the 10m/s crosswind case

0 10 20 30 40 50 60 70 80 90 100500

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1500

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Distance [km]

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[m]

10m/s CrosswindCalm

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35

40

Kin

emat

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ity [m

/s]


−π/8

−3π/32

−π/16

−π/32

0

Hea

ding

[rad

]


Figure 4.2-32 Optimal climb interval in dependence on crosswind

When an optimal profile is flown, the climb interval has to be adapted in dependence on the crosswind as depicted in Figure 4.2-32. In general the adaptation is quite similar as in the moderate headwind case. In Figure 4.2-33, it is shown that also rather strong crosswinds only lead to a moderate increase in cost if the optimal manoeuvre is flown.

Figure 4.2-33 Change in cost function in dependence on crosswind

0 5 10 150

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100C

hang

e in

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b In

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]

Crosswind [m/s]

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ost [

%]

Crosswind [m/s]


4.2.4 Propeller – Electric Engine Concerning the general treatment of electric aircraft with focus on flight performance and range performance properties, two publications of Professor Sachs are to be mentioned. First, Sachs (2012a), presented by this author at the AIAA Atmospheric Flight Mechanics Conference during 13 to 16 August 2012 in Minneapolis, Minnesota, USA and second, Sachs (2013a), presented at the AIAA Atmospheric Flight Mechanics Conference during 19 to 22 August 2013 in Boston, Massachusetts, USA. In case of an electric engine powering a propeller – as for example in the Antares 20E motor glider built by Lange Aviation – the thrust characteristics are modelled in accordance with (2-168) and coefficients from Table 2.2-2. In contrast to the motor glider powered by a combustion engine, the mass of the electric glider now stays constant at 600kg – see Table 4.2-1. Although the Antares 20E is slightly larger than the DG-808C Competition, the data provided by Lange Aviation show that it is sensible to adapt only the thrust characteristics for the generic model. Thereby, a direct comparison between the piston engine powering the propeller and the electric engine powering the propeller is possible. The Antares 20E has a wing span of 20m and reaches at a mass of 600kg a maximum climb rate at sea level of 3.7m/s while the brushless DC/DC motor has a power of 42kW – see Lange (2014a) and appendix H. The maximum climb altitude of the actual Antares 20E is 3500m – see Lange (2013b). All of these values are very close to the respective data published for the DG-808C Competition.

Figure 4.2-34 Altitude profile for saw-tooth flight with electric engine

In Figure 4.2-34, the resulting optimal profiles are shown. The electric engine does not require a cooling phase. Only the propeller has to be stopped before it is retracted into the fuselage. Phase two is thus a 10s stopping phase. In case of the electric engine, the optimal climb altitude is 2719.3m and the optimal cycle length is 125.09km for which the motor glider takes 3769.2s [1h 2min 49s]. The higher climb rate in case of an electric engine installed as compared to a piston engine powering a propeller is shown in Figure 4.2-35. Due to the density independence of the

0 50 100 150500

1250

2000

2750

3500

Distance [km]

Alti

tude

[m]

Electric Engine 10s Stopping PhasePiston Engine 30s Cooling Phase


electric engine, the reduction in climb rate with altitude is also smaller than in case of a naturally aspirated piston engine.

Figure 4.2-35 Climb and sink rate over normalised cycle time (electric engine)

The velocity profiles for the kinematic velocity as well as for the indicated airspeed are given in Figure 4.2-36, where again the beginning and the end of the total cycle, the climb, and the glide are marked by open circles. As in case of the piston engine powering the propeller, the velocity levels of climb and glide clearly differ – but the transition is much shorter. For the pilot, the trajectory is rather easy to fly since during climb and glide he only has to keep a constant IAS.

Figure 4.2-36 Velocity profiles for saw-tooth flight with electric engine

0 0.2 0.4 0.6 0.8 1-4

-2

0

2

4


Clim

b an

d Si

nk R

ate

[m/s

]

Electric Engine 10s Stopping PhasePiston Engine 30s Cooling Phase

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30

35

40

Distance [km]

Velo

city

[m/s

]

Kinematic VelocityIndicated Airspeed


Figure 4.2-37 Climb angle history for saw-tooth flight with electric engine

Although the climb angle reduces during climb – see Figure 4.2-37 – the reduction is smaller than in case of a piston engine powering the propeller. The motor glider equipped with an electric engine may reach a higher ceiling than in case of a naturally aspirated piston engine installed.

Figure 4.2-38 Lift coefficient and commanded thrust for saw-tooth flight with electric engine

The optimal lift coefficient history as well as the commanded thrust are depicted in Figure 4.2-38. Comparing these profiles to Figure 4.2-11 and Figure 4.2-13 shows a very similar optimal control history independent of the type of engine installed to power the propeller.

−π/20

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π/40

π/20

Distance [km]

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ma

[rad

]

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0 50 100 1500.4

0.8

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Lift

Coe

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ent C

L [-]

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50

75

100

Com

man

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Thru

st δ T,

CM

D [%

]


Figure 4.2-39 Altitude and IAS profile during transition to glide for electric engine

In Figure 4.2-39, a closer look is taken at the stopping phase (phase 2) and the retraction of the engine propeller combination (phase 3). In the stopping phase, the altitude loss is 12.3m and the velocity loss is 4.1m/s. The altitude loss over the stopping and retracting phase is 53.6m and the velocity loss is 0.7m/s. As already shown in Figure 4.2-34, the motor glider benefits from the absence of cooling needs before retraction. Due to the very high reachable lift to drag ratio in gliding configuration, every 100m of altitude saved during transition give an additional range of more than 5km. In Table 4.2-6, the results for the electric engine powering a propeller are summarised and compared to an optimised horizontal flight in 600m altitude. With the 10s stopping phase, the optimal climb interval is 2119.3m and the average cycle velocity is 33.19m/s which is higher than that for optimised horizontal flight. In order to enable a comparison with a combustion engine, the electric energy used is transformed into a fuel equivalent also stated in Table 4.2-6. In saw-tooth mode, the electric motor glider passes nearly 14.6km per kWh of electric energy on average during the cycle. Thereby, savings of around 70% are possible where the portion of powered flight during the cycle is 17.4%. Thus, by applying the saw-tooth mode the achievable range is more than 3.3 times as high as in optimised steady state flight.


Altitude [m] 600 (hor.) Δ 2119.3

Average horizontal Velocity [m/s] 29.82 33.19

Equivalent Consumption [kg/ 100km] 5.665 1.700

Energy Use [km/ kWh] 14.59

Savings [%] 69.99

Portion of Powered Flight [%] 17.42 Table 4.2-6 Savings due to saw-tooth mode electric engine calm

18 19 20 212600

2650

2700

2750

2800A

ltitu

de [m

]

Distance [km]18 19 20 2120

25

30

35

40

Indi

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d A

irspe

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/s]

Ph. 2t2=10s

Phase 3t3=15s


4.2.5 Jet Engine In the following, the installation of a jet engine is investigated with the jet engine delivering 75% of the thrust of the propeller-piston engine combination described in paragraph 4.2.3 of which the reference thrust is stated in Table 4.2-3. The jet engine therefore delivers 1053N of reference thrust at a reference velocity of 25m/s.

Figure 4.2-40 Altitude profile for saw-tooth flight with jet engine

In order to maximise range, the motor glider featuring a jet engine climbs up to 3158.0m. The total cycle duration is 4991.7s [1h 23min 12s] where a distance of 172.63km is covered. Again, the climb is performed at full thrust.

Figure 4.2-41 Climb and sink rate over normalised cycle time (jet engine)

In comparison to the piston engine installed, the climb rate reached during the first phase is smaller in case of the jet engine (with lower maximum thrust) installed – see Figure 4.2-41. Due to the higher climb interval necessary for an optimal saw-tooth profile, both, the relative as well as the absolute duration of the climb phase is increased.

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Jet Engine delivering ~1050N of ThrustPiston Engine delivering ~1400N of Thrust

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ate

[m/s

]

Jet Engine delivering ~1050N of ThrustPiston Engine delivering ~1400N of Thrust


In comparison to the propeller piston engine combination, the climb is performed at a higher velocity (as depicted in Figure 4.2-42). This can be expected since the velocity of the maximum climb rate is determined by the power correlation (4-37). In case of a propeller piston engine combination, the velocity at which the maximum climb rate is reached is from

evaluating (4-37) below *V , whilst for a motor glider equipped with a jet engine it is above *V . Since the motor glider tries to maximise range (4-53), its velocity during the climb

differs from the velocity at which the maximum climb rate is reached. As in previous figures, the initial and final values of the whole cycle as well as of the climb and the glide are marked by open circles. The exchange between kinetic and potential energy during the transition phases is rather strong. Apart from the transition phases, the IAS stays in a very narrow bandwidth.

Figure 4.2-42 Velocity profiles for saw-tooth flight with jet engine

In Figure 4.2-43, it is shown that the higher velocity during climb requires smaller climb angles. The glide is, of course, always performed at the best lift to drag ratio.

Figure 4.2-43 Climb angle history for saw-tooth flight with jet engine

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40

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[m/s

]

Kinematic VelocityIndicated Airspeed

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[rad

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As may also be seen in Figure 4.2-43, the climb angle stays negative in phases four to six. Thus the minimum altitude allowed is now reached shortly after the beginning of phase one. The higher velocity level during climb which is at the same level as during glide (see Figure 4.2-42) corresponds to a lift coefficient history depicted in Figure 4.2-44. For the whole trajectory, the lift coefficient is held within a very small bandwidth which is only left during the transition phases – compare also the IAS profile in Figure 4.2-42. The commanded thrust continues to follow the well-known bang-bang structure.

Figure 4.2-44 Lift coefficient and commanded thrust for saw-tooth flight with jet engine

The results for the motor glider with a jet engine installed are summarised in Table 4.2-7. The pilot may fly an optimised horizontal flight at 600m. This results in an average velocity of 35.42m/s and requires 5.77kg of fuel per 100km. But with a jet engine installed, it is advantageous to perform the horizontal flight at a higher altitude, in dependence on thrust available. In 3000m, fuel consumption drops to 4.99kg/100km while the average velocity increases to 39.65m/s. Compared to the optimised flight in 3000m, the saw-tooth manoeuvre saves 57.1% - equal to an increase of achievable range to the factor of more than 2.3. The average velocity, however, is slightly reduced when flying the saw-tooth manoeuvre.


Altitude [m] 600 (hor.) 3000 (hor.) Δ 2558.0

Average horizontal Velocity [m/s] 35.42 39.65 34.58

Consumption [kg/ 100km] 5.77 4.99 2.14

Savings [%] 57.1

Portion of Powered Flight [%] 26.43 Table 4.2-7 Savings due to saw-tooth mode jet engine calm

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]


Variations in Thrust Levels Installed In the following, different levels of installed thrust are compared to each other as well as to optimised horizontal flights performed at different altitudes. In Figure 4.2-45, different thrust levels of the jet engine are compared with the reference trust value in accordance with Table 4.2-3. 100% refer to 1404N of thrust, 75% refer to 1053N of thrust, 60% refer to 842N of thrust, and 50% refer to 702N of thrust. As shown in Figure 4.2-45 and Figure 4.2-46, the more thrust installed, the higher the climb interval and the longer the cycle distance. Savings in comparison to optimised 3000m horizontal flights increase with a higher available thrust.

Figure 4.2-45 Saw-tooth profiles for different levels of thrust installed and savings in comparison to 3000m

horizontal flights

In order to be able to perform the saw-tooth manoeuvre, the motor glider should have at least around 750N of thrust installed. For lower levels of thrust installed, the climb interval reduces, and due to the power limitations only a horizontal flight could be performed. Moreover, such a low thrust level would not give a self-launching capability to the aircraft.

Figure 4.2-46 Optimal cycle length and climb interval in dependence on thrust installed

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~1400N Thrust - 68% Savings~1050N Thrust - 57% Savings ~850N Thrust - 43% Savings ~700N Thrust - <23% Savings

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Dis

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Figure 4.2-47 Savings due to saw-tooth flight and optimised altitude for horizontal flight in dependence on thrust

installed

In Figure 4.2-47, the possible savings due to the saw-tooth mode are given. The optimal saw-tooth cycle is compared to optimised horizontal flights in 600m, 3000m, and the optimal altitude to perform the horizontal flight. This optimal altitude increases strongly with the amount of thrust installed. For 1053N of thrust installed, the optimised altitude, for example, is 7810m. In comparison to optimised horizontal flights in 600m or 3000m, the saw-tooth flight saves for low thrust levels more than 40% of fuel per distance, for medium thrust levels more than 50%, and for higher thrust levels more than 60%. Also if the reference altitude to perform the horizontal flight is optimised, for medium and high thrust levels the saw-tooth manoeuvre saves more than 40% of fuel per distance which means that range is increased by more than 66%.

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4.3 Bounding Flight – Siskin Over a very broad speed range, small birds do not use a steady state level flight style but some kind of intermittent flight technique where velocity, altitude, and wing beat frequency are altered periodically during the cycle. Referring to Rayner et al. (2001), three different modes of intermittent flight may be observed as depicted in Figure 4.3-1, where the respective altitude profiles are shown indicating the flapping and non-flapping parts.

Figure 4.3-1 Different intermittent flight modes (Rayner et al., 2001)

In case of (flap-)bounding flight, the manoeuvre consists of two consecutive arc-segments. First, a pull-up type arc-segment flown with wings flapping, followed by another arc-segment with the closed wings folded completely – or nearly completely – into the body. In this bound phase, the surplus in the energy state achieved in the flapping phase is used to compensate for the energy loss due to drag. Second, an undulating flight which resembles a saw-tooth manoeuvre as described in paragraph 4.2. Here, the second part is a gliding part, where the wings are fully extended. It is very widely spread among birds and occurs in almost any species which can glide. And third, the chattering flight, where the wing beat frequency varies during horizontal flight. This third flight mode is rarely observed and features in contrast to the other two modes no rest phase. A general overview about the first two types may also be found in in the section Intermittent flight in Tobalske (2007) and in paragraph 9.1 Intermittent Flight Styles in Flapping Flight of Pennycuick (2008). In Norberg (1990), the first two intermittent flight modes are discussed in chapter 9.8 Energy-Saving Types of Flight where also additional effects like the ground effect are treated; respective power calculation recipes are provided in the next chapter of Norberg (1990). In the review concerning bounding flight of Keating (2002), the applicability of the bounding flight mode in small and micro UAVs is considered. More general bio-

(Flap-) Bounding Flight

Undulating (Flap-Gliding) Flight

Chattering Flight


inspired technical implementations are described for man-made flying systems in Lentink and Biewener (2010), and the direct reference between hovering and intermittent flight in birds to the development of autonomously flying systems is given in Tobalske (2010). As pointed out by Tobalske (2001), whether the bounding flight mode or the undulating flight mode is used, depends on the body size, the pectoralis composition, the aspect ratio of the wing, and the (aerodynamic) flight velocity. In case of birds of intermediate body mass (35g – 185g), there is a tendency to undulating flight at slow speeds and a switch to flap-bounding flight at higher speeds, regardless of the aspect ratio of the wing (see also Tobalske and Dial (1994). For black-billed magpies (Pica pica), the portion of bounds during the non-flapping intervals increased with speeds above 10ms-1 - see Tobalske and Dial (1996), and also for the European starlings (Sturnus vulgaris) the percentage of bounds increased markedly with increasing flight speed - see Tobalske (1995)). This switching effect is shown in Figure 4.3-2, which is taken from Tobalske and Dial (1994). It shows the wing postures during the non-flapping phase in the budgerigar (Melopsittacus undulatus) in dependence on flight velocity. The pectoralis (primary downstroke) muscle is only active during glides, but not during bounds. The supracoracoideus (primary upstroke) muscle is inactive during both, bounds and glides (Tobalske and Dial, 1994). Citing George and Berger (1966), Tobalske (1995) states that these two muscles constitute up to 35% of the body mass of flying birds – thus, their lack of activity during bounds could represent significant savings in metabolic energy. For a profound description of the bird’s muscle system, see also chapter 11.2 in Norberg (1990) where also the fibre structure is investigated as well as the possible power output.

Figure 4.3-2 Wing postures in non-flapping phase of the budgerigar (Tobalske and Dial, 1994)

Small species below 20g having rounded wings of low aspect ratio will flap-bound at all speeds while such small birds with pointed high aspect ratio wings also use the flap-gliding mode; as exception, the hummingbirds flap continuously at all speeds. Tobalske (2001) points out that this is explained by wing shape rather than pectoralis physiology. The zebra finches use flap-bounding flight at all speeds - even at hovering (Tobalske et al., 1999). He further mentions that swallows (13g–19g) and budgerigars have wings of higher aspect ratio

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Inte

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]

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than the zebra finch (Warrick (1998); Table A1), which may indicate that their wings offer higher lift to drag ratios so that flap-gliding could offer more savings in average mechanical power than flap-bounding at intermediate flight speeds.

Figure 4.3-3 Flapping ratio over speed for zebra finches (Tobalske et al., 1999)

In Figure 4.3-3, which is taken from Tobalske et al. (1999), characteristics of flap-bounding flight in four zebra finches (Taenopygia guttata) at flight speeds from 0 (hovering) to 14ms-1 are shown. Values depicted are means ± standard error of the mean (SEM). The percentage of time spent flapping in a flap-bounding cycle reduces continuously with an increase in flight velocity. The zebra finches weighted 13.2g ± 0.9g.

Figure 4.3-4 Dorsal view of flap-bounding zebra finch (Tobalske et al., 1999)

Also Figure 4.3-4 is taken from Tobalske et al. (1999). It gives the dorsal views of wing and body posture in a zebra finch engaged in flap-bounding flight at 8ms-1 where (A) is the flapping phase, with wing posture at mid-downstroke (dashed line) and at mid-upstroke (solid line) and (B) is the bounding phase, with the wings fully flexed.

