MS Thesis ShashiprakashSingh

Autonomous Landing ofUnmanned Aerial Vehicles

A Thesis

Submitted For the Degree of

Master of Science

in the Faculty of Engineering

by

Shashiprakash Singh

Department of Aerospace Engineering

Indian Institute of Science

BANGALORE – 560 012

February 2009

c©Shashiprakash Singh

February 2009

All rights reserved

DECLARATION

I declare that the thesis entitled Autonomous Landing of Unmanned Aerial

Vehicles submitted by me for the M.Sc.[Engg.] degree of the Indian Institute of Science

did not form the subject matter of any other thesis submitted by me for any outside

degree and the original work done by me and incorporated in this thesis is entirely done

at Indian Institute of Science, Bangalore.

Bangalore Shashiprakash Singh

February 2009

Dedicated to

“Late Dr. S. Pradeep”

For his guidance and teaching me the basics of research.

Acknowledgements

I would like to thank my research advisor Dr. Radhakant Padhi for his constant guidance,

discussions, support and constructive criticism. I am also thankful to Prof. M. S. Bhat and

Prof. D. Ghose for their valuable advice at various instances. I also want to thank all my

course instructors for having enriched my knowledge on various research topics.

My thanks to Mr. V. Surendranath and all the staff at the wind tunnel facility, for carrying

out the wind tunnel test and for letting us use their facility at UAV lab. I also want to thank

Prof. S. P. Govindaraju for answering many of my queries related to aerodynamic data.

I am grateful to all my lab members for creating a constructive research environment in

the lab. It is through the discussions with lab members that I have gained the knowledge on

various problems in the areas of aerospace and control design.

I would like to take this opportunity to thank Prof. B. N. Raghunandan for all the research

facilities at department and his motivating talks. I will not miss to thank the staff at aerospace

main office who have been most humble and have always shown readiness to help.

I am grateful to my family for their patience and support of my decision to pursue higher

studies. They have always understood my position in spite of my less frequent visits to home.

In the last I thank all the trees, plants and flowers of IISc for making the institute a beautiful

and silent place, where the researchers can think freely in the peaceful air.

i

Publications based on this thesis

1. Shashiprakash Singh and Radhakant Padhi,

Automatic Landing of Unmanned Aerial Vehicles Using Dynamic Inversion,

Proceedings of International Conference on Aerospace Science and Technology, Banga-

lore, India, 26-28 June 2008.

2. Shashiprakash Singh and Radhakant Padhi,

Automatic Path Planning and Control Design for Autonomous Landing of UAVs

using Dynamic Inversion, Accepted for 2009 American Control Conference, Missouri,

USA, 10-12 June 2009.

ii

Abstract

In this thesis the problem of autonomous landing of an unmanned aerial vehicle named

AE-2 is addressed. The guidance and control technique is developed and demonstrated

through numerical simulation results. The complete work includes Mathematical mod-

eling, Control design, Guidance and State estimation for AE-2, which is a fixed wing

vehicle with 2m wing span and 6kg weight.

The aerodynamic data for AE-2 is available from static wind tunnel tests. Functional

fit is done on the wind tunnel data with least squares method to find static aerody-

namic coefficients. The aerodynamic forces and moment coefficients are highly nonlinear

some of them are partitioned in two zones based on the angle of attack. The dynamic

derivatives are found with Athena Vortex Lattice software. For the validation of vortex

lattice method the static derivatives obtained by the wind tunnel tests and vortex lattice

method, are compared before finding dynamic derivatives. The dynamics of the servo

actuators for the aerodynamic control surfaces is incorporated in the simulation.

The nonlinear dynamic inversion technique has been used for the guidance and control

design. The control is structured in two loops, outer and inner loop. The goal of outer

loop is to track the guidance commands of altitude, roll angle and yaw angle by converting

them into body rate commands through dynamic inversion. The inner loop than tracks

these commanded roll rate, pitch rate and yaw rate by finding the required deflection of

control surfaces. The forward velocity of the vehicle is controlled by varying the throttle.

A controller for actuator is also designed to reduce the lag.

iii

Abstract iv

The guidance for landing consists of three phases approach, glideslope and flare.

During approach the vehicle is aligned with the runway and guided to a specified height

from where the glideslope can begin. The glideslope is straight line path specified by

a flight path angle which is restricted between 3 to 4 degree. At the end of glideslope

which is marked by flare altitude the flare maneuver begins which is an exponential curve.

The problem of transition between the glideslope and flare has addressed by ensuring

continuity and smoothness at transition. The exponential curve of flare is designed to

end below the ground so that it intersects the ground at a prespecified point. The sink

rate at touchdown is also controlled along with the location of touchdown point.

The state estimation has been done with Extended Kalman Filter in continuous dis-

crete formulation. The external disturbances like wind shear and wind gust are accounted

by appending them in state variables. Further the control design with guidance is tested

from various initial conditions, in presence of wind disturbances. The designed filter has

also been tested for parameter uncertainty.

Contents

Acknowledgements i

Publications based on this thesis ii

Abstract iii

Notation and Abbreviations xiii

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Contribution of present work . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Organization of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Mathematical modeling 6

2.1 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Aerodynamic forces and moments . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Wind tunnel testing . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.2 Curve fitting on wind tunnel data . . . . . . . . . . . . . . . . . . 11

2.2.3 Static stability derivatives . . . . . . . . . . . . . . . . . . . . . . 17

2.2.4 Dynamic stability derivatives . . . . . . . . . . . . . . . . . . . . 24

2.3 Thrust force and moment . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4 Actuator dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5 Trim condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

v

CONTENTS vi

2.6 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.6.1 Open loop simulation . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.6.2 Perturbation around trim condition . . . . . . . . . . . . . . . . . 33

3 Control design 35

3.1 Dynamic inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Outer loop control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.1 Roll angle control . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2.2 Pitch angle control . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2.3 Heading angle control . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3 Inner loop control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3.1 Body rates control . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3.2 Forward velocity control . . . . . . . . . . . . . . . . . . . . . . . 43

3.4 Actuator controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.5 Control structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45


3.6.1 Pitch angle command . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.6.2 Heading angle command . . . . . . . . . . . . . . . . . . . . . . . 48

3.6.3 Bank angle command . . . . . . . . . . . . . . . . . . . . . . . . . 50

4 Path planning and guidance 53

4.1 Path planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.1.1 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.1.2 Glideslope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.1.3 Flare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2 Guidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2.1 Sideward distance control . . . . . . . . . . . . . . . . . . . . . . 58

4.2.2 Altitude control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2.3 Coordinated turn constraint . . . . . . . . . . . . . . . . . . . . . 60

CONTENTS vii


5 State estimation 65

5.1 Extended Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2 Filter equations for UAV . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.2.1 Wind disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.2.2 State sensitivity matrix . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2.3 Output sensitivity matrix . . . . . . . . . . . . . . . . . . . . . . 71

5.2.4 Selection of R, P0 and Q . . . . . . . . . . . . . . . . . . . . . . . 73


6 Conclusions 80

References 82

List of Tables

2.1 Physical data of AE-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Wind tunnel test variables and their range . . . . . . . . . . . . . . . . . 10

2.3 Average of absolute percentage error in curve fitting . . . . . . . . . . . . 28

2.4 Derivatives function coefficient values . . . . . . . . . . . . . . . . . . . 29

2.5 Trim conditions at different velocities . . . . . . . . . . . . . . . . . . . . 31

3.1 Gains for various control loops . . . . . . . . . . . . . . . . . . . . . . . . 45

4.1 Gains for guidance loop . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2 Landing results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.1 Percentage of successful landing . . . . . . . . . . . . . . . . . . . . . . . 79

viii

List of Figures

1.1 AE-2 (All Electric airplane-2) . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Body reference frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 AE-2 in the open circuit wind tunnel . . . . . . . . . . . . . . . . . . . . 10

2.3 Plot of CZstat = f(α, β), data Table1 . . . . . . . . . . . . . . . . . . . . 12

2.4 CZstat vs α for various β . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.5 CZstat vs β for various α . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.6 CZstat vs δr for various α . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.7 CZstat vs δa for various α . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.8 Curve fit on CZstat(α, β) vs α . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.9 Curve fit on CZstat(α, δe) vs α . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.10 Curve fit on CZstat(α, β) vs β . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.11 Curve fit on CZstat(α, δe) vs δe . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.12 Curve fit on CXstat(α, δe) vs α . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.13 Curve fit on CXstat(α, δe) vs δe . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.14 Curve fit on Cmstat(α, β) vs α . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.15 Curve fit on Cmstat(α, δe) vs α . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.16 Curve fit on Cmstat(α, β) vs β . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.17 Curve fit on Cmstat(α, δe) vs δe . . . . . . . . . . . . . . . . . . . . . . . . 20

2.18 Curve fit on CYstat(α, β) vs α . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.19 Curve fit on CYstat(α, β) vs β . . . . . . . . . . . . . . . . . . . . . . . . . 21

ix

LIST OF FIGURES x

2.20 Curve fit on CYstat(α, δr) vs δr . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.21 Curve fit on CYstat(α, δa) vs δa . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.22 Curve fit on Clstat(α, β) vs α . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.23 Curve fit on Clstat(α, δa) vs α . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.24 Curve fit on Clstat(α, β) vs β . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.25 Curve fit on Clstat(α, δa) vs δa . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.26 Curve fit on Cnstat(α, β) vs α . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.27 Curve fit on Cnstat(α, δr) vs α . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.28 Curve fit on Cnstat(α, β) vs β . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.29 Curve fit on Cnstat(α, δr) vs δr . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.30 AE-2 modeling in AVL . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.31 AVL and wind tunnel test results comparison . . . . . . . . . . . . . . . 25

2.32 Pitch rate derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.33 Roll rate derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.34 Yaw rate derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.35 Step response of actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.36 Longitudinal states in trim condition . . . . . . . . . . . . . . . . . . . . 32

2.37 Lateral states in trim condition . . . . . . . . . . . . . . . . . . . . . . . 32

2.38 Effect of velocity and angle of attack perturbation on longitudinal states 33

2.39 Effect of pitch angle and pitch rate perturbation on longitudinal states . 33

2.40 Effect of sideslip and roll angle perturbation on lateral states . . . . . . . 34

2.41 Effect of sideslip and roll angle perturbation on longitudinal states . . . . 34

3.1 Inner and outer loop structure . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Outer loop control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 Inner loop control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.4 Actuator response for 1 deg command of deflection . . . . . . . . . . . . 44

3.5 Control structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

LIST OF FIGURES xi

3.6 Pitch angle variation for 3-2-1-1 command . . . . . . . . . . . . . . . . . 46

3.7 Longitudinal states for pitch angle command . . . . . . . . . . . . . . . . 47

3.8 Control variables for pitch angle command . . . . . . . . . . . . . . . . . 47

3.9 Heading angle variation for 3-2-1-1 command . . . . . . . . . . . . . . . . 48

3.10 Longitudinal states for heading angle command . . . . . . . . . . . . . . 48

3.11 Lateral states for heading angle command . . . . . . . . . . . . . . . . . 49

3.12 Control variables for heading angle command . . . . . . . . . . . . . . . 49

3.13 Roll angle variation for step 3-2-1-1 command . . . . . . . . . . . . . . . 50

3.14 Longitudinal states for roll angle command . . . . . . . . . . . . . . . . . 50

3.15 Lateral states for roll angle command . . . . . . . . . . . . . . . . . . . . 51

3.16 Control variables for roll angle command . . . . . . . . . . . . . . . . . . 51

4.1 Phases during landing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2 Approach geometry in x-y plane (top view) . . . . . . . . . . . . . . . . . 54

4.3 Glideslope and flare geometry in x-h plane . . . . . . . . . . . . . . . . . 55

4.4 Path planning and guidance schematic . . . . . . . . . . . . . . . . . . . 60

4.5 Landing trajectory in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.6 Landing path in x-y plane . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.7 Landing path in x-h plane . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.8 Longitudinal states during landing . . . . . . . . . . . . . . . . . . . . . 62

4.9 Lateral states during landing . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.10 Control values during landing . . . . . . . . . . . . . . . . . . . . . . . . 63

5.1 Landing trajectory in 3D with EKF . . . . . . . . . . . . . . . . . . . . . 74

5.2 Landing path in x-y plane . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.3 Landing path in x-h plane . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.4 Estimation error in longitudinal states . . . . . . . . . . . . . . . . . . . 75

5.5 Estimation error in lateral states . . . . . . . . . . . . . . . . . . . . . . 75

5.6 Wind estimation with EKF . . . . . . . . . . . . . . . . . . . . . . . . . 76

LIST OF FIGURES xii

5.7 Control values during landing with EKF . . . . . . . . . . . . . . . . . . 76

5.8 Touchdown position in x-y plane . . . . . . . . . . . . . . . . . . . . . . 77

5.9 Pitch angle at touch down . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.10 Sink rate at touch down . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.11 Touchdown position in x-y plane with parameter uncertainty . . . . . . . 78

5.12 Pitch angle at touch down . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.13 Sink rate at touch down . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Notation and Abbreviations

English Alphabet

ax − Acceleration in body axis X direction

ay − Acceleration in body axis Y direction

az − Acceleration in body axis Z direction

b − Wing span

c − Mean aerodynamic chord

Cl − Rolling moment coefficient

Cm − Pitching moment coefficient

Cn − Yawing moment coefficient

CX − Axial force coefficient

CY − Side force coefficient

CZ − Normal force coefficient

d − Offset of the thrust line from CG in the Z axis

h − Altitude above ground

Ixx, Iyy, Izz − Moment of inertia around body fixed X, Y and Z axis

Ixz − Product of inertia between body fixed x and z axis

kh − proportional gain for altitude error

kp − proportional gain for rolling moment error

kq − proportional gain for pitching moment error

kr − proportional gain for yawing moment error

ku − proportional gain for forward velocity error

xiii

Notation and Abbreviations xiv

kv − proportional gain for sideward velocity error

ky − proportional gain for sideward distance error

kφ − proportional gain for roll angle error

kθ − proportional gain for pitch angle error

kψ − proportional gain for heading angle error

La − Rolling moment due to aerodynamic effects

m − Mass of the aircraft

M − Mach number

Ma − Pitching moment due to aerodynamic effects

Na − Yawing moment due to aerodynamic effects

p − Roll rate

P − Error covariance matrix

q − Pitch rate

Q − Process noise covariance matrix

q − Dynamic pressure = 12ρV 2

t

r − Yaw rate

R − Measurement noise covariance matrix

Re − Reynolds number

S − Surface area of the wing

u − Forward velocity in body axis X direction

U − Control vector.

Ua − Aerodynamic controls (elevator, aileron and rudder).

