Uavbook Ch5 Partial[1] Copy

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Guidance and Control of

Autonomous Fixed Wing Air Vehicles

Randal W. Beard Timothy W. McLain

Brigham Young University


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Contents

Preface vii

1 Introduction 1

1.1 System Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Design Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Coordinate Frames 5

2.1 Rotation Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 MAV Coordinate Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Equation of Coriolis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Design Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Kinematics and Dynamics 21

3.1 MAV State Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 MAV Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 Rigid Body Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4 Design Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Forces and Moments 29

4.1 Gravitational Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 Aerodynamic Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.3 Propulsion Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.5 Design Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5 Nonlinear Equations of Motion 41

5.1 Six DOF Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.2 Navigation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

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iv CONTENTS

5.3 Wind Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.4 Design Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6 Trim 51

6.1 Turn with a Constant Climb Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6.2 Design Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

7 Open Loop Linear Dynamics 59

7.1 Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

7.2 Linear State Space Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

7.3 Design Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

8 Autopilot Design Using Successive Loop Closure 79

8.1 Lateral Autopilot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

8.2 Longitudinal Autopilot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

8.3 Design Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

9 Micro UAV Sensors 99

9.1 Rate Gyros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

9.2 Accelerometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

9.3 Pressure Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

9.4 Magnetometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

9.5 GPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

9.6 Design Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

10 State Estimation 109

10.1 Low Pass Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

10.2 State Estimation by Inverting the Sensor Model . . . . . . . . . . . . . . . . . . . 110

10.3 Dynamic Observer Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

10.4 Essentials from Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 116

10.5 Derivation of the Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

10.6 Attitude Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

10.7 GPS Smooting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

10.8 Wind Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

10.9 Design Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
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CONTENTS v

11 Waypoint and Orbit Following 133

11.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

11.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15011.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

11.4 Another approach to straight line tracking . . . . . . . . . . . . . . . . . . . . . . 153

11.5 Design Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

12 Path Planning 159

12.1 Point-to-Point Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

12.2 Coverage Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

12.3 Design Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

13 Path Manager 169

13.1 Switching Between Waypoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

13.2 Smooth transitions that satisfy kinematic constraints . . . . . . . . . . . . . . . . . 171

13.3 Smooth transitions through the waypoint . . . . . . . . . . . . . . . . . . . . . . . 171

13.4 Smooth transitions that preserve path length . . . . . . . . . . . . . . . . . . . . . 171

13.5 Path Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

13.6 Dubins Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

13.7 Design Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

14 Cameras on Micro UAVs 199

14.1 Gimbal and Camera Frames and Projective Geometry . . . . . . . . . . . . . . . . 200

14.2 Gimbal Pointing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

1 4 . 3 G e o l o c a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0 4

14.4 Estimating Target Motion in the Image Plane . . . . . . . . . . . . . . . . . . . . 207

14.5 Precision Landing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

14.6 Design Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

A Aviones 223

B Introduction to Modeling in Simulink 225

C Useful Formulas and other Information 227

C.1 Conversion from knots to mph . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

D Graphs Theory 229
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vi CONTENTS

Bibliography 238
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x CONTENTS


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Chapter 5

Nonlinear Equations of Motion

5.1 Six DOF Equations of Motion

5.1.1 General Force and Torque Model

If we use the general force and torque models given by

fx

fy

fz

= mg

sin

cos sin

cos cos

+ qS

CX(x, )

CY(x, )

CZ(x, )

l

m

n

= qS

bCl(x,)

cCm(x,)

bCn(x, )

,

41


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42 5.1 Six DOF Equations of Motion

in the six degree-of-freedom model given by Equations (3.9)-(3.12), we get

pn

pe

h

=

cc ssc cs csc + ss

cs sss + cc css sc

s sc cc

u

v

w

(5.1)

u

v

w

=

rv qw

pw ru

qu pv

+

g sin

g cos sin

g cos cos

+qS

m

CX(x, )

CY(x, )

CZ(x,)

(5.2)

