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.

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

Dedication iii

Preface xiii

1 Kinematics 1

1.1 Physical vectors 1

1.1.1 Scalar Product 2

1.1.2 Vector Cross Product 3

1.1.3 Other Useful Vector Identities 5

1.2 Reference Frames and Physical Vector Coordinates 5

1.2.1 Vector Addition and Scalar Multiplication 7

1.2.2 Scalar Product 7

1.2.3 Vector Cross Product 8

1.2.4 Column Matrix Identities 9

1.3 Rotation Matrices 9

1.3.1 Principal Rotations 12

1.3.2 General Rotations 13

1.3.3 Euler Angles 19

1.3.4 Quaternions 20

1.4 Derivatives of Vectors 27

1.4.1 Angular Velocity 28

1.4.2 Angular Velocity in Terms of Euler Angle Rates 31

1.4.3 Angular Velocity in Terms of Quaternion Rates 32

1.5 Velocity and Acceleration 34

1.6 More Rigorous Definition of Angular Velocity 35

References 37

2 Rigid Body Dynamics 39

2.1 Dynamics of a Single Particle 39

2.2 Dynamics of a System of Particles 41

2.3 Rigid Body Dynamics 44

2.3.1 Translational Dynamics 44

2.3.2 Rotational Dynamics 45

2.4 The Inertia Matrix 47

2.4.1 A Parallel Axis Theorem 48

2.4.2 A Rotational Transformation Theorem 49

2.4.3 Principal Axes 49

2.5 Kinetic Energy of a Rigid Body 51

References 53

3 The Keplerian Two-Body Problem 55

3.1 Equations of motion 55

3.2 Constants of the motion 56

3.2.1 Orbital Angular Momentum 56

3.2.2 Orbital Energy 57

3.2.3 The Eccentricity Vector 58

3.3 Shape of a Keplerian orbit 59

3.3.1 Perifocal Coordinate System 61

3.4 Kepler’s Laws 68

3.5 Time of Flight 71

3.5.1 Circular Orbits 71

3.5.2 Elliptical Orbits 71

3.5.3 Parabolic Orbits 75

3.5.4 Hyperbolic Orbits 75

3.6 Orbital Elements 75

3.6.1 Heliocentric-Ecliptic Coordinate System 76

3.6.2 Geocentric-Equatorial Coordinate System 77

3.7 Orbital Elements given Position and Velocity 78

3.8 Position and Velocity given Orbital Elements 80

References 84

4 Preliminary Orbit Determination 85

4.1 Orbit Determination from Three Position Vectors 85

4.2 Orbit Determination from Three Line-of-Sight Vectors 88

4.3 Orbit Determination from Two Position Vectors and Time (Lambert’s Problem) 94

4.3.1 The Lagrangian Coefficients 94

References 98

5 Orbital Maneuvers 99

5.1 Simple ImpulsiveManeuvers 99

5.2 Coplanar Maneuvers 100

5.2.1 Hohmann Transfers 102

5.2.2 Bi-Elliptic Transfers 104

5.3 Plane Change Maneuvers 106

5.4 Combined Maneuvers 108

5.5 Rendezvous 110

References 111

6 Interplanetary Trajectories 113

6.1 Sphere of Influence 113

6.2 Interplanetary Hohmann Transfers 116

6.3 Patched Conics 120

6.3.1 Departure Hyperbola 121

6.3.2 Arrival Hyperbola 123

6.4 Planetary Flyby 126

6.5 Planetary Capture 127

References 129

7 Orbital Perturbations 131

7.1 Special Perturbations 132

7.1.1 Cowell’s Method 132

7.1.2 Encke’s Method 133

7.2 General Perturbations 134

7.3 Gravitational Perturbations due to a Non-Spherical Primary Body 137

7.3.1 The Perturbative Force Per Unit Mass Due to J2 142

7.4 Effect of J2 on the orbital elements 143

7.5 Special Types of Orbits 146

7.5.1 Sun-synchronous orbits 147

7.5.2 Molniya Orbits 147

7.6 Small Impulse Form of the Gauss Variational Equations 148