Uc − Control variables (throttle, elevator, aileron and rudder).

v − Sideward velocity in body axis Y direction

w − Downward velocity in body axis Z direction

x − Forward distance in inertial frame

X − State vector

Xa − Axial force due to aerodynamic effects in body X axis

Xt − Thrust force in body X axis direction

Notation and Abbreviations xv

Xd − Wind disturbance state vector

y − Sideward distance in inertial frame

Y − Output vector

Ya − Side force due to aerodynamic effects

Za − Normal force due to aerodynamic effects

Greek Alphabet

α − Angle of attack

β − Sideslip angle

γ − Flight path angle

δa − Aileron deflection

δe − Elevator deflection

δr − Rudder deflection

σt − Throttle setting (0-1)

ρ − Density of air

φ − Roll angle

θ − Pitch angle

ψ − Yaw angle

χ − Combined state vector of system and disturbance

Abbreviations

six-DOF − Six degrees of freedom

AE-2 − All Electric airplane - 2

AVL − Athena Vortex Lattice Software

Chapter 1

Introduction

The capabilities of Unmanned Aerial Vehicles(UAVs) as flying machines can be exploited

to carry out surveillance missions and remote operations. The UAVs are becoming more

promising with their applications to the scenarios of disaster management, forest fire

detection, frontier surveillance, power line monitoring and many others. The recovery

of UAVs in landing is one of the key operations in flight which define the overall success

of the mission. The UAVs are recovered through net on naval ships and also on ground,

while some UAVs employ parachutes to decrease its descent rate during landing. Other

methods are hard landing on belly or landing on runway with wheels.

The complete autonomous flight for an UAV without human intervention will include

autonomous take off, waypoint navigation and autonomous landing. The waypoint nav-

igation and take-off are comparatively easier problem than autonomous landing because

they involve lesser constraints and uncertainties. The goals for a successful landing will

require it to touchdown with a bound on sink rate and the pitch angle at touchdown,

also the touchdown location on runway should be within some specified zone. The un-

certainties which make landing a challenging problem the presence of wind disturbances

and ground effects. The ground effects have not been considered in this thesis, because

1“Flying is the second most adventure known to mankind, first is landing”. Unknown

1

Chapter 1. Introduction 2

of the unavailability of reliable models for ground effects for small class of vehicles like

UAVs. However, the uncertainties in forces and moments have been considered.

1.1 Motivation

The dynamics of low speed and light weight vehicles such as UAVs are different compared

to that of conventional aircrafts. The reasons being the coupling between longitudinal

and lateral modes and low Reynolds number effects which is under study [1]. The

wind tunnel tests can give the aerodynamic coefficients for forces and moments. Which

further need to be modeled in global function form [2], [3], [4] in order to reduce the

burden of interpolation from large number of tables. It is found that linearized models

of the aircraft have been used in the literature using separate longitudinal and lateral

dynamics. However, linear system based approaches have a strong limitation that they

work within a small operating range. Gain scheduling can perhaps be used to overcome

this limitation to a limited extent. However gain scheduling is a tedious process and

there is no guarantee that the interpolated gains can assure stability of the closed loop

system [5].

Various control strategies have been adapted to tackle the problem of automatic land-

ing both for manned and unmanned aircrafts. Linear control theory has been heavily

investigated. The concept of Linear Parameter Varying representation with piecewise

affine dependence on the parameters is adopted [6] for modeling and linear matrix in-

equality method for control design . Modern control methods like H2/H∞ have also been

used for landing of UAVs [7], [8]. Dynamic inversion is a popular method of nonlinear

control which relies on the philosophy of feedback linearization [9], [10]. This feedback

control structure cancels the nonlinearities in the plant such that the closed loop plant

behaves like a stable linear system. This method has several advantages, like simplicity in

the control structure, ease of implementation, global exponential stability of the tracking

error etc. A feedback linearization technique has been successfully demonstrated through


simulation for automatic landing of a high performance aircraft [11]. The dynamic in-

version has also been used for the design of longitudinal landing [12] and for the control

of micro aerial vehicles [13]. The robustness with nonlinear control [14], [15] have also

been studied and reported.

The autonomous landing requires an intelligent path planning and guidance. Landing

trajectories of aerial vehicles typically consists of approach, glideslope and flare [11]. A

successful landing would depend upon the good selection of landing trajectory and closely

following it. The problems associated with following glideslope and flare is the change in

slope at the transition. Which put requirement of different gains for following glideslope

and flare. To overcome this blending function [16] for gains at the transition of glideslope

and flare has been reported in literature. It is found in literature that the desired landing

trajectory can be made function of time [11] or forward distance from runway [6], [17].

The other requirements on landing are controlling the sink rate at touchdown and the

location of touchdown.

The landing requires sensors for position location and height with the help of which

it can be achieved successfully. The sensors for height include pressure altitude sensors

or laser altimeter [18]. Vision based algorithms for autonomous landing of UAVs have

been also been designed [19]. The vision sensor is used to extract the relative position of

the UAV from a landing pad than this information is used to direct the UAV to land [20].

The use of vision based precision target detection and recognition along with the GPS for

navigation [21] have also been shown. More advance algorithms have combined the data

from optical flow sensors and pressure based sensor to estimate the HAG [22] for relatively

unknown terrains. The other challenges for landing are uncertainties in measurement

and process noise in form of wind disturbances. The state estimation will be required to

account for non availability for all the sensors. The estimation can be carried out with

Extended Kalman filter [23]. The aircraft landing control and estimation in presence of

wind shear [24], [25] has been reported. The estimation with EKF for landing of UAV

is carried out in presence wind disturbance [26].


1.2 Contribution of present work

In the present research guidance algorithm has been developed for autonomous landing

along with the development of complete mathematical model for the UAV. The nonlinear

control design and estimation have been carried out and demonstrated with simulations.

The aerodynamic coefficients for force and moments are found from the curve fitting

of the wind tunnel data. The nonlinearities found in the functions obtained from curve

fit are presented. The problem of slope discontinuity during transition from glideslope

to flare has been addressed by careful selection of landing trajectory parameters. The

glideslope and flare path is scheduled as a function of forward distance which makes the

trajectory independent of time. The glideslope and flare path parameters are computed

online (i.e. they are not fixed apriori). Further the trajectory parameters are calculated

such that the sink rate at touchdown remains within specified bounds. The control

is designed using dynamic inversion technique, while continuous-discrete formulation of

extended kalman filter has been used for state estimation. The numerical simulations

are carried out using the data of an UAV named AE-2 (All Electric airplane-2) shown in

fig 1.1. The AE-2 was designed and developed [27] at UAV lab of aerospace engineering

department. It is a small fixed wing aircraft with weight of 6kg and wing span of 2m. It

is designed with a capability of autonomous flying and carrying out surveillance missions

over a range of 10 Km.

Figure 1.1: AE-2 (All Electric airplane-2)


1.3 Organization of thesis

The remainder of the thesis is organized in five chapters followed by conclusion and a

list of bibliography. The content of each of these chapters is given below

Chapter 2: Mathematical modeling

In this chapter the mathematical model is derived for the UAV from wind tunnel

data and analytical software. The trim condition are calculated and open loop

flight simulation is done to test the model developed.

Chapter 3: Control design

In this chapter control is designed using Dynamic Inversion in outer and outer loop.

Further it is tested by giving various commands.

Chapter 4: Path planning and guidance

In chapter 4 the guidance algorithm for various phases of landing is developed.

The guidance design along with control is validated by numerical simulation.

Chapter 5: State estimation

In this chapter a state estimator with Extended Kalman Filter in continuous-

discrete form is designed. The estimator is tested with wind disturbances and

parameter uncertainties.

Chapter 2

Mathematical modeling

The dynamics of an object in motion can be given by Newton’s second law which is

applicable in inertial frame. To model the translational dynamics we need the knowledge

of mass and the external forces acting on it. Similarly for the rotational dynamics we

require moment and product of inertias with the external moments acting on it. The

mass can be easily measured with highly accurate weighing machines available. Where as

the inertia properties can be measured in laboratory with bifilar suspension experiment.

The external forces acting on an aircraft are due to gravity, aerodynamics and thrust.

The aerodynamic forces and moments can be measured in wind tunnel by testing the

scaled model or the full model at the expected flight velocities. The thrust force can also

be measured in wind tunnel. The other methods through which aerodynamics forces

and moments can be estimated are empirical calculations or CFD (Computational Fluid

Dynamics) simulation.

The kinematic motion do not involve any parameters but they are also dependent on

the frame of observation. The vectors can be transformed from one frame of reference to

other through three sequential Euler rotations. The three rotations combined form the

transformation matrix between the frames.

2“As far as the propositions of mathematics refer to reality they are not certain, and so far as theyare certain, they do not refer to reality.” Albert Einstein(1921)

6

Chapter 2. Mathematical modeling 7

The dynamic and kinematic equations for rotational and translational motion of

aircraft is discussed in next section.

2.1 Equations of motion

The reference frame for body axis coordinate system is shown in Fig. 2.1. Where positive

X axis is in forward direction through nose of the aircraft. The positive Y axis protrudes

through the right wing. The positive Z axis is perpendicular to X and Y in downward

direction.

Figure 2.1: Body reference frame

The variables defined with respect to body axis frame are given below.

u, v, w are velocity components along the body axis X, Y , Z direction.

p, q, r are roll, pitch and yaw rates respectively about the body axis X, Y , Z direction.

φ, θ, ψ are roll, pitch and yaw angles around the body axis X, Y , Z direction.

The rotational rates and angles are defined with respect to right hand grip rule. The

position coordinates of the aircraft are defined with respect to earth fixed inertial frame.

x, y, h are defined as position coordinates with respect to inertial frame. Where x is

forward distance, y is sideward distance and h is the height above ground.


Six-DOF equations

Under the assumptions of airplane to be a rigid body and earth to be flat, the complete

set of six-DOF equations of motion [28] are given by following differential equations.

Translational dynamic equations

u = rv − qw − gsinθ +Xa + Xt

m(2.1)

v = pw − ru + gsinφcosθ +Ya

m(2.2)

w = qu− pv + gcosφcosθ +Za

m(2.3)

Rotational dynamic equations

p = c1rq + c2pq + c3La + c4Na (2.4)

q = c5pr + c6(p2 − r2) + c7(Ma + Mt) (2.5)

r = c8pq − c2rq + c4La + c9Na (2.6)

Rotational kinematic equations

φ = p + qsinφtanθ + rcosφtanθ (2.7)

θ = qcosφ− rsinφ (2.8)

ψ = qsinφsecθ + rcosφsecθ (2.9)

Translational kinematic equations

x = u cosθcosψ + v(sinφsinθsinψ − cosφsinψ) + w(cosφsinθcosψ + sinφsinψ)(2.10)

y = u cosθsinψ + v(sinφsinθsinψ + cosφcosψ) + w(cosφsinθsinψ − sinφcosψ)(2.11)

h = u sinθ − vsinφcosθ − wcosφcosθ (2.12)

In above equations Xa, Ya, Za are the aerodynamic forces and La, Ma, Na are the

moments about the body axis. m is the mass of the vehicle. Xt is the thrust force in

body axis X direction and Mt is the moment around the Y axis caused by thrust, as it

does not pass through the center of gravity of the aircraft.


The constants c1 − c9 in equations 2.4 - 2.6 are function of inertial properties of aircraft

given as

c1

c2

c3

c4

c8

c9

, 1IxxIzz−I2

xz

Izz(Iyy − Izz)− IxzIxz

Ixz(Ixx − Iyy + Izz)

Izz

Ixz

IxzIxz + Ixx(Ixx − Iyy)

Ixx

c5

c6

c7

, 1

Iyy

(Izz − Ixx)

Ixz

1

The inertia and mass properties with physical data of the UAV AE-2 (Fig. 1.1) is

given in the Table 2.1. Where b is wing span, c is chord length and m is mass of the

vehicle.

Table 2.1: Physical data of AE-2

b c m Ixx Iyy Izz Ixz

2.0 m 0.3 m 6.0 kg 0.5062 kgm2 0.89 kgm2 0.91 kgm2 0.0015 kgm2

2.2 Aerodynamic forces and moments

The aerodynamic forces and moments are given by following equations

[Xa Ya Za] = q S [CX CY CZ ] (2.13)

[La Ma Na] = q S [bCl cCm bCn] (2.14)

where q is dynamic pressure given as 12ρV 2

t . ρ is the density of air and Vt is total velocity

of aircraft relative to air. S is wing planform area. CX , CY , CZ are axial force, side

force and normal force coefficients respectively. Whereas Cl, Cm, Cn are rolling moment,

pitching moment and yawing moment coefficients respectively. Aerodynamic coefficients

are obtained from curve fitting on wind tunnel data which is given in the next section.


2.2.1 Wind tunnel testing

For the study and evaluation of aerodynamic performance, the full scale model AE-

2 was tested [27] in the open circuit wind tunnel (Fig. 2.2) facility available at the

department. The forces and moments were measured and than normalized to find the

coefficients. Only static tests were conducted in wind tunnel with test variables as angle

of attack(α), sideslip angle(β), elevator deflection(δe), aileron deflection(δa) and rudder

deflection(δr). The force and moment coefficients were evaluated as a function of two

variables. During the test one variable was held constant and other varied for its entire

range. For eg. elevator held at -5o and angle of attack varied from -6o to 22o with

incremental angle of +2o, in this process all the six coefficients were measured. The tests

conducted with variables and their full range is given in the table below.

Table 2.2: Wind tunnel test variables and their range

test tablesize I min (step) max II min (step) max1. f(α, β)15×5 α -6o : +2o : 22o β -10o : +5o : 10o

2. f(β, α)11×5 β -100: +2o : 10o α -5o : +5o : 15o

3. f(α, δe)15×7 α -6o : +2o : 22o δe -25o: +5o : 5o

4. f(α, δa)15×7 α -6o : +2o : 22o δa -15o : +5o : 15o

5. f(α, δr)15×7 α -6o : +2o : 22o δr -15o : +5o : 15o

Figure 2.2: AE-2 in the open circuit wind tunnel


From the wind tunnel tests the coefficients are available in 2-dimensional table look

up form. The number of tables is large, which makes the finding of coefficient through

interpolation a time taking process. Also there will be discontinuity in the slope as

the required interpolation point progresses from one region of the table to other. To

overcome these difficulties the curve fitting on aerodynamic coefficients was done to find

a global function [2] which is discussed in the next section. The discussion is carried out

by taking example of normal force coefficient CZ . The same approach has been applied

to other coefficients as well, wherever there is any difference it is discussed. The results

of curve fit with functional form is given in the subsequent sections.

