An extensive text reference includes around an asteroid – a new and important topic

Covers the most updated contents in spacecraft dynamics and control, both in theory and application
Introduces the application to motion around asteroids – a new and important topic
Written by a very experienced researcher in this area

WEIDUO HU, Beihang University, P. R. China

Preface xi

Acknowledgments xiii

About the Author xv

Part I Orbital Mechanics

1 Introduction 3

1.1 History 3

1.1.1 Kepler’s Laws 3

1.1.2 Newton’s Laws 4

1.1.3 Space Missions 5

1.2 Coordinate Systems 6

1.2.1 Earth Reference Frame 6

1.2.2 Sun-centered Frame 8

1.2.3 Right Ascension-Declination System 8

1.2.4 Perifocal Coordinate System 9

1.2.5 Satellite Coordinate System 9

1.2.6 Topo-centric-horizon Coordinate System 9

1.3 Time System 10

1.3.1 Clocks 10

1.3.2 Time 11

1.3.3 Reference Motions 14

1.4 References and Further Reading 16

1.5 Summary and Keywords 16

Problems 17

References 17

2 Keplerian Motion 19

2.1 N-body System 19

2.2 The Two-body Problem 21

2.2.1 Geometry for Two Bodies in an Inertial Reference Frame 21

2.2.2 Relative Motion of Two Bodies 21

2.2.3 Constants of the Motion 23

2.2.4 Orbital Path 27

2.3 Orbital Elements 36

2.3.1 Kepler’s COEs 36

2.3.2 Alternate Orbital Element Quantities 38

2.3.3 A Calculation Example 38

2.4 Coordinate Transformations 40

2.4.1 Rotation 41

2.4.2 From P̂Q̂Ŵ to ÎĴK̂ 41

2.4.3 From ÎĴK̂ to ŜÊẐ 43

2.4.4 Single Radar Observation 45

2.4.5 Summary of the Transformation 48

2.4.6 Three Position Vectors (Gibbs Method) 50

2.5 Time of Flight (TOF) 51

2.5.1 Kepler’s Equation (Elliptical Orbits) 51

2.5.2 Numerical Solution 53

2.5.3 Universal Variable X 56

2.5.4 f and g Expansion 60

2.6 Summary and Keywords 62

Problems 63

References 67

3 Orbit Maneuver 69

3.1 Basic Orbital Transfer 69

3.1.1 In-plane (Coplanar) Changes 69

3.1.2 Out-of-plane (Non-coplanar) Changes 71

3.1.3 The Phase Angle 72

3.2 Ballistic Missiles 73

3.2.1 Ballistic Missile Trajectory 74

3.2.2 Effect of the Earth’s Rotation 77

3.3 Lunar Missions 80

3.3.1 Possibility of Transfer 80

3.3.2 More Practical Scenario 83

3.4 Interplanetary Travel 87

3.4.1 Sphere of Influence (SOI) 88

3.4.2 Scenario 89

3.4.3 Gravity Assist 93

3.5 Launch Issues, Starting the Mission 96

3.5.1 Launch Time 96

3.5.2 When and Where to Launch 97

3.5.3 Launch Velocity 98

3.5.4 Rockets and Launch Vehicles 100

3.5.5 Reentry 101

3.6 Summary and Keywords 102

Problems 102

References 106

4 Special Topics 107

4.1 Relative Motion – CW Equation 107

4.1.1 Equations of Motion 107

4.1.2 Examples 110

4.1.3 J₂ Perturbed No-circular Target Orbit 112

4.2 Lambert Problem 113

4.2.1 Lambert’s Theorem 114

4.2.2 Culp’s Proof of Lambert’s Theorem 117

4.2.3 f , g Function Algorithm 119

4.2.4 Examples 120

4.2.5 Comparison of CW and Nonlinear Lambert Analysis 122

4.3 Orbit Determination 123

4.4 Optimal Control 127

4.5 Three-Body Problem – CRTBP 132

4.5.1 Equations of Motion 134

4.5.2 Lagrange Points 135

4.5.3 Examples of CRTBP 137

4.6 Summary and Keywords 139

Problems 141

References 145

5 Perturbed Orbital Motions 147

5.1 Special Perturbation 147

