folienmaster eth zürich

||Autonomous Systems Lab

151-0851-00 V

Marco Hutter, Michael Blösch, Roland Siegwart, Konrad Rudin and Thomas Stastny

Autonomous Systems Lab

23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 1

Robot DynamicsFixed Wing UAS: Stability and Dynamics


1. Overview

2. Aerodynamic Basics

3. Performance

Considerations

4. Stability

5. Simplified Dynamic

Model

6. UAV Control

Approaches

7. Case Studies

Lecture 2:

Stability and Dynamics

1. Some Notations

2. Stability

Overview: Static and Dynamic

Stability

Criteria for Static Stability

3. Simplified Dynamic Model

Coordinate Frames and

Representation of Orientation

Assumptions and Simplifications

Derivation of the Model

Simulation Results

Contents:

Fixed Wing UAS



The objective of this lecture is:

Not to give you all the detailled theory of flight dynamics

(already the topic of „Flugtechnik“ by Dr. Wildi)

To show the application of the most important theory in order to

analyze stability and to create a mathematical model of an

airplane

Introduction



Additional Notations

Moments

LA : Roll moment

MA : pitch moment

NA : Yaw moment

L/M/NT :Thrust moment

Angles

a : Angle of attack

b : Sideslip angle

e : Thrust-vector angleSpeed (no wind!)

Angular Rates

Important Points

: CoG

: Aerodynamic Center

Forces

FA : Aerodynamic force

FT : Thrust

G : Weight

Background image:

http://upload.wikimedia.org/wikiped

ia/commons/

5/5c/C_172_line_drawing_oblique.

svg

0AC

Ma

zB

xB

yB

FT

e

MA

G

Bv

ab (-)

= yS

xS zS

S: Stability Frame

D=-FA,xs

-FA,zs=L

FA,ys=Y

FA

LA

NA

p

r

q

u

v

w

Bv= u,v,w( )T



Stability Example: LonditudinalDynamic stabilityStatic stability

Disturbance Aerodyn.

reaction

torque

Disturbance No reaction

torque

Stable

Neutral

Disturbance 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



Criteria for Static Stability (1)

Apply in Stability Coordinate Frame

Velocity Stability

u v w

Forces

x

y

z

0 TxAx FFu

0 TyAy FFv

0 TzAz FFw

0 yCb

0 DTx CuCu

0 LCa



Criteria for Static Stability (2)

Directional Stability Rotational Stability

b a p q r

Torques

roll

pitch

yaw

0 TA LLb

0Cl b

0 TA NNb

0CN b

0 TA MMa

0CM a

0 TA LLp

0 lCp

0 TA MMq

0 MCq

0 TA NNr

0 NCr

(Moment „L“ Coeff.)



Example: Longitudinal Static Stability

Equilibrium condition:

Condition for stability: 0CM a

0CM

a

MC

0CM a

0CM a

2

3

1 2 3Aerodyn. Centers

Wing (mean chord) Tail

Zero Lift Line CoG

1

Equi-

librium

(trim

)

Additional Influences:

Fuselage L, D, M

FT, M from

propulsion/slipstream

Stability criterion:

Aerodynamic Center of

the Airplane BEHIND

CoG

Elev.

up

down



Sailplane (glider):

Goal of energy

efficiency and flight

endurance

Large wingspan, low

weight

Low speed

Low payload

Different Airplane Configurations



Fighter aircraft:

Goal of high speed, climbing rate, maneuverability, stealthiness

Strong engines, short wings (swept) with high chord length,

complex geometry, large control surfaces

High fuel consumption (and thus limited operating range)




Tandem plane:

Goal of increased longitudinal stability

With center of gravity between the two wings, the plane is more stable

than a classical design




Biplane:

More compact layout with shorter wingspan

Higher maneuverability

Very popular in the early days of aviation

But: more drag and less lift than a classical design with equal wing

area




Flying wing aircraft:

Most commonly used in the low to medium speed range

High stealth capabilities (low visibility for radar)

Fuel efficient due to low drag

Stability issues: directional and longitudinal

Problem: no passenger windows (in commercial application)

Different Airplane configurations



Search for the limits

SOLARIMPULSE




Airbus A380

Before first flight in 2005, it flew only in Simulation !

Why Simulation?



System analysis:

model allows evaluating future flight characteristics Stability

Controllability

Power required fuel needs

Controllability in the case of actuator failure

Autopilot design and simulation:

model allows comparing different control techniques and

autopilot parameter tuning Gain of time and money

Higher performance of the autopilot

No risk of damage compared to real tests

Pilot training (in Simulator) Allows simulating and training especially emergency situations

Why Model the Dynamics of an Airplane?



