Analytical Mechanics of Space Systems Third Edition Hanspeter Schaub University of Colorado Boulder, Colorado John L. Junkins Texas A&M University College Station, Texas AIM EDUCATION SERIES Joseph A. Schetz, Editor-in-Chief Virginia Polytechnic Institute and State University Blacksburg, Virginia Published by the American Institute of Aeronautics and Astronautics, Inc. tfa AA 1801 Alexander Bell Drive, Reston, Virginia 20191-4344
CONTENTS Preface to the Third Edition Preface to the Second Edition Preface to the First Edition xvii xix xxi PART 1 BASIC MECHANICS Chapter 1 Particle Kinematics 1 1.1 Introduction 1 1.2 Particle Position Description 1 1.3 Vector Differentiation 6 References 23 Problems 23 Chapter 2 Newtonian Mechanics 31 2.1 Introduction 31 2.2 Newton's Laws 31 2.3 Single Particle Dynamics 36 2.4 Dynamics of a System of Particles 47 2.5 Dynamics of a Continuous System 61 2.6 Rocket Problem 66 References 71 Problems 71 Chapter 3 Rigid Body Kinematics 79 3.1 Introduction 79 3.2 Direction Cosine Matrix 80 3.3 Euler Angles 86 3.4 Principal Rotation Vector 95 3.5 Euler Parameters 103 3.6 Classical Rodrigues Parameters 112 xi
xll Analytical Mechanics of Space Systems 3.7 Modified Rodrigues Parameters 117 3.8 Other Attitude Parameters 126 3.9 Homogeneous Transformations 133 3.10 Deterministic Attitude Estimation 136 References 150 Problems 152 Chapter 4 Eulerian Mechanics 159 4.1 Introduction 159 4.2 Rigid Body Dynamics 159 4.3 Torque-Free Rigid Body Rotation 179 4.4 Dual-Spin Spacecraft 189 4.5 Momentum Exchange Devices 195 4.6 Gravity Gradient Satellite 206 References 216 Problems 217 Chapter 5 Generalized Methods of Analytical Dynamics 227 5.1 Introduction 227 5.2 Generalized Coordinates 227 5.3 D'Alembert's Principle 230 5.4 Lagrangian Dynamics 259 5.5 Quasi Coordinates 282 5.6 Cyclic Coordinates 290 5.7 Final Observations 298 References 299 Problems 299 Chapter 6 Variational Methods in Analytical Dynamics 307 6.1 Introduction 307 6.2 Fundamentals of Variational Calculus 307 6.3 Hamilton's Variational Principles 311 6.4 Hamilton's Principal Function 316 6.5 Some Classical Applications of Hamilton's Principle to Distributed Parameter Systems 318 6.6 Explicit Generalizations of Lagrange's Equations for Hybrid Coordinate Systems 326 References 335 Problems 335
Contents xiil Chapter 7 Hamilton's Generalized Formulations of Analytical Dynamics 339 7.1 Introduction 339 7.2 Hamiltonian Function 339 7.3 Relationship of Hamiltonian Function to Work/Energy Integral 344 7.4 Hamilton's Canonical Equations 349 7.5 Poisson's Brackets 353 7.6 Canonical Coordinate Transformations 356 7.7 Perfect Differential Criterion for Canonical Transformations 359 7.8 Transformation Jacobian Perspective on Canonical Transformations 362 References 364 Problems 364 Chapter 8 Nonlinear Spacecraft Stability and Control 367 8.1 Introduction 367 8.2 Nonlinear Stability Analysis 367 8.3 Generating Lyapunov Functions 386 8.4 Nonlinear Feedback Control Laws 405 8.5 Lyapunov Optimal Control Laws 421 8.6 Linear Closed-Loop Dynamics 427 8.7 Reaction Wheel Control Devices 433 8.8 Variable Speed Control Moment Gyroscopes 437 References 464 Problems 466 PART 2 CELESTIAL MECHANICS Chapter 9 Classical Two-Body Problem 471 9.1 Introduction 471 9.2 Geometry of Conic Sections 472 9.3 Coordinate Systems 480 9.4 Relative Two-Body Equations of Motion 488 9.5 Fundamental Integrals 491 9.6 Classical Solutions 503 References 519 Problems 520 Chapter 10 Restricted Three-Body Problem 527 10.1 Introduction 527
xlv Analytical Mechanics of Space Systems 10.2 Lagrange's Three-Body Solution 527 10.3 Circular Restricted Three-Body Problem 542 10.4 Periodic Stationary Orbits 563 10.5 Disturbing Function 564 References 568 Problems 568 Chapter 11 Gravitational Potential Field Models 571 11.1 Introduction 571 11.2 Gravitational Potential of Finite Bodies 572 11.3 MacCullagh's Approximation 575 11.4 Spherical Harmonic Gravity Potential 579 11.5 Multibody Gravitational Acceleration 590 11.6 Spheres of Gravitational Influence 592 References 595 Problems 595 Chapter 12 Perturbation Methods 597 12.1 Introduction 597 12.2 Encke's Method 598 12.3 Variation of Parameters 600 12.4 State Transition and Sensitivity Matrix 632 References 646 Problems 646 Chapter 13 Transfer Orbits 651 13.1 Introduction 651 13.2 Minimum Energy Orbit 651 13.3 Hohmann Transfer Orbit 655 13.4 Lambert's Problem 660 13.5 Rotating the Orbit Plane 672 13.6 Patched-Conic Orbit Solution 677 References 701 Problems 701 Chapter 14 Spacecraft Formation Flying 709 14.1 Introduction 709 14.2 General Relative Orbit Description 710 14.3 Cartesian Coordinate Description 712 14.4 Orbit Element Difference Description 721
Contents xv 14.5 Relative Motion State Transition Matrix 731 14.6 Linearized Relative Orbit Motion 736 14.7 /2-Invariant Relative Orbits 747 14.8 Relative Orbit Control Methods 768 References 789 Problems 790 Appendix A Transport Theorem Derivation Using Linear Algebra 793 Appendix B Various Euler Angle Transformations 797 Appendix C MRP Identity Proof 801 Appendix D Conic Section Transformations 803 Appendix E Numerical Subroutines Library 807 Appendix F First-Order Mapping Between Mean and Osculating Orbit Elements 813 Appendix G Direct Linear Mapping Between Cartesian Hill Frame Coordinates and Orbit Element Differences 817 Appendix H Hamel Coefficients for the Rotational Motion of a Rigid Body 819 Appendix I MRP Kalman Filter 827 Index 835 Supporting Materials 855