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||Autonomous Systems Lab

151-0851-00 V

Marco Hutter, Roland Siegwart, and Thomas Stastny

Autonomous Systems Lab

20.12.2016Robot Dynamics - Fixed Wing UAS: Stability, Control, and Derivation of a Dynamic Model 1

Robot DynamicsFixed Wing UAS: Stability, Control, and Derivation of a

Dynamic Model


1. Introduction/(brief) Historical Overivew

2. Basics of Aerodynamic

3. Aircraft Dynamic Modeling

4. Aircraft Performance (wrap-up)

5. Aircraft Stability

6. Simulation

7. Modeling for Control

8. Fixed-wing Control

Contents | Fixed Wing UAS

20.12.2016 2Robot Dynamics - Fixed Wing UAS: Stability, Control, and Derivation of a Dynamic Model

||Autonomous Systems Lab 20.12.2016 3

Aircraft Performance (Wrap-up)

Robot Dynamics - Fixed Wing UAS: Stability, Control, and Derivation of a Dynamic Model


We previously made some simplifications in the exercise

with assumptions of “steady” and “level” flight.

Steady =>

Level =>

Aircraft Performance | Powered vs. Un-powered

Flight



We previously made some simplifications in the exercise

with assumptions of “steady” and “level” flight.

Steady =>

Level =>


Flight

20.12.2016 5

This obviously does not hold for “gliding”

flight where no thrust can equal out the drag!



Gliding force balance:


Flight

20.12.2016 6

(𝑁, 𝐸)

L

D

mg



Gliding force balance:


Flight

20.12.2016 7

(𝑁, 𝐸)

L

D

mg

Note: glide range

is maximized

when is

maximized, i.e.

with maximum



Given: drag coeff. as a funcion of lift coeff.: 𝐶𝐿 𝑉 =2𝑚𝑔

𝐴𝜌𝑉2 , 𝐶𝐷 𝐶𝐿

Required power:

Specific Excess Power: 𝑆𝐸𝑃 = 𝑃𝑎𝑣𝑎𝑖𝑙𝑎𝑏𝑙𝑒 − 𝑃𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑑 𝑚𝑔 ≈ 𝑉𝑐𝑙𝑖𝑚𝑏,𝑎𝑐ℎ𝑖𝑒𝑣𝑎𝑏𝑙𝑒

20.12.2016 8

Power Required and Available for Level Flight

Drequired ACVVDP 3

2

1

Pavailable

Prequired

Pexcess

Vstall Vmax Vne

Pow

er

True Airspeed

Vemax Vrmax

min

L

D

C

Cmg

L

DmgD

V

Pmax

D

L

C

C

Best glide ratio

max/

P

EVTVs

min/ P

max2

3

D

L

C

C

L

D

LL

D

C

Cmg

CA

mg

C

CmgVVDP

2

Minimum sink in

gliding mode

Max. Range*(vrmax):

Max. Endurance*(vemax):

* Assuming constant propulsive efficiency η



Turning

Demand for coordinated turn: Y=0

L increases with

Vmin increases with

Aircraft Performance | Level Turning Flight

20.12.2016 9

(𝑁, 𝐸)

D

FT

L(𝐷)

mgRobot Dynamics - Fixed Wing UAS: Stability, Control, and Derivation of a Dynamic Model


Turning

Demand for coordinated turn: Y=0

L increases with

Vmin increases with


20.12.2016 10

(𝑁, 𝐸)

D

Assuming NO

sideslip, i.e.

FT

L(𝐷)

mg

Recall Y is composed of only

aerodynamic forces, which must be

zero, thus the lateral force here only

comes from centripetal acceleration.



Maximum required bank

Note, , where is the aircraft’s bank angle, rotated about the

airspeed vector. However, in general flight, the difference is

negligbile. So we will ignore it and only discuss the roll angle.

For a given radius, we can calculate the required roll angle to

maintain level turning flight.


20.12.2016 11

Force balance:Note this assumes we have

thrust force only acting in the

same axis as drag. In reality,

thrust force likely will add a

small vertical component to the

lift (e.g. if we fly at any

pitch/angle of attack).



Aircraft Stability



The aerodynamic forces and moments discussed in the

previous lecture are build up of both static and dynamic

components summed from each part of the aircraft.

E.g.