2.2.2 Curve fitting on wind tunnel data

The aerodynamic coefficients [28] vary with respect to Mach number, Reynolds number,

air relative angles, control surface deflections and body rotational rates. Which can be

represented in mathematical form as

CZ = f(M, Re, α, β, δa, δe, δr, p, q, r) (2.15)

The effect of mach number can be neglected for very small mach numbers. Also due to

small variation range in the flight velocity of the vehicle there will be small changes in

Reynolds number, hence its effect can also be dropped. So we have

CZ = f(α, β, δa, δe, δr, p, q, r) (2.16)

With the assumption that effects of individual variables are additive, we can separate

static and dynamic effects

CZ = fstat(α, β, δa, δe, δr) + fdyn(p, q, r) (2.17)

= CZstat + CZdyn(2.18)

With component build up we can write the static effects,

CZstat = CZ0 +∂CZ

∂αα +

∂CZ

∂ββ +

∂CZ

∂δa

δa +∂CZ

∂δe

δe +∂CZ

∂δr

δr (2.19)


The dynamic effects are discussed in Section 2.2.4. There are five tables available from

the static wind tunnel test for the normal force coefficient which are

Table1 Table2 Table3 Table4 Table5

CZstat = f(α, β) CZstat = f(β, α) CZstat = f(α, δe) CZstat = f(α, δa) CZstat = f(α, δr)

Let us consider the Table1 and Table2 which represent the coefficient as a function of

same variables measured at different values. For these two tables the test was conducted

by setting the value of other variables to zero i.e. δe = 0, δa = 0, δr = 0. If we put these

values in Eq. 2.19 than it will get reduced to give the function which represents Table1

and Table2

CZstat = CZ0 +∂CZ

∂αα +

∂CZ

∂ββ (2.20)

Now the goal is to find CZ0 ,∂CZ

∂α, ∂CZ

∂βsuch that Eq. 2.20 represents the Table1 and

Table2 with minimum error. The plots of data Table1 are shown in Fig. 2.3 - Fig. 2.5

−10−5

05

10

−10

0

10

20

30−0.5

0

0.5

1

1.5

β (deg) α (deg)

CZ

stat

Figure 2.3: Plot of CZstat = f(α, β), data Table1


−5 0 5 10 15 20−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

α (deg)

CZ

sta

t

β=10

β=5

β=0

β=−5

β=−10

Figure 2.4: CZstat vs α for various β

−10 −5 0 5 10 15−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

β (deg)

CZ

sta

t

α=22

α=20

α=18

α=16

α=14

α=12

α=10

α=8

α=6

α=4

α=2

α=0

α=−2

α=−4

α=−6

Figure 2.5: CZstat vs β for various α

The value of CZ0 is a constant, which can be found from looking in the Table1 and

Table2 for α = 0, β = 0. The two values found were very close so average was taken.

The nature of ∂CZ

∂αcan be found qualitatively looking at Fig. 2.4. It is observed from

the Fig. 2.4 that CZstat varies linearly (constant slope) up to 10o of angle of attack and

in the region of higher angles of attack it is nonlinear. The stall angle is near 16o of angle

of attack after which the value of normal coefficient starts decreasing. This observation

on experimental data matches with the theory of lift generation and its behaviour with

angle of attack.

The behaviour of ∂CZ

∂βcan be seen in Fig. 2.5. It is observed that CZstat varies linearly

with β having a small negative slope.

Now to estimate the values ∂CZ

∂α, ∂CZ

∂βsuch that it predicts the measured value with

minimum error, the least squares method is used. The measured data is divided in to

two regions with respect to angle of attack -6o to 10o and 10o to 22o. This is done in order

to account for linear and nonlinear behaviour of the CZstat . So two different functions

are fit in these regions while maintaining continuity and slope at the transition of 10o.


Least squares estimation

Let us denote the measured value of normal force coefficient as CZmij. Which is the

measured value of CZstat at αi and βj. The function which predicts the measured value

is given by Eq. 2.20

CZpij= CZ0 + CZα αi + CZβ

βj (2.21)

where CZpijis the predicted value of CZstat at measured values of αi and βj. The constants

CZα and CZβrepresent ∂CZ

∂αand ∂CZ

∂βrespectively. Here CZα and CZβ

are constants as we

have observed above that the slopes of normal force coefficient vary linearly with α and

β in the range of -6o to 10o of angle of attack. Where the slopes vary, the derivatives are

expanded in power series of the depending variables.

The error in measured value and predicted value can be written as

error = C0Zmij

− C0Zpij

(2.22)

where C0Zmij

, CZmij− CZ0 and

C0Zpij

, CZpij− CZ0 = CZα αi + CZβ

βj (from Eq. 2.21)

The value of CZ0 is subtracted as it is found directly from the table as discussed

above and assumed that is measured to a very good accuracy. The total sum of square

errors will be

J = Σ(C0Zmij

− C0Zpij

)2 (2.23)

Taking the derivative of the above cost function with free variables CZα and CZβand

equating them to zero for minimum

∂J

∂CZα

= 2Σ(C0Zmij

− C0Zpij

)αi = 0 (2.24)

∂J

∂CZβ

= 2Σ(C0Zmij

− C0Zpij

)βj = 0 (2.25)


Carrying out the necessary algebra we get

Σ C0Zmij

αi = CZα Σ α2i + CZβ

Σ αiβj (2.26)

Σ C0Zmij

βj = CZα Σ αiβj + CZβΣ β2

j (2.27)

Which can be written in matrix notation as Σ C0

Zmijαi

Σ C0Zmij

βj

=

Σ α2

i Σ αiβj

Σ αiβj Σ β2j

CZα

CZβ

(2.28)

Performing the inverse matrix operation we get CZα

CZβ

=

Σ α2

i Σ αiβj

Σ αiβj Σ β2j

−1

Σ C0Zmij

αi

Σ C0Zmij

βj

(2.29)

Above equation will give the values of derivatives CZα and CZβwith least square error

from the measured values of coefficients. A similar operation is carried out in the nonlin-

ear zone i.e. 10o to 22o of angle of attack for normal coefficient. In the nonlinear region

the higher order functions from power series are taken in the predicting function.

Now let us consider the Table3 which gives the normal force coefficient as a function

of angle of attack and elevator deflection. This test was conducted by setting the value

of other variables to zero i.e. β = 0, δa = 0, δr = 0. If we put these values in Eq. 2.19

than it will get reduced to give the predicting function which represents Table3

CZpij= CZ0 +

∂CZ

∂ααi +

∂CZ

∂δe

δej(2.30)

We have found the value of CZ0 and ∂CZ

∂α. To find the value of ∂CZ

∂δewe redefine the

predicting function by transferring the known quantities to right hand side

C0Zpij

, CZpij− CZ0 − CZα αi = CZδe

δej(where CZδe

, ∂CZ

∂δe)

The measured value of normal force coefficient will be defined as C0Zmij

given by

C0Zmij

, CZmij− CZ0 − CZα αi

Now we can use least squares method as above to find out the the value of CZδe.


The plots of data Table4 and Table5 are shown in Fig. 2.6 and Fig. 2.7

−15 −10 −5 0 5 10 15 20−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

δr (deg)

CZ

sta

t

α=22

α=20

α=18

α=16

α=14

α=12

α=10

α=8

α=6

α=4

α=2

α=0

α=−2

α=−4

α=−6

Figure 2.6: CZstat vs δr for various α

−15 −10 −5 0 5 10 15 20−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

δa (deg)

CZ

sta

t

α=22

α=20

α=18

α=16

α=14

α=12

α=10

α=8

α=6

α=4

α=2

α=0

α=−2

α=−4

α=−6

Figure 2.7: CZstat vs δa for various α

It is observed from Fig. 2.6 that there is no variation in the value of CZstat with

rudder deflection. In other words the derivative ∂CZ

∂δris nearly equal to zero, hence it can

be neglected from the function. From Fig. 2.7 it is seen that there is small variation

in the CZstat with aileron deflection. The observed behaviour with aileron deflection

is not symmetric on either side of zero. Also measurements at -10o aileron deflection

are contrary to the behaviour, so the effect of aileron is not considered in the normal

coefficient prediction.

Validation of least square

To validate the derivatives found from least squares the measured data from wind tunnel

is divided into sets of 70% and 30% called as training set and testing set respectively. The

training set is used for least squares to estimate the derivatives. The testing set is used

to test the accuracy of the predicting function, if the error in measured value is higher

than 10% in linear range and more than 30% in nonlinear range than the predicting

function is not accepted. The process of least square is again carried with a higher order

predicting function until the error criteria is satisfied.


2.2.3 Static stability derivatives

The results of curve fitting on wind tunnel data is presented in this section. First the

results for normal force coefficient is presented followed by axial force, pitching moment,

side force, rolling moment and yawing moment coefficient. Some of the derivatives are

partitioned with respect to angle of attack into linear region and nonlinear region. The

α value for partition was found to be 10 degree. In expanded derivative functions

α10 , 0 if α <= 10

, α− 10 if α > 10

The values of derivatives and their constants found is given in Table 2.4. The result

plots are shown below where circles in plots show the wind tunnel data and solid lines

are the curves obtained by function fit.

CZstat : Normal force coefficient

Fig 2.8 and Fig. 2.9 show the results for curve fit on CZstat with angle of attack.

−5 0 5 10 15 20−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

α (deg)

CZ

sta

t

β=10

β=5

β=0

β=−5

β=−10

Figure 2.8: Curve fit on CZstat(α, β) vs α

−5 0 5 10 15 20−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

α (deg)

CZ

sta

t

δ e=5

δ e=0

δ e=−5

δ e=−10

δ e=−15

δ e=−20

δ e=−25

Figure 2.9: Curve fit on CZstat(α, δe) vs α


−10 −5 0 5 10 15−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

β (deg)

CZ

sta

t

α=22

α=20

α=18

α=16

α=14

α=12

α=10

α=8

α=6

α=4

α=2

α=0

α=−2

α=−4

α=−6

Figure 2.10: Curve fit on CZstat(α, β) vs β

−25 −20 −15 −10 −5 0 5 10 15−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

δe (deg)

CZ

sta

t

α=22α=20α=18α=16α=14α=12α=10α=8α=6α=4α=2α=0α=−2α=−4α=−6

Figure 2.11: Curve fit on CZstat(α, δe) vs δe

The Fig. 2.10 shows the fit on variation with respect to sideslip angle. Further Fig.

2.11 shows the effect of elevator variation on CZstat . The function obtained after curve

fitting on wind tunnel data for CZstat is given as

CZstat = CZ0 + CZα(α) α + CZββ + CZδe

δe (2.31)

where, CZα(α) α , z10α + z11α210 + z11α

310, CZβ

, z20, and CZδe, z30. The constants

z10 - z30 are given in Table 2.4

CXstat: Axial force coefficient

From the wind tunnel results for axial force coefficient it is observed that the variables

strongly affecting it are angle of attack and elevator deflection. The effects of other

variable being marginal it is not considered in curve fitting. The function obtained from

curve fit is given as

CXstat = CX0 + CXα(α) α + CXδe(α) δe (2.32)


where, CXα(α) α , x10α + x11α2 + x12α

210 + x13α

310 + x14α

410

CXδe, x20 + x21α + x22α

210

The coefficient is divided in two zones on basis of angle of attack as the coefficient

is highly nonlinear after 10o angle of attack, which can also be observed from Fig. 2.12.

The effectiveness of elevator changes in higher angle of attack region which is accounted

in CXδeby making it a function of angle of attack.

−5 0 5 10 15 20−0.2

−0.1

0

0.1

α (deg)

CX

sta

t

δ e=5

δ e=0

δ e=−5

δ e=−10

δ e=−15

δ e=−20

δ e=−25

Figure 2.12: Curve fit on CXstat(α, δe) vs α

−25 −20 −15 −10 −5 0 5 10−0.2

−0.1

0

0.1

δe (deg)

CX

sta

t

α=22

α=20

α=18

α=16

α=14

α=12

α=10

α=8

α=6

α=4

α=2

α=0

α=−2

α=−4

α=−6

Figure 2.13: Curve fit on CXstat(α, δe) vs δe

Cmstat: Pitching moment coefficient

It is observed from wind tunnel data of pitching moment coefficient that aileron and rud-

der have negligible effect, hence not considered for curve fitting. The function obtained

from curve fit is given as

Cmstat = Cm0 + Cmα(α) α + Cmβ(α, β) β + Cmδe

(α) δe (2.33)

where, Cmα , m10 + m11α + m12α2 + m13α

3 + m14α4

Cmβ, m20 + m21β + m22αβ

Cmδe, m30 + m31α


The pitching moment coefficient varies smoothly with angle of attack (Fig 2.14 and

Fig. 2.15) for its entire range, therefore a single function a has been fit for Cmα . While

the effect of sideslip (Fig. 2.16) is quadratic in nature and the effectiveness of elevator

(Fig. 2.17) changes with angle of attack.

−5 0 5 10 15 20−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

α (deg)

Cm

sta

t

beta=10beta=5alpha=0beta=−5beta=−10

Figure 2.14: Curve fit on Cmstat(α, β) vs α

−5 0 5 10 15 20−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

α (deg)

Cm

sta

t

δe=5

δe=0

δe=−5

δe=−10

δe=−15

δe=−20

δe=−25

Figure 2.15: Curve fit on Cmstat(α, δe) vs α

−10 −5 0 5 10 15−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

β (deg)

Cm

sta

t

α=22 α=20 α=18 α=16 α=14 α=12 α=10 α=8 α=6 α=4 α=2 α=0 α=−2 α=−4 α=−6

Figure 2.16: Curve fit on Cmstat(α, β) vs β

−25 −20 −15 −10 −5 0 5 10 15−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

δe (deg)

Cm

sta

t

α=22

α=20

α=18

α=16

α=14

α=12

α=10

α=8

α=6

α=4

α=2

α=0

α=−2

α=−4

α=−6

Figure 2.17: Curve fit on Cmstat(α, δe) vs δe


CYstat: Side force coefficient

The side force coefficient obtained from wind tunnel data had non zero values even when

the sideslip angle was zero along with control surfaces. This non zero side force could

result from asymmetric manufacturing of vehicle or non alignment with wind direction

in wind tunnel. Therefor this error was subtracted from the data table to make the

coefficient symmetric with sideslip.

