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Lehigh Preserve Institutional Repository Lehigh Preserve Institutional Repository Particle Swarm Optimization of Low-Thrust, Geocentric-to-Halo-Orbit Transfers Abraham, Andrew 2014 Find more at https://preserve.lib.lehigh.edu/ This document is brought to you for free and open access by Lehigh Preserve. It has been accepted for inclusion by an authorized administrator of Lehigh Preserve. For more information, please contact [email protected]. Particle Swarm Optimization of Low-Thrust, Geocentric-to-Halo-Orbit Transfers by Andrew J. Abraham Presented to the Graduate and Research Committee of Lehigh University in Candidacy for the Degree of Doctor of Philosophy in Mechanical Engineering Lehigh University May, 2014 . Copyright by Andrew Abraham May, 2014 ii Approved and recommended for acceptance as a dissertation in partial fulfillment of the requirements for the degree of Doctor of Philosophy. ________________ Date _______________________ Professor Terry J. Hart Dissertation Adviser Mechanical Engineering Lehigh University ________________ Accepted Date _______________________ Dr. David B. Spencer Technical Adviser Aerospace Engineering The Pennsylvania State University _______________________ Dr. Nader Motee Committee Chairperson Mechanical Engineering Lehigh University _______________________ Dr. Philip A. Blythe Committee Member Mechanical Engineering Lehigh University _______________________ Dr. Gary G. DeLeo Committee Member Physics Lehigh University iii Acknowledgments Over the course of my graduate studies I have encountered many burdensome obstacles that I could not have overcome without significant help from family, friends, and advisers. I would like to take a moment to personally thank those who have helped make this dissertation a reality. First I would like to thank my parents for their support and encouragement during my graduate studies (especially low-cost housing). I would also like to thank my parents-in-law for their generous support as well. Finally, I would like to thank my wife for all her hard work and support as well as the endless nights of proof-reading my papers, articles, and dissertation. I would also like to recognize significant help that I received from professionals in the Astrodynamics community. Dr. Marty Ozimek and Dr. Chris Scott, from the Johns Hopkins Applied Physics Lab, spent hours via email helping me learn many of the concepts discussed in the first few chapters of this dissertation. To them I owe a large debt of gratitude for the time they have invested in me. I also would like to thank Dr. David B. Spencer, from Penn State University, for acting as a second academic adviser during my dissertation research. To him I owe a great deal of gratitude for lending me his technical expertise, guidance on papers and journal articles, and introducing me to many other experts within the Astrodynamics community. And finally, I would like to thank my adviser, Professor Hart, for all his time, support, and effort during my graduate studies. iv Contents Table of Contents v List of Tables ix List of Figures xii Abstract 1 1 Introduction 3 1.1 History of Lagrange Point Orbits . .3 1.2 History & Background of Low-Thrust Propulsion . .5 1.3 Previous Work . .8 1.4 Present Work . .9 2 Dynamics 12 2.1 Coordinate System and Definitions . 13 2.2 System Dynamics . 15 2.3 Jacobi Energy Integral . 21 2.4 Equilibrium Points . 23 2.5 Jacobi Energy at Equilibrium Points . 26 2.6 Linearization, Stability, and Bifurcation . 28 v Contents 2.7 State Transition Matrix . 34 3 Gradient-Based Optimization Techniques 37 3.1 Generalized Free Variable & Constraint Vectors . 38 3.2 Variable-Time, Single Shooting . 41 3.3 Variable-Time, Multiple Shooting . 43 3.4 Single Parameter Continuation . 46 3.5 Pseudo-Arclength Continuation . 47 3.6 Full Ephemeris . 48 3.7 SPICE/Mice . 49 4 Gradient-Based Optimization of Lagrange Point Orbits 51 4.1 Variable Time, Single Shooting Program . 52 4.2 Pseudo-Arclength Continuation & Single Shooting . 57 4.3 Variable Time, Multiple Shooting Program . 59 4.4 Example of Three-Dimensional Orbits . 64 4.5 Full Ephemeris EOM Program with SPICE . 66 5 Dynamical Systems Theory and Invariant Manifolds 73 5.1 Linearized Motion About A Periodic Orbit . 74 5.2 Floquet’s Theorem & the Non-Autonomous Mapping . 75 5.3 Monodromy Matrix & Stability . 77 5.4 Characteristics of Monodromy Matrix Eigenvalues . 81 5.4.1 Λ = 1 Proof . 81 5.4.2 Reciprocal Pairs of Λj in the CR3BP . 82 5.4.3 The Stability Index in the CR3BP . 84 5.5 Stable Manifold Generation . 85 vi Contents 5.6 Example: L1 Halo Orbit and its Associated Manifold . 87 6 Particle Swarm Optimization 94 6.1 History and Background of PSO . 94 6.2 Search Space & Low-Thrust Control Law . 97 6.3 Parametrization of the Search Space . 99 6.4 Fitness Function . 101 6.5 Particle Swarm Optimization . 