INVESTIGATION OF NONLINEAR CONTROL STRATEGIES USING … · INVESTIGATION OF NONLINEAR CONTROL...

INVESTIGATION OF NONLINEAR CONTROL

STRATEGIES USING GPS SIMULATOR AND

SPACECRAFT ATTITUDE CONTROL SIMULATOR

by

Scott A. Kowalchuk

Dissertation submitted to the Faculty of the

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Doctor of Philosophyin

Aerospace Engineering

Committee Members

Christopher D. Hall, Committee Chair

Wayne Scales, Committee Member

Craig A. Woolsey, Committee Member

Scott L Hendricks, Committee Member

Cornel Sultan, Committee Member

September 7, 2007

Blacksburg, Virginia

Keywords: Spacecraft Attitude Control, Orbit Control, Spacecraft Formation FlyingCopyright 2007, Scott A. Kowalchuk

INVESTIGATION OF NONLINEAR CONTROL STRATEGIES USING GPS

SIMULATOR AND SPACECRAFT ATTITUDE CONTROL SIMULATOR

Scott A. Kowalchuk

Abstract

In this dissertation, we discuss the Distributed Spacecraft Attitude Control System Simulator

(DSACSS) testbed developed at Virginia Polytechnic Institute and State University for the purpose

of investigating various control techniques for single and multiple spacecraft. DSACSS is comprised

of two independent hardware-in-the-loop simulators and one software spacecraft simulator. The two

hardware-in-the-loop spacecraft simulators have similar subsystems as flight-ready spacecraft (e.g.

command and data handling; communications; attitude determination and control; power; payload;

and guidance and navigation). The DSACSS framework is a flexible testbed for investigating a

variety of spacecraft control techniques, especially control scenarios involving coupled attitude and

orbital motion.

The attitude hardware simulators along with numerical simulations assist in the development

and evaluation of Lyapunov based asymptotically stable, nonlinear attitude controllers with three

reaction wheels as the control device. The angular rate controller successfully tracks a time varying

attitude trajectory. The Modified Rodrigues Parmater (MRP) attitude controller results in success-

fully tracking the angular rates and MRP attitude vector for a time-varying attitude trajectory. The

attitude controllers successfully track the reference attitude in real-time with hardware similar to

flight-ready spacecraft.

Numerical simulations and the attitude hardware simulators assist in the development and eval-

uation of a robust, asymptotically stable, nonlinear attitude controller with three reaction wheels

as the actuator for attitude control. The MRPs are chosen to represent the attitude in the develop-

ment of the controller. The robust spacecraft attitude controller successfully tracks a time-varying

reference attitude trajectory while bounding system uncertainties.

The results of a Global Positioning System (GPS) hardware-in-the-loop simulation of two

spacecraft flying in formation are presented. The simulations involve a chief spacecraft in a low

Earth orbit (LEO), while a deputy spacecraft maintains an orbit position relative to the chief space-

craft. In order to maintain the formation an orbit correction maneuver (OCM) for the deputy space-

craft is required. The control of the OCM is accomplished using a classical orbital element (COE)

feedback controller and simulating continual impulsive thrusting for the deputy spacecraft. The

COE controller requires the relative position of the six orbital elements: a, e, I , Ω, ω, M0. The

deputy communicates with the chief spacecraft to obtain the current orbit position of the chief space-

craft, which is determined by a numerical orbit propagator. The position of the deputy spacecraft

is determined from a GPS receiver that is connected to a GPS hardware-in-the-loop simulator. The

GPS simulator creates a radio frequency (RF) signal based on a simulated trajectory, which results

in the GPS receiver calculating the navigation solution for the simulated trajectory. From the relative

positions of the spacecraft the COE controller calculates the OCM for the deputy spacecraft. The

formation flying simulation successfully demonstrates the closed-loop hardware-in-the-loop GPS

simulator.

This dissertation focuses on the development of the DSACSS facility including the development

and implementation of a closed-loop GPS simulator and evaluation of nonlinear feedback attitude

and orbit control laws using real-time hardware-in-the-loop simulators.

iii

To my wife, parents

and

in memory of the 32 members of the Virginia Tech

community that lost their lives April 16, 2007

“We will continue to invent the future through our blood and tears

and through all our sadness... We will prevail...”

- Nikki Giovanni, University Distinguished Professor, poet, activist

Acknowledgments

I express my deep gratitude to Dr. Christopher Hall for his advice, support, and mentoring during

my Ph.D. studies at Virginia Tech. I also express my thanks to Dr. Wayne Scales, Dr. Craig

Woolsey, Dr. Scott Hendricks, and Dr. Cornel Sultan for their valuable input over the past three

years.

Thanks to all that participated in the Space Systems Simulation Laboratory for their time and

dedication as well as the support staff in the Aerospace and Ocean Engineering Department. I have

enjoyed interacting with all the staff and students.

A special thanks to my parents, John and Gerry Kowalchuk, for their sacrifice and support

throughout the years. Finally, I must express my greatest appreciation to my wife, Emily, who has

always provided me with encouragement and support. I have enjoyed my time in Blacksburg and

have made many good friends with many cherished memories.

“Destiny is not a matter of chance. It is a matter of choice.

It’s not a thing to be waited for - it is a thing to be achieved.”

- William Jennings Bryan

v

Contents

Acknowledgments v

List of Figures xi

List of Tables xix

List of Symbols xxi

1 Introduction 1

1.1 Distributed Space Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Dissertation Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Dissertation Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Literature Review 6

2.1 Spacecraft Simulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Nonlinear Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Attitude Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 Orbit Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Distributed Spacecraft Attitude Control System Simulator (DSACSS) 13

3.1 Attitude Hardware-in-the-loop Simulators . . . . . . . . . . . . . . . . . . . . . . 14

vi

Contents

3.1.1 Power Bus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1.2 Flight Computer and Communications . . . . . . . . . . . . . . . . . . . . 18

3.1.3 Attitude Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.3.1 Attitude Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.3.2 Attitude Determination Algorithms . . . . . . . . . . . . . . . . 21

3.1.4 Attitude Control Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Software Simulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 DSACSS Operational Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4 GPS Hardware-in-the-loop Simulator for GN&C Simulations . . . . . . . . . . . . 30

3.4.1 Spirent GPS Simulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.4.2 Ashtech G12 HDMA GPS Receiver . . . . . . . . . . . . . . . . . . . . . 32

3.4.3 Closed-hardware-in-the-loop Spacecraft GPS Simulator . . . . . . . . . . 32

3.4.4 Coordinate Transformations for GPS Receiver . . . . . . . . . . . . . . . 35

3.4.4.1 WGS84 / ECEF Coordinate Transformation . . . . . . . . . . . 35

3.4.4.2 ECEF / ECI Coordinate Transformation . . . . . . . . . . . . . 38

3.4.5 Example of Navigation Determination using GPS Receiver . . . . . . . . 39

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4 Spacecraft Kinematics and Rigid Body Dynamics 46

4.1 Attitude Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.1.1 Quaternion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.1.2 Modified Rodrigues Parameters (MRP) . . . . . . . . . . . . . . . . . . . 48

4.2 Rigid Body Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2.1 Euler’s Rotational Equations of Motion . . . . . . . . . . . . . . . . . . . 50

4.2.2 Euler’s Rotational Equations of Motion with Momentum Exchange Devices 50

4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5 Lyapunov Based Spacecraft Attitude Control Laws 53

vii

Contents

5.1 Nonlinear Stability and Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.2 Angular Rate Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.2.1 Whorl−I Angular Rate Controller Simulation . . . . . . . . . . . . . . . . 57

5.2.1.1 Regulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.2.1.2 Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.2.2 Whorl−II Angular Rate Controller Simulation . . . . . . . . . . . . . . . 70

5.3 Modified Rodrigues Parameter Control . . . . . . . . . . . . . . . . . . . . . . . . 73

5.3.1 Whorl−I MRP Controller Simulation . . . . . . . . . . . . . . . . . . . . 78

5.3.1.1 Regulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.3.1.2 Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.3.2 Whorl−II MRP Controller Simulation . . . . . . . . . . . . . . . . . . . . 91

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6 Sliding Mode Spacecraft Attitude Control Law 97

6.1 Attitude Control Law Development . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.2 Gain Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.3 Parameter Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.4 Sliding Mode Control Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.4.1 Numerical Sliding Mode Control Simulations . . . . . . . . . . . . . . . . 110

6.4.1.1 Equivalent Control Term With No Uncertainties . . . . . . . . . 110

6.4.1.2 Equivalent Control Term With Uncertainties . . . . . . . . . . . 115

6.4.1.3 Robust Attitude Controller . . . . . . . . . . . . . . . . . . . . 119

6.4.2 Whorl−I Sliding Mode Control Simulation . . . . . . . . . . . . . . . . . 124

6.4.2.1 Regulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.4.2.2 Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.4.3 Whorl−II Sliding Mode Control Simulation . . . . . . . . . . . . . . . . 138

6.4.3.1 Regulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

6.4.3.2 Rest to Rest Maneuver . . . . . . . . . . . . . . . . . . . . . . . 144

viii

Contents

6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

7 Classical Orbital Element Controller 150

7.1 Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7.2 GPS Formation Flying Simulation using COE Controller . . . . . . . . . . . . . . 154

7.2.1 Implementation of Classical Orbital Element Controller . . . . . . . . . . 155

7.2.2 LEO Formation Flying Simulation . . . . . . . . . . . . . . . . . . . . . . 156

7.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

8 Sliding Mode Classical Orbital Element Controller 165

8.1 Modification to Gauss’ form of Lagrange’s Planetary Equations . . . . . . . . . . 165

8.2 Sliding Mode Orbit Control Law with Position Uncertainty . . . . . . . . . . . . . 168

8.2.1 Control Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

8.2.2 Gains and Parameter Bound Selection . . . . . . . . . . . . . . . . . . . . 171

8.2.3 LEO Control Simulation using GPS Simulator . . . . . . . . . . . . . . . 174

8.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

9 Conclusions and Recommendations 182

9.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

9.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Bibliography 185

A Appendix: Reaction Wheel Controller 192

A.1 Reaction Wheel Controller Development . . . . . . . . . . . . . . . . . . . . . . . 192

A.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

B Appendix: Implementation of Attitude Control Laws into DSACSS 199

B.1 Implementation Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

B.2 Template Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

ix

Contents

B.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

C Appendix: Implementation of Orbit Control Laws into DSACSS Framework 208

C.1 Implementation Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

C.2 Template Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

C.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

x

List of Figures

1.1 Distributed Space Systems Concept . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Two Spacecraft in Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1 “Tabletop” Style Air Bearing Spacecraft Simulator (Whorl−I) . . . . . . . . . . . 15

3.2 “Dumbbell” Style Air Bearing Spacecraft Simulator (Whorl−II) . . . . . . . . . . 16

3.3 Whorl−I System Block Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.4 Whorl−II System Block Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.5 DSACSS Communication Schematic . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.6 Whorl−I Angular Rates from Extended Kalman Filter During Control Simulation . 23

3.7 Whorl−I Quaternion Vector from Extended Kalman Filter During Control Simulation 23

3.8 Whorl−II Angular Rates from Extended Kalman Filter During Simulation . . . . . 24

3.9 Whorl−II Quaternion Vector from Extended Kalman Filter During Simulation . . . 24

3.10 Reaction Wheel Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.11 Software Simulator (Whorl−III) . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.12 DSACSS-Ops Code Functional Diagram . . . . . . . . . . . . . . . . . . . . . . . 28

3.13 GPS Open-loop Simulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.14 GPS Closed-hardware-in-the-loop Simulator . . . . . . . . . . . . . . . . . . . . . 34

3.15 High-level GN&C Block Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.16 Whorl-I Orbit Radius Determined from the GPS Receiver in the ECI Reference Frame 40

xi

List of Figures

3.17 Whorl−I Orbit Velocity Determined from the GPS Receiver in the ECI Reference

Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.18 Whorl−I Orbit Trajectory Determined from the GPS Receiver in the ECI Reference

Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.19 Semi-Major Axis Time History for Whorl−I . . . . . . . . . . . . . . . . . . . . . 41

3.20 Eccentricity Time History for Whorl−I . . . . . . . . . . . . . . . . . . . . . . . 42

3.21 Inclination Time History for Whorl−I . . . . . . . . . . . . . . . . . . . . . . . . 42

3.22 Longitude of the Ascending Node Time History for Whorl−I . . . . . . . . . . . . 43

3.23 Argument of Perigee Time History for Whorl−I . . . . . . . . . . . . . . . . . . . 43

3.24 True Anomaly Time History for Whorl−I . . . . . . . . . . . . . . . . . . . . . . 44

3.25 Whorl−I GPS Receiver Parameters (PDOP, HDOP, VDOP, and TDOP) . . . . . . 44

3.26 Number of GPS Satellites Used to Determine Navigation Solution . . . . . . . . . 45

4.1 Illustration of Whorl−I and Whorl−II Reaction Wheel . . . . . . . . . . . . . . . 51

5.1 Whorl−I Angular Rates for Angular Rate Controller . . . . . . . . . . . . . . . . 59

5.2 Numerical Simulation Angular Rates for Angular Rate Controller . . . . . . . . . 59

5.3 Whorl−I Control Torque for Angular Rate Controller . . . . . . . . . . . . . . . . 60

5.4 Numerical Simulation Control Torque for Angular Rate Controller . . . . . . . . . 60

5.5 Whorl−I Reaction Wheel Speed for Angular Rate Controller . . . . . . . . . . . . 61

5.6 Numerical Simulation Reaction Wheel Speed for Angular Rate Controller . . . . . 62



5.9 Whorl−I Angular Rate Error for Angular Rate Controller . . . . . . . . . . . . . . 64

5.10 Numerical Simulation Angular Rate Error for Angular Rate Controller . . . . . . . 64



5.13 Whorl−I Reaction Wheel Rates for Angular Rate Controller . . . . . . . . . . . . 66

xii

List of Figures



5.16 Whorl−I Angular Rate Error for Angular Rate Controller . . . . . . . . . . . . . . 68


5.18 Whorl−I Reaction Wheel Speed for Angular Rate Controller . . . . . . . . . . . . 69

5.19 Whorl−II Angular Rates for Angular Rate Controller . . . . . . . . . . . . . . . . 71


5.21 Whorl−II Control Torque for Angular Rate Controller . . . . . . . . . . . . . . . 72


5.23 Whorl−II Reaction Wheel Speed for Angular Rate Controller . . . . . . . . . . . 73


5.25 Whorl−I Angular Rates for MRP Controller . . . . . . . . . . . . . . . . . . . . . 80

5.26 Numerical Simulation Angular Rates for MRP Controller . . . . . . . . . . . . . . 80

5.27 Whorl−I MRP Vector for MRP Controller . . . . . . . . . . . . . . . . . . . . . . 81

5.28 Numerical Simulation MRP Vector for MRP Controller . . . . . . . . . . . . . . . 82

5.29 Whorl−I Control Torque for MRP Controller . . . . . . . . . . . . . . . . . . . . 82

5.30 Numerical Simulation Control Torque for MRP Controller . . . . . . . . . . . . . 83

5.31 Whorl−I Reaction Wheel Speed for MRP Controller . . . . . . . . . . . . . . . . 83

5.32 Numerical Simulation Reaction Wheel Speed for MRP Controller . . . . . . . . . 84

5.33 Whorl−I Angular Rates for MRP Controller . . . . . . . . . . . . . . . . . . . . . 85


5.35 Whorl−I Angular Rate Errors for MRP Controller . . . . . . . . . . . . . . . . . . 86

5.36 Numerical Simulation Angular Rate Errors for MRP Controller . . . . . . . . . . . 87

5.37 Whorl−I MRP Vector for MRP Controller . . . . . . . . . . . . . . . . . . . . . . 87


5.39 Whorl−I MRP Error Vector for MRP Controller . . . . . . . . . . . . . . . . . . . 88

5.40 Numerical Simulation MRP Error Vector for MRP Controller . . . . . . . . . . . . 89

xiii

List of Figures

5.41 Whorl−I Control Torque for MRP Controller . . . . . . . . . . . . . . . . . . . . 89


5.43 Whorl−I Reaction Wheel Speed for MRP Controller . . . . . . . . . . . . . . . . 90


5.45 Whorl−II Angular Rates for MRP Controller . . . . . . . . . . . . . . . . . . . . 92


5.47 Whorl−II MRP Vector for MRP Controller . . . . . . . . . . . . . . . . . . . . . 93


5.49 Whorl−II Control Torque for MRP Controller . . . . . . . . . . . . . . . . . . . . 94


5.51 Whorl−II Reaction Wheel Speed for MRP Controller . . . . . . . . . . . . . . . . 95


6.1 Sliding Mode Control Numerical Simulation Angular Rates with only Equivalent

Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.2 Sliding Mode Control Numerical Simulation Angular Rate Errors with only Equiv-

alent Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.3 Sliding Mode Control Numerical Simulation MRPs with only Equivalent Control . 113

6.4 Sliding Mode Control Numerical Simulation MRP Errors with only Equivalent Con-

trol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.5 Sliding Mode Control Numerical Simulation Equivalent Control Time History . . . 114

6.6 Sliding Mode Control Numerical Simulation Angular Rates with only Equivalent

Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.7 Sliding Mode Control Numerical Simulation Angular Rate Errors with only Equiv-

alent Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.8 Sliding Mode Control Numerical Simulation MRPs with only Equivalent Control . 118

6.9 Sliding Mode Control Numerical Simulation MRP Errors with only Equivalent Con-

trol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

xiv

List of Figures

6.10 Sliding Mode Control Numerical Simulation Equivalent Control Time History . . . 119

6.11 Sliding Mode Control Numerical Simulation Angular Rates . . . . . . . . . . . . . 121

6.12 Sliding Mode Control Numerical Simulation Angular Rate Errors . . . . . . . . . 121

6.13 Sliding Mode Control Numerical Simulation MRPs . . . . . . . . . . . . . . . . . 122

6.14 Sliding Mode Control Numerical Simulation MRP Errors . . . . . . . . . . . . . . 122

6.15 Sliding Mode Control Numerical Simulation Control Time History . . . . . . . . . 123

6.16 Sliding Mode Control Numerical Simulation Reaction Wheel Time History . . . . 123

6.17 Sliding Mode Control Numerical Simulation Reaction Sliding Variable Time History 124

6.18 Whorl−I Angular Rates for MRP Sliding Mode Controller . . . . . . . . . . . . . 126

6.19 Numerical Simulation Angular Rates for MRP Sliding Mode Controller . . . . . . 126

6.20 Whorl−I MRP Vector for MRP Sliding Mode Controller . . . . . . . . . . . . . . 127

6.21 Numerical Simulation MRP Vector for MRP Sliding Mode Controller . . . . . . . 127

6.22 Whorl−I Control Torque for MRP Sliding Mode Controller . . . . . . . . . . . . . 128

6.23 Numerical Simulation Control Torque for MRP Sliding Mode Controller . . . . . . 128

6.24 Whorl−I Sliding Variable for MRP Sliding Mode Controller . . . . . . . . . . . . 129

6.25 Whorl−I Reaction Wheel Speed for MRP Sliding Mode Controller . . . . . . . . . 130

6.26 Numerical Simulation Reaction Wheel Speed for MRP Sliding Mode Controller . . 130

6.27 Whorl−I Angular Rates for MRP Sliding Mode Controller . . . . . . . . . . . . . 131


6.29 Whorl−I Angular Rate Errors for MRP Sliding Mode Controller . . . . . . . . . . 132

6.30 Numerical Simulation Angular Rate Errors for MRP Sliding Mode Controller . . . 133





6.35 Whorl−I Control Torque for MRP Sliding Mode Controller . . . . . . . . . . . . . 135


xv

List of Figures

6.37 Whorl−I Sliding Variable for MRP Sliding Mode Controller . . . . . . . . . . . . 136

6.38 Whorl−I Reaction Wheel Speed for MRP Sliding Mode Controller . . . . . . . . . 137


6.40 Whorl−II Angular Rates for MRP Sliding Mode Controller . . . . . . . . . . . . . 139


6.42 Whorl−II MRP Vector for MRP Sliding Mode Controller . . . . . . . . . . . . . . 140


6.44 Whorl−II Control Torque for MRP Sliding Mode Controller . . . . . . . . . . . . 141


6.46 Whorl−II Sliding Variable for MRP Sliding Mode Controller . . . . . . . . . . . . 142

6.47 Whorl−II Reaction Wheel Speed for MRP Sliding Mode Controller . . . . . . . . 143


6.49 Whorl−II Angular Rates for MRP Sliding Mode Controller . . . . . . . . . . . . . 145


6.51 Whorl−II MRP Vector for MRP Sliding Mode Controller . . . . . . . . . . . . . . 146


6.53 Whorl−II Control Torque for MRP Sliding Mode Controller . . . . . . . . . . . . 147


6.55 Whorl−II Sliding Variable for MRP Sliding Mode Controller . . . . . . . . . . . . 148

6.56 Whorl−II Reaction Wheel Speed for MRP Sliding Mode Controller . . . . . . . . 148


7.1 Orbit Plane Coordinate Frame Illustration . . . . . . . . . . . . . . . . . . . . . . 152

7.2 Spacecraft Formation Flying Simulation Scheme . . . . . . . . . . . . . . . . . . 156

7.3 Mean Semi-Major Axis Time History for Chief Spacecraft . . . . . . . . . . . . . 158

7.4 Mean Eccentricity Time History for Chief Spacecraft . . . . . . . . . . . . . . . . 158

7.5 Mean Inclination Time History for Chief Spacecraft . . . . . . . . . . . . . . . . 159

7.6 Deputy Spacecraft GPS Receiver Parameters (PDOP, HDOP, VDOP, and TDOP) . 159

xvi

List of Figures


7.8 Mean Semi-Major Axis Time History for Deputy Spacecraft . . . . . . . . . . . . 160

7.9 Mean Eccentricity Time History for Deputy Spacecraft . . . . . . . . . . . . . . . 161

7.10 Mean Inclination Time History for Deputy Spacecraft . . . . . . . . . . . . . . . 161

7.11 Mean Semi-Major Axis Error Time History for Deputy Spacecraft . . . . . . . . . 162

7.12 Mean Eccentricity Error Time History for Deputy Spacecraft . . . . . . . . . . . . 162

7.13 Mean Inclination Error Time History for Deputy Spacecraft . . . . . . . . . . . . 163

7.14 Deputy Spacecraft Control Acceleration History . . . . . . . . . . . . . . . . . . . 163

8.1 Orbit Plane Coordinate Frame Illustration . . . . . . . . . . . . . . . . . . . . . . 166

8.2 Spacecraft Orbit Control Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

8.3 Spacecraft GPS Receiver Parameters (PDOP, HDOP, VDOP, and TDOP) . . . . . 176


8.5 Mean Semi-Major Axis Time History for Spacecraft . . . . . . . . . . . . . . . . 177

8.6 Mean Eccentricity Time History for Spacecraft . . . . . . . . . . . . . . . . . . . 178

8.7 Mean Inclination Time History for Spacecraft . . . . . . . . . . . . . . . . . . . . 178

8.8 Mean Semi-Major Axis Error Time History for Spacecraft . . . . . . . . . . . . . 179

8.9 Mean Eccentricity Error Time History for Spacecraft . . . . . . . . . . . . . . . . 179

8.10 Mean Inclination Error Time History for Spacecraft . . . . . . . . . . . . . . . . . 180

8.11 Spacecraft Control Acceleration History . . . . . . . . . . . . . . . . . . . . . . . 180

A.1 Whorl−I Reaction Wheel 3 Steady State Wheel Speed Time Behavior . . . . . . . 193

A.2 Whorl−I Reaction Wheel 3 Steady State Wheel Speed Time Behavior Curve Fit . . 194

A.3 Steady State Reaction Wheel Speed verses Power Commanded to Reaction Wheel

Motor for a Range from -1000 to 1000 motor units for Whorl−I Reaction Wheel 3 195

A.4 Steady State Reaction Wheel Speed verses Power Commanded to Reaction Wheel

Motor for a Range from -100 to 100 motor units for Whorl−I Reaction Wheel 3 . 195

xvii

List of Figures

A.5 Power Commanded to Reaction Wheel Motor verses Scaled Steady State Reaction

Wheel Speed Curve Fit for Whorl−I Reaction Wheel 3 . . . . . . . . . . . . . . . 196

A.6 Power Commanded to Reaction Wheel Motor verses Scaled Steady State Reaction

Wheel Speed Curve Fit for Whorl−I Reaction Wheel 3 . . . . . . . . . . . . . . . 196

xviii

List of Tables

5.1 Angular Rate Controller Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 58




5.5 MRP Controller Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79



6.1 Sliding Mode Control Numerical Simulation Parameters with only Equivalent Con-

trol with no Parameter Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.2 Sliding Mode Control Numerical Simulation Parameters with only Equivalent Con-

trol with Parameter Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.3 Sliding Mode Control Numerical Simulation Parameters . . . . . . . . . . . . . . 120

6.4 Sliding Mode Control Simulation Parameters for Whorl−I . . . . . . . . . . . . . 125

6.5 Sliding Mode Control Simulation Parameters for Whorl−I . . . . . . . . . . . . . 131

6.6 Sliding Mode Control Simulation Parameters for Whorl−II . . . . . . . . . . . . . 139

6.7 Sliding Mode Control Simulation Parameters for Whorl−II . . . . . . . . . . . . . 144

7.1 Chief and Deputy Spacecraft Initial Conditions . . . . . . . . . . . . . . . . . . . 157

xix

List of Tables

8.1 Spacecraft Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 175

xx

List of Symbols

Lower Case Greek

η Positive Constant Vector rad/s2

λ Longitude rad or deg

ω Argument of Perigee rad or deg

ω Spacecraft Angular Rate rad/s

ωn Natural Frequency rad/s

φ Boundary Layer Thickness rad/s

φ Latitude rad or deg

σ Modified Rodrigues Parameters

ψ Heading rad or deg

θrw Maximum Reaction Wheel Fractional Angular Error of Torque

θGMST Greenwich Mean Standard Time Angle rad or deg

θ Argument of Ascending Node rad or deg

ζ Damping Ratio

xxi

List of Symbols

Upper Case Greek

Γ Tracking Control Term rad/s2

Λ MRP Error Gain Matrix rad/s

ΛI MRP Error Integral Gain Matrix rad/s2

Ω Longitude of Ascending Node rad or deg

Φ Principal Rotation Angle rad or deg

Upper Case Roman

B Control Matrix Estimate 1/(kg-m2)

B Control Matrix 1/(kg-m2)

B(σ) MRP Kinematic Expression Matrix

DB Control Matrix Bound

DD Reaction Wheel Angular Momentum Bounding Matrix

DI Inertia Bounding Matrix

Ds Net Reaction Wheel Momentum Estimate kg-m2/s

Ds Net Reaction Wheel Momentum kg-m2/s

Dw Reaction Wheel Spin Axis Orientation Bounding Matrix

F Drift Function Bound Vector rad/s2

H Angular Momentum kg-m2/s

11 Identity Matrix

IW Reaction Wheel Inertia Matrix kg-m2

xxii

List of Symbols

I Inertia Matrix kg-m2

K Gain Matrix rad/s2

L Torque N-m

Lmax Torque Bound N-m

M Mean Anomaly

Ω Reaction Wheel Speed rad/s

P Gain Matrix kg-m2/s

R Rotation Matrix

T Decay Time s

V Lyapunov Function

Ws Reaction Wheel Spin Axis Matrix

Lower Case Roman

a Orbit Semi-Major Axis km

b Semi-Minor Axis km

drw Maximum Reaction Wheel Fractional Error of Torque

e Principal Rotation Axis

e Orbit Eccentricity

f Drift Function Estimate rad/s2

f Drift Function rad/s2

f True Anomaly rad or deg

xxiii

List of Symbols

f Flattening Ratio of Elliptical Earth Model

h Altitude Above Earth Surface km

hs Reaction Wheel Spin Axis Angular Momentum kg-m2/s

i Orbit Inclination rad or deg

p Semi-Latus Rectum km

q Quaternion

rECEF Position Vector in ECEF Coordinates km

rD Chief Spacecraft Orbit Radius km

rD Deputy Spacecraft Orbit Radius km

s Sliding Variable Vector rad/s

t Time s

u Torque Control Vector N-m

ueq Equivalent Control Vector N-m

urob Robust Control Vector N-m

vg Ground Velocity m/s

vv Vertical Velocity m/s

x State Vector

z Integral Term rad/s

xxiv

1 Introduction

Spacecraft formation flying has potential for missions involving coordinated pointing, interferome-

try, stereographic imaging, and synthetic apertures. Many of these scientific missions require precise

relative position and attitude knowledge. A spacecraft formation is a distributed system comprised

of multiple spacecraft. Each spacecraft has an independent attitude and orbit control system that

controls the attitude and orbital position of each spacecraft and cooperates with the other spacecraft

to control the relative attitudes and positions of the spacecraft within the formation. The control

decisions for each spacecraft must be coordinated to ensure the stability and convergence of the

global system.

