università degli studi di napoli ﬁfederico iiﬂ -...

Università degli Studi di Napoli “Federico II”

Dipartimento di Progettazione Aeronautica

Appuntidi

Tecniche di Simulazione di Volo——

Ing. Agostino De Marco

[email protected]

Anno Accademico 2003–2004

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Contents

3 Mathematical models

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §3–1

3.1.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . §3–1

3.1.2 Mathematical modeling versus stability and control analysis . . §3–4

3.2 Elements of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . §3–6

3.2.1 Aerodynamic model . . . . . . . . . . . . . . . . . . . . . . . . . §3–6

3.2.2 Engine model . . . . . . . . . . . . . . . . . . . . . . . . . . . . §3–7

3.2.3 Undercarriage model . . . . . . . . . . . . . . . . . . . . . . . . §3–8

3.2.4 Mass model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §3–8

3.2.5 Atmosferic model . . . . . . . . . . . . . . . . . . . . . . . . . . §3–8

3.2.6 Control system model . . . . . . . . . . . . . . . . . . . . . . . §3–9

3.3 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . §3–10

3.3.1 General equations . . . . . . . . . . . . . . . . . . . . . . . . . . §3–10

3.3.2 Small perturbation equations . . . . . . . . . . . . . . . . . . . §3–12

3.3.3 Aircraft orientation . . . . . . . . . . . . . . . . . . . . . . . . . §3–13

3.3.4 Kinematic relationships for Euler angles . . . . . . . . . . . . . §3–13

3.3.5 The direction cosine matrix . . . . . . . . . . . . . . . . . . . . §3–15

3.3.6 The quaternion approach . . . . . . . . . . . . . . . . . . . . . . §3–17

3.3.7 Axes and frames of reference . . . . . . . . . . . . . . . . . . . . §3–20

3.3.8 Incidence angles . . . . . . . . . . . . . . . . . . . . . . . . . . . §3–29

3.3.9 Direction angles . . . . . . . . . . . . . . . . . . . . . . . . . . . §3–32

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Chapter3

Mathematical models

3.1 Introduction

As stated earlier in these notes, to a simulation engineer, a faithful simulation requires

the following three elements

Model: a complete model, preferably expressed mathematically, of

the response of the aircraft to all inputs, from the pilot and from

the environment;

Solver: a means of solving these equations in “real-time”, or in other

words of animating the model;

Response feedback: a way of presenting the output of this solution

to the pilot by means of mechanical, visual and aural responses.

None of these has yet been completely solved nowadays, if judged by a strict engineering

criterium. Another question is whether or not these requirements have to be completely

met.

3.1.1 Basic concepts

The principal task, that of modelling the dynamic behaviour of the flight vehicle, is

an extension of the well known basic principles of Flight Mechanics. However, simu-

§3–1

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Mathematical models — Introduction §3–2

lation must deal with more than just airframe aerodynamics and broader aspects of

mathematical modelling will be outlined here.

The nature of the problem is that the motion of a flying vehicle is governed by

equations of motion of the generic form

x = F/m , θ = M/I (3.1)

where, respectively x, F , m are the vehicle acceleration, applied force and mass, and

θ, M, I are the angular acceleration, applied moment and moment of inertia.

A more specific form of the equations of motion must be established at this point,

but one can state that the basic mathematical model of the vehicle under simulation is

embodied in the definition of the terms F and M in the eq.s (3.1). For an atmospheric

flying vehicle, this mathematical model is primarily the relationship between the air

reactions (aerodynamic forces and moments) and the motion of the airplane relative

to the air. This is currently known as the aerodynamic model of the aircraft. Other

external forces and moments arise from engine thrust and landing gear contact with the

ground. Definition of all these force and moment components is the key to a realistic

description of an aircraft’s flight characteristics. The performances of an aircraft and

its dynamic behaviour can then be calculated for a wide range of flight conditions,

according to the amount of data available for the evaluation of F (at varying airplane

attitudes). Simulation is fundamentally the generation of those forces and solution of

the equations of motion.

The term “mathematical model” is thus growing in complexity and extent, and

is also applied to other features of the aircraft such as its control system, and many

internal systems. For each feature being modelled, it is necessary to formulate the

model so that the “behaviour” of the whole system, i. e. the response of the system to

a stimulus, may be calculated.

A model is only useful if it can actually be implemented for some application. So

far, the term model has not properly separated the concept from the implementation.

This may usefully be done, with the aid of some definitions given below.

A. De Marco – Appunti di Tecniche di Simulazione di Volo

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Figure 3.1: The simulation space and the definition of “models”.

The so-called simulation space is divided into three basic elements as depicted

in fig. 3.1. Starting with Reality, a conceptual model, described via equations or

other governing relationships, is obtained by analysis. Implementation via computer

programming yields a Computer Model which, through simulation, may be related

to Reality. The credibility of the conceptual model is then evaluated by procedures

which test the adequacy of the model to provide an acceptable level of agreement with

reality, while the computer model, in the form of an operational computer program,

is confirmed as an adequate representation of the conceptual model by procedures of

verification. Finally, model validation demonstrates that the computer model possesses

a satisfactory range of accuracy in comparison with reality and consistent with its

intended application.

In some aircraft simulation applications, the mathematical models may be defined


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by one agency, such as the aircraft manufacturer, for implementation by another, a

simulator manifacturer. A third party, such as a regulatory authority, may require

that the original model be implemented unchanged. Verification would be performed

by the simulator manifacturer using his own internally-generated checks, and validation

by comparison with real-life data redorded from flight tests.

