mathematical models for control of aircraft and satellites slides.pdf · 6! lecture notes ttk 4190...

1 Lecture Notes TTK 4190 Guidance and Control of Vehicles (T. I. Fossen)

Mathematical Models for Control of Aircraft and Satellites 1 Introduc+on 2 Aircra0 Modeling

2.1 Defini+on of Aircra0 State-‐Space Vectors 2.2 Body-‐Fixed Coordinate Systems for Aircra0 2.3 Aircra0 Equa+ons of Mo+on 2.4 Perturba+on Theory (Linear Theory) 2.5 Decoupling in Longitudinal and Lateral Modes 2.6 Aerodynamic Forces and Moments 2.7 Propeller Thrust 2.8 Aerodynamic Coefficients 2.9 Standard Aircra0 Maneuvers 2.10 Aircra0 Stability Proper+es 2.11 Design of flight control systems

3 Satellite Modeling 3.1 AXtude Model 3.2 Actuator Dynamics 3.3 Design of Satellite AXtude Control Systems

4 Matlab Simula+on Models 4.1 Boeing-‐767 4.2 F-‐16 Fighter 4.3 F2B Bristol Fighter 4.4 NTNU Cube Satellite


The lecture notes •  Fossen, T. I. (2011). Mathematical Models for Control of Aircraft and Satellites��

(2nd edition). Department of Engineering Cybernetics, Lecture Notes, January 2011. uses a vectorial notation to describe aircraft and satellites. The notation is similar to the one used for marine craft (ships, high-speed craft and underwater vehicles). The equations of motion are based on: •  Fossen, T. I. (1994). Guidance and Control of Ocean Vehicles ��

Wiley, Chapter 2. •  Fossen, T. I. (2011). Handbook of Marine Craft Hydrodynamics and Motion Control ��

Wiley, Chapters 2 and 3. The kinematic and kinetic equations of a marine craft can be modified to describe aircraft and satellites by minor adjustments of notation and assumptions.

Chapter 1 Introduction


Chapter 2 – Aircraft Modeling

1 Introduc+on 2 Aircra0 Modeling





2.1 Definition of Aircraft State-Space Vectors

Comment 1: No+ce that the capital le[ers L, M, N for the moments are different from those used for marine cra0—that is, K, M, N: The reason for this is that L is reserved as length parameter for ships and underwater vehicles. Comment 2: For aircra0 it is common to use capital le[ers for the states U, V, W, etc. while it is common to use small le[ers u, v, w, etc. for marine cra0.

! :!

UVWPQR

!

longitudinal (forward) velocitylateral (transverse) velocityvertical velocityroll ratepitch rateyaw rate

" :!

XEYEZE ,h!"

#

!

Earth-fixed x-positionEarth-fixed y-positionEarth-fixed z-position (axis downwards), altituderoll anglepitch angleyaw angle

#

#

Figure 2.1: Defini+on of aircra0 body axes, veloci+es, forces, moments and Euler angles (McLean 1990).

XYZLMN

:!

longitudinal forcetransverse forcevertical forceroll momentpitch momentyaw moment

#


2.2 Body-Fixed Coordinate Systems for Aircraft

For aircraft it is common to use the following body-fixed coordinate systems:��

•  Body axes •  Stability axes •  Wind axes

��The axis systems are shown in Figure 2.2 where the angle of attack a and�� sideslip angle b are defined as: Where Is the total speed of the aircra0.

Figure 2.2: Defini+on of stability and wind axes for an aircra0 (Stevens and Lewis 1992).

tan!!" :! WU

sin!"" :! VVT

#

#

VT ! U2 " V2 "W2 #


2.2 Body-Fixed Coordinate Systems for Aircraft Aerodynamic effects are classified according to the Mach number:

where a = 340 m/s = 1224 km/h is the speed of sound in air at a temperature of 20o C on the ocean surface. The following terminology for speed ranges is used:

Subsonic speed M < 1.0 Transonic speed 0.8 ≤ M ≤ 1.2 Supersonic speed 1.0 ≤ M ≤ 5.0 Hypersonic speed 5.0 ≤ M

An aircra0 will break the sound barrier at M = 1.0 and this is clearly heard as a sharp crack. If you fly at low al+tude and break the sound barrier, windows in building will break due to pressure-‐induced waves.

M :! VTa #



pwind ! Rz,!!pstab !

cos!!" sin!!" 0! sin!!" cos!!" 0

0 0 1pstab

pstab ! Ry,"pbody !cos!"" 0 sin!""

0 1 0! sin!"" 0 cos!""

pbody

#

#

Rbodywind ! Rz,!!Ry," #

Rota+on matrices for wind and stability axes The rela+onship between vectors expressed in different coordinate systems can be derived using rota+on matrices. •  The body-‐fixed coordinate system is first rotated a nega+ve sideslip angle -‐β about

the z-‐axis •  The new coordinate system is then rotated a posi+ve angle of a[ack α about the new

y-‐axis such that the resul+ng x-‐axis points in the direc+on of the total speed VT. The first rota+on defines the wind axes while the second rota+on defines the stability axes. This can be mathema+cally expressed as:


pwind ! Rbodywindpbody

"

pwind !

cos!!" sin!!" 0! sin!!" cos!!" 0

0 0 1

cos!"" 0 sin!""0 1 0

! sin!"" 0 cos!""pbody

"

pwind !

cos!""cos!!" sin!!" sin!""cos!!"! cos!"" sin!!" cos!!" ! sin!"" sin!!"

