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||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
||Autonomous Systems Lab
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
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 2
||Autonomous Systems Lab
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
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 3
||Autonomous Systems Lab
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
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 4
||Autonomous Systems Lab
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
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 5
||Autonomous Systems Lab
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
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 6
||Autonomous Systems Lab
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.)
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 7
||Autonomous Systems Lab
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
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 8
||Autonomous Systems Lab
Sailplane (glider):
Goal of energy
efficiency and flight
endurance
Large wingspan, low
weight
Low speed
Low payload
Different Airplane Configurations
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 9
||Autonomous Systems Lab
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)
Different Airplane Configurations
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 10
||Autonomous Systems Lab
Tandem plane:
Goal of increased longitudinal stability
With center of gravity between the two wings, the plane is more stable
than a classical design
Different Airplane Configurations
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 11
||Autonomous Systems Lab
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
Different Airplane Configurations
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 12
||Autonomous Systems Lab
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
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 13
||Autonomous Systems Lab
Search for the limits
SOLARIMPULSE
Different Airplane Configurations
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 14
||Autonomous Systems Lab
Airbus A380
Before first flight in 2005, it flew only in Simulation !
Why Simulation?
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 15
||Autonomous Systems Lab
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?
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 16
||Autonomous Systems Lab
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
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 17
||Autonomous Systems Lab
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…)
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 18
||Autonomous Systems Lab
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
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 19
||Autonomous Systems Lab
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
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 20
||Autonomous Systems Lab
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
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 21
||Autonomous Systems Lab
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
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 22
||Autonomous Systems Lab
On the Rigidity…
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 23
||Autonomous Systems Lab
On the Rigidity…
NASA Helios Crash: www.nasa.gov/centers/dryden/history/pastprojects/Helios
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 24
||Autonomous Systems Lab
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
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 25
||Autonomous Systems Lab
Application of Newton‘s Second Law
Development of the Model
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
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 26
||Autonomous Systems Lab
Summarized equations of motion:
Translation
Development of the Model
aae
aae
coscoscossinsin1
cossin1
sinsincoscos1
gLDFm
pvquw
gYm
rupwv
gLDFm
qwrvu
T
T
w
v
u
z
y
x
EBC
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 27
||Autonomous Systems Lab
Rotation (simplified with Ixz≈0):
Development of the Model
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
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 28
||Autonomous Systems Lab
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
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 29
||Autonomous Systems Lab
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
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 30
||Autonomous Systems Lab
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
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 31
||Autonomous Systems Lab
Simulation
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 32
||Autonomous Systems Lab
Real Prototype
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 33
||Autonomous Systems Lab
Thermal Soaring
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 34
||Autonomous Systems Lab
Can be solved individually
Assistance in HG E 27: 14:00 – 16:00
Today‘s Exercise
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 35
||Autonomous Systems Lab
See you next week!
23.11.2015Robot Dynamics - Fixed Wing UAS: Stability and Dynamic Model 36
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