In creating or deriving mathematical models, it is important that the modeller (the

person or staff doing the modelling) has a clear idea of what the model is for, and that he

states this together with his definition of his model. It is important because the purpose

of the model influences its form and quality. Many systems are strictly governed by

equations which may be extremely complex but which may often be simplified in the

interests of obtaining practical solutions and yet still retain sufficient realism for the

task at hand.

A practical solution in the present context means one of adequate accuracy which

can be achieved in real-time during simulation. Real-time here refers to a solution in

which the calculation of a systen’s behaviour over a given elapsed time interval ∆t

can be achieved in the same ∆t or less of computing time. Adequate accuracy means

that the real-time solution yelds the steady-state performance of the vehicle and its

transient behaviour with accuracy which is acceptable and sufficient for the role of the

simulation.

3.1.2 Mathematical modeling versus stability and control analysis

Deriving mathematical models for simulation has some similarities with the basic tech-

niques of stability and control theory but there are also some differences. Similarities

include mathematical notation, systems of axes and basic nomenclature. Differences,

however are significant, and include the need for a wide range of speed, aircraft confi-

guration and manoeuvre real-time solution, as defined earlier.

The classical approach to stability and control analysis is to start with the complete

equations of motion and make assumptions that enable the equations to be linearised

about some local point of equilibrium (a trimmed state). Once linear equations are


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available, a host of techniques can be applied to the analysis of the system to investigate

the stability of the motion following a disturbance from trim. It will provide answers to

such questions as “will the aircraft return to the original trimmed state and with what

sort of transient behaviour?” While such techniques have been, and are, invaluable in

the design process, only small disturbances from the equilibrium state are permitted

before the model is invalid. Small disturbances means, say, 10% escursions in speed or

not more than 5 degrees in angle of attack. Many simulation tasks wether for training

or for research and development, demand a wide range of flight speed or require large

changes in aircraft configuration in order to accomplish that task. A complete sortie,

from take-off to landing, is an extreme example.

Real-time solution is necessary because, in simulation, there is usually a man in

the control loop (pilot-in-the-loop simulations), who must be supplied with information

in a timely fashion so that he can use his normal control techniques and strategies.

Figure 3.2: The simulation space and the definition of “models”.


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Mathematical models — Elements of the model §3–6

3.2 Elements of the model

As stated, the primary elements of a conceptual/mathematical model for flight sim-

ulation are those which directly produce the values of external forces acting on the

airframe, namely aerodynamics, engine thrust and undercarriage.

Secondary elements of a model contribute significantly to the core, and include

a control system and the model of atmospheric environment. All these features are

summarized in fig. 3.2.

3.2.1 Aerodynamic model

The aerodynamic model has to reproduce the dominant features of the forces and

moments acting on the aircraft: lift, drag – the main contributors to the aircraft

performance –, and moments about all three axes, through which pilot exercises control.

Mathematical models normally consider the physical components of the vehicle, such

as, for conventional aircrafts, wings, body (i.e. the fuselage), and tail, and build

mathematical expressions for the contributions made by these components and their

mutual interferences. A possible expression is the following

Laero = LWing + LBody + ∆LWing−Body + LTail (3.2)

for the total aerodynamic lift Laero (or simply L) of the aircraft, where the term

∆LWing−Body, usually estimated with semi-empirical formulas, takes into account the

interference effects on the wing lift due to the presence of the fuselage, the remaining

terms being calculated as the wing, fuselage and tail were isolated. Similar formulas

are implemented for the other aerodynamic force components and moments.

Additional featuresare included to represent such influences as “ground effect”,

significant during take-off and landing. In these situations, forces acting on the aircraft

differ markedly from those applicable to the same configuration in free air, due to the

presence of the ground close to the aircraft constraining the lower airflow to follow

the runway. This can resul in increased wing lift and possibly in a sharp increase in


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nose-down pitching moment close to the ground. Both these effect are important as

they influence the pilot’s ability to perform a safe and smoth landing. Aerodynamic

models are normally “static” or “quasi-static”, based on the assumption that the airflow

can re-establish steady conditions in a time scale much shorter than the characteristic

time scale of the aircraft manoeuvre itself. Thus the time variable t does not appear

explicitly in the mathematical model, but the aircraft’s transient behaviour is, of course,

a function of time.

3.2.2 Engine model

An engine model, embedded a simulator’s conceptual model, must produce the correct

value of steadt thrust to correspond with pilot’s demand through his power lever. Usu-

ally, thrust is modelled as a net thrust: the net propulsive force is directly calculated,

having allowed for intake momentum drag. More comprehensive models will reproduce

gross thrust and momentum drag as separate entities, taking the difference explicitly

rather than implicitly. These latter models are applied in cases where the simulated

aircraft is capable of flying at high angles of attack or can vector its engine exhaust

nozzles in some way, as the thrust vector and the momentum drag vector may then

deviate by a large angle. Interference between the engine exhaust flow and the tailplane

may further affect the available applied forces.

An additional example of the need for detailed engine modelling can occur with

the simulation of a VSTOL (Vertical or Short Take-Off and Landing) aircraft, which in

the hover, must use a proportion of the airflow through the engine to generate control

forces by blowing out at the extremities. Over-active pilots, trying to keep the hovering

of these type of aircraft, end up with a descending motion as a result of their control

activities. These situation must be simulated correctly.

