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Chapter 1 Primer on Spacecraft Dynamics 1

1.1 Orbital Motion of Spacecraft 2

1.1.1 Gravitational Field of a Particle 2

1.1.2 Gravitational Field of a Rigid Body 2

1.1.3 Dynamical Equations of Two-body System 4

1.1.4 First Integrals 5

1.1.5 Characteristics of Keplerian Orbit 8

1.1.6 Elliptic Orbit 10

1.2 Environmental Torques Acting on Spacecraft 12

1.2.1 Gravitational Torque 12

1.2.2 Magnetic Torque 15

1.3 Attitude Motion of Spacecraft in the Gravitational Field 17

1.3.1 Euler's Equations and Poisson's Equations 17

1.3.2 Planar Libration 19

1.3.3 Stability of Relative Equilibrium 22

1.3.4 Attitude Motion of a Gyrostat 26

1.4 Attitude Motion of Torque-free Spacecraft 27

1.4.1 Torque-free Rigid Body 27

1.4.2 Torque-free Gyrostat 29

1.4.3 Influence of Energy Dissipation on Spinning Spacecraft 31

References 32

Chapter 2 A Survey of Chaos Theory 33

2.1 The Overview of Chaos 34

2.1.1 Descriptions of Chaos 34

2.1.2 Geometrical Structures of Chaos 35

2.1.3 Routes to Chaos 37

2.2 Numerical Identification of Chaos 40

2.2.1 Introduction 40

2.2.2 Lyapunov Exponents 40

2.2.3 Power Spectra 42

2.3 Melnikov Theory 44

2.3.1 Introduction 44

2.3.2 Transversal Homoclinic/Heteroclinic Point 44

2.3.3 Analytical Prediction 47

2.3.4 Interruptions 50

2.4 Chaos in Hamiltonian Systems 51

2.4.1 Hamiltonian Systems,Integrability and KAM Theorem 51

2.4.2 Stochastic Layers and Global Chaos 55

2.4.3 Arnol'd Diffusion 58

2.4.4 Higher-Dimensional Version of Melnikov Theory 59

References 61

Chapter 3 Chaos in Planar Attitude Motion of Spacecraft 63

3.1 Rigid Spacecraft in an Elliptic Orbit 64

3.1.1 Introduction 64

3.1.2 Dynamical Model 65

3.1.3 Melnikov Analysis 66

3.1.4 Numerical Simulations 68

3.2 Tethered Satellite Systems 69

3.2.1 Introduction 69

3.2.2 Dynamical Models 70

3.2.3 Melnikov Analysis of the Uncoupled Case 73

3.2.4 Numerical Simulations 74

3.3 Magnetic Rigid Spacecraft in a Circular Orbit 75

3.3.1 Introduction 75

3.3.2 Dynamical Model 77

3.3.3 Melnikov Analysis 79

3.3.4 Numerical Investigations:Undamped Case 80

3.3.5 Numerical Investigations:Damped Case 83

3.4 Magnetic Rigid Spacecraft in an Elliptic Orbit 89

3.4.1 Introduction 89

3.4.2 Dynamical Model 89

3.4.3 Melnikov Analysis 91

3.4.4 Numerical Simulations 92

References 95

Chapter 4 Chaos in Spatial Attitude Motion of Spacecraft 99

4.1 Attitude Morion Described by Serret-Andoyer Variables 100

4.1.1 Serret-Andoyer Variables 100

4.1.2 Torque-free Rigid Body 103

4.1.3 Torque-free Gyrostat 104

4.1.4 Gyrostat in the Gravitational Field 106

4.1.5 Influence of the Geomagnetic Field 107

4.2 Rigid-body Spacecraft in an Elliptic Orbit 108

4.2.1 Introduction 108

4.2.2 Dynamical Model 109

4.2.3 Melnikov Analysis 111

4.2.4 Numerical Simulations 113

4.3 Rigid-body Spacecraft with an Eccentrically Rotating Mass 113

4.3.1 Introduction 113

4.3.2 Dynamical Model 116

4.3.3 Melnikov Analysis 117

4.3.4 Numerical Simulations 119

4.4 Magnetic Gyrostat Spacecraft in a Circular Orbit 120

4.4.1 Introduction 120

4.4.2 Unperturbed Motion of a Gyrostat 122

4.4.3 Melnikov Analysis 123

4.4.4 Numerical Simulations 125

References 127

Chapter 5 Control of Chaotic Attitude Motion 131

5.1 Control of Chaos:An Overview 131

5.1.1 Introduction 131

5.1.2 Problem Formulations 133

5.1.3 OGY Method and Its Generalization 134

5.1.4 Synchronization:Chaos Control in a Broader Sense 136

5.2 The Parametric Open-plus-closed-loop Method 137

5.2.1 Introduction 137

5.2.2 The Control Law 138

5.2.3 Numerical Examples 140

5.2.4 Discussions 144

5.3 The Stability Criterion Method 145

5.3.1 Introduction 145

5.3.2 The Control Law 146

5.3.3 Numerical Examples 147

5.4 Controlling Chaotic Attitude Motions 153

5.4.1 Introduction 153

5.4.2 Dynamical Model of Controlled Spacecraft 154

5.4.3 Applications of the Parametric Open-plus-closed-loop Method 155

5.4.4 Applications of the Stability Criterion Method 156

References 160