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85% of the asteroid sequences could be eliminated without eliminating any of the ten best sequences. This results in 73 remaining asteroid combinations. This approach, however, requires calculating each metric for all 136 distinct asteroid pairs. Applying the second approach to the sample problem, up to 87% of the asteroid sequences can be eliminated, resulting in 66 remaining asteroid sequences. Using this approach, however, θ wedge must be calculated for all 136 asteroid pairs, but the optimal two-impulse, phase-free ∆V must be calculated for only 62 asteroid pairs. Finally, the third approach can eliminate up to 74% of asteroid sequences, leaving 131 remaining sequences. θ wedge must be calculated for 104 asteroid pairs, while the optimal, two-impulse, phase-free ∆V must be calculated for 77 asteroid pairs. Therefore, all three approaches are comparable in that some require calculating the pruning metrics for more asteroid pairs but enable a greater reduction in the design space, and vice versa. The third approach is selected because for larger problems, a greater reduction in the design space will realize an even greater savings in the number of pruning metrics that must be calculated. This is an important consideration, when the number of asteroid sequences is in the millions or billions. When applying the pruning approach to a problem, the percent of asteroid pairs eliminated in each leg for each pruning metric is chosen based on two factors. First is the desired reduction in the design space – the smaller the resulting number of asteroid sequences, the larger the percent eliminated must be. The more asteroid combinations that are eliminated, however, the greater chance there is of eliminating some of the best sequences. Therefore, a user would choose a desired reduction in the number of asteroid sequences and then find suitable values for the percentages based on that desired final value. Table 3 illustrates how the percentages can be incrementally increased until the desired number of asteroid sequences remains. For a general problem, however, the number of good sequences eliminated is of course unknown. Table 4 summarizes the results when the following percentage values are applied to the sample problem: 30%, 71
aBiB < 25%, and 15% for Leg 1, Leg 2, and Leg 3, respectively. The number of asteroid sequences is reduced from 3072 to 131, and only 19 feasible sequences were eliminated. th The best sequence eliminated has a final mass of 653 kg, which ranks 37P P. Therefore, none of the top solutions are eliminated and the design space is reduced by a factor of 23. Table 4: Pruning methodology applied to sample problem. Pruning Metric Trajectory Leg % Sequences Eliminated # Sequences Eliminated aBi+1B All N/A 2560 θBwedgeB Leg 1 30% 128 θBwedgeB Leg 2 25% 96 θBwedgeB Leg 3 15% 42 ∆VBoptB Leg 1 30% 41 ∆VBoptB Leg 2 25% 46 ∆VBoptB Leg 3 15% 28 2.5.2 Evaluation of Global Optimization Methods on the Sample Problem The global optimization methods under consideration are all tested on the full sample problem (instead of the pruned design space), in order to evaluate them on the largest possible design space. For the genetic algorithm, two approaches are considered, as was presented in Figure 11 and Figure 12: (1) a single-level approach, where all global design variables are solved for simultaneously, and (2) a two-level approach, where an outer loop optimizer solves for the asteroid sequence and an inner loop optimizer solves for the time variables. The branch-and-bound is used solely in the two-level configuration, with the branch-and-bound method solving for the asteroid sequence and a genetic algorithm solving for the time variables. The basic genetic algorithm is applied to the sample problem in a variety of ways. In each case, the following settings must be chosen: 72
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DESIGN SPACE PRUNING HEURISTICS AND
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“The mediocre teacher tells. The
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I am thankful for all of my friends
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2.5 Small Sample Problem...........
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LIST OF TABLES Table 1 Ten best ast
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LIST OF FIGURES Figure 1 Mars round
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Figure 35: Low-thrust optima as a f
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LIST OF SYMBOLS AND ABBREVIATIONS A
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σBXB υ φ ω Ω Sample standard d
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optimization scheme to locate a bro
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provide clues to the nature of the
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additional constraints and objectiv
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Pontryagin’s Minimum Principle, w
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improved accuracy (as compared to d
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Figure 2: Trajectory structure of t
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flyby problems with numerous interm
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Gravity assists are modeled as inst
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Prior to the LTTT effort, the prima
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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
Page 65 and 66: outer loop: a genetic algorithm and
Page 67 and 68: 2.3.2 Branch-and-Bound The branch-a
Page 69 and 70: The order in which the branches are
Page 71 and 72: approach performs best, followed by
Page 73 and 74: Figure 16: Effect of number of segm
Page 75 and 76: Within the sample problem, MALTO wa
Page 77 and 78: Table 2: Orbital elements of astero
Page 79 and 80: final mass. The correlation coeffic
Page 81 and 82: Maximum Final Mass (kg) 1500 1250 L
Page 83 and 84: The next approach is to compare the
Page 85 and 86: Final Mass (kg) 950 900 850 800 750
Page 87 and 88: pruning metric must be calculated f
Page 89: educed-size problem. Furthermore, t
Page 93 and 94: Earth departure date and three time
Page 95 and 96: algorithm, which determines the opt
Page 97 and 98: is calculated using the same specif
Page 99 and 100: metrics which were calculated in th
Page 101 and 102: sorted by final mass. All of the se
Page 103 and 104: impulse optima. Figure 35 plots the
Page 105 and 106: CHAPTER III OVERVIEW OF METHODOLOGY
Page 107 and 108: (1) All asteroid sequences are rank
Page 109 and 110: two-impulse optimum solutions. Ther
Page 111 and 112: identifying another metric that cou
Page 113 and 114: CHAPTER IV VALIDATION OF METHODOLOG
Page 115 and 116: Table 8 lists the 10 best asteroid
Page 117 and 118: In order to further validate the pr
Page 119 and 120: impulse ∆V). The first iteration
Page 121 and 122: Because the impulsive multiplier ha
Page 123 and 124: Table 12: Effectiveness of the meth
Page 125 and 126: function calls to MALTO were requir
Page 127 and 128: order of 0.8 (assuming a final mass
Page 129 and 130: Table 16: Design variables for gene
Page 131 and 132: known solution to 1621 kg. Based on
Page 133 and 134: MBfB (kg) #1, #63, and #16, respect
Page 135 and 136: problem (Table 17). Additionally, t
Page 137 and 138: ≤ Inclination (deg) 60 50 40 30 2
Page 139 and 140: 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
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3283227 "2005 MR5" 0.85281863 0.295
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2000089 Julia 2.5500653 0.18377079
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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
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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
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[20] Hargraves, C. R., and Paris, S
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[38] Petropoulos, A., Kowalkowski,
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[57] Wuerl, A., Crain, T., Braden,
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[78] Cosmic Vision: Space Science f
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time, she enjoys traveling as well
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