1.1 Distributed Space Systems

There are many strategies to control the attitude and orbital position of a single spacecraft. The

control effort required depends on the specific mission objectives. In the past few years much

research has been conducted with regards to controlling the relative position between spacecraft

using classical orbital elements (COE).1,2,3,4,5 Recently, spacecraft formation flying research is

directed toward the relative attitude and orbit control of spacecraft within the formation. Much of

the relative attitude and orbit control research includes development of estimation techniques.6,7,8,9

Spacecraft formation flying is an attractive alternative to the traditional single satellite mission

architecture. Multiple spacecraft provide a higher level of redundancy in the case of a failure as

compared with single spacecraft due to replacing a smaller, cheaper spacecraft or the ability to re-

1

1.1 Distributed Space Systems

configure the formation to achieve mission objectives. Typically the individual spacecraft within

the formation will be smaller in comparison with a single spacecraft which directly leads to cheaper

launch costs. The ability to re-configure the formation allows components to be upgraded within

the formation by replacing specific spacecraft with new spacecraft.

There is currently an initiative referred to as “Responsive Space”.∗ The initiative is in response

to the scenario where a critical military or scientific spacecraft unexpectedly cannot satisfy the

critical mission objectives. Spacecraft component failures from solar flares, meteoroids, or hostile

actions are some situations in which a spacecraft might become nonoperational. The Responsive

Space general concept is small, low-cost spacecraft can be stored on the ground and then can be

rapidly launched. The objective is to reduce cost per satellite and to rapidly add space assets for a

variety of mission objectives.

The Responsive Space initiative is an ideal concept to fulfill needs of formation flying, where

the desire for multiple smaller, cheaper, and rapidly deployed space vehicles satisfy the mission

objectives of large expensive single spacecraft mission. The Responsive Space initiative takes ad-

vantage of mass production techniques to reduce the overall cost per vehicle. The reduction in cost

for individual spacecraft will increase the interest in using a formation of spacecraft to achieve the

objectives typically achieved by a large, expensive, single spacecraft.

The use of spacecraft formation flying to achieve mission objectives is limited by the high visi-

bility of space missions, the high cost associated with space missions, and the higher risk associated

with the implementation of new concepts. Advancements in formation flying technology will re-

duce the higher-risk associated with formation flying as compared with single spacecraft missions.

Simulations provide valuable knowledge for validating concepts and missions. Many of these sim-

ulations are software-based architectures. At Virginia Polytechnic Institute and Sate University we

have developed a hardware-in-the-loop Distributed Spacecraft Attitude Control System Simulator

(DSACSS) to provide a more realistic demonstration of expected performance of distributed space

systems as compared with software-based architectures.

∗Information about Responsive Space may be obtained at www.responsivespace.com

2

1.2 Dissertation Objective

1.2 Dissertation Objective

The DSACSS is developed to experimentally demonstrate formation flying control strategies to

provide a more realistic demonstration of expected performance as compared with numerical simu-

lations.

The DSACSS can investigate linear and nonlinear control strategies in real-time for multiple

spacecraft in a coordinated activity such as shown in Figure 1.1. The two spacecraft in Figure 1.1

are an example of a distributed space concept. The two spacecraft are in low Earth orbit (LEO)

and determine the navigation solution using GPS receivers while independently determining their

respective attitude.

Figure 1.1: Distributed Space Systems Concept

3

1.3 Dissertation Overview

In the distributed space systems concept, Spacecraft 1 has a larger, but less refined sensor

footprint than Spacecraft 2. In this example, Spacecraft 1 is inspecting a large area on the surface of

the Earth and determines an area of interest where further investigation is warranted and additional

sensors are needed. Spacecraft 1 transmits the location of the area of interest to Spacecraft 2.

Mission requirements are such that the relative distance (r12) between Spacecraft 1 and 2 must

be maintained within a specific tolerance throughout the mission. Spacecraft 2 has a small sensor

footprint and must precisely control the pointing of the sensor unit to adequately investigate the area

of interest.

Precise relative orbit and attitude determination and control are required to successfully achieve

mission objectives. The DSACSS has been developed for the previously described scenario, which

can be used to investigate a variety of control schemes. GPS receivers are implemented into the

simulator for navigation control. Magnetometers, accelerometers, and rate gyros are implemented

for attitude determination and reaction wheels and thrusters are implemented for real-time control.

The capability to investigate distributed space system control strategies in a real-time hardware-in-

the-loop simulation is extremely unique and immensely valuable in evaluating the control strategies.


Chapter 2 presents a literature review that includes a discussion on hardware-in-the-loop spacecraft

simulators, nonlinear control, spacecraft attitude control, and spacecraft orbit control. Chapter 3

presents the Distributed Spacecraft Attitude Control System Simulator (DSACSS) and the closed-

loop, hardware-in-the-loop GPS simulator that has been developed to investigate distributed space

system control strategies. Chapter 4 presents the spacecraft kinematic expressions for the quater-

nion and modified Rodrigues parameter attitude representations and derives the spacecraft rigid

body dynamics with momentum exchange devices. Chapter 5 presents the development of nonlin-

ear Lyapunov based spacecraft attitude control laws. The development and demonstration on the

hardware attitude simulators of an asymptotically stable angular rate controller and modified Ro-

drigues parameter attitude controller are shown. Chapter 6 presents the development of a nonlinear

4


sliding mode spacecraft attitude controller that tracks a reference attitude using reaction wheels as

the actuator while bounding system uncertainties. Numerical and hardware-in-the-loop simulations

demonstrate the capabilities of the controller. Chapter 7 presents the results of a GPS hardware-

in-the-loop formation flying simulation using a nonlinear Lyapunov based classical orbital element

feedback controller. Chapter 8 develops a classical orbital element feedback control law based on

the sliding mode methodology while bounding system uncertainties and demonstrates the controller

using the GPS simulator. Chapter 9 summarizes this text and provides suggestions for future work.

5

2 Literature Review

In this chapter we summarize the relevant work associated with distributed space systems and non-

linear attitude and orbit control strategies. A review of air bearing platforms for space applications

is presented. Discussions on nonlinear control, attitude control, and orbit control are presented.

2.1 Spacecraft Simulators

Air bearing platforms have been used since the 1950’s to evaluate hardware components and soft-

ware for space applications. Air bearing platforms typically have a spherical ball floating on air,

which provides a nearly torque free environment between the interaction of the ball and pedestal.

Therefore spherical air bearing platforms are a common device for attitude dynamics research. Var-

ious configurations of the platforms exist depending on the desired application. The air bearing

platforms only provide rotational motion about a fixed point.

A three-axis air bearing platform was developed at NASA Marshall Space Flight Center, which

was used for the NIMBUS second generation weather satellite and the proposed Orbiting Astro-

nomical Observatory. NASA Goddard Space Flight Center developed a spherical air bearing to

investigate energy dissipation, which had plagued missions such as Explorer-1, Applications Tech-

nology Satellite-5, and TACSAT-1.10

In 1975 Stanford University was the first university to develop and use for research an air

bearing platform. Spacecraft contractors such as Lockheed Martin, Boeing, and TRW have small to

large scale air bearing platforms, but very little publicly available information. In 1995 the Naval

6

2.2 Nonlinear Control

Postgraduate School developed an air bearing three axis attitude dynamics and control simulator

to demonstrate the dynamics and control of a twin mirror bifocal relay spacecraft that uses laser

beams. Another air bearing platform is used for evaluating flight hardware-in-the-loop at the Naval

Postgraduate School.10

Utah State University developed an air bearing spacecraft simulator to test the attitude deter-

mination and control system for the Space Dynamics Laboratory’s Skipper spacecraft and was able

to identify a number of integration problems between various spacecraft subsystems. Georgia Tech

has several air bearing platforms for undergraduate and graduate education and nonlinear control

investigations.10

In the early 1990s the Air Force Research Laboratory and Naval Research Laboratory devel-

oped large spherical air bearing platforms to investigate the interaction between control input and

structure. The systems have been used for space based laser research and robust nonlinear control

and model reduction techniques for coupled attitude control and energy storage.10

NASA Goddard Space Flight Center has developed a Formation Flying Testbed (FFTB) to

demonstrate relative navigation technologies for spacecraft.11 The testbed is comprised of oper-

ational GPS receivers and Spirent Federal Systems GPS simulators. The closed-loop, hardware

simulation allows for testing of navigation and control software using real hardware.12 A real-time

autonomous formation flying software has been tested using the FFTB to estimate the absolute and

relative states of the spacecraft within the formation to a control precision in the meter range.5 A

sample low earth orbit demonstration of Precision Formation Flying (PFF) to demonstrate various

guidance navigation and control strategies has been investigated using the FFTB.1

2.2 Nonlinear Control

Nonlinear control deals with the analysis and design of control strategies for nonlinear systems

such as spacecraft attitude and orbit control. The objective of the nonlinear controller is to design a

controller that satisfies the desired response.

Linear control has been successfully implemented for a variety of applications such as missile,

7

2.3 Attitude Control

aircraft, and spacecraft GN&C (guidance, navigation, and control). A system, such as a “collection

of interacting elements which there are cause-and-effect relationships between the variables,”13 is

typically nonlinear in nature and is described through a mathematical model. The nonlinear model

is linearized to obtain the linear model from which the linear controller is derived. Nonlinear con-

trol is preferred over linear control for improvement of existing control systems, analysis of hard

nonlinearities, dealing with model uncertainty, and design simplicity.14

Nonlinear control may be an improvement to existing linear control systems, because linear

control is typically valid for a small operation range. As the range of operation increases the perfor-

mance, accuracy and precision of the linear controller decreases for a highly nonlinear system due

to the contribution of the nonlinear terms. A nonlinear controller can be designed to directly handle

a large range of operation.14

Hard nonlinearities is a term used to describe nonlinearities which are discontinuous in nature

with no linear approximation such as Coulomb friction, saturation, and dead-zones.14 The hard

nonlinearities must be evaluated to prevent adverse control behavior such as system instabilities.14

Linear control assumes the system parameters are well known, which for a real system the

parameters may not be well known. Parametric uncertainties and un-modeled dynamics can have

an adverse effect on controlling a nonlinear system. Nonlinear control provides two control technics

to account for system uncertainties. The two effects can be dealt with through robust or adaptive

control. Robust control is structured such that the controller has a nominal term and additional

terms are added to account for the model uncertainty. Adaptive control updates the model during

operation to account for model uncertainty.14

2.3 Attitude Control

Parametric uncertainties and un-modeled dynamics can have an adverse effect on controlling the

attitude of a nonlinear system. A common robust control technique is the sliding control method-

ology also referred to in the literature as variable structure control. Sliding control ensures stability

and consistent performance with system uncertainties.14 Sliding mode control is a popular closed

8

2.4 Orbit Control

loop technique for robust spacecraft attitude control.15,16,17,18,19,20,21,22,23,24

Reference 19 derives a Cayley-Rodrigues attitude sliding mode controller that is asymptotically

stable. A decoupled linearization of the attitude parameters was used in the control law. The con-

troller demonstrated rest to rest maneuver using jet thrusters for the attitude control. The controller

successfully drove the errors to zero.

Reference 15 derives a general attitude representation sliding mode controller that uses n re-

action wheels as the attitude actuator for a spacecraft. The controller accounts for parameter un-

certainties for the reaction wheels and un-modeled dynamics. The general attitude representation

is the relative rotation between the desired and actual attitude. An issue that arises by using the

attitude representation outlined in the paper is the approximation that θerror = ωerror. The approx-

imation breaks down at large attitude errors. The numerical simulation results show the controller

successfully satisfies the objective of the scenario, which was to track a specified ground station.

Reference 16 derives a Modified Rodrigues Parameter (MRP) sliding mode controller for the

regulation and tracking problem of a spacecraft. The controller is not designed for a specific atti-

tude actuator such as thrusters or reaction wheels. Therefore the uncertainties associated with the

actuators are not accounted for in the controller. The paper omits the discussion for determining

the robust control term gain. Only the numerical simulation results for the regulation problem is

presented, which demonstrates the MRP and angular rate errors converging to zero.

2.4 Orbit Control

Spacecraft formation flying involves controlling the relative position of multiple spacecraft. Current

interests in spacecraft formations include, but are not limited to clusters used for sparse aperture

radar dish in space,25 Earth-based remote sensing, and upper atmospheric sciences.2

Propulsion systems apply a force to the spacecraft, which controls the orbit. A primary concern

with spacecraft formation flying is fuel consumption. Controller design is crucial for the purpose

of conserving fuel. Obviously, a good spacecraft controller will use as little fuel as possible to

accomplish the maneuver. There are a variety of linear control strategies such as linear quadratic

9

2.4 Orbit Control

regulator (LQR), quantitative feedback control (QFC), and H∞ approaches to control the relative

positions of multiple spacecraft.

Figure 2.1 is a diagram of two spacecraft in formation where x, y, and z, are the distances

between the deputy spacecraft and chief spacecraft. The chief orbit radius is specified as rC and the

deputy orbit radius is rD. The true anomaly difference is specified as ∆f .

Figure 2.1: Two Spacecraft in Formation

The nonlinear equations of motion for controlling the relative position of the deputy spacecraft

10

2.4 Orbit Control

based on the chief spacecraft for no environmental perturbations acting on either spacecraft are25,26

x− 2f(y − y rcrc

)− xf2 − µ

r2c

= − µ

r3D

(rc + x) + uDx (2.1)

y + 2f(x− xrcrc

)− yf2 = − µ

r3D

y + uDy (2.2)

z = − µ

r3D

z + uDz (2.3)

where µ is the gravitational constant and uD is the control acceleration for the deputy spacecraft.

Equation (2.3) is simplified by assuming the relative distances between the spacecraft are small

compared to rC and higher order terms are neglected results in the following general form25

x− x(θ2 + 2

µ

r3C

)− yθ − 2yθ = uDx (2.4)

y + xθ + 2xθ − y(θ2 − µ

r3C

)= uDy (2.5)

z +µ

r3C

z = uDz (2.6)

where θ is ω + f .

The Clohessy-Wiltshire (CW) equations,27 which are often referred to as the Hill equations, are

obtained from Equation 2.6 by assuming the chief satellite is in a circular orbit which results in25,28

x− 2ny − 3n2x = uDx (2.7)

y + 2nx = uDy (2.8)

z + n2z = uDz (2.9)

where n is the mean motion of the chief spacecraft.

The CW equations are used to investigate rendezvous and docking and relative position be-

tween two spacecraft using linear control strategies.27,29,30 The CW equations do not take into

account orbital perturbations, large separation distances ( larger than 1 km), and noncircular orbits (

eccentricity greater than 0.00001).31 Reference 32 includes a modified version of the CW equations,

11

2.5 Summary

which allows large separation distances and an elliptical chief orbit. Again, linear control strategies

are implemented.

In the past few years much research has been conducted with regards to controlling the relative

position between spacecraft using nonlinear control strategies. A popular nonlinear orbit control

strategy is a Lyapunov based classical orbital element (COE) feedback controller.1,2,3,4,5,25 The

COE vector is comprised of semi-major axis (a), eccentricity (e), inclination (i), longitude of

ascending node (Ω), argument of perigee (ω), and mean anomaly (M). The controllers control the

relative orbital parameters between the space vehicles.

2.5 Summary

This chapter summarizes the relevant work associated with hardware spacecraft simulators and

nonlinear attitude and orbit control strategies. Many companies and government agencies such

as NASA and AFRL have used air bearing platforms to evaluate hardware and software compo-

nents for space applications. A common nonlinear robust control technique is the sliding control

methodology, which ensures stability and consistent performance with system uncertainties. Sliding

mode control is a popular closed loop technique for robust spacecraft attitude control. Orbit control

for multiple spacecraft involves linear control strategies based on the Clohessy-Wiltshire equations.

Nonlinear orbit control strategies for multiple spacecraft have recently concentrated on relative clas-

sical orbital element feedback control. The controllers control the relative position between multiple

spacecraft.

We have developed a high fidelity solution for Earth-based real-time hardware-in-the-loop sim-

ulations of space vehicles to evaluate nonlinear control strategies for distributed space systems.

Chapter 3 discusses the development and operation of the distributed spacecraft attitude and orbit

control simulators.

12

3 Distributed Spacecraft Attitude Control

System Simulator (DSACSS)

The Distributed Spacecraft Attitude Control System Simulator (DSACSS) facility housed in the

Space Systems Simulation Laboratory (SSSL)∗ provides a high fidelity solution for Earth-based

real-time hardware-in-the-loop simulations of space vehicles. DSACSS allows individual and com-

ponent level development and testing of hardware and software interfaces. DSACSS is comprised

of two hardware simulators and one software spacecraft simulator mounted on separate platforms

that allow the ability to demonstrate decentralized control algorithms. The two hardware simulators

are mounted on individual spherical air bearing platforms, which allow only for rotational motion.

The two hardware simulators have different configurations and consequently a different amounts of

rotational freedom. The third simulator is comprised of software running on a flight computer. The

operating system and DSACSS code for each simulator are maintained as open source.33,34,10,35

The Global Positioning System (GPS) Laboratory at Virginia Tech provides a real-time hardware-

in-the-loop navigation solution by simulating GPS RF signals. The GPS simulator has been inte-

grated into the DSACSS facility to simulate the trajectory of low Earth orbit (LEO) spacecraft.36,37

In the following sections, we discuss the DSACSS system control capabilities, which include

the hardware and software, the control capabilities of the system, and the visualization tools inte-

grated into the DSACSS system to assist with spacecraft related dynamics and control research.

∗Information about the SSSL may be obtained at www.sssl.aoe.vt.edu

13

3.1 Attitude Hardware-in-the-loop Simulators


As previously mentioned DSACSS is comprised of two attitude spacecraft hardware simulators

mounted on individual spherical balls floating on air, which allow only for rotational motion. The

air bearing platforms provide a nearly torque-free environment for simulating attitude control. The

simulators are referred to as Whorl−I and Whorl−II.†

Whorl−I is shown in Figure 3.1, which is a “tabletop” style configuration and provides one

complete axis of freedom. The yaw axis has rotational freedom of 360 while the roll and pitch

axes have± 6.5. The axes for Whorl−I are defined in Figure 3.1. Whorl−I has a 48 inch (122 cm)

by 48 inch footprint with a height of 88 inches (224 cm). The simulator has a maximum payload of

300 pounds (136 kg).33

Whorl−I and Whorl−II have detailed computer aided drawings (CAD), which provide an esti-

mate of the mass and inertia matrix for each simulator. The mass estimate is 88.5 kg and the inertia

matrix estimate is

I =

8.8 0.7 −0.2

0.7 7.7 −0.1

−0.2 −0.1 13.0

kg-m2 (3.1)

for Whorl−I where (·) indicates parameter estimate.

Whorl−II is shown in Figure 3.2, which is a “dumbbell” style configuration and provides two

complete axes of freedom. The yaw and roll axes have rotational freedom of 360 while the pitch

axis has ± 30. The axes for Whorl−II are defined in Figure 3.2. Whorl−II has a 70 inch (179 cm)

by 70 inch footprint with a height of 70 inches (179 cm). The simulator has a maximum payload of

300 lbs (136 kg).33

†The name Whorl comes from The Book of the Long Sun by Gene Wolfe. In the science fiction novel Whorl is the nameof a starship that has been traveling through space for hundreds of years.

14


Figure 3.1: “Tabletop” Style Air Bearing Spacecraft Simulator (Whorl−I)

The mass estimate is 249.6 lbs (113.2 kg) and the inertia matrix estimate is

I =

3.8 1.4 −1.2

1.4 26.1 0.2

−1.2 0.2 26.1

kg-m2 (3.2)

for Whorl−II.

Whorl−I and Whorl−II have similar subsystems as flight ready spacecraft (e.g. command and

data handling; communications; attitude determination and control; power; payload; and guidance

and navigation). The system block diagrams for Whorl−I and Whorl−II are shown in Figures 3.3

and 3.4.

The system block diagrams for Whorl−I and Whorl−II are identical except Whorl−I has a

control moment gyro (CMG). As seen can be seen in Figures 3.1 and 3.2 the various devices for

attitude determination and control are at different locations due to the configuration of each attitude

15


Figure 3.2: “Dumbbell” Style Air Bearing Spacecraft Simulator (Whorl−II)

simulator. Each simulator has attitude determination sensors, attitude control devices, and a flight

computer. The power required for each device is also indicated in Figures 3.3 and 3.4. The follow-

ing subsections discuss the power bus, flight computer, attitude determination, and attitude control

devices for each hardware simulator.

3.1.1 Power Bus

Whorl−I and Whorl−II are each powered by eight 6 volt lead acid batteries divided into 4 battery

banks. Each bank provides 12 volts and 12 amps of power by having two batteries in series. Two

battery banks are wired in parallel resulting in 12 volts and 24 amps and then wired in series to

provide 24 volts and 24 amps of power to the simulator. The batteries are charged using two 12 volt

chargers.

A Mean Well SD-150 switching power supply unit provides 12 and 24 volts of power to the

devices on the simulator. Figures 3.3 and 3.4 indicate the power each device receives from the

power supply unit. The flight computer, reaction wheels, and the CMG on Whorl−I receives 24

volts. The mult-axis inertial sensing system, magnetometer, thrusters, and relay board receive 12

volts of power. All the devices have in-line fuses and a common ground point at the power supply

16


Figure 3.3: Whorl−I System Block Diagram

unit.

A power control scheme has been implemented on Whorl−I to allow the flight computer to

control independently the power to each device. The flight computer turns on and off the devices

through a digital acquisition (DAQ) card by opening or closing a relay switch located on the relay

board. The power control scheme provides the user with the ability to turn off devices that are not

needed for the simulation. The power control scheme is also an added safety measure as it allows

the flight computer to turn off devices (including the flight computer) due to any unexpected event

during a simulation.

17


Figure 3.4: Whorl−II System Block Diagram

3.1.2 Flight Computer and Communications

Each simulator is controlled by its own flight computer as shown in Figures 3.3 and 3.4. Each flight

computer is a PC/104 stack comprised of a 733 MHz processor, 512 MB of RAM, and an 8 GB flash

drive. Each flight computer has an additional 30 GB of space through a private wireless network

connection to a shared drive for the purpose of developing and evaluating code. The operating

system on the PC/104 computers is Scientific Linux 4 (SL4).‡

Each PC104 stack has a HE104 vehicle power supply unit that receives 24 volts from the sim-

ulator batteries. Analog and digital devices interface with the PC/104 through a Diamond Systems

‡Scientific Linux is an open source operating system that is a recompiled version of Enterprise Linux. The primarycontributors are Fermilab and CERN with other labs and universities participating.

18


DMM-32 analog and digital (A/D) board with 32 analog inputs (16 bit resolution), 4 analog outputs

(12 bit resolution), and 32 digital I/O lines. An eight port XDB-9 RS-232 USB serial adapter data

control box manufactured by Vscom is interfaced with the flight computer through a USB port for

communicating with the reaction wheels, linear actuators, and magnetometers.

The flight computers communicate with each other and with other laboratory resources using

a wireless multi-client bridge/access point. The wireless bridge operates at 2.4 GHz and supports

802.11b wireless standard. The bridge uses 64/128-bit WEP data encryption and requires no drivers.

Configuration is accomplished through a web browser interface and the bridge connects to the com-

puter through a RJ45 port. USB wireless adapters were implemented in the past, but replaced with

the wireless bridges due to the lack of drivers available for Linux.

Figure 3.5 is a diagram showing the DSACSS communication scheme. The Whorls communi-

cate between each other through a wireless router. The spacecraft simulators also communicate with

lab computers such as Severian and Typhon. Severian provides orbit information for each simulator

and is discussed in following sections. Typhon is used for development and testing of the simulation

code. Severian and Typhon both use SL4 as the operating system.

3.1.3 Attitude Determination

Whorl−I and Whorl−II determine their respective attitudes using angular rate gyros, accelerome-

ters, and magnetometers, along with an user choice of attitude determination algorithms.

3.1.3.1 Attitude Sensors

The multi-axis inertial sensing system (BEI MotionPak II) has three-axis rate gyros and accelerome-

ters, which are used for attitude determination. The MotionPak senses up to±75 degrees per second

and maximum accelerations of ±1.5 g in their respective axes. The MotionPak is powered by 12

volts and communicates with the flight computer through the analog channels of the DAQ card.

The magnetometer (Honeywell HMR2300 Smart Digital Magnetometer) measures the mag-

netic field in the 3 sensor body axes. The flight computer communicates with the magnetometer

19


Figure 3.5: DSACSS Communication Schematic

through RS-232 serial data interface in binary mode. The magnetometer has a range of ±2 gauss

with less than 70 µgauss resolution and is powered by 12 volts.

The magnetometer on Whorl−I is mounted on a boom 49 inches (124 cm) above the rotation

point of the simulator to reduce the magnetic interference from the air-bearing platform and devices

on the simulator. The magnetometer on Whorl−II is mounted on a boom 10 inches (25 cm) from the

simulator. The boom length is constrained to 10 inches, because Whorl−II is inside an aluminum

and plexiglass cage for safety reasons. Due to the constraint, the magnetometer on Whorl−II expe-

riences more magnetic interference from the air-bearing platform and devices on the simulator than

20


the magnetometer on Whorl−I.

The magnetic interference on Whorl−II was resolved by calibrating the magnetometer mea-

surements to account for hard and soft iron distortion as outlined in Ref. 38. Hard iron distortions

are due to ferrous materials such as magnets and steel that are mounted on the same rigid platform

as the magnetometer. The magnetic distortion adds a constant component along each magnetically

sensed axis independent of the attitude of the magnetometer. For Whorl−II the hard iron distor-

tion arises from the components and structure of the simulator except for the air-bearing platform.

Soft iron distortions come from ferrous materials not mounted on the same rigid platform as the

magnetometer. The distortion component along each sensed axis is a function of attitude of the

magnetometer. The air-bearing platform for Whorl−II is the main contributor to soft iron distor-

tions.

The calibration technique is based on how hard and soft iron distortions affect the measure-

ments differently. Consider only 2-D motion of the magnetometer, such as would be the case for

determining heading of an aircraft. If the magnetometer is rotated 360, and there are no magnetic

distortions, the plot of the x and y sensor components results in a circle centered at the origin. In the

previous scenario, if only hard iron distortions are present, graphing x and y components results in a

circle with an offset from the origin. Subtracting the offset from every measurement eliminates the

hard iron disturbance. If only soft iron distortions are present, graphing x and y components results

in an ellipse rotated by θ. Reference 38 suggests removing the soft iron disturbance by rotating the

ellipse by −θ, scaling the semi-major axis to change the ellipse to a circle, and then rotating the

measurement by θ. The calibration technique works very well for Whorl−II. The magnetometer

calibration is also implemented on Whorl−I.

3.1.3.2 Attitude Determination Algorithms

Two three-axis attitude determination algorithms have been implemented on Whorl−I and Whorl−II.

One is a deterministic method and the other is an extended Kalman filter (EKF).

The TRIAD algorithm implemented on Whorl−I and Whorl−II is a popular three-axis attitude

21


determination algorithm for spacecraft, because the algorithm is simple. The algorithm uses only

two vector observations.39 For Whorl−I and Whorl−II the gravity and magnetic field vectors are

used.

The algorithm creates two “TRIAD” frames from the two body and inertial frame measure-

ments to determine the attitude matrix.40 Any number of attitude representations may be used once

the attitude matrix is known. The development and implementation of the TRIAD algorithm is

found in Refs. 39 and 40.

The Kalman filter represents an optimal method for attitude estimation when uncertainties are

present.41 The filter is a probability-based algorithm that minimizes the error in the estimate of

the state vector.41,42,43 There are numerous types of Kalman filters being used for spacecraft atti-

tude determination. For the attitude determination on Whorl−I and Whorl−II a discrete extended

Kalman filter presented in Ref. 42 is implemented.

The EKF uses a multiplicative error quaternion in the body frame to replace the four compo-

nents of the quaternion vector with three error components. The quaternion is discussed in Sec-

tion 4.1.1. The filter allows for n number of vector observations. The standard multiplicative

quaternion EKF algorithm is computationally expensive due to inverting a 3n× 3n matrix to solve

for the Kalman gains. A modification of the algorithm to decrease the computational load is im-

plemented based on processing one vector observation at a time, which results in inverting a 3 × 3

matrix n times and is typically referred to as Murrell’s Version.42,44

Figures 3.6 and 3.7 demonstrate the attitude determination system of Whorl−I and Whorl−II

by obtaining measurements from the MotionPak and magnetometer and using the EKF to estimate

the quaternion. Figures 3.6 and 3.8 show the angular rates and Figures 3.7 and 3.9 show the quater-

nion during an attitude control simulation, which is discussed in later chapters.

