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OptimalTwo-Impulse J (kg/yr) 200 180 160 140 120 100 80 60 40 2 4 6 8 10 12 14 Time of Flight (years) OptimalTwo-Impulse J (kg/yr) 200 180 160 140 120 100 80 60 40 2 4 6 8 10 12 14 Time of Flight (years) Figure 52: Optimal impulsive solutions, with and without the time of flight restriction, for the modified GTOC2 problem. Table 22 lists the settings used for the genetic algorithm as applied to this problem, and Table 23 lists the design variables, bounds, and discretization. The number of bits for each design variable was chosen so that the discretization on all of the time variables was approximately equal. For the bits chosen, the variables are discretized in approximately seven day increments. Because of the greater number of low-thrust optimizations required in this problem, the genetic algorithm is run only once for each asteroid sequence. At the end of the branch-and-bound algorithm, the genetic algorithm is then run three times on the most promising solutions, in an attempt to improve upon their objective functions. Using these settings, on average, a single run of the genetic algorithm requires 1746 function calls (end-to-end optimizations by MALTO) and 49 generations to converge. With the available computing resources, each genetic algorithm run takes 70 minutes on average. 123
Table 22: Settings for the genetic algorithm as applied to the modified GTOC2 problem. GA Setting Value Population Size 200 Stall Generations 20 Tournament Size 4 Crossover Proability 0.8 Mutation Probablity 0.1 Table 23: Design variables for the genetic algorithm as applied to the modified GTOC2 problem. Design Variable Units # Bits Lower Bound Upper Bound Earth Departure Date JD 10 2457023 2464328 TOF, Leg 1 days 8 200 2000 TOF, Leg 2 days 8 200 2000 TOF, Leg 3 days 8 200 2000 TOF, Leg 4 days 8 200 2000 Stay Time, Ast. 1 days 5 90 360 Stay Time, Ast. 2 days 5 90 360 Stay Time, Ast. 3 days 5 90 360 Although a list of known solutions is available, the branch-and-bound algorithm is started without setting the best solution as the lower bound, since in the competition there were no a priori known solutions. Therefore, the first iteration of the branch-and-bound requires the low-thrust optimum to be computed for the highest ranked asteroid sequence (in terms of the normalized sum of the pruning metrics). The initial lower bound, J min , is 40.29 kg/yr. This iteration is carried out without any multiplier on the impulsive solutions. Overall, the first iteration requires the low-thrust optimization of 809 asteroid sequences, which took approximately 39 days to complete. Additionally, it requires the impulsive optimization of all the asteroid sequences – none of the branches were pruned out until the final trajectory leg. The best solution found corresponds to the best solution found during the GTOC2 competition. Its objective function is 98.64 kg/yr, which ranks 674 th in terms of the normalized sum of the pruning metrics. During the execution of the branch-and-bound algorithm, it is the 378 th asteroid sequence for which the low-thrust 124
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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
Page 91 and 92: aBiB < 25%, and 15% for Leg 1, Leg
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
Page 141: Of the remaining sequences, the 1 s
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 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
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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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