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Rat

io [%

]

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100

0 2 4 6 8 10 12 14

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A

B


Figure 4.3-5 Durations of flapping / bound phase in zebra finches (Tobalske et al., 1999)

In Figure 4.3-5 (from Tobalske et al. (1999)), the durations of the flapping and of the bound phase for the zebra finch is shown over flight velocity. A significant decrease in the duration of the flapping phase and a significant increase in the duration of the bounding phase, as flight speed is increased, may be identified. With an increase in body size, the percentage of flapping increases and birds with a mass strongly exceeding 300g do not flap-bound any longer. The largest birds known to engage in flap-bounding flight are the pileated woodpecker (Dryocopus pileatus) weighing 262.2g ± 49.5g (Tobalske, 1996), and as mentioned in Tobalske (2001), observations of the black woodpecker (Dryocopus martius) weighing 321g ± 30.3g suggest that it also engages in intermittent bounds. The scaling of pectoralis composition may explain this ability (Tobalske, 2001). Figure 4.3-6 is taken from Tobalske (2001) and shows the percent time spent flapping during flap-bounding flight in nine woodpeckers (filled circles) during the breeding season and twelve passerines (open circles) during migration in dependence on their mass. The minimum flapping ratio observed here lies around 25%. In Tobalske and Dial (1994), the range of the flapping ratio for the budgerigars (mean body mass 34.5g ±0.5g, range 29.0g – 40.0g) lied between 23.5% to 100% with a mean of 85.3% ±1.1%. The average flapping phase duration was 727.8 · 10-3 s ± 64.1 · 10-3 s at a range of 50 · 10-3 s to 5,570 · 10-3 s and the average non-flapping duration was 60.7 · 10-3 s ± 3.3 · 10-3 s at a range of zero to 219 · 10-3 s. In Tobalske (1996), six species of woodpeckers (Picidae) varying in body mass and size were investigated: the downy woodpecker (Picoides pubescens, 27.2g), the red-naped sapsucker (Sphyrapicus nuchalis, 47.4g), the hairy woodpecker (Picoides villosus, 70.5g), the Lewis’ woodpecker (Melanerpes Lewis, 106.6g), the northern flicker (Colaptes auratus, 148.1g) and the pileated woodpecker (Dryocopus pileatus, 262.5g). All of the species regularly exhibited flap-bounding. The flapping-phase duration and bounding-phase duration scaled negatively with body mass, whereas the percent cycle time spent flapping and flight speed scaled slightly positively with body mass.

Flight Speed [m/s]

Dur

atio

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0-3s]

0

200

400

0 2 4 6 8 10 12 14

100

300


Figure 4.3-6 Flapping ratio over body mass (Tobalske, 2001)

The upper limit displayed in Figure 4.3-6 may be explained by the ‘adverse-scaling’ hypothesis which is well described in Tobalske and Dial (1996): it is the assumption of adverse scaling of available mass-specific power for flap-bounding flight (see also Rayner (1985), DeJong (1983), and Tobalske (1996)). Rather than being constrained by their body mass to use flap-bounding, as implied by the ‘fixed-gear’ hypothesis (see Rayner (1985)), it may be that only relatively small birds have sufficient mass-specific power available from their flight muscles to use flap-bounding, and that larger birds are unable to use intermittent bounds in spite of the potential that such non-flapping phases offer for reducing metabolic and mechanical power output. Since it is not possible to explain all occurrences of bounding by an argument based solely on external mechanical power – in particular, bounding while hovering must increase both, profile drag and induced drag and is of non-obvious mechanical advantage (Rayner, 1985), in this work, the investigations will be solely concerned with flap-bounding in rather fast forward flight. Here, the integral of mechanical power applied by the bird per distance covered may be used as a cost function and is likely to offer an advantage over continuous flapping flight (see also DeJong (1983), Ward-Smith (1984a), Ward-Smith (1984b), and Rayner (1985)). Numerous TUM-FSD publications and presentations at international conferences are to be mentioned: - The publication termed New modeling approach for bounding flight in birds in the

Journal of Mathematical Biosciences (Sachs et al., 2011b). - Two AIAA publications with presentations given by this author: the presentation at the

AIAA Atmospheric Flight Mechanics (AFM) conference during 8 to 11 August 2011 in Portland, Oregon, USA termed Optimization of Flap-Bounding Flight with the respective

log 2

Flap

ping

Rat

io

1.2

1.6

1.4

1.8

2.0

Flap

ping

Rat

io [%

]

25.1

100

15.8

39.8

63.1

log2 Body Mass [100g](Body Mass [g])

1(10.0)

1.5(31.6)

2(100.0)

2.5(316.2)


paper published as Sachs et al. (2011c), the presentation at the AIAA AFM conference during 13 to 16 August 2012 in Minneapolis, Minnesota, USA termed Performance Enhancements by Bounding Flight with the respective paper published as Sachs et al. (2012c).

- The plenary speech given by Professor Sachs at the VSDIA conference during 5 to 7 November 2012 in Budapest, Hungary with the respective proceedings (Sachs et al., 2012a).

4.3.1 Modelling a Generic Siskin As reported by Tobalske (1995) when describing the intermittent flight of starlings (mean body mass 79.2g ± 2.7g), pull-out phases, when wingspan was increased but wingtip elevation remained near 0cm (see also DeJong (1983)), occurred between two non-flapping postures, within a single non-flapping phase (bound-glide) and between a bound and subsequent flapping flight. These pull-out phases lasted on average 43.8 · 10-3 s ± 2.4 · 10-3 s. In contrast, the zebra finches at the end of bounds simultaneously elevated and extended their wings, taking a shorter time (Tobalske et al., 1999) and immediately started flapping. Such pull-out phases will therefore not be modelled in this work.

Figure 4.3-7 Body lift and drag (including tail) in zebra finches (Tobalske et al., 1999)

Rayner (1985) emphasises the importance of lift and drag during the bound phase. With the wings fully flexed, the body (always including the tail if not otherwise mentioned) of the bird will still produce a certain amount of lift and drag. Figure 4.3-7 which is taken from Tobalske et al. (1999) shows that both, lift and drag increase between 4ms-1 and 10ms-1. At 10ms-1, nearly 16% of the body’s weight is supported during the bounds. The lift to drag ratio by

Flight Speed [m/s]

0 2 4 6 8 10 12 140

2

4

3

1

-0.01

0.01

0.03

0.02

0

-0.01

0.01

0.03

0.02

0-5

10

20

15

5

0

Body

L/D

[-]

Body

Dra

g [N

]Bo

dy L

ift [N

]

Lift/

Wei

ght [

%]


trend reduces with an increase in velocity and corresponds to glide angles of 17.9° to 52.4°. Thus, the finches adapt the aerodynamic function of the bounds in accordance with the speed. At slow speeds, the emphasise is put on lift generation whilst at high speeds, it is put on drag reduction at a slight expense to body lift (Tobalske et al., 1999). Apart from their wings, the birds may also control large parts of their feather arrangement (Tobalske et al., 2009). When analysing five zebra finches (mean body mass 17±3g) at a speed of 6, 8, and 10ms-1, which is their preferred range of flight velocity, the birds generated lift to drag ratios of 1.36 ± 0.03 during bounds. Velocity did not have a significant effect upon lift to drag ratio. Over this velocity range, the percentage of weight supported by body lift during bounds was 20% ± 5%, and body drag was 15% ± 3% of weight (Tobalske et al., 2009). Similar results, that body and tail can support up to 20% of the bird’s weight, are also reported by Csicsáky (1977) who investigates body lift and drag in body-gliding for two differently shaped body contours. For a specialised analysis on the aerodynamic functions of birds’ tails, see Thomas (1997). The emphasis in Maybury et al. (2001) is put on the lift generation of the tail, and the difference between predictions from theoretical models and measurements is pointed out. In Evans (2003), it is shown that delta wing theory to calculate the lift generated by the tail may be applied within certain limits: tail spread less than 60° and at an angle of attack below 20°. In Figure 4.3-8, the results (Tobalske et al., 2009) for mounted zebra finches specimens are shown with data collected by particle image velocimetry (PIV) and force transducers.

Figure 4.3-8 Body lift and drag coefficients for zebra finches (Tobalske et al., 2009)

The measures are in reasonable agreement and comparison versus live birds’ peak lift to drag ratio matched very well. Zebra finches adopt a body posture during bounds that appears to either maximize body lift to drag ratio or, at least, keep it above one (Tobalske et al., 2009). So, the birds are able to generate lift during the bound phase, but at rather high cost (high drag). The dimensionless force coefficients result by using the frontal area (perpendicular to flow) of the bird as reference area. In Hedenström and Liechti (2001), it is advised to Pennycuick (1999) who recommends a minimum body drag coefficient of 0.1. Hedenström and Liechti (2001) themselves measured body drag coefficients ranging from 0.17 to 0.77 for small passerines; the values above 0.4, however, probably result from wing extension and are

CD [-]0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.5

1.5

-0.5

1.0

1.6

0.0

-1.0

CL

[-]

CL=CD


thus too high. For a barn swallow (Hirundo rustica L.) weighing 18.2g, a body drag coefficient of 0.23 was measured and judged as a very reliable estimate since it resulted from a nearly vertical dive with the wings almost completely folded. Pennycuick et al. (2012) show the negative effect of an external transmitter on the measurement of the body drag coefficient (especially the addition of a sloping antenna increased it to nearly twice the clean value) due to separation of the boundary layer over the posterior body. They measured the body drag coefficients of three rose-coloured starlings (Sturnus roseus) weighing between 72.8g and 84.2g. The mean of 20 estimates of the clean body drag coefficient (i.e. with no harness or box) was 0.116 ± 0.0394. Woike and Gewecke (1978) analysed the flap-bounding flight of 30 siskins (Carduelis spinus) weighing on average 14g which flew up to two hours in a wind tunnel. In tethered flight, the flapping intervals lasted about 1s at the beginning of the flap-bounding flight and decreased to about 0.3s after 20min, whereas the durations of the bounding phases were about 0.2s to 0.4s throughout. During the bounding phases, the siskins kept quiet, head stretched forward and wings and legs held closely to the body, the wings in resting position and the feet lifted in flight posture. During the bounding phase, the wing position does not differ from the resting position – consequently, the thrust equals zero, and a drag (and a small lift) is generated passively by the air current of the wind tunnel. Free flying siskins filmed during their autumn migration show a similar bounding flight pattern. The flapping intervals differed between 0.05s and 0.5s, on average they lasted for about 0.25s. The length of the bounds varied from 0.05s to 0.7s with a mean of 0.35s. Based on the data of Woike and Gewecke (1978) and other aerodynamic data for birds, and by applying geometrical scaling laws, Ward-Smith (1984b) obtains estimates for the wing reference area, S, as 100.0 · 10-4 m2, the zero lift drag coefficient for the whole bird (related to the wing reference area and not to the frontal area), CD0, as 0.03 and the lift dependent drag factor, k, as 0.0864 (compare Table 4.3-1 and equation (4-77)). Hedenström et al. (2006) analyse the free flight of two robins (Erithacus rubecula) weighing 16.75g and 16.31g with a wing reference area of 103.8 · 10-4 m2 and 104.0 · 10-4 m2 (including the area of the body between the wings). For a reference flight speed of 9ms-1, the lift to drag ratio is calculated to be 7.5, and by taking the data from Biewener et al. (1992) the effective lift to drag ratio in the starling (Sturnus vulgaris) is estimated to be 9, and by analysing data from Dial et al. (1997) a lift to drag ratio of 10 in the magpie (Pica pica) is estimated. Tobalske et al. (2003) give the power output of the pectoralis muscles of cockatiels (Nymphicus hollandicus) with an average mass of 78.5g in the range of 1.3W to 3.7W. For a general estimation of power curves in birds, see also Rayner (1999). The whole bounding flight cycle is modelled to consist of four phases – see Figure 4.3-9. In the first phase, the bird flaps its wings, flies a curved segment while it accelerates, and finishes at an altitude close to the starting altitude. In the second phase, the configuration changes and the wings are folded into the body. Then, the bound phase follows with no propulsive force acting. The last phase is the wings unfolding phase. At the end, an optimal cycle time is found where only the duration of the wings’ folding and

unfolding phases is fixed to 0.025s, cont∆ , and the duration of the phases number one, 1t ,

and three, 3t , has to be optimised – see also Table 4.3-1.


The time interval thus ranges from 0 to the cycle time, cyct :

conconcyc tttttt ∆++∆+=≤≤ 310 (4-66)

Figure 4.3-9 Bounding flight cycle

The generic siskin has a wing reference area of 0.01 square meters and weights 14 grams with a maximum power to enable thrust generation of one Watt (see Table 4.3-1). This maximum power limit is not violated for the reference cycles presented. For the subsequently following general investigations including high velocities it will be dropped. For all phases the aerodynamics are modelled by quadratic drag polars. In the bounding phase with the wings folded into the body, the lift independent drag component is reduced by 70% (mainly since the bird’s wetted area is drastically reduced) and the k-factor for the induced drag of the body and the tail is increased by the factor of 10. During the folding and unfolding phases, the change between the drag polars occurs linearly.

Parameter Notation Value Unit Mass m 0.014 kg Wing reference area S 100.0 · 10-4 m2

Maximum power to generate thrust TPmax, 1.0 W

Best glide number in flapping *flape 1/10 -

Zero lift coefficient in flapping phase flapDC ,0 0.03 -

Factor for zero lift coefficients flapD

boundD

CC

e,0

,0= 0.3 -

Factor for induced lift coefficients flap

bound

kk

b=

1 10 -

Time to (un-) fold wings cont∆ 0.025 s Table 4.3-1 Model data for siskin

0 1

Alti

tude

tcyc

FlightDirection

FlappingPhase

Phase 1: t1

BoundPhase

Phase 3: tcyc-(t1+2Δtcon)

Phase 2: Δtcon Phase 4: Δtcon


The bird of mass, m , is mathematically modelled as a point mass where it is assumed that the thrust, T , which is generated by flapping the wings is acting in flight direction opposed by the drag, D . The lift, L , thus stands perpendicular on both, thrust and drag, and is the main force to counteract the acceleration due to gravity, g . The whole modelling presented is only valid for forward flight where the thrust characteristics of the bird resemble of a propeller driven aircraft. Investigations on slow flight or hovering would need a completely different model (compare Pennycuick (1968) and its discussion at Hedenström (2009) and the paper of Askew and Ellerby (2007)). It is also advised to chapter eight Hovering Flight in Norberg (1990).

Figure 4.3-10 Forces acting on siskin

The pine siskin in Figure 4.3-10 has been photographed by Sibell (2010). The states of the bird are the kinematic velocity, V , the kinematic climb angle, γ , the altitude, h , and the

distance travelled, s . For better readability the notation is simplified, e.g. V instead of

( )EKK

KV .

The resulting equations of motion (translation from (2-57) and position from (2-8)) are:

γsin⋅−−

= gm

DTV (4-67)

γγ cos⋅−⋅

=Vg

VmL

(4-68)

γsin⋅= Vh (4-69)

γcos⋅= Vs (4-70)

The muscles of the bird generate the mechanical power output, P , to provide thrust. The thrust results from division of this power output by the aerodynamic Velocity, AV . The thrust

L

mg

Dα Vγ

T


may only be generated (and controlled) in the first phase of the cycle. The true muscle

power has to be scaled by an efficiency factor, Muscleη , to give the power available for thrust.

AVPT = (4-71)

MuscleMuscle PP ⋅=η (4-72)

In the aerodynamics model, the dependency of drag and lift from the corresponding coefficients is shown as well as from the density, ρ , the aerodynamic Velocity, AV , and the

wing reference area, S , which is not altered during the different phases.

The lift coefficient, LC , is a control available in all the four phases. The drag coefficient,

DC , is a function of the lift coefficient.

SVCD AD ⋅⋅⋅= 2)2/(ρ (4-73)

SVCL AL ⋅⋅⋅= 2)2/(ρ (4-74)

( )LDD CCC = (4-75)

In accordance with the basic idea of a quadratic drag polar, total drag is composed of a lift independent term and the induced drag component. In the flapping phase, the quadratic drag polar is given by:

2,0, LflapflapDflapD CkCC ⋅+= (4-76)

0833.0/2 ,0

2*

=

= flapDflap Ck e

(4-77)

For better readability, *flape from Table 4.3-1 is, if not explicitly mentioned otherwise, written

as *e .

The velocity for a maximum range stationary flight, *V , is with the density at sea-level,

MSLref,ρ , and the standard acceleration due to gravity, sg , (for both, see Table 2.3-1) which

were also used throughout the optimisations and are thus abbreviated by ρ and g :

sm 112.6

,0

** =

⋅⋅⋅⋅

=flapDCS

gmVρ

e (4-78)

In the bounding phase, lift-independent drag is reduced by 70% mainly since the wetted area of the body with folded wings is much smaller than that for wings extended. At the


same time, the cost to generate lift by the body and the tail is strongly increased (factor 10 – see Table 4.3-1):

2,0, LboundflapDboundD CkCeC ⋅+⋅= (4-79)

The aerodynamic changes in phase two (folding the wings) and phase four (extending the wings) occur linearly, which gives for the drag coefficient – see also Figure 4.3-9:

( ) ( )[ ]con

LflapboundflapDLflapflapDPhD tttCkkCeCkCC

∆−

⋅⋅−+⋅−+⋅+==12

,02

,02, 1 (4-80)

( ) ( )[ ] ( )con

concycLboundflapflapDLboundflapD

PhD

tttt

CkkCeCkCe

C

∆

∆−−⋅⋅−+⋅−+⋅+⋅

==

2,0

2,0

4,

1 (4-81)

The equations were implemented in a single Matlab / Simulink model which is – by using additional model change parameters – able to represent the flapping, bounding, and transition phases. Using the Matlab Real-Time Workshop, the model can be directly coded to C to be used with optimisation software like GESOP. Moreover, the equations were implemented in a Matlab file, where subsequently for the whole model the Jacobian and the Hessian were automatically generated as described in paragraph 3.2.1. The generated code has then been used within the TUM-FSD optimisation framework.