! sin!"" 0 cos!""pbody

#

#

#


Rbodywind ! Rz,!!Ry," #

This gives the following rela+onship between the veloci+es in body and wind axes:

vbody !UVW

! !Rbodywind"!vwind ! Ry,!! Rz,!"!

VT00

!

VT cos!!"cos!""VT sin!""VT sin!!"cos!""

#


2.3 Aircraft Equations of Motion Kinema+c equa+ons for transla+on The kinema+c equa+ons for transla+on and rota+on of a body-‐fixed coordinate system ABC with respect to a local geographic coordinate system NED (North-‐East-‐Down) can be expressed in terms or the Euler angles:

X! EY! EŻE

" Rabcned

UVW

" Rz,#Ry,$Rx,!UVW

#

X! EY! EŻE

"c# !s# 0s# c# 00 0 1

c$ 0 s$0 1 0

!s$ 0 c$

1 0 00 c! !s!0 s! c!

UVW

%

X! EY! EŻE

"c#c$ !s#c! & c#s$s! s#s! & c#c!s$s#c$ c#c! & s!s$s# !c#s! & s$s#c!!s$ c$s! c$c!

UVW

#


2.3 Aircraft Equations of Motion Kinema+c equa+ons for aXtude The aXtude is given by: which gives

PQR

!!"

00

# Rx,!"0$"

0# Rx,!" Ry,$"

00%"

#

!!

"!

#!$

1 s!t" c!t"0 c! !s!0 s!/c" c!/c"

PQR

, c" " 0 #


2.3 Aircraft Equations of Motion Rigid-‐body kine+cs The aircra0 rigid-‐body kine+cs can be expressed as (Fossen 1994, 2011): where ν₁:=[U,V,W]T, ν₂:=[P,Q,R]T, τ₁:=[X,Y,Z]T and τ₂:=[L,M,N]T It is assumed that the coordinate system is located in the aircra0 center of gravity (CG).

m!!" 1 ! !2 ! !1" " #1

ICG!" 2 ! !2 ! !ICG!2" " #2

# #

MRB!" ! CRB!!"! " #RB #

MRB !mI3!3 O3!3

O3!3 ICG, CRB!!" !

mS!!2" O3!3

O3!3 !S!ICG!2" #

ICG :!Ix 0 !Ixz0 Iy 0

!Ixz 0 Iz

#

The iner+a tensor is defined as (assume that Ixy=Iyz=0 which corresponds to xz-‐plane symmetry)


2.3 Aircraft Equations of Motion The forces and moments ac+ng on the aircra0 can be expressed as: where τ is a generalized vector that includes aerodynamic and control forces. The gravita+onal force fG = [0 0 mg]T acts in the CG (origin of the body-‐fixed coordinate system) and this gives the following vector expressed in NED: Hence, the aircra0 model can be wri[en:

!RB ! !g!"" " ! # g!!" ! !!Rabcned"!

fGO3!1

!

mg sin!""

!mg cos!"" sin!""!mg cos!""cos!""

000

#

m!U! " QW ! RV " g sin!#"" $ Xm!V! " UR ! WP ! g cos!#" sin!!"" $ Ym!W! " VP ! QU ! g cos!#"cos!!"" $ Z

IxP! ! Ixz!R! " PQ" " !Iz ! Iy"QR $ LIyQ! " Ixz!P2 ! R2" " !Ix ! Iz"PR $ MIzR! ! IxzP! " !Iy ! Ix"PQ " IxzQR $ N

#

MRB!" ! CRB!!"! ! g!#" " $ #


2.3 Aircraft Equations of Motion Sensors and measurement systems It is common that aircra0 sensor systems are equipped with three accelerometers. If the accelerometers are located in the CG, the measurement equa+ons take the following form: In addi+on to these sensors, an aircra0 is equipped with:

•  Gyros for angular veloci+es •  Magnetometers for heading •  Al+tude sensor measuring h •  Wind speed VT

These sensors are used in iner+al naviga+on systems (INS) which again use a Kalman filter to compute es+mates of U,V,W,P,Q and R as well as the Euler angles Φ,Θ and Ψ. Other measurement systems that are used onboard aircra0 are global naviga+on satellite systems (GNSS), radar and sensors for angle of a[ack.

axCG ! Xm ! U" # QW ! RV # g sin!$"

ayCG ! Ym ! V" # UR ! WP ! g cos!$"sin!!"

azCG ! Zm ! W" # VP ! QU ! g cos!$"cos!!"

#

#

#


2.4 Perturbation Theory (Linear Theory) Defini+on of nominal and perturba+on values According to linear theory it is possible to write the states as the sum of a nominal value (usually constant) and a perturba+on (devia+on from the nominal value). Moreover, The following defini+ons are made: Consequently, a linearized state-‐space model will consist of the states u,v,w,p,q,r,θ,φ and ψ.

Total state ! Nominal value " Perturbation

! :! !0 " !! !

X0

Y0

Z0

L0

M0

N0

"

!X!Y!Z!L!M!N

, " :! "0 " !" !

U0

V0

W0

P0

Q0

R0

"

uvwpqr

#

!