−5 0 5 10 15 20−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

α (deg)

CY

sta

t

β=10

β=5

β=0

β=−5

β=−10

Figure 2.18: Curve fit on CYstat(α, β) vs α

−10 −5 0 5 10 15−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

β (deg)

CY

sta

t

α=22

α=20

α=18

α=16

α=14

α=12

α=10

α=8

α=6

α=4

α=2

α=0

α=−2

α=−4

α=−6

Figure 2.19: Curve fit on CYstat(α, β) vs β

−15 −10 −5 0 5 10 15 20−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

δr (deg)

CY

sta

t

α=22

α=20

α=18

α=16

α=14

α=12

α=10

α=8

α=6

α=4

α=2

α=0

α=−2

α=−4

α=−6

Figure 2.20: Curve fit on CYstat(α, δr) vs δr

−15 −10 −5 0 5 10 15 20−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

δa (deg)

CY

sta

t

α=22

α=20

α=18

α=16

α=14

α=12

α=10

α=8

α=6

α=4

α=2

α=0

α=−2

α=−4

α=−6

Figure 2.21: Curve fit on CYstat(α, δa) vs δa


The side force coefficient is also divided in two regions with angle of attack. The

function obtained from curve fit is given as

CYstat = CYβ(α) β + CYδr

(α) δr + CYδa(α) δa (2.34)

where, CYβ, y10 + y11α + y12α

2 + y13α210 + y14α

310

CYδr, y20 + y21α + y22α

2 + y23α2 + y24α

210 + y25α

310 + y26α

410

CYδa, y30 + y31α

210 + y32α

310

Clstat: Rolling moment coefficient

The rolling moment coefficient is also made symmetric with sideslip angle by subtracting

the non zero values from the data. The effect of elevator and rudder being very small,

hence not considered in curve fitting. The function obtained from curve fit is given as

Clstat = Clβ(α)β + Clδa(α)δa (2.35)

where, Clβ , l10 + l11α + l12α2 + l13α

210 + l14α

310

Clδa, l20 + l21α + l22α

2 + l23α3 + l24α

210 + l25α

310 + l26α

410

−5 0 5 10 15 20−0.03

−0.02

−0.01

0

0.01

0.02

0.03

α (deg)

Cl s

tat

β=10

β=5

β=0

β=−5

β=−10

Figure 2.22: Curve fit on Clstat(α, β) vs α

−5 0 5 10 15 20−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

α (deg)

Cl s

tat

δa=15

δa=10

δa=5

δa=0

δa=−5

δa=−10

δa=−15

Figure 2.23: Curve fit on Clstat(α, δa) vs α


−10 −5 0 5 10 15−0.03

−0.02

−0.01

0

0.01

0.02

0.03

β (deg)

Cl s

tat

α=22

α=20

α=18

α=16

α=14

α=12

α=10

α=8

α=6

α=4

α=2

α=0

α=−2

α=−4

α=−6

Figure 2.24: Curve fit on Clstat(α, β) vs β

−15 −10 −5 0 5 10 15 20−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

δa (deg)

Cl s

tat

α=22

α=20

α=18

α=16

α=14

α=12

α=10

α=8

α=6

α=4

α=2

α=0

α=−2

α=−4

α=−6

Figure 2.25: Curve fit on Clstat(α, δa) vs δa

Cnstat: Yawing moment coefficient

The function obtained from curve fit for yawing moment coefficient is given as

Cnstat = Cnβ(α)β + Cnδr

(α)δr (2.36)

where, Cnβ, n10 + n11α + n12α

2 + n13α210

Cnδr, n20 + n21α

210 + n22α

310

−5 0 5 10 15 20−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

α (deg)

Cn

sta

t

β=10

β=5

β=0

β=−5

β=−10

Figure 2.26: Curve fit on Cnstat(α, β) vs α

−5 0 5 10 15 20−0.015

−0.01

−0.005

0

0.005

0.01

0.015

α (deg)

Cn

sta

t

δr=15

δr=10

δr=5

δr=0

δr=−5

δr=−10

δr=−15

Figure 2.27: Curve fit on Cnstat(α, δr) vs α


−10 −5 0 5 10 15−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

0.025

β (deg)

Cn

sta

t

α=22

α=20

α=18

α=16

α=14

α=12

α=10

α=8

α=6

α=4

α=2

α=0

α=−2

α=−4

α=−6

Figure 2.28: Curve fit on Cnstat(α, β) vs β

−15 −10 −5 0 5 10 15 20−0.015

−0.01

−0.005

0

0.005

0.01

0.015

δr (deg)

Cn

sta

t

α=22

α=20

α=18

α=16

α=14

α=12

α=10

α=8

α=6

α=4

α=2

α=0

α=−2

α=−4

α=−6

Figure 2.29: Curve fit on Cnstat(α, δr) vs δr

The effects of elevator and aileron being small they are neglected. Yawing moment

coefficient is also divided into linear and nonlinear zone with respect to angle of attack.

This completes the discussion of static stability derivatives obtained from curve fitting

on wind tunnel test data. The dynamics derivatives are discussed in next section.

2.2.4 Dynamic stability derivatives

The dynamic effects on coefficients can be written from Eq. 2.17 as CZdyn= fdyn(p, q, r),

Which can be expanded considering the individual effects are additive

CZdyn=

∂CZ

∂pp +

∂CZ

∂qq +

∂CZ

∂rr (2.37)

Where the rotational rates are non dimensionalized with reference length and velocity

which are given as [p q r] = 12Vt

[bp cq br].

The dynamics test were not conducted in wind tunnel, hence the dynamics derivatives

are found with AVL (Athena Vortex Lattice) software [29]. AVL is a program for the

aerodynamic and flight-dynamic analysis of rigid aircraft of arbitrary configuration.


It employs an extended vortex lattice model for the lifting surfaces, together with a

slender-body model for fuselages. AVL requires the mass properties along with geometric

information (Fig. 2.30) of the vehicle. It also requires the 2-dimensional lift and drag

characteristics of the lifting surfaces.

Figure 2.30: AE-2 modeling in AVL

To validate and have confidence in the results given by AVL its static results are

compared with that of wind tunnel, which is shown in Fig. 2.31. It is seen from figure

that AVL gives a good estimate for the static coefficients.

−5 0 5 10 15 20−0.2

0

0.2

α (deg)

CX

stat

AVLWind Tunnel

−5 0 5 10 15 20−2

0

2

α (deg)

CZ

stat

−5 0 5 10 15 20−1

0

1

α (deg)

Cm

stat

Figure 2.31: AVL and wind tunnel test results comparison


The dynamic derivatives are evaluated with respect to angle of attack. While considering

the dynamics effects it is assumed that only pitch rate effects the longitudinal coefficients

of normal force, axial force and pitching moment. Which can be written as

CZdyn= CZq q CXdyn

= CXq q Cmdyn= Cmq q

where as the coefficients effected by roll rate and yaw rate are side force, rolling moment

and yawing moment coefficient. For them the relation can be written as

CYdyn= CYp p + CYr r Cldyn

= CYp p + Clr r Cndyn= CYp p + Cnr r

The following sections give the results obtained for dynamic derivatives from AVL.

Pitch rate derivatives

The pitch rate derivatives are found at various angle of attack. Then a function is fit

which will be able to predict derivative for entire range of angle of attack. The results

are shown in Fig. 2.32 and the functions obtained from curve fit are given as

CZq , z40 + z41α CXq , x30 + x31α Cmq , m40 + m41α + m42α2

−5 0 5 10 15 204

6

8

α (deg)

CZ

q

AVLCurve fit

−5 0 5 10 15 20−5

0

5

α (deg)

CX

q

−5 0 5 10 15 20−14

−13.5

−13

α (deg)

Cm

q

Figure 2.32: Pitch rate derivatives


Roll rate derivatives

The results for roll rate derivatives are shown in Fig. 2.33 and the functions obtained

from curve fit are given as

CYp , y40 + y41α Clp , l30 + l31α + l32α2 Cnp , n30 + n31α + n32α

2

−5 0 5 10 15 20−0.5

0

0.5

α (deg)

CY

p

AVLCurve fit

−5 0 5 10 15 20−0.6

−0.5

−0.4

α (deg)

Clp

−5 0 5 10 15 20−0.5

0

0.5

α (deg)

Cnp

Figure 2.33: Roll rate derivatives

Yaw rate derivatives

The results for yaw rate derivatives are shown in Fig. 2.34 and the functions obtained

from curve fit are given as

CYr , y50 + y51α Clr , l40 + l41α + l42α2 Cnr , n40 + n41α + n42α

2


−5 0 5 10 15 200

0.2

0.4

α (deg)

CY

r

AVLCurve fit

−5 0 5 10 15 200

0.2

0.4

α (deg)

Clr

−5 0 5 10 15 20−0.4

−0.2

0

α (deg)

Cnr

Figure 2.34: Yaw rate derivatives

The aerodynamic coefficients with static and dynamic effects can be summarized as

CZ = CZ0 + CZα(α) α + CZββ + CZδe

δe + CZq(α) q (2.38)

CX = CX0 + CXα(α) α + CXδe(α) δe + CXq(α) q (2.39)

Cm = Cm0 + Cmα(α) α + Cmβ(α, β) β + Cmδe

(α) δe + Cmq(α) q (2.40)

CY = CYβ(α) β + CYδa

(α) δa + CYδr(α) δr + CYp(α)p + CYr(α) r (2.41)

Cl = Clβ(α) β + Clδa(α) δa + Clp(α) p + Clr(α) r (2.42)

Cn = Cnβ(α) β + Cnδr

(α) δr + Cnp(α) p + Cnr(α) r (2.43)

The Table 2.3 gives the value of average of absolute percentage error in the value obtained

by curve fitting and measured data.

Table 2.3: Average of absolute percentage error in curve fitting

α CZ CX Cm CY Cl Cn

−6o to 10o 6% 6% 16% 13% 11% 7%10o to 22o 23% 26% 27% 30% 25% 25%


The values of coefficients of the functions obtained by curve fit is given in Table 2.4.

Table 2.4: Derivatives function coefficient values

CZ0 0.1653 m12 -1.7853e-005 y24 -5.2196e-5 l30 -0.44336z10 0.087138 m13 -2.1109e-006 y25 8.8682e-6 l31 0.00075577z11 -0.0091867 m14 1.1346e-007 y26 -3.2717e-7 l32 -0.00013921z12 0.00024242 m20 0.00024049 y30 -0.0016884 l40 0.076582z20 -0.0020001 m21 -7.8566e-006 y31 -1.3637e-05 l41 0.010019z30 0.0039823 m22 1.0663e-006 y32 1.3214e-06 l42 1.1783e-005z40 6.9303 m23 -7.8866e-007 y40 -0.14504 n10 -0.0015474z41 -0.047657 m30 -0.0145 y41 0.013516 n11 6.1309e-005CX0 0.0386 m31 9.2552e-006 y50 0.13784 n12 -1.8989e-006x10 -0.0040376 m32 9.0437e-006 y51 0.0035514 n13 -5.5706e-006x11 -0.0010525 m40 -13.954 l10 0.0022856 n20 0.00077238x12 0.0027887 m41 0.0017379 l11 6.4827e-005 n21 1.1379e-006x13 0.00010917 m42 0.0016743 l12 -3.0529e-006 n22 -4.1705e-008x14 -5.3586e-6 y10 0.0099319 l13 -2.7687e-005 n30 -0.015512x20 -0.00035832 y11 0.00029462 l14 1.7713e-006 n31 -0.011325x21 -2.2061e-5 y12 1.7831e-005 l20 0.0029091 n32 9.8251e-005x22 -5.7342e-6 y13 -0.00030969 l21 9.0047e-006 n40 -0.085307x30 -0.18476 y14 1.6759e-005 l22 -7.4562e-006 n41 0.00080338x31 -0.10227 y20 0.0022145 l23 3.0423e-007 n42 -0.00026197Cm0 0.0346 y21 0.00041878 l24 -2.5531e-005m10 -0.013841 y22 1.3117e-5 l25 4.1263e-006m11 -0.00026206 y23 -1.1549e-6 l26 -2.0918e-007

2.3 Thrust force and moment

The AE-2 uses electric motor and propeller for thrust generation with lithium polymer

battery as its power source. The thrust force and moment are given as

Xt = (Tmax σt) (2.44)

Mt = −d (Tmax σt) (2.45)

where Tmax is the maximum thrust(15N) which can be produced by the electric motor

and propeller assembly. σt is throttle control which varies from 0 to 1. It is assumed

that thrust produced has linear relation with throttle input. d is offset (0.26m) of the

thrust line from the CG of the vehicle.


2.4 Actuator dynamics

The aerodynamic control surfaces are deflected by actuators. AE-2 employs electrome-

chanical servos for the control surface deflection, all the surfaces have the similar servo.

Here the actuator dynamics [30] for elevator servo is given

δe = bact δe + aact uδe (2.46)

where bact = −9.5, aact = 6.7 and uδe is the width of pulse width modulated (PWM)

signal. The step response of actuator is shown in the Fig. 2.35. The settling time for

the actuator is 0.54 seconds.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

X: 0.54Y: 0.7011

time (sec)

Ser

vo d

efle

ctio

n (d

eg)

Figure 2.35: Step response of actuator

The thrust model can be implemented with first order lag. Where this lag is due

to time delay in increase or decrease of rotational speed of the propeller. This lag is

found from the manual of electric motor and speed controller through which the throttle

command is passed. The equation for thrust generation with first order lag can be

written as

T =1

τ(T − T ∗) (2.47)

where T is actual thrust, T ∗ is the commanded thrust and τ is the time constant.


2.5 Trim condition

The trim condition for steady and level flight can be found at a given velocity and

altitude. In order to find trim conditions we equate the dynamic equations to zero

(X = 0) and solve for the unknown variables. Here the given conditions are velocity

and altitude. The steady and level flight condition demand that the body angular rates

and roll angle be equal to zero (p = q = r = φ = 0). The unknown quantities to be

found are angle of attack, sideslip angle, pitch angle, throttle value and deflections of

elevator, aileron and rudder control. The equations used to solve for the unknowns are

[u v w p q r h] = 0, taken from the six-DOF equation of motion. Since the equations are

nonlinear the fsolve function of Matlab is used for finding solution. The trim conditions

at three different velocities and same altitude are found and given in table below.

Table 2.5: Trim conditions at different velocities

Vt h α β θ σt δe δa δr

m/s m deg deg deg (0-1) deg deg deg15 100 7.21 0 7.21 0.28 -9.81 0 020 100 3.15 0 3.15 0.36 -3.29 0 025 100 1.29 0 1.29 0.36 -1.17 0 0

The trim condition found can be verified with numerical simulation, by initializing the

system states with trim condition and integrating it for some time. The states of the

system will remain in the trim condition (Fig. 2.36 - 2.37) if the solution found is correct.

2.6 Simulation results

The open loop simulation and analysis is done in this section. There are twelve states

in the simulation and four controls

X = [u v w p q r φ θ ψ x y h] Uc = [σt δe δa δr]

The states u, v, w are replace by Vt, α, β in simulation results for better insight and

correlation. The results of the simulation have been clubbed in to three groups.

7→ Longitudinal states = (Vt α q θ x h) 7→ Lateral states = (β p r φ ψ y)

7→ Control variables = (σt δe δa δr)


2.6.1 Open loop simulation

The states are initialized with trim condition and simulated for 60 seconds. It is seen

from Fig. 2.36 and Fig. 2.37 that states continue to remain in trim condition.