4.3.2 Bounding Flight as Periodic Optimal Control Problem In order to investigate a flight of longer duration, due to the uniformity of the occurring bounding flight cycles, it is sufficient to optimise one of these cycles. One cycle consists of four phases as described earlier. The cost function is to minimise the energy used per distance travelled in flap-bounding flight mode:

min)()(

)( !0 =

⋅== ∫

cyc

t

cyc

cycbound

ts

dtP

tstE

Jcyc

(4-82)

The value is given in Joule per 100m of distance travelled [J/100m]. The normalised time interval ranges from 0 to 1:

cyc

concon

ttttt ∆++∆+

=≤≤ 3110 τ (4-83)

of which the durations of phases one and three are to be optimised, while the durations of the phases two and four (folding and unfolding of the wings) are fixed. The controls are the mechanical power output, P , which is only available in the first phase, and the lift coefficient, LC , which is available in all the four phases.


In optimisations not presented here, the times for extending and retracting the wings could be optimised separately with the above time (see Table 4.3-1), given as minimum time. In the convergence case, however, the duration of the configuration changing phases was always as short as possible. Moreover, the mechanical power output, P , had been implemented as a classical control variable but was found to be nearly constant in the optimal case. The bird had no tendency to include additional gliding or semi-gliding phases or to alter the power output. Therefore, in the results presented here, beside the durations of phases one and three, the mechanical power output, P , in the flapping phase is implemented as an optimisation parameter:

[ ]Ptt ,, 31=p (4-84)

All elements of the parameter vector in (4-84) are to be non-negative and for some calculations also upper limits were introduced. For the lift coefficient, LC , upper and lower bounds have to be respected:


With the aim to minimise the energy used per distance travelled (4-82), the flap-bounding flight can only become efficient at an average velocity above that for maximum range

stationary flight, *V (4-78). Therefore, a final boundary constraint is introduced on the

average cycle velocity via the parameter averageVp , :

( )0, =−= averageV

cyc

cycfin p

tts

ψ (4-86)

For some optimisations, also the final distance is set as a boundary constraint:

( ) 0,, =−= sfincycsfin ptsψ (4-87)

The lift coefficient as well as all states apart from the distance travelled are to be periodic:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] 0ψ =−−−−= 0 ,0 ,0 ,0 hthtVtVCtC cyccyccycLcycLper γγ (4-88)

In case of multiple cycles, the periodicity at each cycle bound is to be maintained.

4.3.3 Initial Guess from Parabola In the following, an initial guess for the bounding flight manoeuvre is presented – see also Ward-Smith (1984b). The equations are non-dimensionalised to give results for the bounding flight manoeuvre of generally valid nature. In contrast to the numerically optimised flight, for an analytical optimisation it is assumed, that the whole manoeuvre only consists of a flapping phase which is directly followed by a bound phase – see Figure 4.3-11. Moreover it is assumed that the whole cycle is flown at a

constant velocity, 0V , and the climb angle, γ , is small, to give:

.0 constVV == as well as ( ) 1cos ≈γ and ( ) γγ ≈sin . (4-89)


The load factor, zn , is defined as the lift, L , divided by the bird’s weight:

gmLnz ⋅

= (4-90)

Thereby, (4-68) gives:

( )[ ] gnV z ⋅−=⋅ γγ cos (4-91)

The vertical acceleration, h , under the assumption of (4-89) results as:

( ) gnh z ⋅−= 1 (4-92)

Figure 4.3-11 Initial guess for siskin flight

Performing the integration over time, t , and respecting the initial conditions yields for the

vertical acceleration, h , the vertical velocity, h , the altitude, h , the duration of the flapping

phase, 1t , and the duration of the bound phase, 2t :

Phase 1 Phase 2

( ) gnh z ⋅−= 11,

( ) ( )001, 1 γ−⋅+⋅⋅−= Vtgnh z

( ) ( )

−⋅+⋅⋅−⋅= 001, 2

1 γVtgnth z

( ) gnh z ⋅−= 12,

( ) 002, 1 γ⋅+⋅⋅−= Vtgnh z

( )

⋅+⋅⋅−⋅= 002, 2

1 γVtgnth z

(4-93)

( ) gnV

tz ⋅−

⋅=

12

1,

001

γ ( ) gn

Vt

z ⋅−⋅

=2,

002 1

2 γ (4-94)

FlightDirectionh

0γ−0γ

Phase 1t1

Phase 2t2

Duration tcyc; Distance s


The duration of the bound phase can therefore be expressed in terms of the flapping phase and the applied load factors:

2,

1,12 1

1

z

z

nn

tt−

−⋅= (4-95)

This gives for the total distance, s , of one cycle:

( )2,

2,1,10210 1 z

zz

nnn

tVttVs−−

⋅⋅=+⋅= (4-96)

and the dissipated energy, −E , per distance travelled (for a cycle of distance s ) in bound

flight mode is the averaged drag:

( )( ) cyct

tDtDttV

VtDtDs

E 2211

210

02211 ⋅+⋅=

+⋅⋅⋅+⋅

=− (4-97)

For comparison in stationary horizontal flight, the dissipated energy, horE ,− , per distance

travelled is equal to the drag:

DtV

VtDs

E

cyc

cychor =⋅

⋅⋅=−

0

0, (4-98)

The added energy (energy available from muscle work), +E , during the flapping phase of the

bounding flight manoeuvre referenced to the total cycle distance, s , is with using (4-71):

cyccyc tVtP

tVVtT

sE

⋅⋅

=⋅

⋅⋅=+

0

1

0

01 (4-99)

where the fraction of the flapping phase is:

2,1,

2,1 1

zz

z

cyc nnn

tt

−−

= (4-100)

In the following, non-dimensionalisations are performed to give results of generally valid

nature for the bird’s flight dynamics. The normalised velocity, V , is the flight velocity, V , divided by the velocity for a maximum range stationary flight (wings extended and flapping

as in phase 1), *1V , as given by (4-78):

*1V

VV = (4-101)

Equally, the normalised lift coefficient, LC , is obtained:

*1,L

LL C

CC = (4-102)


In normalised values, the load factor applied in the flapping phase, 1,zn , reads:

21, VCn Lz ⋅= (4-103)

The normalised dissipated energy per distance travelled, −E , results from dividing (4-97) by

the minimum drag, gmD ⋅⋅= *min e :

*

11e

⋅⋅

⋅= −− gms

EE (4-104)

The same normalisation by minD is performed for the dissipated energy during horizontal

flight, (4-98):

*,

,11

e⋅

⋅⋅= −

− gmsE

E horhor (4-105)

And also the normalised energy added during the flapping phase, +E , results from dividing

the energy available from muscle work (4-99) by the minimum drag:

*

11e

⋅⋅

⋅= ++ gms

EE (4-106)

The necessary normalised power, P , results by relating the power applied to the minimum

power at a velocity of 1=V , minP , with the power, P , from solving (4-99):

**1min

11e

⋅⋅

⋅==gmV

PPPP (4-107)

The reference values are always taken from the first phase. Assuming no lift in the 2nd phase, results in a load factor value of zero in the bound phase:

02, =zn . Using the polars (4-76) and (4-79) together with (4-78) and the load factor equation

(4-103), gives for the dissipated energy per distance travelled (4-104):

( )[ ]L

LL

CVCeCE 11

21 22 −⋅⋅++

⋅=− (4-108)

Interestingly, (4-108) is minimised independent of the flight velocity for a lift coefficient of:

eC EL −=−

1min, (4-109)

to give the minimal normalised dissipated energy per distance travelled in bounding flight

mode, min,−E , as:

2min, 2

1 VeeE ⋅+−=− (4-110)

The optimal load factor applied during the flapping phase, 1,zn , results from (4-103) and

(4-109) as:

21, 1 Venz ⋅−= (4-111)


For the stationary horizontal flight, the flapping polar (4-76) together with a load factor of one (4-103) and the relation (4-78) is inserted into (4-105) to give the energy per distance travelled as:

+⋅=

+⋅=− 2

22

,1

211

21

VV

CCEL

Lhor (4-112)

By putting the normalised energy per distance travelled in bounding flight mode (4-110) equal to the normalised energy per distance travelled in horizontal flight (4-112), the

normalised velocity of equal energy, eEV , results. Above this velocity, the bounding flight

stays superior to the horizontal flight:

4 11

eVeE −

= (4-113)

for which the associated normalised energy per distance travelled is:

( )e

eVE eE −−

=− 122

min, (4-114)

At the velocity eEV (4-113), the load factor for the first phase (4-111) equals one and

increases quadratically for higher velocities. The added energy during the flapping phase (4-106) has to equal the dissipated energy during the whole cycle (4-104). Therefore, with using (4-100), the normalised power (4-107) may be written as:

31, VCEVnEP Lz ⋅⋅=⋅⋅= −− (4-115)

Inserting the normalised dissipated energy (4-108) gives for the normalised power during the flapping phase:

( )[ ] 322 1121 VVCeCP LL ⋅−⋅⋅++⋅= (4-116)

This is in the case of the lift coefficient to be chosen as in (4-109):

( ) 53min 2

11 VeeVeP E ⋅−⋅

+⋅−=−

(4-117)

At the normalised velocity of equal energy, eEV , given in (4-113), the required power applied

during the flapping phase equals that in stationary horizontal flight (flapping ratio 100%) and is:

( )3

4min 11

22

−

⋅−

=− e

eVP eEE (4-118)

The normalised power demand (as in (4-107)) in horizontal flight is by using (4-112):

+⋅⋅= 2

2 121

VVVPhor (4-119)


It is important to mention that while the power demand in horizontal flight (4-119) increases

cubically with the normalised velocity, V , the necessary power in bounding flight mode

during the flapping phase increases with the velocity V to the power of five (4-117). Now, to get more realistic approximations, the assumption of no lift during the second (bound) phase is replaced by the assumption that the lift coefficient in the second phase scales as the fraction of the k-factors of the associated polars as presented in (4-76) and (4-79). The lift coefficient fraction is abbreviated by b - see also Table 4.3-1:

1,1,2, Lbound

flapLL C

kk

CbC ⋅=⋅= (4-120)

Since ( ) ( ) *2,

2*2

*1,

2*1 LL CVCV ⋅=⋅ , the load factor in the second (bounding) phase, 2,zn ,

becomes:

22, VCbn Lz ⋅⋅= (4-121)

The assumption (4-120) simplifies the normalised dissipated energy per distance travelled

with lift during the bound phase, LE ,− , where the factor b may be seen as a variable:

( )

( ) L

Lflap

boundL

flap

boundL

L

Cb

VCbbkkCb

kkVCbee

E

⋅−⋅

⋅⋅⋅

−⋅+⋅

⋅−+⋅⋅−+−

=−

12

111 23222

.

(4-122)

to:

( )( ) ( )

L

LLL C

CbVCbeeb

E22

.11

11

21 ⋅−+⋅⋅−+−

⋅−

⋅=− (4-123)

for which in case of ∞→boundk , or equally 0→b , equation (4-108) results.

The energy per distance travelled (4-123) is minimised by the lift coefficient again chosen independently of the flight velocity as:

beC LEL −

−=

− 11

,min, (4-124)

to give:

2min,, 12

111 V

bbe

beE L ⋅

−−

⋅+−−

=− (4-125)

which becomes (4-110) in case of 0→b .

The optimal load factor in the first phase, 1,zn , results from (4-124) and (4-103) as:

21, 1

1 Vbenz ⋅

−

−= (4-126)


The bounding flight with lift (4-125) stays superior in comparison to the continuous horizontal flapping flight (4-112) from the normalised velocity of equal energy with lift in the

bounding phase, LeEV , , on, which is a bit smaller than eEV in (4-113):

4

4

, 11

ebV LeE −

−= (4-127)

The associated normalised energy per distance travelled is:

( )be

beVE LeEL −⋅−−−

⋅=− 112

21

,min,, (4-128)

Again, the added energy during the flapping phase (4-106) has to equal the dissipated energy during the whole cycle (4-104). Therefore, with using (4-100), the normalised power

(4-107) with lift in the bounding phase, LP ,may be written as:

Vnnn

EPz

zzLL ⋅

−−

⋅= −2,

2,1,, 1

(4-129)

Using the load factor relationship (4-121), this normalised power becomes:

( )V

nbbn

EPz

zLL ⋅

⋅−−⋅

⋅= −1,

1,, 1

1 (4-130)

The normalised power to perform the bounding flight with the lift coefficient in accordance with (4-124), and therefore with the load factor of the flapping phase in accordance with (4-126), is thus:

( )

⋅

−−

+−

−⋅⋅

⋅⋅−−−⋅−⋅−

⋅=−

22

3

,min 1112

1111

21 V

bbe

be

VbebVbeP LE (4-131)

At LeEV , (4-127), the required power to perform the bounding flight manoeuvre equals that in

stationary horizontal flight since the flapping ratio is 100% and is:

( )

−−

+−−

⋅−−

⋅=− e

bbe

ebVP LeELE 1

111

11

21

4

4

,,min (4-132)

All the equations (4-120) to (4-132) turn to their pendants described earlier in case of 0→b . As shown in Figure 4.3-12, the normalised energy in flap-bounding flight at velocities above

eEV is clearly lower than for the continuous flapping flight case. The major limitation of the

flap-bounding flight may be identified in Figure 4.3-13, where an exponential increase of the power to be applied during the flapping phase restricts the speed range in which the flap-bounding flight can be performed. It should be remembered that a normalised power of one

( )( )11 ==VP represents the necessary power to generate sufficient thrust to perform a

maximum range stationary horizontal flight. The absolute minimum of the normalised power

(4-119) is 877.027/2 4min, ≈=absP , which is reached at a velocity of 760.03/1 4 ≈=V .


In Figure 4.3-12 as well as in the following figures, the model data for the siskin presented in Table 4.3-1 is used. It is shown, that energy per distance travelled (4-82) in the bounding flight mode can be further reduced when a certain amount of lift (4-120) is provided in the bound phase. Moreover, with lift in the bound phase, the threshold velocity from which on the bounding flight stays superior to the stationary horizontal flapping fight is slightly

lowered: from 0933.1=eEV to 0648.1, =LeEV .

Figure 4.3-12 Normalised energy for flap-bounding flight

Figure 4.3-13 Normalised power for flap-bounding flight

0 1 2 30

1

2

3

4

5

Normalised Velocity V

Nor

mal

ised

Ene

rgy

E

Continuous Flapping FlightFlap-Bounding (no Lift during Bound)Flap-Bounding (Lift during Bound)

eEVLeEV ,

1 2 30

10

20

30

40

50


Nor

mal

ised

Pow

er R

equi

red

P


( )eEE VP _min

( )LeELE VP ,_,min

absPmin,


The power to be applied during the first (flapping) phase is, however, further increased when lift – and thus induced drag – is present in the second (bounding) phase as shown in Figure 4.3-13. So, although a certain amount of lift is advantageous concerning the energy per distance travelled (4-82), the power available may force the bird for high velocities to generate less lift during the bound phase and thus fly at a higher cost which enables lower values of flapping power. Since the applied optimal lift coefficients (4-109) and (4-124) do not change with velocity, an increase in velocity leads to a quadratic increase in the respective load factor; see equations (4-111) and (4-126). The fraction of the flapping phase of the whole cycle (flapping ratio, (4-100)) is transformed using (4-121) and (4-103) to:

( ) bb

bntt

zcyc −−

−⋅=

111

1,

1 (4-133)

The results of (4-133) are shown in Figure 4.3-14 and explain the strong increase in the power needed in the first phase to perform the bounding flight manoeuvre. The load factor in the first phase results for the no-lift case from (4-111) and from (4-126) in case of lift during the bound phase.

Figure 4.3-14 Flapping ratio for flap-bounding flight

In case of wind, especially at headwind conditions it is important that birds have the ability to fly faster against headwinds – see also Pennycuick (2001), the kinematic velocity, KV , is

the vector sum of the aerodynamic velocity, AV , and the wind velocity, WV , (2-183); for

easier understanding the respective indices are used. Also during migration, progress against the wind may play a very important role. There are willow warblers (Phylloscopus trochilus), for example, which travel twice per year the distance between eastern Siberia to South-Africa (~16.000km) weighing only around eight grams (Birdlife, 2014). For an analysis including tailwinds, see Sachs (2013b) where it is shown that the effects of tailwinds yield less differences between bounding flight and continuous flapping flight.

1 1.5 2 2.50

20

40

60

80

100


Flap

ping

Rat

io [%

]

Flap-Bounding (no Lift during Bound)Flap-Bounding (Lift during Bound)


The dissipated energy per distance travelled in bound flight mode is in case of wind under the previous assumptions (4-89):

( )( ) cyc

A

WcycWA

AW

ttDtD

VVtVV

VtDtDs

E 22112211,

1

1 ⋅+⋅⋅

+=

⋅+⋅⋅+⋅

=−

(4-134)

And for comparison, in case of wind in stationary horizontal flight the energy per distance travelled is:

( ) D

VVtVV

VtDs

E

A

WcycWA

AcychorW ⋅+

=⋅+

⋅⋅=−

1

1,,

(4-135)

The normalised kinematic velocity, KV , the normalised aerodynamic velocity, AV , and the

normalised wind velocity, WV , are obtained as in (4-101) by division of the original velocities

through the aerodynamic velocity *1V of the flapping phase:

*1

*1

*1

, ,VVV

VVV

VVV W

WA

AK

K === (4-136)

The vector sum of the normalised kinematic, aerodynamic, and wind velocity therefore reads:

WAK VVV += (4-137)

The normalised dissipated energy per distance travelled (4-104) in case of wind, WE ,− ,

resulting from (4-134) is for no lift during the bound phase:

( )[ ]L

ALL

A

WW C

VCeC

VV

E 11

1

121 22

,−⋅⋅++

⋅+

⋅=− (4-138)

This normalised energy per distance travelled is minimised for the same lift coefficient as in case of calm (4-109), to give:

⋅+−⋅

+=−

2min,, 2

11

1A

A

WW Vee

VV

E (4-139)

Notably, the optimal lift coefficient originally derived in (4-109) does neither change with aerodynamic nor kinematic flight velocity and is therefore independent of wind speed.