!"

:#!0

!0

"0

$

!

"

#

#


2.4 Perturbation Theory (Linear Theory) Lineariza+on of the rigid-‐body kine+cs If the aerodynamic forces and moments, veloci+es, angles and control inputs are expressed as nominal values and perturba+ons τ = τ₀+δτ, ν = ν₀+δν and η = η₀+δη, the aircra0 equilibrium point will sa+sfy (it is assumed that ν₀= 0): This can be expanded according to: The linearized equa+ons of mo+on are usually derived by a 1st-‐order Taylor series expansion about the nominal values. Alterna+vely, it is possible to subs+tute nominal values + perturba+ons into the nonlinear expressions and neglect higher-‐order terms of the perturbed states.

CRB!!0"!0 ! g!"0" " #0 #

m!Q0W0 ! R0V0 ! g sin!"0"" # X0

m!U0R0 ! P0W0 ! g cos!"0" sin!!0"" # Y0

m!P0V0 ! Q0U0 ! g cos!"0"cos!!0"" # Z0

!Iz ! Iy"Q0R0 ! P0Q0Ixz # L0

!P02 ! R0

2"Ixz ! !Ix ! Iz"P0R0 # M0

!Iy ! Ix"P0Q0 ! Q0R0Ixz # N0

#


2.4 Perturbation Theory (Linear Theory) Example 1 (Lineariza+on of surge using perturba+on theory)

Using: In steady-‐state and The equa+ons of mo+on is reduced to: If it is assumed that the 2nd-‐order terms wq and vr are negligible, the linearized model becomes:

m!U! " QW ! RV " g sin"##$ $ X%

m!U! 0 " u! " "Q0 " q#"W0 " w# ! "R0 " r#"V0 " v# " g sin"#0 " !#$ $ X0 " "X #

sin!!0 " !" # sin!!0"cos!!" " cos!!0"sin!!"! small! sin!!0" " cos!!0"! #

U! 0 " 0

m!Q0W0 ! R0V0 ! g sin!"0"" # X0 #

m!u! " Q0w "W0q " wq ! R0v ! V0r ! vr " gcos"#0#!$ $ "X #

m!u! " Q0w "W0q ! R0v ! V0r " gcos"#0#!$ $ "X #


2.4 Perturbation Theory (Linear Theory) Linear state-‐space model for aircra0 If all DOFs are linearized, the following state-‐space model is obtained: Matrix-‐vector form

m!u! " Q0w " W0q ! R0v ! V0r " g cos"#0#!$ $ "Xm!v! " U0r " R0u ! W0p ! P0w ! g cos"#0#cos"!0## " g sin"#0# sin"!0#!$ $ "Ym!w! " V0p " P0v ! U0q ! Q0u " g cos"#0# sin"!0## " g sin"#0#cos"!0#!$ $ "Z

Ixp! ! Ixzr! " "Iz ! Iy#"Q0r " R0q# ! Ixz"P0q " Q0p# $ "LIyq! " "Ix ! Iz#"P0r " R0p# ! 2Ixz"R0r " P0p# $ "M

Izr! ! Ixzp! " "Iy ! Ix#"P0q " Q0p# " Ixz"Q0r " R0q# $ "N

#

MRB!" ! NRB! ! G# " $ #

MRB !

m 0 0m 0 03!3

0 0 mIx 0 !Ixz

03!3 0 Iy 0!Ixz 0 Iz

NRB !

0 !mR0 mQ0 0 mW0 !mV0mR0 0 !P0 !W0 0 U0!mQ0 mP0 0 mV0 !mU0 0

!IxzQ0 !Iz ! Iy"R0 ! IxzP0 !Iz ! Iy"Q003!3 !Ix ! Iz"R0 !2IxzP0 !Ix ! Iz"P0 ! 2IxzR0

!Iy ! Ix"Q0 !Iy ! Ix"P0 " IxzR0 IxzQ0

G !

0 mg cos!"0" 003!3 !mg cos!"0"cos!!0" mg sin!"0" sin!!0" 0

mg cos!"0" sin!!0" mg sin!"0"cos!!0" 0

03!3 03!3


2.4 Perturbation Theory (Linear Theory) Linear state-‐space model using wind and stability axes An alterna+ve state-‐space model is obtained by using α and β as states. Assume that α and β are small such that cos(α) ≈ 1 and sin(β) ≈ β. Hence, The rela+onship between the body-‐fixed velocity vector: ν = [u,v,w,p,q,r]T and the new state-‐space vector x can be wri[en as: If the total speed is VT = U₀ = constant (linear theory), it is seen that:

U ! VT

V ! VT!

W ! VT"

"

U ! VT

! ! VVT

" ! WVT

# x !

u!

"

pqr

!

surge velocitysideslip angleangle of attackroll ratepitch rateyaw rate

#

! ! Tx ! diag!1,VT,VT,1,1,1,1"x #

!! " 1VTw!

"! " 1VTv!

