Aircraft Fossen 2013

MATHEMATICAL MODELS FOR CONTROL OFAIRCRAFT AND SATELLITES

THOR I. FOSSENProfessor of Guidance, Navigation and Control

Centre for Autonomous Marine Operations and SystemsDepartment of Engineering Cybernetics

Norwegian University of Science and Technology

April 20133rd edition

Copyright c©1998-2013 Department of Engineering Cybernetics, NTNU.1st edition published February 1998.

2nd edition published January 2011.

3rd edition published April 2013.

Contents

Figures iii

1 Introduction 1

2 Aircraft Modeling 32.1 Definition of Aircraft State-Space Vectors . . . . . . . . . . . . . . . . . . . . 32.2 Body-Fixed Coordinate Systems for Aircraft . . . . . . . . . . . . . . . . . . 4

2.2.1 Rotation matrices for wind and stability axes . . . . . . . . . . . . . 5

2.3 Aircraft Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3.1 Kinematic equations for translation . . . . . . . . . . . . . . . . . . . 62.3.2 Kinematic equations for attitude . . . . . . . . . . . . . . . . . . . . 7

2.3.3 Rigid-body kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.4 Sensors and measurement systems . . . . . . . . . . . . . . . . . . . . 8

2.4 Perturbation Theory (Linear Theory) . . . . . . . . . . . . . . . . . . . . . . 82.4.1 Definition of nominal and perturbation values . . . . . . . . . . . . . 92.4.2 Linearization of the rigid-body kinetics . . . . . . . . . . . . . . . . . 92.4.3 Linear state-space model based using wind and stability axes . . . . . 11

2.5 Decoupling in Longitudinal and Lateral Modes . . . . . . . . . . . . . . . . . 122.5.1 Longitudinal equations . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.5.2 Lateral equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.6 Aerodynamic Forces and Moments . . . . . . . . . . . . . . . . . . . . . . . 142.6.1 Longitudinal aerodynamic forces and moments . . . . . . . . . . . . . 162.6.2 Lateral aerodynamic forces and moments . . . . . . . . . . . . . . . . 16

2.7 Propeller Thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.8 Aerodynamic Coeffi cients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.8.1 Aerodynamic forces and moments . . . . . . . . . . . . . . . . . . . . 192.8.2 Drag coeffi cient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.8.3 Lift coeffi cient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.8.4 Sideforce coeffi cient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.8.5 Rolling moment coeffi cient . . . . . . . . . . . . . . . . . . . . . . . . 212.8.6 Pitching moment coeffi cient . . . . . . . . . . . . . . . . . . . . . . . 222.8.7 Yawing moment coeffi cient . . . . . . . . . . . . . . . . . . . . . . . . 232.8.8 Equations of motion including aerodynamic coeffi cients . . . . . . . . 23

2.9 Standard Aircraft Maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . 23

i

ii CONTENTS

2.9.1 Dynamic equation for coordinated turn (bank-to-turn) . . . . . . . . 24

2.9.2 Dynamic equation for altitude control . . . . . . . . . . . . . . . . . . 252.10 Aircraft Stability Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.10.1 Longitudinal stability analysis . . . . . . . . . . . . . . . . . . . . . . 272.10.2 Lateral stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.11 Design of flight control systems . . . . . . . . . . . . . . . . . . . . . . . . . 29

3 Satellite Modeling 313.1 Attitude Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.1 Euler’s 2nd Axiom applied to satellites . . . . . . . . . . . . . . . . . 313.1.2 Skew-symmetric representation of the satellite model . . . . . . . . . 32

3.2 Actuator Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3 Design of Satellite Attitude Control Systems . . . . . . . . . . . . . . . . . . 34

3.3.1 Nonlinear quaternion set-point regulator . . . . . . . . . . . . . . . . 343.3.2 PD-controller of Wen and Kreutz-Delago (1991) . . . . . . . . . . . . 35

4 Matlab Simulation Models 374.1 Boeing-767 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.1.1 Longitudinal model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.1.2 Lateral model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2 F-16 Fighter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.2.1 Longitudinal model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3 F2B Bristol Fighter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.3.1 Lateral model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.4 NTNU student cube-satellite . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

List of Figures

1.1 Sketch showing a modern fighter aircraft (Stevens and Lewis 1992). . . . . . 1

2.1 Definition of aircraft body axes, generalized velocities, forces, moments andEuler angles (McLean 1990). . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Definition of stability and wind axes for an aircraft (Stevens and Lewis 1992). 52.3 Control inputs for conventional aircraft. Notice that the two ailerons can be

controlled by using one control input: δA = 1/2(δAL + δAR). . . . . . . . . . 142.4 Lift and drag as a function of angle of attack. . . . . . . . . . . . . . . . . . 222.5 Aircraft longitudinal eigenvalue configuration plotted in the complex plane. . 28

3.1 The NUTS CubeSat project at NTNU- Courtesy to http://nuts.cubesat.no/ 31

4.1 Schematic drawing of the Bristol F.2B Fighter (McRuer et al 1973). . . . . . 40

iii

iv LIST OF FIGURES

Chapter 1

Introduction

This booklet is a collection of lecture notes used in the course TTK4109 Guidance and Con-trol of Vehicles, which is given at the Department of Engineering Cybernetics, NTNU. Thekinematic and kinetic equations of a marine craft (ships, high-speed craft and underwatervehicles) can be modified to describe aircraft and satellites by minor adjustments of nota-tion and assumptions. Consequently, the vectorial notation introduced in the two Wileytextbooks:

Fossen, T. I. (1994). Guidance and Control of Ocean Vehicles

Fossen, T. I. (2011). Handbook of Marine Craft Hydrodynamics and Motion Control

is used to describe the aircraft and satellite equations of motion.

Figure 1.1: Sketch showing a modern fighter aircraft (Stevens and Lewis 1992).

The booklet is organized according to:

• Chapter 2: Aircraft Modeling

• Chapter 3: Satellite Modeling

• Chapter 4: Matlab Simulation Models

1

2 CHAPTER 1. INTRODUCTION

Other useful references on flight control are:

Blakelock, J. H. (1991). Aircraft and Missiles (John Wiley & Sons Ltd.)

Etkin, B. and L. D. Reid (1996). Dynamics of Flight: Stability and Control (JohnWiley & Sons Ltd.)

Fortescue, P., G. Swinerd and J. Stark (2011). Spacecraft Systems Engineering (JohnWiley & Sons Ltd.)

Hughes, P. C. (1986). Spacecraft Attitude Dynamics (John Wiley & Sons Ltd.)

McLean, D. (1990). Automatic Flight Control Systems (Prentice Hall Inc.)

McRuer, D., D. Ashkenas and A. I. Graham (1973). Aircraft Dynamics and Auto-matic Control (Princeton University Press)

Nelson, R. C. (1998). Flight Stability and Automatic Control (McGraw-Hill Int.)

Roskam, J. (1999). Airplane Flight Dynamics and Automatic Flight Controls (Darcor-poration)

Schmidt, D. K. (2012). Modern Flight Dynamics (McGraw-Hill Int.)

Stevens, B. L. and F. L. Lewis (2003). Aircraft Control and Simulation (John Wiley& Sons Ltd.)

Information about the graduate coursesTTK4109 Guidance and Control of Vehicles andTK8109 Advanced Topics in Guidance and Navigation are found on the Wiki-pages

URL: http://www.itk.ntnu.no/emner/ttk4190

URL: http://www.itk.ntnu.no/emner/tk8109

Thor I. FossenTrondheim —3 April 2013

Chapter 2

Aircraft Modeling

This chapter gives an introduction to aircraft modeling. The equations of motion are lin-earized using perturbation theory and the final results are state-space models for the longi-tudinal and lateral motions. The models can be used for aircraft simulation and design offlight control systems.