The required normalised energy per distance travelled, horWE ,,− , to perform the stationary

horizontal flight, results from (4-105) and (4-135) as:

+⋅

+⋅=− 2

2,,

1

1

121

AA

A

WhorW V

V

VV

E (4-140)


Putting the normalised energy per distance travelled in bounding flight mode (4-139) equal to the normalised energy per distance travelled in stationary horizontal flight (4-140), gives

the aerodynamic velocity of equal energy, eEAV , , as originally derived in (4-113). Therefore,

the associated normalised kinematic velocity of equal energy, eEKV , , is:

WeEK Ve

V +−

=4, 1

1 (4-141)

The respective energy per distance travelled (4-139) at the velocity of equal energy, eEKV , ,

is:

( )e

eeV

VEW

eEKW −−

⋅−⋅+

=− 122

111

4,min,, (4-142)

All the above equations (4-138) to (4-142) turn into their pendants at calm (equations (4-108) to (4-114)) if the normalised wind velocity is set to zero. With an increase in headwind ( )0<WV , the kinematic velocity from which the flap-bounding

flight is superior to the stationary horizontal flight (4-141) is shifted to the left, which leads to an earlier superiority in flap-bounding flight compared to stationary horizontal flight when

the kinematic velocity is regarded – see Figure 4.3-15 which is valid for 8.0−=WV .

Figure 4.3-15 Normalised energy for flap-bounding flight in case of headwind ( )8.0−=WV

Moreover, it is shown in Figure 4.3-15, that the energy per distance travelled (4-139) no

longer monotonously increases above the kinematic velocity of equal energy, eEKV , , with an

0 1 2 30

1

2

3

4

5

Normalised Kinematic Velocity VK

Nor

mal

ised

Ene

rgy

E


eEKV ,

( )min,,WEMIN −

LeEKV ,,

( )LWEMIN min,,,−

( )horWEMIN ,,−


increase in kinematic velocity, but that a minimum, ( )min,,WEMIN − , is established which is at

a normalised kinematic velocity found by solving numerically:

01232 223 =

+

−⋅⋅+⋅− WWKWK V

eeVVVV (4-143)

For the headwind case presented ( )8.0−=WV , the kinematic velocity for minimum energy

per distance travelled is 1.0511=KV on the blue curve.

Notably, in the presence of headwind, the flap-bounding flight (4-139) is superior to stationary horizontal flight (4-140) even at values lower than the overall minimum for

stationary horizontal flight, ( )horWEMIN ,,− , which is at a normalised kinematic velocity,

0.6413=KV , found by solving numerically:

( ) 01222872 44322345 =+⋅+−⋅−⋅−⋅+⋅− WWKWKWKWKWKK VVVVVVVVVVVV (4-144)

In case of wind, the possibility to generate lift during the bounding phase so far has been intentionally excluded in the derivation of the equations to show the sheer wind influence. The normalised and – by lift factors chosen as in (4-126) and (4-121) minimised - energy per distance travelled with lift during the bound phase is:

⋅

−−

⋅+−−

⋅+

=−2

min,,, 121

11

1

1A

A

WLW V

bbe

be

VV

E (4-145)

Its minimum, ( )LWEMIN min,,,− , is at a kinematic velocity which is found from solving

numerically:

011232 223 =

+

−−⋅−⋅

⋅+⋅− WWKWK Vbe

beVVVV (4-146)

The energy per distance travelled with lift during the bound phase is shown in Figure 4.3-15 by the green curve, where the lift generation in the bound phase is again abbreviated by ‘L’. It may be seen that the lift producing possibility further enlarges the region of superiority of the bounding flight during headwinds. By lowering the required energy per distance travelled, the green curve shows that the bounding flight becomes superior to continuous flapping flight from an even smaller velocity on. In addition, the new overall minimum energy

per distance travelled is shifted to the right, to a velocity of 1.1965, =LKV .

The normalised kinematic velocity for the overall minima of energy per distance travelled in the presence of headwind are depicted in Figure 4.3-16. At moderate headwinds, the absolute minimum of the energy per distance travelled can only be reached in steady state horizontal flight. But soon, the minima established by the bounding flight manoeuvre allow to fly at lower cost and much faster speed. In the headwind case presented above

( )8.0−=WV , for example, the kinematic velocity in bounding flight mode may be 64% (no

lift) or even 87% (lift in bounding phase) higher than in steady state horizontal flight and still the overall cost is much lower.


Figure 4.3-16 Normalised kinematic velocity for overall minima of energy per distance travelled in dependence

on headwind

The savings are given in Figure 4.3-17. Here, it may be seen, that although the kinematic velocities in bounding flight mode are much higher – enabling far bigger progress against

the wind – the savings increase with headwind. In case of 8.0−=WV savings amount to

17.3% (no lift) or even 23.1% (lift in bounding phase) if the bounding flight mode is applied.

Figure 4.3-17 Savings due to bounding flight mode at overall minima of energy per distance travelled in

dependence on headwind

-1 -0.8 -0.6 -0.4 -0.2 00.6

0.8

1

1.2

1.4

Normalised Wind Velocity VW

Nor

m. K

in. V

el. V

K fo

r MIN

(E)



-1 -0.8 -0.6 -0.4 -0.2 00

10

20

30

40


Savi

ngs

[%]

Flap-Bounding (no Lift during Bound)Flap-Bounding (Lift during Bound)



4.3.4 Reference Trajectories for One Cycle When optimising flap-bounding flight trajectories with the cost function to be minimised being the energy used per distance travelled as stated in (4-82), a minimum limit on the average velocity during the cycle has to be set, since otherwise the flight would be

performed at *V as defined in (4-78) – see also Figure 4.3-12. For a minimum average velocity set, the cycle length (distance travelled) can either be optimised or fixed. All optimisation results presented in paragraph 4.3.4 and paragraph 4.3.5 are obtained using the graphical user interface GESOP (Fischer et al., 2011) from ASTOS. The model of the siskin has been implemented in Simulink and afterwards was coded into C using the Real Time Workshop of Matlab. All other parts like the constraints, the limits of the states and the controls, the cost function, the real parameters including their limits, and the control history for the initial guess were directly implemented in C. These files were then called by GESOP where SNOPT (Gill et al., 2008) has been used as numerical solver for the optimisation process with primal and dual infeasibility set to 1.0 · 10-8. In all four phases, 15 additional multiple shooting nodes were inserted as well as 50 additional control nodes. At every multiple shooting node also an additional control node was inserted. The path constraints were checked on a grid of 100 nodes per phase. The results were stored in Matlab-readable format. All plots are generated in Matlab. For each phase, the duration was normalised to range from zero to one. First, the optimised cycles for different average velocities without limits set on the length are presented. Considering a possible technical application but also due to limits set to the bird in nature for one or a few cycles, it is interesting to see to which extent and cost a cycle may be stretched or shrunken. The same must be kept in mind during bird observation where the bird may perhaps not fly the perfect optimum but accept a certain decrease in cost. First, the distance flown during the cycle is left to be optimised, but the average velocity at the end of the cycle is fixed. The bird always starts and finishes at a reference altitude of 10m.

(i) Average Velocity 11.5 m/s For an average horizontal velocity of 11.5ms-1 for the whole cycle, the optimal altitude profile is shown in Figure 4.3-18. The optimal cycle length is 7.269m and the optimal cycle duration is thus 0.632s of which the flapping phase takes a portion of 22.6%. The maximum load factor reached during the flapping phase is 25.31, ≈zn (see also Figure 4.3-21) by which a flapping ratio of 23.1%

would result using the initial guess approach (4-133). The optimal cycle is periodic in the altitude, the velocity, and the climb angle and thus also in the altitude change which is at both, the beginning and the end of the cycle, minus 1.481ms-1.


Figure 4.3-18 Altitude profile for average cycle velocity of 11.5m/s

As shown in Figure 4.3-19, the average horizontal velocity during the optimised cycle is 11.5ms-1. The reason for the difference at the beginning of the cycle between the airspeed and the average horizontal velocity is the inclination of the trajectory. The velocity itself is periodic, but with the beginning of the flapping phase, the bird experiences an instant acceleration equal to the discrepancy in rate of change of the velocity between the beginning and the end of the cycle.

Figure 4.3-19 Velocity profile for average cycle velocity of 11.5m/s

0 2 4 6 89.75

10

10.25A

ltitu

de [m

]

Distance [m]

Phase 1t1=0.143s

Phase 3t3=0.439s

0 2 4 6 811.2

11.5

11.8

Distance [m]

Velo

city

[m/s

]

Average Hor. Velocity [m/s]Airspeed [m/s]


The climb angle is periodic both in its value as well as in its derivative as shown in Figure 4.3-20. During the flapping phase, the climb angle increases at a nearly constant rate while during the bound phase its reduction is perfectly constant. The respective behaviour may also be seen from the load factor history depicted in Figure 4.3-21 together with the minimum drag coefficient. The minimum drag coefficient history shows the constant values during flapping and during the bound phase with the linear change occurring in between. The load factor stays above three during the flapping phase and is reduced to values around 0.31 during the bound phase.

Figure 4.3-20 Climb angle and its change for average cycle velocity of 11.5m/s

Figure 4.3-21 Minimum drag coefficient and load factor for average cycle velocity of 11.5m/s

−π/20

−π/40

0

π/40

π/20

Distance [m]

Gam

ma

[rad

]

0 2 4 6 8 −π

−π/2

0

π/2

π

Cha

nge

in G

amm

a [r

ad/s

]C

hang

e in

Gam

ma

[rad

/s]

0 2 4 6 80

0.01

0.02

0.03

0.04

Min

imum

Dra

g C

oeffi

cien

t CD

,min

[-]

Distance [m]0 2 4 6 80

1

2

3

4

Load

Fac

tor n

z [-]


In Figure 4.3-22, the control history for the optimised cycle is given. During the flapping phase, the power output is 0.9254W which is a bit more than the initial guess resulting from (4-131) of 0.8225W. Optimisations during which the bird was allowed to alter the power output showed that in the optimal case it was always nearly constant. The lift coefficient slightly decreases during the flapping phase and afterwards reduces at a decreasing rate to the value optimal for the bounding phase during which it stays nearly constant to grow at an increasing rate during the wings extending phase – with underlying linear change in aerodynamics, as depicted in Figure 4.3-21. Thereby, the time of flight at strongly reduced drag is extended. Optimisations in which the bird was allowed to change the duration of the extending and retracting phases showed the minimum time allowed to be always optimal. The values of the lift coefficient resulting from (4-120) and (4-124) are:

0.05292 and 0.5292 2,1, == LL CC (4-147)

As shown in Figure 4.3-22, the optimised lift coefficient deviates only slightly from the initial guess during the flapping and the bound phase.

Figure 4.3-22 Control history for average cycle velocity of 11.5m/s

The change in specific energy of the bird (energy divided by the bird’s weight) during the cycle is depicted in Figure 4.3-23. During the flapping phase, the bird accelerates and continuously increases its kinetic energy while the potential energy at the beginning and at the end of the flapping phase has nearly the same value. During the wings’ folding phase, the potential energy is increased leading to a rather strong reduction in kinetic energy (and thus drag). In the middle of the bounding phase, the potential energy reaches its maximum and afterwards continuously reduces. The kinetic energy stays quasi constant (at the lowest level possible) from the middle of the bounding phase on and also during the wings extending phase.

0 2 4 6 80

0.2

0.4

0.6

0.8

Distance [m]

Lift

Coe

ffici

ent [

-]

0 2 4 6 80

0.25

0.5

0.75

1

Pow

er [W

]

Initial Guess for Lift Coefficient

Pow

er [W

]


Figure 4.3-23 Change in specific energy for average cycle velocity of 11.5m/s

In Figure 4.3-24 and Figure 4.3-25, the flap-bounding flight is compared to a horizontal flight at the same velocity as the average horizontal velocity of the bounding flight manoeuvre at the respective distance. Therefore, the comparative curves show slight changes.

In the optimal case, the cost function (4-82), boundJ , for the siskin is 1.81747J per 100m of

flight at an average velocity of 11.5ms-1. To fly the same distance in horizontal flight would

require 2.624J per 100m abbreviated as horJ . The savings thus amount to:

%7.30=−

=hor

boundhor

JJJSavings (4-148)

Equivalently, the cost increase to avoid the bounding flight would be 44.4%. The power necessary, however, to perform the bounding flight is far higher:

%200max,

max, ≈−

=hor

horbound

PPP

asePowerIncre (4-149)

In stationary horizontal flight only 0.302W are required instead of 0.9254W.

0 2 4 6 8-0.2

0.0

0.2

0.4

0.6∆

Spe

cific

Ene

rgy

[m]

Distance [m]

∆ Spec. Kinetic Energy∆ Spec. Total Energy∆ Spec. Potential Energy


Figure 4.3-24 Energy per distance travelled for average cycle velocity of 11.5m/s

Figure 4.3-25 Power applied for average cycle velocity of 11.5m/s

0 2 4 6 80

2.5

5

7.5

10

Distance [m]

Ener

gy p

er D

ista

nce

[J/1

00m

]

Flap-Bounding FlightContinuous Flapping Flight

0 2 4 6 80

0.25

0.5

0.75

1

Distance [m]

Pow

er [W

]



(ii) Average Velocity 7.2 m/s In case of a reduction in required average velocity to 7.2ms-1, the cycle length is reduced to 6.519m as shown in Figure 4.3-26. The cycle duration is thus 0.905s and has a flapping ratio of 84.4%. As maximum load factor, 21.11, ≈zn is reached from which a flapping ratio

of 80.5% would result if the approximation (4-133) is used. A strong reduction in average velocity only leads to a rather small reduction in optimal cycle distance and thus to a rather strong increase in cycle duration. The optimal cycle is periodic in altitude, velocity, and climb angle and thus also in the change in altitude having the value of minus 0.5814ms-1 at the beginning and at the end of the cycle.

Figure 4.3-26 Altitude profile for average cycle velocity of 7.2m/s

The velocity profile now shows an increase in velocity only up to the middle of the flapping phase, followed by a decrease so that the average cycle velocity diminishes during phases two, three, and four until at the very end of the cycle the required value of 7.2ms-1 is reached – see Figure 4.3-27. The principle behaviour of the climb angle remains unchanged at about half the amplitude when the velocity is reduced from 11.5ms-1 to 7.2ms-1, the positive rate of change is now much smaller in the first phase but the negative rate of change during the bound phase now is clearly lower as depicted in Figure 4.3-28.

0 2 4 6 89.75

10

10.25

Alti

tude

[m]

Distance [m]

Phase 1t1=0.764s

Phase 3t3=0.091s


Figure 4.3-27 Velocity profile for average cycle velocity of 7.2m/s

Figure 4.3-28 Climb angle and its change for average cycle velocity of 7.2m/s

0 2 4 6 87

7.2

7.4

Distance [m]

Velo

city

[m/s

]

Average Hor. Velocity [m/s]Airspeed [m/s]

−π/20

−π/40

0

π/40

π/20

Distance [m]

Gam

ma

[rad

]

0 2 4 6 8 −π

−π/2

0

π/2

π

Cha

nge

in G

amm

a [r

ad/s

]C

hang

e in

Gam

ma

[rad

/s]


The control history is given in Figure 4.3-29. The power during the flapping phase is strongly reduced to 0.123W which is a bit less than 0.1285W resulting from the approximation (4-131). The lift coefficient stays independent from the velocity close to the values derived by (4-120) and (4-124). In case of technical applications, the independency of the lift coefficient from the velocity flown as well as the nearly constant values during the flapping and the bound phase would enable a rather simple design.

Figure 4.3-29 Control history for average cycle velocity of 7.2m/s

Figure 4.3-30 Change in specific energy for average cycle velocity of 7.2m/s

0 2 4 6 80

0.2

0.4

0.6

0.8

Distance [m]

Lift

Coe

ffici

ent [

-]

0 2 4 6 80

0.25

0.5

0.75

1

Pow

er [W

]

Initial Guess for Lift Coefficient

0 2 4 6 8-0.2

0.0

0.2

0.4

0.6

∆ S

peci

fic E

nerg

y [m

]

Distance [m]

∆ Spec. Kinetic Energy∆ Spec. Total Energy∆ Spec. Potential Energy


In the first phase, due to the flapping, the total energy is continuously increased – see Figure 4.3-30. While in the faster case the overall change in kinetic energy was approximately twice as high as the change in potential energy, these two values are now nearly equal. Notable amplitudes in specific energies are only reached during the first phase.

Figure 4.3-31 Energy per distance travelled for average cycle velocity of 7.2m/s

The flapping power applied during the first phase is 0.123W and thus (in absolute values) only a little more than for continuous flapping flight which would require in comparison 0.104W. As depicted in Figure 4.3-31, the bounding flight manoeuvre becomes superior in energy per distance travelled not before the very last phase where the wings are spread.

The required energy per distance travelled in bounding flight mode (4-82), boundJ , is 1.443J

per 100m compared to 1.447J per 100m in the continuous flapping case symbolised by

horJ . The savings thus only amount to:

%3.0=−

=hor

boundhor

JJJSavings (4-150)

Still, the power necessary to perform the bounding flight is clearly higher when relative values are regarded:

18% max,

max, ≈−

=hor

horbound

PPP

asePowerIncre (4-151)

So, for small velocities only rather low savings in cost function (4-82) may be reached if comparatively high flapping power is applied.

0 2 4 6 80

2.5

5

7.5

10

Distance [m]

Ener

gy p

er D

ista

nce

[J/1

00m

]



(iii) Comparison for free Cycle Length In the following, the optimised flap-bounding flight cycles are compared for different average velocities from 7.1ms-1 to 15ms-1. The minimum cycle duration of 0.539s is reached at 8.8ms-1 as depicted in Figure 4.3-32 by an open circle. The duration of the flapping phase continuously decreases with an increase in flight speed while the duration of the bounding phase increases with an increase in flight speed.

Figure 4.3-32 Duration of cycle and phases

As also predicted by the model presented in paragraph 4.3.3 and Figure 4.3-14, the flapping ratio (duration of flapping phase divided by total cycle duration) reduces continuously with an increase in average velocity – see Figure 4.3-33. Also the data from living birds presented at the beginning of paragraph 4.3 is met very satisfactory – see e.g. Figure 4.3-3 and Figure 4.3-5 as well as the durations for the flapping and the bounding intervals stated by Woike and Gewecke (1978).