3.1.4 Attitude Control Devices

The attitudes of Whorl−I and Whorl−II can be controlled using reaction wheels, nitrogen gas (N2)

thrusters, and in the case of Whorl−I, a control moment gyro (CMG). There are three linear actua-

22


Figure 3.6: Whorl−I Angular Rates from Extended Kalman Filter During Control Simulation

Figure 3.7: Whorl−I Quaternion Vector from Extended Kalman Filter During Control Sim-ulation

tors, one along each of three mutually orthogonal axes. The linear actuators are used to change the

center-of-mass of each simulator to be co-located with the center-of-rotation.

Each simulator has a total of three custom reaction wheels. A reaction wheel is mounted on

each axis (x, y, & z also referred to as 1, 2, & 3). Each reaction wheel assembly is comprised of

an aluminum hub and steel rim (0.075 kg·m2) attached to a SM3430 Smart Motor. The motor has

23


Figure 3.8: Whorl−II Angular Rates from Extended Kalman Filter During Simulation

Figure 3.9: Whorl−II Quaternion Vector from Extended Kalman Filter During Simulation

a built-in logic computer with a 4000 count encoder. Figure 3.10 is illustrates the reaction wheel

assembly with the axes identified. The reaction wheels operate on 24 volts and communicate with

the flight computer through RS-232 ports.

Attitude control using the reaction wheels requires knowledge about the physical parameters of

the wheels such as the spin axis orientation and inertia of each wheel. The reaction wheel spin axis

24


orientation matrix estimate for Whorl−I is

Ws =

1 0 0

0 1 0

0 0 1

(3.3)

where (·) indicates a parameter estimate. The reaction wheel spin axis orientation matrix for

Whorl−II is

Ws =

−1 0 0

0 −1 0

0 0 1

(3.4)

The reaction wheel spin axis inertia estimate for each wheel is

IWsi= 0.075 kg-m2 i = 1, 2, 3 (3.5)

An attitude control law determines the necessary torque about each body axis to satisfy the

desired attitude objective. The reaction wheel controller receives the attitude control vector (u)

and determines the necessary wheel speed for each reaction wheel to produce the desired control

torque. A low-level reaction wheel controller has been developed and implemented on Whorl−I

and Whorl−II and is presented in Appendix A.

The CMG assembly consists of a SM3430 Smart Motor, wheel, and a linear actuator assembly

that allows a maximum gimbal angle of ±45. The CMG operates on 24 volts and communicates

with the flight computer through RS-232 ports.

The compressed-air thrusters provide another three-axis control capability. There are 6 thrusters

on Whorl−I and 8 on Whorl−II. The thrusters are supplied by 21 ft3 nitrogen tanks and are con-

trolled by a manual multistage regulator with a maximum delivery pressure of 50 psi. Each thruster

assembly consists of a 1/4 inch (6.35 mm) round nozzle with a 1/16 inch (1.58 mm) orifice diam-

25


Figure 3.10: Reaction Wheel Axes

eter and an Evolutionary Concepts 654-2207 solenoid. The solenoids are controlled by the flight

computer through the relay board.

As previously mentioned, an attitude control law specifies the necessary torque about each body

axis to satisfy the desired attitude objective. A thruster control module for Whorl−I and Whorl−II

is in development which will take the attitude control vector (u) and determines the firing duration

time for each thruster using a pulse width modulation (PWM) scheme.45,46,47

The primary purpose for the linear actuators is to position the center-of-mass of the simulator

at the center of rotation to eliminate the gravitational torque resulting from the offset. The linear

actuators are manufactured by Servo Systems and have a travel distance of 8 inches. The 5023-

122D stepper motor is controlled by a 1240i controller card running on 12 volts. Both devices are

manufactured by Applied Motion Products. Each actuator assembly can support 30 lbs (13.6 kg).

The flight computer communicates with the controller cards through RS-232 ports.

26

3.2 Software Simulator

3.2 Software Simulator

The software simulator is referred to as Whorl−III and is shown in Figure 3.11. Whorl−III is

identical to the flight computers on Whorl−I and Whorl−II, except hardware is not interfaced with

the computer. Whorl−III provides a third spacecraft attitude simulator. DSACSS has the capability

of simulating distributed attitude control between three spacecraft. Whorl−I and Whorl−II provide

realistic attitude motion while Whorl−III provides numerical attitude results.

Whorl−III also serves as a backup flight computer for Whorl−I and Whorl−II. In the event of

a flight computer failure on Whorl−I or Whorl−II the flight computers can be swapped seamlessly

with minimal down time.

Figure 3.11: Software Simulator (Whorl−III)

3.3 DSACSS Operational Code

The DSACSS Operational (DSACSS-Ops) code to operate the attitude simulators is specifically

designed for the hardware in the DSACSS facility. The architecture of DSACSS-Ops is based on

Object-Oriented-Programming (OOP) in C++. DSACSS-Ops is maintained as open-source soft-

ware.

27


Figure 3.12 is a functional diagram of DSACSS-Ops. The dependences directory contains third

person open-source software. The documentation directory contains html documentation generated

using an open-source program (Doxygen§). The source directory contains the drivers for running

the simulators and conducting simulations. The work directory contains the configuration file and

allows users to develop simulation scenarios.

Figure 3.12: DSACSS-Ops Code Functional Diagram

Within the dependences directory there are 4 libraries. The matrix operations directory contains

a library for of matrix operations such as determination of eigenvalues and eigenvectors. The open§Information on Doxygen can be found at www.doxygen.org

28


source computer vision directory contains libraries and sample code for developing computer vision

algorithms. The tiny XML (extensible markup language) directory contains a small XML parser

library for C++. The configuration file for the simulators and simulations are written in XML.

The Open-Source, Extensible Spacecraft Simulation And Modeling Environment (Open - SES-

SAME) framework is primarily a dynamics modeling and simulation tool.48,49 Open-SESSAME is

integrated into DSACSS-Ops for numerically simulating orbit and attitude control and is used to

write controllers for Whorl−I and Whorl−II. Open-SESSAME contains the following items:

• Math: Operations for matrices, vectors, integrators, interpolators, and conversions.

• Utilities: Classes and tools for handling time.

• Rotation: Coordinate transformations.

• Attitude: Spacecraft attitude dynamic equations.

• Orbit: Spacecraft orbital dynamic equations.

• Dynamics: Propagation algorithms for orbit and attitude.

• Environment: Models of space environment disturbances and central bodies.

• Data Handling: Saving and loading applications and also integrating with external applica-

tions.

• Communications: Utilities for setting up network communications.

The source directory contains the drivers for interfacing with the hardware on the simulators.

The actuators directory provides the interface for the reaction wheels, linear actuators, CMG, and

thrusters. The base directory contains the the simulation objects for developing a simulation sce-

nario. The communications directory contains objects to communicate with remote devices. The

controller directory contains the attitude controllers that run during an hardware-in-the-loop atti-

tude simulation. The filtering directory contains classes for using Kalman filtering. The hardware

29

3.4 GPS Hardware-in-the-loop Simulator for GN&C Simulations

directory contains classes for communicating with the DAQ card. The data logging directory pro-

vides a class for recording data during a simulation. The observer directory contains the attitude

determination algorithms that have been developed. The orbit controller directory contains the or-

bit controllers used for the simulators. The orbit observer directory contains the classes to interface

with the GPS simulator, which is discussed in the next section. The power control directory provides

an interface for turning power on and off for devices on Whorl−I. The sensors directory provides

the interface for the MotionPak, magnetometer, and camera. The utilities directory provides classes

to interface with the programs in the dependences directory and simulation time classes. The STK

(Satellite Tool Kit) visualization directory provides classes for displaying attitude and orbit infor-

mation using STK developed by AGI.


The Global Positioning System (GPS) Laboratory at Virginia Tech provides a real-time hardware-

in-the-loop navigation solution capability for a variety of vehicles (e.g., automotive, ship, aircraft,

and spacecraft) by simulating real GPS RF signals. The GPS simulator can run in an open-loop

configuration for navigation simulations. Navigation, Guidance, and Control (GN&C) simulations

can be performed using the GPS simulator in a closed-loop configuration.

The manufacturer of the GPS simulator provides the open-loop configuration for simulating

vehicle trajectories. The closed-loop configuration is typically user specific; therefore the user must

develop the necessary tools for closed-loop GN&C simulations.

The closed-loop hardware-in-the-loop GPS simulator is integrated into the DSACSS facility to

simulate the GN&C of low Earth orbit (LEO) spacecraft. The closed-loop GPS system runs in real-

time and is comprised of a motion propagator, three RF signal generators, and four high dynamic

GPS receivers that provide the capability of simulating GN&C for spacecraft in low Earth orbit.

30


3.4.1 Spirent GPS Simulator

The hardware-in-the-loop GPS simulator is manufactured by Spirent Federal Systems and is com-

prised of a motion propagator and radio frequency (RF) signal generator.

The RF signal generator is a Spirent Federal Systems GSS6560 multi-channel hardware simu-

lator, which has L1 (1.575 GHz) capability and is shown in Figure 3.13. The GSS6560 simulates

the GPS constellation and provides the user with the ability to control the constellation, errors,

and atmospheric effects. The signal generator has two independent L1 channels with 4 RF outputs

to support differential and attitude simulations. The GSS6560 has “high dynamic” capability to

simulate the trajectory of vehicles in LEO in real-time up to 100 Hz.

The GSS6560 is controlled through usb (universal serial bus) by proprietary software from

Spirent Federal Systems. SimGEN is the graphical user interface (GUI) program that allows a user

to specify a specific scenario and send the information to the GSS6560 to generate the appropri-

ate RF signal to simulate the vehicle trajectory. Vehicle motion modeling for aircraft, cars, and

spacecraft is available within SimGEN and allows for interactive run time control. SimGEN has the

capability of creating and running navigation simulations or allowing the user to pass an external

reference signal through TCP/IP to the RF signal generator to generate the scenario.

The open-loop configuration for the GPS simulator is shown in Figure 3.13. The closed-loop

configuration is discussed in Section 3.4.3.

Figure 3.13: GPS Open-loop Simulator

31


3.4.2 Ashtech G12 HDMA GPS Receiver

An Ashtech G12 HDMA GPS receiver is interfaced with the GSS6560 RF signal generator through a

co-axial BNC socket as shown in Figures 3.13 and 3.14. The Ashtech G12 HDMA GPS receiver has

been integrated into systems such as the Super Sonic Naval Ordinance Research Track (SNORT),

NASA’s X-34 vehicle, and used for sounding rocket missions.50 The G12 provides real-time posi-

tion, velocity, and time measurements by processing signals from the simulated GPS constellation

produced by the GSS6560. The receiver has 12 channels for the L1 band and can be used in a

standalone or differential GPS (DGPS) mode.

The standard G12 receiver has limits placed on altitude and velocity up to equivalent dynamics

that an airliner would experience in standard operation. The HDMA version of the G12 turns off

the limits and provides a navigation solution for vehicles with equivalent dynamics as vehicles in

LEO.

The G12 is able to compute the navigation solution at 20 Hz. A position and velocity filter

provided as an option within the G12 software is used to reduce the noise effects. The smoothing

interval is set to 100 seconds. The velocity measurements are computed using doppler values and

all measurements are provided in the World Geodetic System 1984 (WGS-84) reference frame.

The position query command returns the number of satellites used to compute the position, the

current UTC time, latitude, longitude, altitude, heading, ground speed, vertical velocity, positional

dilution of precision (PDOP), horizontal dilution of precision (HDOP), vertical dilution of precision

(VDOP), and time dilution of precision (TDOP). Dilution of precision values between 1 and 3 are

very good, values between 4 and 5 are good, values around 6 are fair, and values greater than 6 are

poor.

3.4.3 Closed-hardware-in-the-loop Spacecraft GPS Simulator

The objective of integrating the GPS hardware-in-the-loop simulator into DSACSS is to provide

GN&C capability to Whorl−I, Whorl−II, and Whorl−III. The open-loop configuration previously

discussed provides navigation, but lacks the ability for real-time control. A real-time closed-loop

32


control strategy using the GPS simulator has been developed for spacecraft GN&C in LEO and is

shown in Figure 3.14.

The closed-loop strategy requires a spacecraft motion propagator that supplies the true position

and velocity of the spacecraft to the GPS simulator and also allows the state vector to be updated

in a real-time situation. For this application the Global Positioning System Enhanced Navigation

Solution (GEONS) flight software is used. GEONS is a high-accuracy real-time on-board orbit

determination software developed by NASA and has many capabilities from on-board satellite op-

erations to ground testing of space systems.51,52

GEONS was developed for on-board spacecraft GN&C operations. For the closed-loop simula-

tions GEONS is being used strictly as an orbit propagator to provide truth data to the GPS simulator.

GEONS has the capability of including orbital perturbations such as J2, radiation pressure, atmo-

spheric drag, and multi-body motion.51 The GPS solution includes secular (linear increase as a

function of time) and harmonic (repetitive as a function of time) corrections to the ephemeris data.

The ephemeris data is the location of each GPS satellite that the GPS receiver has locked onto.

Examples of terms with secular corrections are longitude of ascending node (Ω), mean motion (n),

and inclination (i). The radius and argument of perigee (ω) have harmonic term corrections.53,54

Several C++ classes provide interfaces to GEONS, XML configuration parsing, and communi-

cation with the GPS simulator. Using GEONS as an orbit propagator provides a realistic spacecraft

trajectory. GEONS is also used to calculate the orbit correction maneuver (OCM) during real-time

simulations.

The closed loop real-time simulated GPS spacecraft GN&C solution is obtained through the

following steps as illustrated in Figure 3.14:

• The numerical orbit propagator computer uses GEONS to calculate position, velocity, ac-

celeration, and jerk for a specific time step and sends the data to SimGEN (GPS simulator

computer) through TCP/IP. The truth state includes orbital perturbations for a more realistic

representation of a spacecraft in LEO.

• The GPS simulator computer sends the required information to the GPS RF signal generator

33


Figure 3.14: GPS Closed-hardware-in-the-loop Simulator

(GSS6560) based on the position, velocity, acceleration, jerk, and time through usb.

• The GPS receiver retrieves the RF signal and determines the navigation solution.

• The GPS interface computer is connected to the G12 receiver through a RS-232 port and

provides the interface for communicating with the G12 receiver over TCP/IP from a remote

source. The interface computer allows any flight computer with TCP/IP access to communi-

cate directly with the GPS receiver.

• The spacecraft flight computer retrieves the navigation solution and determines the OCM

based on the orbit control law being investigated. The OCM is sent to the numerical orbit

propagator computer over TCP/IP where GEONS simulates orbit thrusting of the spacecraft.

The high-level scheme for doing GN&C algorithm development using the GPS simulator is

outlined in Figure 3.15. The closed-loop nonlinear orbit control simulations using the scheme shown

in Figure 3.15 are presented in later chapters.

34


Figure 3.15: High-level GN&C Block Diagram

3.4.4 Coordinate Transformations for GPS Receiver

The Ashtech G12 GPS receiver returns WGS84 coordinates, but many of the controllers require the

coordinates to be Earth-center-inertial (ECI) coordinates or classical orbital elements (COE). The

latitude (φ), longitude (λ), altitude (h), heading(ψ), ground speed (vg), vertical velocity (vv) have to

be transformed into the desired coordinates. Mappings from WGS84 to Earth-centered-Earth-fixed

(ECEF) and ECEF to ECI are provided in the following subsections.

3.4.4.1 WGS84 / ECEF Coordinate Transformation

The mapping from latitude, longitude and altitude to ECEF position (rECEF) is54

rECEF =

(n+ h) cosφ cosλ

(n+ h) cosφ sinλ(n(1− 2f − f2

)+ h)

sinφ

(3.6)

35


where n is defined as

n =a√

1− (2f − f2) sin 2φ(3.7)

Note that a (6278.137 km) is the semi-major axis of the elliptical Earth and f (1/298.257) is the

flattening ratio of the elliptical Earth model. The relationship between flattening ratio and semi-

minor axis (b) is54

b = a(1− f) (3.8)

The mapping from north, east, down velocity (rNED) to ECEF velocity (rECEF) is

rECEF = R321(0,−φ− π/2, λ)T rNED (3.9)

where the rotation matrix (R321(0,−φ− π/2, λ)) is given by

R321(0,−φ− π/2, λ) = R2(−φ− π/2)R3(λ) (3.10)

The base rotation matrices about the 1, 2, and 3 axes are

R1(θ) =

1 0 0

0 cos θ sin θ

0 − sin θ cos θ

(3.11)

R2(θ) =

cos θ 0 − sin θ

0 1 0

sin θ 0 cos θ

(3.12)

36


R3(θ) =

cos θ sin θ 0

− sin θ cos θ 0

0 0 1

(3.13)

To compute WGS84 coordinates from ECEF coordinates requires using closed-form or iterative

solutions. The method implemented in DSACSS is by Bowring.55,56 Longitude is given by

λ = tan−1

(rECEFy

rECEFx

)(3.14)

Latitude is determined by an iterative method using

ρ =√r2

ECEFx+ r2

ECEFy(3.15)

βi = tan−1

((1− f) sinφi−1

cosφi−1

)for i = 1, 2, 3, . . . (3.16)

φi = tan−1

rECEFz + (1−f)(2f−f2)1−2f+f2 a sin3 βi

ρ− (2f − f2)a cos3 βi)

for i = 0, 1, 2, . . . (3.17)

where

β0 = tan−1

(rECEFz

(1− f)ρ

)(3.18)

is used on the initial step. For most applications two iterations should be sufficient.55,56 Once the

37


latitude is determined, the altitude is given by

h = ρ cosφ+ (rECEFz + (2f − f2)n sinφ) sinφ− n (3.19)

The mapping from rECEF to rNED is

rNED = R321(0,−φ− π/2, λ)rECEF (3.20)

The next section discusses the mapping between ECEF and ECI coordinates.

3.4.4.2 ECEF / ECI Coordinate Transformation

The mapping from ECEF position (rECEF) to ECI position (rECI) is

rECI = R3(θGMST)TrECEF (3.21)

where θGMST is defined as

θGMST = θGMST2000 + 86, 400ωEarth(Jday + 0.5) (3.22)

Note that GMST stands for Greenwich Mean Sidereal Time. The reference value for Jan 1, 2000

results in θGMST2000 (1.745 rad). The angular rate of the Earth is (ωEarth) 7.292 × 10−5 rad/s. The

Julian time is specified as Jday.

The mapping from ECEF velocity (rECEF) to ECI velocity (rECI) is

rECI =π

43, 200R3(θGMST)

TrECEF + R3(θGMST)T rECEF (3.23)

38


where the derivative of the rotation about the 3 axis is

R3(θ) =

− sin θ cos θ 0

− cos θ − sin θ 0

0 0 0

θ (3.24)

The mapping from rECI to rECEF is

rECEF = R3(θGMST)rECI (3.25)

The mapping from rECI to rECEF is

rECEF =π

43, 200R3(θGMST)rECI + R3(θGMST)rECI (3.26)

The coordinate transformations are used in the next section to determine the classical orbital

elements from the GPS receiver.

3.4.5 Example of Navigation Determination using GPS Receiver

This experiment is an open-loop simulation using the GPS simulator and Whorl−I. The initial

orbital conditions of Whorl-I are semi-major axis, a = 6978.15 km, eccentricity, e = 0.05, incli-

nation, i = 30, right ascension of ascending node, Ω = 0, argument of periapsis, ω = 45, and

Mean anomaly, M = 0.

The GPS receiver takes several minutes to lock onto at least 4 GPS satellites, which is the

minimum necessary to determine the navigation solution. Complete acquisition for a range of 6 to

10 GPS satellites varies from 5 to 30 minutes when simulating a spacecraft LEO trajectory.

The results of a 130 minute real-time open-loop simulation using Whorl−I are presented in

Figures 3.16 – 3.26. Figures 3.16 and 3.17 show the orbit radius and orbit velocity determined by

the GPS receiver in the ECI reference frame. Figure 3.18 shows the trajectory of the spacecraft

determined from the GPS receiver. The classical orbital elements time histories are shown in Fig-

39


ures 3.19 – 3.24. The GPS receiver parameters are shown in Figure 3.25. The PDOP increases to

approximately 5 in the interval of 75 to 80 minutes. The result of the increase in PDOP can be seen

as an increase in noise for the semi-major axis in Figure 3.19. The number of GPS satellites used to

determine the navigation solution is shown in Figure 3.26.

Figure 3.16: Whorl-I Orbit Radius Determined from the GPS Receiver in the ECI ReferenceFrame

Figure 3.17: Whorl−I Orbit Velocity Determined from the GPS Receiver in the ECI ReferenceFrame

40


Figure 3.18: Whorl−I Orbit Trajectory Determined from the GPS Receiver in the ECI Refer-ence Frame

Figure 3.19: Semi-Major Axis Time History for Whorl−I

41


Figure 3.20: Eccentricity Time History for Whorl−I

Figure 3.21: Inclination Time History for Whorl−I

42


Figure 3.22: Longitude of the Ascending Node Time History for Whorl−I

Figure 3.23: Argument of Perigee Time History for Whorl−I

43

3.5 Summary

Figure 3.24: True Anomaly Time History for Whorl−I

Figure 3.25: Whorl−I GPS Receiver Parameters (PDOP, HDOP, VDOP, and TDOP)

3.5 Summary

This chapter discusses the DSACSS control capabilities, which includes the hardware and software,

the control capabilities of the system, and the visualization tools integrated into the DSACSS system

to assist with spacecraft related dynamics and control research. The attitude hardware-in-the-loop

44

3.5 Summary

Figure 3.26: Number of GPS Satellites Used to Determine Navigation Solution

spacecraft simulators are described in detail along with estimates of their mass properties. The

attitude sensors are described in detail along with the attitude determination algorithms. The attitude

control devices for each simulator are described. The development of the reaction wheel controller

for the simulators is provided in Appendix A. The software, DSACSS-Ops, to run the simulators is

described in detail. The implementation of an attitude controller into DSACSS-Ops is described in

Appendix B. The GPS simulator and receiver are described in detail. A closed-loop hardware-in-

the-loop GPS simulation capability is developed and discussed.

The purpose of the attitude spacecraft hardware simulators is to mimic the expected response of

a spacecraft in orbit. The kinematic and rigid body equations that describe the motion of a spacecraft

in orbit are developed in Chapter 4.

45

4 Spacecraft Kinematics and Rigid Body

Dynamics

In this chapter we introduce the quaternion and modified Rodrigues parameters to describe the

attitude of a rigid body. Euler’s rotational equations of motion for a spacecraft with reaction wheels

are also presented.

4.1 Attitude Parameters

Attitude parameters are sets of coordinates that describe the orientation of an objective relative

to a frame of reference. The following sections discuss the quaternion and modified Rodrigues

parameter attitude representations, which are used in the following chapters.

4.1.1 Quaternion

The quaternion is a popular attitude coordinate that is nonsingular and can describe large rotations.

In the literature25,57 the quaternion is also referred to as Euler Parameters. The quaternion is com-

46


posed of 4 elements that correspond to a vector (q) and a scalar (q4) component as

q =

q1

q2

q3

q4

=

q

q4

(4.1)

The principal rotation elements can represent the orientation of a rigid body as a single principal

rotation Φ about a principal axis e.25 The quaternion (q) in terms of the principal rotation elements

is

qi = ei sinΦ2

i = 1, 2, 3

q4 = cosΦ2

(4.2)

The quaternion is a commonly used attitude representation, because there are no singularities. The

ability to describe every attitude orientation makes the quaternion an attractive choice to describe

the attitude orientation of a body. Since the quaternion has 4 components, a constraint is imposed

as

qTq = q21 + q2

2 + q23 + q2

4 = 1 (4.3)

which must be satisfied at all times to be a valid quaternion.

The direction cosine matrix representing the attitude in terms of the quaternion is57

R = (q24 − qT q)11 + 2qqT − 2q4q× (4.4)

47


where 11 is a 3× 3 identity matrix and q× is the skew symmetric matrix defined by

q× =

0 −q3 q2

q3 0 −q1

−q2 q1 0

(4.5)

The kinematic differential equation of the quaternion is57

q =12

(q× + q411)

−qT

ω (4.6)

where ω is the angular rate vector.

This section presents the quaternion attitude representation. The following section presents the

modified Rodrigues parameters, which is an alternate attitude representation.

4.1.2 Modified Rodrigues Parameters (MRP)

The MRP vector can be defined in terms of quaternions, classical Rodrigues parameters, and prin-

cipal rotation elements. The MRP vector (σ) in terms of the principal rotation elements (Φ, e)

is

σ = tanΦ4e (4.7)

The MRP vector has a singularity at a principal rotation angle of ±360. From Eq. (4.7), the

following relationships exist:

|σ| ≤ 1 if Φ ≤ 180

|σ| = 1 if Φ = 180

|σ| > 1 if Φ ≥ 180 (4.8)

48


The relationships between the other attitude representations and the MRP vector can be found in

Ref. 25.

The MRP vector is a convenient attitude representation for attitude feedback control. Using the

relationship

σs1,2,3 =−σ1,2,3

σTσ(4.9)

the MRP vector can always represent a principal rotation angle less than or equal to 180.25 The

MRP mapping is exploited to develop the nonlinear attitude controllers presented in the following

chapters.

The direction cosine matrix in terms of the MRP vector is25

R = 11 +8σ×σ× − 4(1− σTσ)σ×

1 + σTσ(4.10)

where σ× is the skew symmetric matrix defined in Eq. (4.5)

The MRP error vector in terms of two MRP vectors is expressed as25

δσ =(1− |σ′|2)σ − (1− |σ|2)σ′ + 2σ × σ′

1 + |σ′|2|σ|2 + 2σ′ · σ(4.11)

The MRP kinematic differential equation is25

σ =(1− σ2)I + 2σ× + 2σσT

4ω (4.12)

σ =14B(σ)ω (4.13)

B(σ) = (1− σ2)I + 2σ× + 2σσT (4.14)

This section presents the MRP attitude representation. The next section presents the spacecraft

rigid body dynamics.

49

4.2 Rigid Body Dynamics


The rotational rigid body dynamics of a spacecraft with momentum devices is presented in the

following subsections, which will be used in later chapters to develop the attitude control laws.

Euler’s rotational equations of motion for a spacecraft are presented. The equations of motion

for a spacecraft with reaction wheels are presented.

4.2.1 Euler’s Rotational Equations of Motion

Euler’s rotational equations of motion for a rigid body spacecraft are25,28

Iω = −ω×Iω + u+Lext (4.15)

where I is the inertia matrix in the body frame (B), ω is the angular velocity of the spacecraft with

respect to the inertial frame (N ) expressed in the body frame (B),ω× is the skew symmetric matrix,

Lext is the external torque, and u is the control torque.

4.2.2 Euler’s Rotational Equations of Motion with Momentum Exchange

Devices

Each simulator has three orthogonal reaction wheels for attitude control as discussed in Section 3.1.4.

The spacecraft equations of motion with momentum exchange devices are required to develop the

attitude control laws in later chapters.

The equations of motion for a rigid spacecraft with N reaction wheels are25

Iω = −ω× (Iω + Wshs) + Wsu+L (4.16)

where Ws is a 3×N matrix defining the spin axis orientation of the reaction wheels,

Ws = [ws1 · · · wsN ] (4.17)

50


and hs is the reaction wheel angular momentum.

The spacecraft inertia matrix (I) for a rigid body with N reaction wheels is25

I = IB +N∑i=1

(IWti

wtiwTti + IWti

wgiwTgi

)(4.18)

where IB is the spacecraft inertia matrix in the body frame (B), IWt is the wheel inertia in the wgi

and wti directions, and IWs is the wheel inertia in the spin axis wsi . The reaction wheel axes are

illustrated in Figure 4.1.

Figure 4.1: Illustration of Whorl−I and Whorl−II Reaction Wheel

The reaction wheel angular momentum is25

hsi = IWsi

(wTsiω + Ωi

)(4.19)

where IWsiis the reaction wheel spin axis inertia and Ωi is the wheel speed.

The reaction wheel motor torque is25

ui = −IWsi

(wTsiω + Ωi

)(4.20)

51

4.3 Summary

where Ωi is the reaction wheel acceleration.

4.3 Summary

This chapter presents the kinematic and rigid body equations of motion of a spacecraft. The atti-

tude representations, quaternion and modified Rodrigues parameters (MRP), used throughout the

following chapters are presented. The development of the equations of motion for a spacecraft with

N reaction wheels as the control devices are presented.