7.7 Derivation of the Remaining Gauss Variational Equations 149

References 156

8 Low Thrust Trajectory Analysis and Design 157

8.1 Problem Formulation 157

8.2 Coplanar Circle to Circle Transfers 158

8.3 Plane Change Maneuver 160

References 161

9 Spacecraft Formation Flying 163

9.1 Mathematical Description 164

9.2 Relative Motion Solutions 168

9.2.1 Out-of-PlaneMotion 168

9.2.2 In-Plane Motion 168

9.2.3 Alternative Description for In-Plane Relative Motion 170

9.2.4 Further Examination of In-Plane Motion 172

9.2.5 Out-of-PlaneMotion - Revisited 174

9.3 Special Types of Relative Orbits 175

9.3.1 Along-Track Orbits 175

9.3.2 Projected Elliptical Orbits 176

9.3.3 Projected Circular Orbits 178

References 178

10 The Restricted Three-Body Problem 179

10.1 Formulation 179

10.1.1 Equations of Motion 181

10.2 The Lagrangian Points 182

10.2.1 Case (i) 182

10.2.2 Case (ii) 182

10.3 Stability of the Lagrangian Points 183

10.3.1 Comments 184

10.4 Jacobi’s Integral 185

10.4.1 Hill’s Curves 185

10.4.2 Comments on Figure 10.5 187

References 187

11 Introduction to Spacecraft Attitude Stabilization 189

11.1 Introduction to Control Systems 190

11.2 Overview of Attitude Representation and Kinematics 192

11.3 Overview of Spacecraft Attitude Dynamics 193

12 Disturbance Torques on a Spacecraft 195

12.1 Magnetic Torque 195

12.2 Solar Radiation Pressure Torque 195

12.3 Aerodynamic Torque 197

12.4 Gravity-Gradient Torque 199

References 202

13 Torque-Free Attitude Motion 203

13.1 Solution for an Axisymmetric Body 203

13.2 Physical Interpretation of the Motion 209

References 212

14 Spin Stabilization 213

14.1 Stability 213

14.2 Spin Stability of Torque-FreeMotion 215

14.3 Effect of Internal Energy Dissipation 217

References 218

15 Dual-Spin Stabilization 219

15.1 Equations of Motion 219

15.2 Stability of Dual-Spin Torque-FreeMotion 220

15.3 Effect of Internal Energy Dissipation 222

References 228

16 Gravity-Gradient Stabilization 229

16.1 Equations of Motion 230

16.2 Stability Analysis 233

16.2.1 Pitch Motion 233

16.2.2 Roll-Yaw Motion 234

16.2.3 Combined Pitch and Roll/Yaw 237

References 238

17 Active Spacecraft Attitude Control 239

17.1 Attitude Control for a Nominally Inertially Fixed Spacecraft 240

17.2 Transfer Function Representation of a System 241

17.3 System Response to an Impulsive Input 242

17.4 Block Diagrams 243

17.5 The Feedback Control Problem 246

17.6 Typical Control Laws 248

17.6.1 Proportional “P” Control 248

17.6.2 Proportional Derivative “PD” Control 249

17.6.3 Proportional Integral Derivative “PID” Control 250

17.7 Time-Domain Specifications 251

17.7.1 Transient Specifications 252

17.8 Factors that Modify the Transient Behavior 265

17.8.1 Effect of Zeros 265

17.8.2 Effect of Additional Poles 267

17.9 Steady-State Specifications and System Type 268

17.10Effect of Disturbances 274

17.11Actuator Limitations 275

References 277

18 Routh’s Stability Criterion 279

18.1 Proportional-Derivative Control with Actuator Dynamics 280

18.2 Active Dual-Spin Stabilization 282

References 287

19 The Root Locus 289

