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Number of Sequences Requiring Low-Thrust Optimization 5,000 4,000 3,000 2,000 1,000 0 0 20,000 40,000 60,000 80,000 # Sequences Evaluated in Branch-and-Bound Tree Figure 57: Number of sequences requiring low-thrust optimization as a function of the number of sequences evaluated in the branch-and-bound tree, for the full GTOC2 problem, assuming an initial lower bound of 100 kg/yr. Number of Sequences Requiring Low-Thrust Optimization 400 300 200 100 0 0 10,000 20,000 30,000 40,000 # Sequences Evaluated in Branch-and-Bound Tree Figure 58: Number of sequences requiring low-thrust optimization as a function of the number of sequences evaluated in the branch-and-bound tree, for the modified GTOC2 problem, assuming an initial lower bound of 100 kg/yr. Based on the results in Table 26 and Figure 57, in order to locate eight of the top thirteen solutions to the full GTOC2 problem, the low-thrust optimization of 4,620 133
asteroid sequences would be required, from a total of over 41 billion sequences in the original problem (a reduction of seven orders of magnitude). In order to locate the winning solution to GTOC2, the low-thrust optimization of 4,514 asteroid sequences would be required. With improvements in the parallel distribution of the low-thrust optimizations, along with efficiency improvements in the various codes used (e.g., switching from Matlab to Fortran), it is likely possible to complete this number of genetic algorithm executions in the month timeframe. Furthermore, it is likely that a number of additional solutions with J > 80 kg/yr would be located in those 4,620 sequences, based on the fact that three new good solutions were found in just the first 308 sequences in the branch-and-bound tree. 5.3 Sensitivity of Methodology to User-Defined Parameters In the previous two sections, it was shown how the methodology was able to successfully identify the best known solution as well as a family of good solutions for the modified GTOC2 and modified GTOC3 problems. For the modified GTOC2 problem, the design space was pruned by four orders of magnitude and the low-thrust optimization of 809 asteroid sequences was required. For the modified GTOC3 problem, the design space was pruned by two orders of magnitude and the low-thrust optimization of 17 asteroid sequences was required. For the full GTOC2 problem, it was shown that in two weeks, three previously unknown solutions were identified with J > 80 kg/yr, the best of which would have placed third in the competition. Furthermore, for the full GTOC2 problem, it would be possible to identify eight of the top thirteen known solutions with J > 80 kg/yr after the low-thrust optimization of 4,620 asteroid sequences. These results are based on particular values of the pruning metrics, along with additional assumptions such as the initial value of the lower bound for the branch-andbound algorithm. The number of required low-thrust optimizations is primarily dependent on the reduction in the design space that can be achieved by the pruning phase 134
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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
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draws upon the theory of niche and
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the sequence of encounter bodies, t
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1.2.2 Evolutionary Neurocontrollers
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Figure 6: Converting an evolutionar
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which find good solutions but can n
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functions. In this problem, only im
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As aforementioned, branch-and-bound
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consuming, user-intensive, and ofte
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asteroid tour mission design proble
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CHAPTER II DEVELOPMENT OF METHODOLO
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Kˆ ĥ i ν ê v r ω Ĵ Ω nˆ Î
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Finally, both the eccentricity of a
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maximum possible number of revoluti
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outer loop: a genetic algorithm and
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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
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
Page 141 and 142: Of the remaining sequences, the 1 s
Page 143 and 144: Table 22: Settings for the genetic
Page 145 and 146: 120 Impulsive Solutions Low-Thrust
Page 147 and 148: of exactly four weeks. As a benchma
Page 149 and 150: Objective Function (kg/yr) 120 100
Page 151: these two problems, the best known
Page 155 and 156: Table 27: Best known solutions rema
Page 157 and 158: Table 29: Best known solutions rema
Page 159 and 160: good solutions exist in the design
Page 161 and 162: heuristics chosen were based on the
Page 163 and 164: problem, the best set of solutions
Page 165 and 166: Finally, for each set of inner loop
Page 167 and 168: solutions from the GTOC2 competitio
Page 169 and 170: sequences were optimized in low-thr
Page 171 and 172: application is for the conceptual d
Page 173 and 174: types of the TSP, they have not bee
Page 175 and 176: methodology would be applied in the
Page 177 and 178: APPENDIX A SET OF GTOC2 ASTEROIDS T
Page 179 and 180: 3054373 "2000 UK11" 0.88325596 0.24
Page 181 and 182: 3152317 "2003 GQ22" 0.87232869 0.18
Page 183 and 184: 3283227 "2005 MR5" 0.85281863 0.295
Page 185 and 186: 2000089 Julia 2.5500653 0.18377079
Page 187 and 188: 2000496 Gryphia 2.1987751 0.079568
Page 189 and 190: 2001216 Askania 2.2322234 0.1793551
Page 191 and 192: 2000134 Sophrosyne 2.5632069 0.1166
Page 193 and 194: 2000712 Boliviana 2.5738464 0.18812
Page 195 and 196: 2005209 "1989 CW1" 5.1533221 0.0495
Page 197 and 198: 36 2006 RJ1 0.9508113 0.30070707 1.
Page 199 and 200: REFERENCES [1] Rayman, M.D., Willia
Page 201 and 202: [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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