V! T " 0

#

#

#

!! " UW! ! WU!U2 # W2

"! " V! VT ! VV! TVT2 cos"

V! T " UU! # VV! # WW!VT

#

#

#

lineariza+on


2.5 Decoupling in Longitudinal and Lateral Modes For an aircra0 it is common to assume that the longitudinal modes (DOFs 1, 3 and 5) are decoupled from the lateral modes (DOFs 2, 4 and 6) by assuming: •  The fuselage is slender (length is much larger than the width and the height) •  The longitudinal velocity is much larger than the ver+cal and transversal veloci+es. In order to decouple the rigid-‐body kine+cs in longitudinal and lateral modes it will be assumed that the states v,p,r and φ are negligible in the longitudinal channel while u,w,q and θ are negligible when considering the lateral channel. This gives two decoupled sub-‐systems:

•  Longitudinal equa+ons •  Lateral equa+ons


2.5 Decoupling in Longitudinal and Lateral Modes Longitudinal equa+ons Kine2cs Kinema2cs

m!u! " Q0w " W0q " g cos"#0#!$ $ "Xm!w! ! U0q ! Q0u " g sin"#0#cos"!0#!$ $ "Z

Iyq! $ "M #

m 0 00 m 00 0 Iy

u!w!q!

"

0 mQ0 mW0

!mQ0 0 !mU0

0 0 0

uwq

"

mg cos!#0"

mg sin!#0"cos!!0"

0! $

"X"Z"M

#

!! " q #

DOFs 1, 3 and 5


2.5 Decoupling in Longitudinal and Lateral Modes Lateral equa+ons Kine2cs Kinema2cs

m!v! " U0r ! W0p ! g cos"#0#cos"!0#!$ $ "YIxp! ! Ixzr! " "Iz ! Iy#Q0r ! IxzQ0p $ "LIzr! ! Ixzp! " "Iy ! Ix#Q0p " IxzQ0r $ "N

#

m 0 00 Ix !Ixz0 !Ixz Iz

v!p!r!

"

0 !mW0 mU0

0 !IxzQ0 !Iz ! Iy"Q0

0 !Iy ! Ix"Q0 IxzQ0

vpr

"

!mg cos!#0"cos!!0"

00

! $

"Y"L"N

#

!!

"!"

1 tan!#0"

0 1/cos!#0"

pr

#

DOFs 2, 4 and 6


2.6 Aerodynamic Forces and Moments Aerodynamic coefficients The following abbrevia+ons and nota+on will be used for the aerodynamic coefficients: In order to illustrate how control surfaces influence the aircra0, an aircra0 equipped with the following control inputs will be considered

X index ! !X! index

L index ! !L! index

Yindex ! !Y! index

M index !!M

! index

Z index ! !Z! index

Nindex ! !N! index

!T Thrust Jet/propeller

!E ElevatorControl surfaces on the rear of the aircraft used for pitch andaltitude control

!A AileronHinged control surfaces attached to the trailing edge of the wing usedfor roll/bank control

!F FlapsHinged surfaces on the trailing edge of the wings used for brakingand bank-to-turn

!R Rudder Vertical control surface at the rear of the aircraft used for turning


2.6 Aerodynamic Forces and Moments

Control inputs for conven+onal aircra0. No+ce that the two ailerons can be treated as one control input:

!A ! 1/2!!AL " !AR"


2.6 Aerodynamic Forces and Moments Linear theory will be assumed in order to reduce the number of aerodynamic coefficients. Control inputs and aerodynamic forces and moments are wri[en as: where -‐MF is aerodynamic added mass, -‐NF is aerodynamic li0 and drag and B is a matrix describing the actuator configura+on (thrust and control surfaces). The actuator dynamics is modeled by a 1st-‐order system: where uc is commanded input, u is the actual control input produced by the actuators and T = diag{T₁,T₂,...,Tr} is a diagonal matrix of posi+ve +me constants. This gives: The matrices M and N are defined as M = MRB+MF and N = NRB+NF. The linearized kinema+cs takes the following form:

! ! !MF"# ! NF" " Bu #

u! ! T!1!uc ! u" #

!MRB !MF"!" ! !NRB ! NF "! ! G# " Bu#

M!" ! N! ! G# " Bu #

!" ! J!# " J"! #


2.6 Aerodynamic Forces and Moments Linear state-‐space model with actuator dynamics Linear state-‐space model neglec+ng the actuator dynamics

!"

#"

u"!

J! J" 0!M!1G !M!1N M!1B0 0 !T!1

!

#

u"

00T!1

uc #

!"

#"!

J! J"!M!1G !M!1N

!

#"

0M!1B

u #


2.6 Aerodynamic Forces and Moments Longitudinal aerodynamic forces and moments McLean (1990) expresses the longitudinal forces and moments as: This corresponds to the matrices -‐MF , -‐NF and B . If the cruise speed U₀= constant, then δT = 0. Al+tude can be controlled by using the elevators δE. Flaps δF can be used to reduce the speed during landing. The flaps can also be used to turn harder for instance by moving one flap while the other is kept at the zero posi+on. This is common in bank-‐to-‐turn maneuvers. For conven+onal aircra0 the following aerodynamic coefficients can be neglected: such that

dXdZdM

!

Xu" Xw" Xq"Zu" Zw" Zq"Mu" Mw" Mq"

u"w"q"

#

Xu Xw XqZu Zw ZqMq Mw Mq

uwq

#

X!T X!E X!F

Z !TZ!E Z!F

M!T M!E M!F

!T!E!F

#

Xu! ,Xq,Xw! ,X!E ,Zu! ,Zw! ,Mu! #

dXdZdM

!