19.1 Rules for Constructing the Root Locus 290

19.2 PD Attitude Control with Actuator Dynamics - Revisited 297

19.3 Derivation of the Rules for Constructing the Root Locus 301

References 309

20 Control Design by the Root Locus Method 311

20.1 Typical Types of Controllers 313

20.2 PID Design for Spacecraft Attitude Control 317

References 324

21 Frequency Response 327

21.1 Frequency Response and Bode Plots 328

21.1.1 Plotting the Frequency Response as a Function of ω (Bode Plots) 330

21.2 Low-Pass Filter Design 338

References 339

22 Relative Stability 341

22.1 Polar Plots 341

22.2 Nyquist Stability Criterion 343

22.2.1 Argument Principle 344

22.2.2 Stability Analysis of the Closed-Loop System 346

22.3 Stability Margins 352

22.3.1 Stability Margin Definitions 354

References 362

23 Control Design in the Frequency Domain 363

23.1 Feedback Control Problem - Revisited 368

23.1.1 Closed-Loop Tracking Error 369

23.1.2 Closed-Loop Control Effort 370

23.1.3 Modified Control Implementation 371

23.2 Control Design 372

23.2.1 Frequency Responses for Common Controllers 375

23.3 Example - PID Design for Spacecraft Attitude Control 380

References 385

24 Nonlinear Spacecraft Attitude Control 387

24.1 State-Space Representation of the Spacecraft Attitude Equations 387

24.2 Stability Definitions 390

24.2.1 Equilibrium Points 390

24.2.2 Stability of Equilibria 390

24.3 Stability Analysis 392

24.3.1 Detumbling of a Rigid Spacecraft 392

24.3.2 Lyapunov Stability Theorems 395

24.4 LaSalle’s Theorem 397

24.5 Spacecraft Attitude Control with Quaternion and Angular Rate Feedback 399

24.5.1 Controller Gain Selection 401

References 404

25 Spacecraft Navigation 405

25.1 Review of Probability Theory 405

25.1.1 Continuous Random Variables and Probability Density Functions 405

25.1.2 Mean and Covariance 407

25.1.3 Gaussian Probability Density Functions 409

25.1.4 Discrete-TimeWhite Noise 411

25.1.5 Simulating Noise 411

25.2 Batch Approaches for Spacecraft Attitude Estimation 412

25.2.1 Wahba’s Problem 413

25.2.2 Davenport’s q-Method 413

25.2.3 The QUEST Algorithm 416

25.2.4 The TRIAD Algorithm 418

25.2.5 Example 419

25.3 The Kalman Filter 421

25.3.1 The Discrete-Time Kalman Filter 421

25.3.2 The Norm-Constrained Kalman Filter 425

25.3.3 Spacecraft Attitude Estimation Using the Norm-Constrained

Extended Kalman Filter 431

References 438

26 Practical Spacecraft Attitude Control Design Issues 441

26.1 Attitude Sensors 441

26.1.1 Sun-Sensors 441

26.1.2 Three-AxisMagnetometers 443

26.1.3 Earth Sensors 444

26.1.4 Star Trackers 446

26.1.5 Rate Sensors 447

26.2 Attitude Actuators 447

26.2.1 Thrusters 448

26.2.2 Magnetic Torquers 450

26.2.3 ReactionWheels 450

26.2.4 MomentumWheels 452

26.2.5 Control Moment Gyroscopes 452

26.3 Control Law Implementation 453

26.3.1 Time-Domain Representation of a Transfer Function 453

26.3.2 Control Law Digitization 455

26.3.3 Closed-Loop Stability Analysis 458

26.3.4 Sampling Considerations 460

26.4 Unmodeled dynamics 464

26.4.1 Effects of Spacecraft Flexibility 464

26.4.2 Effects of Propellant Sloshing 477

References 478

INDEX 495