103 6.6 Local PSO . 104 6.7 Search Space Boundary Conditions: . 106 7 Application of One-Dimensional PSO 107 7.1 Study A: Eccentricity-Only Fitness Function . 108 7.1.1 Fitness Function . 108 7.1.2 Data . 109 7.1.3 Optimization of an Arbitrary kfixed ............ 116 7.2 Study B: Eccentricity and Fuel Optimizing Fitness Function . 119 7.2.1 Fitness Function . 119 7.2.2 Data . 119 7.3 Study C: Eccentricity, Propellant, and Time of Flight Optimizing Fitness Function . 123 7.3.1 Fitness Function . 123 7.3.2 Data . 124 8 Application of Two-Dimensional PSO 128 8.1 Shape of the Fitness Function . 129 8.2 Sizing N p .............................. 131 vii Contents 8.3 Sizing jmax via the Convergence Metric, γ ............ 132 8.4 Effect of the Product Np jmax on the Optimal Solution . 136 8.5 Effect of rlocal on the Optimal Solution . 137 8.6 Optimal Low-Thrust Trajectory . 139 9 Two-Maneuver, Impulsive Transfers via a Hybrid PSO/Shooting Method 144 9.1 Single Shooting a Two-Maneuver, Impulsive Transfer . 145 9.2 PSO Algorithm . 149 9.2.1 Fitness Function . 149 9.2.2 Fast Transfer . 150 9.2.3 Slow Transfer . 151 9.3 Application of the Fast Transfer Case . 152 9.4 Application of the Slow Transfer Case . 155 10 Conclusions & Future Work 158 10.1 Conclusions for One-Dimensional PSO Applications . 159 10.2 Conclusions for Two-Dimensional PSO Applications . 160 10.3 Conclusions for Two-Impulsive Burn PSO Applications . 161 10.4 Conclusions From This Work . 163 10.5 Future Work . 165 References 167 Appendix 178 Vita 185 viii List of Tables 2.1 Summary of Jacobi energies of the Lagrange points in the Earth- Moon system. 26 2.2 Top: Coefficients of A matrix for collinear Lagrange points in the Earth-Moon system. Bottom: Eigenvalues of A matrix for colinear Lagrange points in the Earth-Moon system. 32 4.1 Ten known patch points. 59 4.2 MATLAB custom propagator error. 68 5.1 Summary of eigenmode stability characteristics where Λj is an eigenvalue of the monodromy matrix, M............. 81 7.1 Top: Results from eight Monte Carlo trials. Both the negative perturbation (-) and positive perturbation (+) are investigated. Bottom: Results from eight PSO trials. Both perturbations are also investigated. The optimal case is shown in bold. 110 ix List of Tables 7.2 Comparison of PSO vs. Monte Carlo (MC) methods. Both the negative perturbation (-) and positive perturbation (+) are in- vestigated. The PSO and MC columns indicates the number of solutions found with an eccentricity less than 0.01 using the Particle Swarm Optimization and Monte Carlo methods, respec- tively. 111 7.3 Optimal patch points found for nine trajectories (228+-233+) us- ing PSO. 116 7.4 Top: Results from eight Monte Carlo trials. Both the negative perturbation (-) and positive perturbation (+) are investigated. Bottom: Results from eight PSO trials. Both perturbations are also investigated. The optimal case is shown in bold. A 3GHz Core 2 Duo processor was used for the computations. 120 7.5 Comparison of PSO vs. Monte Carlo (MC) methods. Both the negative perturbation (-) and positive perturbation (+) are in- vestigated. The PSO and MC column indicates the number of solutions found with a fitness function less than J = 0.1 us- ing the Particle Swarm Optimization and Monte Carlo methods, respectively. 121 7.6 Top: Results from eight Monte Carlo trials. Both the negative perturbation (-) and positive perturbation (+) are investigated. Bottom: Results from eight PSO trials. Both perturbations are also investigated. The optimal case is shown in bold and has been independently identified twice by two separate trials with differing amounts of particles. This demonstrates the robustness of the PSO method. 125 x List of Tables 7.7 Comparison of PSO vs. Monte Carlo (MC) methods. Both the negative perturbation (-) and positive perturbation (+) are in- vestigated. The PSO and MC column indicates the number of solutions found with a fitness function less than J = 0.1 us- ing the Particle Swarm Optimization and Monte Carlo methods, respectively. 126 8.1 Cost, convergence, and CPU time for jmax = 15 iterations and various number of particles (Np). Each entry consists of the average of 10 identical trials with error bounds of ±1σ standard deviation. 131 8.2 Summary of the maximum, minimum, and median values of the fitness function. 140 9.1 Summary of fast transfer to (a) any inclination and (b) a 28o inclination. 153 9.2 Summary of slow transfer. 156 xi List of Figures 2.1 CR3BP coordinate system. 13 2.2 Five Lagrange points for the Earth-Moon system. 23 2.3 Forbidden regions with Jacobi energies (from upper left to lower right) corresponding to L1, L2, L3, and L4,5............ 27 2.4 Top (L1), middle (L2), and bottom (L3) plots of the real part of the eigenvalues of A vs. µ. All six eigenvalues are plotted for each case (four are on the x−axis).
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