Analytical Mechanics. of Space Systems. tfa AA. Hanspeter Schaub. College Station, Texas. University of Colorado Boulder, Colorado.

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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.

1 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

2 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 Introduction Particle Position Description Vector Differentiation 6 References 23 Problems 23 Chapter 2 Newtonian Mechanics Introduction Newton's Laws Single Particle Dynamics Dynamics of a System of Particles Dynamics of a Continuous System Rocket Problem 66 References 71 Problems 71 Chapter 3 Rigid Body Kinematics Introduction Direction Cosine Matrix Euler Angles Principal Rotation Vector Euler Parameters Classical Rodrigues Parameters 112 xi

3 xll Analytical Mechanics of Space Systems 3.7 Modified Rodrigues Parameters Other Attitude Parameters Homogeneous Transformations Deterministic Attitude Estimation 136 References 150 Problems 152 Chapter 4 Eulerian Mechanics Introduction Rigid Body Dynamics Torque-Free Rigid Body Rotation Dual-Spin Spacecraft Momentum Exchange Devices Gravity Gradient Satellite 206 References 216 Problems 217 Chapter 5 Generalized Methods of Analytical Dynamics Introduction Generalized Coordinates D'Alembert's Principle Lagrangian Dynamics Quasi Coordinates Cyclic Coordinates Final Observations 298 References 299 Problems 299 Chapter 6 Variational Methods in Analytical Dynamics Introduction Fundamentals of Variational Calculus Hamilton's Variational Principles Hamilton's Principal Function Some Classical Applications of Hamilton's Principle to Distributed Parameter Systems Explicit Generalizations of Lagrange's Equations for Hybrid Coordinate Systems 326 References 335 Problems 335

4 Contents xiil Chapter 7 Hamilton's Generalized Formulations of Analytical Dynamics Introduction Hamiltonian Function Relationship of Hamiltonian Function to Work/Energy Integral Hamilton's Canonical Equations Poisson's Brackets Canonical Coordinate Transformations Perfect Differential Criterion for Canonical Transformations Transformation Jacobian Perspective on Canonical Transformations 362 References 364 Problems 364 Chapter 8 Nonlinear Spacecraft Stability and Control Introduction Nonlinear Stability Analysis Generating Lyapunov Functions Nonlinear Feedback Control Laws Lyapunov Optimal Control Laws Linear Closed-Loop Dynamics Reaction Wheel Control Devices Variable Speed Control Moment Gyroscopes 437 References 464 Problems 466 PART 2 CELESTIAL MECHANICS Chapter 9 Classical Two-Body Problem Introduction Geometry of Conic Sections Coordinate Systems Relative Two-Body Equations of Motion Fundamental Integrals Classical Solutions 503 References 519 Problems 520 Chapter 10 Restricted Three-Body Problem Introduction 527

5 xlv Analytical Mechanics of Space Systems 10.2 Lagrange's Three-Body Solution Circular Restricted Three-Body Problem Periodic Stationary Orbits Disturbing Function 564 References 568 Problems 568 Chapter 11 Gravitational Potential Field Models Introduction Gravitational Potential of Finite Bodies MacCullagh's Approximation Spherical Harmonic Gravity Potential Multibody Gravitational Acceleration Spheres of Gravitational Influence 592 References 595 Problems 595 Chapter 12 Perturbation Methods Introduction Encke's Method Variation of Parameters State Transition and Sensitivity Matrix 632 References 646 Problems 646 Chapter 13 Transfer Orbits Introduction Minimum Energy Orbit Hohmann Transfer Orbit Lambert's Problem Rotating the Orbit Plane Patched-Conic Orbit Solution 677 References 701 Problems 701 Chapter 14 Spacecraft Formation Flying Introduction General Relative Orbit Description Cartesian Coordinate Description Orbit Element Difference Description 721

6 Contents xv 14.5 Relative Motion State Transition Matrix Linearized Relative Orbit Motion /2-Invariant Relative Orbits 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

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