The kinematic equations and rotational equations of motion developed in this chapter are used

in the development of the Lyapunov based attitude controllers in Chapter 5 and the sliding mode

controller in Chapter 6.

52

5 Lyapunov Based Spacecraft Attitude

Control Laws

In this chapter we develop and demonstrate Lyapunov based spacecraft attitude control laws using

numerical simulations, Whorl−I, and Whorl−II. The chapter starts by discussing nonlinear stability

and control, then presents the development of the angular rate controllers, and concludes with the

development of the modified Rodrigues parameter controllers.

5.1 Nonlinear Stability and Control

Lyapunov’s direct method is used to design the nonlinear spacecraft attitude controllers in this chap-

ter. Lyapunov’s direct method allows a control designer to make rigorous, analytical stability claims

for nonlinear systems.14,25,58 The nonlinear system stability and controller design are accomplished

by studying the behavior of an energy-like function referred to as a Lyapunov function (V (x)).25

Summarizing Refs. 14 and 25, a nonlinear system (x = f(x)) has a Lyapunov function if the

scalar function V (x) is continuous and has a continuous partial derivative while also satisfying the

following conditions:

V (x 6= 0) > 0 positive definite function (5.1)

V (x = 0) = 0 (5.2)

V (x) ≤ 0 negative semi-definite function (5.3)

53

5.2 Angular Rate Control

A nonlinear system is locally stable about the origin if a Lyapunov function exists. If the

Lyapunov function is

V (x 6= 0) < 0 negative definite function (5.4)

then the stability is asymptotic.14,25,58 Global stability claims require the Lyapunov function to be

radially unbounded (i.e. V (x)→∞ as ‖x‖ → ∞).14

If V (x) ≤ 0, asymptotic stability can still be determined by investigating higher order deriva-

tives of the Lyapunov function for autonomous systems (systems that do not explicitly depend on

time). In the case where V (x) ≤ 0 for an autonomous system, Theorem 8.5 in Ref. 25 states that

a sufficient condition for asymptotic stability is the first non-zero, odd higher order derivative of

V (x) must be negative definite (i.e. diV (x)/dxi < 0 where i is odd). Global stability claims can

be made if the Lyapunov function is radially unbounded. An alternate method to prove asymptotic

stability for an autonomous system where V (x) ≤ 0 uses LaSalle’s Invariance Principle.14,58


In this section we develop a nonlinear spacecraft attitude control law that asymptotically drives the

angular velocity errors to zero. The angular velocity error is

δω = ω − ωr (5.5)

where ω is the angular velocity of the spacecraft and ωr is the reference angular velocity. The

spacecraft angular velocity (ω) has components in the body frame (B) and typically ωr has com-

ponents in the reference frame (R). The angular velocity error expressed in the body frame (B)

is

Bδω = Bω −RBR Rωr (5.6)

54


where(RBR

)is the rotation matrix from the reference frame to the body frame.

The derivative of the rotation matrix is

RBR = −ωB/R×RBR

= −( Bω −RBR Rωr

)×RBR (5.7)

Differentiating Eq. (5.6) and using Eq. (5.7) results in

ddt

(δω) =ddt

(ω)−RBR ddt

( Rωr)−ddt(RBR

) Rωr=

ddt

(ω)−RBR ddt

( Rωr) +( Bω −RBR Rωr

)×(RBR Rωr

)= ω −RBR Rωr + ω×RBR Rωr (5.8)

where for simplicity the following notation is implemented ω = Bω.

Lyapunov’s direct method is used to design the angular rate controller. Assume the candidate

Lyapunov function provided is25

V (δω) =12δωT Iδω (5.9)

where I is the inertia matrix in the body frame. Taking the derivative of Eq. (5.9) and substituting

Eq. (4.16) results in

V (δω) = δωT Iδω

= δωT I(ω −RBR Rωr + ω×RBR Rωr

)= δωT

(Iω − I

(RBR Rωr + ω×RBR Rωr

))= δωT

(−ω× (Iω + Wshs) + Wsu+L

−I(RBR Rωr − ω×RBR Rωr

))(5.10)

If the objective of the angular rate controller is to drive the angular rates to zero, Eq. (5.10)

55


simplifies to

V (ω) = ωT(−ω× (Iω + Wshs) + Wsu+L

)(5.11)

The reference angular velocity is zero for the desired situation where the angular rates go to zero

and Eq. (5.5) simplifies to δω = ω.

To guarantee asymptotic stability, we set Eq. (5.11) equal to the following negative definite

function:

V (δω) = −δωTPδω

= −ωTPω if Rωr = 0 (5.12)

where P is a positive definite gain matrix. The result is

−ωTPω = ωT (−ω× (Iω + Wshs) + Wsu+L) (5.13)

Rearranging Eq. (5.13) results in the following asymptotically stable angular rate control law

for driving the angular rates to zero:

u = Ws−1(−Pω + ω× (Iω + Wshs)−L

)(5.14)

For a real system the inertia matrix (I) may have a high uncertainty. Therefore a control law

that does not depend on the inertia matrix is desirable. Equation (5.13) is simplified by noting that

ωTω× = 0, which results in

−ωTPω = −ωTω×Iω + ωT (−ω×Wshs + Wsu+L)

= ωT (−ω×Wshs + Wsu+L)(5.15)

Rearranging Eq. (5.15) results in the following asymptotically stable angular rate control law for

56


driving the angular rates to zero with no knowledge of the inertia matrix:

u = Ws−1 (−Pω + ω×Wshs −L) (5.16)

If the reaction wheel spin axis inertia (IWS) is not well known, the control law is simplified as

u = Ws−1 (−Pω −L) (5.17)

which results in driving the angular rates to zero with no knowledge of the wheel inertia.

If the objective of the controller is to track a reference angular rate (ωr), Eq. (5.10) is set equal

to Eq. (5.12) to guarantee asymptotic stability:

−δωTPδω = δωT(− ω× (Iω + Wshs) + Wsu+L− I

(RBR Rωr + ω×RBR Rωr

))−Pδω = −ω× (Iω + Wshs) + Wsu+L− I

(RBR Rωr − ω×RBR Rωr

)(5.18)

where the gain matrix (P) is positive definite.

Rearranging Eq. (5.18) results in the following asymptotically stable angular rate control law

for tracking a reference angular velocity:

u = Ws−1(−Pδω + ω× (Iω + Wshs)−L+ I


))(5.19)

This section developed four angular rate control laws. The following section presents hardware

simulator and numerical simulation results using the angular rate controllers.

5.2.1 Whorl−I Angular Rate Controller Simulation

The following subsections present the simulation results for the angular rate controller using nu-

merical simulations and Whorl−I. The simulation results for a regulator (ω → 0) and tracking

(ω → ωr) maneuver are shown.

57


5.2.1.1 Regulator

Assuming no external torque and substituting the reaction wheel spin axis orientation matrix (Ws)

for Whorl−I into the control law (Eq. (5.16)) simplifies to

u = −Pω + ω×hs

where the angular rates are driven to zero. The estimates for the simulator parameters are provided

in Section 3.1.

The angular rate controller parameters are provided in Table 5.1. The angular rate controller

results are shown in Figures 5.1 – 5.6. Whorl−I was given an initial angular rate about the 3

body axis and then the angular rate controller was turned on. The 16 second delay in the start of

the controller at the beginning of the simulation is due to the time necessary to initialize the three

reaction wheels.

Table 5.1: Angular Rate Controller ParametersParameter ValueControl Rate 2 HzObserve Rate 2 HzWheel Speed Query 10 Hzω(t0) (-0.002, -0.009, 0.088) rad/sP diag( 3.64, 3.40, 4.41 ) kg-m2/sΩ1 0 rad/sΩ2 0 rad/sΩ3 0 rad/s

The control results from Whorl−I are compared to numerical simulation control results. Fig-

ures 5.1 and 5.2 provide the angular rate time history for Whorl−I and the numerical simulation

respectively. Whorl−I has approximately zero angular rate in the 3 axis after 20 seconds into the

simulation. The numerical simulation has approximately zero angular rate after 15 seconds into

the simulation. The angular rates for Whorl−I have the same decreasing trend as predicted by the

numerical simulation. After 20 seconds the noise from the angular rate measurements can be clearly

58


seen.

Figure 5.1: Whorl−I Angular Rates for Angular Rate Controller

Figure 5.2: Numerical Simulation Angular Rates for Angular Rate Controller

Figures 5.3 and 5.4 show the required control torque specified by the angular rate controller for

Whorl−I and the numerical simulation respectively. The control torques specified by the angular

rate controller for Whorl−I have the same control torque trend to zero as the numerical simulation.

Figures 5.5 and 5.6 show the reaction wheel speed time history for the controller running on

Whorl−I and the numerical simulation respectively. The reaction wheels have the same speed

trend as the numerical simulation. The reaction wheel aligned along the 3 axis for Whorl−I has a

59


Figure 5.3: Whorl−I Control Torque for Angular Rate Controller

Figure 5.4: Numerical Simulation Control Torque for Angular Rate Controller

60


similar trend as the numerical simulation. The numerical simulation ends with an angular velocity

of approximately 15 rad/s as compared with 17 rad/s for the reaction wheel aligned along the 3 axis

on Whorl−I. The discrepancies in the wheel speeds between Whorl−I and the numerical simulation

are due to the simplified reaction wheel motor torque model used in the numerical simulation and

shown in Eq. (4.20). Reference 59 provides a high fidelity dynamic model of a reaction wheel motor.

The model takes into account the electrical resistance of the motor armature, voltage limits, current

limits, viscosity damping, and static friction of the rotor, which are not considered in the simplified

reaction wheel model in Eq. (4.20). Even with the discrepancies, the numerical simulation and

Whorl−I results match.

Figure 5.5: Whorl−I Reaction Wheel Speed for Angular Rate Controller

5.2.1.2 Tracking

Assuming no external torque Eq. (5.16) simplifies to

u = Ws−1(−Pδω + ω× (Iω + Wshs) + I


))where the angular rate errors are driven to zero. The estimates for the simulator parameters are

provided in Section 3.1.

61


Figure 5.6: Numerical Simulation Reaction Wheel Speed for Angular Rate Controller


results are shown in Figures 5.7 – 5.14. Whorl−I is initially at rest and then the angular rate

controller is turned on to track the desired angular rates. The 16 second delay in the start of the

controller at the beginning of the simulation is due to the time necessary to initialize the three

reaction wheels.

Table 5.2: Angular Rate Controller ParametersParameter ValueControl Rate 2 HzObserve Rate 2 HzWheel Speed Query 10 Hzω(t0) (-0.005, -0.004, 0.001) rad/sP diag( 3.64, 3.40, 4.41 ) kg-m2/sΩ1 0 rad/sΩ2 0 rad/sΩ3 0 rad/s



respectively. Figures 5.9 and 5.10 provide the angular rate error time history for Whorl−I and the

numerical simulation respectively. Whorl−I angular rate errors track to zero except in the interval

62


of 80 to 100 seconds. After 100 seconds the angular rate errors converge to zero. The numerical

simulation tracks the angular rates perfectly. The reaction wheels are the reason for the discrepan-

cies between Whorl−I and the numerical simulation. In the interval of 80 to 100 seconds Ω3 is close

to zero and is in the dead band zone for the electric motor, which results in the control authority

accuracy decreasing.



Figures 5.11 and 5.12 show the required control torque specified by the angular rate controller

for Whorl−I and the numerical simulation respectively. The control torques specified by the angular

63


rate controller for Whorl−I have the same control torque trend as the numerical simulation, except

between 80 and 100 seconds into the simulation.

Figure 5.9: Whorl−I Angular Rate Error for Angular Rate Controller

Figure 5.10: Numerical Simulation Angular Rate Error for Angular Rate Controller


Whorl−I and the numerical simulation respectively. The reaction wheels have the same speed trend

as the numerical simulation, except between 80 and 100 seconds. The numerical simulation and

Whorl−I results match, except between 80 and 100 seconds.

64




65


Figure 5.13: Whorl−I Reaction Wheel Rates for Angular Rate Controller


66


The following Whorl−I angular rate tracking simulation avoids the dead band region for Ω3 by

setting the initial wheel speed to -40 rad/s. The angular rate controller parameters are provided in

Table 5.3. The angular rate controller results are shown in Figures 5.15 – 5.18. Whorl−I is initially

at rest then the angular rate controller is turned on to track the desired angular rates. The 25 second

delay in the start of the controller at the beginning of the simulation is due to the time necessary to

initialize the three reaction wheels.

Table 5.3: Angular Rate Controller ParametersParameter ValueControl Rate 2 HzObserve Rate 2 HzWheel Speed Query 10 Hzω(t0) (-0.003, -0.001, 0.007) rad/s[P ] diag( 3.64, 3.40, 4.41 ) kg-m2/sΩ1 0 rad/sΩ2 0 rad/sΩ3 -40 rad/s

Figure 5.15 provides the angular rate time history for Whorl−I. Figure 5.16 provides the an-

gular rate error time history for Whorl−I . The angular rate errors are tracking to zero throughout

the simulation, which is an improvement compared to the previous tracking simulation where Ω3 is

initially set to zero.

Figure 5.17 shows the required control torque specified by the angular rate controller for

Whorl−I . The control torques specified by the angular rate controller for Whorl−I have the same

control torque trend as the previous numerical simulation.

Figure 5.18 shows the reaction wheel speed time history for the controller running on Whorl−I.

Avoiding the electric motor deadband region results in the expected performance for the angular rate

controller.

67



Figure 5.16: Whorl−I Angular Rate Error for Angular Rate Controller

68



Figure 5.18: Whorl−I Reaction Wheel Speed for Angular Rate Controller

69


5.2.2 Whorl−II Angular Rate Controller Simulation

The simulation results for the angular rate controller (Eq. (5.16)) using numerical simulations and

Whorl−II are presented for a regulator (ω → 0) problem.

Assuming no external torque the control law (Eq. (5.16)) simplifies to

u = Ws−1 (−Pω + ω×Wshs)

where the angular rates are driven to zero. The estimates for the simulator parameters are provided

in Section 3.1.


results are shown in Figures 5.19 – 5.24. Whorl−II is given an initial angular rate about the 3

body axis and then the angular rate controller is turned on. The 16 second delay in the start of


reaction wheels.

Table 5.4: Angular Rate Controller ParametersParameter ValueControl Rate 2 HzObserve Rate 2 HzWheel Speed Query 10 Hzω(t0) (0.013, -0.007, 0.222) rad/sP diag( 3.64, 3.40, 4.41 ) kg-m2/sΩ1 0 rad/sΩ2 0 rad/sΩ3 0 rad/s

The control results from Whorl−II are compared to numerical simulation control results. Fig-

ure 5.19 and 5.20 provide the angular rate time history for Whorl−I and the numerical simulation

respectively. Whorl−II has approximately zero angular rate after 20 seconds into the simulation.

The numerical simulation has approximately zero angular rate after 20 seconds into the simula-

tion. The angular rates for Whorl−II have the same decreasing trend as predicted by the numerical

70


simulation. After 20 seconds the noise from the angular rate measurements can be clearly seen.

Figure 5.19: Whorl−II Angular Rates for Angular Rate Controller



for Whorl−II and the numerical simulation respectively. The control torques specified by the an-

gular rate controller for Whorl−II have the same control torque trend to zero as the numerical

simulation.

Figures 5.23 and 5.24 show the reaction wheel speed time history for the controller running

71


Figure 5.21: Whorl−II Control Torque for Angular Rate Controller


72

5.3 Modified Rodrigues Parameter Control

on Whorl−II and the numerical simulation respectively. The reaction wheels have the same speed

trends as the numerical simulation. Whorl−I and the numerical simulation results closely match.

Figure 5.23: Whorl−II Reaction Wheel Speed for Angular Rate Controller



In this section we develop a nonlinear spacecraft attitude control law that asymptotically drive the

angular velocity errors and MRP errors to zero. Lyapunov’s direct method is used to design the MRP

73


control law for controlling the attitude of a spacecraft. Assume the candidate Lyapunov function

is25

V (δω, δσ) =12δωT Iδω︸︷︷︸V (δω)

+ 2K ln(1 + δσT δσ

)︸︷︷︸V (δσ)

(5.20)

where K is a positive scalar gain quantity. Eq. (5.20) is represented in terms of V (δω) and V (δσ).

Taking the derivative of Eq. (5.20) results in

V (δω, δσ) = V (δω) + V (δσ) (5.21)

where V (δω) is previously derived in Section 5.2.

The MRP Lyapunov term in Eq. (5.20) is

V (δσ) = 2K ln (1 + δσT δσ) (5.22)

The derivative of Eq. (5.22) is

V (δσ) = 4K1

1 + δσT δσδσT δσ

= 4K1

1 + δσT δσδσT

(1− δσ2)I3×3 + 2δσ× + 2δσδσT

4δω

= K1

1 + δσT δσ

(1− δσ2)δσT + 2 δσT δσ×︸︷︷︸0

+2 (δσT δσ)︸︷︷︸δσ2

δσT

δω

= K1

1 + δσT δσ

(1− δσ2 + 2δσ2

)δσT δω

= K1

1 + δσTσ

(1 + δσT δσ

)δσT δω

= KδσT δω

= δωT (Kδσ) (5.23)

74


Substituting Eq. (5.10) and Eq. (5.23) into Eq. (5.21) results in

V (δω, δσ) = δωT(−ω× (Iω + Wshs) + Wsu+L


)+Kδσ

)(5.24)

If the objective of the MRP controller is to drive the MRP attitude vector to zero, Eq. (5.24)

simplifies to

V (δω, δσ) = ωT(−ω× (Iω + Wshs) + Wsu+L+Kσ

)(5.25)

where the reference MRP and reference angular velocity are zero and Eq. (5.5) simplifies to δω = ω

and δσ = σ.

To guarantee asymptotic stability, we set Eq. (5.25) equal to the following semi-negative defi-

nite function:

V (δω) = −δωTPδω

= −ωTPω if Rωr = 0 (5.26)

where P is a positive definite gain matrix. The result is

−ωTPω = ωT(−ω× (Iω + Wshs) + Wsu+L+Kσ

)(5.27)

Rearranging Eq. (5.27) results in the following stable angular MRP control law for driving σ

and ω to zero:

u = Ws−1 (−Kσ −Pω + ω× (Iω + Wshs)−L) (5.28)

Asymptotic stability is investigated for the control law in Eq. (5.28) by taking higher order

derivatives of V (ω,σ) and satisfying the sufficient conditions for asymptotic stability as discussed

at the beginning of this chapter.

75


The higher order derivatives for V (ω,σ) are

V (ω,σ) = −ωTPω

= 0

V (ω,σ) = −2ωTPω

= 0...V (ω,σ) = −2ωTPω − 2ωTPω

= −2ωTPω (5.29)

where ω is solved for by substituting the control law into the equations of motion resulting in

ω = I−1 (−Kσ −Pω) (5.30)

The third derivative of the Lyapunov function is

...V (ω,σ) = −2K2

(I−1σ

)T P(I−1σ

)< 0 (5.31)

where P and I−1 are positive definite. The sufficient conditions for asymptotic stability are satisfied

and the control law in Eq. (5.28) is globally asymptotically stable.

It may be desirable to have a control law that does not depend on the inertia matrix as mentioned

in Section 5.2. Equation (5.27) is simplified by noting that ωTω× = 0, which results in

−ωTPω = −ωTω×Iω + ωT(−ω×Wshs + Wsu+L+Kσ

)= ωT

(−ω×Wshs + Wsu+L+Kσ

)(5.32)

Rearranging Eq. (5.32) results in the following stable MRP control law for driving the MRPs and

76


angular rates to zero with no knowledge of the inertia matrix:

u = Ws−1(−Kσ −Pω + ω×Wshs −L

)(5.33)

By investigating the higher order derivatives of V (ω,σ) and satisfying the sufficient conditions for

asymptotic stability the control law in Eq. (5.33) is shown to be globally asymptotically stable.

If the reaction wheel spin axis inertia (IWs) is not well known, the control law is simplified as

u = Ws−1 (−Kσ −Pω −L) (5.34)

which drives the MRP vector and angular rates to zero with no knowledge of the wheel inertia.

Again, the control law in Eq. (5.34) is shown to be globally asymptotically stable by investigating the

higher order derivatives of V (ω,σ) and satisfying the sufficient conditions for asymptotic stability

as discussed for the control law in Eq. (5.28).

If the objective of the controller is to track a reference MRP vector (σr), Eq. (5.24) is set equal

to Eq. (5.12) to guarantee asymptotic stability:

−δωTPδω = δωT(−ω× (Iω + Wshs) + Wsu+L


)+Kσ

)−Pδω = −ω× (Iω + Wshs) + Wsu+L


)+Kσ (5.35)

where the gain matrix (P) is positive definite.

Rearranging Eq. (5.35) results in the following stable angular rate control law for tracking a

reference MRP and angular rate vector:

u = Ws−1(−Kσ −Pδω + ω× (Iω + Wshs)−L+ I


))(5.36)

77


Asymptotic stability is investigated for the control law in Eq. (5.36) by taking higher order

derivatives of V (δω, δσ) and satisfying the sufficient conditions for asymptotic stability as dis-

cussed at the beginning of this chapter.

The higher order derivatives for V (δω, δσ) are

V (δω, δσ) = −δωTPδω

= 0

V (δω, δσ) = −2δωTPδω

= 0...V (δω, δσ) = −2δωTPδω − 2δωTPδω

= −2δωTPδω (5.37)

where δω is solved for by substituting the control law into the error dynamics.

δω = I−1 (−Kδσ −Pδω) (5.38)

Therefore the third derivative of the Lyapunov function is

...V (δω = 0, δσ) = −2K2

(I−1δσ

)T P(I−1δσ

)< 0 (5.39)

where P and I−1 are positive definite. The sufficient conditions for asymptotic stability are satisfied

and the control law in Eq. (5.36) is globally asymptotically stable.

5.3.1 Whorl−I MRP Controller Simulation

The following subsections present the simulation results for the MRP controller using numerical

simulations and Whorl−I. The simulation results for a regulator (ω → 0,σ → 0) and tracking

(ω → ωr,σ → σr) problem are shown.

78


5.3.1.1 Regulator


u = Ws−1(−Kσ −Pω + ω×Wshs

)where the angular rates and MRPs are driven to zero. The estimates for the simulator parameters

are provided in Section 3.1.

The results of running the controller on Whorl−I along with a numerical simulation comparing

the results are presented. The MRP controller parameters are provided in Table 5.5. The MRP

controller results are shown in Figures 5.25 – 5.32. Whorl−I is given an initial angular rate about

the 3 body axis and then the MRP controller is turned on. The 16 second delay in the start of


reaction wheels.

Table 5.5: MRP Controller ParametersParameter ValueControl Rate 2 HzObserve Rate 2 HzWheel Speed Query 10 Hzσ(t0) (-0.003, 0.015, 0.482 )ω(t0) (-0.007, -0.005, 0.118) rad/sK 1.5 kg-m2/s2

P diag( 3.64, 3.40, 4.41 ) kg-m2/sΩ1 0 rad/sΩ2 0 rad/sΩ3 0 rad/s



respectively. Whorl−I has approximately zero angular rate in the 3 axis after 45 seconds into the

control simulation. The numerical simulation has approximately zero angular rate after 42 seconds

into the simulation. The angular rates for Whorl−I have the same decreasing trend as predicted by

79


the numerical simulation.

Figure 5.25: Whorl−I Angular Rates for MRP Controller

Figure 5.26: Numerical Simulation Angular Rates for MRP Controller

Figures 5.27 and 5.28 provide the MRP vector time history for Whorl−I and the numerical

simulation respectively. Whorl−I has approximately a constant offset from zero for σ3 after 50

seconds into the simulation. The numerical simulation has approximately a zero MRP vector after

42 seconds into the simulation. The constant offset seen in Figure 5.27 is due to atmospheric drag

torque acting on Whorl−I. The steady state offset for σ3 is approximately 0.02. The external torque

80


vector expression is25

σsteady state =LextK

(5.40)

where Lext is the external torque. The external torque acting on Whorl−I is 0.03 N-m from

Eq. (5.40). The external torque is in agreement with the atmospheric drag lower bound analysis

presented in Ref. 34 for Whorl−I. The MRP vector for Whorl−I has the same decreasing trend as

predicted by the numerical simulation.

Figure 5.27: Whorl−I MRP Vector for MRP Controller

Figures 5.3 and 5.4 show the required control torque specified by the angular rate controller for

Whorl−I and the numerical simulation respectively. The control torques specified by the angular

rate controller for Whorl−I have the same control torque trend to zero as the numerical simulation.


Whorl−I and the numerical simulation respectively. The reaction wheels have the same speed

trend as the numerical simulation. The reaction wheel aligned along the 3 axis for Whorl−I has a

similar trend as the numerical simulation. The numerical simulation ends with an angular velocity

of approximately 20 rad/s as compared with 23 rad/s for the reaction wheel aligned along the 3 axis

of Whorl−I. The discrepancies in the wheel speeds between Whorl−I and the numerical simulation

81


are due to the simplified reaction wheel motor torque model used in the numerical simulation.

Figure 5.28: Numerical Simulation MRP Vector for MRP Controller

Figure 5.29: Whorl−I Control Torque for MRP Controller

82


Figure 5.30: Numerical Simulation Control Torque for MRP Controller

Figure 5.31: Whorl−I Reaction Wheel Speed for MRP Controller

83


Figure 5.32: Numerical Simulation Reaction Wheel Speed for MRP Controller

5.3.1.2 Tracking


u = Ws−1(−Kσ −Pδω + ω× (Iω + Wshs) + I


))where the MRP and angular rate errors are driven to zero. The estimates for the simulator parameters


The MRP controller parameters are provided in Table 5.6. The MRP controller results are

shown in Figures 5.33 – 5.44. Whorl−I is initially at rest and then the MRP controller is turned on

to track the desired attitude. The 40 second delay in the start of the controller at the beginning of

the simulation is due to the time necessary to initialize the three reaction wheels.



respectively. Figures 5.35 and 5.36 provide the angular rate error time history. Whorl−I has ap-

proximately zero angular rate error in the 3 axis after 65 seconds into the control simulation. The

numerical simulation has approximately zero angular rate after 55 seconds into the simulation. The

angular rates and angular rate errors for Whorl−I have the same decreasing trend as predicted by

84


Table 5.6: MRP Controller ParametersParameter ValueControl Rate 2 HzObserve Rate 2 HzWheel Speed Query 10 Hzσ(t0) (0.003, -0.018, -0.854 )ω(t0) (-0.006, -0.001, 0.000) rad/sK 2.5 kg-m2/s2

P diag( 4.70, 4.39, 5.70 ) kg-m2/sΩ1 0 rad/sΩ2 0 rad/sΩ3 -50 rad/s


Figure 5.33: Whorl−I Angular Rates for MRP Controller

Figures 5.37 and 5.38 provide the MRP vector time history for Whorl−I and the numerical sim-

ulation respectively. The MRP switching previously discussed is clearly seen for σ3. Figures 5.39

and 5.40 provide the MRP error vector time history. Whorl−I has approximately a constant offset

from zero error for σ3 after 130 seconds into the simulation. The numerical simulation has approx-

imately a zero MRP error vector after 120 seconds into the simulation. The steady state offset for

σ3 is approximately 0.02. The external torque acting on Whorl−I is approximately 0.05 N-m from

85



Figure 5.35: Whorl−I Angular Rate Errors for MRP Controller

86


Figure 5.36: Numerical Simulation Angular Rate Errors for MRP Controller

Eq. (5.40). The external torque is in agreement with the atmospheric drag lower bound analysis

presented in Ref. 34 for Whorl−I. The MRP error vector for Whorl−I has the same decreasing

trend as predicted by the numerical simulation.

Figure 5.37: Whorl−I MRP Vector for MRP Controller


for Whorl−I and the numerical simulation respectively. The control torques specified by the MRP

controller for Whorl−I have the same control torque trend to zero as the numerical simulation.

87



Figure 5.39: Whorl−I MRP Error Vector for MRP Controller

88


Figure 5.40: Numerical Simulation MRP Error Vector for MRP Controller

Figure 5.41: Whorl−I Control Torque for MRP Controller

89




on Whorl−I and the numerical simulation respectively. The reaction wheels have the same speed

trend as the numerical simulation. The numerical simulation ends with an angular velocity of ap-

proximately -30 rad/s as compared with -25 rad/s for the reaction wheel aligned in the 3 axis. The

discrepancy in wheel speeds between Whorl−I and the numerical simulation is attributed to the

simplified reaction wheel motor torque model used in the numerical simulation.

