OPTIMAL SPACECRAF1 ROTATIONAL MANEUVERS

Transcription

1 STUDIES IN ASTRONAUTICS 3 OPTIMAL SPACECRAF1 ROTATIONAL MANEUVERS JOHNL.JUNKINS Texas A&M University, College Station, Texas, U.S.A. and JAMES D.TURNER Cambridge Research, Division of PRA, Inc., Cambridge, Massachusetts, U.S.A. Amsterdam Oxford New York Tokyo 1986

2 ix CONTENTS Page PREFACE v CHAPTER 1 INTRODUCTION 1 2 GEOMETRY AND KINEMATICS OF ROTATIONAL MOTION 5 Special Vector Kinematic Notations 6 Direction Cosines and Orthogonal Projections 9 Rotations About a Fixed Axis 13 Eulerian Angles 16 Euler's Principal Rotation Theorem 26 Euler Parameters 28 Other Orientation Parameters 37 References 39 3 BASIC PRINCIPLES OF DYNAMICS 41 Newtonian/Eulerian Equations of Motion 42 Translational Motion 42 Angular Momentum of Continuous Systems 43 Translational Equations of Motion for Rigid Spacecraft 46 Rotational Equations of Motion for Rigid Spacecraft 49 Generalized Methods in Dynamics 56 Kinetic Energy 56 D'Alembert's Principle 58 Lagrange's Equations for Holonomic Systems 59 Lagrange's Equations for Rigid Bodies and Multiple Body Systems 64

3 X Lagrange's Equations for Non-Holonomic Constraints 71 Hamilton' s Principles 73 References 79 4 ROTATIONAL DYNAMICS OF RIGID AND MULTIPLE RIGID BODY SPACECRAFT 81 Overview 81 Torque-Free Motion of a Single Rigid Body 81 Energy and Momentum Integrals 81 Nonlinear Oscillator Analog of Rigid Body Motion 88 Dynamics of a Rigid Space Structure under the Influence of Grav i ty Torques 100 Dynamics of Multi-Wheel Configurations 113 Dual Spin Configuration 114 Maneuvers of a Dual Spin Configuration 119 Equations of Motion for an n-rotor Spacecraft 127 References DYNAMICS OF FLEXIBLE SPACECRAFT 134 Introduction 134 The Hybrid Coordinate Method 135 Example Application of the Hybrid Coordinate Method 138 Virtual Work and Generalized Forces 140 Kinetic Energy 144 Potential Energy 146 Approximate Discretization Methods for Distributed Parameter Systems 147 The Assumed Modes Method 148

4 xi Assumed Modes Application for a Simple Structure 150 Assumed Mode Application for a Rotating Spacecraft 153 The Finite Element Method 156 Multibody Spacecraft Equations of Motions Requiring Sub-Structure Finite Element Models 158 Fin '. e Element Models: A Simplified Example 163 References ELEMENTS OF OPTIMAL CONTROL THEORY 171 The Variational Problem of Lagrange 171 Statement of the Problem 171 Applications of the Calculus of Variations in Analytical Dynamics 177 Functional Optimization with Differential Equation Constraints 180 Pontryagin's Optimal Control Necessary Conditions 182 A Smooth Control Example: A Single-Axis Rotational Maneuver 184 Free Time and Free Final Angle 189 Discontinuous Bounded Control: Minimum Time Bang-Bang Maneuvers 193 Derivative Penalty Performance Indices 201 Optimal Feedback Control 209 Motivation for Feedback Control 209 The Hamilton-Jacobi-BelIman Equations and Bellman's Principle of Optimality 215 Tuning Optimal Quadratic Regulators Vis-a-Vis Closed Loop Eigenvalue Placement and Sensitivity 218 References 222

5 xi i 7 NUMERICAL SOLUTION OF TWO POINT BOUNDARY VALUE PROBLEMS 224 Introduction 224 Statement of the Boundary-Value Problem 225 Quasi-Linearization 226 Shooting Methods 228 Method of Particular Solutions 228 Method of Differential Corrections 230 Polynomial Approximation Methods 231 Continuation 236 Multipoint Methods 237 Two Examples Solved by Three Methods 238 Concluding Remarks 247 References OPTIMAL MANEUVERS OF RIGID SPACECRAFT 250 Introductory Remarks 250 Minimum-Time Magnetic Attitude Maneuvers 250 Background 250 Kinematics and Dynamics 252 Optimal Control Formulation 256 Example Calculations of NOVA Optimal Maneuvers 260 Concluding Remarks on Minimum Time Magnetic Maneuvers 266 Optimal Momentum Transfer Maneuvers 267 Background 267 Formulation of the Necessary Conditions 268 Three Wheel Momentum Transfer Maneuvers 273 Equations of Motion 273

