Analytical Mechanics of Space Systems, Second Edition Volume 7417 (Second Edition) ||

This book provides a comprehensive treatment of dynamics of space systems, starting with the fundamentals and covering topics from basic kinematics and dynamics to more advanced celestial mechanics. All material is presented in a consistent manner, and the reader is guided through the various derivations and proofs in a tutorial way. Cookbook formulas are avoided; instead, the reader is led to understand the principles underlying the equations at issue, and shown how to apply them to various dynamical systems. The book is divided into two parts. Part I covers analytical treatment of topics such as basic dynamic principles up to advanced energy concepts. Special attention is paid to the use of rotating reference frames that often occur in aerospace systems. Part II covers basic celestial mechanics, treating the two-body problem, restricted three-body problem, gravity field modeling, perturbation methods, spacecraft formation flying, and orbit transfers. MATLAB[registered], Mathematica[registered] and C-Code toolboxes are provided for the rigid body kinematics routines discussed in chapter 3, and the basic orbital 2-body orbital mechanics routines discussed in chapter 9. A solutions manual is also available for professors. MATLAB[registered] is a registered trademark of The Math Works, Inc.; Mathematica[registered] is a registered trademark of Wolfram Research, Inc. Cover Front Matter S Title Analytical Mechanics of Space Systems Copyright 2009 by the American Institute of Aeronautics and Astronautics ISBN 9781600867217 QB350.5.S33 2009 521 dc22 LCCN 2009032378 Dedication Editorial Board Foreword Table of Contents Preface Part 1: Basic Mechanics 1. Particle Kinematics 1.1 Introduction 1.2 Particle Position Description 1.2.1 Basic Geometry 1.2.2 Cylindrical and Spherical Coordinate Systems 1.3 Vector Differentiation 1.3.1 Angular Velocity Vector 1.3.2 Rotation About a Fixed Axis 1.3.3 Transport Theorem 1.3.4 Particle Kinematics with Moving Frames References Problems 2. Newtonian Mechanics 2.1 Introduction 2.2 Newton’s Laws 2.3 Single Particle Dynamics 2.3.1 Constant Force 2.3.2 Time-Varying Force 2.3.3 Kinetic Energy 2.3.4 Linear Momentum 2.3.5 Angular Momentum 2.4 Dynamics of a System of Particles 2.4.1 Equations of Motion 2.4.2 Kinetic Energy 2.4.3 Linear Momentum 2.4.4 Angular Momentum 2.5 Dynamics of a Continuous System 2.5.1 Equations of Motion 2.5.2 Kinetic Energy 2.5.3 Linear Momentum 2.5.4 Angular Momentum 2.6 Rocket Problem Reference Problems 3. Rigid Body Kinematics 3.1 Introduction 3.2 Direction Cosine Matrix 3.3 Euler Angles 3.4 Principal Rotation Vector 3.5 Euler Parameters 3.6 Classical Rodrigues Parameters 3.7 Modified Rodrigues Parameters 3.8 Other Attitude Parameters 3.8.1 Stereographic Orientation Parameters 3.8.2 Higher Order Rodrigues Parameters 3.8.3 The (w, z) Coordinates 3.8.4 Cayley–Klein Parameters 3.9 Homogeneous Transformations References Problems 4. Eulerian Mechanics 4.1 Introduction 4.2 Rigid Body Dynamics 4.2.1 Angular Momentum 4.2.2 Inertia Matrix Properties 4.2.3 Euler’s Rotational Equations of Motion 4.2.4 Kinetic Energy 4.3 Torque-Free Rigid Body Rotation 4.3.1 Energy and Momentum Integrals 4.3.2 General Free Rigid Body Motion 4.3.3 Axisymmetric Rigid Body Motion 4.4 Dual-Spin Spacecraft 4.4.1 System Equations of Motion 4.4.2 Linear Stability Conditions 4.5 Momentum Exchange Devices 4.5.1 Spacecraft with Single VSCMG 4.5.2 Spacecraft with Multiple VSCMGs 4.6 Gravity Gradient Satellite 4.6.1 Gravity Gradient Torque 4.6.2 Rotational–Translational Motion Coupling 4.6.3 Small Departure Motion About Equilibrium Attitudes References Problems 5. Generalized Methods of Analytical Dynamics 5.1 Introduction 5.2 Generalized Coordinates 5.3 D’Alembert’s Principle 5.3.1 Virtual Displacements and Virtual Work 5.3.2 Classical Development of d’Alembert’s Principle 5.3.3 Holonomic Constraints 5.3.4 Newtonian Constrained Dynamics of N Particles 5.3.5 Lagrange Multiplier Rule for Constrained Optimization 5.4 Lagrangian Dynamics 5.4.1 Minimal Coordinate Systems and Unconstrained Motion 5.4.2 Lagrange’s Equations for Conservative Forces 5.4.3 Redundant Coordinate