0 0 Xq"0 0 Zq"0 Mw" Mq"

u"w"q"

#

Xu Xw 0Zu Zw ZqMq Mw Mq

uwq

#

X!E

Z!E

M!E

!E #


2.6 Aerodynamic Forces and Moments Lateral aerodynamic forces and moments McLean (1990) expresses the longitudinal forces and moments as: This corresponds to the matrices -‐MF , -‐NF and B . For conven+onal aircra0 the following aerodynamic coefficients can be neglected: This gives

dYdLdN

!

Yv" Yp" Yr"Lv" Lp" Lr"Nv" Np" Nr"

v"p"r"

#

Yv Yp YrLv Lp LrNv Np Nr

vpr

#

Y!A Y!RL!A L!R

N!A N!R

!A!R

#

Yv! ,Yp,Yp! ,Yr,Yr! ,Y!A ,Lv! ,Lr! ,Nv! ,Nr! #

dYdLdN

!

0 0 00 Lp" 00 Np" 0

v"p"r"

#

Yv 0 0Lv Lp LrNv Np Nr

vpr

#

0 Y!RL !A

L!R

N!A N!R

!A!R

#


2.7 Propeller Thrust

T ! Kt!n2D4 #

T Thrust force [N]Kt Thrust coefficient! Density of air [kg/m3]n Propeller angular velocity [rad/s]D Propeller diameter [m]

ftb !

XtYtZt

!

T00

# mtb !

LtMt

Nt

! rt/bb ! ftb ! S!rt/bb "ft

b

!

0Trtz!Trty

#

rt/bb ! !rtx ,rty ,rtz"! is the location of the propeller with respect to CO


2.8 Aerodynamic Coefficients

fab !

XaYaZa

! q"SCXCYCz

mab !

LaMa

Na

! q"SbClc"CmbCn

#

#

The most important parameters used to describe an airfoil are:

b Wing span (tip to tip)c Wing chord (varies along span)c! Mean aerodynamic chordS Total wing areaAR " b2/S Aspect ratio

q! " 12 !aVT2 #

Dynamic pressure: Aerodynamic forces and moments:



The aerodynamic coefficients are in general functions of angle of attack α, sideslip angle β, Mach number M, altitude h, control surface deflections δS and the thrust coefficient: where SD is the area of the disc swept out by a propeller blade. Therefore, the aerodynamic coefficients are functions: Consequently, the aerodynamic coefficients may have complicated dependences, which are hard to model with great accuracy. If the aircraft has restricted flight modes, that is restricted to low Mach numbers and small angles of attack and sideslip, the aerodynamic coefficients may be greatly simplified. The complicated dependency may then be broken into a sum of simpler linear terms.

Ci, i ! D,Y,L, l,m,n

TC ! Tq"SD

#

Ci ! Ci!!,",M,h,#S,TC" #



CD0CL

! Rbs!CX

0!CZ

!

cos!!" 0 sin!!"0 1 0

! sin!!" 0 cos!!"

!CX0

!CZ

#

fab !

XaYaZa

! q"S0CY0

# q"SRsb!CD

0!CL

! q"S!CD cos!!" # CL sin!!"

CY!CD sin!!" ! CL cos!!"

#

Relationship between CX, CZ and CD, CL (drag/lift):



CD ! CD0 " CD!! " CD"E"E " CD"R

"R " CD"F"F #

CY ! CY0 " CY!! " CY"R"R " CY"A"A #

CL ! CL0 " CL!! " CL"F"F #

Linear Theory Lift and drag coefficients Sideforce coefficient Rolling moment coefficient Pitching moment coefficient Yawing moment coefficient

Cl ! Cl0 " Cl!! " Cl"A"A " Cl"R"R " Clpb

2VTP " Clr b

2VTR #

Cm ! Cm0 " Cm!! " Cm"e"e " Cmqc#

2VTQ #

Cn ! Cn0 " Cn!! " Cn"R"R " Cn"A"A " Cnpb

2VTP " Cnr b

2VTR #



Equations of motion including aerodynamic coefficients U! " VR ! WQ ! g sin!#" $ T

m $q%Sm CX

V! " WP ! UR $ g cos!#"sin!!" $ q%Sm CY

W! " UQ ! VP $ g cos!#"cos!!" $ q%Sm CZ

P! ! IxzIxR! " ! Iz ! IyIx

QR $ IxzIxPQ $

q%bSIxCl

Q! " ! Ix ! IzIyPR ! IxzIy

!P2 ! R2" $ TrtzIy$IpIy

&pR $q%Sc%IyCm

R! ! IxzIzP! " ! Iy ! IxIz

PQ ! IxzIzQR !

