A M NALYTICAL ECHANICS of A S EROSPACE YSTEMS Hanspeter Schaub and John L. Junkins January 1, 2002 Contents Preface ix I BASIC MECHANICS 1 1 Particle Kinematics 3 1.1 ParticlePositionDescription . . . . . . . . . . . . . . . . . . . . 3 1.1.1 Basic Geometry. . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.2 Cylindrical and Spherical Coordinate Systems . . . . . . . 6 1.2 Vector Di(cid:11)erentiation . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.1 Angular Velocity Vector . . . . . . . . . . . . . . . . . . . 8 1.2.2 Rotation about a Fixed Axis . . . . . . . . . . . . . . . . 10 1.2.3 TransportTheorem . . . . . . . . . . . . . . . . . . . . . 11 1.2.4 ParticleKinematics with MovingFrames . . . . . . . . . 15 2 Newtonian Mechanics 25 2.1 Newton’s Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 Single Particle Dynamics. . . . . . . . . . . . . . . . . . . . . . . 29 2.2.1 Constant Force . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2.2 Time-VaryingForce . . . . . . . . . . . . . . . . . . . . . 32 2.2.3 Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . 34 2.2.4 Linear Momentum . . . . . . . . . . . . . . . . . . . . . . 35 2.2.5 Angular Momentum . . . . . . . . . . . . . . . . . . . . . 35 2.3 Dynamics of a System of Particles . . . . . . . . . . . . . . . . . 38 2.3.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . 38 2.3.2 Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . 41 2.3.3 Linear Momentum . . . . . . . . . . . . . . . . . . . . . . 43 2.3.4 Angular Momentum . . . . . . . . . . . . . . . . . . . . . 45 2.4 Dynamics of a ContinuousSystem . . . . . . . . . . . . . . . . . 47 2.4.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . 47 2.4.2 Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . 49 2.4.3 Linear Momentum . . . . . . . . . . . . . . . . . . . . . . 50 2.4.4 Angular Momentum . . . . . . . . . . . . . . . . . . . . . 51 2.5 The RocketProblem . . . . . . . . . . . . . . . . . . . . . . . . . 52 iii iv CONTENTS 3 Rigid Body Kinematics 63 3.1 Direction Cosine Matrix . . . . . . . . . . . . . . . . . . . . . . . 64 3.2 Euler Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.3 Principal Rotation Vector . . . . . . . . . . . . . . . . . . . . . . 78 3.4 Euler Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.5 Classical Rodrigues Parameters . . . . . . . . . . . . . . . . . . . 91 3.6 Modi(cid:12)ed Rodrigues Parameters . . . . . . . . . . . . . . . . . . . 96 3.7 Other Attitude Parameters . . . . . . . . . . . . . . . . . . . . . 103 3.7.1 Stereographic OrientationParameters . . . . . . . . . . . 103 3.7.2 Higher Order Rodrigues Parameters . . . . . . . . . . . . 105 3.7.3 The (w,z) Coordinates . . . . . . . . . . . . . . . . . . . . 106 3.7.4 Cayley-KleinParameters . . . . . . . . . . . . . . . . . . 107 3.8 Homogeneous Transformations . . . . . . . . . . . . . . . . . . . 107 4 Eulerian Mechanics 115 4.1 Rigid Body Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 115 4.1.1 Angular Momentum . . . . . . . . . . . . . . . . . . . . . 115 4.1.2 Inertia Matrix Properties . . . . . . . . . . . . . . . . . . 118 4.1.3 Euler’s Rotational Equations of Motion . . . . . . . . . . 123 4.1.4 Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . 