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

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Table 22 Settings for the genetic algorithm as applied to the modified GTOC2 problem ........................................................................................................................124 Table 23 Design variables for the genetic algorithm as applied to the modified GTOC2 problem ..........................................................................................................124 Table 24 Top seven solutions found by the methodology for the modified GTOC2 problem ..........................................................................................................127 Table 25 Top ten known solutions to full GTOC2 problem.........................................128 Table 26 Known solutions to GTOC2 problem with J > 80 kg/yr................................131 Table 27 Best known solutions remaining in design space for varying orders of magnitude reduction during the pruning phase, for the modified GTOC2 problem ..........................................................................................................136 Table 28 Best known solutions remaining in the design space for varying orders of magnitude reduction during the pruning phase, for the full GTOC2 problem ........................................................................................................................137 Table 29 Best known solutions remaining in design space for varying orders of magnitude reduction during the pruning phase, for the modified GTOC3 problem ..........................................................................................................138 Table 30 Summary of pruning phase applied to each problem.....................................144 Table 31 Summary of global optimization phase applied to each problem..................147 Table 32 Summary of overall performance of methodology as applied to each problem ........................................................................................................................149 Table 33 GTOC2 Asteroids. .........................................................................................158 Table 34 GTOC3 Asteroids. .........................................................................................177 x
LIST OF FIGURES Figure 1 Mars roundtrip trajectory results from Ranieri: variable Isp (left), constant Isp (right). ................................................................................................................8 Figure 2 Trajectory structure of the Sims and Flanagan direct method. ........................10 Figure 3 Figure 4 Results of shape-based method design space exploration for Earth-Mars-Ceres (left) and EVEMJ (right)..................................................................................15 Results of a genetic algorithm with a sharing function applied to an Earth- Mars impulsive transfer ...................................................................................21 Figure 5 Example of a layered, feed-forward neural network .......................................24 Figure 6 Converting an evolutionary algorithm chromosome into a spacecraft trajectory (left); example neurocontroller that implements a spacecraft trajectory (right) ...............................................................................................................26 Figure 7 Pareto frontier for satellite rendezvous problem with six targets66. ...............30 Figure 8 Portion of a branch-and-bound search tree. .....................................................32 Figure 9 Classical orbital elements. ...............................................................................40 Figure 10 (a) Delta-V as a function of the transfer orbit semi-major axis for a twoimpulse transfer (left); (b) contour plot of the minimum two-impulse delta-V transfers over all departure and arrival true anomalies (right).........................43 Figure 11 Single-level global optimization scheme.........................................................45 Figure 12 Multi-level global optimization scheme. .........................................................45 Figure 13 Example branch-and-bound tree......................................................................48 Figure 14 Optimum final mass for Earth – 2006 QQ56 with a 600-day time of flight, using three different approaches for the initial guess of the MALTO variables. ..........................................................................................................52 Figure 15 Optimum final mass for Medusa – Kostinsky with a 1200-day time of flight, using three different approaches for the initial guess of the MALTO variables. ..........................................................................................................52 Figure 16 Effect of number of segments on the optimum final mass for Earth – 2006 QQ56, with a 600-day time of flight................................................................54 Figure 17 Effect of number of segments on computation time for Earth – 2006 QQ56, with a 600-day time of flight ...........................................................................54 xi
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: LIST OF TABLES Table 1 Ten best ast
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 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
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maximum possible number of revoluti
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outer loop: a genetic algorithm and
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2.3.2 Branch-and-Bound The branch-a
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The order in which the branches are
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approach performs best, followed by
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Figure 16: Effect of number of segm
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Within the sample problem, MALTO wa
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Table 2: Orbital elements of astero
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final mass. The correlation coeffic
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Maximum Final Mass (kg) 1500 1250 L
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The next approach is to compare the
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Final Mass (kg) 950 900 850 800 750
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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
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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