Figure 5.43: Whorl−I Reaction Wheel Speed for MRP Controller

90



5.3.2 Whorl−II MRP Controller Simulation

In this section we present the simulation results of the MRP controller using numerical simulations

and Whorl−II. The simulation results for a regulator (ω → 0,σ → 0) problem are shown.


u = Ws−1(−Kσ −Pω + ω×Wshs

)where the angular rates and MRPs are driven to zero. The estimates for the simulator parameters


The results of running the controller on Whorl−II along with a numerical simulation comparing

the results are presented. The MRP controller parameters are provided in Table 5.7. The MRP

controller results are shown in Figures 5.45 – 5.52. Whorl−II is given an initial angular rate about

the 3 body axis and then the MRP controller is activated. The 16 second delay in the start of


reaction wheels.

Figures 5.45 and 5.46 provide the angular rate time history for Whorl−II and the numerical

simulation respectively. Whorl−II has approximately zero angular rate in the 3 axis after 45 seconds

91


Table 5.7: MRP Controller ParametersParameter ValueControl Rate 2 HzObserve Rate 2 HzWheel Speed Query 10 Hzσ(t0) (-0.003, 0.015, 0.482 )ω(t0) (-0.007, -0.005, 0.118) rad/sK 1.5 kg-m2/s2

P diag( 3.64, 3.40, 4.41 ) kg-m2/sΩ1 0 rad/sΩ2 0 rad/sΩ3 0 rad/s

into the control simulation. The numerical simulation has approximately zero angular rate after 45

seconds into the simulation. The angular rates for Whorl−II have the same decreasing trend as


Figure 5.45: Whorl−II Angular Rates for MRP Controller

Figures 5.47 and 5.48 provide the MRP vector time history for Whorl−II and the numerical

simulation respectively. Whorl−II has approximately a zero MRP vector after 50 seconds into the

simulation. The numerical simulation also has approximately a zero MRP vector after 50 seconds

into the simulation. The MRP vector for Whorl−II has the same decreasing trend as predicted by

92




Figure 5.47: Whorl−II MRP Vector for MRP Controller

Figures 5.49 and 5.50 show the required control torque specified by the MRP controller for

Whorl−II and the numerical simulation respectively. The control torques specified by the MRP

controller for Whorl−II have the same control torque trend as the numerical simulation.


on Whorl−II and the numerical simulation respectively. The reaction wheels have the same speed

93



Figure 5.49: Whorl−II Control Torque for MRP Controller

94




proximately 58 rad/s as compared with 55 rad/s for the reaction wheel aligned along the 3 axis. The

discrepancy in wheel speeds between Whorl−II and the numerical simulation is attributed to the

simplified reaction wheel motor torque model used in the numerical simulation. The results from

Whorl−II and the numerical simulation closely match.

Figure 5.51: Whorl−II Reaction Wheel Speed for MRP Controller

95

5.4 Summary


5.4 Summary

In this chapter we develop nonlinear asymptotically stable angular rate control laws. The attitude

hardware simulators successfully demonstrate the angular rate controllers. We also develop nonlin-

ear asymptotically stable MRP attitude control laws that drive the attitude to zero and track a time

varying reference attitude. The attitude hardware simulators successfully demonstrate the MRP

attitude controllers.

The attitude controllers developed in this chapter assume perfect knowledge of system param-

eters such as spacecraft mass properties, reaction wheel alignment, and perfect attitude knowledge.

Real systems only have estimates of these quantities. The next chapter develops a robust attitude

control law that tracks a reference attitude while bounding system uncertainties.

96

6 Sliding Mode Spacecraft Attitude Control

Law

The motivation of this section is to develop a spacecraft attitude controller that tracks a reference

attitude using reaction wheels as the actuator while bounding system uncertainties. The Modified

Rodrigues Parameters (MRP) are chosen as the attitude representation for the controller to take

advantage of the properties discussed in Section 4.1.2 such as rotations up to ±360, shortest path

to zero error (i.e. avoid windup), and minimal parameterization.

6.1 Attitude Control Law Development

Assume a second order system structure such as

x = f + Bu (6.1)

where f is the drift function, B is the input matrix, and u is the control input.14,15 Let f and B be

the estimates of f and B, respectively. The matrices are required to be nonsingular and assume the

model uncertainties are bounded by a column matrix F and matrix DB as14,15

|f − f | ≤ F (6.2)

97


B = (11 +4B) B, for |4B| ≤ DB (6.3)

where 11 is a 3×3 identity matrix. Note that with no uncertainty F = 0, f = f , 4B = 0, and

B = B.

The spacecraft equations of motion with reaction wheels (Eq. (4.16)) are provided as

ω = −I−1ω× (Iω + Wshs) + I−1L︸︷︷︸f

+ I−1Ws︸︷︷︸B

u (6.4)

where f is 3× 1 matrix and B is 3× n matrix where n is the number of reaction wheels.

The design procedure for sliding mode control involves choosing the sliding manifold that

achieves the control objective and designing the sliding conditions that satisfy the sliding mani-

fold.21

The sliding variable (s) is defined as a class of linear sliding vectors as

s(δω, δσ) , δω + Λδσ (6.5)

where Λ is a 3 × 3 symmetric positive-definite constant matrix, δω is the difference in angular

velocity, and δσ is the difference in the MRP vector between the spacecraft and reference model.

The system will track the desired state if the system stays on the sliding manifold (s = 0).21

The attitude tracking errors (δσ) when on the sliding manifold are investigated by making use of

the nonlinear transformation in Eq. (4.14)(

14B(δσ)

)and Eq. (6.5):21

14B(δσ) (δω + Λδσ) = 1

4B(δσ)s

δσ + 14B(δσ)Λδσ = 1

4B(δσ)s

δσ + 14B(δσ)Λδσ = 0 (6.6)

98


Consider the following candidate Lyapunov function for the system in Eq. (6.6).

V (δσ) = 2δσT δσ (6.7)

Differentiating Eq. (6.7) and substituting Eq. (6.6) results in

V (δσ) = 4δσT δσ

= −4δσT(1− δσ2)I3×3 + 2δσ× + 2δσδσT

4Λδσ

= −

(1− δσ2)δσT + 2 δσT δσ×︸︷︷︸0

+2 (δσT δσ)︸︷︷︸δσ2

δσT

Λδσ

= −(1− δσ2 + 2δσ2

)δσTΛδσ

= −(1 + δσ2

)δσTΛδσ (6.8)

Equation (6.7) is a Lyapunov function due to satisfying the necessary conditions by the function

being continuously differentiable, V (δσ = 0) = 0 and V (δσ 6= 0) > 0. The gain matrix Λ is

positive definite and V (δσ = 0) = 0, therefore V (δσ 6= 0) ≤ 0. The negative semi-definite

function indicates that the error between the true state and desired state will converge to zero as

t→∞.

The control is designed such that the sliding manifold is reached and sliding on the manifold

occurs. The best approximation of a continuous control law (ueq) that will result in s(δω, δσ) = 0

is developed by substituting the error dynamics (Eq. (5.8) and Eq. (4.14)) and system equations of

motion (Eq. (6.4)) into s(δω, δσ), which results in

s(δω, δσ) = δω + Λδσ

= ω −RBR Rωr + ω×RBR Rωr +14ΛB(δσ)δω

= f + Bu−RBR Rωr + ω×RBR Rωr +14ΛB(δσ)δω (6.9)

99


Setting s(δω, δσ) = 0, the equivalent control expression (ueq) is

ueq = B+

(−f + RBR Rωr − ω×RBR Rωr −

14ΛB(δσ)δω

)(6.10)

where the estimates of B and f are used to determine the control and B+ indicates the pseudo

inverse. The equivalent control keeps the sliding variable on the sliding surface in the absence of

errors (i.e. s(δω, δσ) = 0 if the dynamics are precisely known such that B = B and f = f ).

The equivalent control (ueq) shown in Eq. (6.10) results in driving the angular rate error to

zero. The MRP error results in a constant offset from zero. Integral control is implemented into the

equivalent control law to drive the MRP error to zero. The sliding variable (s) is now defined as

s(δω, δσ, z) , δω + Λδσ + z (6.11)

z = ΛI

∫ t

0δσdt (6.12)

where Λ and ΛI are 3× 3 symmetric positive-definite constant matrices.

The system will track the desired state if the system stays on the sliding manifold (s = 0).21

The attitude tracking errors (δσ) when on the sliding manifold are investigated by making use of

the nonlinear transformation in Eq. (4.14)(

14B(δσ)

)and Eq. (6.11):

14B(δσ) (δω + Λδσ + z) = 1

4B(δσ)s

δσ + 14B(δσ) (Λδσ + z) = 1

4B(δσ)s

δσ + 14B(δσ) (Λδσ + z) = 0 (6.13)

Consider the following candidate Lyapunov function for the system in Eq. (6.13):

V (δσ, z) = 2 log (1 + δσT δσ) +12zTΛI

−1z (6.14)

100


Differentiating Eq. (6.14) and substituting Eq. (6.13) results in

V (δσ, z) = 41

1 + δσ2δσT δσ + zTΛI

−1z

= −41

1 + δσ2δσT

(1− δσ2)I3×3 + 2δσ× + 2δσδσT

4. . .

(Λδσ + z) + zTΛI−1ΛIδσ

= − 11 + δσ2

(1− δσ2)δσT + 2 δσT δσ×︸︷︷︸0

+2 (δσT δσ)︸︷︷︸δσ2

δσT

. . .

(Λδσ + z) + zT δσ

= − 11 + δσ2

(1− δσ2 + 2δσ2

)δσT (Λδσ + z) + δσTz

= − 11 + δσ2

(1 + δσ2

)δσT (Λδσ + z) + δσTz

= −δσT (Λδσ + z) + δσTz

= −δσTΛδσ (6.15)

Eq. (6.14) is a Lyapunov function due to satisfying the necessary conditions by the function

being continuously differentiable, V (δσ = 0) = 0, and V (δσ 6= 0) > 0. The gain matrix Λ is

positive definite, V (δσ = 0) = 0, therefore V (δσ 6= 0) ≤ 0. The negative semi-definite function

indicates the error between the true state and desired state will converge to zero as t→∞.

The next step is to design the control such that the sliding manifold is reached and sliding on

the manifold occurs. The best approximation of a continuous control law (ueq) that will result in

s(δω, δσ, z) = 0 is developed by substituting the error dynamics (Eq. (5.8) and Eq. (4.14)) and

101


system equations of motion (Eq. (6.4)) into s(δω, δσ, z), which results in

s(δω, δσ, z) = δω + Λδσ + z

= ω −RBR Rωr + ω×RBR Rωr

+14ΛB(δσ)δω + ΛIδσ

= f + Bu−RBR Rωr + ω×RBR Rωr

+14ΛB(δσ)δω + ΛIδσ (6.16)

Setting s(δω, δσ, z) = 0, the equivalent control expression (ueq) is

ueq = B+(−f + RBR Rωr − ω×RBR Rωr

−14ΛB(δσ)δω −ΛIδσ

)(6.17)

where the estimates of B and f are used to determine the control. The equivalent control keeps the

sliding variable on the sliding surface in the absence of errors (i.e. s(δω, δσ, z) = 0 if the dynamics

are precisely known such that B = B and f = f ).

The condition where the sliding variable (s) is not on the sliding surface must be addressed.

Adding a robustness term (urob) to the control guarantees that s converges to the sliding surface.

Ensuring s converges to the sliding surface can be achieved if each element of the sliding variable

si satisfies the sliding condition (also known as the reachability condition60) specified as

12

ddts(δω, δσ, z)2

i ≤ −ηi|s(δω, δσ, z)i| (6.18)

where ηi is a strictly positive constant. The sliding surface condition ensures that the system trajec-

tory points to the surface and that once the system trajectory reaches the sliding surface it remains

on that surface.14

The robust control term (urob) ensures that the sliding condition given in Eq. (6.18) is met in

102


the presence of model uncertainty. The robust control term is given as a sign or saturation function,

urob = B+K sign (s(δω, δσ, z)) (6.19)

urob = B+K sat(si(δω, δσ, z)

φi

)(6.20)

where K is a diagonal matrix whose values are not yet identified.

The sign function is defined as14

sign (si(δω, δσ, z)) =

−1 if si(δω, δσ, z) < 0

1 if si(δω, δσ, z) ≥ 0(6.21)

and the saturation function is defined as14

sat(si(δω, δσ, z)

φi

)=

si(δω, δσ, z)

φiif |si(δω, δσ, z)| ≤ φi

sign(si(δω, δσ, z)) if |si(δω, δσ, z)| > φi

(6.22)

where φi is a boundary layer thickness that will eliminate chatter if appropriately chosen so that the

un-modeled high-frequency dynamics are not excited.

The final control law includes the equivalent and robust control terms as

u = ueq − urob (6.23)

For the purpose of this analysis the continuous saturation function is desired over the discon-

tinuous sign function to prevent chattering of the controller. The final robust tracking control law

is

u = B+(− f + RBR Rωr − ω×RBR Rωr

−14ΛB(δσ)δω −ΛIδσ −K sat

(si(δω, δσ, z)

φi

))(6.24)

103

6.2 Gain Selection

where the control gains are presented in the next section.

6.2 Gain Selection

This section determines the appropriate gains for the sliding mode controller shown in Eq. (6.24).

The gains associated with the equivalent control are Λ and ΛI with the only stipulation being that

the matrices are symmetric and positive definite. The robust control gain K is required to be a

diagonal matrix.

The relationship between the two feedback gains for the equivalent control part of the controller

are determined by investigating the closed-loop dynamics of the system using the equivalent control

with no uncertainties and perfect dynamics.

Substituting the equivalent control (Eq. (6.17)) into Euler’s rotational equation of motion (Eq. (6.4))

and using the expression in Eq. (5.8) results in the closed-loop dynamics as

δω = −Λδσ −ΛIδσ (6.25)

Linearizing the MRP kinematic differential equation (Eq. (4.14)) about 0 results in

δσ ≈ 14δω (6.26)

which is valid for large rotations.25

The linear closed-loop dynamics are rewritten as

δσ

δω

=

0 1411

−ΛI −14Λ

δσ

δω

(6.27)

The linear closed-loop dynamics can be decoupled by requiring Λ and ΛI to be diagonal. Now

104

6.2 Gain Selection

the decoupled dynamics are expressed as

δσi

δωi

=

0 14

−ΛIi −14Λi

δσ(t)i

δω(t)i

i = 1, 2, 3 (6.28)

The linear closed-loop natural frequencies (ωni) for the system in Eq. (6.28) are determined from

the square root of the determinant of the matrix, which results in

ωni =12

√ΛIi i = 1, 2, 3 (6.29)

The roots for the system in Eq. (6.28) are

λi = −18

(Λi ±

√Λ2i − 16ΛIi

)i = 1, 2, 3 (6.30)

The damping ratio (ζi) is the parameter that determines whether the responses oscillate and

decay or purely decay. If ζi < 1 the responses will have decaying oscillations and the system is

underdamped, if ζi > 1 the responses will have exponential decay and the system is overdamped,

and if ζi = 1 the system is critically damped. The damping ratio is determined from the roots and

natural frequency based on the following general form:61

λ = −ζωn ±√ζ2 − 1 (6.31)

From the previous equation, the damping ratio is

ζi =Λi

4√

ΛIii = 1, 2, 3 (6.32)

The time it takes for the state errors to decay to 1/e based on the initial values is provided in

the following expression:61,25

Ti =8Λi

i = 1, 2, 3 (6.33)

105

6.2 Gain Selection

The gains Λ and ΛI can now be selected based on the desired system response by selecting ωni , ζi,

and Ti.

The gain expression (K) in the robustness term is developed by substituting the control signal

(u) given in Eq. (6.24) into Eq. (6.16) and solving for the sliding surface constraint in Eq. (6.18).

The result is shown as

12

ddt(sTs

)= s(δω, δσ, z)Ts(δω, δσ, z)

= (Γ + f + Bu)T s(δω, δσ, z)

=(f −BB+f

)Ts(δω, δσ, z)

+((

I−BB+)

Γ)Ts(δω, δσ, z)

−(

BB+K sat(si(δω, δσ, z)

φi

))Ts(δω, δσ, z) (6.34)

where

Γ = −RBR Rωr + ω×RBR Rωr +14ΛB(δσ)δω + ΛIδσ (6.35)

The diagonal matrix terms (diag(K)) are determined by satisfying the sliding condition in Eq. (6.18),

which results in

−η ≥ f −BB+f +(11−BB+

)Γ−BB+K sat

(si(δω, δσ, z)

φi

)(6.36)

K sat(si(δω, δσ, z)

φi

)≥ BB+f − f +

(BB+ − 11

)Γ + BB+η

diag(K) ≥∣∣∣BB+f − f +

(BB+ − 11

)Γ + BB+η

∣∣∣ (6.37)

106

6.3 Parameter Bounds

The gain expression is simplified as

diag(K) ≥ BB+F +∣∣∣(BB+ − 11

)(f + Γ

)+ BB+η

∣∣∣ (6.38)

by noting that f = f + (f − f) and |f − f | ≤ F.

Using Eq. (6.3) results in

BB−1 = (11 +4B)−1BB−1

= (11 +4B)−1

≤ (11−DB)−1 (6.39)

The expression for the diagonal matrix terms (diag(K)) are rewritten as

diag(K) ≥ (11−DB)−1 F +∣∣∣((11−DB)−1 − 11

)(f + Γ

)∣∣∣+ (11−DB)−1 η

= (11−DB)−1(F +

∣∣∣DB

(f + Γ

)∣∣∣+ η)

=(

11−DB

)−1 (F +

∣∣∣DB

(f −RBR Rωr + ω×RBR Rωr

+14ΛB(δσ)δω + ΛIδσ

)∣∣∣∣+ η)

(6.40)

In summary the sliding mode controller is specified in Eq. (6.24) and the gain expression is

provided in Eq. (6.40).


The various parameter bounds that complete the control definition are determined in the following

analysis. Modifying Eq. (6.4) such that Ds = Wshs results in

f = −I−1ω× (Iω + Ds) + I−1L

B = I−1Ws (6.41)

107


The estimates of f and B are

f = −I−1ω×(Iω + Ds

)B = I−1Ws (6.42)

assuming no knowledge of the external torque.

The estimate of the inertia matrix I is defined as

I = (11 +4I)I, with |4I | ≤ DI

I−1 ≈ I−1(11−4I), first order approximation (6.43)

where DI is the bounding matrix. The external torque is assumed to have a bounded magnitude

given by

||L|| ≤ Lmax (6.44)

The reaction wheel spin axis orientation matrix Ws is the size 3 × n, where n is the number

of reaction wheels. The hardware simulators each have three reaction wheels, which results in a

square matrix. For generality consider Ws as 3× n. A pseudo inverse for Ws is indicated as W+s .

The estimate of the wheel spin axis orientation matrix is

Ws = (11 +4w) Ws, with |4w| ≤ Dw (6.45)

where the reaction wheel spin axis orientation matrix is bounded by Dw. The matrix Dw is

Dw =

drw (1 + drw)θrw (1 + drw)θrw

(1 + drw)θrw drw (1 + drw)θrw

(1 + drw)θrw (1 + drw)θrw drw

(6.46)

where the matrix is defined by assuming the maximum fractional error of torque and maximum

angular error of torque to be drw and θrw.15

108


The actual net reaction wheel momentumDs is assumed to be

Ds = (11 +4D)Ds, with |4D| ≤ DD (6.47)

where the matrix DD bounds the reaction wheel angular momentum and is assumed to have a

similar form as matrix Dw.15

The difference between the actual and estimated drift is

f − f ≈ I−1(4Iω

× − ω×4I +4Iω×4I

)Iω

+I−1(4Iω

× − ω×4D +4Iω×4D

)Ds

+I−1 (11−4I)L (6.48)

The bound F is determined as

|f − f | ≤ F = I−1∣∣∣(DIω

× − ω×DI + DIω×DI

)Iω

+(DIω

× − ω×DD + DIω×DD

)Ds

∣∣∣+I−1 (11 + DI)Lmax1 (6.49)

where 1 is a vector comprised of ones(1 = [1, 1, 1]T

).

The input matrix (B) is approximated from Eq. (6.41) and Eq. (6.43) as

B ≈ I−1(11−4I)(11 +4w)Ws

≈ I−1(11−4I)(11 +4w)I︸︷︷︸(11+4B)

B (6.50)

109

6.4 Sliding Mode Control Simulation

The uncertainty in the input matrix (4B) is

4B = I−1(11−4I)(11 +4w)I− 11

= I−1 (11 +4w −4I −4I4w) I− 11

= I−1 (4w −4I −4I4w) I (6.51)

The bound DB results in

DB =∣∣∣I−1 (4w −4I −4w4I) I

∣∣∣= I−1 (Dw + DI + DIDw) I (6.52)

All the parameter uncertainties are bounded for the spacecraft hardware-in-the-loop simulators.


This section demonstrates the robust nonlinear attitude controller using numerical simulation, Whorl−I,

and Whorl−II.

6.4.1 Numerical Sliding Mode Control Simulations

There are three numerical simulations presented in this section. The first and second simulation

demonstrates the performance of only the equivalent control part of the controller. The third simu-

lation demonstrates the performance of the complete sliding mode controller. The configuration and

parameters of the spacecraft closely match Whorl−I to get a better idea of expected performance of

the controller on the hardware simulator.

6.4.1.1 Equivalent Control Term With No Uncertainties

The first numerical simulation evaluates the equivalent control term (ueq) with no parameter uncer-

tainties and no un-modeled external torques. As previously mentioned Whorl−I has three reaction

110


wheels aligned along each body axis. The actual and estimated parameters are set equal to each

other for the simulation.

The reaction wheel spin axis orientation matrix is provided as

Ws = Ws =

1 0 0

0 1 0

0 0 1

The inertia matrix is provided as

I = I =

8.8 0.1 −0.2

0.1 7.7 −0.1

−0.2 −0.1 13.0

kg-m2

The spin axis inertia for each reaction wheel is

IWs1,2,3= IWs1,2,3

= 0.075 kg-m2

The numerical simulation results for the sliding mode controller with only the equivalent con-

trol term (i.e. u = ueq) with no parameter uncertainties are presented in Figures 6.1–6.5. The

controller is repeated here for convenience:

ueq = B+

(− f + RBR Rωr − ω×RBR Rωr −

14ΛB(δσ)δω −ΛIδσ

)(6.53)

The objective of the simulation is to see how well the sliding mode controller will track a time

varying reference spacecraft model using only the equivalent control term with no parameter uncer-

tainties and no external torques. The simulation parameters are shown in Table 6.1.

The equivalent control term is designed to drive the errors to zero with no parameter uncertain-

ties and no un-modeled external torques, which is demonstrated by the numerical simulation shown

111


Table 6.1: Sliding Mode Control Numerical Simulation Parameters with only Equivalent Con-trol with no Parameter Uncertainties

Parameter Valueω(t0) (0.017, 0.017, 0.087) rad/sσ(t0) (0.05, 0.01, -0.01)Λ diag( 0.4, 0.4, 0.4 ) rad/sΛI diag( 0.01, 0.01, 0.01 ) rad/s2

Lunmodeled ( 0, 0, 0 ) N-m

in Figures 6.1 – 6.5. Figures 6.2 and 6.4 show the angular rate and MRP errors go to zero after

approximately 140 seconds. Figure 6.3 shows the MRP vector where the MRP switching condition

is activated at approximately 70 seconds. The required control torque to drive the angular rate and

MRP errors to zero has a maximum amplitude of 0.1 N-m as shown in Figure 6.5.

Figure 6.1: Sliding Mode Control Numerical Simulation Angular Rates with only EquivalentControl

112


Figure 6.2: Sliding Mode Control Numerical Simulation Angular Rate Errors with onlyEquivalent Control

Figure 6.3: Sliding Mode Control Numerical Simulation MRPs with only Equivalent Control

113


Figure 6.4: Sliding Mode Control Numerical Simulation MRP Errors with only EquivalentControl

Figure 6.5: Sliding Mode Control Numerical Simulation Equivalent Control Time History

114


6.4.1.2 Equivalent Control Term With Uncertainties

The second numerical simulation evaluates the equivalent control term (ueq) with parameter un-

certainties and un-modeled external torques. The actual and estimated reaction wheel spin axis

orientation matrices are provided as

Ws =

0.9951 0.0697 0.0697

0.0697 0.9951 0.0697

0.0697 0.0756 0.0697

Ws =

1 0 0

0 1 0

0 0 1

The inertia matrix estimate is

I =

8.8 0.1 −0.2

0.1 7.7 −0.1

−0.2 −0.1 13.0

kg-m2

The actual inertia matrix is specified as

I =

10.6 0.3 0.2

0.3 7.7 0.2

0.2 0.2 15.6

kg-m2

The spin axis inertia for each reaction wheel is

IWs1,2,3= 0.075 kg-m2

115


The actual spin axis inertia for each reaction wheel is

IWs1,2,3= 0.072 kg-m2

The numerical simulation results for the sliding mode controller with only the equivalent con-

trol term (i.e. u = ueq) are presented in Figures 6.6 – 6.10. The controller is repeated for conve-

nience in Eq. (6.53). The objective of the simulation is to see how well the sliding mode controller

will track a time varying reference spacecraft model without using the robust control term. The

estimates of the parameters are used to determine the equivalent control and the actual values are

used in determining the response of the simulated spacecraft. The simulation parameters are shown

in Table 6.2. An un-modeled external torque is included in the simulation.

Table 6.2: Sliding Mode Control Numerical Simulation Parameters with only Equivalent Con-trol with Parameter Uncertainties

Parameter Valueω(t0) (0.017, 0.017, 0.087) rad/sσ(t0) (0.05, 0.01, -0.01)Λ diag( 0.4, 0.4, 0.4 ) rad/sΛI diag( 0.01, 0.01, 0.01 ) rad/sLunmodeled ( 0.05, -0.03, 0.01 ) N-m

Figures 6.7 and 6.9 show the angular rate and MRP errors do not go to zero. The expectation

is that the equivalent control term would not drive the errors to zero with the inclusion of parameter

uncertainties and un-modeled external torques in the simulation. In the development of the sliding

mode controller, the robust control term is included to account for the parameter uncertainties and

un-modeled torques.

116


Figure 6.6: Sliding Mode Control Numerical Simulation Angular Rates with only EquivalentControl

Figure 6.7: Sliding Mode Control Numerical Simulation Angular Rate Errors with onlyEquivalent Control

117


Figure 6.8: Sliding Mode Control Numerical Simulation MRPs with only Equivalent Control

Figure 6.9: Sliding Mode Control Numerical Simulation MRP Errors with only EquivalentControl

118


Figure 6.10: Sliding Mode Control Numerical Simulation Equivalent Control Time History

6.4.1.3 Robust Attitude Controller

The last numerical simulation involves the complete sliding mode controller (i.e. u = ueq − urob).

The results are shown in Figures 6.11 – 6.17. The control law is repeated here for convenience:

u = B+(− f + RBR Rωr − ω×RBR Rωr

−14ΛB(δσ)δω −ΛIδσ −K sat

(si(δω, δσ)

φi

))(6.54)

The objective of the simulation is to evaluate how well the sliding mode controller will track a time

varying reference spacecraft model with parameter uncertainties and un-modeled external torques.

The simulation parameters are shown in Table 6.3.

The bounding matrix for the inertia is

DI =

0.25 0.05 0.05

0.05 0.25 0.05

0.05 0.05 0.25

119


Table 6.3: Sliding Mode Control Numerical Simulation ParametersParameter Valueω(t0) (0.017, 0.017, 0.087) rad/sσ(t0) (0.05, 0.01, -0.01)Λ diag( 0.4, 0.4, 0.4 ) rad/sΛI diag( 0.01, 0.01, 0.01 ) rad/sφ 0.016 rad/sη ( 0.01, 0.01, 0.01 ) rad/s2

Lunmodeled ( 0.05, -0.03, 0.01 ) N-mΩ1 0 rad/sΩ2 0 rad/sΩ3 0 rad/s

The bounding matrix for the reaction wheels is

Dw =

0.10 0.09 0.09

0.09 0.10 0.09

0.09 0.09 0.10

The bounding matrix for the reaction wheel angular momentum (DD) is assumed to be the same as

the bounding matrix for the reaction wheels (Dw).

Figures 6.12 and 6.14 show the angular rate and MRP errors, which go to zero after approx-

imately 45 seconds. The robust control term does result in accounting for parameter uncertainties

and un-modeled external torques as clearly seen by comparing the previous simulations. Figure 6.17

shows the norm of the sliding variable (‖s‖2) time history, which goes to the sliding surface within

1 second. Figure 6.16 shows the maximum reaction wheel rate is 20 rad/s. The required control

torque shown in the numerical simulation indicates that the initial angular rates and gain selections

for the controller are appropriate for Whorl−I and Whorl−II, since the hardware simulators have a

maximum wheel rate of 100 rad/s.