Pilot’s ability to control the aircraft also depend on the time needed by the engine

to produce the request level of thrust. An engine can not change its thrust instanta-

neously: burning more fuel to increase thrust must first accelerate the rotating machin-

ery of the engine and this takes time. The timescale varies in the range of few tents of


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a second to several seconds. This implies, besides the static, performance-orientated

thrust modelling, the need for an adequate modelling of the growth or decay of thrust

with time.

Engine modelling is an area where a wide variety of techniques is used, according

to the fidelity required to the engine dynamic response. A description of some engine

models adopted in flight simulation practice will be given elsewhere in these notes.

3.2.3 Undercarriage model

The simulation of take-off or landing require some means to produce the force which

supports the aircraft in contact with the ground. This force is given by the undercar-

riage model. The real undercarriage is a complex mechanical, hydraulic and in some

cases pneumatic assembly, which can be modelled in a variety of ways. A model of a

spring coupled with a damper can provide the ground reaction in a simple way. An

extra damping can be added for the recoil phase after the impact in landing.

Detailed models are necessary in some simulation tasks, which demand the repro-

duction of tyres and brakes, anti-skid systems and nose-wheel steering.

3.2.4 Mass model

The discussion so far has been concerned with the contributions to the external forces,

broken down as

F = Faero + Fengine + Fundercarriage (3.3)

The other item which also need modelling, recalling the generic equations (3.1),

is the set of vehicle’s mass properties. These comprises the its mass, its moment of

inertia and center of gravity, usually all function of payload, fuel state, and geometry,

e. g. wing sweep, or stores configuration.

3.2.5 Atmosferic model

A full atmospheric model is needed to represent a number of properties. The starting

point is the variation of density and temperature with altitude, as defined by the


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International Standard Atnosphere (ISA). Non-standard ambient ambient conditions

may also be needed, such as in cases when the training for “hot day” is of interest or

when design checks have to be accomplished.

These properties are usually known as atmospheric statics as opposed to wind

environments, or atmospheric dynamics, which are needed in terms of mean surface

wind for take-off and landing, and winds at altitude. Atmospheric dynamics, i. e.

turbulence associated with wind, affect performance on a route schedule, for example,

or can cause distraction and disturbance on pilot in a precise task, or sometimes gives

rise to a design case for regulation authority or crew ride comfort.

Methods of modelling atmospheric turbulence are nowadays quite advanced as a

consequence of active research in the real world environment. Extreme phenomena,

such as wind shear, responsible for several major accidents, are now modelled and are

a legitimate item to be simulated, either for research into aircraft design or for training

pilots to recognise and counter it effectively.

Many training tasks demand a full range of metereological phenomena to be sim-

ulated, including weather radar returns and runway state.

3.2.6 Control system model

The control system of an aircraft also needs modelling. It is interposed, in the real

aircraft, between the pilot’s control stick and the vehicle control surfaces, and consists

typically of a system of levers, cables and bell-cranks, terminating in a hydraulic-

actuated jack. The mechanical deficiencies of these system have to be modelled, such

as cable stretch, control jams etc.

Current military aircraft and projected future civil transport aircraft are repla-

cing mechanical connectionc by electrical ones, coupled with computer-driven systems

shaping the pilot’s demands and processing feedback signals. These developments make

the modelling needs growing dramatically as more and more functions are included in

the flight control computer. In this respect, incorporating real flight hardware in the

simulator may be the best solution.


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Mathematical models — Equations of motion §3–10

3.3 Equations of motion

To the general discussion of mathematical models given above, follows here a presenta-

tion of some explicit mathematical formulations. Equation of motion will be outlined

in this section while some forms of data representation and managing will be discussed

later.

3.3.1 General equations

The following exposition assumes that the reader is familiar with the simpler concepts

of aerodynamics and the basic theory of aircraft stability and control.

The mathematical formulation used here adopts the classical notation of alpha-

betical triads to represent the three components of force, moment, linear and angular

velocity. These are illustrated in fig. 3.7 for the conventional right-handed orthogonal

axes with origin fixed in the aircraft. No particular set of vehicled carried axes is yet

assumed.

The equations of motion of a rigid aircraft flying in still air, when referred to any

system of axes fixed in the aircraft and rotatong with it, are

Wg (u+ qw − rv) = X +

Wg gx

Wg (v + ru− pw) = Y +

Wg gy

Wg (w + pv − qu) = Z +

Wg gz

(3.4)

Ixxp− Ixy(q2 − r2) − Izx (r + pq) − Ixy (q − rp) − (Iyy − Izz) qr = L

Iyyq − Izx(r2 − p2) − Ixy (p+ qr) − Iyz (r − pq) − (Izz − Ixx) rp = M

Izz r − Ixy(p2 − q2) − Iyz (q + rp) − Izx (p− qr) − (Ixx − Iyy) pq = N

(3.5)

These are sometimes known as the “total force” equations, as opposed to the small

perturbation equations of classical stability and control.


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Precisely, the first three equations (3.4) are the force equations while the second

three (3.5) are the moment equations in their “complete” form. The full non-linear

set equations is sufficient to cope with general, large-scale motion of an aircraft and is

soluble in these form by numerical integration techniques only once the forcing term

X, Y , Z, L, M, and N on the right-hand side can be prescribed at each time step.

