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

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the reduced design space. The goal of the pruning step is to quickly reduce the size of the problem by several orders of magnitude. This is accomplished using heuristics specific to the physics of the underlying problem, in order to identify areas of the design space that will not likely yield favorable solutions in terms of the objective function. Heuristic methods, however, cannot guarantee that only poor solutions will be eliminated from the design space. The goal of this first phase is to ensure that a large percentage of the best solutions remain for the second phase. In this second phase, a global optimization algorithm is applied to the reduced design space to locate the best set of solutions. The global optimizer is responsible for solving for the following design variables: asteroid combination, launch date, times of flight, and stay times. This system-level optimization is coupled with a local low-thrust trajectory optimization scheme that determines the optimal control history of the spacecraft in order to minimize propellant for a given set of global optimization variables. The methodology developed is used to predict the solution to a range of test problems with a known optimal soltution, and once verified, is applied to several larger asteroid tour problems, including versions of the GTOC2 and GTOC3 problems. The GTOC competition problems were chosen because a set of solutions is available from the competition results which can be used as a benchmark in evaluating the performance of the methodology. The following are the key contributions of this work: (1) A three-level heuristic sequence is developed based on the physics of the problem that allows for efficient pruning of the design space. In reducing the size of the design space, a majority of the better solutions are maintained. This pruning methodology is verified through solution of an intermediate-size sample problem whose solution is obtained through complete enumeration of the design space. The pruning methodology is shown to apply well across a range of low-thrust 35
asteroid tour mission design problems and relies on user-defined parameters to effectively tailor the degree of design space pruning based on the available computational resources. (2) A global optimizer is combined with a low-thrust trajectory optimization method to locate a broad suite of good solutions for the reduced problem. This approach combines an innovative branch-and-bound algorithm (to solve for the optimal asteroid sequence) with a genetic algorithm (which solves for the optimal departure date, times of flight, and stay times for a given asteroid sequence), and finally with a low-thrust trajectory optimization program (which determines the optimal thrust profile that maximizes final mass). The global optimization scheme is able to consistently locate the best known solution, along with a suite of good solutions across the design space. A strategy was developed to set the initial lower bound in the branch-and-bound algorithm (a user-defined parameter) as a means of controlling the number of required low-thrust optimizations required and the number of good solutions found. (3) When the global optimization scheme is combined with the heuristic screening process, a systematic methodology for identification of a broad suite of good solutions to low-thrust, multiple asteroid rendezvous, conceptual mission design problems is achieved. In addition to a wide range on each of the continuous variables, the problems to which the methodology is applied in this investigation are characterized by as many as 41 billion discrete asteroid sequences. The key contribution of this methodology is the ability to locate a suite of good solutions, as opposed to just a single optimum solution. In locating these good solutions, the overall methodology is able to reduce the number of asteroid sequences that 36
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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 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: consuming, user-intensive, and ofte
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
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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,
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[57] Wuerl, A., Crain, T., Braden,
Page 207 and 208:
[78] Cosmic Vision: Space Science f
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time, she enjoys traveling as well
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