Figure 4.3-33 Flapping ratio and cycle duration

7 8 9 10 11 12 13 14 150

0.25

0.5

0.75

1

Average Velocity for Cycle [m/s]

Dur

atio

n [s

]

Cycle DurationFlapping DurationBounding Duration

7 8 9 10 11 12 13 14 150

25

50

75

100

Flap

ping

Rat

io [%

]

Average Velocity for Cycle [m/s]8 0

0.25

0.5

0.75

1

Cyc

le D

urat

ion

[s]


The minimum cycle length of 4.6m is reached at a velocity of 8.3ms-1 – see Figure 4.3-34 where the minima are again indicated by open circles. In the velocity range presented, the variation in distance with velocity (sensitivity) is around twice the variation in cycle duration with velocity.

Figure 4.3-34 Optimal distance and optimal cycle duration

Figure 4.3-35 shows, that the strong increase in energy consumed per distance travelled at high velocities can be avoided when the bird switches from continuous flapping flight mode to bounding flight mode. By applying a flap-bounding flight technique, the energy consumed per distance travelled increases at a far lower rate. Moreover, the possible savings are shown. The faster the flight, the higher are the savings in energy per distance. It is to be remembered that 35% savings correspond to a range increase by nearly 54%.

Figure 4.3-35 Cost function and savings for bounding flight

7 8 9 10 11 12 13 14 150

4

8

12

16

Opt

imal

Dis

tanc

e [m

]

Average Velocity for Cycle [m/s]7 8 9 10 11 12 13 14 150

0.25

0.5

0.75

1

Cyc

le D

urat

ion

[s]

7 8 9 10 11 12 13 14 150

1

2

3

4


Ener

gy p

er D

ista

nce

[J/1

00m

]

7 8 9 10 11 12 13 14 150

25

50

75

100

Savi

ngs

[%]

Continuous Flapping FlightFlap-Bounding FlightSavings


However, the flapping power required to fly the manoeuvre exponentially increases with average velocity as shown in Figure 4.3-36. At 7.1ms-1, the necessary flapping power in bounding flight mode is 0.1152W to which the power factor represented by the red dashed line gives reference. From 7.2 ms-1 to 11.5 ms-1, the flapping power required for the bounding flight manoeuvre increases by more than the factor of seven.

Figure 4.3-36 Power required for bounding flight and horizontal flight

The power available is thus the limiting factor concerning high velocity flight. With an increase in velocity, more power has to be applied during a continuously decreasing / shorter time span. In case of the bird, the preparation to flap is modelled by the wings extending phase and the flapping finishes with the wings attaching phase. For a technical implementation, phases two and four would be the engine start-up and the engine turn-off. Due to the maximum power available, at high aerodynamic velocities, both the bird as well as a man-made system might be forced to switch back from a bounding flight mode into a continuous thrust mode where the cost in terms of energy per distance travelled is higher, but with a fixed power limit installed also the maximum aerodynamic velocity is higher.

(iv) Fixed Cycle Length In the following, the average cycle velocity is fixed to 11.5ms-1. In different optimisations, altered values for the distance travelled are set as final boundary constraints. Length variations between 5.0m and 15.0m for a single cycle are analysed. As reference, the cycle with a freely optimised length of 7.239m is used. Due to the fixed average velocity implemented as final boundary constraint, the cycle durations scale directly with the distance set, and for 5.0m, 7.269m, and 15.0m they are 0.4348s, 0.6321s, and 1.3043s. As seen from the altitude profiles (Figure 4.3-37), the velocity profiles (Figure 4.3-38), and the climb angle histories (Figure 4.3-39), the range used of all state variables increases with an increased cycle length while the principle shape is preserved. The durations of the single

7 8 9 10 11 12 13 14 150

1

2

3

4


Pow

er R

equi

red

[W]

Flap-Bounding FlightContinuous Flapping FlightPower Factor: 1, 5, 10, 20 & 30


phases of each cycle scale closely with the cycle durations and thus the distances covered. Only in case of 15m cycle length, the initial and final velocity is above the average cycle velocity.

Figure 4.3-37 Altitude profiles in dependence on cycle length

Figure 4.3-38 Velocity profiles in dependence on cycle length

0 0.2 0.4 0.6 0.8 19

10

11


Alti

tude

[m]

Cycle Length: 5.0mCycle Length: 7.269mCycle Length: 15.0m

0 0.2 0.4 0.6 0.8 110.5

11

11.5

12

12.5


Velo

city

[m/s

]



Figure 4.3-39 History of climb angles in dependence on cycle length

Figure 4.3-40 History of lift coefficients in dependence on cycle length

In contrast to the variation of the states in dependence on the cycle length, the time normalised controls hardly vary when the cycle length is changed – see Figure 4.3-40. The flapping power applied also varies only very little: for the length of 5.0m, 7.269m, and 15.0m, the respective flapping power is 0.9437W, 0.9254W, and 0.9985W – see also Figure 4.3-45. Even when the cycle length is extended to 22.0m, the required power stays below 1.2W. The energy consumed per distance travelled for the length of 5.0m is 1.8257J/100m, for 7.269m it is 1.8175J/100m (reference cost function), for 15.0m it is 1.8639J/100m, and for 22.0m it is 1.9516J/100m – thus a variation of less than 7.5%, which may also be seen in Figure 4.3-44. The associated altitude profiles are given in Figure 4.3-41. This may also explain why the precise analysis of bounding flights in living birds is very laborious. All the

−π/10

−π/20

0

π/20

π/10


Gam

ma

[rad

]

0 0.2 0.4 0.6 0.8 1


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8


Lift

Coe

ffici

ent [

-]



trajectories shown in Figure 4.3-41 vary by less than a tenth in energy per distance, and if velocity is additionally allowed to alter, this variation is further reduced. The cost functions over distance travelled for one to three cycles are presented in the next paragraph (see Figure 4.3-44) and compared to horizontal flight. There it is shown, that even with a 7.5% increase in cost function (CF) for the bounding flight it is still clearly superior to continuous flapping flight. But it should be kept in mind that a tripling of cycle length from 5.0m to 15.0m only equals an increase in cost function of around 2%.

Figure 4.3-41 Altitude profiles over distance

From a technical point of view, the fact that power is not the limiting factor for cycle length variation, offers great possibilities due to increased flexibility. If, for example, a system is not able to build up thrust extremely fast, the whole cycle may be stretched to meet the capabilities of the system. This stretching does, like already mentioned, not require additional thrust capacities, and the thrust phase as well as the bounding phase durations scale directly with cycle length.

4.3.5 Multiple Cycles In the following, distances travelled between 5.0m and 25.0m at an average speed of 11.5ms-1 are analysed. The bird may fly this range using one, two or three cycles. Exemplarily, for a distance of 15.0m, which is around twice the optimal single cycle distance, the altitude profiles for the optimised cycles are shown in Figure 4.3-42, and the velocity profiles are shown in Figure 4.3-43. As value of the reference cost function, 1.8175J/100m is taken again and as value of the reference power 0.9254W is taken again. As stated in the legend of Figure 4.3-42, the cost function value for an optimised trajectory will only increase marginally if instead of two cycles three cycles have to be flown to cover a distance of 15.0m. If the whole distance has to be covered flying a single cycle, the cost function is increased by 2.55%. This is – like already mentioned – remarkable, since the

0 5 10 15 20 259.5

10

10.5

11

11.5

12

Distance [m]

Alti

tude

[m]

5.0m: + 0.45% in CF7.269m: Reference CF15.0m: + 2.55% in CF22.0m: + 7.38% in CF


reference distance of 7.269m is more than doubled and, as indicated in the figures, the manoeuvre is quite dynamic. The rather strong velocity increase during the flapping phase when the whole distance of 15.0m shall be covered by a single cycle requires an increase in power by 7.91% above reference power - see the legend of Figure 4.3-43. Less power increase is required if three cycles are chosen to cover the distance, and in case of two cycles the necessary power applied during the flapping phase may even be slightly lowered beneath the reference value resulting from flight at maximum range. Only in case of a single cycle chosen to fly the complete distance, the initial and final velocity is above the average cycle velocity.

Figure 4.3-42 Altitude profiles for one to three cycles

Figure 4.3-43 Velocity profiles for one to three cycles

In Figure 4.3-44, the cost functions for one to three cycles for distances between 5.0m and 25.0m are given. The three minima are marked by dashed black lines. All of the minima are

0 5 10 159

10

11

Distance [m]

Alti

tude

[m]

One Cycle: + 2.55% in CFTwo Cycles: <+ 0.01% in CFThree Cycles: + 0.45% in CF

0 5 10 1510.5

11

11.5

12

12.5

Distance [m]

Velo

city

[m/s

]

One Cycle: + 7.91% in PowerTwo Cycles: >- 0.03% in PowerThree Cycles: + 1.98% in Power


very flat and the smooth shape of the cost functions shows that the optimisation procedures applied work fine. As mentioned in (3-104), at the minimum of the cost function its sensitivity with respect to the parameter determining the final distance has to be zero. The cost function may thus – see later – be only approximated by a quadratic function without a linear term. The three open marker dots at 15.0m represent the cost function values for the cycles shown in Figure 4.3-42 and Figure 4.3-43.

Figure 4.3-44 Cost functions for one to three cycles

Figure 4.3-45 Power during flapping phase for one to three cycles

The power behaviour is similar to that of the cost functions, though more sensitive to changes in cycle length set – see Figure 4.3-45. If the number of cycles is chosen correctly, basically any distance may be covered without a strong increase in necessary flapping power. Moreover, the diagram shows by the red dashed lines that the cycle length for

5 10 15 20 251.8

1.85

1.9

1.95

Distance [m]

Ener

gy p

er D

ista

nce

[J/1

00m

]

One CycleTwo CyclesThree CyclesMinima

5 10 15 20 250.9

0.95

1

1.05

1.1

Distance [m]

Pow

er [W

]

One CycleTwo CyclesThree CyclesMinima


minimum power is a bit larger than the cycle length for minimum energy per distance (indicated by black dashed lines as in Figure 4.3-44). For the single cycle flown at 11.5ms-1, the distance for minimum power is 7.671m of which the other power minima are multiples.

Figure 4.3-46 Savings due to flap-bounding flight for one to three cycles

Figure 4.3-46 shows that for any distance the possible savings due to flap-bounding flight in comparison to stationary horizontal flight can be kept above 30% (energy per distance travelled) if the number of cycles is chosen correctly. More precisely: an optimized bounding flight at an average cycle velocity of 11.5ms-1 enables the bird to save at most 30.7% and at least 30.4%. For a higher number of cycles, the gap between maximum and minimum saving narrows as the lower limit approaches 30.7%; this is equivalent to a cost increase of 44.4% if the bounding flight mode is avoided.

4.3.6 Bilevel Optimisation for Solution Structure Analysis As just shown, the correct selection of the number of cycles flown is of high importance to truly minimise cost and to keep the necessary power applied at a reasonable level. Exemplarily, the task to determine the parameter to set the constraint on the final distance, where one and two cycles are equally efficient with respect to the cost function (4-82), is solved via a bilevel optimisation. It offers the fastest way to determine at which point it should be switched from one to two cycles and is thus in addition interesting for technical applications. For comparability, the average velocity is again set to 11.5ms-1. All calculations are performed within the TUM-FSD optimisation framework.

First, the optimisation is performed for a single cycle by which point I results – see Figure 4.3-47. Then, by two times consecutively stringing together the optimal control history for one cycle, a very good initial guess for two cycles is provided. The successful optimisation

for two cycles gives point II in Figure 4.3-47. During both optimisations, the first and second order sensitivity of the cost function to the parameter determining the final distance

5 10 15 20 2529

29.5

30

30.5

31

Distance [m]

Savi

ngs

[%]

One CycleTwo CyclesThree Cycles


are obtained. And, as stated in (3-104), since the cycle length may be freely determined, the first order sensitivity has to be zero. For the first order sensitivity the numeric results are:

m100mJ10-3.7547 10-

1,,

1

1,,,⋅

⋅=sfinoptpsfindp

dJ (4-152)

m100mJ101.8580- 10-

2,,

2

2,,,⋅

⋅=sfinoptpsfindp

dJ (4-153)

The second order sensitivities are obtained as described in (3-83) and due to the linear change also (3-86) and (3-87) are to be respected:

( )

[ ] [ ] [ ]TT

ddL

ddL

ddL

dJd

⋅∇⋅+

⋅∇⋅

+∇

=

== 000

0

000

200

2

2

,

p0

pzp

zzp0

pp

p

pzp

pzp

pzp

ppz

)())()

(4-154)

This gives numerically:

mm100mJ102.4496 3-

21,,

12

1,,,⋅⋅

⋅=sfinoptpsfindp

Jd (4-155)

mm100mJ106.1241 4-

22,,

22

2,,,⋅⋅

⋅=sfinoptpsfindp

Jd (4-156)

Equally, (3-91) may be used: calculating the sensitivity of the Lagrange multipliers, μ , to the

parameter vector, p , then (4-155) and (4-156) result with the numerical differences of

9.9 · 10-11 and 5.6 · 10-11.

Figure 4.3-47 Optimisations for one and two cycles and sensitivities of cost functions to final distance

5 10 15 20 251.8

1.84

1.88

Distance [m]

Ener

gy p

er D

ista

nce

[J/1

00m

]

Approximations of Cost FunctionsOptimisation Results for one CycleOptimisation Results for two Cycles

I II


The above results, namely the optimisations for one and two optimal cycles, enable the calculations for the intersection of the approximated cost functions as described in (3-109) and (3-110), to give:

m 9.6911, =guessswitchp (4-157)

( )100m

J 1.82466,2,1 =guessswitchpJ (4-158)

The first guess for the switching distance stated in (4-157) is used as initial guess for the respective parameter in the bilevel optimisation problem. The region of interest is enlarged in Figure 4.3-48 where the values of (4-157) and (4-158)

are marked as point III . It may be seen that in this first step the true intersection is already well approximated. The true intersection is represented by the open circle at the top of the green dashed vertical line. The approximated distance differs around 40cm from the true distance at which the cost functions intersect and the approximated cost function value differs only by 6.82 · 10-4 J/100m.

Figure 4.3-48 Optimisations and sensitivities of cost functions - zoom level

In the bilevel optimisation, the cost function of the upper level problem is the difference between the cost functions of the two lower level problems. The upper level cost function is thus described by (3-102) and its derivative by (3-103). The first lower level problem is the optimisation of a single cycle and the second lower problem is the optimisation of two consecutive cycles. Both lower level problems have a cost function as stated in (4-82). The task of the upper level parameter optimisation problem is thus to determine the final distance at which both lower level cost functions are equal. Thereby, the switching point is determined from which distance on two cycles should be flown rather than one cycle. The upper level problem is solved using SNOPT (Gill et al., 2008) with primal and dual infeasibility set to 1.0 · 10-8. The lower level problems are fully discretised (collocation) featuring 200 discretisation points in the state and control variables per phase using Hermite-Simpson discretisation and are solved using IPOPT (Wächter and Vigerske, 2013)

7 9 11 13 151.81

1.82

1.83

1.84

1.85

1.86

Distance [m]

Ener

gy p

er D

ista

nce

[J/1

00m

]

Approximations of Cost FunctionsOptimisation Results for one CycleOptimisation Results for two Cycles

I IIIII


with primal and dual infeasibility set to 1.0 · 10-11. IPOPT has been chosen since it is able to include the Hessian into the optimisation process. Compared to the mere use of the Jacobian, calculation time was reduced between 50% to 90% when the analytic Hessian was included. The Jacobian and the Hessian of the model were generated automatically as described in paragraph 3.2.1. Thus, a maximum of knowledge about the system is delivered to and exploited by the solver. Performing the bilevel optimisation returns 10.093366m as switching distance and 1.825343 J/100m as cost function. Above this distance at least two cycles have to be flown and in the optimised case the cost function will remain below 1.825343 J/100m. At the switching point, the sensitivity of the cost functions with respect to final distance flown, is:

m100mJ105.081 3-1

⋅⋅=

switchpswitchdpdJ

(4-159)

m100mJ104.132- 3-2

⋅⋅=

switchpswitchdpdJ

(4-160)

Checking these values against the negative of the respective Lagrange multipliers, μ , as

described by (3-90), shows numerical differences below 2.5 · 10-9.