TrtyIz

! IpIz&pQ $

q%SbIzCn

#

#

#

#

#

#


2.9 Standard Aircraft Maneuvers The nominal values depends on the aircra0 maneuver. For instance:

1.  Straight flight: Θ₀ = Q₀ = R₀ = 0 2.  Symmetric flight: Ψ₀ = V₀ = 0 3.  Flying with wings level: Φ₀ = P₀ = 0

Constant angular rate maneuvers can be classified according to:

1.  Steady turn: R₀ = constant 2.  Steady pitching flight: Q₀= constant 3.  Steady rolling/spinning flight: P₀ = constant


2.9 Standard Aircraft Maneuvers Dynamic equa+on for coordinated turn (bank-‐to-‐turn) A frequently used maneuver is coordinated turn where the accelera+on in the y-‐direc+on is zero, no sideslip and zero steady-‐state pitch and roll angles: Furthermore it is assumed that VT = U₀= constant Assume that the longitudinal and lateral mo+ons are decoupled (u=w=q=θ=0). If perturba+on theory is applied under the assump+on that the 2nd-‐order terms ur=pw=0,we get: Steady-‐state condi+on

! ! !" ! 0#0 ! !0 ! 0

# #

m!V! " UR ! WP ! g cos"##sin"!#$ $ Y%

m!"U0 " u#"R0 " r# ! "W0 " w#"P0 " p# ! g cos"!#sin""#$ $ Y0

#

#

! ! V/U0 V ! V" ! 0

!Y ! 0

m!U0R0 ! U0r !W0P0 !W0p ! g sin!!"" " Y0 #

m!U0R0 !W0P0" ! Y0 # m!U0r !W0p ! g sin!!"" ! 0 #


2.9 Standard Aircraft Maneuvers Bank-‐to-‐turn The aircra0 is o0en trimmed such that the angle of a[ack α₀=W₀/U₀ = 0. This implies that the yaw rate can be expressed as: This is a very important result since it states that a roll angle φ different from zero will induce a yaw rate r which again turns the aircra0 (bank-‐to-‐turn). With other words, we can use a moment in roll, for instance generated by the ailerons, to turn the aircra0. The yaw angle is given by: An alterna+ve method is of course to turn the aircra0 by using the rear rudder to generate a yaw moment. The bank-‐to-‐turn principle is used in many missile control systems since it improves maneuverability, in par+cular in combina+on with a rudder controlled system.

m!U0r !W0p ! g sin!!"" ! 0 #

r ! W0U0p " g

U0sin!!" #

r ! gU0

sin!!"! small! g

U0! #

!! " r #


2.9 Standard Aircraft Maneuvers Example 2 (Augmented turning model using rear rudders) Turning autopilots using the rudder δR as control input is based on the lateral state-‐space model. The differen+al equa+on for ψ is augmented on the lateral model as shown below:

v!p!r!!!

"!

"

a11 a12 a13 a14 0a21 a22 a23 0 0a31 a32 a33 0 00 1 0 0 00 0 1 0 0

vpr!

"

#

b11

b21

b31

00

#R #


2.9 Standard Aircraft Maneuvers Example 3 (Augmented bank-‐to-‐turn model using ailerons) Bank-‐to-‐turn autopilots are designed using the lateral state-‐space model with ailerons as control inputs, for instance: This is done by augmen+ng the bank-‐to-‐turn equa+on to the lateral state-‐space model according to:

!A ! 1/2!!AL " !AR"

v!p!r!!!

"!

"

a11 a12 a13 a14 0a21 a22 a23 0 0a31 a32 a33 0 00 1 0 0 00 0 0 g

U00

vpr!

"

#

b11

b21

b31

00

#A #

r ! gU0

sin!!"! small! g

U0! #


Dynamic equa+on for al+tude control Aircra0 al+tude control systems are designed by considering the equa+on for the ver+cal accelera+on in the center of gravity expressed in NED coordinates: If the accelera+on is perturbed according to azCG = az0 + δaz and d/dt(W₀) = 0, the following equilibrium condi+on is obtained: Symmetric straight-‐line flight with horizontal wings V₀ = Φ₀= Θ₀ = P₀ = Q₀ = 0 gives: Assume that the 2nd-‐order terms vp and uq can be neglected such that Differen+a+ng the al+tude twice with respect to +me gives the rela+onship: If we integrate this expression under the assump+on that we get:

Flight path: where α = w/U₀

2.9 Standard Aircraft Maneuvers

azCG ! W" # VP ! QU ! gcos!$"cos!!" #

az0 ! V0P0 ! Q0U0 ! gcos!"0"cos!!0" #

az0 ! !az " w# ! vp ! q!U0 ! u" ! gcos!""cos!#" # !az ! w" ! U0q #

h! " !!az " U0q ! w# #

h! !0" " U0!!0" ! w!0" " 0

h! " U0! ! w # ! :! " ! # #

h! " U0! #


2.9 Standard Aircraft Maneuvers Example 4 (Augmented model for al+tude control using elevator) An autopilot model for al+tude control based on the longitudinal state-‐space model with states u,w (alt. α), q and θ is obtained by augmen+ng the ODE for h to the state-‐space model:

u!w!q!!!

h!

"

a11 a12 a13 a14 0a21 a22 a23 a24 0a31 a32 a33 0 00 0 1 0 00 !1 0 U0 0

uwq!

h

"

b11 b12

b21 b22

b31 b32

0 00 0

"T"E

#


Hardware-in-the-loop testing— UAV Simulators


Simulator Architecture


Hardware-in-the-loop Testing using Matlab/Simulink and X-Plane

Ref. Lucio R. Ribeiro and Neusa Maria F. Oliveira. UAV Autopilot Controllers Test Platform Using Matlab/Simulink and X-Plane. 40th ASEE/IEEE Frontiers in Education Conference, Washington DC, 2010


Hardware-in-the-loop Testing using Matlab/Simulink and X-Plane

Manual: Brede Børhaug (2011). UAV Simulink and X-Plane simulator

Cirrus Thejet low flyby at OSL Gardermoen using the demo flight control system in Simulink


Hardware-in-the-loop Testing using Matlab/Simulink and FlightGear

Running UAV Models in Aerospace blockset with FlightGear

The FlightGear flight simulator project is an open-source, multi-platform, cooperative flight simulator development project. Source code for the entire project is available and licensed under the GNU General Public License.

