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Figure 18 Set of asteroids for sample problem.................................................................55 Figure 19 Optimal solution for the small sample problem...............................................56 Figure 20 Maximum final mass for each asteroid sequence as a function of inclination change. .............................................................................................................59 Figure 21 Maximum final mass for each asteroid combination as a function of wedge angle, the angle between the two angular momentum vectors. .......................61 Figure 23 Maximum final mass for each asteroid pair as a function of the minimum, phase-free, two-impulse ∆V. ...........................................................................62 Figure 23 Comparison of two-impulse and low-thrust mass-optimal solutions for Earth – 2006 QQ56 with a 600-day time of flight........................................................63 Figure 24 Comparison of two-impulse and low-thrust mass-optimal solutions for Chicago – Kostinsky with a 1200-day time of flight.......................................63 Figure 25 Comparison of two-impulse and low-thrust mass-optimal solutions for Earth – Apophis with a time of flight up to 600 days...................................................64 Figure 26 Final mass of all feasible trajectories as a function of arrival true anomaly at final asteroid.....................................................................................................65 Figure 27 Maximum final mass for top twenty asteroid sequences as a function of arrival true anomaly at final asteroid...........................................................................66 Figure 28 Final mass as a function of the summed pruning metric (Equation 17) for each asteroid sequence remaining in the small sample problem..............................67 Figure 29 Final mass as a function of θwedge, summed over all legs for each asteroid sequence remaining in the small sample problem. ..........................................68 Figure 30 Final mass as a function of optimal, phase-free, two-impulse ∆V summed over all legs for each asteroid sequence remaining in the small sample problem. ..69 Figure 31 Comparison of mass-optimal low-thrust and two-impulse solutions for all Earth – Asteroid 1 sequences...........................................................................78 Figure 32 Comparison of mass-optimal low-thrust and two impulse solutions for all Earth – Asteroid 1 – Asteroid 2 sequences. .....................................................78 Figure 33 Comparison of mass-optimal low-thrust and two-impulse solutions for all Earth – Asteroid 1 – Asteroid 2 – Asteroid 3 sequences. ................................79 Figure 34 Asteroid sequences identified by applying branch-and-bound algorithm to small sample problem. .....................................................................................82 xii
Figure 35: Low-thrust optima as a function of the normalized sum of the pruning metrics (branch-and-bound ranking), for the small sample problem ...........................84 Figure 36 Set of asteroids for intermediate sample problem............................................94 Figure 37 Asteroid sequences remaining in design space after 1st pruning metric (blue) and 2nd & 3rd pruning metrics (pink). ............................................................97 Figure 38 Maximum final mass for each asteroid pairing as a function of the angle between the two angular momentum vectors...................................................98 Figure 39 Maximum final mass for each asteroid combination as a function of the minimum, phase-free, two-impulse ∆V...........................................................98 Figure 40 Branch-and-bound tree enumerating all asteroid sequences remaining in the intermediate sample problem after the pruning phase. ..................................100 Figure 41 Branch-and-bound tree illustrating asteroid sequences pruned out by the first iteration of the branch-and-bound algorithm on the sample problem............100 Figure 42 Results of the 1st iteration of the branch-and-bound algorithm applied to the intermediate sample problem.........................................................................101 Figure 43 Results of the 2nd iteration of the branch-and-bound algorithm applied to the intermediate sample problem.........................................................................102 Figure 44 Asteroid sequences identified by applying branch-and-bound algorithm to intermediate problem. ....................................................................................103 Figure 45 GTOC3 set of asteroids..................................................................................106 Figure 46 Branch-and-bound results, iteration #1 (impulsive multiplier = 1). ..............111 Figure 47 Branch-and-bound results, iteration #2 (impulsive multiplier = 1.077). .......112 Figure 48 Branch-and-bound results, iteration #3 (impulsive multiplier = 1.096). .......113 Figure 49 Comparison of optimum impulsive final mass (multiplied by 1.096) and optimum low-thrust final mass for modified GTOC3 problem.....................115 Figure 50 GTOC2 set of asteroids..................................................................................118 Figure 51 Set of asteroids for modified GTOC2 problem..............................................120 Figure 52 Optimal impulsive solutions, with and without the time of flight restriction, for the modified GTOC2 problem. ................................................................123 Figure 53 Results of branch-and-bound algorithm applied to modified GTOC2 problem. .........................................................................................................125 xiii
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: LIST OF FIGURES Figure 1 Mars round
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
Page 63 and 64:
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
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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,
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[78] Cosmic Vision: Space Science f
Page 209:
time, she enjoys traveling as well
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