2.1 Definition of Aircraft State-Space Vectors

The aircraft generalized velocity vector is defined according to (see Figure 2.1):

ν :=

UVWPQR

=

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

(2.1)

η :=

XE

YEZE, hΦΘΨ

=

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

(2.2)

Forces and moments are defined in a similar manner:XYZLMN

:=

longitudinal forcetransverse forcevertical forceroll momentpitch momentyaw moment

(2.3)

Comment 1: Notice that the capital letters L,M,N for the moments are different from

3

4 CHAPTER 2. AIRCRAFT MODELING

Figure 2.1: Definition of aircraft body axes, generalized velocities, forces, moments and Eulerangles (McLean 1990).

those used for marine craft—that is, K,M,N. The reason for this is that L is reserved aslength parameter for ships and underwater vehicles.

Comment 2: For aircraft it is common to use capital letters for the states U, V,W, etc.while it is common to use small letters for marine craft.

2.2 Body-Fixed Coordinate Systems for Aircraft

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

• BODY axes, superscript b

• STABILITY axes, superscript s

• WIND axes, superscript w

The axis systems are shown in Figure 2.2 where the angle of attack α and sideslip angle βare defined as:

tan(α) :=W

U(2.4)

sin(β) :=V

VT(2.5)

whereVT =

√U2 + V 2 +W 2 (2.6)

2.2. BODY-FIXED COORDINATE SYSTEMS FOR AIRCRAFT 5

Figure 2.2: Definition of stability and wind axes for an aircraft (Stevens and Lewis 1992).

is the total speed of the aircraft. Aerodynamic effects are classified according to the Machnumber :

M :=VTa

(2.7)

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

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 aircraft will break the sound barrier at M = 1.0 and this is clearly heard as a sharpcrack. If you fly at low altitude and break the sound barrier, windows in building will breakdue to pressure-induced waves.

2.2.1 Rotation matrices for wind and stability axes

The relationship between vectors expressed in different coordinate systems can be derivedusing rotation matrices. The BODY axes are first rotated a negative sideslip angle −β aboutthe z-axis. The new coordinate system is then rotated a positive angle of attack α aboutthe new y-axis such that the resulting x-axis points in the direction of the total speed VT .The first rotation defines theWIND axes while the second rotation defines the STABILITY


axes. This can be mathematically expressed as:

pw = Rz,−βps =

cos(β) sin(β) 0− sin(β) cos(β) 0

0 0 1

ps (2.8)

ps = Ry,αpb =

cos(α) 0 sin(α)0 1 0

− sin(α) 0 cos(α)

pb (2.9)

The rotation matrix becomes:Rwb = Rz,−βRy,α (2.10)

Hence,

pw = Rwb p

b (2.11)

m

pw =

cos(β) sin(β) 0− sin(β) cos(β) 0

0 0 1



pb (2.12)

m

pw =

cos(α) cos(β) sin(β) sin(α) cos(β)− cos(α) sin(β) cos(β) − sin(α) sin(β)− sin(α) 0 cos(α)

pb (2.13)

This gives the following relationship between the velocities in body and wind axes:

vb =

UVW

= (Rwb )>vw = R>y,αR

>z,−β

VT00

=

VT cos(α) cos(β)VT sin(β)VT sin(α) cos(β)

(2.14)

Consequently,U = VT cos(α) cos(β)V = VT sin(β)W = VT sin(α) cos(β)

(2.15)

2.3 Aircraft Equations of Motion

2.3.1 Kinematic equations for translation

The kinematic equations for translation and rotation of a body-fixed coordinate systemBODY with respect to a local geographic coordinate system NED (North-East-Down) canbe expressed in terms or the Euler angles: XE

YEZE

= Rnb

UVW

= Rz,ΨRy,ΘRx,Φ

UVW

(2.16)

2.3. AIRCRAFT EQUATIONS OF MOTION 7

Expanding this expression gives: XE

YEZE

=

cΨ −sΨ 0sΨ cΨ 00 0 1

cΘ 0 sΘ0 1 0−sΘ 0 cΘ

1 0 00 cΦ −sΦ0 sΦ cΦ

UVW

m (2.17) XE

YEZE

=

cΨcΘ −sΨcΦ + cΨsΘsΦ sΨsΦ + cΨcΦsΘsΨcΘ cΨcΦ + sΦsΘsΨ −cΨsΦ + sΘsΨcΦ−sΘ cΘsΦ cΘcΦ

UVW

2.3.2 Kinematic equations for attitude

The attitude is given by: PQR

=

Φ00

+R>x,Φ

0

Θ0

+R>x,ΦR>y,Θ

00

Ψ

(2.18)

which gives: Φ

Θ

Ψ

=

1 sΦtΘ cΦtΘ0 cΦ −sΦ0 sΦ/cΘ cΦ/cΘ

PQR

, cΘ 6= 0 (2.19)

2.3.3 Rigid-body kinetics

The aircraft rigid-body kinetics can be expressed as (Fossen 1994, 2011):

m(ν1 + ν2 × ν1) = τ 1 (2.20)

ICGν2 + ν2 × (ICGν2) = τ 2 (2.21)

where ν1 := [U, V,W ]T ,ν2 := [P,Q,R]T , τ 1 := [X, Y, Z]T and τ 2 := [L,M,N ]T . It is as-sumed that the coordinate system is located in the aircraft center of gravity (CG). Theresulting model is written:

MRBν +CRB(ν)ν = τRB (2.22)

where

MRB =

[mI3×3 O3×3

O3×3 ICG

], CRB(ν) =

[mS(ν2) O3×3

O3×3 −S(ICGν2)

](2.23)

The inertia tensor is defined as (assume that Ixy = Iyz = 0 which corresponds to xz−planesymmetry):

ICG :=

Ix 0 −Ixz0 Iy 0−Ixz 0 Iz

(2.24)

The forces and moments acting on the aircraft can be expressed as:

τRB = −g(η) + τ (2.25)


where τ is a generalized vector that includes aerodynamic and control forces. The gravita-

tional force fG = [0 0 mg]T acts in the CG (origin of the body-fixed coordinate system) andthis gives the following vector expressed in NED:

g(η) = −(Rnb)>[

fGO3×1

]=

mg sin(Θ)

−mg cos(Θ) sin(Φ)−mg cos(Θ) cos(Φ)

000

(2.26)

Hence, the aircraft model can be written in matrix form as:

MRBν +CRB(ν)ν + g(η) = τ (2.27)

or in component form:

m(U +QW −RV + g sin(Θ)) = X

m(V + UR−WP − g cos(Θ) sin(Φ)) = Y

m(W + V P −QU − g cos(Θ) cos(Φ)) = Z

IxP − Ixz(R + PQ) + (Iz − Iy)QR = L (2.28)

IyQ+ Ixz(P2 −R2) + (Ix − Iz)PR = M

IzR− IxzP + (Iy − Ix)PQ+ IxzQR = N

2.3.4 Sensors and measurement systems

It is common that aircraft sensor systems are equipped with three accelerometers. If theaccelerometers are located in the CG, the measurement equations take the following form:

axCG =X

m= U +QW −RV + g sin(Θ) (2.29)

ayCG =Y

m= V + UR−WP − g cos(Θ) sin(Φ) (2.30)

azCG =Z

m= W + V P −QU − g cos(Θ) cos(Φ) (2.31)

In addition to these sensors, an aircraft is equipped with gyros, magnetometers and a sensorfor altitude h and wind speed VT . These sensors are used in inertial navigation systems (INS)which again use a Kalman filter to compute estimates of U, V,W, P,Q and R as well as theEuler angles Φ,Θ and Ψ. Other measurement systems that are used onboard aircraft areglobal navigation satellite systems (GNSS), radar and sensors for angle of attack.

2.4 Perturbation Theory (Linear Theory)

The nonlinear equations of motion can be linearized by using perturbation theory. This isillustrated below.

2.4. PERTURBATION THEORY (LINEAR THEORY) 9

2.4.1 Definition of nominal and perturbation values

According to linear theory it is possible to write the states as the sum of a nominal value(usually constant) and a perturbation (deviation from the nominal value). Moreover,

Total state = Nominal value + Perturbation

The following definitions are made:

τ := τ 0 + δτ =

X0

Y0

Z0

L0

M0

N0

+

δXδYδZδLδMδN

, ν := ν0 + δν =

U0

V0

W0

P0

Q0

R0

+

uvwpqr

(2.32)

Similar, the angles are defined according to: ΘΦΨ

:=

Θ0

Φ0

Ψ0

+

θφψ

(2.33)

Consequently, a linearized state-space model will consist of the following states u, v, w, p, q, r, θ, φand ψ.

