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1.2.3 Combinatorial Optimization Choosing the optimal asteroid sequence in an asteroid tour mission design problem is by nature a combinatorial and integer optimization problem. The distinguishing feature of such problems is that the variables belong to a discrete set where there is not a continuum of alternatives. One can write the linear integer programming problem, which has no continuous variables, as follows 65 : { cx : Ax b,x ∈ Z n } max ≤ + (12) In Equation 12, Z n + is the set of nonnegative integral n-dimensional vectors, and x = (x 1 ,…,x n ) are the variables or unknowns. An instance of the problem is specified by the data (c, A, b), where c is an n-dimensional vector, A is an m x n matrix, and b is an m- dimensional vector. While this represents the general problem, it can be altered in a number of ways to represent myriad types of discrete optimization problems. One example of a formulation of Equation 12 is the 0-1 knapsack problem, which is one of the most basic and common problems in combinatorial optimization. This problem deals with choosing a subset of projects to maximize the sum of their values while not exceeding some budget constraint. From Equation 12, the projects are represented by the variables x 1 ,…x n , where a value of 1 indicates that project j is in the subset and a value of 0 indicates that it is not. The jth project has a cost of a j and a value of c j , and b represents the overall budget constraint. Another common type of combinatorial optimization problem, which is more directly applicable to the asteroid tour problem, is the traveling salesman problem. The classic version of this problem deals with solving the following scenario: given a set of cities and the distance between each pair of cities, determine the shortest route to visit each city exactly once while returning to the city of origin. 66 Solution methods generally fall into two categories: algorithms for finding exact solutions and heuristic algorithms, 27
which find good solutions but can not be proven to be optimal. Exact solutions can typically be implemented successfully only for small problems, while heuristic methods are used for larger problems where computation time of an exact method would become prohibitive. A commonly used exact algorithm is branch-and-bound, which branches the original problem into successively smaller sub-problems. Each subset contains a relaxed version of the original problem, which is easier to solve. The procedure continues until each branch has resulted in either a feasible solution or is shown to contain no solution better than one already obtained. Branch-and-bound methods result in locating the global optimum66. Similarly, dynamic programming also takes advantage of problem decomposition, where the optimal solution to a given problem is expressed in terms of optimal solutions of smaller sub-problems. 67 One of the most commonly known heuristic algorithms is the nearest neighbor algorithm, also referred to as the greedy algorithm. In this method, the local optimum is chosen at each step. For the classic TSP, for example, this would equate to choosing the closest city at each step, until all of the cities have been visited. Another common heuristic method is to use minimum spanning trees. A spanning tree is a collection of (n- 1) edges which join all n cities into a tree-structure. This can then be extrapolated to create a tour, where each city is only visited once. While the heuristic methods do not solve for the optimum solution, they can at least provide lower and upper bounds on the optimum. However, one of the biggest challenges of heuristic methods is establishing performance guarantees – i.e., bounds on how far the solution will be from the optimum in the worst case66. The classic TSP has many analogous features to the asteroid rendezvous problem, where the “distance” between each asteroid is instead a combination of propellant consumption and time of flight. Some major differences do exist, however, between the classic TSP and the asteroid rendezvous problem. First, the asteroid problem does not 28
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 and 38: the shape of the trajectory and ana
Page 39 and 40: draws upon the theory of niche and
Page 41 and 42: the sequence of encounter bodies, t
Page 43 and 44: 1.2.2 Evolutionary Neurocontrollers
Page 45: Figure 6: Converting an evolutionar
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
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
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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
Page 201 and 202:
[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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