design space pruning heuristics and global optimization method for ...

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TABLE OF CONTENTS ACKNOWLEDGEMENTS............................................................................................... iv LIST OF TABLES............................................................................................................. ix LIST OF FIGURES .............................................................................................................x LIST OF SYMBOLS AND ABBREVIATIONS ..............................................................xv SUMMARY................................................................................................................... xviii I INTRODUCTION ...................................................................................................1 1.1 Low-Thrust Trajectory Optimization Methods and Tools...........................4 1.1.1 Improvements to Indirect Methods..................................................8 1.1.2 Improvements to Direct Methods ....................................................9 1.1.3 Shape-Based Analytic Methods.....................................................13 1.1.4 Low-Thrust Trajectory Optimization Tools...................................15 1.2 Global Optimization Methods and Applications .......................................18 1.2.1 Evolutionary Algorithms ...............................................................18 1.2.2 Evolutionary Neurocontrollers.......................................................24 1.2.3 Combinatorial Optimization ..........................................................27 1.3 Research Objectives and Contributions.....................................................33 II DEVELOPMENT OF METHODOLOGY............................................................38 2.1 Approach....................................................................................................38 2.2 Candidate Pruning Methods.......................................................................39 2.2.1 Ephemeris-Based Pruning Techniques ..........................................39 2.2.2 Phase-Free, Ballistic Approximations ...........................................42 2.2.3 Pruning Techniques Based on Phasing..........................................43 2.3 Candidate Global Optimization Methods ..................................................44 2.3.1 Genetic Algorithm .........................................................................46 2.3.2 Branch-and-Bound.........................................................................48 2.4 Low-Thrust Trajectory Optimization.........................................................50 vi
2.5 Small Sample Problem...............................................................................54 2.5.1 Evaluation of Pruning Techniques on the Sample Problem ..........58 2.5.2 Evaluation of Global Optimization Methods on the Sample Problem..........................................................................................72 III OVERVIEW OF METHODOLOGY....................................................................86 3.1 Overview of Methodology.........................................................................86 3.2 Assumptions and Scope .............................................................................91 IV VALIDATION OF METHODOLOGY ................................................................94 4.1 Intermediate Sample Problem....................................................................94 4.2 Application of Methodology to Intermediate Problem..............................96 4.2.1 Pruning Phase.................................................................................96 4.2.2 Global Optimization Phase ............................................................99 V APPLICATION OF METHODOLOGY TO LARGER PROBLEMS................105 5.1 Modified GTOC3 Problem ......................................................................106 5.2 GTOC2 Problem ......................................................................................117 5.2.1 Modified GTOC2 Problem ..........................................................119 5.2.2 Full GTOC2 Problem...................................................................127 5.3 Sensitivity of Methodology to User-Defined Parameters........................134 5.3.1 Pruning Phase Sensitivity to Selection of Leg Pruning Percentages ..................................................................................135 5.3.2 Global Optimization Phase Sensitivity to Selection of Initial Lower Bound.....................................................................138 VI CONCLUSIONS AND RECOMMENDATIONS ..............................................141 6.1 Performance of Methodology ..................................................................141 6.1.1 Pruning Phase...............................................................................141 6.1.2 Global Optimization Phase ..........................................................145 6.1.3 Overall Performance ....................................................................148 vii
Page 1 and 2: DESIGN SPACE PRUNING HEURISTICS AND
Page 3 and 4: “The mediocre teacher tells. The
Page 5: I am thankful for all of my friends
Page 9 and 10: LIST OF TABLES Table 1 Ten best ast
Page 11 and 12: LIST OF FIGURES Figure 1 Mars round
Page 13 and 14: Figure 35: Low-thrust optima as a f
Page 15 and 16: LIST OF SYMBOLS AND ABBREVIATIONS A
Page 17 and 18: σBXB υ φ ω Ω Sample standard d
Page 19 and 20: optimization scheme to locate a bro
Page 21 and 22: provide clues to the nature of the
Page 23 and 24: additional constraints and objectiv
Page 25 and 26: Pontryagin’s Minimum Principle, w
Page 27 and 28: improved accuracy (as compared to d
Page 29 and 30: Figure 2: Trajectory structure of t
Page 31 and 32: flyby problems with numerous interm
Page 33 and 34: Gravity assists are modeled as inst
Page 35 and 36: Prior to the LTTT effort, the prima
Page 37 and 38: the shape of the trajectory and ana
Page 39 and 40: draws upon the theory of niche and
Page 41 and 42: the sequence of encounter bodies, t
Page 43 and 44: 1.2.2 Evolutionary Neurocontrollers
Page 45 and 46: Figure 6: Converting an evolutionar
Page 47 and 48: which find good solutions but can n
Page 49 and 50: functions. In this problem, only im
Page 51 and 52: As aforementioned, branch-and-bound