259

Chapter 5

Conclusion and Perspectives

The thesis starts with a chapter dedicated to flight mechanics where the derivation of a solar model suitable for flight trajectory optimisation is included. In the chapter on applied optimal control, the parameter dependent optimal control problem is described and its solution process is explained. A special class of bilevel optimal control tasks is regarded for which the post optimal sensitivity analysis forms the basis. In the next chapter, the trajectories are optimised for a solar powered aircraft circling the world with the least possible weight penalty resulting from the batteries, for a motor glider featuring a retractable propulsive unit aiming to maximise range, and for a siskin trying to minimise the energy used per distance travelled by applying the bounding flight technique. In all the three cases periodic trajectories result. For the solar aircraft the non-steady periodic trajectory enables to reach the capability to circle the world. In case of the motor glider performing a saw-tooth flight, for all types of engines investigated, the periodic trajectories increase range not only by a difference but by a factor as compared to optimised steady state flight. For the siskin it is analytically shown in normalised form that the bounding flight manoeuvre in case of a headwind at the same time enables faster progress against the wind and less energy consumed per distance travelled as compared to steady state horizontal flight. Also the numeric optimisations show how over a very broad range of velocity and cycle distance strong savings may be obtained. Finally, the bilevel optimisation is used for a solution structure analysis where the distance is determined at which the switching between one and two cycles should occur to minimise cost. In Chapter 2 Fundamentals of Flight Mechanics, Newton’s second law is applied on a rigid body to derive its equations of motion. A rigid body is defined as to preserve its external as well as internal shape but may vary its mass by locally varying its density. The only restriction on density variation imposed is that the position of the centre of gravity within the rigid body must not be shifted. Thereby, mass flows crossing the system boundary are included in the translatory and rotatory equations and their effect on the movement of the system can be precisely investigated making vague statements on their relevance obsolete. An analytic gravity model for the WGS 84 ellipsoid of revolution is derived. It is clearly distinguished between gravitation and the influence of the centrifugal force. By the model the change in both, magnitude and direction of the gravity vector for a change in altitude is determined. It is interesting to see how small the differences between this analytical model and the highly sophisticated EGM96 and EGM2008 models are. For the vast majority of flight applications they can be considered to be irrelevant. The gravity model derived also allows for the evaluation of simpler gravity models. The basis for the paragraph on the atmosphere forms the International Standard Atmosphere. It is shown how static and convective wind fields may be included in the equations of motion. For most applications the influence of wind can be easily averaged

260 Conclusion and Perspectives

over the aircraft. Here, in many cases the changes in wind the aircraft experiences is not due to the temporal changes in the wind field but due to spatial changes within the wind field the aircraft is crossing at high speed. If the wind influence may not be averaged, the respective resultant aerodynamic rates are derived. The model to determine the receivable solar radiation within the earth’s atmosphere uses a Keplerian orbit for the earth moon barycentre to calculate in the first step the receivable radiation on that orbit for a given date. The deviations of the true earth orbit from that elliptical orbit are mainly due to the moon and much weaker deviations are caused by other planets. The influence of these deviations concerning the receivable radiation are judged and found to be very small. In the next step, the position of the sun as seen from the earth and thus also the apparent movement of the sun is calculated. For a given receiver position the location of the sun may be described by the declination and the hour angle. While for the declination a sufficient precision is obtained by a calculation in dependence on date, the quickly changing hour angle is computed as a function of date and actual local time. Within the modelling process it is described that the earth’s rotational rate is nearly perfectly constant. The averaged apparent solar movement (24h per day) deviates from the true apparent solar movement mainly for two reasons: first, the eccentricity of the earth orbit, and second the obliquity of the ecliptic. Both of these effects are described by the equation of time which determines the time for the cancelling of these effects by relating the occurring angular differences to the earth’s rotational rate. The resulting amplitude is more than 30 minutes equivalent to approximately 8° in hour angle. Then, the attenuation of the solar radiation by the atmosphere is regarded. First, it is important to know the atmospheric conditions, and second the length of the path the ray of light has to travel through the atmosphere before reaching the receiver. It is shown how the total radiation receivable is combined of the direct and the diffuse radiation. Thereby, the radiation receivable for any receiver position and any receiver orientation can be calculated. Also shading can be easily included. The daily integrated irradiance during the year in dependence on latitude is shown for a horizontal receiver at sea level altitude. Two maxima are identified at around 40° northern and 40° southern latitude. Both differ in intensity due to the earth’s elliptical movement around the sun. Thereby, a time period and a latitude region can be determined where it is most favourable to fly an earth circling mission. Further, it is shown by how much this integrated irradiance may be increased if the receiver is oriented towards the sun. Also the insolation per year at sea level altitude is calculated and it is shown that it can be increased by orientation of the receiver the stronger the farer the position from the equator. Finally, the maximum and minimum daily insolation during the year is depicted. There it is shown that throughout the surface of the earth (from 60° southern to 60° northern latitude) the variation in the maximal receivable insolation per day is rather small, but the differences in minima of receivable insolation per day – always assuming a clear sky – strongly increase with deviations of the receiver position from the equator. Therefore an earth circling mission may be flown at any latitude if the date is appropriately chosen. In Chapter 3 Applied Optimal Control, the transcription of an infinite-dimensional parameter dependent optimal control problem to a finite-dimensional parameter dependent optimisation problem is described. The focus is put on direct transcription using multiple shooting since it largely increases flexibility. The necessity of mesh refinement or mesh


adaptation within the numeric solution process is pointed out. An optimisation framework is described of which the NLP solver may be associated with the discipline of mathematics while the function evaluator is associated with the discipline of engineering. The importance of the distinction between the generation of information and the processing of information is outlined. Processing may to a large extent be automated. The information which is no longer directly accessible due to the discretisation, for example, can be restored up to the second order by automatically generated Jacobian and Hessian matrices of the model. In the post optimal sensitivity analysis, it is investigated how a change in the parameter vector influences the optimal trajectory. Effects of parameter changes on the cost function are investigated up to the second order. Thereby, also the basis for a special class of bilevel optimisation problems is laid. On the upper level a parameter optimisation problem delivers a set of parameters to two periodic optimal control problems on the lower level. The aim is to perform a solution structure analysis by determining the conditions under which the two lower level problems fulfil the given task equally well. Since in the periodic optimal case the relative cost functions (e.g. energy per distance) for one and two cycles are equal and their derivatives are zero, the second order sensitivities of the cost functions have to be evaluated. The cost functions are thus approximated by parabolas. The intersection of these parabolas gives the initial guess for the parameter of the upper level problem delivered to the two lower level problems. In Chapter 4 Periodic Optimal Control for Flight Applications, several periodic optimisations are performed. First, unlimited endurance missions for solar powered aircraft are investigated. The task of the aircraft is to circle the world in sustained flight. In order to handle the night with no solar radiation present, and to satisfy the permanently requested on-board demand for power, the aircraft is equipped with batteries. The aim of the optimisation is to minimise the maximum battery capacity since the batteries represent a weight penalty. Thus a Tschebyscheff problem is to be solved. First, a suitable region on earth may be selected using the solar modelling presented above. In such a region, during the day, the power available from the solar cells is far higher than the power required for horizontal flight. The aircraft basically has two possibilities of energy storage: first in terms of mechanical energy and second by charging the batteries. If no pressure cabin is installed, the maximum altitude allowed is set to 8500m and therefore the maximum of potential energy is set. Since for the solar aircraft the amount of energy storable in kinetic energy is negligibly small, the altitude limit defines the maximum of mechanical energy storable. To this kind of energy storage no weight penalty is associated. Due to the large energy surplus during the day, the aircraft has many degrees of freedom not affecting the required battery capacity which is determined by the night segments. It is, however, absolutely important that at the end of segment 1, the aircraft has reached the maximum altitude allowed of 8500m and the batteries are fully charged. At the end of segment 1 the available solar power is just at the level required to perform a minimum power horizontal flight in 8500m. During segment 2, all solar power available for the engines is put into thrust. After sunset a mere glide follows representing segment 3. Segment 4 starts when the minimum altitude allowed is reached and the engines are switched on again to generate thrust for a steady state horizontal flight which is a little bit faster than that for minimum power. The cycle finishes when the true local solar time, and thus the hour angle of the sun, from the very beginning is reached again. The periodic optimal trajectory thus enables a circling of the world by the solar aircraft; a steady state trajectory would not.


Since the determining factor for the required battery capacity is the night during which the lift coefficient variations are rather small and due to the large energy surplus during the day representing many degrees of freedom, it is possible to fly a very similar trajectory at a fixed lift coefficient. The value for the fixed lift coefficient is optimised and lies very close to the lift coefficient values used during the night in the variable lift coefficient configuration. The cost function is only marginally increased. Thereby, a possibility for sustained flight of the solar aircraft with extremely reduced control effort is presented. In case of a pressure cabin installed, the upper altitude limit may be dropped. In the optimal case, the aircraft then climbs to much higher altitudes whereby the energy stored in potential energy is largely increased. As before, the maximum altitude is reached at the very end of segment 1 since it is used for energy storage. The whole flight is performed at little lift coefficient variation and a velocity very close to that for minimum power required. If no stronger engines are installed, an unusable amount of solar energy remains since the battery capacity required is very small. This small required battery capacity allows for around 15% of the aircraft’s total mass to be used for a pressure cabin. If the required mass for a pressure cabin is lower, it is advantageous to drop the altitude limit. Installing stronger engines could further reduce the required battery capacity. Due to the small variations in lift coefficient, it is very likely that by an optimised fixed lift coefficient, resulting in a drastic reduction in control effort, a very similar trajectory could be flown avoiding a notable increase in necessary battery capacity. In case of unmanned solar systems, all installations, and thus weight concerning the pilot may be dropped. The aircraft’s weight thereby is significantly reduced. The aircraft now can store so much energy in potential energy that no power for the engines is ever taken from the batteries. Thus, the horizontal flight phase represented by segment 4 is completely omitted. Due to the engine power limitations a non-usable energy surplus during the day results. It could therefore be advantageous to equip the solar UAS with stronger engines and shift the whole trajectory to higher altitude levels to increase safety. In conclusion for the optimisation of solar aircraft trajectories, it is important to mention that an altitude limitation restricts the maximum mechanical energy storable but offers many degrees of freedom for flight during the day. For all optimisations performed it could be interesting to investigate the influence of an optimisable maximum engine power on the trajectories. With strong enough engines it can be avoided that a certain amount of solar power is lost. For the configurations investigated, the periodic trajectories enable the aircraft to fulfil its task – a steady state trajectory could not. In case of manned solar flight, the resulting trajectories always require rather strong batteries. At the same time, for most trajectories an energy surplus exists, which is already established for the solar cells installed. Therefore, technical progress in the battery sector will have a major impact on manned solar flight. In case of a motor glider featuring a completely retractable propulsive unit, the key to maximise range is the saw-tooth manoeuvre. First, a motor glider equipped with a piston engine powering a propeller is regarded. For comparison, an optimised horizontal flight in the minimum altitude allowed is calculated. By flying the periodic saw-tooth cycle, the motor glider can more than triple range and at the same time is around 10% faster than at fuel optimal steady state horizontal flight. Then, the influence of head- and tailwinds on the optimal trajectory is investigated. In case of a headwind, the pilot has to increase the climb interval and also the indicated airspeed (IAS) during glide has to be increased. For tailwind the opposite holds. As concerning the


cost function, the glider can profit to a large extent from tailwinds and may handle moderate headwinds well. At stronger headwinds, power limitations restrict the performance of the glider. In case of a crosswind, the permanently adapted heading differs from the course as the glider travels in perfect perpendicularity to the wind avoiding any aerodynamic angle of sideslip. In dependence on the crosswind, the climb interval has to be slightly increased. By adapting the manoeuvre the impact on the cost function is rather low. The glider thus can handle cross winds very well. In case of an electric engine powering the propeller, the necessity of cooling is omitted and the respective phase therefore takes only 10s for stopping and orientating the propeller. In contrast to the naturally aspirated piston engine, the power available of the electric engine does not depend on the density respectively the altitude flown in. The saw-tooth manoeuvre is again compared to an optimised steady state horizontal flight. In case of the electric engine installed, the average velocity during the saw-tooth manoeuvre is again around 10% higher than that at optimised horizontal flight. As concerning range, the motor glider featuring an electric engine can increase range by flying the saw-tooth manoeuvre even further than in case of a piston engine installed. The achievable range is now increased to the factor of 3.3. For a jet engine installed, different maximum thrust levels are investigated. To assess the superiority of the saw-tooth flight, the comparative horizontal steady state trajectories are flown in 600m, 3000m, and at an altitude optimised for the respective thrust level installed. It is found that a certain amount of thrust should be available to enable a considerable climb rate. The optimal periodic trajectory for a jet engine delivering 1053N of thrust is regarded in more detail. Although now the average velocity during the saw-tooth manoeuvre is around 10% lower than the velocity for optimal steady state horizontal flight in 3000m, the achievable range is increased to the factor of 2.3. The more thrust is installed, the higher is the optimal altitude to perform the horizontal flights in. The optimal altitude with1053N of thrust installed to perform the horizontal flight would be 7810m. In comparison to the flight in this altitude, the saw-tooth manoeuvre saves 43% of fuel which means that range is increased to the factor of 1.75. For the thrust levels investigated, it is shown that the savings by performing the saw-tooth manoeuvre increase with an increase in thrust available. This holds for all altitudes in which the horizontal flights for comparison are optimised. In conclusion, it can be stated for the saw-tooth flight manoeuvre that it largely improves range. In many cases range is increased up to the factor three and above as compared to an optimised steady state horizontal flight. The manoeuvre is rather easily flyable by the pilot as he can keep the IAS at a (nearly) constant level during climb and glide. Such a simple control strategy is also advantageous to be implemented in an unmanned system. In the presence of wind, the manoeuvre needs to be moderately adapted. The clear superiority of the saw-tooth flight with respect to range is independent of the type of engine installed. In the section on bounding flight, the thorough investigations are directed to a wide speed range of forward flight. Here, first, important information on the observed flights of birds is presented. The flapping ratio (percentage of time spent flapping during the cycle), for example, reduces with an increase in flight speed while the measured duration of the bounding phase increases with an increase in flight speed. During the bound phase, the bird’s body with the wings fully attached still can produce a certain amount of lift which is actively controlled by the bird. Using this information, a realistic model of the bird and the underlying equations is derived. The drag coefficient is modelled by a quadratic polar. Two


factors for the zero drag coefficients and the induced drag coefficients are introduced relating the drag polar during flapping to the drag polar during bounding. The cost function to be minimised is the energy consumed per distance travelled for a given average cycle velocity. In the first step, an analytical investigation of the manoeuvre is conducted. A constant flight velocity is assumed and the transition phases where the wings are extended or retracted are omitted. In order to derive results of generally valid nature, all underlying equations are normalised. Thereby also a connection to small unmanned aerial systems is established. In the analytical optimisation, the influence of the drag coefficient factors in all the equations is clearly shown. As result of the analytical optimisation, the energetic superiority of the bounding flight for higher flight velocities becomes evident and it is advantageous for the bird to apply a certain amount of lift during the bound phase. An important result is that the optimal lift coefficients during the flapping phase and also during the bounding phase are independent of the flight velocity. The superiority of the bounding flight manoeuvre starts soon above the velocity for minimum drag. As in case of the observed bird flights, the flapping ratio diminishes with an increase in flight velocity. This also explains the only drawback – namely a strong increase in power required with a higher flight velocity. In nature, the possibility of birds to make progress against the wind is of high importance, e.g. during migration. Therefore, the influence of a headwind is analytically investigated. The energetic superiority of the bounding flight manoeuvre is even increased in the presence of a headwind. The bounding flight now is already superior to a steady state horizontal flight for flight velocities far lower than that for minimum drag. Also the overall energetic minimum of the bounding flight lies far below that of steady state flight – and the bounding flight minimum is located at a far higher kinematic velocity. At the flight presented, for example, the kinematic velocity is 87% (with lift in bounding phase) higher than in optimal steady state horizontal flight, and at the same time the overall cost is 23% lower. The model of the siskin is implemented in Matlab and its Jacobian and Hessian matrices are automatically generated. The analytically derived results are used as initial guess for the numeric optimisations including all the four phases. The results of the numeric optimisations support all assumptions made for the analytical optimisations proving that the physical background has been adequately reflected. The numeric optimisations show the decrease in flapping duration and the increase in bounding duration with flight speed. The flapping ratio decreases with an increase in flight speed. By applying the bounding flight mode, the savings as compared to steady state horizontal flight increase with higher flight speeds. At medium flight speeds – which are definitely possible concerning the available power – already savings of around 30% are reached. Then, the variability of a single cycle is analysed. At fixed average velocity the cycle distance is increased from 5m to 15m. It is shown that in normalised time the structure of the state trajectories is preserved and that also the controls namely the lift coefficient and the power set are nearly unaffected. Although the distance is tripled, the cost function only increases by around 2%. And even if the cycle length is set to 22m, the variation in cost function is only around 7%. The bounding flight manoeuvre thus shows a great flexibility which may be a bit problematic for observations in nature but is very favourable concerning technical implementations. One of the biggest advantages for technical implementations is the quasi-independence of the lift coefficient values from both, flight velocity and cycle distance.


In the last section, the trajectories for one, two, and three cycles are optimised over distances from 5m to 25m and the respective savings are shown. In order to maximise savings, the correct switching between the numbers of cycles flown is of high importance. The bilevel optimisation is used to determine the distance at which the switching from one to two cycles must occur, in order to minimise cost. The initial guess for the upper level parameter optimisation problem (definition of the switching point by setting the cost functions of the lower level problems equal) is a result of the post optimal sensitivity analysis of the two lower level optimal control problems (one and two optimal bounding flight cycles). As initial guess, the intersection of the second order sensitivities of the respective cost function with regard to the distance travelled is determined. For the bilevel optimisation, the solver fully exploits the knowledge provided by the Jacobian and the Hessian matrices and thus, within very short calculation time, the required switching point is precisely delivered. Therefore, a bilevel optimisation is efficient means to perform a solution structure analysis.

