Analytical mechanics of aerospace systems

588 Pages·2002·6.26 MB·English

by  Schaub H.

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