=

1 sin()tan() cos()tan()

0 cos() sin()

0 sin()sec() cos()sec()

p

q

r

(5.3)

p

q

r

=

1pq 2qr

5pr 4(p2 r2)+

6pq 1qr

+ qS

b [Gamma3Cl(x, ) + 4Cn(x, )]cJy

Cm(x, )

b [4Cl(x, ) + 7Cn(x, )]

. (5.4)

5.1.2 Linear Aerodynamic Model

A variety of different models for the aerodynamic forces and moments appear in the literature. In

particular, if we use the linear model described in Chapter 4 we get the following equations of
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Nonlinear Equations of Motion 43

motion:

pn = (cos cos )u + (sin sin cos cos sin )v + (cos sin cos + sin sin )w

(5.5)

pe = (cos sin )u + (sin sin sin + cos cos )v + (cos sin sin sin cos )w

(5.6)

h = u sin v sin cos w cos cos (5.7)

u = rv qw g sin +qS

m

CX0 + CX + CXqcq

Va + CXee

+

Sprop2m Cprop

(kt)

2 V

2a

(5.8)

v = pw ru + g cos sin +qS

m

CY0 + CY+ CYp

bp

2Va+ CYr

br

2VaCYaa + CYr r

(5.9)

w = qu pv + g cos cos +qS

m

CZ0 + CZ + CZq

cq

Va+ CZee

(5.10)

= p + qsin tan + r cos tan (5.11)

= qcos r sin (5.12)

= qsin sec + r cos sec (5.13)

p = 1pq 2qr + qSb

Cp0 + Cp+ Cpp

bp

2Va+ Cpr

br

2Va+ Cpaa + Cpr r

(5.14)

q = 5pr 4(p2 r2) +

qSc

Jy

Cm0 + Cm + Cmq

cq

2Va+ Cmee

(5.15)

r = 6pq 1qr + qSb

Cr0 + Cr+ Crp

bp

2Va+ Crr

br

2Va+ Craa + Crr r

, (5.16)


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44 5.2 Navigation Models

where

Cp0 = 3Cl0 + 4Cn0

Cp = 3Cl + 4Cn

Cpp = 3Clp + 4Cnp

Cpr = 3Clr + 4Cnr

Cpa = 3Cla + 4Cna

Cpr = 3Clr + 4Cnr

Cr0 = 4Cl0 + 7Cn0

Cr = 4Cl + 7Cn

Crp = 4Clp + 7Cnp

Crr = 4Clr + 7Cnr

Cra = 4Cla + 7Cna

Crr = 4Clr + 7Cnr .

5.2 Navigation Models

5.2.1 8 State Navigation Equations

The full 12-state equations of motion given in Equations (5.1)(5.4) are usually not needed to

derive good navigation models. In this section we will show a useful method for reducing these

equations.

The first simplification is to note that the velocity vector in the wind frame, i.e. (Va, 0, 0)T,

where Va is the airspeed, can be related to the inertial coordinates by two angles: the flight path

angle and the heading , as shown in Figure 5.1. The heading is obtained by rotating the inertial

coordinate from by until the x-axis is aligned with the projection of the velocity vector on the

x-y plane. The appropriate transformations are given by

pnpeh

= cos

sin 0sin cos 0

0 0 1

cos 0 sin 0 1 0 sin 0 cos

Va00

= cos cos sin cos sin

Va.

Therefore

pn

pe

h

=

cos cos

sin cos

sin

Va.
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Flight path

Figure 5.1: The flight path angles and .

Coordinated Turn

The following derivation draws on the discussion [2, p. 224226]. From Equation (5.13) we get

that

= sin cos

q+ cos cos

r.

