Spacecraft Dynamics and Control
An Introduction

Provides the basics of spacecraft orbital dynamics plus attitude dynamics and control, using vectrix notation

Spacecraft Dynamics and Control: An Introduction presents the fundamentals of classical control in the context of spacecraft attitude control. This approach is particularly beneficial for the training of students in both of the subjects of classical control as well as its application to spacecraft attitude control. By using a physical system (a spacecraft) that the reader can visualize (rather than arbitrary transfer functions), it is easier to grasp the motivation for why topics in control theory are important, as well as the theory behind them.  The entire treatment of both orbital and attitude dynamics makes use of vectrix notation, which is a tool that allows the user to write down any vector equation of motion without consideration of a reference frame. This is particularly suited to the treatment of multiple reference frames. Vectrix notation also makes a very clear distinction between a physical vector and its coordinate representation in a reference frame. This is very important in spacecraft dynamics and control problems, where often multiple coordinate representations are used (in different reference frames) for the same physical vector.

• Provides an accessible, practical aid for teaching and self-study with a layout enabling a fundamental understanding of the subject
• Fills a gap in the existing literature by providing an analytical toolbox offering the reader a lasting, rigorous methodology for approaching vector mechanics, a key element vital to new graduates and practicing engineers alike
• Delivers an outstanding resource for aerospace engineering students, and all those involved in the technical aspects of design and engineering in the space sector
• Contains numerous illustrations to accompany the written text. Problems are included to apply and extend the material in each chapter

Essential reading for graduate level aerospace engineering students, aerospace professionals, researchers and engineers.