266

Appendix

A. Modelling of the Earth (flat – round – rotating) Translational Dynamics For endo-atmospheric flight systems, when modelling the shape of the surface of the earth and deciding whether or not to include its rotation, the kinematic velocity flown is the determining factor apart from accurately calculated long distance flights requiring WGS 84 position representation. Starting from (2-50) with the quasi-steady mass assumption (2-79) and a stationary horizontal flight to make an analytic analysis accessible:

( )( )

( ) ( ) ( ) ( ) ( ) ( ) ( )[ ][ ][ E

EIE

EIE

OEEO

KO

EOEO

KO

IEKO

EO

KK

II

KK

T

m

rωωMVωVωM

V

VF

0

(

××⋅+×+×⋅⋅

+

==∑

2

(A-1)

When presuming ( ) 0V =

EO

KK , the last line of (A-1) directly gives the difference between the

non-inertial and the true (inertial) acceleration. The single terms of the last line of (A-1) are used to analyse this difference. Looking at the last term of (A-1) first, which represents the position dependent centrifugal acceleration due to the rotation of the earth:

( ) ( ) ( )[ ][ ]

( )( )

( )[ ]

⋅+−⋅⋅⋅+⋅⋅+

×

⋅×

⋅

⋅

−⋅−⋅−−

⋅−⋅−

=××⋅

EE

IE

E

IE

OE

E

EIE

EIE

OE

heNhNhN

ϕλϕλϕ

ωω

ϕλϕλϕλλ

ϕλϕλϕ

ϕ

ϕ

ϕ

sin1sincoscoscos

100

100

sinsincoscoscos0cossin

cossinsincossin

2

rωωM

(A-2)

Modelling of the Earth (flat – round – rotating) 267

For the centrifugal acceleration follows:

( ) ( ) ( )[ ][ ] ( ) ( )O

IEE

E

IEE

IEOE hN

⋅⋅+⋅=××⋅

ϕ

ϕϕω ϕ

2

2

cos0cossin

rωωM (A-3)

The centrifugal acceleration reaches its maximum at the equator and vanishes at the poles – see also Figure 2.2-4. At the equator on the surface of the earth using the WGS 84 definition from the National Imagery and Mapping Agency NIMA (2000) the centrifugal acceleration is:

( ) ( ) ( )[ ][ ] 22-

0

100

103.39157 sm

O

E

EIE

EIE

OE

⋅⋅=××⋅

=ϕrωωM

(A-4)

The second term of the last line of (A-1), representing the acceleration due to the transport

rate, EOω , is:

( ) ( )E

ONE

N

EE

O

E

N

O

EO

KO

EO

VVV

VVV

⋅+⋅⋅⋅⋅−

⋅⋅=

×

⋅−−⋅

=×ϕϕλ

ϕλϕλ

ϕλϕ

ϕλ

cossin

sin

0sin

cosVω (A-5)

This gives with (2-24):

( ) ( )

E

O

NE

NE

E

EO

KO

EO

hMV

hNV

hNVVhN

V

++

+

⋅+

⋅−

⋅+

=×

ϕϕ

ϕ

ϕ

ϕ

ϕ

22

2

tan

tan

Vω

(A-6)

The above equation is not defined on the poles. To analyse its influence, it is evaluated for a horizontal flight above and parallel to the equator neglecting the altitude term, h :

27-2 1101.5678559 V

mNV

⋅⋅=ϕ

(A-7)

What means that for very high flight velocities, the acceleration due to the transport rate may reach the order of one or even above. The first term of the last line of (A-1), representing the Coriolis acceleration is:

( ) ( )E

OE

N

EIE

E

O

E

N

O

IEEO

KO

IE

VV

VVV

⋅⋅−

⋅⋅⋅=

×

−⋅⋅=×⋅

ϕϕ

ϕω

ϕ

ϕω

cossin

sin2

0sin0

cos22 Vω

(A-8)

268 Appendix

Its order of magnitude is:

Vs

VIE ⋅⋅=⋅⋅110 1.458423 2 4-ω (A-9)

See also Goldstein et al. (2001) in the paragraph 4.10 The Coriolis Effect and Stevens and Lewis (2003) in Significance of the Earth-Rotation Terms, and Etkin (2005) in Discussion of the System of Equations. Equation (A-1) may therefore be written as:

( )

( ) ( )

⋅⋅+⋅+

++

+

⋅+⋅

−

⋅+

+

⋅⋅−

⋅⋅⋅

⋅−= ∑

O

IE

E

O

NE

NE

E

E

OE

N

EIE

KO

TEO

KK

hN

hMV

hNV

hNVVhN

V

VV

V

m

ϕ

ϕϕωϕ

ϕ

ϕϕ

ϕω ϕ

ϕϕ

ϕ

ϕ

2

2

22

2

cos0cossin

tan

tan

cossin

sin2

MF

V

(A-10)

The Coriolis acceleration which scales linearly with the flight velocity is larger than the acceleration due to the transport rate for velocities smaller than around 930m/s where values of approximately 0.14m/s2 are reached. At a flight velocity of 100m/s this

acceleration is less than 2‰ of standard acceleration due to gravity at sea level, sg , from

Table 2.3-1. The results are presented in Figure A-1.

Figure A-1 Centrifugal, Coriolis, and acceleration due to transport rate in dependence on kinematic velocity

The Coriolis acceleration reaches 1‰ of sg at a velocity of around 68m/s while the

acceleration due to the transport rate reaches 1‰ of sg at a velocity of around 251m/s.

0 10 100 1000 1000010-8

10-6

10-4

10-3

10-2

10-1

100

102

Kinematic Flight Velocity [m/s]

Acc

eler

atio

n [g

s]

Max CentrifugalCoriolisTransport

Modelling of the Earth (flat – round – rotating) 269

The Coriolis acceleration reaches 1% of sg at a velocity of around 673m/s while the

acceleration due to the transport rate reaches 1% of sg at a velocity of around 791m/s. The

acceleration due to the transport rate reaches 10% of sg at a velocity of around 2501m/s

while the Coriolis acceleration reaches 10% of sg at a velocity of around 6725m/s. At a

velocity of 7909m/s the acceleration due to the transport rate reaches 100% of sg - this is

the first cosmic velocity for the earth (the velocity corresponding to the circular orbit). Rotational Dynamics In this part, the difference between the full rotational dynamics equation (2-86) and the simplified equation (2-87) for a flat non-rotating earth is analysed. Taking an aircraft performing a standard turn of 360° in two minutes time, gives with (2-61):

( )s

rad10 5.2359877 1202 2-⋅== sBOB πω (A-11)

Even this very small rotational rate is more than the factor of 700(!) higher than the earth’s rotational rate (see Table 2.1-1) of:

( )s

radB

IE 5-10 7.292115 ⋅=ω (A-12)

or the transport rate (see (A-7) ) which is dependent on the kinematic velocity:

( ) VmN

VB

EO ⋅⋅==110 1.5678559 7-

ϕ

ω (A-13)

So, even at high velocities the terms corresponding to the transport rate and the rotational rate of the earth may be neglected in the rotational dynamics. The corresponding control surface deflection could hardly be measured.

270 Appendix

B. Quaternion Representation of Attitude By using the four unit quaternions instead of the three Euler angles, the singularity in the attitude equations of motion (2-62) may be avoided. The system, however, becomes over-determined. Therefore an additional constraint has to be introduced: the sum of the squares of the quaternions has to equal one:

123

22

21

20 =+++ qqqq (B-1)

The first order time derivative of this constraint gives:

033221100 =+++ qqqqqqqq (B-2)

The transformation from Euler angles to quaternions may be found in Shoemake (1985):

2sin

2sin

2cos

2cos

2cos

2sin

2sin

2cos

2sin

2cos

2sin

2cos

2cos

2sin

2sin

2sin

2cos

2cos

2sin

2sin

2sin

2cos

2cos

2cos

3

2

1

0

Φ⋅

Θ⋅

Ψ−

Φ⋅

Θ⋅

Ψ=

Φ⋅

Θ⋅

Ψ+

Φ⋅

Θ⋅

Ψ=

Φ⋅

Θ⋅

Ψ−

Φ⋅

Θ⋅

Ψ=

Φ⋅

Θ⋅

Ψ+

Φ⋅

Θ⋅

Ψ=

q

q

q

q

(B-3)

The transformation matrix from the NED-frame to the body-fixed frame in quaternions reads (Stevens and Lewis, 2003):

( )( ) ( )

( ) ( )( ) ( )

BOqqqqqqqqqqqq


qqqq

+−−−⋅+⋅+⋅−+−−⋅−⋅+⋅−−+

=23

22

21

2010322031

103223

22

21

203021

2031302123

22

21

20

3210B0

222222

,,,M

(B-4)

Thereby the backward transformation from quaternions to Euler angles is straight forward. By comparing elements of the first line and last column of (B-4) to their pendants in (2-59) results:

( )

−−+

+⋅=Ψ 2

322

21

20

30212arctanqqqq

qqqq (B-5)

( ) ( )[ ]20312arcsin qqqq −⋅−=Θ (B-6)

( )

+−−

+⋅=Φ 2

322

21

20

10322arctanqqqq

qqqq (B-7)

Quaternion Representation of Attitude 271

Equation (B-4) allows by means of the strapdown equation the representation of the angular rates in quaternions (Stevens and Lewis, 2003):

( )

⋅

−−−−

−−⋅=

3

2

1

0

0123

1032

2301

2

qqqq

qqqqqqqqqqqq

BOB

ω (B-8)

By inserting the above constraint (B-2) in the last row, the above matrix easily becomes invertible:

⋅

−−−−

−−

⋅=

3

2

1

0

3210

0123

1032

2301

2

0 qqqq

qqqqqqqqqqqqqqqq

rqp

OBK

OBK

OBK

(B-9)

Performing the inversion gives the first order time derivative of the quaternions as a function of the body angular rates:

⋅

−−

−−−−

⋅=

021

3012

2103

1230

0321

3

2

1

0

OBK

OBK

OBK

rqp

qqqqqqqqqqqqqqqq

qqqq

(B-10)

Or in short (Holzapfel et al., 2006):

B

OBK

OBK

OBK

rqp

qqqqqq

qqqqqq

qqqq

⋅

−−

−−−−

⋅=

012

103

230

321

3

2

1

0

21

(B-11)

In plain quaternions the attitude equations read (Stevens and Lewis, 2003):

⋅

−−−−

−−⋅−=

3

2

1

0

3

2

1

0

00

00

21

qqqq

pqrprq

qrprqp

qqqq

OBK

OBK

OBK

OBK

OBK

OBK

OBK

OBK

OBK

OBK

OBK

OBK

(B-12)

In Rolfe and Staples (2008) it is suggested to meet the constraint (B-2) by calculating the accuracy to which (B-1) is fulfilled:

( )23

22

21

201 qqqq +++−=λ (B-13)

272 Appendix

and replace the exact zero in (B-10) by λ⋅k2 where k has to be:

hk 1

≤ (B-14)

With h being the integration stepsize. Equally the exact zero in (B-12) may be replaced by λ⋅k to establish the Corbett-Wright orthogonality control. It may, dependent on the integration scheme and stepsize, be better to renormalize the quaternions by dividing through the magnitude of the vector. The renormalized quaternion vector is found from:

( ) ( )( )

( )23

22

21

20 qqqq

r+++

==q

qqq (B-15)

For a comparison see Phillips (2009) in the chapter 11.10 Numerical Integration of the Quaternion Formulation.

Vector Field Classifications 273

C. Vector Field Classifications In this paragraph some short information about vector fields and their classification as conservative, solenoidal (incompressible), and harmonic are provided. In the following the definitions and theorems as stated in Barack (2004) are given in a slightly modified way:

Let ( )zyx ,,φ be a differentiable function in a domain D . The vector field ( )zyx ,,F

defined

by:

( )

∂∂∂∂∂∂

=∇=

z

y

xzyx

φ

φ

φ

φ ,,F

(C-1)

is called a conservative vector field in D . The function φ is called a potential for F

in D .

Sometimes a minus sign is used in the above definition as a matter of convention.

The curl of a differentiable vector field, F

, is the vector field F

×∇ . The curl can also be expressed using the potential φ :

∂∂

∂∂

−

∂∂

∂∂

∂∂

∂∂

−

∂∂

∂∂

∂∂

∂∂

−

∂∂

∂∂

=

∂∂

−∂∂

∂∂

−∂∂

∂∂

−∂∂

=×∇

xyyx

zxxz

yzzy

yF

xF

xF

zF

zF

yF

φφ

φφ

φφ

12

31

23

F

(C-2)

The curl thus contains all the second order mixed derivatives of the potential φ . If the curl of

a vector field is zero, then F

is said to be irrotational. One important property of a conservative vector field is that its curl is zero:

F

conservative 0F

=×∇⇒ (C-3)

If D is an open connected domain and F

is a smooth vector field defined in D , the following statements are equivalent:

a.) F

is conservative in D

b.) 0=⋅∫csF

d for every piecewise smooth closed curve in D

c.) Given any two points 1P and 2P in D , the line integral of F

, ∫ ⋅P

dsF , has the same

value for all piecewise smooth curves (rectifiable paths P ) in D starting at 1P and

ending at 2P .

274 Appendix

The divergence of a differentiable vector field, F

, is the scalar function F

⋅∇ . Also the divergence can be expressed using the potential φ :

2

2

2

2

2

2321

zyxzF

yF

xF

∂∂

+∂∂

+∂∂

=∂∂

+∂∂

+∂∂

=⋅∇φφφF

(C-4)

The divergence thus contains all the second order unmixed derivatives of the potential φ . If

the divergence of a vector field is zero, then F

is said to be divergence free or solenoidal (in the study of electricity and magnetism) or incompressible (in hydrodynamics). As stated for example in Eisenmann (2005):

φφ ∆=∇⋅∇ (C-5)

is called the Laplacian of the potential φ and the equation:

0=∆φ (C-6)

is called Laplace’s equation. The potential φ is called harmonic if 0=∆φ .

In Eisenmann (2005) it is proven, that the scalar potential φ of every solenoidal,

conservative vector field F

must be harmonic. In Ivanov (1994) also the simultaneously

irrotational and solenoidal vector field F

is called harmonic. All of the elements necessary to compute the curl (C-2) and the divergence (C-4) are

elements of the Jacobian of the vector field F

, FJ , which is the Hessian of the potential φ :

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

=

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

=

2

2

2

2

2

2

333

222

111

zzyzx

yzyyx

xzxyx

zF

yF

xF

zF

yF

xF

zF

yF

xF

φφφ

φφφ

φφφ

FJ (C-7)

This gives for a conservative vector field F

using (C-3) that the Jacobian (C-7) has to be symmetric:

F conservative [ ]TFF JJ =⇒ (C-8)

For a solenoidal vector field F

using (C-4) and (C-6) the trace of the Jacobian (C-7) has to be zero:

F solenoidal ( ) 0 tr =⇒ FJ (C-9)

If both (C-8) and (C-9) hold, the scalar potential φ is harmonic.

Vector Field Classifications 275

Example Relating to the potential of gravitation of a point mass (2-93) the simplest harmonic function (Hofmann-Wellenhof and Moritz, 2006) is the reciprocal distance:

222

11zyxl ++

= (C-10)

The Jacobian of the associated vector field reads ( )0x ≠ :

−−

−⋅=

22

22

22

5

333333333

1

lzyzxzyzlyxyxzxylx

lFJ (C-11)

The above matrix is symmetric and its trace is zero, thus (C-10) is harmonic. In case of gravitation the 1 in the numerator of (C-10) may be replaced by 21 mm ⋅⋅ to give the

cumulative attraction of the point masses 1m and 2m separated by the distance, l , with

being the Newtonian gravitational constant. To find the electrostatic force of interaction between two point charges 1q and 2q , the 1 in the numerator of (C-10) may be replaced by

21 qqke ⋅⋅− with the Coulomb's constant ek .

The determinant of (C-11) is ( ) 9/2det l=FJ . In case of gravity or electrostatic force it has to

be multiplied by the respective factor cubed.

276 Appendix

D. Matrix Differential Calculus The vec operator (vectorisation) stacks the columns of a matrix A one after the other – see also Magnus and Neudecker (1980):

[ ]( ) [ ]( )[ ]1

1

1 ,,

×

×

==

mnn

nnm vecveca

aaaA (D-1)

The further basic calculations presented in this paragraph refer to Magnus and Neudecker (1985) and the Kronecker product definition may be found in Magnus and Neudecker (1979). Given are the matrices M and A as well as the vector b :

nm×∈ R M (D-2)

pn×∈ R A (D-3)

1 ×∈ sRb (D-4)

The Kronecker product BA ⊗ for a matrix A (D-3) and a matrix ts×∈ R B is defined as the

[ ]ptns × matrix:

( )BBA ⋅=⊗ ija (D-5)

The first order derivative of the matrix product AM ⋅ to the vector b using theorem 9 of Magnus and Neudecker (1985) is:

( ) [ ] ( ) [ ] ( )TpTm

TT

vecvecvecb

AMIb

MIAb

AM∂

∂⋅⊗+

∂∂

⋅⊗=∂

⋅∂ (D-6)

If the matrix A has only one column, the vector a results:

1 ×∈ nRa (D-7)

Then the first order derivative of the matrix-vector product aM ⋅ to the vector b using theorem 9 of Magnus and Neudecker (1985) is:

( ) [ ] ( )TTm

TT

vecvecbaM

bMIa

baM

∂∂

⋅+∂

∂⋅⊗=

∂⋅∂

(D-8)

Matrix Differential Calculus 277

The second order derivative of the matrix-vector product aM ⋅ to the vector b using additionally theorem 11 of Magnus and Neudecker (1985) for the Kronecker product is:

( )

[ ] ( )

( )( ) ( )( )

( ) ( ) [ ][ ]( )

( ) [ ]

( ) [ ] ( )[ ][ ]

[ ][ ]( )

( ) [ ] T

T

sTm

T

T

T

T

mT

s

Tmmmnm

T

T

T

T

sTm

T

T

T

T

mT

sTm

T

m

T

T

TT

TTTmT

T

TT

vecvec

vecvec

vecvec

vecvec

vecvecvecvec

vecvec

vecvecvec

vecvec

bba

MIb

MIba

bb

M

IaI

baIIIII

bM

bba

MIb

MIba

bb

M

IaIb

IaIb

M

bAM

bMA

baM

bbMIa

b

baM

b

∂

∂∂

∂⋅⊗+

∂∂

⋅

⊗

∂∂

+∂

∂∂

∂⋅⊗⊗

+∂∂

⋅⋅⊗⊗⋅

⊗

∂∂

=∂

∂∂

∂⋅⊗+

∂∂

⋅

⊗

∂∂

+∂

∂∂

∂⋅⊗⊗+

∂⊗∂

⋅

⊗

∂∂

=∂

⋅∂+

∂⋅∂

=

∂∂

⋅∂

∂+

∂∂

⋅⊗∂

∂

=

∂⋅∂

∂∂

ˆˆ

(D-9)

with the abbreviations:

[ ] mnmm

T ×∈⊗= R ˆ IaA (D-10)

( ) smnT

vec ×∈∂

∂= R ˆ

bMM (D-11)

snT

×∈∂∂

= R baA

(D-12)

Moreover, in (D-9) it has been used, that a commutation matrix of type:

mmm IKK == 11 (D-13)

is the unity matrix – see Magnus and Neudecker (1979).