Manual: Ingrid Hagen Johansen (2011). Simulink to FlightGear Interface. A User guide


X-Plane Flying over and landing in Gardermoen


2.10 Aircraft Stability Properties The aircra0 stability proper+es can be inves+gated by compu+ng the eigenvalues of the system matrix A using: For a matrix A of dimension n × n there are n solu+ons of λ.

det!!I ! A" ! 0 #


2.10 Aircraft Stability Properties Longitudinal stability analysis For conven+onal aircra0 the characteris+c equa+on is of 4th order: where the subscripts ph and sp denote the following modes:

•  Phugoid mode •  Short-‐period mode

!!2 ! 2"ph#ph! ! #ph2 "!!2 ! 2"sp#sp! ! #sp

2 " " 0 #

The phugoid mode is observed as a long period oscilla+on with li[le damping. In some cases the Phugoid mode can be unstable such that the oscilla+ons increase with +me. The Phugoid mode is characterized by the natural frequency ωph and rela+ve damping ra+o ζph. The short-‐period mode is a fast mode given by the natural frequency ωsp and rela+ve damping factor ζsp. The short-‐period mode is usually well damped.


2.10 Aircraft Stability Properties

Phugoid mode demonstrated in simulator by pulsing throttle. Cockpit and external views.��Uploaded by Professor Eric Johnsen on Apr 30, 2010

Play YouTube video Play YouTube video

Two-view video showing a elevator deflection and the resulting free response oscillation known as the “long period” or phugoid. Uploaded by GuvnK on Feb 16, 2009



Short period mode demonstrated in simulator by pulsing elevator. Velocity vector also shown. Uploaded by Professor Eric Johnsen on Apr 30, 2010

Play YouTube video


2.10 Aircraft Stability Properties Classifica+on of eigenvalues Conven+onal aircra0 have usually two complex conjugated pairs of Phugoid and short-‐period types. For the B-‐767 model in Chapter 4, the eigenvalues can be computed using damp.m in Matlab: From this is seen that:

ωph =0.0596 and ζph = 0.1070 ωsp =2.0943 and ζsp = 0.4143

Time constants: Tph = 105.4 s and Tsp = 3.0 s


2.10 Aircraft Stability Properties Tuck mode: Supersonic aircra0 may have a very large aerodynamic coefficient Mu. This implies that the oscillatory Phugoid equa+on gives two real solu+ons where one is posi+ve (unstable) and one is nega+ve (stable). This is referred to as the tuck mode since the phenomenon is observed as a downwards poin+ng nose (tucking under) for increasing speed.

A third oscillatory mode: For fighter aircra0 the center of gravity is o0en located behind the neutral point or the aerodynamic center (the point where the trim moment Mw w is zero). When this happens, the aerodynamic coefficient Mw takes a value such that the roots of the characteris+c equa+on has four real solu+ons. When the center of gravity is moved backwards one of the roots of the Phugoid and short-‐period modes become imaginary and they form a new complex conjugated pair. This is usually referred to as the 3rd oscillatory mode.


2.10 Aircraft Stability Properties Lateral stability analysis The lateral characteris+c equa+on is usually of 5th order: where the term λ in the last equa+on corresponds to a pure integrator in roll (dψ/dt = r). The term (λ+e) is the aircra0 spiral/divergence mode. This is usually a very slow mode. Spiral/divergence corresponds to horizontally leveled wings followed by roll and a diverging spiral maneuver. The term (λ+f) describes the subsidiary roll mode while the 2nd-‐order system is referred to as Dutch roll. This is an oscillatory system with a small rela+ve damping factor ζD. The natural frequency in Dutch roll is denoted ωD

!5 ! "4!4 ! "3!3 ! "2!2 ! "1! ! "0 " 0#

!!! ! e"!! ! f"!!2 ! 2#D$D! ! $D2 " " 0

#

#



Dutch roll mode demonstrated in simulator by pulsing rudder. Velocity vector shown Uploaded by Professor Eric Johnsen on Apr 30, 2010

Dutch roll on landing a B-747, well i managed to edit the plane, if it flies with no yaw damper it goes 9-10 times on low speed into a Dutch roll ��Uploaded by cba999casey on Jan 9, 2012

Play YouTube video

Play YouTube video


2.10 Aircraft Stability Properties Spiral roll mode demonstrated in simulator. Cockpit and map views. Uploaded by Professor Eric Johnsen on Apr 30, 2010

Play YouTube video

Roll mode demonstrated in simulator by pulsing aileron. Angular velocity vector shown. Uploaded by Professor Eric Johnsen on Apr 30, 2010

Play YouTube video


2.10 Aircraft Stability Properties Lateral stability analysis If the lateral B-‐767 Matlab model in Chapter 4 is considered, the Matlab command damp.m gives: In this example we only get four eigenvalues since the pure integrator in yaw is not include in the system matrix. It is seen that the spiral mode is given by e= -‐0.0143 while the subsidiary roll mode is given by f= -‐2.0863. Dutch roll is recognized by ωD = 1.5038 and ζD = 0.0745.


