Machine generated contents note: 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