The performance of the sliding mode controller using reaction wheels for attitude control will

vary due to the selection of Λ, ΛI, φ, and η. A critical consideration for determining the afore-

mentioned parameters is saturation of the reaction wheels. As Λ and ΛI increase the commanded

120


Figure 6.11: Sliding Mode Control Numerical Simulation Angular Rates

Figure 6.12: Sliding Mode Control Numerical Simulation Angular Rate Errors

121


control increases, which results in the reaction wheels increasing speed. The desirable trend is to

set Λ and ΛI to as small of values as possible, but it will take longer for the errors to converge to

zero. Obviously, there is a tradeoff between reaction wheel speeds and control time. Numerical

simulations indicate that η ≤ φ results in the best performance of the controller.

Figure 6.13: Sliding Mode Control Numerical Simulation MRPs

Figure 6.14: Sliding Mode Control Numerical Simulation MRP Errors

122


Figure 6.15: Sliding Mode Control Numerical Simulation Control Time History

Figure 6.16: Sliding Mode Control Numerical Simulation Reaction Wheel Time History

123


Figure 6.17: Sliding Mode Control Numerical Simulation Reaction Sliding Variable TimeHistory

6.4.2 Whorl−I Sliding Mode Control Simulation

The following subsections present the simulation results for the MRP sliding mode controller using

numerical simulations and Whorl−I. The simulation results for a regulator (ω → 0,σ → 0) and

tracking (ω → ωr,σ → σr) problem are shown. The estimates for the simulator parameters are



DI =

0.35 0.05 0.05

0.05 0.35 0.05

0.05 0.05 0.35

(6.55)


Dw =

0.10 0.08 0.08

0.08 0.10 0.08

0.08 0.08 0.10

(6.56)

124


The bounding matrix for the reaction wheel angular momentum (DD) is assumed to be the

same as the bounding matrix for the reaction wheels (Dw).

The actual simulator parameters used in the numerical simulation are not necessarily the true

parameters for the simulator, but chosen so as to satisfy the bounding matrices. Therefore there will

be expected differences between Whorl−I and the numerical simulations.

6.4.2.1 Regulator

The results of running the controller on Whorl−I along with a numerical simulation comparing the

results are presented. The MRP sliding mode controller parameters are provided in Table 6.4. The

controller results are shown in Figures 6.18 – 6.26. Whorl−I is given an initial angular rate about

the 3 body axis and then the controller is activated. The 18 second delay in the start of the controller

at the beginning of the simulation is due to the time necessary to initialize the three reaction wheels.

Table 6.4: Sliding Mode Control Simulation Parameters for Whorl−IParameter ValueControl Rate 2 HzObserve Rate 2 HzWheel Speed Query 10 Hzω(t0) (-0.006, 0.003, 0.144) rad/sσ(t0) (-0.006, -0.004, -0.229)Λ diag( 0.89, 0.89, 1.26 ) rad/sΛI diag( 0.05, 0.05, 0.1 ) rad/s2

φ 0.5 rad/sη ( 0.04, 0.04, 0.04 ) rad/s2

Lmax 0.02 N-mΩ1 0 rad/sΩ2 0 rad/sΩ3 0 rad/s

Figures 6.18 and 6.19 provide the angular rate time history for Whorl−I and the numerical

simulation respectively. Whorl−I has approximately zero angular rate about the 3 axis after 12

seconds into the control simulation. The numerical simulation has approximately zero angular

rate after 12 seconds into the simulation. The angular rates for Whorl−I have the same trends as

125



Figure 6.18: Whorl−I Angular Rates for MRP Sliding Mode Controller

Figure 6.19: Numerical Simulation Angular Rates for MRP Sliding Mode Controller

Figures 6.20 and 6.21 provide the MRP vector time history for Whorl−I and the numerical

simulation respectively. Whorl−I has approximately a zero MRP vector after 12 seconds into the

simulation. The numerical simulation has approximately a zero MRP error vector after 12 seconds

into the simulation. The MRP vector for Whorl−I has the same decreasing trend as predicted by


126


Figure 6.20: Whorl−I MRP Vector for MRP Sliding Mode Controller

Figure 6.21: Numerical Simulation MRP Vector for MRP Sliding Mode Controller

127


Figures 6.22 and 6.23 show the required control torque specified by the MRP sliding mode con-

troller for Whorl−I and the numerical simulation respectively. The control torques specified by the

controller for Whorl−I have a similar control torque trend as the numerical simulation. Figure 6.20

indicates a larger over shoot as compared with the numerical simulation. More control torque is

commanded to compensate for the over shoot for σ3.

Figure 6.22: Whorl−I Control Torque for MRP Sliding Mode Controller

Figure 6.23: Numerical Simulation Control Torque for MRP Sliding Mode Controller

Figure 6.24 shows the sliding variable specified by the MRP sliding mode controller for Whorl−I.

128


The sliding variable for Whorl−I goes to approximately zero after 10 seconds.

Figure 6.24: Whorl−I Sliding Variable for MRP Sliding Mode Controller


on Whorl−I and the numerical simulation respectively. The reaction wheels have the same wheel

speed trend as the numerical simulation. The final wheel speeds for Whorl−I are 0 rad/s for Ω1, -3

rad/s for Ω2, and 30 rad/s for Ω3. The final wheel speeds for the numerical simulation are -0.5 rad/s

for Ω1, -2 rad/s for Ω2, and 32 rad/s for Ω3.

6.4.2.2 Tracking

The results of running the controller on Whorl−I along with a numerical simulation for tracking a

time varying attitude are presented. The MRP sliding mode controller parameters are provided in

Table 6.5.

The controller results are shown in Figures 6.27 – 6.26. Whorl−I is at rest and then the con-

troller is turned on. The 48 second delay in the start of the controller at the beginning of the

simulation is due to the time necessary to initialize the three reaction wheels and to spin up Ω3 to a

steady state condition of -50 rad/s.

Figures 6.27 and 6.28 provide the angular rate time history for Whorl−I and the numerical sim-

129


Figure 6.25: Whorl−I Reaction Wheel Speed for MRP Sliding Mode Controller

Figure 6.26: Numerical Simulation Reaction Wheel Speed for MRP Sliding Mode Controller

130


Table 6.5: Sliding Mode Control Simulation Parameters for Whorl−IParameter ValueControl Rate 2 HzObserve Rate 2 HzWheel Speed Query 10 Hzω(t0) (0.000, 0.002, 0.004) rad/sσ(t0) (-0.005, 0.006, 0.247)Λ diag( 0.89, 0.89, 1.26 ) rad/sΛI diag( 0.05, 0.05, 0.1 ) rad/s2

φ 0.5 rad/sη ( 0.04, 0.04, 0.04 ) rad/s2

Lmax 0.02 N-mΩ1 0 rad/sΩ2 0 rad/sΩ3 -50 rad/s

ulation respectively. Figures 6.29 and 6.30 provide the angular rate error time history for Whorl−I

and the numerical simulation respectively. Whorl−I has approximately zero angular rate error after

25 seconds into the control simulation. The numerical simulation has approximately zero angular

rate error after 25 seconds into the simulation. The angular rate errors for Whorl−I have the same

decreasing trend as predicted by the numerical simulation.

Figure 6.27: Whorl−I Angular Rates for MRP Sliding Mode Controller

131



Figure 6.29: Whorl−I Angular Rate Errors for MRP Sliding Mode Controller

132


Figure 6.30: Numerical Simulation Angular Rate Errors for MRP Sliding Mode Controller

Figures 6.31 and 6.32 provide the MRP vector time history for Whorl−I and the numerical sim-

ulation respectively. Figures 6.33 and 6.34 provide the MRP error vector time history for Whorl−I

and the numerical simulation respectively. Whorl−I has approximately a zero MRP error vector

offset for σ3 of 0.02 after 80 seconds into the control simulation. The numerical simulation has

approximately a zero MRP error vector after 70 seconds into the simulation. The MRP vector for

Whorl−I has the same decreasing trend as predicted by the numerical simulation.


133




134



Figures 6.35 and 6.36 show the required control torque specified by the MRP sliding mode

controller for Whorl−I and the numerical simulation respectively. The control torques specified by

the controller for Whorl−I have a similar control torque trend as the numerical simulation.

Figure 6.35: Whorl−I Control Torque for MRP Sliding Mode Controller

Figure 6.37 shows the sliding variable specified by the MRP sliding mode controller for Whorl−I.

The sliding variable for Whorl−I goes to approximately zero after 40 seconds.

135



Figure 6.37: Whorl−I Sliding Variable for MRP Sliding Mode Controller

136



Whorl−I and the numerical simulation respectively. The reaction wheels have a similar speed trend

as the numerical simulation. The final wheel speeds for Whorl−I are 0 rad/s for Ω1, 0 rad/s for Ω2,

and -25 rad/s for Ω3. The final wheel speeds for the numerical simulation are 0 rad/s for Ω1, 0 rad/s

for Ω2, and -27 rad/s for Ω3.

Figure 6.38: Whorl−I Reaction Wheel Speed for MRP Sliding Mode Controller


137


6.4.3 Whorl−II Sliding Mode Control Simulation

The following subsections present the simulation results for the MRP sliding mode controller using

numerical simulations and Whorl−II. The simulation results for a regulator (ω → 0,σ → 0) and

rest to rest (ω → 0,σ → 0) maneuver are shown. The estimates for the simulator parameters are



DI =

0.35 0.05 0.05

0.05 0.35 0.05

0.05 0.05 0.35

(6.57)


Dw =

0.10 0.08 0.08

0.08 0.10 0.08

0.08 0.08 0.10

(6.58)

The bounding matrix for the reaction wheel angular momentum (DD) is assumed to be the

same as the bounding matrix for the reaction wheels (Dw).

6.4.3.1 Regulator

The results of running the controller on Whorl−II along with a numerical simulation comparing the


controller results are shown in Figures 6.40 – 6.48. Whorl−II is given an initial angular rate about

the 3 body axis and then the controller is activated. The 45 second delay in the start of the controller

at the beginning of the simulation is due to the time necessary to initialize the three reaction wheels.



138


Table 6.6: Sliding Mode Control Simulation Parameters for Whorl−IIParameter Valueω(t0) (-0.040, -0.047, -0.183) rad/sσ(t0) (0.026, 0.033, 0.773)Λ diag( 0.90, 0.90, 0.28 ) rad/sΛI diag( 0.05, 0.05, 0.005 ) rad/sφ 0.5 rad/sη ( 0.04, 0.04, 0.04 ) rad/s2

Lmax 0.02 N-mΩ1 10 rad/sΩ2 -10 rad/sΩ3 10 rad/s




Figure 6.40: Whorl−II Angular Rates for MRP Sliding Mode Controller


simulation respectively. Whorl−II has approximately a constant offset from zero error for σ3 after

80 seconds into the simulation. The numerical simulation has approximately a zero MRP error

vector after 60 seconds into the simulation. The steady state offset for σ3 is approximately 0.02. The

139



MRP vector for Whorl−II has the same decreasing trend as predicted by the numerical simulation.

Figure 6.42: Whorl−II MRP Vector for MRP Sliding Mode Controller


controller for Whorl−II and the numerical simulation respectively. The control torques specified by

the controller for Whorl−II have a similar control torque trend as the numerical simulation.

Figure 6.46 shows the sliding variable specified by the MRP sliding mode controller for Whorl−II.

The sliding variable for Whorl−II goes to approximately 0 after 40 seconds.

140



Figure 6.44: Whorl−II Control Torque for MRP Sliding Mode Controller

141



Figure 6.46: Whorl−II Sliding Variable for MRP Sliding Mode Controller

142



on Whorl−II and the numerical simulation respectively. The reaction wheels have a similar speed


proximately -60 rad/s as compared with -55 rad/s for the reaction wheel aligned along the 3 axis.

The discrepancy in wheel speeds between Whorl−II and the numerical simulation is attributed to

the simplified reaction wheel motor torque model used in the numerical simulation.

Figure 6.47: Whorl−II Reaction Wheel Speed for MRP Sliding Mode Controller


143


6.4.3.2 Rest to Rest Maneuver

The results of running the controller on Whorl−II along with a numerical simulation comparing the


controller results are shown in Figures 6.49 – 6.57. Whorl−II is at rest and then the controller is

activated. The 22 second delay in the start of the controller at the beginning of the simulation is due

to the time necessary to initialize the three reaction wheels.

Table 6.7: Sliding Mode Control Simulation Parameters for Whorl−IIParameter Valueω(t0) (-0.003, 0.003, -0.007) rad/sσ(t0) (0.002, 0.002, 0.621)Λ diag( 0.90, 0.90, 0.28 ) rad/sΛI diag( 0.05, 0.05, 0.005 ) rad/sφ 0.5 rad/sη ( 0.04, 0.04, 0.04 ) rad/s2

Lmax 0.02 N-mΩ1 10 rad/sΩ2 -10 rad/sΩ3 10 rad/s







simulation respectively. Whorl−II has approximately a zero MRP vector after 70 seconds into the

simulation. The numerical simulation has approximately a zero MRP vector after 70 seconds into

the simulation. The MRP vector for Whorl−II has the same decreasing trend as predicted by the

numerical simulation.


144


Figure 6.49: Whorl−II Angular Rates for MRP Sliding Mode Controller


145


Figure 6.51: Whorl−II MRP Vector for MRP Sliding Mode Controller


146


controller for Whorl−II and the numerical simulation respectively. The control torque specified by

the controller for Whorl−II has a similar control torque trend as the numerical simulation.

Figure 6.53: Whorl−II Control Torque for MRP Sliding Mode Controller


Figure 6.55 shows the sliding variable specified by the MRP sliding mode controller for Whorl−II.

The sliding variable for Whorl−II goes to approximately zero after 55 seconds.


on Whorl−II and the numerical simulation respectively. The reaction wheels have a similar speed

147


Figure 6.55: Whorl−II Sliding Variable for MRP Sliding Mode Controller


proximately 8 rad/s as compared with 12 rad/s for the reaction wheel aligned along the 3 axis. The

discrepancy in wheel speeds between Whorl−II and the numerical simulation is attributed to the

simplified reaction wheel motor torque model used in the numerical simulation.

Figure 6.56: Whorl−II Reaction Wheel Speed for MRP Sliding Mode Controller

148

6.5 Summary


6.5 Summary

In this chapter we develop a robust nonlinear attitude control law that tracks a reference attitude

while bounding system uncertainties. The development of the controller is described along with a

method to determine the equivalent control term gain matrices and robust control term gain matrix.

Bounding the spacecraft system uncertainties is discussed. Numerical simulations, Whorl−I, and

Whorl−II successfully demonstrate the spacecraft MRP attitude sliding mode controller.

The hardware-in-the-loop spacecraft attitude control simulators have demonstrated the perfor-

mance of several angular rate, MRP, and robust nonlinear attitude controllers. The following chap-

ter investigates the relative position control between spacecraft in low Earth orbit. A spacecraft

formation flying control simulation using a GPS simulator and the flight computers on the hardware

simulators is demonstrated using a classical orbital element feedback controller.

149

7 Classical Orbital Element Controller

A nonlinear Lyapunov based classical orbital element (COE) feedback controller based on the work

of Naasz, Karlgaard, and Hall in Refs. 2 and 62 is presented in the following sections along with a

Global Positioning System (GPS) hardware-in-the-loop simulation of multiple spacecraft flying in

formation using the COE controller.

Six pieces of information are necessary to identify the location of a spacecraft within an orbit

relative to an inertial reference frame (N ). The position (r) and velocity (r) vectors are one rep-

resentation. Another representation is the classical orbital elements. The classical orbital elements

have six elements. The semi-major axis (a) and eccentricity (e) define the spacecraft orbit and

shape. The inclination (i), longitude of ascending node (Ω), and argument of perigee (ω) define the

orbit plane orientation. The mean anomaly (M) provides the location of the spacecraft on the orbit

trajectory at time t.25,63,64

7.1 Controller

The COE feedback controller is a technique to control the six orbital element (a, e, I , Ω, ω, M0)

relative motion of a spacecraft in low Earth orbit (LEO). The development of the controller from

Ref. 62 is summarized in this section.

150

7.1 Controller

Gauss’s variational equations are25,63

dadt

=2a2

h

(e sin f ur +

p

ruθ

)(7.1)

dedt

=1h

(p sin f ur +

((p+ r) cos f + re

)uθ

)(7.2)

didt

=r cos θh

uh (7.3)

dΩdt

=r sin θh sin i

uh (7.4)

dωdt

=1he

(− p cos f ur + (p+ r) sin f uθ

)− r sin θ cos i

h sin iuh (7.5)

dMdt

= n+b

ahe

((p cos f − 2re) ur − (p+ r) sin f uθ

)(7.6)

where h is the angular momentum, f is the true anomaly, p is the semi-latus rectum, r is the scalar

orbit radius, θ is the argument of latitude (ω + f), b is the semi-minor orbit axis, and ur, uθ, uh are

the control acceleration components in the local vertical, local horizon (LVLH) frame. The LVLH

frame is a rotating coordinate frame where ur is the control in the orbit radial direction (er), uh

is the control in the orbit normal direction (eh), and uθ is the control in the direction (eθ) that is

perpendicular to the radial and orbit normal directions as shown in Figure 7.1.25,63

Gauss’ variational equations have the nonlinear structure as62

x = f (x) + Bu (7.7)

where x has the elements a, e, i, Ω, ω, and M . The mean anomaly is

M = M0 + nt (7.8)

where n is the mean motion and is equal to√µ

a3where µ is the gravitational parameter of the Earth

with a value of 3.986× 105 km3/s2.

151

7.1 Controller

Figure 7.1: Orbit Plane Coordinate Frame Illustration

Replacing M in Eq. (7.6) with M0 results in

x = Bu (7.9)

where x has the elements a, e, i, Ω, ω, and M0. The control matrix (B) is

B =

2a2e sin fh

2a2p

hr0

p sin fh

(p+ r) cos f + re

h0

0 0r cos θh

0 0r sin θh sin i

−p cos fhe

(p+ r) sin fhe

−r sin θ cos ih sin i

b(p cos f − 2re)ahe

−b(p+ r) sin fahe

0

(7.10)

152

7.1 Controller

Reference 2 developed the following Lyapunov based COE feedback controller by expanding

Ilgen’s work in Ref. 65:

u = −

2a2e sin fh

2a2p

hr0

p sin fh

(p+ r) cos f + re

h0

0 0r cos θh

0 0r sin θh sin i

−p cos fhe

(p+ r) sin fhe

−r sin θ cos ih sin i

b(p cos f − 2re)ahe

−b(p+ r) sin fahe

0

T

Kaδa

Keδe

Kiδi

KΩδΩ

Kωδω

KMδMo

(7.11)

where Kδ(·) are the gains and δ(·) is defined as the relative orbital element values of the controlled

spacecraft with respect to the reference spacecraft as

δ(·) = (·)controlled − (·)reference (7.12)

The symbol (·) indicates the appropriate orbital element. Reference 2 proved the control law in

Eq. (7.11) is globally asymptotically stable.

153

7.2 GPS Formation Flying Simulation using COE Controller

Reference 2 developed the following expressions for the full state feedback gains:

Ka =h2

4a4(1 + e)2∆t(7.13)

Ke =h2

4p2∆t(7.14)

Ki =(h+ eh cos(ω + arcsin(e sinω))

p(−1 + e2 sin2 ω)

)2 1∆t

(7.15)

KΩ =(h sin(i) (−1 + e sin(ω + arcsin(e sinω)))

p(1− e2 cos2 ω)

)2 1∆t

(7.16)

Kω =e2h2

4p2

(1− e2

4

)1

∆t(7.17)

KM =a2e2h2

4b2p2

(1− e2

4

)∆t (7.18)

where ∆t is the the thruster duration and the gains are calculated at the beginning of the maneuver

as a function of the desired orbit and held constant.2


The results of a GPS hardware-in-the-loop simulation of two spacecraft flying in formation are

presented. The simulations involve a chief spacecraft in a low Earth orbit (LEO), while a deputy

spacecraft maintains an orbit position relative to the chief spacecraft. In order to maintain the for-

mation an orbit correction maneuver (OCM) for the deputy spacecraft is required. The control of

the OCM is accomplished using the COE controller presented in the previous section and simulat-

ing continual impulsive thrusting for only the deputy spacecraft. The COE controller requires the

relative position of the six orbital elements (a, e, I , Ω, ω, M0). The deputy communicates with the

chief spacecraft to obtain the current orbit position of the chief spacecraft, which is determined by

a numerical orbit propagator. The position of the deputy is determined from a GPS receiver that is

connected to a GPS hardware-in-the-loop simulator. The GPS simulator creates an RF signal based

on a simulated trajectory, which results in the GPS receiver calculating the navigation solution for

154


the simulated trajectory. From the relative positions of the spacecraft the COE controller calculates

the OCM for the deputy spacecraft.

The following sections discuss the implementation of the COE controller and the results of the

LEO formation flying simulation.

7.2.1 Implementation of Classical Orbital Element Controller

For this study the controller is used to control the relative orbit of the deputy spacecraft with re-

spect to the chief spacecraft. The deputy spacecraft retrieves the navigation solution from the GPS

receiver, which includes orbital perturbations. The mean orbital elements are determined by trans-

forming WGS-84 coordinates to Earth Centered Earth Fixed (ECEF) coordinates to Earth Centered

Inertial (ECI) coordinates to osculating orbit elements and then mapped to mean orbital elements.

The mapping between osculating and mean orbital elements is a first-order mapping algorithm out-

lined in Ref. 25. Mean elements are orbital elements with the short and long period motion removed.

The COE controller uses the mean orbital elements to determine the control acceleration.

The chief spacecraft determines its position by retrieving the ECI coordinates from the numeri-

cal orbit propagator through TCP/IP and converts the coordinates to osculating orbital elements and

maps the elements to mean orbital elements. The controller scheme is illustrated in Figure 7.2.

A desired relative orbit between the chief and deputy spacecraft is specified and the COE con-

troller determines the appropriate ∆V for the deputy spacecraft. Most spacecraft have fixed magni-

tude thrusters. For this study the thrust magnitude is constrained by the available thrust magnitude

(Tavailable) with instantaneous thrust direction. The deputy spacecraft will thrust if and only if the

requested thrust is greater than the available thrust(Trequested > Tavailable

). The ∆V in the instanta-

neous requested direction is determined from

∆V =Tavailable

m∆t (7.19)

where m is the mass and ∆t is the thrust duration of the deputy spacecraft.

The available thrust magnitude and thrust direction are assumed to be known perfectly. The

155


Figure 7.2: Spacecraft Formation Flying Simulation Scheme

∆V is sent to the numerical orbit propagator computer through TCP/IP as shown in Figure 3.14.

7.2.2 LEO Formation Flying Simulation

The real-time hardware-in-the-loop LEO formation flying simulation presented in this section demon-

strates the ability of the GPS simulator and the DSACSS facility to simulate controlling the semi-

major axis of the deputy spacecraft relative to the chief spacecraft.

The chief spacecraft has a mass of 100 kg and has no orbital control capability. The spacecraft

obtains its position and velocity by querying a numerical orbit propagator, which includes orbital

perturbations. The position and velocity are returned through TCP/IP in the ECI reference frame

and assumed to be known perfectly. The osculating orbital elements are determined from the po-

sition and velocity and mapped into mean orbital elements. The mean orbital elements are sent to

the deputy spacecraft. The initial mean orbital elements for the chief spacecraft are provided in

Table 7.1. The chief spacecraft position and velocity is queried at 0.5 Hz.

The deputy spacecraft has a mass of 100 kg with an orbital thrust magnitude capability of 1.0 N

with an impulse thrust duration of 2.0 seconds. As previously mentioned the deputy spacecraft

156


Table 7.1: Chief and Deputy Spacecraft Initial ConditionsParameter Chief Spacecraft Deputy Spacecrafta 7116.0 km 7113.2 kme 0.0486 0.0487i 5.0 5.0

Ω -0.004 39.329

ω 0.001 0.352

M0 0.006 0.321Mass 100 kg 100 kgThruster Magnitude NA 1.0 NThrust Duration NA 2.0 secNote: (·) indicates mean element.

obtains its position and velocity by querying the Ashtech G12 HDMA GPS receiver through TCP/IP.

The returned filtered position and velocity in the WGS-84 reference frame is converted to ECEF

coordinates and then to ECI coordinates as illustrated in Figure 7.2. The mean orbital elements are

mapped from the osculating orbital elements, which are determined from the position and velocity

in the ECI reference frame. The initial mean orbital elements for the deputy spacecraft are provided

in Table 7.1. The deputy spacecraft queries the GPS receiver at 0.5 Hz.

The desired semi-major axis difference between the deputy spacecraft and chief spacecraft is

15 km. The initial relative semi-major axis between the chief spacecraft and deputy spacecraft

is 2.7 km. The deputy spacecraft desired mean semi-major axis is approximately 7131 km. The

simulation results are shown in Figures 7.3 – 7.14.

The chief spacecraft mean semi-major axis, eccentricity, and inclination are shown in Fig-

ures 7.3 – 7.5. The mean semi-major axis is approximately 7116.0 km throughout the simulation.

The mean eccentricity is approximately 0.048 and the mean inclination is approximately 5.0 degrees

throughout the simulation.

The PDOP, HDOP, VDOP, and TDOP values for the deputy spacecraft GPS receiver are shown

in Figure 7.6. The DOP values range from 1 to 3 throughout the simulation. The number of GPS

satellites used to determine the navigation solution range from 7 to 10 throughout the simulation as


157


Figure 7.3: Mean Semi-Major Axis Time History for Chief Spacecraft

Figure 7.4: Mean Eccentricity Time History for Chief Spacecraft

158


Figure 7.5: Mean Inclination Time History for Chief Spacecraft

Figure 7.6: Deputy Spacecraft GPS Receiver Parameters (PDOP, HDOP, VDOP, and TDOP)

159



Figures 7.8 – 7.10 show the mean semi-major axis, eccentricity, and inclination of the deputy

spacecraft during the simulation. The mean semi-major axis starts at approximately 7113 km and

increases to approximately 7131 km over 19 minutes. The mean eccentricity is approximately 0.05

and the inclination remains approximately 5 degrees throughout the simulation.

Figure 7.8: Mean Semi-Major Axis Time History for Deputy Spacecraft

160


Figure 7.9: Mean Eccentricity Time History for Deputy Spacecraft

Figure 7.10: Mean Inclination Time History for Deputy Spacecraft

161


The deputy spacecraft mean semi-major axis, mean eccentricity, and mean inclination errors

are shown in Figures 7.11 – 7.13. The error is defined as the desired orbital element subtracted

from the current orbital element. The mean semi-major axis error starts at approximately −18 km

and is approximately zero after 19 minutes. The mean eccentricity and inclination errors remain

approximately zero throughout the simulation.

Figure 7.11: Mean Semi-Major Axis Error Time History for Deputy Spacecraft

Figure 7.12: Mean Eccentricity Error Time History for Deputy Spacecraft

162


Figure 7.13: Mean Inclination Error Time History for Deputy Spacecraft

The control required to drive the mean orbital element errors to zero is shown in Figure 7.14.

The control components correspond to the radial, in-track, and cross-track directions. The total ∆V

to accomplish the 19 minute orbit maneuver is 10.52 m/s.

Figure 7.14: Deputy Spacecraft Control Acceleration History

The formation flying simulation demonstrates the successful implementation of the GPS hardware-

in-the-loop simulator into DSACSS. The system provides real-time navigational data for investigat-

ing a variety of control strategies.

163

7.3 Summary

7.3 Summary

In this chapter we summarize the classical orbital element feedback controller developed in Ref. 2.

We implement the orbit controller into DSACSS and successfully demonstrate the performance of

the controller using a space qualified GPS receiver and GPS simulator. A low Earth orbit formation

flying simulation is conducted using the flight computers from the attitude hardware simulators.

The COE controller successfully controls the relative orbital elements between the chief and deputy

spacecraft while increasing the semi-major axis of the deputy spacecraft to the desired relative value.

164

8 Sliding Mode Classical Orbital Element

Controller

In this chapter we develop a classical orbital element (COE) feedback control law based on the slid-

ing mode methodology while bounding system uncertainties. Gauss’ form of Lagrange’s Planetary

Equations are modified to provide the control vector in the orbit velocity direction, orbit normal di-

rection, and direction perpendicular to the previous two directions. A COE control law is developed

and a low Earth orbit (LEO) simulation is presented using the GPS hardware-in-the-loop simulator.

