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

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Problem Description Small Sample Problem Intermediate Sample Problem Modified GTOC3 Problem Modified GTOC2 Problem Full GTOC2 Problem Table 31: Summary of global optimization phase applied to each problem. # Branchand-Bound Iterations # Sequences Requiring Low-Thrust Optimization 2 14 2 55 3 17 1 809 N/A 308 (in 2 weeks) Best Set of Sequences Found − Optimal solution found − 5 of the top 10 remaining solutions found − Optimal solution found − 5 of top 9 remaining solutions found − Best known solution found − 8 of the top 10 remaining solutions found − Best known solution found − 7 solutions found with J>80 kg/yr: − 4 of 4 competition solutions remaining after pruning phase − 3 new solutions not reported in competition − Three new solutions with J>80 kg/yr found that were not reported in competition − Better solution would have ranked 3 rd in GTOC2 For the full GTOC2 problem, the branch-and-bound algorithm was allowed to run for two weeks to simulate the time constraints from the competition. In those two weeks, 308 asteroid sequences were optimized in low-thrust. Of these, three previously unknown asteroid sequences were identified with J > 80 kg/yr, the best of which would have placed third in the competition. Furthermore, Section 5.2.2 analyzed how many low-thrust optimizations would be required to locate the additional known good solutions. It was determined that using an initial guess for the lower bound of 100 kg/yr (same assumption as was made in the two weeks of analysis), 4,620 asteroid sequences would have to be optimized in low-thrust in order to locate eight of the thirteen known solutions with J > 80 kg/yr. The thirteen known solutions include the seven reported 147
solutions from the GTOC2 competition along with the six additional solutions found in this work. 6.1.3 Overall Performance Table 32 summarizes the overall performance of the proposed methodology on each of the problems presented in this study (the full GTOC2 problem is not included in these results as the methodology was not run to completion for this problem). Two metrics are of importance here. First is the ratio of the number of asteroid sequences that require low-thrust optimization to the number of sequences in the initial design space. This is a measure of the efficiency that can be achieved by the overall methodology. For the modified GTOC3 and modified GTOC2 problems, this results in an overall reduction of between 6 and 7 orders of magnitude. Second is the number of good solutions found by the overall methodology. Because the best solutions are not known for the GTOC2 and GTOC3 problem, the solutions previously identified during the competitions are used as a benchmark. Tailoring of the methodology to the amount of computing time available has also been demonstrated. For the large problems examined, it is possible to reduce the number of asteroid combinations by up to four orders of magnitude and still keep a majority of the best known solutions in the design space. Clearly, based on user-assigned values, the design space could be pruned less aggressively at the expense of more required lowthrust optimizations during the global optimization phase. Another user-defined setting, the initial lower bound in the branch-and-bound algorithm, was also shown to directly affect the number of required low-thrust optimizations required and the number of good solutions found. An aggressive estimate can serve to greatly reduce the number of lowthrust optimization required; however, it can also prune out some of the good solutions if set too high. A strategy was developed in which this value could be set high based on the 148
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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
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sorted by final mass. All of the se
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impulse optima. Figure 35 plots the
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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
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 and 152: these two problems, the best known
Page 153 and 154: asteroid sequences would be require
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: Finally, for each set of inner loop
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
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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