6 xi i i Optimal Control Formulation 278 De-Tumble With Momentum Transfer 283 Optimal Large Angle Maneuvers of a Single Rigid Body 287 Introductory Remarks 287 Kinematics and Dynamics 287 Optimal Maneuver Necessary Conditions 289 Analytical Solution for a Special Case: Single Axis Maneuvers 291 A Continuation Process for Solution of the Two-Point Boundary-Value Problem 293 An Example Maneuver 301 Concluding Remarks 306 References OPTIMAL LARGE-ANGLE SINGLE-AXIS MANEUVERS OF FLEXIBLE SPACECRAFT 309 Introduction 309 Equations of Motion 312 State Space Formulation 315 Optimal Control Problem 316 Statement of the Problem 316 Derivation of Necessary Conditions from Pontryagin's Principle 317 Solution for the Initial Co-States 318 Free Final Angle Transversal ity Conditions 320 Solution for the Initial Co-States for the Free Final Angle Maneuver Problem 323 Nonlinear Optimal Large-Angle Maneuvers for Flexible Spacecraft 324

7 xiv Equations of Motion and Optimal Control Formulation 324 Continuation Method for the Solution of the Nonlinear TPBVP 326 Nonlinear State Transition Matrix 333 Example Maneuvers 335 References FREQUENCY-SHAPED LARGE-ANGLE MANEUVERS OF FLEXIBLE SPACECRAFT 358 Introduction 358 Equation of Motion 361 State Space Formation 362 Optimal Control Problem Using the Control-Rate Penalty Technique 363 Statement of the Problem 363 Derivation of Necessary Conditions from Pontryagin's Principle 364 Solution for the Initial Co-states 365 Free Final Time Target Tracking Maneuvers 369 Free Final Angle Maneuvers Using the Control-Rate Penalty Method 374 Solution for the Initial Co-States and Control Time Derivatives for the Free Final Angle Maneuver Problem 376 Weighting Matrix Specification in the Control-Rate Penalty Formulation 376 Evaluation of the Response of a Residual Plant Model 377 Example Maneuvers 379 Concluding Remarks 401 References COMPUTATIONAL METHODS FOR CLOSED-LOOP CONTROL PROBLEMS 404 Introduction 404

8 XV The Optimal Control Problem 405 Linear Regulator Problem: Necessary Conditions and Sol ut ions 406 State Trajectory Equation 408 Recursion Relationship for Evaluating the State at Discrete Time Steps 410 Residual State Trajectory Equations 411 State Trajectory Sensitivity Calculations 413 Terminal Control Problem: Necessary Conditions and Solutions 416 Exponential Solutions for the Riccati-Matrix Equations 418 Change of Variables for the Riccati-Like Matrix Equations 420 State Trajectory Equation 423 Recursion Relationship for Evaluating the State and Control at Discrete Time Steps 424 State Trajectory Sensitivity Calculations 426 Disturbance-Accommodating Tracking Problem: Necessary Conditions and Solutions 428 Closed-Form Solution for the Prefilter Equations 431 State Trajectory Equation 433 Recursion Relationship for Evaluating the State and Control at Discrete Time Steps 435 Constant Reference State and Disturbance State Special Cases 441 Residual State Trajectory Equation 441 State Trajectory Sensitivity Calculations 443 A Short Integral Table for Riccati-Like Matrix Differential Equations 452 Conclusions and Recommendations 452

9 xv i References 454 APPENDIX A: AUTONOMOUS SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS 456 Overview 456 Solutions of x = Ax + Bu 456 Homogeneous Solution 456 Forced Solution 460 Solutions of Ay + By = f 461 Homogeneous Solutions 461 Forced Solution 463 Solutions of Mx + Cx + Kx = f 464 Solutions of X = A,X + XA + B, X(t ) = С * o ; Preliminary Remarks 471 Coordinate Transformation for the Differential Equation Variation of Parameters Solution of the Transformed Differential Equation 473 Solution for the Initial Condition Matrix T(t Q ) 475 Summary of Solution 476 Solution of the Matrix Riccati Equation 476 Prel iminary Remarks 476 Steady State Solution (Potter's Method) 477 Change of Variables for the Matrix Riccati Differential Equation 479 Coordinate Transformation 480 Variation of Parameters Solution of the Transformed Differential Equation 481 Solution for the Initial Condition Matrix 482 Solution for the Transformed Matrix Elements 483

10 xvi i Evaluation of the Matrix Riccati Equation at Discrete Time Steps 484 Summary of Solution 48Ф Calculation of the Optimal Control 485 References 485 APPENDIX B: APPENDIX C: CLOSED-FORM SOLUTION FOR AN INTEGRAL CONTAINING MATRIX EXPONENTIALS 487 CLOSED-FORM SOLUTIONS FOR CONVOLUTION MATRIX INTEGRALS AND SENSITIVITY CALCULATIONS 490 Convolution Matrix Integral Calculations 490 Convolution Matrix Integral Sensitivity Calculations 492 Ait Sensitivity Calculation for e 492 Sensitivity Partial Derivatives for Gj(t) 495 Sensitivity Calculation for I^(t) 499 APPENDIX D: ANALYTICAL SOLUTION OF THE TWO BODY PROBLEM (KEPLERIAN MOTION) 501 Reference 502 APPENDIX E: AN ANALYTIC FOURIER TRANSFORM FOR A CLASS OF FINITE-TIME CONTROL PROBLEMS 503 Prob 1 em Formu 1 at i on 503 Reducing Subspace Coordinate Transformation 504 Solution for the Uncoupling Transformation Vector 505 Example Application 506 Conclusions 508 References 508 INDEX 510