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.
Table of Contents:
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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x
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
About the Author :
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
Review :
“In conclusion, this book covers a broad range of areas – including some more in-depth content (stabilisation techniques, practical design issues) – and is best used as an introductory text to the field for latter year undergraduates.” (The Aeronautical Journal, 1 November 2014)
“Overall, this book provides a good, comprehensive examination of the fundamentals of translational and rotational dynamics, determination, and control of spacecraft. Summing Up: Recommended. All academic and professional aerospace engineering collections.” (Choice, 1 September 2013)