2.4.2 Linearization of the rigid-body kinetics

The rigid-body kinetics can be linearized by using perturbation theory.

Equilibrium condition

If the aerodynamic forces and moments, velocities, angles and control inputs are expressedas nominal values and perturbations τ = τ 0 + δτ , ν = ν0 + δν and η = η0 + δη, the aircraftequilibrium point will satisfy (it is assumed that ν0 = 0):

CRB(ν0)ν0 + g(η0) = τ 0 (2.34)

This can be expanded according to:

m(Q0W0 −R0V0 + g sin(Θ0)) = X0

m(U0R0 − P0W0 − g cos(Θ0) sin(Φ0)) = Y0

m(P0V0 −Q0U0 − g cos(Θ0) cos(Φ0)) = Z0

(Iz − Iy)Q0R0 − P0Q0Ixz = L0 (2.35)

(P 20 −R2

0)Ixz + (Ix − Iz)P0R0 = M0

(Iy − Ix)P0Q0 +Q0R0Ixz = N0


Perturbed equations

The perturbed equations—that is, the linearized equations of motion are usually derived bya 1st-order Taylor series expansion about the nominal values. Alternatively, it is possibleto substitute (2.32) and (2.35) into (2.27) and neglect higher-order terms of the perturbedstates. This is illustrated for the first degree of freedom (DOF):

Example 1 (Linearization of surge using perturbation theory)

m[U +QW −RV + g sin(Θ)] = X

⇓ (2.36)

m[U0 + u+ (Q0 + q)(W0 + w)− (R0 + r)(V0 + v) + g sin(Θ0 + θ)] = X0 + δX

This can be written:

sin(Θ0 + θ) = sin(Θ0) cos(θ) + cos(Θ0) sin(θ)θ small≈ sin(Θ0) + cos(Θ0)θ (2.37)

Since U0 = 0 and

m(Q0W0 −R0V0 + g sin(Θ0)) = X0 (2.38)

Equation (2.36) is reduced to:

m[u+Q0w +W0q + wq −R0v − V0r − vr + g cos(Θ0)θ] = δX (2.39)

If it is assumed that the 2nd-order terms wq and vr are negligible, the linearized modelbecomes:

m[u+Q0w +W0q −R0v − V0r + g cos(Θ0)θ] = δX (2.40)

Linear state-space model for aircraftIf all DOFs are linearized, the following state-space model is obtained:

m[u+Q0w +W0q −R0v − V0r + g cos(Θ0)θ] = δX

m[v + U0r +R0u−W0p− P0w − g cos(Θ0) cos(Φ0)φ+ g sin(Θ0) sin(Φ0)θ] = δY

m[w + V0p+ P0v − U0q −Q0u+ g cos(Θ0) sin(Φ0)φ+ g sin(Θ0) cos(Φ0)θ] = δZ

Ixp− Ixz r + (Iz − Iy)(Q0r +R0q)− Ixz(P0q +Q0p) = δL (2.41)

Iy q + (Ix − Iz)(P0r +R0p)− 2Ixz(R0r + P0p) = δM

Iz r − Ixzp+ (Iy − Ix)(P0q +Q0p) + Ixz(Q0r +R0q) = δN

This can be expressed in matrix form as:

MRBν +NRBν +Gη = τ (2.42)

2.4. PERTURBATION THEORY (LINEAR THEORY) 11

where

MRB =

m 0 0

m 0 03×3

0 0 mIx 0 −Ixz

03×3 0 Iy 0−Ixz 0 Iz

NRB =

0 −mR0 mQ0 0 mW0 −mV0

mR0 0 −P0 −W0 0 U0

−mQ0 mP0 0 mV0 −mU0 0−IxzQ0 (Iz − Iy)R0 − IxzP0 (Iz − Iy)Q0

03×3 (Ix − Iz)R0 −2IxzP0 (Ix − Iz)P0 − 2IxzR0

(Iy − Ix)Q0 (Iy − Ix)P0 + IxzR0 IxzQ0

G =

0 mg cos(Θ0) 0

03×3 −mg cos(Θ0) cos(Φ0) mg sin(Θ0) sin(Φ0) 0mg cos(Θ0) sin(Φ0) mg sin(Θ0) cos(Φ0) 0

03×3 03×3

In addition to this, the kinematic equations must be linearized.

2.4.3 Linear state-space model based using wind and stability axes

An alternative state-space model is obtained by using α and β as states. If it is assumedthat α and β are small such that cos(α) ≈ 1 and sin(β) ≈ β, Equation (2.15) can be writtenas:

U = VT

V = VTβ

W = VTα

⇒

U = VT

β = VVT

α = WVT

(2.43)

Furthermore, the state-space vector:

x =

uβαpqr

=

surge velocitysideslip angleangle of attackroll ratepitch rateyaw rate

(2.44)

is chosen to describe motions in 6 DOF. The relationship between the body-fixed velocityvector:

ν = [u, v, w, p, q, r]T (2.45)

and the new state-space vector x can be written as:

ν = Tx = diag1, VT , VT , 1, 1, 1, 1x (2.46)


where VT > 0. If the total speed is VT = U0 = constant (linear theory), it is seen that:

α =1

VTw (2.47)

β =1

VTv (2.48)

VT = 0 (2.49)

If nonlinear theory is applied, the following differential equations are obtained:

α =UW −WU

U2 +W 2(2.50)

β =V VT − V VTV 2T cos β

(2.51)

VT =UU + V V +WW

VT(2.52)

In the linear case it is possible to transform the body-fixed state-space model:

ν = Fν +Gu (2.53)

tox = Ax+Bu (2.54)

whereA = T−1FT, B = T−1G (2.55)

For VT 6= 0 this transformation is much more complicated. The linear state-space transfor-mation is commonly used by aircraft manufactures. An example is the Boeing B-767 model(see Chapter 4).

2.5 Decoupling in Longitudinal and Lateral Modes

For an aircraft 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). The key assumption is that thefuselage is slender—that is, the length is much larger than the width and the height of theaircraft. It is also assumed that the longitudinal velocity is much larger than the verticaland transversal velocities.In order to decouple the rigid-body kinetics (2.41) in longitudinal and lateral modes it

will be assumed that the states v, p, r and φ are negligible in the longitudinal channel whileu,w, q and θ are negligible when considering the lateral channel. This gives two sub-systems:

2.5. DECOUPLING IN LONGITUDINAL AND LATERAL MODES 13

2.5.1 Longitudinal equations

Kinetics:

m[u+Q0w +W0q + g cos(Θ0)θ] = δX

m[w − U0q −Q0u+ g sin(Θ0) cos(Φ0)θ] = δZ (2.56)

Iy q = δM m 0 00 m 00 0 Iy

uwq

+

0 mQ0 mW0

−mQ0 0 −mU0

0 0 0

uwq

+

mg cos(Θ0)mg sin(Θ0) cos(Φ0)

0

θ =

δXδZδM

(2.57)

Kinematics:θ = q (2.58)

2.5.2 Lateral equations

Kinetics:

m[v + U0r −W0p− g cos(Θ0) cos(Φ0)φ] = δY

Ixp− Ixz r + (Iz − Iy)Q0r − IxzQ0p = δL (2.59)

Iz r − Ixzp+ (Iy − Ix)Q0p+ IxzQ0r = δN m 0 00 Ix −Ixz0 −Ixz Iz

vpr

+

0 −mW0 mU0

0 −IxzQ0 (Iz − Iy)Q0

0 (Iy − Ix)Q0 IxzQ0

vpr

+

−mg cos(Θ0) cos(Φ0)00

φ =

δYδLδN

(2.60)

Kinematics: [φ

ψ

]=

[1 tan(Θ0)0 1/ cos(Θ0)

] [pr

](2.61)


Figure 2.3: Control inputs for conventional aircraft. Notice that the two ailerons can becontrolled by using one control input: δA = 1/2(δAL + δAR).