0 20 4019

20

21

t (sec)

Vt (m

/s)

0 20 40 602

3

4

t (sec)

α (d

eg)

0 20 40 60−1

0

1

t (sec)

Q (d

eg/s

)

0 20 40 602

3

4

t (sec)

θ (d

eg)

0 20 40 600

500

1000

t (sec)

x (m

)

0 20 40 6099

100

101

t (sec)

h (m

)

Figure 2.36: Longitudinal states in trim condition

0 20 40 60−1

0

1

t (sec)

β (d

eg)

0 20 40 60−1

0

1

t (sec)

P (d

eg/s

)

0 20 40 60−1

0

1

t (sec)

R (d

eg/s

)

0 20 40 60−1

0

1

t (sec)

φ (de

g)

0 20 40 60−1

0

1

t (sec)

ψ (d

eg)

0 20 40 60−1

0

1

t (sec)

y (m

)

Figure 2.37: Lateral states in trim condition


2.6.2 Perturbation around trim condition

To study the stability of the system various perturbations are given around trim condi-

tion. A stable aircraft should return to the same or a near by equilibrium point after

oscillations. Fig. 2.38 and Fig. 2.39 show the stability of the vehicle with respect to

velocity, angle of attack, pitch angle and pitch rate perturbations.

0 50 10018

20

22

t (sec)

Vt (m

/s)

0 50 1002

4

6

t (sec) α

(deg

)

V

t(+2 m/s)

α(+2 deg)

0 50 100−10

0

10

t (sec)

Q (d

eg/s

)

0 50 100−20

0

20

t (sec)

θ (d

eg)

0 50 1000

2000

4000

t (sec)

x (m

)

0 50 10090

100

110

t (sec)

h (m

)

Figure 2.38: Effect of velocity and angle of attack perturbation on longitudinal states

0 50 10019

20

21

t (sec)

Vt (m

/s)

0 50 1002.5

3

3.5

t (sec)

α (d

eg)

θ(+3 deg)Q(−5 deg/s)

0 50 100−5

0

5

t (sec)

Q (d

eg/s

)

0 50 1000

5

10

t (sec)

θ (d

eg)

0 50 1000

2000

4000

t (sec)

x (m

)

0 50 10098

100

102

t (sec)

h (m

)

Figure 2.39: Effect of pitch angle and pitch rate perturbation on longitudinal states


Fig. 2.38 and Fig. 2.39 show the effect of sideslip and roll angle perturbation on the

states. It can be observed from the figures that the perturbation effects of sideslip are

highly damped compared to roll angle and they return to equilibrium very fast.

0 50 100−5

0

5

t (sec)

β (d

eg)

0 50 100−20

0

20

t (sec)

P (d

eg/s

)

β(+5 deg)

φ(+5 deg)

0 50 100−20

0

20

t (sec)

R (d

eg/s

)

0 50 100−5

0

5

t (sec)φ (

deg)

0 50 1000

100

200

t (sec)

ψ (d

eg)

0 50 100−2000

0

2000

t (sec)

y (m

)

Figure 2.40: Effect of sideslip and roll angle perturbation on lateral states

0 50 10019

20

21

t (sec)

Vt (m

/s)

0 50 1002

3

4

t (sec)

α (d

eg)

β(+5 deg)

φ(+5 deg)

0 50 100−2

0

2

t (sec)

Q (d

eg/s

)

0 50 1002

3

4

5

t (sec)

θ (d

eg)

0 50 1000

1000

2000

t (sec)

x (m

)

0 50 10099

100

101

t (sec)

h (m

)

Figure 2.41: Effect of sideslip and roll angle perturbation on longitudinal states

Summary: In this chapter the mathematical model for AE-2 has been developed. The

next chapter discusses the control design for inner and outer loop to track the guidance

commands along with velocity control.

Chapter 3

Control design

Various control strategies have been attempted to achieve the flight stabilization and

autonomous control for small unmanned aerial vehicles. Linear control theory has been

heavily investigated. Linear control theory has limitation of operation in linearization

range. To encompass the whole flight regime many such linear models would be required

with gain scheduling. Due to the limitations of linear control theory and the approxima-

tions involved in it, there has been a lot of research interest in nonlinear control design.

A popular method of nonlinear control design for tracking is the technique of dynamic

inversion. Which is based on the philosophy of feedback linearization. In this approach

an appropriate coordinate transformation is carried out followed by the application of

linear control theory. Here the feedback control cancels the nonlinearities in the plant

and closed loop plant behaves as a linear system. The limitation of this method is that

it requires an accurate knowledge of the plant model, in the absence of which the track-

ing will not be perfect. Dynamic inversion concept also involves the presence of hidden

zero dynamics. They must be examined separately to make certain that they are sta-

ble and well-behaved (analysis included at the end of this chapter). The mathematical

background for dynamic inversion [9], [10] is discussed in next Section.

3“The battlefield is a scene of constant chaos. The winner will be the one who controls that chaos,both his own and the enemies.” Napoleon Bonaparte(1769-1821)

35

Chapter 3. Control design 36

3.1 Dynamic inversion

Let us consider a nonlinear dynamical system which is affine in control and given by

following equations

X = f(X) + g(X)U (3.1)

Y = h(X) (3.2)

where X ∈ <n, U ∈ <m, Y ∈ <p are the state, control and output vectors of the nominal

system respectively. We assume the system is pointwise controllable. The objective is

to design a control U so that Y → Y ∗ as t →∞, where Y ∗ is the commanded signal for

the Y to track. We assume Y ∗ is bounded, smooth and slowly varying.

To achieve the above objective, it is noticed from Eq. 3.2, that using the chain rule

of derivative the expression for Y can be written as

Y = fY (X) + gY (X)U (3.3)

where fY , [ ∂h∂X

]f(X) and gY , [ ∂h∂X

]g(X). Next, defining E , (Y − Y ∗) the

controller is synthesized such that the following stable linear error dynamics is satisfied

E + KE = 0 (3.4)

The solution of the above differential equation is given as E = E0 exp−Kt. Where

K is chosen to be positive definite matrix. It can be chosen as diagonal matrix with

positive elements, where elements represent the ‘settling time constant’. Next, using the

definition of E and substituting the expression for Y from 3.3 in 3.4, we can obtain by

carrying out the algebra

gY (X)U = −fY (X)−K(Y − Y ∗) + Y ∗ (3.5)


If p = m i.e. system has same number of controls as output and gY (X) is nonsingular

than we can obtain the control solution as

U = [gY (X)]−1[−fY (X)−K(Y − Y ∗) + Y ∗] (3.6)

The salient features of this method are that it provides a close form solution for the

controller and it can be implemented online without any computational difficulties. The

control solution also ensures that E → 0 as t → ∞, i.e. the asymptotic tracking

is achieved. However, the approach looks simple and it results in powerful nonlinear

controller, there are some important issues with this technique. First, is that it requires

p = m which may not hold good for all the dynamical systems. The second limitation

requires an accurate knowledge of the plant model, in the absence of which the tracking

will not be perfect. This deficiency can however be overcome by using the concepts of

robust control [31] or adaptive control [32].

The implementation of this method will require appearance of control term in dy-

namics after differentiation of output. The number of times output is differentiated to

get control term is known as ‘relative degree’. Sometimes this relative degree can be of

higher order. The other method to achieve this is through forming first order dynamics

in cascaded form using the concept of ‘time scale separation’. Here, the system is divided

into an inner and an outer loop (Fig. 3.1), where the dynamics in the inner loop has to

be faster than the dynamics in the outer loop.

Figure 3.1: Inner and outer loop structure


A cascaded form of inner loop and outer loop also cancels the dynamics of a system,

but the cancellation is not exact. It relies on the approximation that the inner loop is so

fast, that the tracking in inner loop is achieved by the time new commands come from

outer loop. This can achieved by choosing the settling time constants of inner loop lower

than outer loop. The outer and inner loop control for the UAV is discussed in following

sections.

3.2 Outer loop control

The goal of outer loop control (Fig. 3.2) is to track the commands generated by guidance

loop. Where the guidance commands come from path planning which is discussed in next

Chapter. The commands from guidance are desired pitch angle (θ∗), desired roll angle

(φ∗) and desired heading angle (ψ∗). The outer loop achieves the tracking by converting

the guidance commands into desired body rates which are roll rate (p∗), pitch rate (q∗)

and yaw rate (r∗). These desired body rate signals are than fed to inner loop for tracking.

Figure 3.2: Outer loop control

The transformation of guidance commands to body rate commands is achieved by

enforcing first order error dynamics, which is discussed in following sections.


3.2.1 Roll angle control

The roll angle can be controlled through roll rate command. The relative degree between

roll rate and roll angle is one. Hence to generate the roll rate command from error in

roll angle, we enforce the following first order error dynamics for roll angle error

(φ− φ∗) + kφ(φ− φ∗) = 0 (3.7)

where error = φ − φ∗ and φ∗ is the desired roll angle. kφ is chosen positive, given in

Table 3.1. Rearranging above equation we get

φ = φ∗ − kφ(φ− φ∗) (3.8)

From six-DOF equation of motion we have

φ = p + qsinφtanθ + rcosφtanθ (3.9)

Substituting φ equation in Eq. 3.8 and rearranging we get

p∗ = φ∗ − kφ(φ− φ∗)− (qsinφ + rcosφ)tanθ (3.10)

We have p∗ which is the required roll rate to achieve the desired roll angle.

3.2.2 Pitch angle control

The pitch angle can be controlled through pitch rate command. The relative degree

between pitch rate and pitch angle is one. Hence to generate the pitch rate command

from error in pitch angle, we enforce the following first order error dynamics for pitch

angle error

(θ − θ∗) + kθ(θ − θ∗) = 0 (3.11)

where error = θ − θ∗ and θ∗ is the desired pitch angle. kθ is chosen positive, given in

Table 3.1.


Rearranging previous equation, we get

θ = θ∗ − kθ(θ − θ∗) (3.12)


θ = qcosφ− rsinφ (3.13)

Substituting θ equation in Eq. 3.12 and rearranging we get

q∗ = secφ(θ∗ −Kθ(θ − θ∗) + rsinφ) (3.14)

We have q∗ which is the required pitch rate to achieve the desired pitch angle.

3.2.3 Heading angle control

The heading angle can be controlled through yaw rate command. The relative degree

between yaw rate and heading angle is one. Hence to generate the yaw rate command

from error in heading angle, we enforce the first order error dynamics for heading angle

(ψ − ψ∗) + kψ(ψ − ψ∗) = 0 (3.15)

where error = ψ − ψ∗ and ψ∗ is the desired heading angle. kψ is chosen positive, given

in Table 3.1. Rearranging above equation we get

ψ = ψ∗ − kψ(ψ − ψ∗) (3.16)


ψ = qsinφsecθ + rcosφsecθ (3.17)

Substituting ψ equation in Eq. 3.16 and rearranging we get

r∗ = secφ cosθ(ψ∗ − kψ(ψ − ψ∗))− qtanφ (3.18)

We have r∗ which is the required yaw rate to achieve the desired heading angle.


3.3 Inner loop control

The goal of inner loop (Fig. 3.3) is to track the body rate commands generated by outer

loop. The inner loop achieves the tracking by transforming the body rate commands

into aerodynamic controls. The body rate commands generated by outer loop control

are desired roll rate (p∗), desired pitch rate (q∗), and desired yaw rate (r∗). Where as

the aerodynamic controls are aileron (δa), elevator (δe) and rudder (δr) deflections.

Figure 3.3: Inner loop control

There is a separate loop for velocity control where velocity is maintained constant

through throttle control. The body rates control and velocity control are discussed in

following sections.

3.3.1 Body rates control

The aerodynamics controls should be calculated such that it tracks the body angular

rates desired by the outer loop. The relative degree between aerodynamic controls and

body rates is one. Hence enforce first order error dynamics for error in body rates

p− p∗

q − q∗

r − r∗

+

kp 0 0

0 kq 0

0 0 kr

p− p∗

q − q∗

r − r∗

= 0 (3.19)

where error = [(p− p∗) (q− q∗) (r− r∗)]T . kp, kq and kr are chosen positive (Table 3.1).


Rearranging previous equation we get

p

q

r

=

p∗ − kp(p− p∗)

q∗ − kq(q − q∗)

r∗ − kr(r − r∗)

(3.20)


p = c1rq + c2pq + c3La + c4Na (3.21)

q = c5pr + c6(p2 − r2) + c7(Ma + Mt) (3.22)

r = c8pq − c2rq + c4La + c9Na (3.23)

Separating the state and control terms in p, q, r equations and rearranging Eq. 3.20 we

can writefr + grUa = br (3.24)

where Ua = [δa δe δr]T and other terms are defined as follows

fr ,

c1rq + c2pq + c3Lax + c4Nax

c5pr + c6(p2 − r2) + c7(Max −Mt)

c8pq − c2rq + c4Lax + c9Nax

gr ,

c3Lau 0 c4Nau

0 c7Mau 0

c4Lau 0 c9Nau

br ,

p∗ − kp(p− p∗)

q∗ − kq(q − q∗)

r∗ − kr(r − r∗)

Lax , qSb[Clβ(α) β + Clp(α) p + Clr(α) r] Lau , qSbClδa

Max , qSc[Cm0 + Cmα(α) α + Cmβ(α, β) β + Cmq(α) q] Mau , qScCmδe

Nax , qSb[Cnβ(α) β + Cnp(α) p + Cnr(α) r] Nau , qSbCnδr

Carrying out the necessary algebra in Eq. (3.24), the control solution is

Ua = g−1r (br − fr) (3.25)

The above solution for aerodynamic controls will track the desired body rates generated

by outer loop control.


3.3.2 Forward velocity control

The forward velocity can be controlled by varying the thrust through throttle control.

The relative degree between throttle control and forward velocity is one. Enforcing the

following first order error dynamics for error in forward velocity

(u− u∗) + ku(u− u∗) = 0 (3.26)

where error = (u − u∗) and u∗ is the velocity which has to be maintained during the

flight. ku is chosen positive, given in Table 3.1. Rearranging above equation we get

u = u∗ − ku(u− u∗) (3.27)


u = rv − qw − gsinθ +Xa + Xt

m(3.28)

Substituting u equation in Eq. (3.27) and rearranging

fu + guσt = bu (3.29)

Where σt is throttle control value and other terms are defined as follows

fu , rv − qw − gsinθ +Xa

m

gu , Tmax

m

bu , u∗ − ku(u− u∗)

Rearranging the Eq. (3.29) we get the control solution as

σt = g−1u [bu − fu] (3.30)

The above solution for throttle control will track the desired velocity.