Page 53 and 54: consuming, user-intensive, and ofte
Page 55 and 56: asteroid tour mission design proble
Page 57 and 58:
CHAPTER II DEVELOPMENT OF METHODOLO
Page 59 and 60:
Kˆ ĥ i ν ê v r ω Ĵ Ω nˆ Î
Page 61 and 62:
Finally, both the eccentricity of a
Page 63 and 64:
maximum possible number of revoluti
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outer loop: a genetic algorithm and
Page 67 and 68:
2.3.2 Branch-and-Bound The branch-a
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The order in which the branches are
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approach performs best, followed by
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Figure 16: Effect of number of segm
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Within the sample problem, MALTO wa
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Table 2: Orbital elements of astero
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final mass. The correlation coeffic
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Maximum Final Mass (kg) 1500 1250 L
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The next approach is to compare the
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Final Mass (kg) 950 900 850 800 750
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pruning metric must be calculated f
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educed-size problem. Furthermore, t
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aBiB < 25%, and 15% for Leg 1, Leg
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Earth departure date and three time
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algorithm, which determines the opt
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is calculated using the same specif
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metrics which were calculated in th
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sorted by final mass. All of the se
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impulse optima. Figure 35 plots the
Page 105 and 106:
CHAPTER III OVERVIEW OF METHODOLOGY
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(1) All asteroid sequences are rank
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two-impulse optimum solutions. Ther
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identifying another metric that cou
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CHAPTER IV VALIDATION OF METHODOLOG
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Table 8 lists the 10 best asteroid
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In order to further validate the pr
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impulse ∆V). The first iteration
Page 121 and 122:
Because the impulsive multiplier ha
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Table 12: Effectiveness of the meth
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function calls to MALTO were requir
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order of 0.8 (assuming a final mass
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Table 16: Design variables for gene
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known solution to 1621 kg. Based on
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MBfB (kg) #1, #63, and #16, respect
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problem (Table 17). Additionally, t
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≤ Inclination (deg) 60 50 40 30 2
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order to have a benchmark with whic
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Of the remaining sequences, the 1 s
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Table 22: Settings for the genetic
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120 Impulsive Solutions Low-Thrust
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of exactly four weeks. As a benchma
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Objective Function (kg/yr) 120 100
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these two problems, the best known
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asteroid sequences would be require
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Table 27: Best known solutions rema
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Table 29: Best known solutions rema
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good solutions exist in the design
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heuristics chosen were based on the
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problem, the best set of solutions
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Finally, for each set of inner loop
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solutions from the GTOC2 competitio
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sequences were optimized in low-thr
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application is for the conceptual d
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types of the TSP, they have not bee
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methodology would be applied in the
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APPENDIX A SET OF GTOC2 ASTEROIDS T
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3054373 "2000 UK11" 0.88325596 0.24
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3152317 "2003 GQ22" 0.87232869 0.18
Page 183 and 184:
3283227 "2005 MR5" 0.85281863 0.295
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2000089 Julia 2.5500653 0.18377079
Page 187 and 188:
2000496 Gryphia 2.1987751 0.079568
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2001216 Askania 2.2322234 0.1793551
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2000134 Sophrosyne 2.5632069 0.1166
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2000712 Boliviana 2.5738464 0.18812
Page 195 and 196:
2005209 "1989 CW1" 5.1533221 0.0495
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36 2006 RJ1 0.9508113 0.30070707 1.
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REFERENCES [1] Rayman, M.D., Willia
Page 201 and 202:
[20] Hargraves, C. R., and Paris, S
Page 203 and 204:
[38] Petropoulos, A., Kowalkowski,
Page 205 and 206:
[57] Wuerl, A., Crain, T., Braden,
Page 207 and 208:
[78] Cosmic Vision: Space Science f
Page 209:
time, she enjoys traveling as well
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