Figure 5.2 shows a free body diagram of the UAV indicating forces in the x z plane during a

Figure 5.2: Free body diagram indicating forces on the UAV in the y z plane. The nose of the

UAV is out of the page. The UAV is assumed to be pitched at the flight path angle .
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cork-screw maneuver. Writing the force equations we get

L cos cos = mg cos (5.17)

L sin cos = mVa. (5.18)

Dividing (5.18) by (5.17) and solving for gives

=g

Vatan , (5.19)

which is the equation for a coordinated turn. Given that the turning radius is given by Rt = Va/

we get

Rt =V2a

g tan . (5.20)

From Figure 5.2 we also see that

q = sin (5.21)

r = cos . (5.22)

Plugging Eq. (5.19) into Eq. (5.21) and (5.22) gives

q =g

Va

sin2

cos (5.23)

r =g

Vasin . (5.24)

Plugging into Eq. (??) gives

=g

Va

sin

cos

1

cos .

Noting from Eq. (5.17) that1

cos =

L

mg,

and defining the Load Factoras n= L/mg gives

=g

Va

sin

cos n. (5.25)

Our derivation of the dynamic equation for draws on the discussion in [2, p. 227228]. The

free body diagram of the UAV in the x zplane is shown in Figure 5.3. Since the UAV has a roll

angle of , the projection of the lift vector onto the x z plane is L cos . The centripetal force

due to to the pull-up maneuver is mVa. Therefore, summing the forces in the x zplane gives

L cos = mVa+ mg cos .
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Figure 5.3: Free body diagram indicating forces on the UAV in the x z plane. The left wing is

out of the page. The UAV is assumed to be in a roll angle of.

Letting n = L/mg and solving for gives

=g

Va(n cos cos ) . (5.26)

We will assume that the mini-UAV is equipped with an autopilot that implements the following

feedback loops: (1) airspeed hold, (2) roll-attitude hold, and (3) load-factor hold. In addition,

we will assume that the autopilot is tuned such that the closed-loop behavior of these loops is

essentially first order. Therefore, the closed loop behavior of the autopilot is given by

Va =1

V(Vca Va)

= 1

(c ) (5.27)

n =1

n(nc n),

where > 0 are positive autopilot time constants, and Vca ,

c, and nc are the inputs to the autopilot.


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In summary, the equations of motion for the UAV are given by

pn = Va cos cos + wx (5.28)

pe = Va sin cos + wy (5.29)

h = Va sin + wh (5.30)

=g

Va

sin

cos n (5.31)

=g

Va(n cos cos ) (5.32)

Va =1

V(Vca Va) (5.33)

=1

(c ) (5.34)

n =1

n(nc n). (5.35)

An equivalent, but useful alternative to these equations is to assume that the load factor is

selected as

nc = n =cos

cos . (5.36)

In this case equations (5.31) and (5.32) become

=g

Vatan() (5.37)

=

g cos

Va (

1). (5.38)Note that when = 1, the airframe is experiencing a level flight coordinated turn.


If we assume that the airspeed is constant and that the load factor is given by Equation (5.36), then

the six state navigation equations become

pn = Va cos cos + wx (5.39)

pe = Va sin cos + wy (5.40)

h = Va sin + wh (5.41)

=g

Vatan() (5.42)

=g cos

Va( 1) (5.43)

=1

(c ), (5.44)
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where the inputs are and c.


There are times when we are interested in 2D path planning. In that case we select = 1 to obtain

pn = Va cos + wx (5.45)

pe = Va sin + wy (5.46)

=g

Vatan (5.47)

=1

(c ), (5.48)

where c is the input.

5.3 Wind Models

5.3.1 Wind Speed Above Ground

According to [9], the wind-speed increases with altitude. Up to an altitude of few hundred meters,

the wind speed profile vw(z) is approximately a logarithmic function of the altitude (z):

vw(z) vo() ln

zz0

, for z > 10z0,

where is the thickness of the surface boundary layer, z0 = 0.1 m in a low density urban zone,

hence

z(vw) = z0evw/v0().
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50 5.4 Design Project

5.4 Design Project

5.1 Homework problem 1.


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