The moment equations (3.5) are “complete” in the sense that all the cross-inertia

terms have been retained. If, as for most cases, it can be assumed that the aircraft has

a symmetrical distribution of mass with respect to the fore-and-aft plane of synnetry,

then Iyz = Ixy = 0 and the equations become more compact

Ixxp− Izx (r + pq) − (Iyy − Izz) qr = L

Iyyq − Izx(r2 − p2) − (Izz − Ixx) rp = M

Izz r − Izx (p− qr) − (Ixx − Iyy) pq = N

(3.6)

Such an assumption is reasonable for conventional aircraft, such as a transport

or executive aircraft, but a fast-jet aircraft could well have, either intentionally or a

result of a failure, an asymmetric distribution of stores beneaths its wing, in which case

the complete equations (3.5) are necessary. It is well known the major influence that

inertia distribution can have on the flying qualities of a fast-jet aircraft and must be

included with care in a simulator’s matematical model.

Regarding the force equations (3.4), while simple in appearence, they pose some

practical difficulties in their solution. These difficulties are associated firstly with prod-

uct terms like qw, rv, etc. which, although nominally second-order, can become dom-

inant in large-scale, vigorous manoeuvres, secondly, with the “still air” assumption.

Some simulator mathematical models are implemented according to a formulation re-

ferred to a earth-based frame, to eliminate both the above difficulties. In an inertial

frame of reference, the product terms disappear, as (i) the axis system is not rotating

and (ii) the treatment of winds is simpler.


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3.3.2 Small perturbation equations

In many simulated situations, there is no alternative to use of the total force equations

and fortunately modern computer techniques now permit this. However, in some cases

it may still be desiderable to deal with a linearized form of the equations of motion, since

they become amenable to treatment by a wide range of analytical techniques. Such

procedures are important at a design stage of a new aircraft and also in research, when

experiments are conducted to establish data on the fundamental handling qualities of

aircraft based on simplified situations and models.

If the aircraft’s motion deviates to only a small extent from a datum, trimmed

state, then the well-known small perturbation equations of motion are applicable. The

datum state is also called “equilibrium” state in which flight quantities, say f assume a

value feq, and evolve in time being perturbed by a small variation ∆f , which becomes

the unknown, viz. f = feq + ∆f .

Below are reported, as an example of formulation, the linearized equations of

longitudinal motion. Assuming level flight (θeq = 0), using aerodynamic axes (such

that weq = 0), and omitting some derivatives that are commonly neglected, one can

write the following equations in matrix form

d

dt

∆u

∆w

∆q

∆θ

=

xu xw xq xθ

zu zw zq zθ

mu mw mq mθ

·

∆u

∆w

∆q

∆θ

+

xη

zη

mη

0

η (3.7)

where η (sometimes denoted also with δe) is the elevator deflection, and the coefficients,

or aerodynamic stability derivatives are constants matrix elements for a given aircraft

configuration and datum flight condition, reported here in their “concise” form.

A corresponding set of equations can be written for the latero-directional motion,

but are not reported here.


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The small perturbation equations are rarely used in full-blown simulations, but

can have relevance in research studies of handling qualities or to preliminary studies

of a new design, when fundamental properties of an aircraft’s dynamic behaviour may

be examined.

3.3.3 Aircraft orientation

The aircraft’s orientation in space is given by its attitude, which may be defined in a

number of ways. The most common method consists in defining a sequence of three

angles, known as the Euler attitude angles.

Starting with a set of axes, with origin C fixed in the aircraft, initially aligned

with those of a fixed inertial, or anyway datum, reference frame Ox0y0z0, and with

C ≡ O, the first set of axes is brought into alignment with the body-fixed axes Cxyz

by rotating the first set successively through each angle in turn. The usual trio of

angles consists of: À the heading or azimuth or yaw angle ψ, Á the inclination or pitch

angle θ, and Â the roll angle φ.

The sequence of rotations is illustrated in fig. 3.3. In fig. 3.3(b) the intermediate

axes set between successive rotations are labeled x0y0z0, x1y1z1, x2y2z2, xyz. The se-

quence is significant, as a different order from that defined here will produce a different

final orientation.

With the definitions given here, the attitude angles can take the following range

of values:

− π < ψ ≤ +π , −π

2< θ ≤ +

π

2, −π < φ ≤ +π (3.8)

3.3.4 Kinematic relationships for Euler angles

Given these definitions of Euler attitude angles, derivation of the angles in practice

requires a relationship between the rates of change of the attitude angles and the

components of angular velocity of the aircraft’s body axes. When the latter is referred


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(a) Euler angles as a definition of aircraft orientation with respect to a datum

frame Ox0y0z0. Frame Cxyz is body-fixed and represents the aircraft.

(b) The sequence of rotations bringing the datum frame into alignment with the

body-fixed axes. The classical color sequence Red-Green-Blue is applied accordingly.

Figure 3.3: Euler attitude angles (φ, θ, ψ) and rotation sequence.


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to the body-fixed frame these relationships take the form

φ

θ

ψ

=

1 sinφ tan θ cosφ tan θ

0 cosφ −sinφ

0sinφ

cos θ

cosφ

cos θ

·

p

q

r

(3.9)

the inverse being

p

q

r

=

1 0 −sin θ

0 cosφ sinφ cos θ

0 −sinφ cosφ cos θ

·

φ

θ

ψ

(3.10)

Note that φ ≡ p only when θ = 0 and θ ≡ q only when φ = 0.

Equations (3.9), also known as gimbal equations are widely used, but do contain

a singularity, when the x body axis becomes vertical (θ = ±90◦), tan θ and 1/ cos θ

(= ±∞) make the explressions for φ and ψ indeterminate.