8.1 Modification to Gauss’ form of Lagrange’s Planetary

Equations

Gauss’ variational equations are shown in Eqs. 7.1 – 7.6 and are repeated here for convenience:

dadt

=2a2

h

(e sin fur +

p

ruθ

)dedt

=1h

(p sin fur + ((p+ r) cos f + re)uθ)

didt

=r cos θh

uh

165

8.1 Modification to Gauss’ form of Lagrange’s Planetary Equations

dΩdt

=r sin θh sin i

uh

dωdt

=1he

(−p cos far + (p+ r) sin faθ)−r sin θ cos ih sin i

uh

dMdt

= n+b

ahe((p cos f − 2re) ar − (p+ r) sin faθ)

where h is the angular momentum, f is the true anomaly, p is the semi-latus rectum, r is the scalar

orbit radius, θ is the argument of latitude (ω + f), b is the semi-minor orbit axis, and ur, uθ, uh are

the control acceleration components in the local vertical, local horizon (LVLH) frame. The LVLH

frame is a rotating coordinate frame where ur is the control in the orbit radial direction (er), uh is

the control in the orbit plane normal direction (eh), and uθ is the control in the direction (eθ) that

is perpendicular to the radial and orbit normal directions as shown in Figure 8.1.25,63

We express the control (u) in terms of the orbit velocity direction (ev), orbit normal direction

(eh), and the direction (en) that is perpendicular to the velocity and orbit normal directions as


Figure 8.1: Orbit Plane Coordinate Frame Illustration

166

8.1 Modification to Gauss’ form of Lagrange’s Planetary Equations

As can be seen in Figure 8.1, going from er, eθ, eh to en, ev, eh involves a rotation about

the eh axis. The orbit plane control accelerations in the er and eθ directions are25

ur

uθ

=h

pv

p/r e sin f

−e sin f p/r

un

uv

(8.1)

Substituting Eq. (8.1) into Eqs. (7.1) – (7.6) results in the following form of Gauss’ variational

equations:25

dadt

=2a2v

µuv (8.2)

dedt

=1v

(ra

sin fun + 2 (e+ cos f)uv)

(8.3)

didt

=r cos θh

uh (8.4)

dΩdt

=r sin θh sin i

uh (8.5)

dωdt

=1ev

(−(

2e+r

a

)cos fan + 2 sin fav

)− r sin θ cos i

h sin iuh (8.6)

dMdt

= n+b

aev

(r

acos fan − 2

(1 + e2 r

p

)sin fav

)(8.7)

The modified form of Gauss’ variational equations has the nonlinear structure of

x = f (x) + Bu (8.8)

where x has the elements a, e, i, Ω, ω, and M . The mean anomaly is

M = M0 + nt (8.9)

where n is the mean motion and is equal to√µ

a3where µ is the gravitational parameter for the

Earth with a value of 3.986× 105 km3/s2.

167

8.2 Sliding Mode Orbit Control Law with Position Uncertainty

Replacing M in Eq. (8.7) with M0 results in

x = H(oe)u (8.10)

where x has the elements a, e, i, Ω, ω, and M0. The elements of the control matrix (H) are

H(oe) =

02a2v

µ0

r

avsin f

2v

(e+ cos f) 0

0 0r cos θh

0 0r sin θh sin i

− 1ev

(2e+

r

a

)cos f

2ev

sin f −r sin θ cos ih sin i

br

a2evcos f − 2b

aev

(1 + e2 r

p

)sin f 0

(8.11)

The following sections develop COE control laws based on the sliding mode methodology

while bounding system uncertainties.


The desire is to design a robust nonlinear orbit controller that tracks a time-varying reference tra-

jectory while bounding position uncertainty.

8.2.1 Control Law

The desired structured form for sliding mode control is66

x = Bu (8.12)

168


From Eq. (8.10), the structured form is

x = H(oe)︸︷︷︸B

u (8.13)

The sliding variable (s) is defined as

s(δoe) , Λδoe (8.14)

where Λ is a 6× 6 positive-definite gain matrix, oe is the 6× 1 matrix of orbital elements, and δoe

is defined as the relative orbital elements with respect to the reference orbital elements:

δoe = oe− oer (8.15)

Ensuring s converges to the sliding surface can be achieved if each element of the sliding

variable si satisfies the sliding condition (also known as the reachability condition60) specified as

12

ddts(δoe)2

i ≤ −ηi|s(δoe)i| (8.16)

where ηi is a positive constant. The sliding surface condition ensures that the system trajectory

point to the surface and that once the system trajectory reaches the sliding surface it remains on that

surface.14

The control acceleration vector u ensures that the sliding condition given in Eq. (8.16) is met

in the presence of model uncertainty. The controller that satisfies the sliding condition is given as a

sign or saturation function:66

u = −B+K sign (s(δoe)) (8.17)

u = −B+K sat(si(δoe)φi

)(8.18)

where K is a 6×6 diagonal matrix whose values are not yet identified and B+ is the pseudo-inverse

169


for the estimate of the control matrix.

The control matrix, B, is a 6 × 3 matrix and the inverse is determined by a pseudo-inverse or

weighted pseudo-inverse. The pseudo-inverse is25

B+ = BT(BBT

)−1(8.19)

The pseudo-inverse can also be found using singular value decomposition:

B = USVT (8.20)

where U and V are orthogonal m× n matrices and S is a positive diagonal n× n matrix. Then the

pseudo-inverse is67

B+ = V(STS

)−1STUT (8.21)

The weighted pseudo-inverse is25

B+ = WBT(BWBT

)−1(8.22)

where W is a diagonal matrix with user desired weighted values.

The sign function is defined as14

sign (si(δoe)) =

−1 if si(δoe) < 0

1 if si(δoe) ≥ 0(8.23)

The saturation function is defined as14

sat(si(δoe)φi

)=

si(δoe)φi

if |si(δoe)| ≤ φi

sign(si(δoe)) if |si(δoe)| > φi

(8.24)

170


where φi is a boundary layer thickness that will eliminate chatter if appropriately chosen so that the

un-modeled high-frequency dynamics are not excited.

8.2.2 Gains and Parameter Bound Selection

This section develops the gain matrix K, the gain matrix Λ, and the control bound DB. The

derivative of the sliding variable is

s(δoe) = Λδoe

= Λoe

= ΛBu (8.25)

The gain (K) expression is solved for by substituting the control signal (u) given in Eq. (8.17)

into s(δoe) and solving for the sliding surface constraint in Eq. (8.16). The result is

12

ddt(sTs

)= s(δoe)Ts(δoe)

= (ΛBu)T s(δoe)

= −(ΛBB+Ksign (s(δoe))

)Ts(δoe) (8.26)

The diagonal matrix terms (diag(K)) are determined by satisfying the sliding condition in

Eq. (8.16), which results in

−η ≥ −ΛBB+Ksign (s(δoe)) (8.27)

171


The matrix diag(K) expression is solved for by

ΛBB−1Ksign (s(δoe)) ≥ η

Ksign (s(δoe)) ≥(ΛBB+

)−1η

Ksign (s(δoe)) ≥ BB+Λ−1η

diag(K) ≥∣∣∣BB+Λ−1η

∣∣∣ (8.28)

The control matrix is bounded as

B = (11 +4B) B (8.29)

where |4B| ≤ DB. From Eq. (8.29), BB−1 results in

BB+ = (11 +4B)−1BB+

= (11 +4B)−1BBT(BBT

)−1

= (11 +4B)−1

≤ (11−DB)−1 (8.30)

The diagonal matrix terms, diag(K), are rewritten as

diag(K) =∣∣∣(11−DB)−1 Λ−1η

∣∣∣ (8.31)

The full state feedback gains (Λ) developed in Ref. 2 and shown in Eqs. (7.13)–(7.18) are

developed based on the response of each orbital element to thrust impulses. These gain expressions

172


are chosen for the sliding mode orbit controller and repeated here for convenience:

Λa =h2

4a4(1 + e)2∆t(8.32)

Λe =h2

4p2∆t(8.33)

Λi =(h+ eh cos(ω + arcsin(e sinω))

p(−1 + e2 sin2 ω)

)2 1∆t

(8.34)

ΛΩ =(h sin(i) (−1 + e sin(ω + arcsin(e sinω)))

p(1− e2 cos2 ω)

)2 1∆t

(8.35)

Λω =e2h2

4p2

(1− e2

4

)1

∆t(8.36)

ΛM =a2e2h2

4b2p2

(1− e2

4

)∆t (8.37)

where ∆t is the the thruster duration and Λ = diag(Λa,Λe,Λi,ΛΩ,Λω,ΛM ). The gains are calcu-

lated at the beginning of the maneuver as a function of the desired orbit and held constant throughout

the maneuver.2

The actual and estimated control matrices are

B = H(oe) (8.38)

B = H(oe) (8.39)

where H(oe) is bounded as

H(oe) = (11 +4H) H(oe) for |4H| ≤ DH (8.40)

173


The control matrix bound DB is

B = H(oe)

= (11 +4H) H(oe)

= (11 +4H)︸︷︷︸11+4B

B

|4B| = |4H|

DB = DH (8.41)

In summary the robust orbit controller with the saturation function selected is

u = −B+K sat(si(δoe)φi

)(8.42)

s(δoe) = Λδoe (8.43)

diag(K) =∣∣∣(11−DB)−1 Λ−1η

∣∣∣ (8.44)

where B is provided in Eq. (8.11).

8.2.3 LEO Control Simulation using GPS Simulator

The real-time hardware-in-the-loop simulation presented in this section demonstrates the ability of

the sliding mode control law shown in Eq. (8.42) to increase the semi-major axis of a spacecraft

in LEO. The Ashtech G12 GPS receiver is used for the simulation along with the GPS simulator

described in Section 3.4.

The spacecraft obtains its position and velocity by querying the Ashtech G12 HDMA GPS

receiver through TCP/IP. The returned filtered position and velocity in the WGS-84 reference frame

are converted to ECEF coordinates and then to ECI coordinates as illustrated in Figure 8.2. The

mean orbital elements are mapped from the osculating orbital elements, which are determined from

the position and velocity in the ECI reference frame.

The initial mean orbital elements of the spacecraft, and other simulation parameters, are pro-

174


Figure 8.2: Spacecraft Orbit Control Scheme

vided in Table 8.1. The spacecraft queries the GPS receiver at 0.5 Hz. The control matrix bound

is

DB =

0.05 0 0 0 0 0

0 0.05 0 0 0 0

0 0 0.05 0 0 0

0 0 0 0.05 0 0

0 0 0 0 0.05 0

0 0 0 0 0 0.05

(8.45)

Table 8.1: Spacecraft Simulation ParametersParameter Spacecrafta 7109.1 kme 0.0485i 5.0

Ω 39.41

ω 0.06

M0 1.23Λ diag( 1.26×10−7, 7.02, 30.9, 0.21, 0.017, 0.067 )φ ( 0.2, 0.2, 0.3, 0.9, 0.9, 0.9 )η ( 0.01, 1×10−5, 0.01, 1×10−7, 1×10−6, 1×10−6 )Mass 100 kgNote: (·) indicates mean element.

The desired semi-major axis increase is 20 km for the spacecraft. The spacecraft desired mean

semi-major axis is approximately 7129 km. The simulation results are shown in Figures 8.3 – 8.11.

175


The PDOP, HDOP, VDOP, and TDOP values for the spacecraft GPS receiver are shown in

Figure 8.3. The DOP values range from 0.6 to 2.8 during the simulation. The number of GPS

satellites used to determine the navigation solution range from 8 to 10 during the simulation as


Figure 8.3: Spacecraft GPS Receiver Parameters (PDOP, HDOP, VDOP, and TDOP)


176


Figures 8.5 – 8.7 show the mean semi-major axis, eccentricity, and inclination of the orbit

during the simulation. The mean semi-major axis starts at approximately 7109 km and increases

to approximately 7129 km over 19 minutes. The mean eccentricity is approximately 0.05 and the

inclination remains approximately 5 degrees throughout the simulation.

Figure 8.5: Mean Semi-Major Axis Time History for Spacecraft

The orbit mean semi-major axis, mean eccentricity, and mean inclination errors are shown in

Figures 8.8 – 8.10. The mean semi-major axis error starts at approximately −20 km and is approx-

imately zero after 19 minutes. The mean inclination error is approximately zero after 32 minutes

into the simulation.

The control required to drive the mean orbital element errors to zero is shown in Figure 8.11.

The control components correspond to the ev, eh, and en directions. The total ∆V to accomplish

the successful semi-major axis increase maneuver is 19.6 m/s in 19 minutes .

177


Figure 8.6: Mean Eccentricity Time History for Spacecraft

Figure 8.7: Mean Inclination Time History for Spacecraft

178


Figure 8.8: Mean Semi-Major Axis Error Time History for Spacecraft

Figure 8.9: Mean Eccentricity Error Time History for Spacecraft

179


Figure 8.10: Mean Inclination Error Time History for Spacecraft

Figure 8.11: Spacecraft Control Acceleration History

180

8.3 Summary

8.3 Summary

In this chapter we develop a classical orbital element feedback controller based on the sliding mode

methodology to account for uncertainties in the control matrix. The development of the controller is

described along with a method to determine the gain matrices. The orbit controller is implemented

into DSACSS. A low Earth orbit spacecraft control simulation is conducted using the flight com-

puters from the attitude hardware simulators and the GPS simulator. The controller successfully

increases the mean semi-major axis of the spacecraft.

181

9 Conclusions and Recommendations

This dissertation presents the development of the Distributed Spacecraft Attitude Control System

Simulator (DSACSS) for evaluating nonlinear attitude and orbit control strategies. The system is

comprised of two attitude hardware-in-the-loop simulators floating on individual spherical air bear-

ing platforms for a nearly torque free environment, a numerical attitude and orbit control simulator,

and a closed-loop hardware-in-the-loop GPS simulator.

9.1 Conclusions

The DSACSS is capable of real-time closed-loop orbit control using an operational space vehicle

GPS receiver. The closed-loop GPS simulator provides a realistic orbit trajectory by including

orbital perturbations, solar radiation pressure, and atmospheric drag. A classical orbital element

feedback controller is successfully evaluated using the closed-loop simulator.

The real-time hardware-in-the-loop attitude simulators determine attitude and have three-axis

attitude control capability through the use of reaction wheels. The development and implementation

on the simulators of angular rate and MRP nonlinear attitude controllers are shown. The attitude

controllers successfully drive the errors to zero and match the trends expected from numerical sim-

ulations.

A robust, asymptotically stable, nonlinear MRP attitude controller is developed and evaluated

using numerical simulations and both real-time hardware-in-the-loop attitude simulators. The robust

spacecraft attitude control law is based on the sliding mode methodology by bounding un-modeled

182

9.2 Recommendations

dynamics and system uncertainties. The robust spacecraft MRP attitude controller successfully

tracks a time varying reference attitude trajectory for both real-time hardware-in-the-loop attitude

simulators.

A classical orbital element feedback controller based on the sliding mode methodology is de-

veloped and evaluated using the closed-loop GPS simulator. The development of the controller is

described along with a method to determine the gain matrices. The controller successfully controls

the classical orbital elements of the spacecraft.

The DSACSS framework is a flexible testbed for investigating a variety of spacecraft control

techniques, especially control scenarios involving coupled attitude and orbital motion. DSACSS

allows individual or component level development and testing of hardware and software interfaces.

The DSACSS facility also allows for investigation into spacecraft formation flying such as con-

trolling the formation and pointing accuracy. The advantage of DSACSS over software simulator

architectures is that hardware-in-the-loop simulations for testing and validation of concepts provide

a more realistic demonstration of expected performance of space systems.

9.2 Recommendations

The DSACSS can be easily upgraded and modified to suit the needs of the specific research. The

hardware components are modular in nature, which allows for quick and easy replacement in the

case of failure or poor performance. The hardware-in-the-loop simulation software is also modular

in nature, which allows for easy continued development of DSACSS-Ops.

A document describing C++ coding conventions has been adopted for DSACSS-Ops.∗ The

style rules outlined in the document are for the purpose of making the process of reviewing and

scanning for errors of the source code more efficient. An important point is to ensure that the code

is thoroughly evaluated through testing and group reviews to ensure continued successful opera-

tion of DSACSS. Approximately 85% of DSACSS-Ops has been written or rewritten based on the

coding conventions. A strong recommendation is to rewrite the rest of DSACSS-Ops to the coding

∗http://www.sssl.aoe.vt.edu/documentation/software/codingdocument.txt

183

9.2 Recommendations

standards.

Each air bearing platform has a fixed rotation point (f.r.p.). If the center of gravity (c.g.) of the

simulator is not co-located with the f.r.p., a torque is introduced into the system. The desire is to

have the c.g. and f.r.p. co-located. The linear actuators can be used to move the c.g. to the f.r.p.

The recommendation is to develop an estimation technique similar to the one outlined in Ref. 34

to estimate the c.g. offset (rc.g.). Once an estimate of rc.g. is determined, a control scheme can be

devised to drive rc.g. to zero by using the linear actuators.

The reaction wheel interface for controlling the attitude of the hardware simulators is fully

developed. The recommendation is to develop a pulse width modulation (PWM) scheme for the

thrusters to control the attitude of the hardware simulators.

The attitude determination for Whorl−II needs improvement. The magnetic field in the North

East corner of the Whorl−II enclosure is not in a consistent direction. Therefore investigating

attitude control pointing within the North East direction is not recommended at this time. Possible

solutions to improve attitude determination for Whorl−II are: 1) improve estimation techniques, 2)

use a distributed magnetometer configuration, and 3) develop a visual attitude sensor.

The Ashtech G12 GPS receiver is implemented into the spacecraft low Earth orbit closed-

loop hardware-in-the-loop GPS simulator. Three Ashtech DG14 HDMA GPS receivers have been

procured for the purpose of investigating spacecraft formation flying using multiple GPS receivers.

The DG14 receivers have been successfully tested using the GPS simulator in an open-loop aircraft

scenario. The recommendation is to test and implement the DG14 receivers into the spacecraft low

Earth orbit closed-loop hardware-in-the-loop GPS simulation.

184

Bibliography

[1] B. J. Naasz, R. D. Burns, D. Gaylor, and J. Higinbotham, “Hardware-in-the-loop Testing of

Continuous Conrol Algorithms for a Precision Formation Flying Demonstration Mission,”

18th International Symposium on Space Flight Dynamics, Haus der Bayerischen Wirtschaft,

Munich, Germany, October 11–15, 2004.

[2] B. J. Naasz, C. D. Karlgaard, and C. D. Hall, “Application of Several Control Techniques for

the Ionospheric Observation Nanosatellite Formation,” 2002 AAS/AIAA Space Flight Mechan-

ics Conference, San Antonio, Texas, January 27–30, 2002.

[3] H. Schaub, S. R. Vadali, J. L. Junkins, and K. T. Alfriend, “Spacecraft Formation Flying

Control Using Mean Orbit Elements,” Journal of the Astronautical Sciences, Vol. 48, No. 1,

2000, pp. 69–89.

[4] P. Gurfil, “Control-Theorectic Analysis of Low-Thrust Orbital Transfer using Orbital Ele-

ments,” Journal of Guidance, Control, and Dynamics, Vol. 26, No. 6, 2003, pp. 979–1011.

[5] S. Leung, E. Gill, O. Montenbruck, and S. Montenegro, “A Navigation Processor for Flexible

Real-Time Formation Flying Applications,” International Symposium Formation Flying Mis-

sions and Technologies, Centre National d’Etudes Spatiales Toulouse Space Centre - France,

October 29–31, 2002.

[6] S. Kim, J. L. Crassidis, Y. Cheng, A. M. Fosbury, and J. L. Junkins, “Kalman Filtering for Rel-

ative Spacecraft Attitude and Position Estimation,” AIAA Guidance, Navigation, and Control

Conference and Exhibit, San Francisco, California, August 15–18, 2005.

185

Bibliography

[7] M. C. VanDyke and C. D. Hall, “Decentralized Coordinated Attitude Control of a Formation

of Spacecraft,” Journal of Guidance, Control, and Dynamics, Vol. 29, No. 5, 2006, pp. 1101–

1109.

[8] G. Q. Xing and S. A. Parvez, “Relative Attitude Kinematics and Dynamics Equations and

Its Applications to Spacecraft Attitude State Capture and Tracking in Large Angle Slewing

Maneuvers,” 1999 Space Control Conference, MIT Lincoln Laboratory, 1999. April 13-15.

[9] F. L. Markley, J. L. Crassidis, and Y. Cheng, “Nonlinear Attitude Filtering Methods,” AIAA

Guidance, Navigation, and Control Conference and Exhibit, San Francisco, California, August

15–18, 2005.

[10] J. L. Schwartz, M. A. Peck, and C. D. Hall, “Historical Survey of Air-Bearing Spacecraft

Simulators,” Journal of Guidance, Control, and Dynamics, Vol. 26, No. 4, 2003, pp. 513–522.

[11] J. Leitner, “A Hardware-in-the-loop Testbed for Spacecraft Formation Flying Applications,”

IEEE Aerospace Conference, Big Sky, Montana, 2001.

[12] R. D. Burns and B. J. Naasz, “An Environment for Hardware-in-the-Loop Formation Naviga-

tion and Control,” AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Providence,

Rhode Island, 16–19 August 2004.

[13] C. M. Close and D. K. Frederick, Modeling and Analysis of Dynamic Systems. Boston, MA:

Houghton Mifflin, 2nd ed., 1993.

[14] J. E. Slotine and W. Li, Applied Nonlinear Control. Upper Saddle River, New Jersey: Prentice

Hall, 1991.

[15] B. B. Goeree and E. D. Faase, “Sliding Mode Attitude Control of a Small Satellite for Ground

Tracking Maneuvers,” American Control Conference, Chicago, Illinois, June 2000.

[16] J. L. Crassidis and F. L. Markley, “Sliding Mode Control Using Modified Rodrigues Parame-

ters,” Journal of Guidance, Control, and Dynamics Control, Vol. 19, No. 6, 1996, pp. 1381–

1383.

[17] Y.-J. Cheon, “Sliding Mode Control for Attitude Tracking of Thruster-Controlled Spacecraft,”

Transactions on Control, Automation, and Systems Engineering, Vol. 3, December 2001,

186

Bibliography

pp. 257–261.

[18] T. A. Dwyer and H. Sira-Ramirez, “Variable-Structure Control of Spacecraft Attitude Maneu-

vers,” Journal of Guidance, Control, and Dynamics, Vol. 11, May 1988, pp. 262–270.

[19] H. Sira-Ramirez and T. A. Dwyer, “Variable Structure Control of Spacecraft Reorientation Ma-

neuvers,” Guidance, Navigation and Control Conference, AIAA, August 18–20 1986, pp. 86–

96.

[20] Y. B. Shtessel, “Decentralized Sliding Mode Control in Three-Axis Inertial Platforms,” Jour-

nal of Guidance, Control, and Dynamics, Vol. 18, July 1995, pp. 773–781.

[21] S.-C. Lo and Y.-P. Chen, “Smooth Sliding-Mode Control for Spacecraft Attitude Tracking Ma-

neuvers,” Journal of Guidance, Control, and Dynamics, Vol. 18, December 1995, pp. 1345–

1349.

[22] H. Bang and Y. H. Lho, “Sliding Mode Control for Spacecraft Containing Rotating Wheels,”

AIAA Guidance, Navigation, and Control Conference and Exhibit, Montreal, Canada, AIAA,

August 6–9 2001.

[23] J. C. Chiou, M.-C. Hwang, and S.-D. Wu, “Robust Control of Nonlinear Dynamic Systems Us-

ing Sliding Mode Control and Productive Networks,” American Control Conference, Seattle,

Washington, June 1995.

[24] D. Shin and J. Kim, “Robust Spacecraft Attitude Control Using Sliding Mode Control,” AIAA

Guidance, Navigation, and Control Conference and Exhibit, AIAA, August 10–12 1998.

[25] H. Schaub and J. L. Junkins, Analytical Mechanics of Space Systems. Reston, VA: AIAA

Education Series, 2003.

[26] D. Irvin, “A Study of Linear vs. Nonlinear Control Techniques for the Reconfiguration of

Satellite Formations,” Master’s thesis, Air Force Institute of Technology, Wright-Patterson

AFB, OH, 2001.

[27] W. H. Clohessey and R. S. Wiltshire, “Terminal Guidance System for Satellite Rendezvous,”

Journal of the Aerospace Sciences, Vol. 27, 1960, pp. 653–658.

[28] B. Wie, Space Vehicle Dynamics and Control. Reston, VA: AIAA Education Series, 1998.

187

Bibliography

[29] S. E. Lennox, “Coupled Attitude and Orbital Control System using Spacecraft Simulators,”

Master’s thesis, Virginia Polytechnic Institute and State University, Blacksburg, Virginia,

Blacksburg, VA, 2004.

[30] S. R. Vadali, K. T. Alfriend, and S. Vaddi, “Hill’s Equations, Mean Orbit Elements, and Forma-

tion Flying of Satellites,” The Richard H. Battin Astrodynamics Conference, College Station,

TX, AAS, March 2000.

[31] S. Vaddi, Modeling and Control of Satellite Formations. PhD thesis, Texas A&M University,

College Station, TX, May 2003.

[32] K. T. Alfriend, H. Yan, and S. R. Vadali, “Nonlinear Considerations in Satellite Formation

Flying,” AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Monterey, California,

5–8 August 2002.

[33] J. L. Schwartz and C. D. Hall, “The Distributed Spacecraft Attitude Control System Simulator:

Development, Progress, Plans,” 2003 Flight Mechanics Symposium, NASA Goddard Space

Flight Center, Greenbelt, Maryland, October 28–30, 2003.

[34] J. L. Schwartz, Distributed Spacecraft AttitudeControl System Simulator From Design to De-

centralized Control. PhD thesis, Virginia Polytechnic Institute and State University, Blacks-

burg, VA, 2004.

[35] S. A. Kowalchuk and C. D. Hall, “Distributed Spacecraft Attitude Control System Simulator:

Feedback Control Capabilities and Visualization Techniques,” 7th International Quantitative

Feedback Theory and Robust Frequency Domain Symposium, University of Kansas, Lawrence,

Kansas, August 5–7, 2005.

[36] S. A. Kowalchuk and C. D. Hall, “Hardware-in-the-Loop Simulation of Classical Element

Feedback Controller,” Goddard Flight Mechanics Symposium, Greenbelt, Maryland, October

18–20, 2005.

[37] S. A. Kowalchuk and C. D. Hall, “GPS Hardware-in-the-loop Spacecraft Formation Flying

Simulation,” AAS/AIAA Space Flight Mechanics Meeting, Sedona, Arizona, Jan. 28–Feb. 1,

2007.

188

Bibliography

[38] M. J. Caruso, “Applications of Magnetic Sensors for Low Cost Compass Systems,” tech. rep.,

Honeywell Solid State Electronics Center, Morristown, NJ, 2006.

[39] M. D. Shuster and S. D. Oh, “Three-Axis Attitude Determination From Vector Observations,”

Journal of Guidance and Control, Vol. 4, January 1981, pp. 70–77.

[40] C. D. Hall, “AOE 4140 Spacecraft Dynamics and Control Lecture Notes,” Available at www.

aoe.vt.edu/˜cdhall/courses/aoe4140/, Blacksburg, VA 2003.

[41] R. F. Stengel, Optimal Control and Estimation. New York: Dover Publications, 1994.

[42] J. L. Crassidis and J. L. Junkins, Optimal Estimation of Dynamic Systems. New York: CRC

Press Company, 2004.

[43] D. Simon, Optimal State Estimation. New Jersey: Wiley, 2006.

[44] J. W. Murrell, “Precision Attitude Determination for Multimission Spacecraft,” AIAA Guid-

ance, Navigation, and Control Conference and Exhibit, Palo Alto, California, AIAA, August

1978, pp. 70–87.

[45] H. Sira-Ramirez, “On the Design of Nonlinar Pulse-Width-Modulation Controllers,” Control

and Applications, IEEE, April 1989, pp. 397–402.

[46] K. H. Kienitz and J. Bals, “Pulse Modulation for Attitude Control with Thrusters Subject

to Switching Restrictions,” German Aerospace Congress, Vol. III, Friedrichshafen, Germany,

2005.

[47] D. J. Zimpfer, L. S. Shieh, and J. W. Sunkel, “Digitally Redesigned Pulse-Width Modulation

Spacecraft Control,” AIAA Guidance, Navigation and Control Conference, Balitmore, MD,

AIAA, August 7–10 1995.

[48] A. J. Turner and C. D. Hall, “An Open Source, Extensible Spacecraft Simulation and Model-

ing Environment Framework,” Proceedings of the AAS/AIAA Astrodynamics Specialist Con-

ference, Big Sky, Montana, August 3 -7 2003.