2.6 Aerodynamic Forces and Moments

In the forthcoming sections, the following abbreviations and notation will be used to describethe aerodynamic coeffi cients:

Xindex = ∂X∂ index Lindex = ∂L

∂ index

Yindex = ∂Y∂ index Mindex = ∂M

∂ index

Zindex = ∂Z∂ index Nindex = ∂N

∂ index

In order to illustrate how control surfaces influence the aircraft, an aircraft equipped withthe following control inputs will be considered (see Figure 2.3):

2.6. AERODYNAMIC FORCES AND MOMENTS 15

δT Thrust Jet/propeller

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

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

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

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

A conventional aircraft has control surfaces such as ailerons δA, elevators δE, flaps δF andrudders δR. A positive deflection of the control surfaces are to give a negative aerodynamicmoment on the aircraft. Some of the control surfaces, like the ailerons, deflects simultane-ously in an asymmetric manner. If an aircraft has two ailerons δAL and δAR , the ailerondeflection is computed by:

δA =1

2(δAR − δAL) (2.62)

Other control surfaces, such as elevators, deflect symmetrically. If an aircraft has two eleva-tors δEL and δER , the elevator deflections become:

δE =1

2(δER + δEL) (2.63)

Linear theory

The number of aerodynamic coeffi cients is reduced if linear theory is assumed. The controlinputs and aerodynamic forces and moments can be written as:

τ = −MF ν −NFν +Bu (2.64)

where −MF is aerodynamic added mass, −NF is aerodynamic damping and B is a matrixdescribing the actuator configuration including the force coeffi cients. The actuator dynamicsis modeled by a 1st-order system:

u = T−1(uc − u) (2.65)

where uc is commanded input, u is the actual control input produced by the actuators andT = diagT1, T2, ..., Tr is a diagonal matrix of positive time constants. Substitution of (2.64)into the model (2.42) gives:

(MRB +MF )ν + (NRB +NF

)ν +Gη = Bu

m (2.66)

Mν +Nν +Gη = Bu

The matricesM and N are defined asM = MRB +MF and N = NRB +NF. The linearized

kinematics takes the following form:

η = Jνν + Jηη (2.67)

The resulting state-space models are:


Linear state-space model with actuator dynamics ηνu

=

Jη Jν 0−M−1G −M−1N M−1B

0 0 −T−1

ηνu

+

00T−1

uc (2.68)

Linear state-space model neglecting the actuator dynamics[ην

]=

[Jη Jν

−M−1G −M−1N

] [ην

]+

[0

M−1B

]u (2.69)

2.6.1 Longitudinal aerodynamic forces and moments

McLean [7] expresses the longitudinal forces and moments as:

dXdZdM

=

Xu Xw Xq

Zu Zw ZqMu Mw Mq

uwq

+

Xu Xw Xq

Zu Zw ZqMq Mw Mq

uwq

+

XδT XδE XδF

ZδT

ZδE ZδFMδT MδE MδF

δTδEδF

(2.70)

which corresponds to the matrices −MF ,−NF and B in (2.64). If the aircraft cruise speedU0 = constant, then δT = 0. Altitude can be controlled by using the elevators δE. Flaps δFcan be used to reduce the speed during landing. The flaps can also be used to turn harderfor instance by moving one flap while the other is kept at the zero position. This is commonin bank-to-turn maneuvers. For conventional aircraft the following aerodynamic coeffi cientscan be neglected:

Xu, Xq, Xw, XδE , Zu, Zw,Mu (2.71)

Hence, the model for altitude control reduces to: dXdZdM

=

0 0 Xq

0 0 Zq0 Mw Mq

uwq

+

Xu Xw 0Zu Zw ZqMq Mw Mq

uwq

+

XδE

ZδEMδE

δE (2.72)

If the actuator dynamics is important, aerodynamic coeffi cients such as XδT, XδE

, ... mustbe included in the model.

2.6.2 Lateral aerodynamic forces and moments

The lateral model takes the form [7]:

2.7. PROPELLER THRUST 17

dYdLdN

=

Yv Yp YrLv Lp LrNv Np Nr

vpr

+

Yv Yp YrLv Lp LrNv Np Nr

vpr

+

YδA YδRLδA LδRNδA NδR

[ δAδR

](2.73)

which corresponds to the matrices −MF ,−NF and B in (2.64). For conventional aircraftthe following aerodynamic coeffi cients can be neglected:

Yv, Yp, Yp, Yr, Yr, YδA,Lv, Lr, Nv, Nr (2.74)

This gives:

dYdLdN

=

0 0 00 Lp 00 Np 0

vpr

+

Yv 0 0Lv Lp LrNv Np Nr

vpr

+

0 YδRLδA

LδRNδA NδR

[ δAδR

](2.75)

2.7 Propeller Thrust

The thrust from a propeller can be modeled as:

T = Ktρn2D4 (2.76)

whereT Thrust force [N]Kt Thrust coeffi cientρ Density of air [kg/m3]n Propeller angular velocity [rad/s]D Propeller diameter [m]

Assuming that the trust force acts in the horizontal plane of b, and parallel with thexb-axis, it can be expressed in b as:

f bt =

Xt

YtZt

=

T00

(2.77)

If the line of action of the thrust is not directed through CO, it will in addition generate anapplied moment given by

mbt =

LtMt

Nt

= rbt/b × f bt = S(rbt/b)fbt

=

0Trtz−Trty

(2.78)


where rbt/b = [rtx , rty , rtz ]> is the location of the propeller with respect to CO.

Sometimes the effects due to the rotating mass caused by the propeller must be consideredas an applied moment. This is because the equations of motion has been derived by assumingthat the aircraft is a rigid body with no internal moving parts. The applied moment due tothe rotating mass caused by the propeller is given by:

mbp =

id

dthbp (2.79)

where hbp is the angular momentum of the rotating mass, and is given in b as:

hbp =

IpΩp

00

(2.80)

where Ip is the inertia of the rotating mass and Ωp is the angular velocity. Assuming theangular velocity of the rotating mass is constant:

mbp =

LpMp

Np

=id

dthbp =

bd

dthbp + ωbb/n × hbp

= ωbb/n × hbp = S(ωbb/n)hbp

=

0IpΩpR−IpΩpQ

(2.81)

The total forces and moments due to thrust, τ t are:

τ t =

[f bt

S(rbt/g)fbt + S(ωbb/n)hbp

](2.82)

2.8 Aerodynamic Coeffi cients

Different methods can be used to estimate the aerodynamic coeffi cients such as wind tunnelexperiments or system identification based on logged experimental data from the aircraft.The aircraft equations of motion can be quite simple if the aircraft is not highly maneu-verable. The more maneuverable aircraft, the more coeffi cients are needed to accuratelydescribe the aerodynamic forces and moments.The aerodynamic coeffi cients are usually linear at small angles of angle of attack and

sideslip. 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

2.8. AERODYNAMIC COEFFICIENTS 19

Aircraft with delta-shaped wings often have a low-aspect ratio, AR. A low-aspect ratiopermits very high roll rates, but have a greatly reduced lift-over-drag. High lift-over-draggives an aircraft that has effective cruise performance, that is passenger jets.The aerodynamic forces and moments are often proportional to the freestream mass

density, ρa, and the square of the freestream airspeed, VT . It is thus convenient to define thedynamic pressure, q, which is later used to calculate the aerodynamic forces and moments:

q =1

2ρaV

2T (2.83)

2.8.1 Aerodynamic forces and moments

The aerodynamic forces and moments acting on an aircraft can be parametrized as:

f ba =

Xa

YaZa

= qS

CXCYCz

(2.84)

mba =

LaMa

Na

= qS

bClcCmbCn

(2.85)

where the aerodynamic coeffi cients CX and CZ often are replaced by expressions that arefunctions of the drag coeffi cient CD and the lift coeffi cient CL (see Figure 2.4). Positive dragand lift coeffi cients are directed along the −xs and −zs stability axes, respectively, such that: CD

0CL

= Rsb

−CX0−CZ

=



−CX0−CZ

(2.86)

Hence, the aerodynamic forces can then be rewritten as:

f ba =

Xa

YaZa

= qS

0CY0

+ qSRbs

−CD0−CL

= qS

−CD cos(α) + CL sin(α)CY

−CD sin(α)− CL cos(α)

(2.87)

The generalized aerodynamic forces become:

τ a =

[f bamb

a

](2.88)

The aerodynamic coeffi cients Ci, i = D, Y, L, l,m, n, are in general functions of angle ofattack α, sideslip angle β, Mach number M , altitude h, control surface deflections δS andthe thrust coeffi cient:

TC =T

qSD(2.89)


where SD is the area of the disc swept out by a propeller blade. Therefore, the aerodynamiccoeffi cients are functions:

Ci = Ci(α, β,M, h, δS, TC) (2.90)

where i = D, Y, L, l,m, n. Consequently, the aerodynamic coeffi cients may have complicateddependences, which are hard to model with great accuracy. If the aircraft has restrictedflight modes, thts is restricted to low Mach numbers and small angles of attack and sideslip,the aerodynamic coeffi cients may be greatly simplified. The complicated dependency maythen be broken into a sum of simpler linear terms.