The situations in which this occur are clear. If such manoeuvres are avoided, then

the equations may be used without difficulty. For the simulation of aircraft which

perform aerobatics or similar gross manoeuvres, an alternative formulation is required

so that the derivation of the aircraft attitude angles is always trouble-free.

3.3.5 The direction cosine matrix

Transformation of variables between pairs of reference systems is often required in

order, for example, to reformulate the equations of motion in a form suitable for the

specific treatment of some terms.

The expression providing the airplane center of mass velocity components referred

to the body axes as a function of the corresponding components in earth axes is reported


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below. It is put in a form similar to the classical

x

y

z

=

l1 l2 l3

m1 m2 m3

n1 n2 n3

·

x0

y0

z0

(3.11)

where {x} = {x, y, z}T represents a generic vector quantity, (of velocity, say, or force)

with components referred to body axes, and {x}0

= {x0, y0, z0}T, the same entity but

with components in the earth axes. finally the matrix in eq. (3.11) is denoted here as

[R] is known as direction cosine matrix.

The inverse relationship, giving earth axes components in terms of body axes

components, is

x0

y0

z0

=

l1 m1 n1

l2 m2 n2

l3 m3 n3

·

x

y

z

(3.12)

in which the transformation matrix is written as the transpose of [R], as it is also its

inverse: [R]T = [R]−1.

When direction cosines are expressed in terms of the Euler attitude angles one has

[R] =

cos θ cosψ cos θ sinψ −sin θ

sinφ sin θ cosψ − cosφ sinψ sinφ sin θ sinψ + cosφ cosψ sinφ cos θ

cosφ sin θ cosψ + sinφ sinψ cosφ sin θ cosψ − sinφ cosψ cosφ cos θ

(3.13)

The direction cosine may be defined and derived in other ways, but the transfor-

mation matrices written above are still needed and used.


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3.3.6 The quaternion approach

A practical alternative to the use of Euler equations for defining the orientation of an

aircraft, which avoids singularities when pitch attitude reaches 90 degrees, is to use the

so-called four parameter method or quaternion approach.

It is well known that when a frame of axes Oxyz may be brought into coincidence

with a reference frame Ox0y0z0 by a single rotation D about a fixed axis in space,

making angles A, B, C with reference axes Ox0, Oy0, Oz0, respectively. These four

parameters A, B, C, D can then define the orientation of the frame Oxyz with respect

to the reference frame.

It can be shown that the transformation matrix R relating a vector (x0, y0, z0) in

the reference frame to its representation in the new frame (x, y, z) according to (3.11)

is given by

l1 l2 l3

m1 m2 m3

n1 n2 n3

=

e20+ e2

1− e2

2− e2

32 (e1e2 + e0e3) 2 (e1e3 − e0e2)

2 (e1e2 − e0e3) e2

0− e2

1+ e2

2− e2

32 (e2e3 + e0e1)

2 (e0e2 + e1e3) 2 (e2e3 − e0e1) e2

0− e2

1− e2

2+ e2

3

(3.14)

where

e0 = cosD

2

e1 = cosA sinD

2

e2 = cosB sinD

2

e3 = cosC sinD

2

(3.15)

are a new set of four parameters called quaternions. It can be shown that they are

constrained by the equation

e20+ e2

1+ e2

2+ e2

3= 1 (3.16)

Furthermore, it can be shown that the following espression are valid for quaternion


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rates

e0 = −1

2(e1p+ e2q + e3r)

e1 =1

2(e0p+ e2r − e3q)

e2 =1

2(e0q + e3p− e1r)

e3 =1

2(e0r + e1q − e2p)

(3.17)

These equations provide the mean to generate the quadruplet (e0, . . . , e3) from the

body axis components of angular velocity p, q, r. Considering the constraint expressed

by eqs. (3.16) above, one of the values obtained from time integration of eqs. (3.17) is

effectively redundant.

Implementation in a computer code of these equations has to take into account this

constraint. One of the most widely diffused way to do this is to rewrite the quaternion

rates as

e0 = −1

2(e1p+ e2q + e3r) + k λ e0

e1 =1

2(e0p+ e2r − e3q) + k λ e1

e2 =1

2(e0q + e3p− e1r) + k λ e2

e3 =1

2(e0r + e1q − e2p) + k λ e3

(3.18)

where k is a constant chosen such that k∆t ≤ 1 for an integration time step ∆t and

λ = 1 −(

e20+ e2

1+ e2

2+ e2

3

)

(3.19)

This is known as algebraic constraint method.


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Finally, expressed in terms of the three Euler angles, the four quaternions are

e0 = cosψ

2cos

θ2

cosφ

2+ sin

ψ

2sin

θ2

sinφ

2

e1 = cosψ

2cos

θ2

sinφ

2− sin

ψ

2sin

θ2

cosφ

2

e2 = cosψ

2sin

θ2

cosφ

2+ sin

ψ

2cos

θ2

sinφ

2

e3 = −cosψ

2sin

θ2

sinφ

2+ sin

ψ

2cos

θ2

cosφ

2

(3.20)

Espressions (3.20) are necessary to derive initial values for (e0, . . . , e3) when attitude

angles (ψ, θ, φ) are known.

In the context of aircraft simulation, the direction cosines are still required for the

transformation of variables from body axes to earth axes (and vice versa), and equation

(3.11) may be used to derive them from quaternion parameters.