Systems and Constrained Motion 5.4.4 Vector-Matrix Form of the Lagrangian Equations of Motion 5.5 Quasi Coordinates 5.6 Cyclic Coordinates 5.6.1 Some Introductory Observations 5.6.2 Routh’s Approach to Ignorable Coordinates 5.6.3 Concluding Remarks on Routh’s Approach to Ignorable Coordinates 5.7 Final Observations References Problems 6. Variational Methods in Analytical Dynamics 6.1 Introduction 6.2 Fundamentals of Variational Calculus 6.3 Hamilton’s Variational Principles 6.3.1 Virtual Displacements vs Path Variations 6.3.2 Hamilton’s Principle Derived from d’Alembert’s Principle 6.4 Hamilton’s Principal Function 6.5 Some Classical Applications of Hamilton’s Principle to Distributed Parameter Systems 6.6 Explicit Generalizations of Lagrange’s Equations for Hybrid Coordinate Systems References Problems 7. Hamilton’s Generalized Formulations ofAnalytical Dynamics 7.1 Introduction 7.2 Hamiltonian Function 7.3 Relationship of Hamiltonian Function to Work=Energy Integral 7.4 Hamilton’s Canonical Equations 7.5 Poisson’s Brackets 7.6 Canonical Coordinate Transformations 7.7 Perfect Differential Criterion for Canonical Transformations 7.8 Transformation Jacobian Perspective on Canonical Transformations References Problems 8. Nonlinear Spacecraft Stability and Control 8.1 Introduction 8.2 Nonlinear Stability Analysis 8.2.1 Stability Definitions 8.2.2 Linearization of Dynamical Systems 8.2.3 Lyapunov’s Direct Method 8.3 Generating Lyapunov Functions 8.3.1 Elemental Velocity–Based Lyapunov Functions 8.3.2 Elemental Position–Based Lyapunov Functions 8.4 Nonlinear Feedback Control Laws 8.4.1 Unconstrained Control Law 8.4.2 Asymptotic Stability Analysis 8.4.3 Feedback Gain Selection 8.5 Lyapunov Optimal Control Laws 8.6 Linear Closed-Loop Dynamics 8.7 Reaction Wheel Control Devices 8.8 Variable Speed Control Moment Gyroscopes 8.8.1 Control Law 8.8.2 Velocity-Based Steering Law 8.8.3 VSCMG Null Motion References Problems Part 2: Celestial Mechanics 9. Classical Two-Body Problem 9.1 Introduction 9.2 Geometry of Conic Sections 9.3 Coordinate Systems 9.4 Relative Two-Body Equations of Motion 9.5 Fundamental Integrals 9.5.1 Conservation of Angular Momentum 9.5.2 Eccentricity Vector Integral 9.5.3 Conservation of Energy 9.6 Classical Solutions 9.6.1 Kepler’s Equation 9.6.2 Orbit Elements 9.6.3 Lagrange=Gibbs F and G Solution References Problems 10. Restricted Three-Body Problem 10.1 Introduction 10.2 Lagrange’s Three-Body Solution 10.2.1 General Conic Solutions 10.2.2 Circular Orbits 10.3 Circular Restricted Three-Body Problem 10.3.1 Jacobi Integral 10.3.2 Zero-Relative-Velocity Surfaces 10.3.3 Lagrange Libration Point Stability 10.4 Periodic Stationary Orbits 10.5 Disturbing Function References Problems 11. Gravitational Potential Field Models 11.1 Introduction 11.2 Gravitational Potential of Finite Bodies 11.3 MacCullagh’s Approximation 11.4 Spherical Harmonic Gravity Potential 11.5 Multibody Gravitational Acceleration 11.6 Spheres of Gravitational Influence References Problems 12. Perturbation Methods 12.1 Introduction 12.2 Encke’s Method 12.3 Variation of Parameters 12.3.1 General Methodology 12.3.2 Lagrangian Brackets 12.3.3 Lagrange’s Planetary Equations 12.3.4 Poisson Brackets 12.3.5 Gauss’ Variational Equations 12.4 State Transition and Sensitivity Matrix 12.4.1 Linear Dynamic Systems 12.4.2 Nonlinear Dynamic Systems 12.4.3 Symplectic State Transition Matrix 12.4.4 State Transition Matrix of Keplerian Motion References Problems 13. Transfer Orbits 13.1 Introduction 13.2 Minimum Energy Orbit 13.3 Hohmann Transfer Orbit 13.4 Lambert’s Problem 13.4.1 General Problem Solution 13.4.2 p-Iteration Method 13.4.3 Elegant Velocity Properties 13.5 Rotating the Orbit Plane 13.6 Patched-Conic Orbit Solution 13.6.1 Canonical Units 13.6.2 Establishing the Heliocentric Departure Velocity 13.6.3 Escaping the Departure Planet’s Sphere of Influence 13.6.4 Entering the Target Planet’s Sphere of Influence 13.6.5 Planetary Flybys References Problems 14. Spacecraft Formation Flying 14.1 Introduction 14.2 General Relative Orbit Description 14.3 Cartesian Coordinate Description 14.3.1 Clohessy–Wiltshire Equations 14.3.2 Closed Relative Orbits in the Hill Reference Frame 14.4 Orbit Element