2.8.2 Drag coeffi cient

The drag of an aircraft is one of the most important features to model. By reducing drag,the fuel consumption is reduced, thus giving higher range, and in addition higher maximumcruise speed. The total drag of an aircraft is usually taken as the sum of the following dragcomponents:

• Parasite drag = friction drag + form drag

• Induced drag (effect of wing-tip vortices)

• Wave drag (effect of shock waves)

where

Friction drag also known as skin friction, describes the friction of the fluid against thesurface of the aircraft. The friction drag is proportional to the area in contact withthe fluid, also known as the wetted area.

Form drag is the pressure drag caused by flow separation at high alpha.

Induced Drag also known as vortex drag, is the pressure drag caused by the tip vorticesof a finite wing when it is producing lift.

Wave drag is the pressure drag caused if shock waves are present over the surface of theaircraft. The wave drag becomes significant at Mach numbers M > 0.4 for a fighteraircraft, where the flow reaches supersonic speed.

The friction drag often varies parabolically with the lift coeffi cient, while the induceddrag can often be modeled as:

CDi =C2L

πeAR(2.91)

where e is the effi ciency factor, close to unity, and AR is the aspect ratio. The total drag ofan aircraft takes the form:

CD(CL,M) = k(M) (CL − CLDM )2 + CDM(M) (2.92)

2.8. AERODYNAMIC COEFFICIENTS 21

where k(M) is a proportionality constant that varies with the Mach number. In addition,the drag coeffi cient may be a function of the control surfaces and landing gear. A moregeneral equation for the drag coeffi cient is:

CD = CD(α, β,M, h) + ∆CD(M, δE) + ∆CD(M, δR) + ∆CD(δF ) + ∆CD(gear) + · · · (2.93)

A linear approximation of this expressions is:

CD = CD0 + CDαα + CDδE δE + CDδRδR + CDδF δF (2.94)

2.8.3 Lift coeffi cient

The lift coeffi cient, CL, is in general mainly determined by the fuselage, wings and theirinterference effects. It is also a function of Mach and the angle of attack. The lift coeffi cientis usually quite linear with a until near stall, where it drops sharply, and then may rise againbefore falling to zero at large angles of attack (see Figure 2.4). A general equation for thelift coeffi cient is given in

CL = CL(α, β,M, TC) + ∆CL(δF ) + ∆CLge(h) (2.95)

where ∆CLge(h) is the increment of lift due to ground effect. A typical simple linear modelstructure for the lift coeffi cient is given in

CL = CL0 + CLαα + CLδF δF (2.96)

2.8.4 Sideforce coeffi cient

The sideforce coeffi cient, CY , is in general mainly determined by sideslip and rudder deflec-tion. It is often quite linear in sideslip angle β, and a positive sideslip will give a negativesideforce. A typical expression for the sideforce coeffi cient is:

CY = CY (α, β,M)+∆CYδR (α, β,M, δR)+∆CYδA (α, β,M, δA)+b

2VT

[CYp(α,M)P + CYr(α,M)R

](2.97)

A typical simple linear model structure for the sideforce coeffi cient is given in

CY = CY0 + CYββ + CYδRδR + CYδAδA (2.98)

2.8.5 Rolling moment coeffi cient

The rolling moment coeffi cient, Cl, is in general mainly determined by sideslip and by controlaction of the ailerons and the rudder. It is often quite linear in sideslip angle and a positivesideslip will give a negative rolling moment. A typical expression for the rolling momentcoeffi cient is:

Cl = Cl(α, β,M)+∆ClδA (α, β,M, δA)+∆ClδR (α, β,M, δR)+b

2VT

[Clp(α,M)P + Clr(α,M)R

](2.99)


Figure 2.4: Lift and drag as a function of angle of attack.

A linear model for the rolling moment coeffi cient is:

Cl = Cl0 + Clββ + ClδAδA + ClδRδR + Clpb

2VTP + Clr

b

2VTR (2.100)

2.8.6 Pitching moment coeffi cient

The pitching moment coeffi cient, Cm, is in general mainly determined by angle of attackα and by control action of the elevators. It may be quite linear in angle of attack, anda positive angle of attack will give a negative pitching moment. A typical model for thepitching moment coeffi cient is:

Cm = Cm(α,M, h, δF , TC) + ∆Cmδe (α,M, h, δE)

+c

2VT

(Cmq(α,M, h)Q+ Cmα(α,M, h)α

)+xRcCL

+ ∆Cmthrust (δT ,M, h) + ∆Cmgear (h) (2.101)

The term xRcCL is used to correct for any x-displacement of the aircraft CG from the aerody-

namic data reference position. The second last term represents the effect of the engine thrustvector not passing through the aircraft CG, while the last term represents the moment dueto the landing-gear doors and the landing gear. A linear model for the pitching momentcoeffi cient takes the form:

Cm = Cm0 + Cmαα + Cmδeδe + Cmqc

2VTQ (2.102)

2.9. STANDARD AIRCRAFT MANEUVERS 23

2.8.7 Yawing moment coeffi cient

The yawing moment coeffi cient, Cn, is in general mainly determined by the sideslip angle βand by control action of the rudders. It is quite linear in sideslip β for small angles of attackα, where a positive sideslip β will give a positive yawing moment. A general equation forthe yawing moment coeffi cient is given by:

Cn = Cn(α, β,M, TC) + ∆CnδR (α, β,M, δR) + ∆CnδA (α, β,M, δA)

+b

2VT

[Cnp(α,M)P + Cnr(α,M)R

](2.103)

A linear model for the yawing moment coeffi cient is gven by:

Cn = Cn0 + Cnββ + CnδRδR + CnδAδA + Cnpb

2VTP + Cnr

b

2VTR (2.104)

2.8.8 Equations of motion including aerodynamic coeffi cients

The resulting equations of motion including the aerodynamic coeffi cients are:

U = V R−WQ− g sin(Θ) +T

m+qS

mCX (2.105)

V = WP − UR + g cos(Θ) sin(Φ) +qS

mCY (2.106)

W = UQ− V P + g cos(Θ) cos(Φ) +qS

mCZ (2.107)

P − IxzIxR = −Iz − Iy

IxQR +

IxzIxPQ+

qbS

IxCl (2.108)

Q = −Ix − IzIy

PR− IxzIy

(P 2 −R2) +TrtzIy

+IpIy

ΩpR +qSc

IyCm (2.109)

R− IxzIzP = −Iy − Ix

IzPQ− Ixz

IzQR−

TrtyIz− IpIz

ΩpQ+qSb

IzCn (2.110)

2.9 Standard Aircraft Maneuvers

The nominal values depends on the aircraft 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


2.9.1 Dynamic equation for coordinated turn (bank-to-turn)

A frequently used maneuver is coordinated turn where the acceleration in the y-direction iszero (β = 0), sideslip β = 0 and zero steady-state pitch and roll angles—that is,

β = β = 0 (2.111)

Θ0 = Φ0 = 0 (2.112)

Furthermore it is assumed that VT = U0 = constant. Since β = β = 0 and β = V/U0 itfollows that V = V = 0. This implies that the external forces δY = 0. From (2.28) it is seenthat:

m[V + UR−WP − g cos(Θ) sin(Φ)] = Y (2.113)