Dynamics of an airplane

... Are very different from an acrobatic aircraft to a line jet airplane

... but the principles remain the same for all

Wings, stabilizers

control surfaces (ailerons, rudder, elevator, flaps,spoilers)

propulsion group (motor-gearbox-propeller, turbine, rocket,…)

In this lecture, we will model the solar airplane Sky-Sailor

Steps for the creation of the model

1. Define the coordinate frames

2. List all the physical effects acting on the airplane

3. Set assumptions, make simplifications

4. Express the physical effects into equations

5. Derive the equations of motion (here: Newton)

Introduction



Coordinate FramesEarth fixed frame

(regarded as inertial): Body fixed frame:exB,eyB,ezBexE,eyE,ezE

eyE

ezE

exEψ

ey1

=ez1

ex1

θ

ey2=

ez2

ex

2φ

=ex

B

eyB

ezB

eyE

ezE

exEψ

ey1

=ez1

ex1

θ

ey2=

ez

2

ex2

eyE

ezE

exEψ

ey1

=ez1

ex1

Rotation Matrix (B to E) is parametrized with 3 successive rotations using

the zyx Tait-Brian Angles (specific kind of Euler Angles):CEB

Roll:

φ around ex2:

Frame B

3 )(2 BCPitch:

θ around ey1:

Frame 2

2 )(12 CYaw:

ψ around ezE:

Frame 1

1 )(1 EC

EBC )(1 EC )(12 C )(2 BC (post-multiply for rotations

around new axes…)



The rotation matrix calculated:

Be careful with the boundaries:

The inverse transformation:

Coordinate Frames

Roll (-<<) Pitch (-/2<</2) Yaw (-<<)

)()(0

)()(0

001

)(2

cs

scBC

)(0)(

010

)(0)(

)(12

cs

sc

C

100

0)()(

0)()(

)(1

cs

sc

EC

)()()()()(

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

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

2121

ccscs

sccssccssscs

sscsccsssccc

EEEB CCCC

TEBEBBE CCC 1



Angular Rates:

Time variation of Tait-Bryan angles

Body angular rates

Singularity: for (Jr becomes singular)

« Gimbal Lock »

Coordinates System

≠ ,,

rqp ,,

coscossin0

cossincos0

sin01

rJ

r

r

q

p

J

2



Forces and moments acting on the airplane

Weight at the center of gravity

Thrust of propeller: complex task will not be presented here

Aerodynamic forces on each

part of the airplane:

see previous lecture…

Wing

Tail

Fuselage

Forces and Moments

z

x

y

T

M

L

N

G

V

a

D

L

Y



Definitions

Remember: origin of body-fixed coordinate frame set into center of gravity

Assumptions and simplifications

Rigid and symmetric structure:constant, (almost) diagonal inertia matrix

Constant mass

Motor without dynamics and without gyroscopic effects (can be adapted)

Aerodynamics (list not complete):

We don’t enter stall (operation in the linear cl domain)

Neglect fuselage lift/sideslip force (may be easily included, if modeled correctly)

Inputs/Outputs/States

Definitions, Assumptions and Simplifications

Velocities (Body Fr.): u,v,w

Turn rates (Body Fr.): p,q,r

Position (Earth Fr.): x,y,z

Tait-Bryan angles: ,,

Nonlinear

Aircraft

Dynamics

Forces

Moments

u,v,w;

p,q,r

x,y,z;

,,

Propulsion,

Mechanics,

Aerodynamics

Elevator

Aileron

Rudder

Throttle



On the Rigidity…



On the Rigidity…

NASA Helios Crash: www.nasa.gov/centers/dryden/history/pastprojects/Helios



Forces and momentsrepresented in body frame, attacking at the CoG:

Development of the Model

aae

aae

e

e

aa

aa

coscoscossinsin

cossin

sinsincoscos

0

0

sin

0

cos

cossin

sincos

mgLDF

mgY

mgLDF

g

m

F

F

LD

Y

LD

T

T

BE

T

T

tot CF

T

T

T

tot

N

M

L

N

M

L

M



Application of Newton‘s Second Law


Euler

Derivatives

xzxxyyzzxz

xzzzxxyy

xzyyzzxzxx

zzxz

yy

xzxx

zzxz

yy

xzxx

qrIIIpqrIpI

IprIIprqI

qpIIIqrrIpI

r

q

p

II

I

II

r

q

p

r

q

p

II

I

II

22

0

00

0

0

00

0

Ftot =d

dtm

b Bv( ) =

qupvw

pwruv

rvqwu

m

w

v

u

r

q

p

w

v

u

m

r

q

p

II

I

II

dt

d

dt

d

zzxz

yy

xzxx

Btot

0

00

0

IMTypically

small



Summarized equations of motion:

Translation


aae

aae

coscoscossinsin1

cossin1

sinsincoscos1

gLDFm

pvquw

gYm

rupwv

gLDFm

qwrvu

T

T

w

v

u

z

y

x

EBC



Rotation (simplified with Ixz≈0):


xxyyT

zz

zzxxT

yy

yyzzT

xx

IIpqNNI

r

IIprMMI

q

IIqrLLI

p

1

1

1

cos

cos

cos

sinsincos

costansintan1

rq

rq

rqp

r

q

p

rJ



c

Turning

Demand for coordinated turn:

L increases with

Vmin increases with

Stationary Flight

constc

a

0Y

R

22

mRR

mV

D

≈FT

L

G

ccos

1

ccos

1



10m260m

Simulation

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

For the airplane

m measured

Ixx, Iyy, Izz calculated using CAD model

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


Simulation



Real Prototype



Thermal Soaring



Can be solved individually

Assistance in HG E 27: 14:00 – 16:00

Today‘s Exercise



See you next week!


folienmaster eth zürich

Documents