3.4 Actuator controller

The control solutions found by inner loop are fed to actuator for deflection of aerodynamic

control surfaces. The actuator dynamics is given in Section 2.4 which is same for all

control surfaces. It is observed from the step response (Fig. 2.35) of actuator that it

has a settling time of 0.54 seconds. Which is higher than the desired settling time for

the inner loop, hence a first order controller is designed for the actuator. Enforcing first

order error dynamics for error in actuator deflection

(δe − δ∗e) + kδe(δe − δ∗e) = 0 (3.31)

where error = (δe − δ∗e) and δ∗e is the desired deflection for elevator. Substituting the

actuator dynamics given by Eq. 2.46 in above equation and rearranging we get

uδe =1

6.7[−9.5 δe + δ∗e − kδe(δe − δ∗e)] (3.32)

where uδe is the command to be fed to the actuator to achieve the desired deflection.

The response of actuator to 1 deg deflection command is shown in Fig. 3.4.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

X: 0.22Y: 0.9931

time (sec)

Ser

vo d

efle

ctio

n (d

eg)

X: 0.54Y: 1

without controlwith control

Figure 3.4: Actuator response for 1 deg command of deflection


3.5 Control structure

The Fig. 3.5 shows the combined inner and outer loop structure. Where roll angle, pitch

angle and heading angle command come from guidance. The commands from guidance

are converted into desired body rates by the outer loop. Further inner loop transforms

these desired body rates into required aerodynamic control surface deflections.

Figure 3.5: Control structure

The velocity control loop maintains the desired velocity by throttle control, not shown

in above figure. The various gains for control loops chosen after iterative simulation are

given in the table below.

Table 3.1: Gains for various control loops

kφ kθ kψ kp kq kr ku kδe

sec−1 sec−1 sec−1 sec−1 sec−1 sec−1 sec−1 sec−1

2 2 2 5 5 5 1 20

To test the control design done in above sections, simulations are carried out by

giving various commands. The simulations and resulting performance are presented in

next section.



The simulation is initialized with the trim condition obtained in Section 2.5 as initial

conditions. After 5 seconds of simulation a sequence of 3-2-1-1 command is given where

the width of one unit has been kept as 5 seconds. The initial conditions are given below.

Initial states

X0 = [ Vt0 α0 β0 p0 q0 r0 φ0 θ0 ψ0 x0 y0 h0 ]

= [ 20 3.15o 0 0 0 0 0 3.15o 0 − 1200 0 100 ]

Initial control

Ua0 = [ σt0 δe0 δa0 δr0 ]

= [ 0.36 − 3.29o 0 0 ]

3.6.1 Pitch angle command

A 3-2-1-1 sequence for pitch angle command is given with ±5 deg around trim pitch

angle. The velocity control loop maintains the forward velocity as in trim condition.

The Fig. 3.6 shows the response of pitch angle to the command.

0 10 20 30 40 50−4

−2

0

2

4

6

8

10

t (sec)

θ (d

eg)

actualdesired

Figure 3.6: Pitch angle variation for 3-2-1-1 command


It is seen from Fig 3.7 that the velocity is maintained constant and the altitude is

increasing at constant rate due to increase in pitch angle from the trim condition. The

Fig. 3.8 shows the control requirements. There is increased demand in negative elevator

for the positive pitching motion. The increase in elevator also increase drag hence more

of throttle is required to compensate for maintaining constant velocity.

0 10 20 30 4019

20

21

t (sec)

Vt (m

/s)

0 10 20 30 400

5

t (sec)

α (d

eg)

0 10 20 30 40

−10

0

10

t (sec)

Q (d

eg/s

)

0 10 20 30 40−2

02468

t (sec)

θ (d

eg)

0 10 20 30 40−1200−1000−800−600−400−200

t (sec)

x (m

)

0 10 20 30 40100

110

120

t (sec)

h (m

)

Figure 3.7: Longitudinal states for pitch angle command

0 10 20 30 400

0.2

0.4

0.6

0.8

1

t (sec)

σt (

0 −

1 )

0 10 20 30 40

−6

−4

−2

0

t (sec)

δe (d

eg)

0 10 20 30 40−1

−0.5

0

0.5

1

t (sec)

δa (d

eg)

0 10 20 30 40−1

−0.5

0

0.5

1

t (sec)

δr (d

eg)

Figure 3.8: Control variables for pitch angle command


3.6.2 Heading angle command

A 3-2-1-1 sequence command for heading angle is given after 5 seconds of simulation

while the desire roll angle is zero and trim pitch angle is maintained. The Fig. 3.9 shows

the response of heading angle to ±5 deg command. It is seen from Fig. 3.10 that there

is small decrease in pitch angle this is due to reduction in lift during the turn.

0 10 20 30 40 50−6

−4

−2

0

2

4

6

t (sec)

ψ (d

eg)

actualdesired

Figure 3.9: Heading angle variation for 3-2-1-1 command

0 10 20 30 4019

20

21

t (sec)

Vt (m

/s)

0 10 20 30 402

3

4

t (sec)

α (d

eg)

0 10 20 30 40−1

0

1

t (sec)

Q (d

eg/s

)

0 10 20 30 402

3

4

t (sec)

θ (d

eg)

0 10 20 30 40−1200−1000−800−600−400−200

t (sec)

x (m

)

0 10 20 30 4099

100

101

t (sec)

h (m

)

Figure 3.10: Longitudinal states for heading angle command


Fig. 3.11 shows the yaw rate required for the heading angle command. Also the

sideslip angle is non zero for some time as the coordinated turn constraint has not been

enforced. The control plots in Fig. 3.12 show the requirement in rudder deflection and

also aileron in an effort to maintain zero roll angle.

0 10 20 30 40

−5

0

5

t (sec)

β (d

eg)

0 10 20 30 40−2

0

2

t (sec)

P (d

eg/s

)

0 10 20 30 40

−10

0

10

t (sec)

R (d

eg/s

)

0 10 20 30 40−2

0

2

t (sec)

φ (d

eg)

0 10 20 30 40−5

0

5

t (sec)

ψ (d

eg)

0 10 20 30 400

10

20

t (sec)

y (m

)

Figure 3.11: Lateral states for heading angle command

0 10 20 30 400

0.2

0.4

0.6

0.8

1

t (sec)

σt (

0 −

1 )

0 10 20 30 40−5

−4

−3

−2

−1

t (sec)

δe (d

eg)

0 10 20 30 40

−5

0

5

t (sec)

δa (d

eg)

0 10 20 30 40

−10

−5

0

5

10

t (sec)

δr (d

eg)

Figure 3.12: Control variables for heading angle command


3.6.3 Bank angle command

A 3-2-1-1 sequence command for roll angle is given while the desire heading angle is zero

and trim pitch angle is maintained. The response of roll angle to ±5 deg command is

shown in Fig. 3.13. It is seen from Fig. 3.14 though the pitch angle is constant, there is

constant decrease in altitude. It is due to reduction in lift due to non zero roll angle.

0 10 20 30 40 50−6

−4

−2

0

2

4

6

t (sec)

φ (d

eg)

actualdesired

Figure 3.13: Roll angle variation for step 3-2-1-1 command

0 10 20 30 4019

20

21

t (sec)

Vt (m

/s)

0 10 20 30 402

3

4

t (sec)

α (d

eg)

0 10 20 30 40−1

0

1

t (sec)

Q (d

eg/s

)

0 10 20 30 402

3

4

t (sec)

θ (d

eg)

0 10 20 30 40−1200−1000−800−600−400−200

t (sec)

x (m

)

0 10 20 30 4096

98

100

t (sec)

h (m

)

Figure 3.14: Longitudinal states for roll angle command


Fig. 3.15 shows the roll angle is maintained without changing the heading angle.

However, this comes at a price of constant control effort for aileron and rudder deflections.

0 10 20 30 40

−5

0

5

t (sec)

β (d

eg)

0 10 20 30 40

−10

0

10

t (sec)

P (d

eg/s

)0 10 20 30 40

−1

0

1

t (sec)

R (d

eg/s

)

0 10 20 30 40−5

0

5

t (sec)

φ (d

eg)

0 10 20 30 40−1

0

1

t (sec)

ψ (d

eg)

0 10 20 30 400

10

20

30

t (sec)

y (m

)

Figure 3.15: Lateral states for roll angle command

0 10 20 30 400

0.2

0.4

0.6

0.8

1

t (sec)

σt (

0 −

1 )

0 10 20 30 40−4

−3

−2

−1

t (sec)

δe (d

eg)

0 10 20 30 40

−5

0

5

t (sec)

δa (d

eg)

0 10 20 30 40−15

−10

−5

0

5

10

t (sec)

δr (d

eg)

Figure 3.16: Control variables for roll angle command

Summary: The control has been designed in this chapter for inner and outer loop to

track the guidance commands along with velocity control. Next chapter discusses the

path planning and guidance command generation for autonomous landing.


Appendix: Zero dynamics

The zero-dynamics is defined to be the internal dynamics of the system when the system

output is kept at zero by the input. The states of the system taken as output are

u, p, q, r, φ, θ, ψ, x, y, h. The remaining internal states are v and w. There dynamics is

given by following equations


m(3.33)

w = qu− pv + gcosφcosθ +Za

m(3.34)

The input which keeps the outputs to zero can be calculated from Eq. 3.25 by putting

br = 0

Ua = g−1r (−fr) (3.35)

Now replacing the outputs as zero and the input given by above equation, we will get

v

w

=

[Avw

] v

w

Where Avw is linearized matrix around the initial condition and is given as

Avw =

∂f1

∂v∂f2

∂w

∂f1

∂v∂f2

∂w

The eigen values of the Avw matrix are tabulated for various operating points

Vt(m/s) h(m) λ1 λ2

15 100 -0.5397 -0.011120 100 -0.3148 -0.012525 100 -0.0128 -0.0011

It is observed from the above table that the eigen values of Avw matrix lie in left half

plane for the operating range of the UAV, hence the zero dynamics is locally stable.

Chapter 4

Path planning and guidance

The autonomous landing of UAVs require an intelligent path planning and guidance.

Here the landing is divided into three phases which are approach, glideslope and flare

(Fig. 4.1). The approach is initial phase of landing where UAV descents to a height from

which glideslope can begin while aligning itself with runway. The next is glideslope, a

straight line path where it looses most of its height. The last segment of landing is flare

which is an exponential curve at the end of which UAV touches down on ground. The

design and selection of the landing trajectory is discussed in next section.

Figure 4.1: Phases during landing

4“Our scientific power has outrun our spiritual power. We have guided missiles and misguided men.”Martin Luther King Jr.(1963)

53

Chapter 4. Path planning and guidance 54

4.1 Path planning

To guide the UAV it is required to specify the desired trajectory. The desired trajectory

can be made a function of time or distance in space. The trajectory tracking with respect

to time requires UAV to be in a particular location at particular point of time. There will

be problems in case of tracking a trajectory varying with time, when the UAV is flying

into wind disturbances [33]. To overcome this the effects of wind should be accounted

properly to move the desired trajectory slower or faster. To remove the dependence of

time a path in space can be specified, where UAV will be required to be on the path

rather than at a certain point on a particular time. Here the desired trajectory is made

a function of forward distance which is discussed in next sections.

4.1.1 Approach

During approach the vehicle should come and align with the runway at a specified height

from where the glideslope can be started. Fig. 4.2 shows the geometry for alignment

where (x0, y0) is the initial location of UAV at the beginning of landing. (xg, yg) is point

from where glideslope begins. ψ0 is the initial orientation of the vehicle.

y

runway x (0,0)

(x0,y

0)

(xg, y

g)

ψ0

Figure 4.2: Approach geometry in x-y plane (top view)


The initial condition (x0) for landing has bounds from which landing can be started

though there is no such constraint for the initial orientation (ψ0) and sideward distance

(y0). This constraint is due to the fact that UAV will require a minimum distance to take

turn and align before reaching the glideslope. The margin for initial condition has been

kept as (x0 − xg) > 400m, i.e. the alignment should begin 400m before the glideslope

initiation point. The equation for path of approach can be written as a straight line

function of forward distance.

y∗ = y0 +

(yg − y0

xg − x0

)(x− x0) (4.1)

This is the desired value of sideward distance during approach which UAV has to follow.

The desired value of sideward distance during glideslope and flare is zero (y∗ = 0) i.e.

aligned with the runway centerline. The UAV may also require to loose height during

approach to reach the glideslope height.

4.1.2 Glideslope

The glideslope is a straight line path whose slope is defined by the flight path angle. The

desired height is also scheduled as a function of forward distance.

(0, 0)

h(x

g, h

g)

(xf, h

f)

(x∞,h

c)

x γ∗

(xg0

, hg0

)

(xtd

, htd

)

Figure 4.3: Glideslope and flare geometry in x-h plane


In Fig. 4.3 (xg, hg) is the point where glideslope begins. (xf , hf ) is point where flare

begins. (xtd, htd = 0) is the touchdown point. (0, 0) is the origin of inertial frame at the

beginning of runway. (xg0 , hg0 = 0) is a fictitious point considered on ground from where

glideslope is projected. (x∞, hc) is the final point of flare path, this point is chosen to be

below ground so that exponential flare path intersects the ground at touchdown point.

From the geometry of the Fig. 4.3 the required flight path angle which vehicle has to

follow is calculated as follows

γ∗ = tan−1

(hg − hg0

xg − xg0

)(4.2)

The desired flight path angle (γ∗) which UAV has to follow is limited between 3o to 5o.

At lower flight path angle the position from which the glideslope has to begin (xg) will

become very large for a given glideslope altitude. At higher values of flight path angle

the vehicle will have limitation as the component of gravity will cause an increase in

velocity during landing. The desired height at every point of glideslope can be written

as a function of forward distance given by following equation

h∗ = hg0 + (x− xg0) tanγ∗ (4.3)

The UAV will follow the path specified by above equation while maintaining lateral error

to zero until the flare height is reached. The flare height is calculated online and it is

not fixed apriori which is discussed in next section.

4.1.3 Flare

The flare path is an exponential curve shown in Fig. 4.3. The exponential curve needs to

be chosen to end below the ground so that it touches the ground at a finite location. The

problem of slope discontinuity at the transition from glideslope to flare can be addressed

by appropriately choosing the parameters which specify the flare path. The advantage

of scheduling desired height as a function of forward distance is that we can specify the

location of touchdown. It is also required to control the sink rate at the touchdown in

order to avoid a hard landing.


The desired height during exponential flare can be scheduled as a function of forward

distance, given by following equation

h∗ = hc + (hf − hc)e−kx(x−xf ) (4.4)

The unknowns in above equation are flare height hf , distance at which to begin flare xf ,

final height below the ground where flare trajectory should end hc, and constant kx. We

can solve for these four unknowns under the following four constraints,

Initial condition: The point where glideslope ends and flare begins be coincident. Sub-

stituting x = xf in (4.3) and (4.4) than equating

hf = −(xf − xg0) tanγ∗ (4.5)

Initial slope: The slope at the beginning of flare and at the end of glideslope be same.