⇓m[(U0 + u)(R0 + r)− (W0 + w)(P0 + p)− g cos(θ) sin(φ)] = Y0 (2.114)

Assume that the longitudinal and lateral motions are decoupled—that is, u = w = q = θ = 0.If perturbation theory is applied under the assumption that the 2nd-order terms ur = pw =0,we get:

m(U0R0 + U0r −W0P0 −W0p− g sin(φ)) = Y0 (2.115)

The equilibrium equation (2.35) gives the steady-state condition:

m(U0R0 −W0P0) = Y0 (2.116)

Substitution of (2.116) into (2.115) gives:

m(U0r −W0p− g sin(φ)) = 0 (2.117)

or

r =W0

U0

p+g

U0

sin(φ) (2.118)

The aircraft is often trimmed such that the angle of attack α0 = W0/U0 = 0. This impliesthat the yaw rate can be expressed as:

r =g

U0

sin(φ)φ small≈ g

U0

φ (2.119)

which is a very important result since it states that a roll angle angle φ different from zerowill induce a yaw rate r which again turns the aircraft (bank-to-turn). With other words, wecan use a moment in roll, for instance generated by the ailerons, to turn the aircraft. Theyaw angle is given by:

ψ = r (2.120)

An alternative method is of course to turn the aircraft by using the rear rudder to generatea yaw moment. The bank-to-turn principle is used in many missile control systems since itimproves maneuverability, in particular in combination with a rudder controlled system.

2.9. STANDARD AIRCRAFT MANEUVERS 25

Example 2 (Augmented turning model using rear rudders)Turning autopilots using the rudder δR as control input is based on the lateral state-spacemodel. The differential equation for ψ is augmented on the lateral model as shown below:

vpr

φ

ψ

=

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

vprφψ

+

b11

b21

b31

00

δR (2.121)

Example 3 (Augmented bank-to-turn model using ailerons)Bank-to-turn autopilots are designed using the lateral state-space model with ailerons ascontrol inputs, for instance δA = 1/2(δAL + δAR). This is done by augmenting the bank-to-turn equation (2.119) to the state-space model according to:

vpr

φ

ψ

=

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

U00

vprφψ

+

b11

b21

b31

00

δA (2.122)

2.9.2 Dynamic equation for altitude control

Aircraft altitude control systems are designed by considering the equation for the verticalacceleration in the center of gravity expressed in NED coordinates; see (2.28). This gives:

azCG = W + V P −QU − g cos(Θ) cos(Φ) (2.123)

If the acceleration is perturbed according to azCG = az0 + δaz and we assume that W0 = 0,the following equilibrium condition is obtained:

az0 = V0P0 −Q0U0 − g cos(Θ0) cos(Φ0) (2.124)

Equation (2.123) can be perturbed as:

az0 + δaz = w + (V0 + v)(P0 + p)− (Q0 + q)(U0 + u)− g cos(Θ0 + θ) cos(Φ0 + φ) (2.125)

Furthermore, assume that the altitude is changed by symmetric straight-line flight withhorizontal wings such that V0 = Φ0 = Θ0 = P0 = Q0 = 0. This gives:

az0 + δaz = w + vp− q(U0 + u)− g cos(θ) cos(φ) (2.126)


Assume that the 2nd-order terms vp and uq can be neglected and subtract the equilibriumcondition (2.124) from (2.126) such that:

δaz = w − U0q (2.127)

Differentiating the altitude twice with respect to time gives the relationship:

h = −δaz = U0q − w (2.128)

If we integrate this expression under the assumption that h(0) = U0θ(0)−w(0) = 0, we get:

h = U0θ − w (2.129)

The flight path is defined as:

γ := θ − α (2.130)

where α = w/U0. This gives the resulting differential equation for altitude control:

h = U0γ (2.131)

Example 4 (Augmented model for altitude control using ailerons)An autopilot model for altitude control based on the longitudinal state-space model withstates u,w(alt. α), q and θ is obtained by augmenting the differential equation for h to thestate-space model according to:

uwq

θ

h

+

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

uwqθh

+

b11 b12

b21 b22

b31 b32

0 00 0

[δTδE

](2.132)

2.10 Aircraft Stability Properties

The aircraft stability properties can be investigated by computing the eigenvalues of thesystem matrix A using:

det(λI−A) = 0 (2.133)

For a matrix A of dimension n× n there is n solutions of λ.

2.10. AIRCRAFT STABILITY PROPERTIES 27

2.10.1 Longitudinal stability analysis

For conventional aircraft the characteristic equation is of 4th order. Moreover,

(λ2 + 2ζphωphλ+ ω2ph)(λ

2 + 2ζspωspλ+ ω2sp) = 0 (2.134)

where the subscripts ph and sp denote the following modi:

• Phugoid mode

• Short-period mode

The phugoid mode is observed as a long period oscillation with little damping. In somecases the Phugoid mode can be unstable such that the oscillations increase with time. ThePhugoid mode is characterized by the natural frequency ωph and relative damping ratio ζph.The short-period mode is a fast mode given by the natural frequency ωsp and relative

damping factor ζsp. The short-period mode is usually well damped.

Classification of eigenvalues

• Conventional aircraft: Conventional aircraft have usually two complex conjugatedpairs of Phugoid and short-period types. For the B-767 model in Chapter 4, theeigenvalues can be computed using damp.m in Matlab:

a = [-0.0168 0.1121 0.0003 -0.5608-0.0164 -0.7771 0.9945 0.0015-0.0417 -3.6595 -0.9544 0

0 0 1.0000 0]

damp(a)Eigenvalue Damping Freq. (rad/sec)-0.0064 + 0.0593i 0.1070 0.0596-0.0064 - 0.0593i 0.1070 0.0596-0.8678 + 1.9061i 0.4143 2.0943-0.8678 - 1.9061i 0.4143 2.0943

From this is seen that ωph = 0.0596, ζph = 0.1070, ωsp = 2.0943 and ζsp = 0.4143.

• Tuck Mode: Supersonic aircraft may have a very large aerodynamic coeffi cient Mu.This implies that the oscillatory Phugoid equation gives two real solutions where oneis positive (unstable) and one is negative (stable). This is referred to as the tuck modesince the phenomenon is observed as a downwards pointing nose (tucking under) forincreasing speed.


Figure 2.5: Aircraft longitudinal eigenvalue configuration plotted in the complex plane.

• A third oscillatory mode: For fighter aircraft the center of gravity is often locatedbehind the neutral point or the aerodynamic center—that is, the point where the trimmoment Mww is zero. When this happens, the aerodynamic coeffi cient Mw takes avalue such that the roots of the characteristic equation has four real solutions. Whenthe 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. Thisis usually referred to as the 3rd oscillatory mode. The locations of the eigenvalues areillustrated in Figure 2.5.

2.10.2 Lateral stability analysis

The lateral characteristic equation is usually of 5th order:

λ5 + α4λ4 + α3λ

3 + α2λ2 + α1λ+ α0 = 0 (2.135)

⇓λ(λ+ e)(λ+ f)(λ2 + 2ζDωDλ+ ω2

D) = 0 (2.136)

where the term λ in the last equation corresponds to a pure integrator in roll—that is, ψ = r.The term (λ+e) is the aircraft spiral/divergence mode. This is usually a very slow mode.

Spiral/divergence corresponds to horizontally leveled wings followed by roll and a divergingspiral 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 relative damping factor ζD. Thenatural frequency in Dutch roll is denoted ωD.If the lateral B-767 Matlab model in Chapter 4 is considered, the Matlab command

damp.m gives:

2.11. DESIGN OF FLIGHT CONTROL SYSTEMS 29

a = [-0.1245 0.0350 0.0414 -0.9962-15.2138 -2.0587 0.0032 0.6450

0 1.0000 0 0.03571.6447 -0.0447 -0.0022 -0.1416

damp(a)Eigenvalue Damping Freq. (rad/sec)-0.0143 1.0000 0.0143-0.1121 + 1.4996i 0.0745 1.5038-0.1121 - 1.4996i 0.0745 1.5038-2.0863 1.0000 2.0863

In this example we only get four eigenvalues since the pure integrator in yaw is notinclude in the system matrix. It is seen that the spiral mode is given by e = −0.0143 whilethe subsidiary roll mode is given by f = −2.0863. Dutch roll is recognized by ωD = 0.0745and ζD = 1.5038.