124 4.2 Torque-FreeRigid Body Rotation . . . . . . . . . . . . . . . . . . 128 4.2.1 Energy and Momentum Integrals . . . . . . . . . . . . . . 128 4.2.2 General Free Rigid Body Motion . . . . . . . . . . . . . . 133 4.2.3 Axisymmetric Rigid Body Motion . . . . . . . . . . . . . 135 4.3 Momentum ExchangeDevices . . . . . . . . . . . . . . . . . . . . 137 4.3.1 Spacecraft with Single VSCMG . . . . . . . . . . . . . . . 138 4.3.2 Spacecraft with Multiple VSCMGs . . . . . . . . . . . . . 143 4.4 GravityGradient Satellite . . . . . . . . . . . . . . . . . . . . . . 145 4.4.1 GravityGradient Torque . . . . . . . . . . . . . . . . . . 145 4.4.2 Rotational - TranslationalMotion Coupling . . . . . . . . 148 4.4.3 Small Departure Motion about Equilibrium Attitudes . . 149 5 Generalized Methods of Analytical Dynamics 159 5.1 Generalized Coordinates . . . . . . . . . . . . . . . . . . . . . . . 159 5.2 D’Alembert’s Principle . . . . . . . . . . . . . . . . . . . . . . . . 162 5.2.1 Virtual Displacements and Virtual Work. . . . . . . . . . 163 5.2.2 Classical Developmentsof D’Alembert’s Principle . . . . . 164 5.2.3 Holonomic Constraints . . . . . . . . . . . . . . . . . . . . 170 5.2.4 Newtonian Constrained Dynamics of N Particles . . . . . 177 5.2.5 LagrangeMultiplier Rule for Constrained Optimization . 178 5.3 LagrangianDynamics . . . . . . . . . . . . . . . . . . . . . . . . 182 5.3.1 Minimal Coordinate Systems and Unconstrained Motion . 183 5.3.2 Lagrange’sEquations for ConservativeForces . . . . . . . 187 5.3.3 Redundant Coordinate Systems and Constrained Motion 190 5.3.4 Vector-MatrixFormofthe LagrangianEquationsofMotion195 CONTENTS v 6 Advanced Methods of Analytical Dynamics 203 6.1 The Hamiltonian Function . . . . . . . . . . . . . . . . . . . . . . 203 6.1.1 Some Special Properties of The Hamiltonian . . . . . . . 203 6.1.2 RelationshipoftheHamiltoniantoTotalEnergyandWork Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 6.1.3 Hamilton’s Canonical Equations . . . . . . . . . . . . . . 203 6.1.4 Hamilton’s Principal Function and the Hamilton-Jacobi Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 6.2 Hamilton’s Principles. . . . . . . . . . . . . . . . . . . . . . . . . 203 6.2.1 VariationalCalculus Fundamentals . . . . . . . . . . . . . 204 6.2.2 Path Variations versusVirtual Displacements . . . . . . 204 6.2.3 Hamilton’s Principles from D’Alembert’s Principle . . . . 204 6.3 Dynamics of Distributed ParameterSystems. . . . . . . . . . . . 204 6.3.1 ElementaryDPS: Newton-Euler Methods . . . . . . . . . 204 6.3.2 Energy Functions for Elastic Rods and Beams. . . . . . . 204 6.3.3 Hamilton’s Principle Applied for DPS . . . . . . . . . . . 204 6.3.4 Generalized Lagrange’sEquations for Multi-Body DPS . 204 7 Nonlinear Spacecraft Stability and Control 205 7.1 Nonlinear Stability Analysis . . . . . . . . . . . . . . . . . . . . . 206 7.1.1 Stability De(cid:12)nitions . . . . . . . . . . . . . . . . . . . . . 206 7.1.2 Linearization of Dynamical Systems . . . . . . . . . . . . 210 7.1.3 Lyapunov’sDirect Method . . . . . . . . . . . . . . . . . 212 7.2 Generating LyapunovFunctions. . . . . . . . . . . . . . . . . . . 219 7.2.1 Elemental Velocity-BasedLyapunovFunctions . . . . . . 