The Euler angle derivatives are also required for displays on pilot’s instruments

and elsewhere. A suitable selection from eq. (3.14) gives

sin θ = − l3 =2 (e0e2 + e1e3)

=: θ = sin−1(− l3)(3.21)

Since, by definition, the elevation angle θ of the x- body axis above the horizontal plane

lies in the range ±π2,

−π

2≤ θ ≤ +

π

2(3.22)

the inverse sine process will then yeld a unique value of θ.

Next, from eq. (3.14), one has

cosψ = l1/cos θ (3.23)

and since cos θis always positive for the defined range of θ, then

cos θ sinψ = l2 (3.24)

yelds

sgn [sinψ] = sgn [l2] (3.25)


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and hence

ψ = cos−1 (l1/cos θ) · sgn [l2] (3.26)

Similarly

cosφ = n3/cos θ (3.27)

φ = cos−1 (n3/cos θ) · sgn [m3] (3.28)

To summarize the two approaches to aircraft orientation, the Euler technique is

simple, widely known and widely used. It has the weakness consisting in the singularity

at zenith and nadir. The quaternion technique is more complex and less familiar but

is robust and will cope with all kinds of manoeuvres.

Here, for the sake of comparison, are reported the solution sequences in each case:

Euler angle approach Quaternion approach

Ê calculate body rates calculate body rates

Ë calculate Euler rates calculate quaternion rates

Ì calculate Euler angles calculate quaternions

Í calculate direction cosines calculate direction cosines

Î apply transformation matrixÀ calculate Euler angles

Á apply transformation matrix

3.3.7 Axes and frames of reference

In Flight Simulation, and generally in the study of Flight Mechanics problems, it is

necessary to define and take into account some ets of axes as frames of reference (or

coordinate frames) for many purposes and reasons. They are necessary to define the

position of a point in space or to quantify the orientation of a body relative to another

one. The general need is that of characterizing the motion of a body in terms of its


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position and orientation versus time. Of course by saying “orientation of a body” one

means the orientation of a reference frame tied to it.

Moreover, equations of motions of a body subject to external action are written

down with respect to a specified coordinate frame, usually chosen to simplify the spec-

ification of particular groups, such as the aerodynamic forces and moments; or to take

advantage of circumstances in which some terms vanish.

The definition of reference frames is required also when there is the need to read

or measure a vectorial quantity in terms of its scalar counterparts and to transform

these entities from a frame to another one.

There are many possible axis systems, but for simulation they fall into two broad

classes: À inertial or, under certain hypoteses earth axes, and Á body axes. Of partic-

ular interest also is the class of wind axes, which will be described below. Wind axes,

as well as some earth axes, are “vehicle-carried”, i.e. they all have a moving origin

superimposed to the vehicle’s center of mass, but they are not rotating with the body,

like a body axis system rigorously does. They are defined with respect to the relative

wind, or to the trajectory in general, or to the local geographic position.

All these frames are orthogonal, right-handed coordinate frames. Generally, the

earth is supposed to be a sphere having its center of mass superimposed to its geometric

center.

Inertial frame {Oxyz}i

This is a reference frame needed in every dynamics’ prob-

lem, explicitly defined or lurking implicitly in the background of theoretical formula-

tions. An inertial frame is defined as a frame which is non-accelerated with respect to

the “fixed stars”. In Navigation, a coordinate frame that has origin at the center of the

earth and is non-rotating relative to the stars can be considered an inertial frame for

measurements made in the vicinity of the earth.

Earth frames The following earth frames are of interest in Flight Mechanics and

Flight Simulation.


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· Earth-Center Frame {Oxyz}ec

This frame has its origin at the earth’s center. Its axes

are fixed in the earth. This frame is thus rotating with respect to the inertial frame

{Oxyz}i previously defined. The zec-axis points North and is supposed coincident with

the earth’s angular velocity vector, while the plane {Oxy}ec is equatorial. The axes

of this plane can be arranged such that this earth frame and the inertial frame are

superimposed at some initial time t = 0. The angular rate of the earth-centered frame,

i.e. the angular rate of the earth relative to the inertial frame, is denoted by ωE.

This frame is needed when the earth’s rotation has to be necessarily taken into

account, as in problems concerning, for instance, inertial navigation or space trajecoties.

· Tangent Frame {Oxyz}t

This is an earth-fixed frame with origin near the vehicle. Its

axes are aligned with the north, east, and down directions. Here “down”, or vertical,

means the normal to the earth’s surface. The north axis is in the direction of the

projection of the earth’s angular velocity vector into the local horizontal plane (this

plane being perpendicular to the down direction). The east direction completes the

right-handed orthogonal set. The definition of the tangent frame follows the so-called

North-East-Down (NED) convention.

Sometimes the origin of the tangent frame is chosen so that it is superimposed to

the vehicle mass center at the start of a particular manoeuvre under study, or at its

projection on the ground.

When the earth is supposed to be spherical, the tangent frame is also a geocentric

frame, in the geodesic jargon. The zt-axis has the same direction of the geocentric

position vector from the earth’s center to the origin Ot, and opposite versus.

The origin of a tangent frame is positioned, relative to the earth-center frame,

through its geographic coordinates, i.e. the two angles: λt, latitude of Ot, and µt

latitude of Ot.

