This is a long-overdue volume dedicated to space trajectory optimization. Interest in the subject has grown, as space missions of increasing levels of sophistication, complexity, and scientific return - hardly imaginable in the 1960s - have been designed and flown. Although the basic tools of optimization theory remain an accepted canon, there has been a revolution in the manner in which they are applied and in the development of numerical optimization. This volume purposely includes a variety of both analytical and numerical approaches to trajectory optimization. The choice of authors has been guided by the editor's intention to assemble the most expert and active researchers in the various specialties presented. The authors were given considerable freedom to choose their subjects, and although this may yield a somewhat eclectic volume, it also yields chapters written with palpable enthusiasm and relevant to contemporary problems.
Author(s): Bruce Conway
Edition: 1
Year: 2010
Language: English
Pages: 310
Half-title......Page 3
Series title......Page 5
Title......Page 7
Copyright......Page 8
Contents......Page 9
Preface......Page 13
1.1 Introduction......Page 15
1.2 Solution Methods......Page 17
1.2.1 Analytical Solutions......Page 18
1.2.2 Numerical Solutions via Discretization......Page 21
1.2.3 Evolutionary Algorithms......Page 23
1.3 The Situation Today with Regard to Solving Optimal Control Problems......Page 26
2.1 Introduction......Page 30
2.2.1 Optimal Constant-Specific-Impulse Trajectory......Page 31
2.2.2 Optimal Impulsive Trajectory......Page 35
2.2.3 Optimal Variable-Specific-Impulse Trajectory......Page 36
2.3 Solution to the Primer Vector Equation......Page 37
2.4 Application of Primer Vector Theory to an Optimal Impulsive Trajectory......Page 38
2.4.1 Criterion for a Terminal Coast......Page 40
2.4.2 Criterion for Addition of a Midcourse Impulse......Page 44
2.4.3 Iteration on a Midcourse Impulse Position and Time......Page 47
3.1 Introduction......Page 51
3.2.1 A Basic Collocation Method (Using Hermite Polynomials)......Page 54
3.2.2 Pseudospectral Methods......Page 59
3.2.3 A Direct Method Not Using Collocation: R-K Parallel-Shooting......Page 61
3.2.4 Comparison of Direct Transcription Methods......Page 63
3.3.1 Motivation for Choice of Coordinate System......Page 66
3.3.2 Coordinate Transformations......Page 70
3.3.3 Interplanetary Trajectories......Page 73
3.4.1 Impulsive Thrust Case......Page 74
3.4.3 Power-Limited Case......Page 75
3.5 Generating an Initial Guess......Page 76
3.5.1 Using a Known Optimal Control Strategy......Page 77
3.5.3 Using Evolutionary Methods to Generate an Initial Guess......Page 78
3.6.1 Equation Scaling......Page 79
3.6.2 Grid Refinement......Page 81
3.6.3 Other Grid Refinement Strategies......Page 84
3.7.1 Optimality of Assumed Control Switching Structures and the DiscreteSwitch Function......Page 85
3.7.2 Example: Two and Three Thrust-Arc Rendezvous......Page 87
3.7.3 Verifying Optimality by Integration of the Euler-Lagrange Equations......Page 88
4.1 Introduction......Page 93
4.2 Trajectory Model......Page 94
4.4 Finite Burn Control Models......Page 99
4.4.1 Thrust Vector Parameter Model......Page 100
4.4.2 Thrust Vector Optimal Control Model......Page 101
4.5 Solution Methods......Page 104
4.5.3 Nonlinear Programming Problem: mineq0,0meq4.6.1 Free-Return Lunar Flyby Mission......Page 107
4.6.2 Lunar Orbiter and Lunar Impacter Mission......Page 110
4.7 Concluding Remarks......Page 124
5.1 Introduction and Background......Page 126
5.2.1 Problem Statement......Page 127
5.2.2 Thrust-Steering Control Laws......Page 130
5.2.3 Analysis of the Control Laws......Page 133
5.2.4 Solution Method......Page 138
5.3.1 Minimum-Time LEO-GEO Transfer......Page 139