221 7.2.2 Elemental Position-BasedLyapunovFunctions . . . . . . 227 7.3 Nonlinear FeedbackControlLaws . . . . . . . . . . . . . . . . . . 233 7.3.1 Unconstrained ControlLaw . . . . . . . . . . . . . . . . . 233 7.3.2 Asymptotic Stability Analysis . . . . . . . . . . . . . . . . 236 7.3.3 FeedbackGain Selection . . . . . . . . . . . . . . . . . . . 242 7.4 LyapunovOptimal Control Laws . . . . . . . . . . . . . . . . . . 247 7.5 Linear Closed-Loop Dynamics . . . . . . . . . . . . . . . . . . . . 253 7.6 Reaction Wheel ControlDevices . . . . . . . . . . . . . . . . . . 258 7.7 Variable Speed ControlMoment Gyroscopes . . . . . . . . . . . . 260 7.7.1 ControlLaw . . . . . . . . . . . . . . . . . . . . . . . . . 261 7.7.2 VelocityBased Steering Law . . . . . . . . . . . . . . . . 264 7.7.3 VSCMG Null Motion . . . . . . . . . . . . . . . . . . . . 269 II CELESTIAL MECHANICS 283 8 Classical Two-Body Problem 285 8.1 Geometry of Conic Sections . . . . . . . . . . . . . . . . . . . . . 286 8.2 Relative Two-BodyEquations of Motion . . . . . . . . . . . . . . 294 8.3 Fundamental Integrals . . . . . . . . . . . . . . . . . . . . . . . . 296 8.3.1 Conservationof Angular Momentum . . . . . . . . . . . . 296 vi CONTENTS 8.3.2 The EccentricityVector Integral . . . . . . . . . . . . . . 297 8.3.3 Conservationof Energy . . . . . . . . . . . . . . . . . . . 300 8.4 Classical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 306 8.4.1 Kepler’s Equation . . . . . . . . . . . . . . . . . . . . . . 307 8.4.2 Orbit Elements . . . . . . . . . . . . . . . . . . . . . . . . 310 8.4.3 Lagrange/GibbsF and G Solution . . . . . . . . . . . . . 316 9 Restricted Three-Body Problem 325 9.1 Lagrange’sThree-Body Solution . . . . . . . . . . . . . . . . . . 326 9.1.1 General Conic Solutions . . . . . . . . . . . . . . . . . . . 326 9.1.2 Circular Orbits . . . . . . . . . . . . . . . . . . . . . . . . 335 9.2 Circular Restricted Three-Body Problem. . . . . . . . . . . . . . 339 9.2.1 Jacobi Integral . . . . . . . . . . . . . . . . . . . . . . . . 341 9.2.2 Zero Relative Velocity Surfaces . . . . . . . . . . . . . . . 346 9.2.3 LagrangeLibration PointStability . . . . . . . . . . . . . 353 9.3 Periodic Stationary Orbits . . . . . . . . . . . . . . . . . . . . . . 357 9.4 The Disturbing Function . . . . . . . . . . . . . . . . . . . . . . . 358 10 Gravitational Potential Field Models 365 10.1 GravitationalPotentialof Finite Bodies . . . . . . . . . . . . . . 366 10.2 MacCullagh’s Approximation . . . . . . . . . . . . . . . . . . . . 369 10.3 Spherical Harmonic GravityPotential . . . . . . . . . . . . . . . 372 10.4 Multi-Body GravitationalAcceleration . . . . . . . . . . . . . . . 381 10.5 Spheres of GravitationalIn(cid:13)uence . . . . . . . . . . . . . . . . . 383 11 Perturbation Methods 389 11.1 Encke’sMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 11.2 Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . . 392 11.2.1 General Methodology . . . . . . . . . . . . . . . . . . . . 393 11.2.2 LagrangianBrackets . . . . . . . . . . . . . . . . . . . . . 395 11.2.3 Lagrange’sPlanetary Equations . . . . . . . . . . . . . . 401 11.2.4 PoissonBrackets . . . . . . . . . . . . . . . . . . . . . . . 