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ii. If the optimal, multi-rev, two-impulse solution is greater than the current lower bound, continue to next segment. c. Above process is continued until the end of the branch has been reached, i.e., Earth – 1 st asteroid – 2 nd asteroid - … - n th asteroid. i. If the optimal, multi-rev, two-impulse solution is greater than the current lower bound, compute the optimal low-thrust solution for the entire sequence (by running the genetic algorithm N GA times). ii. If the optimal low-thrust solution is greater than the current lower bound, update the current lower bound. d. Continue above steps until all branches have been solved or pruned. (4) Once all branches have been evaluated, compare the known low-thrust optimal solutions with their corresponding optimal, multi-rev, two-impulse solutions. Update the impulsive multiplier such that all known low-thrust solutions are bounded by the two-impulse solutions. a. If the impulsive multiplier needs to be updated, repeat Step 3 with the new value of impulsive multiplier. b. Otherwise, terminate branch-and-bound. The algorithm specifies calculating the optimal two-impulse solutions for a number of asteroid sequences. In this work, this is calculated using a grid search, but a genetic algorithm, a gradient-based method with multiple starting points, or another method could also be employed. If after the first iteration of the branch-and-bound, no low-thrust optimum solutions are calculated (other than the first optimum calculated to set the lower bound, if applicable), it is possible that the impulsive multiplier need not be updated. Instead of terminating the branch-and-bound with only the lower bound computed, a small set of additional low-thrust optima can be calculated based on the sequences that have the best 89
two-impulse optimum solutions. Therefore, a larger sample size is available to more accurately determine the next required value for the impulsive multiplier. This procedure can also be carried out at the end of the branch-and-bound algorithm (after multiple iterations) to ensure that the correct value of the impulsive multiplier has been converged on. This process also serves to potentially locate additional good solutions that were not calculated during the branch-and-bound algorithm. This is especially important if the optimal solution is located early in the branch-and-bound algorithm, since the high value of the lower bound will eliminate a large number of potentially good solutions. While locating the optimum solution early in the algorithm will minimize the number of lowthrust optimizations that must be carried out, it also results in a larger number of good solutions being pruned out. The number of good solutions desired (in addition to the optimum) will determine the number of additional low-thrust optimizations to be carried out after the branch-and-bound algorithm as terminated. Furthermore, if the branch-and-bound algorithm is started with an estimate made for the initial value of the lower bound (step 1c), and a satisfactory set of low-thrust solutions is not found, the lower bound can be incrementally decreased and the branchand-bound algorithm re-run with the new initial lower bound. Both in this scenario and in the iterative approach to setting the impulsive multiplier, it is not necessary to re-run the two-impulse or low-thrust optimizations that have already been calculated. In order to minimize computation time, both the two-impulse and low-thrust optima should be saved for each asteroid sequence for which they are evaluated. A simple table look-up can then be used for these sequences as opposed to rerunning the optimizations for each iteration through the branch-and-bound algorithm. In this manner, only the new asteroid sequences requiring optimization need to be evaluated. A similar iterative approach can also be taken with regards to the pruning phase. If time permits upon completion of the methodology, the pruning percentages could then also be relaxed, increasing the number of asteroid sequences passed to the global 90
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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
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 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: (1) All asteroid sequences are rank
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 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
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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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