Other model structures could be used, but the build-up approach

typically takes a polynomial form

Aircraft Stability | Component Build-up

20.12.2016 13

1st order Taylor Expansion

(only linear terms)

2nd order Taylor Expansion

(some coupled terms)



It is standard practice to non-dimensionalize rate

coefficients per operating airspeed and aircraft geometry

For example, using span in the lateral-directional derivatives:

…and mean geometric chord in the longitudinal derivatives:

Aircraft Stability | Non-dimensionalization



Aircraft Stability | Criteria for Static Stability

Velocity Stability

u v w

Forces

x

y

z

20.12.2016 15

Note: is the summed aerodynamic forces acting in the x-body direction. The

same applies to y and z directions.



Aircraft Stability | Criteria for Static Stability

Directional Stability Rotational Stability

b a p q r

Torques

roll

pitch

yaw



Aircraft Stability | Longitudinal Static Stability

Equilibrium condition:

Condition for stability:

a

MC

2

3

1 2 3Aerodyn. Centers

Wing (mean chord) Tail

Zero Lift Line CoG

1

Equilibrium (trim)

Additional Influences:

Fuselage L, D, M

FT, Mm from

propulsion/slipstream

Stability criterion:

Center of gravity IN

FRONT of aerodynamic

center.

Elev.

up

down

20.12.2016 17

resulting static moment



Aircraft Stability | Statics vs. DynamicsDynamic stabilityStatic stability

Disturbance Aerodyn.

reaction

torque

No reaction

torque

Stable

Neutral

Aerodyn.

reaction

torqueUnstable

Stable

Neutral

Unstable

Tre

ate

d w

ith

aero

dyn

am

ic d

eri

vati

va

Mo

delin

g o

f th

e d

yn

am

ics r

eq

uir

ed

20.12.2016 18

Disturbance

Disturbance



Simulation



Simulation

For the airplane

m measured

Ixx, Iyy, Izz calculated using CAD model or “swing” test

CL CD CM, … calculated with CFD software

measured in wind tunnel tests

Precision of physical

parameters in the modelQuality of the model


http://en.wikipedia.org/wiki/Image:Windkanal.jpg

http://en.wikipedia.org/wiki/Image:Windkanal.jpg


10m260m

Simulation | Sky Sailor

Behaviour in open-loop:

• Natural Stability

• Flight speed, glide slope

very close to reality

Initial condition:

• Roll 0°, Pitch -12°, Yaw 0°

• Speed 8.2 m/s

• Control surfaces at 0°

• Motor off

Stabilized after ~50 s

Flight speed ~8.2 m/s

Glide Ratio ~26



Simulation | Sky Sailor

We will investigate more of Sky Sailor’s dynamics in

tomorrow’s exercise.



Modeling for Control



Control of airplanes is not easy:

Inherently non-linear

Low control authority

Actuator saturation

„double integrator“ characteristics

MIMO: 4 inputs, 6 DoF, thus underactuated

Modeling for Control | Introduction



Some remarks about the conventions used in this lecture:

Input limits/units:

Aileron:

Down deflection / left = positive deflection

positive deflections will induce negative moments!!

Modeling for Control | The Plant

Velocities (Body Fr.):

Turn rates (Body Fr.):

Position (Earth Fr.):

Tait-Bryan angles:

Nonlinear

Aircraft

Dynamics

Forces

Moments

Propulsion,

Mechanics,

Aerodynamics

Elevator

Aileron

Rudder

Throttle

State vector:

1,0;1,1;1,1;1,1 Trae

rightaleftaa ,,

20.12.2016 25

Output: e.g.Input vector:



Modeling for Control | The (Linearized) Plant

Δu, Δw;

Δq;

ΔT

e

Longitudinal

Plantr

a

Lateral

Plant

im

re

2

-2

-2

Short Period

Mode:

ω = 5 rad/s

Phugoid

Mode:

ω = 0.6 rad/s

im

re

4

-4

-4

Roll Subsidence

Mode

Spiral Mode

Dutch Roll

Mode

ω = 5 rad/s

Corresponding Poles (Aerobatic Model Airplane)

Subsystem



Modeling for Control | The (Linearized) Plant

Short Period Mode: oscillation of angle of attack. FAST

Phugoid mode: exchange between kinetic and potential energy. SLOW

Spiral Divergence: often

slightly unstable, but

controllable.