408 11.2.5 Gauss’ VariationalEquations . . . . . . . . . . . . . . . . 415 11.3 State Transition and Sensitivity Matrix . . . . . . . . . . . . . . 417 11.3.1 Linear Dynamic Systems . . . . . . . . . . . . . . . . . . 418 11.3.2 Nonlinear Dynamic Systems . . . . . . . . . . . . . . . . . 422 11.3.3 Symplectic State TransitionMatrix. . . . . . . . . . . . . 425 11.3.4 State TransitionMatrix of Keplerian Motion . . . . . . . 427 12 Transfer Orbits 433 12.1 Minimum Energy Orbit . . . . . . . . . . . . . . . . . . . . . . . 434 12.2 The Hohmann TransferOrbit . . . . . . . . . . . . . . . . . . . . 437 12.3 Lambert’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 442 12.3.1 General Problem Solution . . . . . . . . . . . . . . . . . . 443 12.3.2 ElegantVelocity Properties . . . . . . . . . . . . . . . . . 447 12.4 Rotating the Orbit Plane . . . . . . . . . . . . . . . . . . . . . . 450 CONTENTS vii 12.5 Patched-ConicOrbit Solution . . . . . . . . . . . . . . . . . . . . 455 12.5.1 Establishing the Heliocentric Departure Velocity . . . . . 457 12.5.2 Escaping the Departure Planet’s Sphere of In(cid:13)uence . . . 461 12.5.3 Enterthe Target Planet’s Sphere of In(cid:13)uence . . . . . . . 467 12.5.4 Planetary Fly-By’s . . . . . . . . . . . . . . . . . . . . . . 472 13 Spacecraft Formation Flying 477 13.1 General Relative Orbit Description . . . . . . . . . . . . . . . . . 479 13.2 Cartesian Coordinate Description . . . . . . . . . . . . . . . . . . 480 13.2.1 Clohessy-Wiltshire Equations . . . . . . . . . . . . . . . . 481 13.2.2 Closed Relative Orbits in the Hill Reference Frame . . . . 484 13.3 Orbit Element Di(cid:11)erence Description . . . . . . . . . . . . . . . . 487 13.3.1 LinearMappingBetweenHillFrameCoordinatesandOr- bit Element Di(cid:11)erences . . . . . . . . . . . . . . . . . . . 489 13.3.2 Bounded Relative Motion Constraint . . . . . . . . . . . . 495 13.4 Relative Motion State Transition Matrix . . . . . . . . . . . . . . 497 13.5 Linearized Relative Orbit Motion . . . . . . . . . . . . . . . . . . 502 13.5.1 General Elliptic Orbits . . . . . . . . . . . . . . . . . . . . 502 13.5.2 Chief Orbits with Small Eccentricity . . . . . . . . . . . . 506 13.5.3 Near-CircularChief Orbit . . . . . . . . . . . . . . . . . . 508 13.6 J2-InvariantRelative Orbits . . . . . . . . . . . . . . . . . . . . . 511 13.6.1 Ideal Constraints . . . . . . . . . . . . . . . . . . . . . . . 512 13.6.2 Energy Levels between J2-InvariantRelative Orbits . . . 519 13.6.3 ConstraintRelaxation Near PolarOrbits. . . . . . . . . . 520 13.6.4 Near-CircularChief Orbit . . . . . . . . . . . . . . . . . . 524 13.6.5 Relative Argument of Perigeeand Mean Anomaly Drift . 526 13.6.6 Fuel Consumption Prediction . . . . . . . . . . . . . . . . 528 13.7 Relative Orbit Control Methods. . . . . . . . . . . . . . . . . . . 531 13.7.1 Mean Orbit Element ContinuousFeedbackControlLaws 532 13.7.2 Cartesian Coordinate ContinuousFeedbackControlLaw . 539 13.7.3 Impulsive FeedbackControlLaw . . . . . . . . . . . . . . 542 13.7.4 Hybrid FeedbackControl Law. . . . . . . . . . . . . . . . 546 APPENDIX A 553 APPENDIX B 557 APPENDIX C 559 APPENDIX D 563 APPENDIX E 565 APPENDIX F 569 APPENDIX G 573 viii CONTENTS Preface ix