Differentiating (4.3) and (4.4) than equating at x = xf

(hf − hc)kx = tanγ∗ (4.6)

Touchdown condition: The flare trajectory should intersect the ground at touchdown

point. Replacing x = xtd and h∗ = 0 in (4.4) we get

0 = hc + (hf − hc)e−kx(xtd−xf ) (4.7)

Sink rate at touchdown: The descent rate at touchdown should be equal to specified sink

rate. Differentiating (4.4) and evaluating at x = xtd. Putting h∗ = h∗td, where h∗td is the

desired sink rate at touchdown. The ground velocity at touchdown (xtd) is taken as equal

to the air velocity controlled at touchdown.

h∗t = −(hf − hc)kxxtde−kx(xtd−xf ) (4.8)

Now we can solve the Eq. 4.5 - 4.8 for the four unknowns of the flare path. The solution

will ensure a smooth transition from glideslope to flare path. We also have the direct

control over the touchdown point and sink rate at touchdown. Which can be the design

parameters and tuned as per the requirement.


To closely follow the path specified by path planning, appropriate guidance commands

need to be generated which is discussed in next section.

4.2 Guidance

The desired values from path planning to be tracked are sideward distance (y∗ = 0) and

altitude (h∗ = 0). The kinematic equations for sideward distance and altitude can be

inverted to find the desired values of heading angle and pitch angle required for tracking.

While taking turn during approach the coordinated turn constraint is imposed from

which the desired roll angle can be found.

4.2.1 Sideward distance control

To track the desired sideward distance (y∗) we enforce a first order error differential

equation as

(y − y∗) + ky(y − y∗) = 0 (4.9)

where error = y − y∗. The gain ky is chosen positive, given in Table 4.1. Rearranging

above equation we get

y = y∗ − ky(y − y∗) (4.10)

From six-DOF equations we can write

y = ay sinψ + by cosψ (4.11)

where ay , u cosθ + v sinφsinθ + w cosφsinθ

by , v cosφ− w sinφ

Substituting Eq. (4.11) in Eq. (4.10) than solving for ψ

ψ∗ = sin−1

(y∗ − ky(y − y∗)√

a2y + b2

y

)− tan−1

(by

ay

)(4.12)

We have ψ∗ which is the required heading angle to track the path for sideward distance.


4.2.2 Altitude control

The altitude can be controlled by commanding pitch angle. The relative degree between

pitch angle and altitude is one. To generate the pitch angle command from error in

altitude, we enforce the first order error dynamics for altitude error

(h− h∗) + kh(h− h∗) = 0 (4.13)

where error = h−h∗ and h∗ is the desired altitude. The gain kh is chosen positive, given

in Table 4.1. Rearranging above equation we get

h = h∗ − kh(h− h∗) (4.14)

From six-DOF equations we can write

h = ahsinθ − bhcosθ (4.15)

where ah = u and bh = v sinφ + w cosφ

Substituting Eq. (4.15) in Eq. (4.14) than solving for θ

θ∗ = sin−1

(h∗ − kh(h− h∗)√

a2h + b2

h

)+ tan−1

(bh

ah

)(4.16)

Now θ∗ is the desired pitch angle required to track the desired altitude. The UAV should

touchdown on the ground at the end of flare with bounds on pitch angle. This is ensured

by passing the desired pitch angle through a limiter (20 5 θ∗ 5 80) when near to ground.

The other approach taken is to find the trim condition for a given pitch angle at the

touchdown. This trim condition will provide us with a desired velocity at the end of flare

which can be fed to velocity control loop for tracking.

It is seen from Eq. 4.12 and Eq. 4.16 that they involve inverse sine functions. The value

of quantities inside the inverse function need to be ensured less than equal to 1, else the

solutions will be imaginary. This can be achieved by careful selection of ky and kh.


4.2.3 Coordinated turn constraint

The coordinated turn requires the sideslip angle be equal to zero, equivalently it can

enforced by maintaining side velocity in body frame to be zero. This can be achieved

by finding an equivalent roll angle from the enforced first order error dynamics for side

velocity.

(v − v∗) + kv(v − v∗) = 0 (4.17)

where error = v − v∗ and v∗ = 0 is the desired side velocity. From six-DOF equations

of motion we have


m(4.18)

Substituting Eq. 4.18 in Eq. 4.17 and than solving for φ, we will get

φ∗ = sin−1

(v∗ − kv(v − v∗) + ru− pw − Ya

m

gcosθ

)(4.19)

This desired roll angle will ensure the coordinated turn and sideslip angle to be zero.

Schematic of path planning and guidance

Fig. 4.4 shows the complete structure of path planning and guidance.

Figure 4.4: Path planning and guidance schematic


The gains used for path planning and guidance are given in the table below.

Table 4.1: Gains for guidance loop

ky kh kv

1 0.5 1


In this section the simulation results in a disturbance free environment are given for a

complete sequence of landing including approach, glideslope and flare. The ground roll

phase after the touchdown is not simulated. The results for three cases with different

initial conditions is given in Table 4.2. The plots 4.5 - 4.10 correspond to case I.

The simulation is initialized with trim condition and landing is commanded. The UAV

first aligns itself with runway than follows glideslope and flare as seen in Fig. 4.5.

−1500−1000

−5000

−100

−50

0

50

100

150

0

10

20

30

40

50

60

Down range (m) Cross range (m)

Alt

itu

de

(m)

Case 1Case 2

Figure 4.5: Landing trajectory in 3D


The Fig. 4.6 and Fig. 4.7 show the landing path in lateral and longitudinal plane.

−1500 −1000 −500 0−80

−60

−40

−20

0

20

40

60

80

100

120

x (m)

y (

m)

Case 1Case 2

Figure 4.6: Landing path in x-y plane

−1500 −1000 −500 0−10

0

10

20

30

40

50

60

70

x (m)

h (

m)

Case 1Case 2

Figure 4.7: Landing path in x-h plane

Fig. 4.8 shows the evolution of longitudinal states during landing. There are oscil-

lations observed in initial approach phase due to the coupling between longitudinal and

lateral dynamics, also the constant altitude is commanded during the turning maneuver.

0 20 40 60 8015

20

25

t (sec)

Vt (m

/s)

0 20 40 60 800

5

10

t (sec)

α (d

eg)

Case 1Case 2

0 20 40 60 80−20

0

20

t (sec)

Q (d

eg/s

)

0 20 40 60 80−10

0

10

t (sec)

θ (d

eg)

0 20 40 60 80−2000

0

2000

t (sec)

x (m

)

0 20 40 60 80−100

0

100

t (sec)

h (m

)

Figure 4.8: Longitudinal states during landing


Fig. 4.9 shows the evolution of lateral states during landing. The coordinated turn

constraint brings the sideslip angle during the turn close to zero.

0 20 40 60 80−5

0

5

t (sec)

β (d

eg)

0 20 40 60 80−100

0

100

t (sec)

P (d

eg/s

)

Case 1Case 2

0 20 40 60 80−20

0

20

t (sec)

R (d

eg/s

)

0 20 40 60 80−50

0

50

t (sec)φ

(deg

)

0 20 40 60 80−200

0

200

t (sec)

ψ (d

eg)

0 20 40 60 80−200

0

200

t (sec)

y (m

)

Figure 4.9: Lateral states during landing

Fig. 4.10 shows the control values during landing plotted with respect to forward dis-

tance. It is seen that there are oscillations in the initial approach phase while maintaining

constant altitude. The throttle requirement during glideslope has gone to idle position

(5%) as the commanded velocity is decreased (Fig. 4.8).

−1500 −1000 −500 00

0.2

0.4

0.6

0.8

1

x (m)

δt (

0 −

1 )

Case 1Case 2

−1500 −1000 −500 0

−20

−10

0

x (m)

δe (d

eg)

−1500 −1000 −500 0

−10

0

10

x (m)

δa (d

eg)

−1500 −1000 −500 0

−10

0

10

x (m)

δr (d

eg)

Figure 4.10: Control values during landing


The numerical results for the two cases of landing which have different initial conditions

is given in Table 4.2. The commanded values for path planning are given below.

(x∗g, y∗g , h

∗g) = (−1000, 0, 50) Location for end of approach and beginning of glideslope.

(x∗td, y∗td, h

∗td) = (50, 0, 0) Location for touchdown.

h∗td = −0.1 m/s Value of sink rate at touchdown.

The values achieved by UAV during simulation are given in table below. Where (xf , hf )

is location at which flare begins and θtd is the pitch angle at touchdown.

Table 4.2: Landing results

Initial Flare Touchdown

(x0, y0, h0) ψ0 (xf , hf ) xtd htd θtd

m deg m m m/s degCase 1 (-1500, 10, 50) 120o (-5.24, 14.04) 48.74 -0.10 5.55o

Case 2 (-1500, -50, 60) -45o (-4.88, 13.44) 49.06 -0.08 5.63o

It can be observed from the Table 4.2 that the commanded values of touchdown

location and sink rate at touchdown are closely followed. The pitch angle at touch down

was not enforced directly but it was solved for trim condition at touchdown with 6o of

pitch angle. The resulting solution for velocity was controlled at touchdown which has

resulted in pitch angles (θtd) close to 6o.

Summary: The path planning and guidance command generation for autonomous land-

ing has been discussed in this chapter and demonstrated through simulation. In next

chapter the state estimation has been done with Extended Kalman Filter, further the

design is tested in presence of wind disturbance.

Chapter 5

State estimation

The state feedback control laws require an estimate of the true state. There are many

uncertainties associated with the state estimation which make it a difficult problem. The

state estimation will depend upon the accuracy of mathematical modeling. The aero-

dynamic force and moment coefficients can not be modeled accurately even after wind

tunnel tests. There will also be process noise due to the presence of wind disturbances

which are not easy to measure. The sensors have measurement noise which will not give

the deterministic output. Kalman filter is an optimal estimator which minimizes the

variance of states error in the presence of process and measurement noise.

Kalman filter uses the knowledge of system and measurement dynamics, assumed

statistics of system disturbance and measurement errors with initial condition informa-

tion to arrive at optimal estimate. The Kalman filter was initially designed for linear

systems, further it has been modified for nonlinear systems called as Extended Kalman

Filter (EKF). The EKF minimizes the state estimation error with respect to the nonlin-

ear system’s linearized model. The mathematical background for EKF [23], [34] is given

in the next Section.

4“Position and velocity of a particle can not be measured accurately at the same time.” WernerHeisenberg(1927)

65

Chapter 5. State estimation 66

5.1 Extended Kalman Filter

In a dynamical system states are continuously evolving and the measurements arrive at

discrete intervals. Therefore continuous-discrete formulation of EKF best describes a

dynamical system. The nonlinear system model for estimation can be given by following

equations for states and outputs

X = f(X(t), Uc(t), t) + G(t)w(t) (5.1)

Yk = h(Xk) + vk (5.2)

Where, the vector f represents the system dynamics which is a nonlinear function of

the state X, the deterministic control input Uc and varies with time t. The disturbance

input or process noise w is white, zero mean gaussian random process. The statistical

properties of process noise can be written as,

E[w(t)] = 0 (5.3)

E[w(t)wT (t)] = Q (5.4)

The vector h represents the output equation which is a nonlinear function of the state

X. The measurement noise v is also white, zero mean gaussian random process that

is uncorrelated with disturbance input. The statistical properties of measurement noise

can be written as,

E[vk] = 0 (5.5)

E[vkvTk ] = R (5.6)

E[w(t)vTk ] = 0 (5.7)

The basic assumption in EKF is that the true state is sufficiently close to the estimated

state. Therefore, the error dynamics can be represented fairly accurately by a linearized

first-order Taylor series expansion. So we linearize the nonlinear system around the


Kalman filter estimate of the state. The linearized matrices also known as sensitivity

matrices are given as

F (t) =∂f

∂X|X(t)=X(t) (5.8)

H(t) =∂h

∂X|X(t)=X(t) (5.9)

The expected values of the initial state and its covariance are assumed known and given

by following equations

E(X0) = X0 (5.10)

E[(X0 − X0)(X0 − X0)T ] = P0 (5.11)

A weighting factor called the Kalman filter gain is computed using the covariance infor-

mation, output sensitivity matrix and the measurement covariance matrix . The Kalman

filter gain is used to combine the extrapolated estimate with the new measurement. This

gain is defined in such a way that it minimizes the estimation error covariance after the

update. The Kalman filter gain [23] is given by following equation

Kk = P−k HT

k (t−k ) [Hk(t−k ) P−

k HTk (t−k ) + Rk]

−1 (5.12)

New measurements are combined with the extrapolated estimate to generate an updated

estimated state. The state covariance also is updated to reflect the information contained

in the measurement. The state and covariance update equation are

X+k = X−

k + Kk [Yk − h(t−k )] (5.13)

P+k = [I −Kk Hk(t

−k )] P− (5.14)


The EKF propagates both the state estimate and its covariance using the system model

given by following equations

˙X(t) = f(X(t), Uc(t), t) (5.15)

P (t) = F (t) P (t) + P (t) F T (t) + G(t) Q(t) GT (t) (5.16)

The EKF algorithm involves initialization of states and covariance, computation of

Kalman filter gain, update of states and covariance followed by propagation of states and

covariance. This process is followed in a cyclic manner excluding initialization. Equa-

tions (5.9)-(5.16) constitute the EKF algorithm for continuous-time nonlinear systems

with discrete measurements.

5.2 Filter equations for UAV

The equations for UAV need to be modified in order to take in account the wind distur-

bance. This can be achieved by making the wind disturbances as a part of state vector,

which is discussed in following sections.

5.2.1 Wind disturbances

The wind disturbance is considered in inertial frame x, y and h directions. The natural

wind tends to be constant or slowly varying in horizontal plane where as there are wind

gusts in vertical direction. Therefore the wind in x and y direction are assumed to be

constant and wind in h direction accelerating. It is necessary to make these disturbances

part of the system state vector. The wind state vector is defined as

Xd = [wx wy wh wh] (5.17)

Any mathematical model of the wind dynamics represent an approximation to the con-

ditions that exist. Error in the modeling can lead to biases in the estimates and even


divergence from the actual values. Hence the dynamics of the wind [25] can be repre-

sented by using a simple Integral state model [36] given by following equations

Xd = FdXd + wd (5.18)

where

Fd =

0 0 0 0

0 0 0 0

0 0 0 1

0 0 0 0

wd =

0

0

0

1

To account the wind acceleration we transform it from inertial frame to body frame

aBw = [HB

I ]aIw

Where HBI is the transformation matrix from inertial frame to body frame and the

acceleration of wind disturbance is given as aIw , [ 0 0 wh ]. The transformation of

wind acceleration to body axis will give

aBwx

aBwy

aBwh

=

cosθcosψ cosθsinψ −sinθ

sinφsinθsinψ − cosφsinψ sinφsinθsinψ + cosφcosψ sinφcosθ

cosφsinθcosψ + sinφsinψ cosφsinθsinψ − sinφcosψ cosφcosθ

0

0

−wh

Simplifying above equation we will get

aBwx

= wh sinθ aBwy

= −wh sinφcosθ aBwh

= −wh cosφcosθ

5.2.2 State sensitivity matrix

To account for the wind and wind acceleration the translational dynamic and kinematic

equations are revised to wind relative reference frame [35].