5.3.2 Minimum-Time GTO-GEO Transfer......Page 144
5.3.3 Minimum-Propellant LEO-GEO Transfer......Page 146
5.3.4 Minimum-Propellant LEO-GEO Transfer with Variable Isp......Page 148
5.4 Conclusions......Page 150
6.1 Introduction......Page 153
6.2.1 The Linearized Reduced Equations of Motion......Page 155
6.2.3 The Hamiltonian and Euler-Lagrange Equations......Page 156
6.2.4 The Analytic Form of the State and Control Variables......Page 157
6.3.1 The Augmented Hamiltonian......Page 159
6.3.3 The Jump Conditions at the Constraint Entry Point......Page 160
6.3.4 The Non-Existence of a Corner at the Constraint Entry and Exit Points......Page 163
6.3.5 Evaluation of the Constraint Arc Exit and Final Transfer Times......Page 165
6.4 The Split-Sequence Transfers......Page 171
6.4.1 Analytic (V, i) Transfer......Page 172
6.4.2 Analytic (V, bold0mu mumu Raw) transfer......Page 173
6.4.3 Analytic Split (V, i), (V,bold0mu mumu Raw) Sequence Transfer......Page 175
6.4.4 Analytic split (V, bold0mu mumu Raw ), (V, i) sequence transfer......Page 176
6.4.5 Comparison with Precision Integration......Page 178
6.4.6 Maximization of the Equatorial Inclination in Circular OrbitUsing Analytic Averaging under J2 Influence and Minimum-TimeTransfer for Fixed Boundary Conditions......Page 183
7.1 Introduction......Page 192
7.3 Problem Transcription......Page 193
7.4 The MGA Problem......Page 195
7.4.1 Spacecraft Position and Velocity......Page 196
7.5 The MGA-1DSM Problem......Page 197
7.5.1 Spacecraft Position and Velocity......Page 198
7.5.3 The Final Form......Page 199
7.6 Benchmark Problems......Page 200
7.6.2 GTOC1......Page 201
7.6.3 Rosetta......Page 202
7.7 Global Optimization......Page 204
7.7.1 Discussion......Page 207
7.8.2 Pruning the MGA-1DSM Problem: Cluster Pruning......Page 208
7.9 Concluding Remarks......Page 211
8.1 Introduction......Page 216
8.2.1 The Linked Conic Approximation......Page 217
8.2.2 Velocity Formulation......Page 219
8.2.3 Position Formulation......Page 221
8.3 The Incremental Approach......Page 223
8.3.1 Solution of the Subproblems......Page 225
8.3.3 Exploration of the Pruned Space......Page 227
8.3.4 Discussion......Page 229
8.4 Testing Procedure and Performance Indicators......Page 230
8.5 Case Studies......Page 235
8.5.1 EEM Test Case......Page 236
8.5.2 EVVMeMe Transfer......Page 241
8.6 Conclusions......Page 248
9.1 Introduction......Page 252
9.2 System Dynamics......Page 254
9.2.1 Orbit Elements......Page 256
9.2.2 Equinoctial Elements......Page 257
9.2.4 Periodic Orbits......Page 259
9.2.5 Stable and Unstable Manifolds......Page 260
9.3 Basics of Trajectory Optimization......Page 261
9.4 Generation of Periodic Orbit Constructed as an Optimization Problem......Page 264
9.4.1 Results......Page 266
9.5 Optimal Earth Orbit to Lunar Orbit Transfer: Part 1–GTO to Periodic Orbit......Page 267
9.6 Optimal Earth Orbit to Lunar Orbit Transfer: Part 2 – Periodic Orbit to Low-Lunar Orbit......Page 270
9.6.1 Halo Orbit-Moon Trajectory......Page 271
9.6.2 Combined Earth-Moon Trajectory......Page 272
9.7 Extension of the Work to Interplanetary Flight......Page 273
9.8 Conclusions......Page 274
10.1 Introduction......Page 277
10.2.1 Unconstrained Optimization......Page 280
10.2.2 Constrained Optimization......Page 282
10.3.1 Problem Definition......Page 283
10.3.2 Numerical Results......Page 286
10.4.1 Problem Definition......Page 288
10.4.2 Numerical Results......Page 289
10.5 Optimal Four-Impulse Orbital Rendezvous......Page 291
10.5.1 Problem Definition......Page 292
10.5.2 Numerical Results......Page 296
10.6.1 Problem Definition......Page 298
10.6.2 Constraint Reduction......Page 300
10.6.3 Numerical Solution......Page 301
10.7 Concluding Remarks......Page 304
Index......Page 309