· Vehicle-Carried Vertical Frame {Oxyz}v

A simply defined moving reference frame is

the one whose origin is fixed at the vehicle’s center of mass C (whose origin is “carried”


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by the vehicle and follows it), but has its three axes aligned in such a way that the

zv-axis is continuously directed vertically and the plane {Oxy}v is locally tangent to

a sphere having radius RE + h, where RE is the earth’s radius and h is the vehicle’s

altitude. For atmospheric flight one has h� RE.

Note that the vehicle-carried vertical frame is a frame moving with the body but

is not a body frame (cfr. below).

This frame follows the NED convention, i.e. the xv-axis points to the north and

the yv-axis points to the east.

Since the orientations of the tangent frame and of the vehicle-carried vertical frame

are defined similarly, the latter can be transported over the former by two consecutive

rotations in the sense of latitude and longitude, respectively. This means that the the

moving origin of the vehicle carried vertical frame is positioned, at the generic time

instant, in a point in space having{

latitude: λv = λt + ∆λ

longitude: µv = µt + ∆µ

where ∆µ is a rotation around the earth-centered frame’s zec-axis and −∆λ is a rotation

around an axis which is parallel to the tangent frame’s yt-axis and passing through the

earth’s center.

The angular rate of the vehicle-carried vertical frame with respect to the inertial

frame is usually denoted by ωv. It is useful to point out that in inertial navigation

problems the earth’s surface is supposed to be an ellipsoid. This is an analytically

defined surfacewhich is an approximation to the mean sea-level gravity equipotential

surface, also known as geoid. When the earth is supposed even spherical, the reference

frame used for navigation coincides with the geocentric frame and the tangent frame

is aligned with a geographic frame at a fixed location on the earth.

The relative orientation of earth frames defined so far can be derived from fig. 3.4.

· Datum-Path Earth/Vehicle-Carried Frame {Oxyz}0

In flight dynamics applications, an-

other adopted definition for earth axes exists, useful when short term motion is of


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interest, and perfectly adequate. It is a reference frame moving with the earth, or with

the aircraft, known as datum-path earth frame, or datum-path vehicle-carried frame.

Flight above a flat earth is still assumed, initially straight and level. Straight and

level flight implies a motion in a horizontal plane at a constant altitude, and whatever

the subsequent motion of the airplane might be, the attude is determined with respect

to the horizontal.

Referring to the tangent frame defined above, a horizontal plane {Oxy}0is defined,

being parallel to the plane {Oxy}t. The only difference is that the x0-axis points in the

arbitrary initial direction of flight rather than to the north. The z0-axis points down

as in the earth frame. It is only necessary to place the origin O0 in the atmosphere at

the most convenient point.

Sometimes this reference frame is considered as moving with the aircraft by assum-

ing its origin coincident with the vehicle mass center. But, unlike the vehicle-carried

vertical frame, the vehicle-carried datum-path frame x0-axis gives constantly a unique

reference direction, which is the initial flight direction, while keeping its {Oxy}0

plane

horizontal and its {Oxz}0

plane vertical. It provides an inertial frame for short term

airplane motion when the flight speed varies under the hypoteses of the small pertur-

bation theory.

· Practical usage of earth frames When atmospheric flight only is of interest, it is usual to

measure aeroplane motion with reference to an earth fixed framework like the tangent

frame. Clearly the plane {Oxy}t defines the local horizontal plane, thus the flight path

of an airplane flying in the atmosphere in the vicinity of the reference point Ot may be

completely described by its coordinates in the axis system, assuming a flat earth, where

the vertical is “tied” to the gravity vector. This model is quite adequate to localized

flight, where the effects of earth’s rotation on vehicle motion are negligible, and the

tangent frame becomes an inertial, or fixed, frame for the particular problem under

study or motion under simulation. The widely used notation for the tangent frame in

such cases is {Oxyz}e, which stands for earth frame, fig. 3.5. The earth frame is best


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suited to navigation and performances applications, and to flight simulations where

flight path trajectories are of primary interest.

For investigations involving trans-global navigation. the earth frame described

above is inappropriate, the earth-centered ec frame being preferred.

Body axes Body axes consist of an axis system fixed in the aircraft, and normally

with origin C at its center of gravity. Body axes move in space and rotate with the

aircraft. For a specific set of body axes it essential to know how the orientation of the

axis system is defined with respect to the aircraft.

· Geometric-body axes {Oxyz}b

Among possible body axis systems, geometric-body

axes {Oxyz}b, or simply {Oxyz}, are usually those defined such that the x-axis is

aligned with a geometric feature, such as the fuselage reference line (FRL) or a wing

datum plane (WDP). Often such axes are known just as body axes. Both kinds of

reference may be used for one mathematical model of the aircraft, for example WDP

to define the aerodynamic forces and FRL for all other uses, including axis transforma-

tions. Tipically the inclination of the FRL with respect to the WDP is a small angle

(one or two degrees).

· Principal-body axes {Oxyz}P

An alternative set of body axes is the set of principal-

body axes, defined such that the xP-axis is aligned with the principal inertia axis, thus

making the cross-products of inertia Izx zero. This simplifies the lateral equations of

motion, as can be seen by inspecting eq. (3.6), and is useful in theoretical studies,

but is not particularly helpful in simulation, as the location of the principal axis could

change with aircraft configuration. Geometric-body axes offer sufficient advantage in

giving constant moments of inertia.

Wind axes The following definitions introduce wind axes as they are generally ac-

cepted in the aeronautical practice. The subtle differences that arise between different


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Figure 3.4: Earth reference frames. ec: earth-centered frame; t: tangent frame;v: vehicle-carried vertical frame.