Translational dynamics

ua = rva − qwa − gsinθ +Xa + Xt

m− aB

wx

va = pwa − rua + gsinφcosθ +Ya

m− aB

wy

wa = qua − pva + gcosφcosθ +Za

m− aB

wh


Translational kinematics

x = uacosθcosψ + va(sinφsinθsinψ − cosφsinψ) + wa(cosφsinθcosψ + sinφsinψ) + wx

y = uacosθsinψ + va(sinφsinθsinψ + cosφcosψ) + wa(cosφsinθsinψ − sinφcosψ) + wy

h = uasinθ − vasinφcosθ − wacosφcosθ + wh

The rotational dynamic and kinematic equations remain unchanged as given Chapter 2.

In above equations the ua, va, and wa are the wind relative body frame velocities. Now

the the system state vector is defined as

X = [ua va wa p q r φ θ ψ x y h] (5.19)

Combining the system and wind disturbance states, we can define

χ ,

X

Xd

(5.20)

The combined dynamics can be represented as

χ =

X

Xd

=

f(X,Uc, Xd)

FdXd

+

0

wd

(5.21)

The linearized matrix of state can be found as

F (t) =∂f

∂χ(5.22)

The expressions for linearized matrix is given in below equations

∂(ua, va, wa)∂χ =

0 r −q 0 −wa va 0 −(g − wh)cθ sθ

−r 0 p wa 0 −ua (g − wh)cθcφ −(g − wh)sθsφ 03×7 −sφcθ

q −p 0 −va ua 0 −(g − wh)cθsφ −(g − wh)sθcφ −cφcθ

3×16


∂(p, q, r)∂χ =

c2q c1r + c2p c1q

03×3 c5r − 2c6p 0 c5p + 2c6r 03×10

c8q c8p− c2r −c2q

3×16

∂(φ, θ, ψ)∂χ =

1 sφtθ cφtθ (qcφ − rsφ)tθ (qsφ + rcφ)se2θ

03×3 0 cφ −sφ −qsφ − rcφ 0 03×8

0 sφseθ cφseθ (qcφ − rsφ)seθ (qsφ + rcφ)seθtθ

3×16

∂(x, y, h)∂χ =

cθcψ sφsθcψ − cφsψ cφsθsψ − sφcψ

cθsψ sφsθsψ + cφcψ cφsθsψ − sφcψ 03×3 03×3 I3×3 03×1

sθ −sφcθ −cφcθ

3×16

∂(wx, wy , wh, wh)∂χ =

0

04×15 0

1

0

4×16

where, sφ , sinφ, sθ , sinθ, sψ , sinψ tθ , tanθ,

cφ , cosφ, cθ , cosθ, cψ , cosψ seθ , secθ

The state sensitivity will be given as

F (t) = ∂f∂χ

=

∂(ua, va, wa)∂X

∂(p, q, r)∂X

∂(φ, θ, ψ)∂X

∂(x, y, h)∂X

∂(wx, wy, wh, wh)

∂X

16×16

5.2.3 Output sensitivity matrix

The sensors available for measurement are accelerometers, rate gyros, GPS, magnetome-

ter and pressure sensors which provide air velocity and altitude.


The output vector can be written as

Y = [vt ax ay az p q r ψ x y h] (5.23)

The output sensitivity matrix can be found as

H(t) =∂h

∂χ(5.24)

The expressions for linearized matrix is given in below equations

∂(vt, ax, ay az)∂χ =

ua

vt

ua

Vt

ua

Vt0 0 0 0 0 0

0 −r q 0 wa −va 0 (g − wh)cθ −sθ

r 0 −p −wa 0 ua −(g − wh)cθcφ (g − wh)sθsφ 04×7 sφcθ

−q p 0 va −ua 0 (g − wh)cθsφ (g − wh)sθcφ cφcθ

4×16

∂(p, q, r)∂χ =

03×3 I3×3 03×10

3×16

∂(ψ, x, y, h)∂χ =

04×8 I4×4 04×4

4×16

The output sensitivity will be given as

H(t) = ∂h∂X

=

∂(vt, ax, ay az)

∂χ

∂(p, q, r)∂X

∂(ψ, x, y, h)∂X

11×16


5.2.4 Selection of R, P0 and Q

The outputs given by the onboard sensors are accelerations, body rates, air velocity,

altitude and ground position given by GPS. The sensors give output at a frequency of

50hz other than GPS whose data arrive at 5hz. Hence the algorithm uses the inertial data

until the GPS measurements arrive, than the positions are corrected. The measurement

noise covariance matrix obtained from the given specifications of the sensors is

R = diag(22, 12, 12, 12, 0.022, 0.022, 0.022, 0.052, 52, 52, 52) (5.25)

The initial error covariance was selected by trial and error and finally set to P0 = I. If the

initial covariance was made large, than states showed large oscillations in transient and

filter diverged in some cases. The smaller values of covariance delayed the convergence

of estimated states to the actual states.

The process noise covariance is set to

Q = diag(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.1, 0.1, 0, 0.001) (5.26)

The zero elements in the diagonal terms represent there is no process noise in that channel

of state. The nonzero diagonal elements were determined through trial-and-error. If they

were made too large, the EKF attenuated the measurement noise and very small diagonal

elements introduced significant lags into the estimates of the states.


The simulation results are shown here in presence of wind disturbance. The plots show

the comparison of performance between ideal conditions when there is no wind and the

states are known perfectly and the actual conditions when the state is estimated in

presence of wind disturbances and measurement noise. The landing command is given

after 5 seconds of simulation. This lead time is required for the settling of filter gains.


Fig. 5.1 shows the comparison of landing path when there is no wind and states are

known perfectly and when there are wind disturbances and state is estimated.

−1500−1000

−5000

0

50

100

0

10

20

30

40

50

Forward distance (m) Sideward distance (m)

Alti

tude

(m

)

IdealActual

Figure 5.1: Landing trajectory in 3D with EKF

The Fig. 5.2 and Fig. 5.3 show the landing path in lateral and longitudinal plane. In

Fig. 5.3 it is seen that altitude drops initially as the wind is introduced, this leads to

decrease in air relative velocity hence the lift generated decreases.

−1600 −1400 −1200 −1000 −800 −600 −400 −200 0

0

20

40

60

80

100

120

X (m)

Y (

m)

IdealActual

Figure 5.2: Landing path in x-y plane

−1600 −1400 −1200 −1000 −800 −600 −400 −200 00

5

10

15

20

25

30

35

40

45

50

X (m)

h (

m)

IdealActual

Figure 5.3: Landing path in x-h plane


The Fig. 5.4 shows the estimation error in longitudinal states. It is seen that all the

states remain in the bound other than pitch rate, which is due to the wind acceleration

in pitch plane.

0 20 40 60−2

0

2

t (sec)

U (m

/s)

0 20 40 60−2

0

2

t (sec)

W (m

/s)

0 20 40 60−5

0

5

t (sec)

θ (d

eg)

0 20 40 60−5

0

5

t (sec) Q

(deg

/s)

error

3 σ

0 20 40 60−5

0

5

t (sec)

x (m

)

0 20 40 60−5

0

5

t (sec)

h (m

)

Figure 5.4: Estimation error in longitudinal states

The Fig. 5.5 shows the estimation error in lateral states. It is seen that all the states

remain in the bound other than heading angle which takes some time.

0 20 40 60−2

0

2

t (sec)

β (de

g)

0 20 40 60−10

0

10

t (sec)

P (d

eg/s

)

error3 σ

0 20 40 60−10

0

10

t (sec)

R (d

eg/s

)

0 20 40 60−10

0

10

t (sec)

φ (de

g)

0 20 40 60−1

0

1

t (sec)

ψ (d

eg)

0 20 40 60−5

0

5

t (sec)

y (m

)

Figure 5.5: Estimation error in lateral states


The Fig. 5.6 shows the estimation of wind disturbances. It can be observed that the

estimated states follow the actual wind with minimum error.

0 20 40 60−5

0

5

10

t (sec)

Wx (m

/s)

error3 σ

0 20 40 60−4

−2

0

2

4

t (sec)

Wy (m

/s)

0 20 40 60−4

−2

0

2

4

t (sec)

Wh (m

/s)

0 20 40 60−4

−2

0

2

4

t (sec)

Whd

ot (m

/s2 )

Figure 5.6: Wind estimation with EKF

The control values required with and without the disturbance are shown in Fig.5.7.

It is seen that there is continuous demand in control even during glideslope and flare

this is due to changing wind and wind gust.

0 20 40 600

0.5

1

t (sec)

δt (

0 −

1 )

IdealActual

0 20 40 60−30

−20

−10

0

10

t (sec)

δe

(deg

)

0 20 40 60−10

−5

0

5

10

t (sec)

δa

(deg

)

0 20 40 60−10

−5

0

5

10

t (sec)

δr (

deg)

Figure 5.7: Control values during landing with EKF


To test the algorithm a large number (100) of simulations were carried out. The

Fig. 5.8 shows the touchdown locations for various cases. To measure the performance

a successful landing zone is defined as shown in Fig. 5.8.

0 10 20 30 40 50 60 70 80−5

−4

−3

−2

−1

0

1

2

3

4

5

X (m)

Y (m

)

50m

6m

Successful landing zone

Figure 5.8: Touchdown position in x-y plane

The sink rate at touchdown and pitch angle bounds are specified for successful land-

ing. The large sink rate can cause damage to structure and lower sink rate can be carried

away by wind gust. At negative pitch angle the nose will touch first and at higher angle

the tail boom will touch the ground before wheels.

0 20 40 60 80 1000

1

2

3

4

5

6

7

8

9

10

no. of simulations

pitch

an

gle

at

tou

ch

do

wn

(d

eg

)

Figure 5.9: Pitch angle at touch down

0 20 40 60 80 100

−1

−0.8

−0.6

−0.4

−0.2

0

no. of simulations

sin

k r

ate

at to

uch

do

wn

(m

/s)

Figure 5.10: Sink rate at touch down


Parameter uncertainty

The uncertainty in the aerodynamic model is one of the key difficulties in the imple-

mentation of the dynamic inversion and extended kalman filter. The uncertainty in

parameters can be accommodated by considering fictitious process noise wf in the sys-

tem model [25]. The uncertainty in the system model is considered as gaussian random

input. The system dynamic equations which use the aerodynamic coefficients are added

with fictitious random noise. The revised model can be expressed as

χ =

X

Xd

=

f(X, Uc, Xd)

FdXd

+

wf

wd

(5.27)

The elements of process noise covariance matrix (Q) will have to be revised and chosen to

reflect the uncertainty in aerodynamic coefficients. The process noise covariance matrix

was chosen by trial and error while increasing the uncertainty in the coefficients. The

maximum percentage uncertainty accommodated by EKF was found to be 4%. Further

this was found to increase up to 6% with inclusion of integrator in the inner most loop.

The corresponding process noise covariance matrix is found as

Q = diag(0.001, 0.001, 0.2, 0.02, 0.05, 0.02, 0, 0, 0, 0, 0, 0, 0.1, 0.1, 0, 0.001)

The Fig. 5.11 shows the touchdown location for hundred simulations cases with the

uncertainty in parameters. It can be observed that the dispersion of landing location

has increased compared to when the uncertainty was not considered.

−10 0 10 20 30 40 50 60 70 80−6

−4

−2

0

2

4

6

X (m)

Y (m

)

Successful landing zone

Figure 5.11: Touchdown position in x-y plane with parameter uncertainty


The Fig. 5.12 and Fig. 5.13 show the pitch angle and sink rate at touchdown when

the uncertainty in the parameters is considered.

0 20 40 60 80 1000

1

2

3

4

5

6

7

8

9

10

no. of simulations

pitch

an

gle

at

tou

ch

do

wn

(d

eg

)

Figure 5.12: Pitch angle at touch down

0 20 40 60 80 100−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

no. of simulations

sin

k r

ate

at to

uch

do

wn

(m

/s)

Figure 5.13: Sink rate at touch down

The results obtained from simulations with the percentage of successful landing cases

is given in Table 5.1. The success criteria is defined as below

7→ For touch down location, 20m 5 xtd 5 70m and −3m 5 ytd 5 3m

7→ For pitch angle, 2odeg 5 θtd 5 8o

7→ For sink rate, −0.1m/s 5 htd 5 −1m/s

Table 5.1: Percentage of successful landing

touchdown pitch angle sink rate overallEKF with only disturbance 76% 79% 85% 71%

EKF with disturbance and uncertainty 69% 63% 64% 60%

It can be observed from the table that with parameter uncertainty in the aerodynamic

coefficients the percentage of successful landing has decreased. The success criteria can

be further relaxed with respect to sink rate and touchdown location.

Summary: In this chapter the state estimation has been done with Extended Kalman

Filter. Further the design is tested in presence of wind disturbances and parameter

uncertainty. The next chapter gives the conclusion of the research work done.

Chapter 6

Conclusions

In this thesis the problem of autonomous landing of unmanned aerial vehicles has been

addressed. The complete work has involved Mathematical modeling, Control design,

Guidance and State estimation. The guidance and control technique developed has been

demonstrated through numerical simulations using the data of an UAV named AE-2.

The curve fitting is done on the wind tunnel data of AE-2 obtained by static tests.

The method of least squares is used for minimizing the error in prediction and the

nonlinearities in the function obtained are reported. Further the dynamic derivatives are

found through AVL software which has been validated by comparison of static coefficients

obtained by the wind tunnel tests and vortex lattice method. The dynamics of thrust

and servo actuators have been incorporated in the simulation.

The control has been designed using nonlinear dynamic inversion technique. The

control is structured in outer loop and inner loop. The goal of outer loop is to track

the guidance commands by converting them into body rates. The inner loop tracks the

body rate commands by finding the required deflection of control surfaces. The forward

velocity of the vehicle is controlled by throttle. A controller for actuator is also designed

to reduce the lag.

80

Chapter 6. Conclusions 81

The guidance for landing is divided into three phases which are approach, glideslope

and flare. The desired trajectory for the landing has been made independent of time by

scheduling the trajectory as a function of forward distance from runway. The problem

of transition between the glideslope and flare has been addressed by ensuring continuity

and smoothness at transition. The location of touchdown on runway and the sink rate

at touchdown are controlled.

The state estimation has been done with EKF in continuous discrete formulation.

The external disturbances like wind shear and wind gust are accounted by appending

them in state variables. Further the control design with guidance is tested from various

initial conditions, in presence of wind disturbances. The designed filter has also been

tested for parameter uncertainty.

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MS Thesis ShashiprakashSingh

Engineering

Transcript of MS Thesis ShashiprakashSingh