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and upper bounds. In each generation, the population is subjected to certain genetic operators such that the population will “evolve” and improve its fitness (objective function). The typical genetic operators are reproduction, crossover, and mutation. The purpose of reproduction is to weed out the members of the population with low fitness values, and to keep those with high fitness values. Crossover combines two “parents” by switching parts of their chromosome strings with each other to create two “children”. Mutation is responsible for switching individual bits in a chromosome string. Because there is no necessary condition for optimality, the convergence criteria is generally chosen either as a maximum number of generations (iterations) or a certain number of generations with no change in the objective function and/or design variables. As the generations progress, there should be a steady improvement in both the average fitness of the population as well as the fitness of the best member. In general, at the termination of the GA, the population will be clustered around the global optimum. One of the main advantages of genetic algorithms is their ability to find a global optimum in a discrete, multi-modal design space. They can also handle a large number of variables, and require no initial guesses for the design variables. Genetic algorithms, however, do have some disadvantages. Because of the probabilistic nature of the algorithm, there is no guarantee that the optimal solution will be found. Therefore, the GA must generally be run more than once to ensure optimality. Genetic algorithms also require a large number of iterations, and therefore function calls, in comparison to a gradient-based method. Finally, if the original design space is comprised of continuous design variables, the discretized solution will generally not correspond to the precise global optimum. A common practice is to use the solution obtained by the GA as an initial guess to a gradient-based optimizer, in order to improve the accuracy of the solution. Several variations on the standard genetic algorithm have also been developed. In order to locate multiple local optima, a sharing function can be added to the GA, which 19
draws upon the theory of niche and speciation in Darwinian evolution. 49 The purpose of the sharing function is to degrade an individual’s fitness function based on its proximity to neighboring individuals. As a result, the largest number of individuals will converge to the local optimum with the best fitness value, with fewer individuals converging to optima with lesser fitness. The number of optima found by a genetic algorithm with sharing is a function of the size of the population. Another genetic algorithm variant addresses multi-objective problems. 50 One way of dealing with multiple objectives is to solve for the Pareto-optimal set, which encompasses the set of non-dominated solutions. When comparing two solutions, x 1 and x 2 , x 1 dominates x 2 if (1) x 1 is no worse than x 2 in all objectives and (2) x 1 is strictly better than x 2 in at least one objective. Therefore, the Pareto-optimal set contains all the solutions that are not dominated by any other solutions. This concept has been implemented in NSGA-II (non-dominated sorting genetic algorithm), developed by the Kanpur Genetic Algorithm Laboratory. 51 Genetic algorithms have been applied to a number of different trajectory optimization problems, beginning with their application to ballistic (high thrust) transfers and gravity assist problems. 52,53,54,55 For the high-thrust case, solving for a single trajectory is much less time-consuming and is generally done using a Lambert Solver. Therefore, a genetic algorithm, even with the large number of required function calls, is appropriate for global optimization. Gage and Braun applied a genetic algorithm with a sharing function to impulsive Earth-Mars trajectories in order to optimize for launch date and time of flight54. Figure 4 plots ∆V as a function of departure date and transfer time. As can be seen, this is a multimodal space, and for conceptual design, it is desirable to locate each of the local minima, which was successfully accomplished using a sharing function. 20
Page 1 and 2: DESIGN SPACE PRUNING HEURISTICS AND
Page 3 and 4: “The mediocre teacher tells. The
Page 5 and 6: I am thankful for all of my friends
Page 7 and 8: 2.5 Small Sample Problem...........
Page 9 and 10: LIST OF TABLES Table 1 Ten best ast
Page 11 and 12: LIST OF FIGURES Figure 1 Mars round
Page 13 and 14: Figure 35: Low-thrust optima as a f
Page 15 and 16: LIST OF SYMBOLS AND ABBREVIATIONS A
Page 17 and 18: σBXB υ φ ω Ω Sample standard d
Page 19 and 20: optimization scheme to locate a bro
Page 21 and 22: provide clues to the nature of the
Page 23 and 24: additional constraints and objectiv
Page 25 and 26: Pontryagin’s Minimum Principle, w
Page 27 and 28: improved accuracy (as compared to d
Page 29 and 30: Figure 2: Trajectory structure of t
Page 31 and 32: flyby problems with numerous interm
Page 33 and 34: Gravity assists are modeled as inst
Page 35 and 36: Prior to the LTTT effort, the prima
Page 37: the shape of the trajectory and ana
Page 41 and 42: the sequence of encounter bodies, t
Page 43 and 44: 1.2.2 Evolutionary Neurocontrollers
Page 45 and 46: Figure 6: Converting an evolutionar
Page 47 and 48: which find good solutions but can n
Page 49 and 50: functions. In this problem, only im
Page 51 and 52: As aforementioned, branch-and-bound
Page 53 and 54: consuming, user-intensive, and ofte
Page 55 and 56: 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
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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
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Table 8 lists the 10 best asteroid
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In order to further validate the pr
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impulse ∆V). The first iteration
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Because the impulsive multiplier ha
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Table 12: Effectiveness of the meth
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function calls to MALTO were requir
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order of 0.8 (assuming a final mass
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Table 16: Design variables for gene
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known solution to 1621 kg. Based on
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MBfB (kg) #1, #63, and #16, respect
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problem (Table 17). Additionally, t
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≤ Inclination (deg) 60 50 40 30 2
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order to have a benchmark with whic
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Of the remaining sequences, the 1 s
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Table 22: Settings for the genetic
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120 Impulsive Solutions Low-Thrust
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of exactly four weeks. As a benchma
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Objective Function (kg/yr) 120 100
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these two problems, the best known
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asteroid sequences would be require
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Table 27: Best known solutions rema
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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,
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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