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First, the branch-and-bound algorithm is applied without any multiplier on the optimal two-impulse mass solutions. Figure 46 plots the results of this first iteration. Each asteroid sequence is plotted as a function of its optimal impulsive ∆V, with the red line indicating the best known low-thrust solution at the end of the iteration. Any points that fall below this red line are pruned from the branch-and-bound tree. During the first iteration, all of the asteroid sequences are pruned, since none of the optimal impulsive solutions are greater than 1590 kg. The optimal impulsive final mass for this asteroid sequence is 1477 kg, leading to a required multiplier of 1.077 for the next branch-andbound iteration. Optimal Impulsive Mf (kg) Optimal Impulsive Mf (kg) 2000 1800 1600 1400 1200 Earth-Ast.1 1000 0 5 10 Asteroid Sequence # Earth-Ast.1-Ast.2-Ast.3 2000 1800 1600 1400 1200 1000 0 5000 10000 Asteroid Sequence # Optimal Impulsive Mf (kg) Optimal Impulsive Mf (kg) 2000 1800 1600 1400 1200 Earth-Ast.1-Ast.2 1000 0 100 200 Asteroid Sequence # Earth-Ast.1-Ast.2-Ast.3-Earth 2000 1800 1600 1400 1200 1000 0 5000 10000 Asteroid Sequence # Figure 46: Branch-and-bound results, iteration #1 (impulsive multiplier = 1). Figure 47 plots the results of the second iteration. The low-thrust optimum solution is calculated for an additional seven asteroid sequences. In the process, an asteroid sequence with a higher low-thrust final mass is located, raising the current best 111
known solution to 1621 kg. Based on comparing the eight known low-thrust final masses to their corresponding impulsive optimal solutions, the multiplication factor must be increased again, to a value of 1.096. Optimal Impulsive Mf (kg) Optimal Impulsive Mf (kg) 2000 1800 1600 1400 1200 Earth-Ast.1 1000 0 5 10 Asteroid Sequence # Earth-Ast.1-Ast.2-Ast.3 2000 1800 1600 1400 1200 1000 0 5000 10000 Asteroid Sequence # Optimal Impulsive Mf (kg) Optimal Impulsive Mf (kg) 2000 1800 1600 1400 1200 Earth-Ast.1-Ast.2 1000 0 100 200 Asteroid Sequence # Earth-Ast.1-Ast.2-Ast.3-Earth 2000 1800 1600 1400 1200 1000 0 5000 10000 Asteroid Sequence # Figure 47: Branch-and-bound results, iteration #2 (impulsive multiplier = 1.077). Figure 48 plots the results of the third iteration. Nine additional low-thrust optima are computed during this iteration. The best known solution, however, is not improved upon. Furthermore, the impulsive multiplier does not have to be updated, as it is sufficient to bound all of the known low-thrust solutions. Therefore, the branch-andbound algorithm is considered converged after three iterations, with a final impulsive multiplier of 1.096 and a best low-thrust final mass of 1621 kg. 112
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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
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 and 90: 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: Table 16: Design variables for gene
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 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
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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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