3 Satellite Modeling






3.1 Attitude Model Euler's 2nd Axiom Applied to Satellites The rigid-‐body kine+cs were ω = [p,q,r]T and ICG =ICGT > 0 is the iner+a tensor about the center of gravity given by: If the Euler angles are used to represent aXtude, the kinema+cs become: where The satellite has three controllable moments τ = [τ₁,τ₂,τ₃]T which can be generated using different actuators.

ICG!" ! ! ! !ICG!" " # #

ICG !

Ix !Ixy !Ixz!Ixy Iy !Iyz!Ixz !Iyz Iz

#

!" ! J!!"# # J!!" !

1 sin!!" tan!"" cos!!" tan!""

0 cos!!" ! sin!!"0 sin!!"/ cos!"" cos!!"/ cos!""

cos!!" ! 0

! ! !!,",#""


3.1 Attitude Model

The dynamic model can also be wri[en: where we have used the fact that a×b = -‐b×a. Furthermore, it is possible to find a matrix S(ω) such that S(ω) = -‐ST(ω) is skew-‐symmetric. With other words: The matrix S(ω) must sa+sfy the condi+on: A matrix sa+sfying this condi+on is:

ICG!" ! !ICG!" ! ! ! # #

ICG!" ! S!!"! " # #

S!!"! ! "!ICG!" ! ! #

S!!" !0 !Iyzq ! Ixzp " Izr Iyzr " Ixyp ! Iyq

Iyzq " Ixzp ! Izr 0 !Ixzr ! Ixyq " Ixp!Iyzr ! Ixyp " Iyq Ixzr " Ixyq ! Ixp 0

#


3.3 Design of Satellite Attitude Control Systems Consider the Lyapunov func+on candidate: where h(θ) is a posi+ve definite func+on depending of the aXtude. Differen+a+on of V with respect to +me gives:

Since S(ω) = -‐ST(ω) , it follows that: This suggests that the control input τ should be chosen as: where Kd > 0. This finally gives: and stability and convergence follow from standard Lyapunov techniques.

!!S!!"! ! 0 !! #

V! " !!!Kd! " 0 #

! ! !Kd" ! Tq!!q" "h"q#! !Kd" ! Tq!!q"Kpq# #

Nonlinear quaternion-‐based PD-‐controller for set-‐point regula+on

V ! 12 !!ICG! " h!q"" #

V! " !TICG!" # q" !!h!q#

" !!!"S!!"! # $" # !!Tq!!q" !h!q# #


3.3 Design of Satellite Attitude Control Systems PD-‐controller of Wen and Kreutz-‐Delago (1991)

!!" # 12 !!"#

"!" # ! 12 !!! ! 1

2 ! " "#

#

#

! ! kp"# ! kd$ #

! ! kp"# ! kd$ #

V! " !ckp#!"#2 ! kd2###2 ! c ! 1

2 !$# ! 12 # " !"

!ICG# ! c!"! !# " ICG#

" ! !" #! ckp 0

0 kd ! 2c#I

!"

#

!

#

V ! !kp " ckd"!!!# ! 1"2 " !"!!"" " 12 #!ICG# ! c!!ICG# #


4 Matlab Simulation Models






The longitudinal and lateral B-‐767 state vectors are: Equilibrium point:

4.1 Boeing B-767

xlang !

u (ft/s)! (deg)q (deg/s)" (deg)

,xlat !

# (deg)p (deg/s)$ (deg/s)r (deg)

#

ulang !%E (deg)%T (%)

,ulat !%A (deg)%R (deg)

#

Speed VT ! 890 ft/s ! 980 km/hAltitude h ! 35 000 ftMass m ! 184 000 lbsMach-number M ! 0.8


4.1 Boeing B-767


4.2 F-16 Fighter The lateral F-‐16 state vectors are: Equilibrium point:

xlat !

! (ft/s)" (ft/s)p (rad/s)r (rad)#A (rad)#R (rad)rw (rad)

,ulat !uA (rad)uR (rad)

,ylat !

rw (deg)p (deg/s)! (deg)" (deg)

#

Speed VT ! 502 ft/s ! 552 km/hMach-number M ! 0.45


4.2 F-16 Fighter

No+ce that the last two eigenvalues correspond to the actuator states.


The lateral model of the F2B Bristol fighter aircra0 is given below. This is a Bri+sh aircra0 from World War I. The aircra0 model is for β=0 (coordinated turn) and the state-‐space vector is: where the dynamics for ψ sa+sfies the bank-‐to-‐turn equa+on: Equilibrium point:

4.3 F2B Bristol Fighter

xlat !

p (deg/s)r (deg/s)! (deg)" (deg)

,ulat ! !#A (deg)" #

!! " gU0

" " 9.81138 ! 0.3048 " " 0. 233" #

Speed VT ! 138 ft/s ! 151.4 km/hAltitude h ! 6 000 ftMach-number M ! 0.126


4.3 F2B Bristol Fighter


4.4 NTNU Student Cube Satellite

I_CG ! diag{[0.0108,0.0113,0.0048]}; % kg m^2

mathematical models for control of aircraft and satellites slides.pdf · 6! lecture notes ttk 4190...

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