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Figure 3.5: Earth frame in flat earth hypothesis.

types of wind frames of reference, which sometimes are referred to with the same name

in different contexts, are also pointed out in this section.

· General Wind-axis Frame {Oxyz}w

A general wind-axis system is defined such that

the origin of a rectangular Cartesian system is at the center of gravity of the aircraft,

having the xw-axis pointing into the direction of the oncoming free-stream velocity

vector.

The general wind zw-axis is more rigorously defined with unique reference to the

trajectory of the aircraft mass center. it is defined as the intersection of the plane

locally normal to the trajectory and the vertical plane containing xw. The remaining

axis yw completes the system.

According to this definition, the geometric-body axis zb is at an angle ν, known

as the bank angle, with respect to the vertical plane {Oxz}w. The relative orientation


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of the earth, body, and wind frames defined here is illustrated in fig. 3.6.

Figure 3.6: Relative orientation of the general wind axes, earth axes, and bodyaxes.

· UK Wind-axis Frame {Oxyz}w

According to this convention, the xw-axis is defined

as in the general case. The zw-axis, instead, lies this time in the aircraft plane of

symmetry, is perpendicular to the xw-axis and is directed generally downward. The

yw-axis completes the system and is thus coincident with the body yb-axis.

In many theoretical problems airplane motion is in the geometric plane of sym-

metry (no yawing motion) so that the xw-axis also lies in the plane of symmetry. The

system then is termed the simplified wind-axis system, but is not of practical interest

in flight simulation applications.


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· Aerodynamic-axis Frame {Oxyz}a

Aerodynamic axes, also called aerodynamic-body

axes, have their origin at the aircraft center of gravity and have the xa-axis aligned

with the projection of the velocity vector, on to the plane of symmetry.

The za-axis belongs to the aircraft plane of symmetry, the ya-axis completing the

axis system accordingly. Note that the latter axis coincides with the body axis yb.

According to this definition, aerodynamic axis xa is therefore at an angle α relative

to the geometric body axis xb. Moreover, xa is an angle β relative to the geometric

wind axis xw, i.e. to the direction of the relative wind.

In the analysis of short-term motion, aerodynamic axes are also known as stability

axes in USA, corresponding to the UK wind axes, if the xa-axis is the projection of the

velocity vector on to the plane of symmetry in a datum flight condition.

3.3.8 Incidence angles

Unlike attitude angles, which define the orientation of an aircraft with respect to a

set of earth axes, incidence angles define the direction of the airflow relative to the

vehicle’s body, and are needed in the derivation of the aerodynamic forces acting on

the airframe.

Denoted with ~Vr the opposite of the relative-wind velocity vector, the angle of

attack α is the angle between À the projection of ~Vr on to the aircraft plane symmetry

and Á the body x-axis.

Note that the definitions given above for the reference frames consider the velocity

vector of the aircraft mass center in a fixed reference frame. Nothing is said about the

local wind. One could also redefine the wind frames starting from the direction of the

total relative-wind velocity vector, including the local atmospheric wind. In that case

α is the angle between the body axis xb and the aerodynamic axis xa.

The sideslip angle formed by À velocity vector ~Vr with Á the aircraft’s symmetry

plane. It is the angle between the wind axis xw and aerodynamic axis xa.

If the velocity components are known in the body axis system, the incidence angles


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Figure 3.7: The body- and aerodynamic- frames of references. The three com-ponents of force, moment, linear and angular velocity. Aerody-namic angles.


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Figure 3.8: The Euler angles (ψ, θ, φ) with respect to the definition of generalwind axes, earth axes, and body axes.


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are defined as (fig. 3.7 and 3.9)

tanα =w

u, sin β =

v

Vr

(3.29)

where V 2

r = u2 + v2 + w2. The ranges of variations of α and β are

− π < α ≤ +π , −π

2< β ≤ +

π

2(3.30)

3.3.9 Direction angles

Direction angles define the direction of the flight path velocity vector ~V with respect

to a fixed reference frame, e.g. the earth axes. The angles, shown in fig. 3.9, are the

climb angle γ and the track angle χ.

The climb angle is that formed by the wind axis xw with the horizontal plane

{Oxy}e, positive for a climbing path.

The track angle is that formed by the wind axis xw with the reference vertical plane

{Oxz}e, i.e. the angle between the local tangent to the ground track and the north-

pointing axis xe. It is positive for an east-travelling aircraft. The difference between

track angle and heading ψ is called drift and is due to the presence of winds along

the flight path. The summation of the relative wind due to the motion of the aircraft

with respect to the earth reference frame, and of the local atmospheric wind, given or

modelled in terms of wind velocity components in earth axes, gives the instantaneus

relative wind experienced by the aircraft, the same relative wind which defines the

incidence angles introduced above.

If the flight path velocity vector is ~V C, i.e. the velocity of the aircraft mass center

C, its components in earth axes ue, ve, we are

uCe = V C cos γ cosχ

vCe = V C cos γ sinχ

vCe = −V C sin γ

Incidence angle, attitude angles and direction angles can be related but the ex-

pressions are complex and omitted here. In the absence of winds, in steady, straight


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symmetric (i.e. wings-level) flight, it can be shown that

γ = θ − α (3.31)

but it must be emphasized that this helpful relationship is only true in the restricted

but common circumstances quoted.

Figure 3.9: Direction angles. Climb γ, track χ.


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