Dutch Roll

Mode:

combined yaw-

roll oscillation

Grafics adapted from:

http://history.nasa.gov/SP-367/chapt9.htm and

http://www.fzt.haw-hamburg.de/pers/Scholz/Flugerprobung.html

20.12.2016 27

Longitudinal

Modes

Lateral-

directional

Modes



Fixed-wing Control



A popular concept: cascaded control loops

Control = low level part

Stabilize attitude and speed

Guidance = high level part

Follow paths or trajectory

Effect: Reject constant low frequency

perturbation (constant wind)

Fixed-wing Control | Control & Guidance

Guidance

SKY-SAILORLLCHLC

Inner Loop

Outer Loop



Many control techniques :

Cascaded PID loops

Optimal Control

Robust Control

…

The chosen control techniques determined according to:

Computational Power

Type of flight (aerobatics - level flight)

Fixed-wing Control | Control Concepts



Fixed-wing Control | Optimal Control: LQR

I Linearize the system

around the operating point

xCy

uBxAx

uuxxx

uxfA

,

,

uuxxu

uxfB

,

),(

),(

,

uygy

uxfx

uuxxx

uxgC

,

),(

x,u

where Δx, Δy and Δu constitute differences to the linearization point20.12.2016 31Robot Dynamics - Fixed Wing UAS: Stability, Control, and Derivation of a Dynamic Model



I Linearize the system

around the operating point

xCy

uBxAx

uuxxx

uxfA

,

,

uuxxu

uxfB

,

),(

),(

,

uygy

uxfx

uuxxx

uxgC

,

),(

x,u

where Δx, Δy and Δu constitute differences to the linearization point20.12.2016 32Robot Dynamics - Fixed Wing UAS: Stability, Control, and Derivation of a Dynamic Model



II Define the cost integral

Choose the Matrices Q and R:

Q punishes deviations of the states from the set-point

R punishes deviations of the control inputs from the set-point

0

)()()()( dtttttJ TTRuuQxx

Considerations for the choice of Q and R

• Diagonal Q and R

• Minimal lateral velocity v (coordinated turn, increased drag

otherwise)

• Small variation on airspeed

• Action on ailerons as small as possible (drag!)

• Fast control on roll and pitch




III )()( tt xKu Find the corresponding control law

By solving the (algebraic) Matrix-Riccatti Equation

(for P and K):

(use MATLAB…)

PBRK

0QPBPBRPAPA

T

TT

1

1



Problems:

Non-linear effects when further away from operating point

Computation Costs arising from:

Linearization

Solution to Riccatti Equation:

Too expensive, cannot be done on-line

Way out: compute gains off-line as a look-up table

for discretized state space: Gain-Scheduling




Attitude

Controller

1 PI with anti-reset wind-up2

Gain scaled with 1/VT2

Body Rate

Controller

rd

qd

pdD

2

D2

D2

Fixed-wing Control | Simple Cascaded Control

Airplane

Dynamics

rudd

elev

ail

y

PI1

PI1d

Constrain to

coordinated turn:

Jr

thr

• Bandwidths of inner Loops must

be sufficiently larger!

Trajectory

Generation

and

Guidance



Fixed-wing Control | Guidance

20.12.2016 37

Theroy and Graphics from:

S. Park, J. Deyst, and J. P. How, “A New Nonlinear Guidance Logic for Trajectory Tracking”, Proceedings of the AIAA Guidance, Navigation and Control Conference, Aug

2004. AIAA-2004-4900

Following a Trajectory on the Horizontal Plane



Fixed-wing Control | TECS (Total Energy Control

System)

Control Altitude and Airspeed



Fixed-wing Control | TECS (Total Energy Control

System)



In lower-level loops, dynamics are modeled from

actuators→attitude/airspeed

Note that aside from computaitonal costs, these dynamics are

challenging to globally identify in a nonlinear, high fidelity form.

Thus linearizations are often made.

Higher-level loops often model the aircraft in a three-

degrees-of-freedom (3DoF) sense, mapping

attitude/airspeed→position

Modeling of high-level dynamics does not require identification, as

typically only kinematics are used.

Fixed-wing Control | Note the model abstraction



What could be on the exam?

20.12.2016 41

Basic aerodynamic concepts / definitions

Aircraft reference frames

Model derivation (decoupling)

Performance analysis (calculations)

Stability criteria

Model abstractions for control (high- vs. low-level)



See you tomorrow for the exercise!

20.12.2016 42

Next week – Fixed-wing case studies!


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