Preface xvii

1 Kinematics 1

1.1 Physical Vectors 1

1.2 Reference Frames and Physical Vector Coordinates 6

1.3 Rotation Matrices 11

1.4 Derivatives of Vectors 32

1.5 Velocity and Acceleration 41

1.6 More Rigorous Definition of Angular Velocity 42

Notes 44

References 45

2 Rigid Body Dynamics 47

2.1 Dynamics of a Single Particle 47

2.2 Dynamics of a System of Particles 49

2.3 Rigid Body Dynamics 52

2.4 The Inertia Matrix 56

2.5 Kinetic Energy of a Rigid Body 60

Notes 63

References 63

3 The Keplerian Two-Body Problem 65

3.1 Equations of Motion 65

3.2 Constants of the Motion 67

3.3 Shape of a Keplerian Orbit 69

3.4 Kepler’s Laws 80

3.5 Time of Flight 83

3.6 Orbital Elements 89

3.7 Orbital Elements given Position and Velocity 92

3.8 Position and Velocity given Orbital Elements 94

Notes 98

References 98

4 Preliminary Orbit Determination 99

4.1 Orbit Determination from Three Position Vectors 99

4.2 Orbit Determination from Three Line-of-Sight Vectors 103

4.3 Orbit Determination from Two Position Vectors and Time (Lambert’s Problem) 109

Notes 114

References 114

5 Orbital Maneuvers 115

5.1 Simple Impulsive Maneuvers 115

5.2 Coplanar Maneuvers 116

5.3 Plane Change Maneuvers 123

5.4 Combined Maneuvers 125

5.5 Rendezvous 127

Notes 128

Reference 128

6 Interplanetary Trajectories 129

6.1 Sphere of Influence 129

6.2 Interplanetary Hohmann Transfers 133

6.3 Patched Conics 137

6.4 Planetary Flyby 143

6.5 Planetary Capture 145

Notes 146

References 147

7 Orbital Perturbations 149

7.1 Special Perturbations 150

7.1.1 Cowell’s Method 151

7.2 General Perturbations 154

7.3 Gravitational Perturbations due to a Non-Spherical Primary Body 156

7.4 Effect of J2 on the Orbital Elements 164

7.5 Special Types of Orbits 168

7.6 Small Impulse Form of the Gauss Variational Equations 169

7.7 Derivation of the Remaining Gauss Variational Equations 171

Notes 180

References 181

8 Low Thrust Trajectory Analysis and Design 183

8.1 Problem Formulation 183

8.2 Coplanar Circle to Circle Transfers 184

8.3 Plane Change Maneuver 186

Notes 188

References 188

9 Spacecraft Formation Flying 189

9.1 Mathematical Description 190

9.2 Relative Motion Solutions 194

9.3 Special Types of Relative Orbits 203

Notes 207

Reference 207

10 The Restricted Three-Body Problem 209

10.1 Formulation 209

10.2 The Lagrangian Points 212

10.3 Stability of the Lagrangian Points 214

10.4 Jacobi’s Integral 215

Notes 218

References 218

11 Introduction to Spacecraft Attitude Stabilization 219

11.1 Introduction to Control Systems 220

11.2 Overview of Attitude Representation and Kinematics 222

11.3 Overview of Spacecraft Attitude Dynamics 223

12 Disturbance Torques on a Spacecraft 227

12.1 Magnetic Torque 227

12.2 Solar Radiation Pressure Torque 228

12.3 Aerodynamic Torque 230

12.4 Gravity-Gradient Torque 231

Notes 234

Reference 234

13 Torque-Free Attitude Motion 235

13.1 Solution for an Axisymmetric Body 235

13.2 Physical Interpretation of the Motion 242

Notes 245

References 245

14 Spin Stabilization 247

14.1 Stability 247

14.2 Spin Stability of Torque-Free Motion 249

14.3 Effect of Internal Energy Dissipation 252

Notes 253

References 253

15 Dual-Spin Stabilization 255

15.1 Equations of Motion 255

15.2 Stability of Dual-Spin Torque-Free Motion 257

15.3 Effect of Internal Energy Dissipation 259

Notes 266

References 266

16 Gravity-Gradient Stabilization 267

16.1 Equations of Motion 268

16.2 Stability Analysis 272

Notes 277

References 277

17 Active Spacecraft Attitude Control 279

17.1 Attitude Control for a Nominally Inertially Fixed Spacecraft 280

17.2 Transfer Function Representation of a System 281

17.3 System Response to an Impulsive Input 282

17.4 Block Diagrams 284

17.5 The Feedback Control Problem 286

17.6 Typical Control Laws 289

17.7 Time-Domain Specifications 292

17.8 Factors that Modify the Transient Behavior 308

17.9 Steady-State Specifications and System Type 311

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viii

Contents

2.4 The Inertia Matrix 56

2.4.1 A Parallel Axis Theorem

57

2.4.2 A Rotational Transformation Theorem

58

2.4.3 Principal Axes

59

2.5 Kinetic Energy of a Rigid Body 60

Notes

63

References 63

3 The Keplerian Two-Body Problem 65

3.1 Equations of Motion 65

3.2 Constants of the Motion 67