[49] A. J. Turner, “An Open Source, Extensible Spacecraft Simulation and Modeling Environment

Framework,” Master’s thesis, Virginia Polytechnic Institute and State University, Blacksburg,

VA, 2003.

189

Bibliography

[50] L. Haas, “The Ashtech G12-HDMA: A Low Cost, High Performance GPS Space Receiver,”

Institute of Navigation International Technical Meeting ION GPS-2000, Salt Lake City, 2000.

September 19-22.

[51] Goddard Space Flight Center, Guidance Navigation and Control, CSC-5570-16R0UD0, “Posi-

tioning System (GPS) Enhanced Onboard Navigation System, System Description and User’s

Guide, Version 2, Release 2.1.A,” tech. rep., Computer Sciences Corporation, December 2003.

[52] Goddard Space Flight Center, Guidance Navigation and Control, CSC-5570-14R0UD0,

“Global Positioning System (GPS) Enhanced Onboard Navigation System, Mathematical

Specifications, Version 2, Release 2.1, Update 1,” tech. rep., Computer Sciences Corporation,

December 2003.

[53] B. W. Parkinson and J. J. Spilker, Global Positionion System: Theory and Applications, Vol. 1.

Washington, D.C.: AIAA Education Series, 1996.

[54] E. D. Kaplan and C. J. Hegarty, Understanding GPS Principles and Applications. Boston,

MA: Artech House Inc, 2nd ed., 2005.

[55] B. R. Bowring, “The Accuracy of Geodetic Latitude and Height Equations,” Survey review -

Directorate of Overseas Surveys, Surrey, England, October 28 1985, pp. 202–206.

[56] R. Burtch, “A Comparison of Methods used in Rectangular to Geodetic Coordinate Trans-

formations,” ACSM Annual Conference and Technology Exhibition, Orlando, Florida, ACSM,

April 21–26 2006.

[57] P. C. Hughes, Spacecraft Attitude Dynamics. Mineola, New York: Dover Publications, 2004.

[58] H. K. Khalil, Nonlinear Systems. New Jersey: Prentice Hall, third ed., 2002.

[59] M. J. Sidi, Spacecraft Dynamics and Control. New York: Cambridge University Press, 1997.

[60] W. Perruquetti and J. P. Barbot, Sliding Mode Control in Engineering. Control Engineering

Series, New York: Marcel Dekker, Inc, 2002.

[61] B. H. Tongue, Principles of Vibration. New York: Oxford University Press, 1996.

[62] B. J. Naasz, “Classical Element Feedback Control for Spacecraft Orbital Maneuvers,” Master’s

thesis, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, Blacksburg,

190

Bibliography

VA, 2002.

[63] R. H. Battin, An Introduction to the Mathematics and Methods of Astrodynamics, Revised

Edition. Reston, VA: AIAA Education Series, 1999.

[64] R. R. Bate, D. D. Mueller, and J. E. White, Fundamentals of Astrodynamics. New York: Dover

Publications, 1971.

[65] M. R. Ilgen, “Low Thrust OTV Guidance using Lyapunov Optimal Feedback Control Tech-

niques,” Advances in the Astronautical Sciences, Vol. 85, No. 2, 1993, pp. 1527–1545.

[66] F. Garofalo and L. Glielmo, Robust Control via Variable Structure and Lyapunov Techniques.

New York: Springer, 1996.

[67] A. J. Laub, Matrix Analysis for Scientists and Engineers. Philadelphia, PA: Society of Indus-

trial and Applied Mathematics, 2005.

191

A Appendix: Reaction Wheel Controller

An attitude control law determines the necessary torque about each body axis to satisfy the de-

sired attitude objective. The reaction wheel controller receives the attitude control vector (u)

and determines the necessary wheel speed for each reaction wheel to produce the desired control

torque. A low-level reaction wheel controller has been developed and implemented on Whorl−I

and Whorl−II. This appendix outlines the development of the reaction wheel controller.

A.1 Reaction Wheel Controller Development

The amount of control torque (ui) produced about the reaction wheel spin axis as a function of

wheel acceleration (Ωi) is

hi = ui

IWsΩi = ui (A.1)

where IWs is the reaction wheel spin axis inertia. The angular acceleration of the wheel is approxi-

mated as

Ωi ≈Ωfinali − Ωcurrenti

∆t(A.2)

and substituting into Eq. (A.1) results in an expression for the final wheel speed (Ωfinali) as

192


Ωfinali ≈ Ωcurrenti +uiIWs

∆t (A.3)

Eq. (A.3) provides an expression for the final wheel speed given a desired torque about the

reaction wheel spin axis, current wheel speed, and desired time (∆t) to produce the desired torque.

Figure A.1 shows Whorl−I reaction wheel 3 steady state wheel speed for various motor power

settings. The wheel speeds reach a steady state wheel speed in different amounts of time based on

the power setting specified to the motor.

Figure A.1: Whorl−I Reaction Wheel 3 Steady State Wheel Speed Time Behavior

Figure A.2 plots the same data as provided in Figure A.1 with the following ratio:

Ωratio =Ωcurrent − Ωinitial

Ωsteady state − Ωinitial(A.4)

As shown in Figure A.2, Ωratio converges to 1 in approximately the same time independent of the

power settings. The wheel ratio curve is expressed as a function of time using a polynomial curve

193


Figure A.2: Whorl−I Reaction Wheel 3 Steady State Wheel Speed Time Behavior Curve Fit

fit. The expression for Ωratio is

Ωratio = −6.000× 10−15t10 + 2.074× 10−12t9 − 2.887× 10−10t8

+2.271× 10−8t7 − 1.111× 10−6t6 + 3.512× 10−6t5 − 7.216× 10−4t4

+9.515× 10−3t3 − 7.825× 10−2t2 + 3.839× 10−1t+ 2.609× 10−2 (A.5)

The steady state wheel speed (Ωsteady state) is developed using Eq. (A.5) and is

Ωsteady state = Ωcurrenti +Ωfinali − Ωcurrenti

Ωratio(A.6)

The steady state wheel speed corresponding power setting is the necessary power setting to result in

Ωfinali after a set amount of time (∆t).

The command power (Pcommand) to the reaction wheel motors that result in Ωsteady state must be

determined. Each reaction wheel for Whorl−I and Whorl−II are characterized such that Pcommand is

determined from a desired Ωsteady state. For brevity, only Whorl−I reaction wheel 3 characterization

procedure is provided.

Figures A.3 and A.4 shows the steady state wheel speed verses command power for Whorl−I

194


reaction wheel 3. Figure A.3 shows a highly nonlinear behavior with an absolute command power

greater than 800 motor units. A dead-band for the reaction wheel from -22 to 22 motor units is

clearly seen in Figure A.4. The nonlinearity and dead-band regions have to be considered to design

an effective wheel controller.

Figure A.3: Steady State Reaction Wheel Speed verses Power Commanded to Reaction WheelMotor for a Range from -1000 to 1000 motor units for Whorl−I Reaction Wheel 3

Figure A.4: Steady State Reaction Wheel Speed verses Power Commanded to Reaction WheelMotor for a Range from -100 to 100 motor units for Whorl−I Reaction Wheel 3

Figures A.5 and A.6 show the axes switched and Ωsteady state divided by the motor voltage

195


(Ωscale = Ωsteady state/V ) recorded at each steady state sample as compared with Figures A.3 and

A.4. The command power curve as a function of Ωscale is developed using a polynomial curve fit

in four discrete regions. The four regions allow for a more accurate curve fit while excluding the

dead-band and nonlinearity zones.

Figure A.5: Power Commanded to Reaction Wheel Motor verses Scaled Steady State ReactionWheel Speed Curve Fit for Whorl−I Reaction Wheel 3

Figure A.6: Power Commanded to Reaction Wheel Motor verses Scaled Steady State ReactionWheel Speed Curve Fit for Whorl−I Reaction Wheel 3

196


The curve fit equations for Ω3 Whorl−I are

P =

0.3085Ω3scale + 4.3691Ω2

scale − 94.8364Ωscale − 15.6830 −7.6 ≤ Ωscale < −1.2

25.9997Ω2scale − 31.1477Ωscale + 25.9680 −1.2 ≤ Ωscale < 0

−26.1740Ω2scale − 31.1465Ωscale − 25.9364 0 ≤ Ωscale ≤ 1.2

0.1015Ω3scale − 1.9227Ω2

scale − 103.3392Ωscale + 24.3888 1.2 < Ωscale ≤ 7.6

(A.7)

and the curve fits are shown in Figures A.5 and A.6.

All the reaction wheels have the same trends, but with different coefficients in the power curve

fit expression. The coefficients in the power curve expression change as the reaction wheel electric

motors have more use. The change is gradual and is resolved by re-characterizing the reaction

wheels off-line to obtain new coefficients for the power curve expression at the users discretion.

The power curve fit equations for Ω1 Whorl−I are

P =

−0.0292Ω3scale + 0.4310Ω2




0.3702Ω3scale − 5.0961Ω2


(A.8)

The power curve fit equations for Ω2 Whorl−I are

P =

−0.1300Ω3scale − 0.3346Ω2




0.0614Ω3scale − 1.9880Ω2


(A.9)

197

A.2 Summary

The power curve fit equations for Ω1 Whorl−II are

P =

0.0073Ω3scale + 0.9891Ω2




0.1442Ω3scale − 2.4665Ω2


(A.10)


P =

0.0937Ω3scale + 2.0285Ω2




0.1841Ω3scale − 2.8970Ω2

scale − 99.9913Ωscale + 26.7459 1.2 < ωscale ≤ 7.6

(A.11)


P =

0.1721Ω3scale + 2.4755Ω2




0.2112Ω3scale − 3.3315Ω2


(A.12)

A.2 Summary

This appendix discusses the development and implementation of a low-level reaction wheel con-

troller that determines the necessary wheel speed for each reaction wheel to produce the desired

torque. The power curve equations for each reaction wheel are provided, but will gradually change

as the reaction wheels are used. The power curve changes are resolved by re-characterizing the

reaction wheels off-line to obtain new coefficients for the power curve expression at the users dis-

cretion.

198

B Appendix: Implementation of Attitude

Control Laws into DSACSS

In this appendix we present the method for implementing attitude control laws into DSACSS-Ops

and running the controllers on Whorl−I and Whorl−II. The software structure is module, which

assists with quick and easy implementation of control laws. The following sections present the

method for implementing an attitude control law and template files to assist in the implementation.

B.1 Implementation Process

The process for implementing an attitude control law into DSACSS-Ops is as follows:

1. Copy and rename testController.h and testController.cpp. For convenience the new file name

is referred to as testController. Rename the testController class.

2. Implement the attitude control law in testController.cpp and modify testController.h

3. Modify the Makefile in the Controller directory to compile the new .cpp file.

4. Include testController.h in the WhorlSim.h file.

5. Add the object name for the controller to WhorlSim.h and WhorlSim.cpp.

6. Compile DSACSS-Ops and check for any warnings or errors during compilation.

199

B.2 Template Files

7. Modify the configuration file (DSACSSConfig.xml) in the work directory for the desired

simulation parameters.

8. Modify and rename templateController.cpp in the work directory for the desired simulation.

9. Modify the Makefile in the work directory to compile templateController.cpp.

10. Run the simulation executable file in the work directory.

B.2 Template Files

There are three template files to assist with implementing attitude control laws into DSACSS-Ops∗

to run on the hardware simulators. The testController.h and testController.cpp files are located in the

Controller directory in DSACSS-Ops. The testController.h file is where the functions and variables

are defined. The testController.cpp file on lines 99–101 is where the attitude control law is specified.

The templateController.cpp file is located in the work directory in DSACSS-Ops. Compiling

the templateController.cpp file produces the executable simulation file. The simulator name and

simulation duration are the parameters the user is required to modify.

1 //////////////////////////////////////////////////////////////////////////////////////////////////

2 /*! \file testController.h

3 * \brief Template file for implementation of an attitude controller using the hardware simulators.

4 * \author $Author: $

5 * \version $Revision: 1.1 $

6 * \date $Date: $

7 *//////////////////////////////////////////////////////////////////////////////////////////////////

8 /*!

9 */

10 //////////////////////////////////////////////////////////////////////////////////////////////////

1112 #ifndef __SSSL_TESTCONTROL_H__

13 #define __SSSL_TESTCONTROL_H__

1415 #include "Controller.h"

1617 using namespace std;

∗Complete documentation for DSACSS-Ops is provided on the web at www.sssl.aoe.vt.edu.

200

B.2 Template Files

18 using namespace O_SESSAME;

1920 class testController : public Controller

2122 public:

23 /** Constructors */

24 testController( );

25 testController( Whorl* ptr_whorl );

2627 /** Destructor */

28 virtual ˜testController( );

2930 /*! Runs the default controller */

31 int Run( );

3233 /*! Initializes the default controller */

34 int Initialize( );

3536 protected:

3738 Matrix m_gainMatrix;

39 Vector m_controlTorque;

40 ;

4142 #endif

4344 // Do not change the comments below - they will be added automatically by CVS

45 /*****************************************************************************

46 * $Log: testController.h,v $

47 *

48 ******************************************************************************/

1 //////////////////////////////////////////////////////////////////////////////////////////////////

2 /*! \file testController.cpp

3 * \brief Template file for implementation of an attitude controller using the hardware simulators.



6 * \date $Date: $

7 *//////////////////////////////////////////////////////////////////////////////////////////////////

8 /*!

9 */

10 //////////////////////////////////////////////////////////////////////////////////////////////////

1112 #include "testController.h"

13

201

B.2 Template Files

14 /*! /brief Constructor for testController */

15 testController::testController( )

16

17

1819 /*! /brief Constructor for testController

20 *

21 * @param ptr_whorl is a pointer to the Whorl object.

22 */

23 testController::testController( Whorl* ptr_whorl )

24

25 m_whorl = ptr_whorl;

2627 Initialize( );

2829

3031 /*! /brief De-Constructor for testController */

32 testController::˜testController( )

33

34

3536 /*! /brief Initialize testController */

37 int testController::Initialize( )

38

39 /* Initialize control vector */

40 m_controlTorque.initialize(3); // [N-m]

4142 /* Specify Max Wheel Torque */

43 m_uMax = 1.5; // [N-m]

4445 /* Initialize contorl gain matrix */

46 m_gainMatrix.initialize(3,3);

47 m_gainMatrix(1,1) = 0;



5051 /* Get orientation of Momentum Wheels. Defined in Controller Class. */

52 A = FindA( );

5354 /* Get Momentum Wheel spin axis MOI. Defined in Controller Class */

55 MOI_sw = FindMOI_sw( );

5657 /* Reference Model. Initialized in Controller.cpp */

202

B.2 Template Files

58 Quaternion qri( 0.0, 0.0, 0.0, 1.0 );

59 m_quaternionReference(1) = qri(1);




63 m_mrpReference = ModifiedRodriguezParameters( qri );

64 m_angularRateReference(1) = 0.0; // [rad/s]



6768 return( 0 );

69

7071 int testController::Run()

72

7374 cerr << "Start of testController algorithm" << endl;

7576 /* Attitude Reference Model */

7778 /* Error Vectors */

79 Vector angularRateError(3);

80 ModifiedRodriguezParameters mrpError;

8182 /* Set Whorl Error Values */

83 m_whorl->SetMRPError( mrpError );

84 m_whorl->SetAngularRateError( angularRateError );

8586 /* Set Whorl Reference Parameters */

87 m_whorl->SetReferenceOmegaBL( m_angularRateReference );

88 m_whorl->SetReferenceQuaternion( m_quaternionReference );

8990 /* Reaction Wheel Speed */

91 static Vector wheelSpeed(3);

92 double measurementTime;

93 m_whorl->GetMomentumWheel("REACTION_X")->GetWheelSpeed( wheelSpeed(1), measurementTime );

94 m_whorl->GetMomentumWheel("REACTION_Y")->GetWheelSpeed( wheelSpeed(2), measurementTime );

95 m_whorl->GetMomentumWheel("REACTION_Z")->GetWheelSpeed( wheelSpeed(3), measurementTime );

9697 /* Control law */

98 m_controlTorque(1) = 0; // [N-m]



101

203

B.2 Template Files

102 /* Wheel Saturation Function. Max control specified in constructor. */

103 m_controlTorque = WheelSaturation( m_controlTorque ); // [N-m]

104105 /* Set the desird contorl torque in the whorl object */

106 m_whorl->SetControl( m_controlTorque ); // [N-m]

107108 /* Set the torque for the momentum wheels to produce */

109 SetWheelTorque( m_controlTorque ); // [N-m]

110111 cerr << "End of testController algorithm" << endl;

112113 return( 0 );

114


117 /*****************************************************************************

118 * $Log: testController.cpp,v $

119 *

120 ******************************************************************************/

1 /************************************************************************************************/

2 /*! \file templateController.cpp

3 * \brief A template file to run an attitude control simulation on Whrol-I and II. The specific

4 * controller and observer are specified in the configuration file (.xml).



7 * \date $Date: $

8 ************************************************************************************************/

9 /*!

10 *

11 ************************************************************************************************/

1213 // Standard System includes

14 #include <iostream.h>

15 #include <iomanip.h>

16 #include <fstream.h>

1718 // Simulator Interface Includes

19 #include "Base/Whorl.h"

20 #include "Base/WhorlSim.h"

21 #include "Logging/WhorlHistory.h"

2223 // XML Parser Include

24 #include <dsacssinterface.h>

25

204

B.2 Template Files


2728 // Specify Xml File to Parse

29 const char *fileName = "./DSACSSConfig.xml";

3031 int main()

32

33 /******************************************************************

34 * Load Configuration Parameters from xml file

35 *******************************************************************/

36 // Declare TiXmlDocument

37 TiXmlDocument config( fileName );

38 // Load Xml File

39 bool loadOkay = config.LoadFile();

40 checkLoadFile(loadOkay, fileName, config);

41 // Declare TiXmlHandle

42 TiXmlHandle docHandle( &config );

4344 /******************************************************************

45 * Setup Hardware-in-the-loop Attitude Simualation

46 *******************************************************************/

4748 /* Setup WhorlHistory to collect data. Note must be changed for

49 each simulator. */

50 WhorlHistory WhorlData( docHandle, "WHORL-I" );

5152 /* Create Whorl Object */

53 Whorl whorlOne;

54 /* Initialize Whorl Object */

55 whorlOne.Initialize( docHandle );

5657 /* Create WhorlSim Object */

58 WhorlSim whorlOneSim( &whorlOne );

59 /* Initialize WhorlSim Object */

60 whorlOneSim.Initialize( docHandle );

6162 /* Setup Time values */

63 ssfTime currentTime;

6465 /* Setup Vectors */

66 Vector attitudeState(7);

67 Vector attitudeControl(3);

68 Vector MRP(3);

69 Vector referenceAttitudeState(7);

205

B.2 Template Files

70 Vector referenceMRP(3);

71 Vector controlError(6);

7273 /******************************************************************

74 * Run Attitude Control Simulation

75 *******************************************************************/

76 cerr << "\n--------------------------------------------------------------" << endl;

77 cerr << "Start of Program " << endl;

78 cerr << "--------------------------------------------------------------" << endl;

7980 for ( int ii = 0; ii <= 200 ; ii++ )

81

82 /* Get Current attitude state */

83 whorlOneSim.Observe();

8485 /* Control based on attitude state */

86 whorlOneSim.Control();

8788 /* Get Current Time */

89 currentTime = Now( );

9091 /* Get attitude state vector (7x1) [ quaternion, angular rates [rad/s] ] */

92 attitudeState = whorlOne.GetState( );

9394 /* Get attitude Control vector (3x1) [N-m]*/

95 attitudeControl = whorlOne.GetControl( );

9697 /* Get MRP vector */

98 MRP = whorlOne.GetMRP( );

99100 /* Get reference attitude state (7x1) */

101 referenceAttitudeState( _(1,4) ) = whorlOne.GetReferenceQuaternion( );

102 referenceAttitudeState( _(5,7) ) = whorlOne.GetReferenceOmegaBL( );

103 referenceMRP = whorlOne.GetReferenceMRP( );

104105 /* Get control error */

106 controlError = whorlOne.GetControlError( );

107108 /* Save data to file specified in configuration file */

109 WhorlData.AppendHistory( currentTime, attitudeState, attitudeControl, MRP,

110 referenceAttitudeState, referenceMRP, controlError );

111112 /* Pause */

113 usleep( 500000 );

206

B.3 Summary

114 ;

115116 cout << "END of program " << endl;

117118 return( 0 );

119


122 /*****************************************************************************

123 * $Log: templateController.cpp,v $

124 *

125 ******************************************************************************/

B.3 Summary

In this appendix we present the method for implementing attitude control laws into DSACSS-Ops

and running the controllers on Whorl−I and Whorl−II. The next appendix presents the method for

implementing orbit control laws into DSACSS-Ops.

207

C Appendix: Implementation of Orbit

Control Laws into DSACSS Framework

In this appendix we present the method for implementing orbit control laws into DSACSS-Ops and

running the controllers on Whorl−I and Whorl−II. The software structure allows for quick and easy

implementation of orbit control laws. The following sections present the method for implementing

orbit control laws and template files to assist in the implementation.

C.1 Implementation Process

The process for implementing an orbit control law into DSACSS-Ops is as follows:

1. Copy and rename testOrbitController.h and testOrbitController.cpp. For convenience the new

file name is referred to as testOrbitController. Rename the testOrbitController class.

2. Implement the orbit control law in testOrbitController.cpp and modify testOrbitController.h

3. Modify the Makefile in the Orbit Controller directory to compile the new .cpp file.

4. Include testOrbitController.h in the WhorlSim.h file.

5. Add the object name for the controller to WhorlSim.h and WhorlSim.cpp.

6. Compile DSACSS-Ops and check for any warnings or errors during compilation.

208

C.2 Template Files

7. Modify the configuration file (DSACSSConfig.xml) in the work directory for the desired

simulation parameters.

8. Modify and rename templateOrbitController.cpp in the work directory for the desired simu-

lation.

9. Modify the Makefile in the work directory to compile templateOrbitController.cpp.

10. Run the simulation executable file in the work directory.

C.2 Template Files

There are three template files to assist with implementing orbit control laws into DSACSS-Ops∗

to run on the hardware simulators. The testOrbitController.h and testOrbitController.cpp files are

located in the Orbit Controller directory in DSACSS-Ops. The testOrbitController.h file is where

the functions and variables are defined. The testOrbitController.cpp file on lines 61–63 is where the

orbit control law is specified.

The templateOrbitController.cpp file is located in the work directory in DSACSS-Ops. Com-

piling the templateOrbitController.cpp file produces the executable simulation file. The simulator

name and simulation duration are the parameters the user is required to modify.

1 //////////////////////////////////////////////////////////////////////////////////////////////////

2 /*! \file testOrbitController.h

3 * \brief Template file for implenting orbit control laws into DSACSS-Ops.



6 * \date $Date: $

7 *//////////////////////////////////////////////////////////////////////////////////////////////////

8 /*!

9 */

10 //////////////////////////////////////////////////////////////////////////////////////////////////

1112 #ifndef __SSSL_TESTORBITCONTROL_H__

13 #define __SSSL_TESTORBITCONTROL_H__

∗Complete documentation for DSACSS-Ops is provided on the web at www.sssl.aoe.vt.edu.

209

C.2 Template Files

1415 #include "OrbitController.h"

1617 using namespace std;


1920 class testOrbitController : public OrbitController

2122 public:

23 /*! Constructors */

24 testOrbitController( );

25 testOrbitController( Whorl* ptr_whorl );

2627 /*! Destructor */

28 virtual ˜testOrbitController( );

2930 /*! Runs the default controller */

31 int Run( );

3233 /*! Initializes the default controller */

34 int Initialize( );

3536 protected:

3738 Vector m_Gains;

39 ;

4041 #endif


44 /*****************************************************************************

45 * $Log: testOrbitController.h,v $

46 *

47 ******************************************************************************/

48

1 //////////////////////////////////////////////////////////////////////////////////////////////////

2 /*! \file testOrbitController.cpp

3 * \brief Template file for implenting orbit control laws into DSACSS-Ops.



6 * \date $Date: $

7 *//////////////////////////////////////////////////////////////////////////////////////////////////

8 /*!

9 */

210

C.2 Template Files

10 //////////////////////////////////////////////////////////////////////////////////////////////////

1112 #include "testOrbitController.h"

1314 /* Constructor */

15 testOrbitController::testOrbitController( Whorl* ptr_whorl )

16

17 m_whorl = ptr_whorl;

1819 Initialize( );

20

2122 /* Deconstructor */

23 testOrbitController::˜testOrbitController()

24

2526 int testOrbitController::Initialize( )

27

28 /* Control Acceleratin Gains */

29 m_Gains.initialize(6);

30 m_Gains(1) = 0;

31 m_Gains(2) = 0;

32 m_Gains(3) = 0;

33 m_Gains(4) = 0;

34 m_Gains(5) = 0;

35 m_Gains(6) = 0;

3637 /* Space Vehicle Id */

38 m_vehicleID = 1;

3940 /* Acceleration Duration [s] */

41 m_duration = 2;

4243 /* Space Vehicle Mass [kg] */

44 m_mass = 100;

4546 /* Current Time */

47 m_vehicleTime = Now( );

4849 /* Reset delta V time history from orbit propagator */

50 m_accelerationVector.ResetValues( );

5152 return( 0 );

53

211

C.2 Template Files

5455 int testOrbitController::Run( )

56

57 /* Get Classical Orbital Elements */

58 m_whorl->GetCOE( );

5960 /* Radial, In-Track, Cross-Track control acceleration [m/sˆ2] */

61 u(1) = 0;

62 u(2) = 0;

63 u(3) = 0;

6465 /* Current Time */

66 m_vehicleTime = Now( );

6768 /* Set Orbit Control Acceleration */

69 m_whorl->SetOrbitControl( u );

7071 /* Send control accleration to GPS simulator propagator (running on Severian) */

72 m_accelerationVector.SendAccelerationMessage( m_vehicleID, m_vehicleTime.GetSeconds( ),

73 m_duration, u, m_mass );

7475 return( 0 );

76


79 /*****************************************************************************

80 * $Log: testOrbitController.cpp,v $

81 *

82 ******************************************************************************/

1 /************************************************************************************************/

2 /*! \file templateOrbitController.cpp

3 * \brief Orbit Control Simulation using GPS Receiver and Simulator



6 * \date $Date: $

7 ************************************************************************************************/

8 /*!

9 *

10 ************************************************************************************************/

1112 // Standard System includes

13 #include <iostream.h>

14 #include <iomanip.h>

15 #include <fstream.h>

212

C.2 Template Files

1617 // Simulator Interface Includes

18 #include "Base/Whorl.h"

19 #include "Base/WhorlSim.h"

2021 // XML Parser Include

22 #include <dsacssinterface.h>

2324 #include "Logging/WhorlHistory.h"

2526 #include "OrbitObserver/GPSObserver.h"


2930 // Specify Xml File to Parse

31 const char *fileName = "../DSACSSConfig.xml";

3233 int main( )

34

3536 /******************************************************************

37 * Load Configuration Parameters from xml file

38 *******************************************************************/

39 // Declare TiXmlDocument

40 TiXmlDocument config( fileName );

41 // Load Xml File

42 bool loadOkay = config.LoadFile();

43 checkLoadFile(loadOkay, fileName, config);

44 // Declare TiXmlHandle

45 TiXmlHandle docHandle( &config );

4647 /******************************************************************

48 *

49 *******************************************************************/

5051 WhorlHistory WhorlData( docHandle, "WHORL-III" );

5253 Whorl whorlOne;

5455 whorlOne.Initialize( docHandle );

5657 WhorlSim whorlOneSim(&whorlOne);

5859 whorlOneSim.Initialize( docHandle );

213

C.3 Summary

6061 ssfTime currentTime;

6263 Vector orbitState(6);

6465 Vector orbitControl(3);

6667 Vector orbitError(6);

6869 for ( int ii = 0; ii < 10; ii++ )

70

71 whorlOneSim.OrbitObserve( );

7273 whorlOneSim.OrbitControl( );

7475 currentTime = Now( );

7677 orbitState = whorlOne.GetOrbitState( );

7879 orbitControl = whorlOne.GetOrbitControl( );

8081 orbitError = whorlOne.GetOrbitError( );

8283 WhorlData.AppendOrbitHistory( currentTime, orbitState, orbitControl, orbitError );

8485 sleep(1);

86

8788 return(0);

89


92 /*****************************************************************************

93 * $Log: templateOrbitController.cpp,v $

94 *

95 ******************************************************************************/

C.3 Summary

In this appendix we present the method for implementing orbit control laws into DSACSS-Ops and

running the controllers on Whorl−I and Whorl−II.

214

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