Difference Description 14.4.1 Linear Mapping Between Hill Frame Coordinates and Orbit Element Differences 14.4.2 Bounded Relative Motion Constraint 14.5 Relative Motion State Transition Matrix 14.6 Linearized Relative Orbit Motion 14.6.1 General Elliptic Orbits 14.6.2 Chief Orbits with Small Eccentricity 14.6.3 Near-Circular Chief Orbit 14.7 J2-Invariant Relative Orbits 14.7.1 Ideal Constraints 14.7.2 Energy Levels between J2-Invariant Relative Orbits 14.7.3 Constraint Relaxation for Near-Polar Orbits 14.7.4 Near-Circular Chief Orbit 14.7.5 Relative Argument of Perigee and Mean Anomaly Drift 14.7.6 Fuel Consumption Prediction 14.8 Relative Orbit Control Methods 14.8.1 Mean Orbit Element Continuous Feedback Control Laws 14.8.2 Cartesian Coordinate Continuous Feedback Control Law 14.8.3 Impulsive Feedback Control Law 14.8.4 Hybrid Feedback Control Law References Problems Back Matter Appendices Appendix A: Transport Theorem DerivationUsing Linear Algebra Appendix B: Various Euler Angle Transformations B.1 Direction Cosine Matrix in Terms of the 12 Euler Angle Sets B.2 Mapping Between Body Angular Velocity Vector and the Euler Angle Rates Appendix C: MRP Identity Proof Appendix D: Conic Section Transformations D.1 Elliptic Orbit Elements D.2 Elliptic Anomaly Mapping and Sensitivities D.3 Hyperbolic Orbit Parameter Transformations D.4 Hyperbolic Anomaly Mapping and Sensitivities Appendix E: Numerical Subroutines Library E.1 Matlab Functions E.2 Mathematica Packages E.3 C-Code Subroutines Appendix F: First-Order Mapping Between Mean andOsculating Orbit Elements References Appendix G: Direct Linear Mapping BetweenCartesian Hill Frame Coordinatesand Orbit Element Differences Appendix H: Hamel Coefficients for theRotational Motion of aRigid Body Bibliography Index Supporting Materials Errata "This book provides a comprehensive treatment of dynamics of space systems, starting with the fundamentals and covering topics from basic kinematics and dynamics to more advanced celestial mechanics. All material is presented in a consistent manner, and the reader is guided through the various derivations and proofs in a tutorial way. Cookbook formulas are avoided; instead, the reader is led to understand the principles underlying the equations at issue, and shown how to apply them to various dynamical systems. The book is divided into two parts. Part I covers analytical treatment of topics such as basic dynamic principles up to advanced energy concepts. Special attention is paid to the use of rotating reference frames that often occur in aerospace systems. Part II covers basic celestial mechanics, treating the two-body problem, restricted three-body problem, gravity field modeling, perturbation methods, spacecraft formation flying, and orbit transfers. MATLAB®, Mathematica® and C-Code toolboxes are provided for the rigid body kinematics routines discussed in chapter 3, and the basic orbital 2-body orbital mechanics routines discussed in chapter 9. A solutions manual is also available for professors. MATLAB® is a registered trademark of The MathWorks, Inc.; Mathematica® is a registered trademark of Wolfram Research, Inc."--Publisher's website Particle Kinematics -- Newtonian Mechanics -- Rigid Body Kinematics -- Eulerian Mechanics -- Generalized Methods Of Analytical Dynamics -- Variational Methods In Analytical Dynamics -- Hamilton's Generalized Formulations Of Analytical Dynamics -- Nonlinear Spacecraft Stability And Control -- Classical Two-body Problem -- Restricted Three-body Problem -- Gravitational Potential Field Models -- Perturbation Methods -- Transfer Orbits -- Spacecraft Formation Flying -- Appendix A. Transport Theorem Derivation Using Linear Algebra -- Appendix B. Various Euler Angle Transformations -- Appendix C. Mrp Identity Proof -- Appendix D. Conic Section Transformations -- Appendix E. Numerical Subroutines Library -- Appendix F. First-order Mapping Between Mean And Osculating Orbit Elements -- Appendix G. Direct Linear Mapping Between Cartesian Hill Frame Coordinates And Orbit Element Differences -- Appendix H. Hamel Coefficients For The Rotational Motion Of A Rigid Body. Hanspeter Schaub, John L. Junkins. Includes Bibliographical References And Index.