3.2.1 Orbital Angular Momentum

67

3.2.2 Orbital Energy

67

3.2.3 The Eccentricity Vector

68

3.3 Shape of a Keplerian Orbit 69

3.3.1 Perifocal Coordinate System

72

3.4 Kepler’s Laws 80

3.5 Time of Flight 83

3.5.1 Circular Orbits

83

3.5.2 Elliptical Orbits

84

3.5.3 Parabolic Orbits

88

3.5.4 Hyperbolic Orbits

89

3.6 Orbital Elements 89

3.6.1 Heliocentric-Ecliptic Coordinate System

89

3.6.2 Geocentric-Equatorial Coordinate System

90

3.7 Orbital Elements given Position and Velocity 92

3.8 Position and Velocity given Orbital Elements 94

Notes

98

References 98

4 Preliminary Orbit Determination 99

4.1 Orbit Determination from Three Position Vectors 99

4.2 Orbit Determination from Three Line-of-Sight Vectors 103

4.3 Orbit Determination from Two Position Vectors and Time (Lambert’s

Problem) 109

4.3.1 The Lagrangian Coefficients

110

Notes

114

References 114

5 Orbital Maneuvers 115

5.1 Simple Impulsive Maneuvers 115

5.2 Coplanar Maneuvers 116

5.2.1 Hohmann Transfers

118

5.2.2 Bi-Elliptic Transfers

120

5.3 Plane Change Maneuvers 123

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Contents

ix

5.4 Combined Maneuvers 125

5.5 Rendezvous 127

Notes

128

Reference 128

6 Interplanetary Trajectories 129

6.1 Sphere of Influence 129

6.2 Interplanetary Hohmann Transfers 133

6.3 Patched Conics 137

6.3.1 Departure Hyperbola

139

6.3.2 Arrival Hyperbola

141

6.4 Planetary Flyby 143

6.5 Planetary Capture 145

Notes

146

References 147

7 Orbital Perturbations 149

7.1 Special Perturbations 150

7.1.1 Cowell’s Method

151

7.1.2 Encke’s Method

151

7.2 General Perturbations 154

7.3 Gravitational Perturbations due to a Non-Spherical Primary Body 156

7.3.1 The Perturbative Force Per Unit Mass Due to J

2

163

7.4 Effect of

J

2

on the Orbital Elements 164

7.5 Special Types of Orbits 168

7.5.1 Sun-Synchronous Orbits

168

7.5.2 Molniya Orbits

169

7.6 Small Impulse Form of the Gauss Variational Equations 169

7.7 Derivation of the Remaining Gauss Variational Equations 171

Notes

180

References 181

8 Low Thrust Trajectory Analysis and Design 183

8.1 Problem Formulation 183

8.2 Coplanar Circle to Circle Transfers 184

8.3 Plane Change Maneuver 186

Notes

188

References 188

9 Spacecraft Formation Flying 189

9.1 Mathematical Description 190

9.2 Relative Motion Solutions 194

9.2.1 Out-of-Plane Motion

195

9.2.2 In-Plane Motion

195

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Contents

9.2.3 Alternative Description for In-Plane Relative Motion

198

9.2.4 Further Examination of In-Plane Motion

200

9.2.5 Out-of-Plane Motion - Revisited

202

9.3 Special Types of Relative Orbits 203

9.3.1 Along-Track Orbits

203

9.3.2 Projected Elliptical Orbits

204

9.3.3 Projected Circular Orbits

207

Notes

207

Reference 207

10 The Restricted Three-Body Problem 209

10.1 Formulation 209

10.1.1 Equations of Motion

211

10.2 The Lagrangian Points 212

10.2.1 Case (i)

212

10.2.2 Case (ii)

213

10.3 Stability of the Lagrangian Points 214

10.3.1 Comments

215

10.4 Jacobi’s Integral 215

10.4.1 Hill’s Curves

216

10.4.2 Comments on Figure 10.5

218

Notes

218

References 218

11 Introduction to Spacecraft Attitude Stabilization 219

11.1 Introduction to Control Systems 220

11.1.1 Open-loop versus Closed-loop

220

11.1.2 Typical Feedback Control Structure

221

11.2 Overview of Attitude Representation and Kinematics 222

11.3 Overview of Spacecraft Attitude Dynamics 223

11.3.1 Properties of the Inertia Matrix - A Summary

224

12 Disturbance Torques on a Spacecraft 227

12.1 Magnetic Torque 227

12.2 Solar Radiation Pressure Torque 228

12.3 Aerodynamic Torque 230

12.4 Gravity-Gradient Torque 231

Notes

234

Reference 234

13 Torque-Free Attitude Motion 235

13.1 Solution for an Axisymmetric Body 235

13.2 Physical Interpretation of the Motion 242

Notes

245

References 245

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Contents

xi

14 Spin Stabilization 247

14.1 Stability 247

14.2 Spin Stability of Torque-Free Motion 249

14.3 Effect of Internal Energy Dissipation 252

14.3.1 Energy Sink Hypothesis

252

14.3.2 Major Axis Rule

253

Notes

253

References 253

15 Dual-Spin Stabilization 255

15.1 Equations of Motion 255

15.2 Stability of Dual-Spin Torque-Free Motion 257

15.3 Effect of Internal Energy Dissipation 259

Notes

266

