design space pruning heuristics and global optimization method for ...

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equire visiting every asteroid; only one asteroid must be visited in each defined group. Furthermore, the spacecraft does not have to return to Earth, the point of origin. More importantly, the cost function (“distance”) between each pair of asteroids is not known a priori; instead, it changes with time and is not easily computable. For each instance in time, for example, calculating the cost to go from asteroid i to asteroid i+1 requires solving a low-thrust trajectory optimization problem to determine the optimal thrust profile that minimizes propellant expenditure for the given time of flight. Several variations on the classic traveling salesman problem have been studied, which address some of the asteroid tour problem complexities. In the time-dependent (or moving-target) TSP, the cost of traveling from city i to city j changes as a function of time. Work has been done on developing reliable heuristic methods with provable performance bounds for restricted versions of the time-dependent problem, such as where each target moves with a constant speed and direction and the pursuer has a maximum speed greater than the speed of each of the targets. 68 In the Generalized Traveling Salesman Problem (GTSP), all of the targets are partitioned into clusters, and the problem is transformed into finding the shortest route while visiting at least one target in each cluster. Another version of the GTSP requires that exactly one target must be visited in each cluster. 69,70 Finally, the wandering salesman problem, also known as the messenger problem, does not require returning to the point of origin, but instead deals with finding the least cost route from u to v. 70 Several authors have formulated trajectory optimization problems as traveling salesman problems. Stodgell and Spencer posed the problem of autonomous satellite servicing as a multi-objective wandering salesman problem with dynamically moving vertices. 71 Their specific problem can be defined as follows: given a set of target satellites, find tours that visit each target exactly once, while rendezvousing with the target for some minimum stay time and minimizing total flight time and ∆V. In order to solve this problem, NSGA-II was used in order to deal with the multiple objective 29
functions. In this problem, only impulsive maneuvers were considered, and a multirevolution Lambert algorithm was used to compute the required ∆V to transfer between two orbits for a specified time of flight. The genetic algorithm was then used to determine the following global design variables: tour order, initial departure time, flight time between each target satellite, and stay time at each target. This formulation was applied to several different target satellite configurations, with up to six target satellites. Figure 7 plots the resulting Pareto frontier for six target satellites in six different orbit planes separated by one degree each. For this test case, the optimal tour order and corresponding time variables were successfully found by the genetic algorithm for all three random initial populations considered. For some of the other test cases, however, the genetic algorithm would prematurely converge to a sub-optimal tour order depending on the initial population. Figure 7: Pareto frontier for satellite rendezvous problem with six targets. 71 Wall and Conway posed the optimal control problem as a motorized traveling salesman problem, where the salesman drives a car with two bounded controls: steering angle velocity and acceleration. 72 As in the original TSP, the salesman must still visit 30
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 and 46: Figure 6: Converting an evolutionar
Page 47: which find good solutions but can n
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
Page 97 and 98: is calculated using the same specif
Page 99 and 100:
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
Page 105 and 106:
CHAPTER III OVERVIEW OF METHODOLOGY
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(1) All asteroid sequences are rank
Page 109 and 110:
two-impulse optimum solutions. Ther
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identifying another metric that cou
Page 113 and 114:
CHAPTER IV VALIDATION OF METHODOLOG
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Table 8 lists the 10 best asteroid
Page 117 and 118:
In order to further validate the pr
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impulse ∆V). The first iteration
Page 121 and 122:
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
Page 129 and 130:
Table 16: Design variables for gene
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known solution to 1621 kg. Based on
Page 133 and 134:
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
Page 143 and 144:
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
Page 155 and 156:
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
Page 179 and 180:
3054373 "2000 UK11" 0.88325596 0.24
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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
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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.
Page 199 and 200:
REFERENCES [1] Rayman, M.D., Willia
Page 201 and 202:
[20] Hargraves, C. R., and Paris, S
Page 203 and 204:
[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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