5.1.1 General Concept of Perturbation 147

5.1.2 Cowell’s Method 148

5.1.3 Encke’s Method 148

5.1.4 Variation of Parameters (COEs) 149

5.2 Systematic Method to Derive VOP 151

5.2.1 Variation of Orbital Elements 151

5.2.2 Variation of r and 𝑣 156

5.3 General Perturbations (GA) 157

5.3.1 An Example of GP 158

5.3.2 General Perturbation Techniques 158

5.3.3 Total Perturbation 161

5.3.4 Analytical Perturbation Formulation 162

5.3.5 Gravity Potential 165

5.4 Numerical Methods 168

5.5 Summary and Keywords 169

Problems 169

References 171

6 Orbital Motion Around Asteroids 173

6.1 Introduction 173

6.2 Problem Formulation 174

6.3 Motion Equations 177

6.4 Progress and Some Conclusions 178

6.4.1 Secular Motion 178

6.4.2 Resonance 180

6.4.3 Periodic Orbits 180

6.5 Conclusions 182

6.6 Summary and Keywords 182

Acknowledgments 183

Problems 183

References 183

7 Application 185

7.1 Two Line Elements (TLE) 185

7.2 GPS RINEX 187

7.2.1 Distribution of Ephemerides 187

7.2.2 A GPS RINEX Navigation File 188

7.3 Remote Observation 188

7.3.1 Orbital Coverage 188

7.3.2 Ground Track 189

7.3.3 Ground Station 193

7.4 Summary and Keywords 196

Problems 198

References 198

Part II Attitude Dynamics

8 Rigid Body Kinematics 201

8.1 Attitude Notions 201

8.1.1 Attitude 201

8.1.2 Frames 202

8.1.3 Vector 204

8.2 Attitude Parameters 205

8.2.1 Direct Cosine Matrix 205

8.2.2 Euler Angles 207

8.3 Differential Equations of Kinematics 210

8.3.1 Typical Problem Involving Angular Velocity and Attitude 213

8.4 Euler’s Theorem 214

8.4.1 Another Four-Parameter Set 215

8.4.2 Summary and Extension of Kinematics Notation 216

8.5 Attitude Determination 219

8.5.1 Sensors 219

8.5.2 Attitude Determination 222

8.6 Summary and Keywords 228

Problems 228

References 230

9 Attitude Dynamics 231

9.1 Rigid Body Models 231

9.1.1 Particle Model 232

9.1.2 Continuous Model 232

9.1.3 Angular Momentum 236

9.2 Inertia Tensor 238

9.2.1 Calculation of Inertia Tensor (𝐈) 238

9.2.2 Parallel Axis Theorem (PAT) 240

9.2.3 Change of Vector Basis Theorem 242

9.2.4 Inertia with a Rotating Panel 244

9.3 Equation of S/C Motion 245

9.3.1 Angular Momentum Principle 245

9.3.2 Rotational Equation of Motion 246

9.3.3 Coupled Equations of Motion 247

9.4 Euler’s Equation 247

9.4.1 Axisymmetric, Torque-Free Rigid Body 248

9.4.2 Axisymmetric, Torque-Free Rigid Body Summary 252

9.4.3 Asymmetric, Torque-Free Rigid Body 253

9.5 Summary and Keywords 255

Problems 255

References 255

10 Attitude Stabilization and Control 257

10.1 Free Body Motion Stability 257

10.2 Dual-Spin Stabilization 259

10.3 Gravity Gradient Stabilization 260

10.4 Three-axis Stabilization 263

10.5 Complete Dynamical Modeling of Spacecraft 266

10.5.1 Flexible Panels Example 267

10.5.2 Liquid Sloshing Example 268

10.5.3 A Mass Spring S/C Example 268

10.6 Summary and Keywords 269

Problems 269

References 271

Appendix A Math Review 273

A.1 Power, Taylor Series 273

A.2 Differential Correction 276

Algorithm Process 277

A.3 Spherical Trigonometry 278

A.4 Summary and Keywords 281

References 281

Index 283

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