278 Appendix

E. Constraints and Optimisation Parameter Vector

In (3-30) the discretised constraints vector, discC , is included. Subsequently it is given in

more detail exemplarily for three phases. In SNOPT, the inactive inequality constraints are not excluded and therefore appear in the vector:

( )( )( )( )( )( )( )( )( )( )

( )( )

( )( )( )( )( )( )( )( )

( )( )

( )( )( )( )( )( )( )( )

≤≤==

=≤≤==

=≤≤==≤

=

EndsConstraintBoundary Final,

sConstraintPath ,sConstraintPath ,

Residuals,III Phase

sConstraintConnect Phase,sConstraintBoundary Final,


Residuals,II Phase

sConstraintConnect Phase,sConstraintBoundary Final,


Residuals,sConstraintBoundary Initial,

I Phase

0zzxψ0zzxC0zzxC0zzxC

0zzxC0zzxψ0zzxC0zzxC0zzxC

0zzxC0zzxψ0zzxC0zzxC0zzxC0zzxψ

C

fin

ineq

eq

def

connect

fin

ineq

eq

def

connect

fin

ineq

eq

def

ini

disc

(E-1)

In (E-1) the residuals and the path constraints are implemented in strict chronological order in τ for 10 ≤≤ τ . Thereby a mixing of the above order: residuals, equality path constraints

followed by inequality path constraints may occur in discC .

The optimisation parameter vector, z , from (3-63) in more detail is given in (E-2). In case of a phase of variable length, the first element of the parameter vector, p , contains the phase

duration.

Constraints and Optimisation Parameter Vector 279

phaseeach for , ... ,0 and , ... ,1

III Phase

II Phase

I Phase

,

,

,

cMS

i

jMS

ini

i

jMS

ini

i

jMS

ini

ninj ==

=

uxxp

uxxp

uxxp

z

(E-2)

Here, the multiple shooting nodes and the control nodes are to be implemented in strict chronological order in τ for 10 ≤≤ τ where in this work all multiple shooting nodes are also a control node. Again a mixing of multiple shooting nodes and control odes may thus occur in the optimisation parameter vector, z .

280 Appendix

F. Jacobian and Hessian Matrix The Jacobian matrix, J , for a non- autonomous system as described in Chapter 3 is:

( )

( ) ( )[ ]1111

11

1

11

1

11

1

1

111

11

1

11

1

11

1

1

+++×+

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

=

puxyx

y

p

yy

u

yy

x

yy

pux

x

p

xx

u

xx

x

xx

pux

nnnnn

n

n

nn

n

nn

n

nn

nnn

n

n

nn

n

nn

n

nn

nnn

t

y

p

y

p

y

u

y

u

y

x

y

x

y

ty

py

py

uy

uy

xy

xy

tx

px

px

ux

ux

xx

xx

tx

px

px

ux

ux

xx

xx

t

J

(F-1)

The Hessian matrix contains the second order derivatives of the outputs ( )yx, with respect

to the inputs. It contains ( )yx nn + layers, each has the dimension of ( ) ( )[ ]11 +++×+++ puxpux nnnnnn .

Exemplarily, the 1st layer is shown:

( )

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

=

tx

tpx

tpx

tux

tux

txx

txx

t

tx

ppx

ppx

pux

pux

pxx

pxx

p

tx

ppx

ppx

pux

pux

pxx

pxx

p

tx

upx

upx

uux

uux

uxx

uxx

u

tx

upx

upx

uux

uux

uxx

uxx

u

tx

xpx

xpx

xux

xux

xxx

xxx

x

tx

xpx

xpx

xux

xux

xxx

xxx

x

t

pux

ppppuppxpp

pux

upuuuuuxuu

pux

xpxxuxxxxx

pux

nnn

nnnnnnnnnn

nnn

nnnnnnnnnn

nnn

nnnnnnnnnn

nnn

11

1

11

1

11

1

1

11

1

11

1

11

1

1

1

1

1

11

1

1

1

11

1

1

1

11

1

1

11

1

11

1

11

1

1

1

1

1

11

1

1

1

11

1

1

1

11

1

1

11

1

11

1

11

1

1

1

1

1

11

1

1

1

11

1

1

1

11

1

1

H

(F-2)

The following equations are based on Magnus and Neudecker (1980) to enable calculations using the symmetry of the Hessian.

Jacobian and Hessian Matrix 281

In case of a matrix A being a square [ ]nn × matrix, ( )A v denotes the ( )[ ] 112/1 ×+⋅⋅ nn

vector that is obtained from ( )A vec by eliminating all supra-diagonal elements of A , thus

all elements ijA for which ji ≥ . The duplication matrix, D , gives for a symmetric matrix A :

( ) ( )AAD vecv =⋅ (F-3)

Whilst the elimination matrix, L , gives for every square [ ]nn × matrix A :

( ) ( )AAL vvec =⋅ (F-4)

282 Appendix

G. Solar Cells and Maximum Power Point Trackers In the following some background information is provided on solar cells and the maximum power point tracker (MPPT). An overview of the materials and methods used for fabricating photovoltaic solar cells may for example be found in Miles et al. (2005). There, solar cells based on the use of silicon (in the monocrystalline, multicrystalline, amorphous, and micro-crystalline forms), the III to V compounds (e.g. gallium arsenide (GaAs), indium phosphide (InP), and gallium antimonide (GaSb)), the polycrystalline compounds (e.g. cadmium telluride (CdTe), copper gallium indium diselenide and copper indium disulphide (CuInS2)), and organic materials (e.g. dyes, polymers, and fullerenes) are investigated and compared. In case of silicon solar cells the best modules made using multicrystalline silicon generally have efficiencies which are 2 to 3% less than those of monocrystalline silicon but cost approximately 20% less to produce. Monocristalline silicon solar cells of PERL (passivated emitter rear locally diffused) design reach efficiencies of up to 24%. Moreover, it is concluded that the III to V compounds are excellent photovoltaic materials but are extremely expensive to produce and contain either toxic or not very abundant elements. They also offer advantages for space applications, improved power/weight ratios and better radiation resistances compared to silicon devices. The National Center for Photovoltaics (NCPV) at the National Renewable Energy Laboratory (NREL) of the U.S. Department of Energy (DOE) maintains a plot of compiled values of highest confirmed conversion efficiencies for research cells, from 1976 to the present – see Figure G-1. The most recent world record for each technology is highlighted along the right edge in a flag that contains the efficiency and the symbol of the technology. The company or group that fabricated the device for each most-recent record is bolded on the plot (National Renewable Energy Laboratory, 2014). The Soitec (4-J, 297x) is the most efficient solar cell shown in Figure G-1 and is a joint research effort of The Fraunhofer Institute for Solar Energy Systems ISE, Soitec, CEA-Leti, and the Helmholtz Center Berlin. They jointly announced in September 2013 having achieved a new world record for the conversion of sunlight into electricity using a solar cell structure with four solar subcells (Fraunhofer Institute for Solar Energy Systems ISE, 2013); 4-J abbreviates the four junctions. The new record efficiency of 44.7% was measured at a concentration of 297 suns, therefore the abbreviation 297x. 44.7% of the solar spectrum's energy, from ultraviolet through to the infrared, is converted into electrical energy. These solar cells are used in concentrator photovoltaics (CPV), a technology which achieves more than twice the efficiency of conventional photovoltaics (PV) power plants in sun-rich locations. The terrestrial use of so-called III-V multi-junction solar cells, which originally came from space technology, has prevailed to realize highest efficiencies for the conversion of sunlight to electricity. In this multi-junction solar cell, several cells made out of different III-V semiconductor materials are stacked on top of each other. The single subcells absorb different wavelength ranges of the solar spectrum.

Solar Cells and Maximum Power Point Trackers 283

Figure G-1 Best research cell efficiencies (National Renewable Energy Laboratory, 2014)

The advantages of a multi-junction cell are demonstrated by Figure G-2 which is taken from Yastrebova (2007). On the left it shows the solar energy that can be theoretically used by single- and (on the right) by III-V triple-junction cells if the air mass (AM) is 1.5 – compare also (2-277) and Figure 2.4-21.

Figure G-2 Parts of the solar spectrum that can, in theory, be used by solar cells (Yastrebova, 2007)

Referring to the above presented chart, Weber (2013) states that the theoretical efficiency limit for a silicon solar cell is 28% whilst the theoretical efficiency limit for a triple-junction concentrator cell is 61%. The current to voltage (IV) curve of a solar cell has a characteristic shape as well as the resulting power curve (product of voltage times current). The IV curve results from basic physical considerations as for example presented in chapter 4 Solar Cell Device Physics of Fraas and Partain (2010), in Green (1981), or in Part II: Solar Cells of Markvart and Castaner (2003).

Spec

tral I

rradi

ance

[W/(m

2 ·µm

)]

Spec

tral I

rradi

ance

[W/(m

2 ·µm

)]

Wavelength λ [nm] Wavelength λ [nm]

500 1000 1500 2000 2500 500 1000 1500 2000 25000

200

400

600

800

1000

1200

1400

1600

0

200

400

600

800

1000

1200

1400

1600

Thermalization losses

Energy that can beused by a Si solar cell

Transmission losses

AM1.5 spectrumSi (1.12 eV)

AM1.5 spectrumGaInP (1.70 eV)GaInAs (1.18 eV)Ge (0.67 eV)

284 Appendix

Figure G-3 Current to voltage curves of two different solar cells.

Left: Soitec (Fraunhofer Institute for Solar Energy Systems ISE, 2013). Right: Silicon RWE-S-32 (Noth, 2008)

In Figure G-3 the measured current to voltage curve for the Soitec (4-J, 297x) – see description above – is depicted on the left (Fraunhofer Institute for Solar Energy Systems ISE, 2013). When the cell is short circuited, the voltage is zero and the current is ISC. In case of the unconnected cell, the current is zero and the voltage is the open circuit voltage, VOC. The right picture of Figure G-3 is taken from Noth (2008) and shows the characteristics of a Silicon RWE-S-32 cell including the power receivable. The working point is ideally set to the maximum power point (MPP) where the cell should be operated. At the maximum power

point, the power is MPPMPPMPP VIPP ⋅==max . The maximum power is represented by the

maximum rectangle which may be fitted within the current to voltage diagram.

Figure G-4 Variation of the current to voltage curve of a Silicon RWE-S-32 cell with irradiance (left) and

temperature (right); (Noth, 2008)

In Figure G-4 which is taken from Noth (2008), the variation of the current to voltage curve – and thus the variation of power receivable – of a Silicon RWE-S-32 cell in dependence on irradiance and temperature is depicted. As may be identified in the left chart of Figure G-4, the current scales nearly linearly with the irradiance while the dependency of the voltage on irradiance is rather limited. On the right chart of Figure G-4 it may be seen that a variation of temperature has little effect on the current but a comparatively strong effect on the voltage. The single cells are connected to modules. If they are connected in parallel, the current is increased, and if they are connected in series, the voltage is increased. The principle shape of the current to voltage curve is preserved.

VMPP, IMPP

MPP

Solar Cells and Maximum Power Point Trackers 285

The task of the maximum power point tracker (MPPT) is to operate the solar cells at a current to voltage ratio where maximum power is delivered to the system. Since the solar cells in operation are facing continuously changing environmental conditions (irradiance, temperature), the working point has to be permanently adapted. The MPPT is basically a DC/DC converter featuring an adjustable gain between the input voltage (from the solar cells) and the output voltage. In their comparative study of maximum power point tracking algorithms, Hohm and Ropp (2003) point out that the perturb-and-observe (P&O) method, claimed by many in the literature to be inferior to others, continues to be by far the most widely used method in commercial PV MPPTs. They show that the P&O method, when properly optimized, can have MPPT efficiencies well in excess of 97%, and is thus highly competitive against other MPPT algorithms. For a general classification of MPPT algorithms see Salas et al. (2006) where it is distinguished between indirect methods (‘quasi seeks’), which are estimating the maximum power from either irradiance or temperature measures together with evaluating system data and direct methods (‘true seeking methods’). Direct methods offer the advantage that they need no large database but may obtain the actual maximum power from the measures of the PV generator’s voltage and current. In Houssamo et al. (2010) the experimental comparison for the same given set of conditions (irradiance, cell temperature, and running conditions) of two MPPT algorithms is conducted. The incremental conductance (INC) method uses the fact that the tangent of the power curve is zero at the MPP. The measurements of extracted powers by both algorithms, P&O and INC, implemented into an experimental system composed of two PV identical systems and a DC/DC converter, show that the choice of the couple incremental step - calculation step, is very important. When the value of this couple is well chosen, the differences in power and response time between both the algorithms become negligible.

286 Appendix

H. The Propulsion System of the Antares 20E Motor Glider In the following, some short information on the propulsion system of the Antares 20E motor glider built by Lange Aviation is given. All of the information provided in this paragraph is published by Lange (2014b).

Figure H-1 The propulsion system of the Antares 20E (Lange, 2014b)

In Figure H-1 the propulsion system of the Antares 20E motor glider with the brushless 42 kW external rotor electric motor in the centre is depicted. As may be seen in the figure, the propeller has no variable pitch. A variable pitch propeller would offer the ability to maintain an optimal angle of attack on the blades as operating conditions change. The efficiency presented later on could therefore be improved by a variable pitch propeller. The Antares 20E has a brushless fixed-shaft electric motor running on DC/DC current named EM42. It delivers maximum torque over a wide RPM range (revolutions per minute) – see Figure H-2. The motor has a total efficiency of 90% and a maximum torque of 216 Nm.

The Propulsion System of the Antares 20E Motor Glider 287

Figure H-2 Torque versus rotational speed of EM42 (Lange, 2014b)

The large propeller diameter of 2m results in a high propeller efficiency. The electric motor is unaffected by the air density, and leaves the propeller as the only part of the propulsion system affected by altitude. At 3000m (9840ft.), the propeller has a maximum efficiency loss of 4% compared to sea-level. The higher the aircraft gets, the faster the propeller has to turn in order to deliver the required thrust. At very high altitudes the maximum thrust available will be limited by the max RPM of the motor.

Figure H-3 Propeller efficiency in dependence on power and altitude (Lange, 2014b)

Rotational Speed [1 / min]100 200 300 400 5000 600 700 800 900 1000 1200 1300 1400 1500 16001100 1700

100

200

50

150

250

0

Torq

ue [N

m]

5 10 15 20 250 30 35 40 45

Engine Power [kW]50

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Prop

elle

r Effi

cien

cy η

[ ]

Altitude: 0mAltitude: 1000mAltitude: 2000mAltitude: 3000m



288

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SACHS, G., LENZ, J. & HOLZAPFEL, F. 2009e. Unlimited Endurance Performance of Solar UAVs with Minimal or Zero Electrical Energy Storage. AIAA Guidance, Navigation, and Control Conference. American Institute of Aeronautics and Astronautics.

SACHS, G., LENZ, J. & HOLZAPFEL, F. 2010a. Maximum Range Performance of Electric Motor Gliders with Retractable Engine. XXX OSTIV Congress, Szeged, Hungary.

SACHS, G., LENZ, J. & HOLZAPFEL, F. 2010b. Periodic Optimal Flight of Solar Aircraft with Unlimited Endurance Performance. Applied Mathematical Sciences, 4, 3761.

SACHS, G., LENZ, J. & HOLZAPFEL, F. 2010c. Range Maximization by Saw-Tooth Mode Optimization for Motor Gliders with Retractable Engines. Technical Soaring, 34, 68-74.

SACHS, G., LENZ, J. & HOLZAPFEL, F. 2010d. Wind Effects on Maximum-Range Sawtooth Flight. XXX OSTIV Congress, Szeged, Hungary.

SACHS, G., LENZ, J. & HOLZAPFEL, F. 2012. Wind Effects on Periodic Optimal Flight of Motor Gliders with Retractable Engine. In: ZOBORY, I., ed. 12th MINI Conference on Vehicle System Dynamics, Identification and Anomalies, 8 - 10 November 2010e, Budapest, Hungary. 321-332.

SACHS, G., LENZ, J. & HOLZAPFEL, F. 2010f. Wind Effects on the Maximum Range Performance of Motor Gliders with Retractable Engine. AIAA Guidance, Navigation, and Control Conference. American Institute of Aeronautics and Astronautics.

SACHS, G., LENZ, J. & HOLZAPFEL, F. 2011a. Head-, Tail- and Crosswind Effects on the Maximum Range of Powered Sailplanes with Retractable Engine. AIAA Guidance, Navigation, and Control Conference. American Institute of Aeronautics and Astronautics.

SACHS, G., LENZ, J. & HOLZAPFEL, F. 2011b. New modeling approach for bounding flight in birds. Mathematical Biosciences, 234, 75-83.

SACHS, G., LENZ, J. & HOLZAPFEL, F. 2011c. Optimization of Flap-Bounding Flight. AIAA Atmospheric Flight Mechanics Conference. American Institute of Aeronautics and Astronautics.

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SACHS, G., LENZ, J. & HOLZAPFEL, F. 2014. Bounding Flight in Birds – A Problem of Periodic Optimal Control. In: ZOBORY, I., ed. 13th MINI Conference on Vehicle System Dynamics, Identification and Anomalies, 5 - 7 November 2012a, Budapest, Hungary. 15 - 28.

SACHS, G., LENZ, J. & HOLZAPFEL, F. 2012b. Maximum Range Performance of Motor Gliders with Retractable Jet Engine. AIAA Guidance, Navigation, and Control Conference. American Institute of Aeronautics and Astronautics.

SACHS, G., LENZ, J. & HOLZAPFEL, F. 2012c. Performance Enhancements by Bounding Flight. AIAA Atmospheric Flight Mechanics Conference. American Institute of Aeronautics and Astronautics.

SACHS, G., LENZ, J. & HOLZAPFEL, F. 2012d. Periodic Optimal Control for Range Maximization of Powered Sailplanes with Retractable Electric Motor. AIP Conference Proceedings, 1493, 830-837.

SACHS, G., LENZ, J. & HOLZAPFEL, F. Periodic Optimal Flight of Motor Gliders with Retractable Jet Engine. In: ZOBORY, I., ed. 13th MINI Conference on Vehicle System Dynamics, Identification and Anomalies, 5 - 7 November 2012e, Budapest, Hungary. 433-445.

SACHS, G., LENZ, J. & HOLZAPFEL, F. 2012f. Saw-Tooth Flight Performance of Motor Gliders with Retractable Jet Engine. XXXI OSTIV Congress, Uvalde, Texas, USA.

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SACHS, G., TRAUGOTT, J. & HOLZAPFEL, F. 2011d. Progress against the Wind with Dynamic Soaring - Results from In-Flight Measurements of Albatrosses. AIAA Guidance, Navigation, and Control Conference. American Institute of Aeronautics and Astronautics.

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