2.11 Design of flight control systems

A detailed introduction to design of flight control systems are given by [7], [11], [8] and [12].The control system are based on linear design techniques using linearized models similar tothose discussed in the sections above.

Chapter 3

Satellite Modeling

When stabilizing satellites in geostationary orbits only the attitude of the satellite is ofinterest since the position is given by the Earth’s rotation. Figure 3.1 shows the NUTSCubeSat project at NTNU.

3.1 Attitude Model

3.1.1 Euler’s 2nd Axiom applied to satellites

The rigid-body kinetics (2.21) gives:

ICGω + ω × (ICGω) = τ (3.1)

were ω = [p, q, r]> and ICG = I>CG > 0 is the inertia tensor about the center of gravity givenby:

ICG =

Ix −Ixy −Ixz−Ixy Iy −Iyz−Ixz −Iyz Iz

(3.2)

Both the Euler angles and quaternions can be used to represent attitude. Moreover,

q = Tq(q)ω (3.3)

Figure 3.1: The NUTS CubeSat project at NTNU- Courtesy to http://nuts.cubesat.no/

31

32 CHAPTER 3. SATELLITE MODELING

θ = Tθ(θ)ω (3.4)

where q = [η, ε1, ε2, ε3]>,θ = [Φ,Θ,Ψ]> and

Tθ(θ) =

1 sin(Φ) tan(Θ) cos(Φ) tan(Θ)0 cos(Φ) − sin(Φ)0 sin(Φ)/ cos(Θ) cos(Φ)/ cos(Θ)

, cos(Θ) 6= 0 (3.5)

Tq(q) =1

2

−ε1 −ε2 −ε3

η −ε3 ε2

ε3 η −ε1

−ε2 ε1 η

(3.6)

The satellite has three controllable moments τ = [τ 1, τ 2, τ 3]>, which can be generated usingdifferent actuators. Magnetic actuators are described in Section 3.2.

3.1.2 Skew-symmetric representation of the satellite model

The dynamic model (3.1) can also be written:

ICGω − (ICGω)× ω = τ (3.7)

where we have used the fact that a × b = −b × a. Furthermore, it is possible to find amatrix S(ω) such that S(ω) = −S>(ω) is skew-symmetric. With other words:

ICGω + S(ω)ω = τ (3.8)

The matrix S(ω) must satisfy the condition:

S(ω)ω ≡ −(ICGω)× ω (3.9)

A matrix satisfying this condition is:

S(ω) =

0 −Iyzq − Ixzp+ Izr Iyzr + Ixyp− IyqIyzq + Ixzp− Izr 0 −Ixzr − Ixyq + Ixp−Iyzr − Ixyp+ Iyq Ixzr + Ixyq − Ixp 0

(3.10)

3.2 Actuator Dynamics

Magnetic actuators are used to control the NUTS cube-satellite, which is a student satelliteproject at NTNU. The magnetic actuators are cheap, solid-state and energy-effi cient. Dueto variations of the magnetic field for a moving satellite, modeling and control may bechallenging. An analogy could be an airplane moving through a various density atmospheres,suddenly leaving you with limited control of either roll, pitch or yaw.When applying an electrical current through the windings of the coils, a magnetic dipole

moment is generated. When this field reacts with the magnetic field surrounding the Earth,a magnetic torque is generated, which makes the satellite move. Let the Earth´s magnetic

3.2. ACTUATOR DYNAMICS 33

field in BODY coordinates be denoted by Bb = [Bx, By, Bz]> while the magnetic field mb =

[mx,my,mz]> set up by the actuators is:

mb =

NxAxVxRx

NyAyVyRy

NzAzVzRz

(3.11)

= KcoilV (3.12)

where

Kcoil =

NxAxRx

0 0

0 NyAyRy

0

0 0 NzAzRz

, V =

VxVyVz

(3.13)

and for n = x, y, z,

Nn is the number of turns with copper thread in the coil

An is the area of the coil

Vn is the voltage over the coil

Rn is the resistance of the coil

Then, the torque becomes:

τ = mb ×Bb

= S(−Bb)mb (3.14)

where

S(−Bb) =

0 Bz −By

−Bz 0 Bx

By −Bx 0

(3.15)

For control purposes, we usually assign τ , however, the satellite only accepts voltage signalsas input, and not torque. It is straightforward to find the magnetic field mb due to theactuators:

mb = τ ×Bb (3.16)

In view of (3.13), one may find the relation between voltages and the torque vector:

V = K−1coil

(τ ×Bb

)(3.17)

The matrix Kcoil, usually referred to as the control allocation matrix, is a constant matrix.Note that Bb will change throughout the orbit.According to (3.14), τ b always lies in the plane perpendicular to Bb. Suppose that τ d is

the torque that the controller produces. It is required that τ d lies in the plane perpendicular


to Bb. It implies that τ only represents the component of τ d that is perpendicular to Bb. Toavoid unnecessary use of electricity, the length of τ d is scaled down by 1/

∥∥Bb∥∥2. Accordingly,

mb =1

‖Bb‖2 (τ d ×Bb) (3.18)

⇓

τ =1

‖Bb‖2 (τ d ×Bb)×Bb (3.19)

Let γ be the angle between τ d and Bb. The magnitude of τ is given by:

‖τ‖ =1

‖Bb‖2 ‖τ d‖∥∥Bb

∥∥ sin(γ)∥∥Bb

∥∥= ‖τ d‖ sin(γ) (3.20)

3.3 Design of Satellite Attitude Control Systems

Design of nonlinear trajectory-tracking control systems for Hamiltonian systems in the form(3.8) is quite common in the literature. One example is the nonlinear and passive adap-tive attitude control system of Slotine and Di Benedetto [10]. An alternative representa-tion is proposed by Fossen [3], which simplifies the representation of the regressor matrix.Trajectory-tracking controllers require that the model parameters are known or estimatedusing parameter adaptation. For set-point regulation is possible to derive controllers thatdo not require model parameters. This is presented below.

3.3.1 Nonlinear quaternion set-point regulator

Consider the Lyapunov function candidate:

V =1

2ω>ICGω + h(q) (3.21)

where h(q) is a positive definite function depending of the attitude error, for instance:

h(q) =1

2q>Kpq, q = q− qd (3.22)

where qd = constant and Kp > 0 is a design matrix. This means that the desired attitudemust be specified as a unit quaternion, which usually is computed by specifying the desiredEuler angles and transforming these to a desired unit quaternion.Differentiation of V with respect to time gives:

V = ωT ICGω + q>∂h

∂q

= ω>(−S(ω)ω + τ ) + ω>T>q (q)

∂h

∂q(3.23)

3.3. DESIGN OF SATELLITE ATTITUDE CONTROL SYSTEMS 35

Since S(ω) = −S>(ω), it follows that:

ω>S(ω)ω = 0 ∀ω (3.24)

This suggests that the control input τ should be chosen as:

τ = −Kdω −T>q (q)∂h

∂q

= −Kdω −T>q (q)Kpq (3.25)

where Kd > 0. such that

V = ω>(τ +T>q (q)

∂h

∂q

)= −ω>Kdω ≤ 0 (3.26)

and stability and convergence follow from standard Lyapunov techniques, for instance byusing Barbalat’s lemma.