References 266

16 Gravity-Gradient Stabilization 267

16.1 Equations of Motion 268

16.2 Stability Analysis 272

16.2.1 Pitch Motion

272

16.2.2 Roll-Yaw Motion

273

16.2.3 Combined Pitch and Roll/Yaw

277

Notes

277

References 277

17 Active Spacecraft Attitude Control 279

17.1 Attitude Control for a Nominally Inertially Fixed Spacecraft 280

17.2 Transfer Function Representation of a System 281

17.3 System Response to an Impulsive Input 282

17.4 Block Diagrams 284

17.5 The Feedback Control Problem 286

17.6 Typical Control Laws 289

17.7 Time-Domain Specifications 292

17.8 Factors that Modify the Transient Behavior 308

17.9 Steady-State Specifications and System Type 311

17.10 Effect of Disturbances 316

17.11 Actuator Limitations 319

Notes 320

References 320

18 Routh’s Stability Criterion 321

18.1 Proportional-Derivative Control with Actuator Dynamics 322

18.2 Active Dual-Spin Stabilization 325

Notes 330

References 330

19 The Root Locus 331

19.1 Rules for Constructing the Root Locus 332

19.2 PD Attitude Control with Actuator Dynamics - Revisited 341

19.3 Derivation of the Rules for Constructing the Root Locus 345

Notes 353

References 353

20 Control Design by the Root Locus Method 355

20.1 Typical Types of Controllers 357

20.2 PID Design for Spacecraft Attitude Control 361

Notes 369

References 369

21 Frequency Response 371

21.1 Frequency Response and Bode Plots 372

21.2 Low-Pass Filter Design 383

Notes 385

References 385

22 Relative Stability 387

22.1 Polar Plots 387

22.2 Nyquist Stability Criterion 390

22.3 Stability Margins 399

Notes 410

References 410

23 Control Design in the Frequency Domain 411

23.1 Feedback Control Problem - Revisited 416

23.2 Control Design 422

23.3 Example - PID Design for Spacecraft Attitude Control 430

Notes 435

References 435

24 Nonlinear Spacecraft Attitude Control 437

24.1 State-Space Representation of the Spacecraft Attitude Equations 437

24.2 Stability Definitions 440

24.3 Stability Analysis 442

24.4 LaSalle’s Theorem 448

24.5 Spacecraft Attitude Control with Quaternion and Angular Rate Feedback 451

Notes 456

References 457

25 Spacecraft Navigation 459

25.1 Review of Probability Theory 459

25.2 Batch Approaches for Spacecraft Attitude Estimation 467

25.3 The Kalman Filter 477

Notes 496

References 497

26 Practical Spacecraft Attitude Control Design Issues 499

26.1 Attitude Sensors 499

26.2 Attitude Actuators 506

26.3 Control Law Implementation 511

26.4 Unmodeled Dynamics 523

Notes 539

References

Appendix A: Review of Complex Variables 541

Appendix B: Numerical Simulation of Spacecraft Motion 557

Notes 561

Reference 561

Index 563

Anton de Ruiter, Assistant Professor, Mechanical and Aerospace Engineering Department, Carleton University, Ottawa, Canada.
Obtained his PhD in Aerospace Engineering from the University of Toronto in 2005.? Until 2006 he was a Visiting Research Fellow at the Space Technologies Branch of the Canadian Space Agency.?His interests include Nano-Satellite Technologies, Interplanetary Missions, Spacecraft Formation Flying, Spacecraft Attitude and Orbit Determination and Control, GPS-based Spacecraft Navigation, Control Systems, and Optimization Theory and Applications.?Professor De Ruiter has written extensively on spacecraft dynamics and related topics for journals, articled papers and conference proceedings.

Christopher J. Damaren, Professor, University of Toronto Institute for Aerospace Studies.
Obtained his doctorate at UTIAS in 1990 in the area of control systems for flexible spacecraft. In the 1990's most of his research concentrated on control system design for large structurally flexible robot manipulator systems such as the Space Station robotic systems developed by Canada. Since joining the faculty of UTIAS in 1999, his research group has been involved in the dynamics and control of spacecraft including the orbital, attitude, and structural motions of these systems.

James R. Forbes, Assistant Professor, Department of Mechanical Engineering, McGill University.
Obtained his doctorate at UTIAS in 2011 in the area of control system design with applications to aerospace systems, including spacecraft attitude control. His teaching duties at McGill University include spacecraft dynamics and control courses at the upper undergraduate/beginning graduate level