3.3.2 PD-controller of Wen and Kreutz-Delago (1991)

The classical PD-controller for satellite attitude regulation uses feedback from the imaginarypart ε = [ε1, ε2, ε3]> of the unit quaternion error q = [η, ε1, ε2, ε3]> where η denotes thereal part of the unit quaternion. The attitude error dynamics q can be described by unitquaternions according to:

˙η =1

2ω>ε (3.27)

˙ε = −1

2ηω − 1

2ω × ε (3.28)

The following PD-controller is proposed by Wen and Kreutz-Delgado [13]:

τ = kpε− kdω (3.29)

where kp > 0 and kd > 0 are the controller gains. Notice that the control law does notdepends on the model of the system, and it makes ω and ε asymptotically converge to zeroand hence η converges to zero. Consider the following scalar function:

V = (kp + ckd)((η − 1)2 + ε>ε

)+

1

2ω>ICGω − cε>ICGω (3.30)

One may show that there exists a suffi ciently small c > 0 such that V > 0 for all η 6= 1,ε 6= 0, and ω 6= 0. In fact, V is bounded from below as

V ≥ (kp + ckd)(η − 1)2 +1

2

[ε ω

] [ 2(kp + ckd) cγIcγI µI

] [εω

](3.31)


where γI = ‖ICG‖ and µI = inf‖v‖=1 v>ICGv. µI > 0 since the inertia matrix ICG is positive

definite. We take the time derivative of V :

V = 2(kp + ckd)((η − 1) ˙η + ε> ˙ε

)+ ω>ICGω − c ˙ε>ICGω − cε>ICGω (3.32)

It is straightforward to find that (η − 1) ˙η + ε> ˙ε = −(1/2)ω>ε since ε>(ω × ε) = 0. Also,ω>ICGω = ω>(τ − ω × ICGω) = ω>τ . Moreover, we can find that

−c ˙ε>ICGω − cε>ICGω = −c(−1

2ηω − 1

2ω × ε

)>ICGω − cε> (τ − ω × ICGω) (3.33)

Thus, together with the control law ((3.29)), the time derivative will be

V = −ckp ‖ε‖2 − k2d ‖ω‖

2 − c−(

1

2ηω − 1

2ω × ε

)>ICGω − cε> (−ω × ICGω)

≤ −[ε ω

]> [ ckp 00 kd − 2cγI

] [εω

]>(3.34)

V ≤ 0; then, from Barbalat’s lemma, it follows that ε → 0 and ω → 0 as time tends toinfinity.As explained by (3.11)—(3.13), the input signal is voltage V of the coil. Thus, we need

to find the relation between τ and V. In practical situations, scaling is also required so asto avoid unnecessary electricity load. Therefore, one may derive the following input signal:

V = K−1coil

(τ ×Bb

)=

kp

‖Bb‖2K−1coil

(ε×Bb

)− kd

‖Bb‖2K−1coil

(ω ×Bb

)(3.35)

Chapter 4

Matlab Simulation Models

4.1 Boeing-767

The longitudinal and lateral B-767 state-space models are given below. The state vectorsare:

xlang =

u (ft/s)α (deg)q (deg/s)θ (deg)

,xlat =

β (deg)p (deg/s)φ (deg/s)r (deg)

(4.1)

ulang =

[δE (deg)δT (%)

],ulat =

[δA (deg)δR (deg)

](4.2)

Equilibrium point:

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

4.1.1 Longitudinal model

a = [ -0.0168 0.1121 0.0003 -0.5608-0.0164 -0.7771 0.9945 0.0015-0.0417 -3.6595 -0.9544 0

0 0 1.0000 0];

b = [ -0.0243 0.0519-0.0634 -0.0005-3.6942 0.0243

0 0 ];

37

38 CHAPTER 4. MATLAB SIMULATION MODELS

Eigenvalues:

lam = [-0.8678 + 1.9061i-0.8678 - 1.9061i-0.0064 + 0.0593i-0.0064 - 0.0593i];

4.1.2 Lateral model

a = [-0.1245 0.0350 0.0414 -0.9962-15.2138 -2.0587 0.0032 0.6458

0 1.0000 0 0.03571.6447 -0.0447 -0.0022 -0.1416];

b = [-0.0049 0.0237-4.0379 0.9613

0 0-0.0568 -1.2168];

Eigenvalues:

lam = [-0.1121 + 1.4996i-0.1121 - 1.4996i-2.0863-0.0143];

4.2 F-16 Fighter

The lateral model of the F-16 fighter aircraft is based on [11], pages 370—371. The statevectors are:

xlat =

β (ft/s)φ (ft/s)p (rad/s)r (rad)δA (rad)δR (rad)rw (rad)

,ulat =

[uA (rad)uR (rad)

],ylat =

rw (deg)p (deg/s)β (deg)φ (deg)

(4.3)

Equilibrium point:

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

4.2. F-16 FIGHTER 39

4.2.1 Longitudinal model

a = [-0.3220 0.0640 0.0364 -0.9917 0.0003 0.0008 00 0 1 -0.0037 0 0 0

-30.6492 0 -3.6784 0.6646 -0.7333 0.1315 08.5396 0 -0.0254 -0.4764 -0.0319 -0.0620 0

0 0 0 0 -20.2 0 00 0 0 0 0 -20.2 00 0 0 57.2958 0 0 -1 ];

b = [0 00 00 00 0

20.2 00 20.20 0 ];

c= [0 0 0 57.2958 0 0 -10 0 57.2958 0 0 0 0

57.2958 0 0 0 0 0 00 57.2958 0 0 0 0 0 ];

Eigenvalues:

lam = [-1.0000-0.4224+ 3.0633i-0.4224- 3.0633i-0.0167-3.6152-20.2000-20.2000 ];

Notice that the last two eigenvalues correspond to the actuator states.


Figure 4.1: Schematic drawing of the Bristol F.2B Fighter (McRuer et al 1973).

4.3 F2B Bristol Fighter

The lateral model of the F2B Bristol fighter aircraft is given below [8]. This is a Britishaircraft from World War I (see Figure 4.1). The aircraft model is for β = 0 (coordinatedturn). The state-space vector is:

xlat =

p (deg/s)r (deg/s)φ (deg)ψ (deg)

,ulat = [δA (deg)] (4.4)

where the dynamics for ψ satisfies the bank-to-turn equation:

ψ =g

U0

φ =9.81

138 · 0.3048φ = 0. 233φ (4.5)

Equilibrium point:

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

4.4. NTNU STUDENT CUBE-SATELLITE 41

4.3.1 Lateral model

a = [-7.1700 2.0600 0 0-0.4360 -0.3410 0 01.0000 0 0 0

0 0 0.2330 0];b = [

26.1000-1.6600

00];

Eigenvalues:

lam = [00

-0.4752-7.0358];

4.4 NTNU student cube-satellite

The NTNU student cube-satellite shown in Figure 3.1 has the following inertia matrix:

I_CG = diag[0.0108,0.0113,0.0048]; % kg m^2

Bibliography

[1] Blakelock, J. H. (1991). Aircraft and Missiles (John Wiley & Sons Ltd.)

[2] Etkin, B. and L. D. Reid (1996). Dynamics of Flight: Stability and Control (JohnWiley & Sons Ltd.)

[3] Fossen, T. I. (1993). Comments on ”Hamiltonian Adaptive Control of Spacecraft”,IEEE Transactions on Automatic Control, TAC-38(5):671—672.

[4] Fossen, T. I. (1994). Guidance and Control of Ocean Vehicles (John Wiley & SonsLtd.)

[5] Fossen, T. I. (2011). Handbook of Marine Craft Hydrodynamics and Motion Control.(John Wiley & Sons Ltd.)

[6] Hughes, P. C. (1986). Spacecraft Attitude Dynamics (John Wiley & Sons Ltd.)

[7] McLean, D. (1990). Automatic Flight Control Systems (Prentice Hall Inc.)

[8] McRuer, D., D. Ashkenas og A. I. Graham (1973). Aircraft Dynamics andAutomatic Control. (Princeton University Press, New Jersey 1973).

[9] Nelson R. C. (1998). Flight Stability and Automatic Control (McGraw-Hill Int.)

[10] Slotine, J. J. E. og M. D. Di Benedetto (1990). Hamiltonian Adaptive Controlof Spacecraft, IEEE Transactions on Automatic Control, TAC-35(7):848—852.

[11] Stevens, B. L. og F. L. Lewis (1992). Aircraft Control and Simulation (John Wiley& Sons Ltd.)

[12] Roskam, J. (1999). Airplane Flight Dynamics and Automatic Flight Controls (Dar-corporation)

[13] Wen, J. T.-Y. and K. Kreutz-Delgado (1991). The Attitude Control Problem.IEEE Transactions on Automatic Control, Vol. 36, No. 10. October 1991.

43

Aircraft Fossen 2013

Documents

Transcript of Aircraft Fossen 2013