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

More documents

Recommendations

Info

1000 900 Final Mass (kg) 800 700 600 500 400 300 200 100 Ballistic x 1.25 Low-Thrust 0 0 100 200 300 400 500 600 Asteroid Sequence # Figure 33: Comparison of mass-optimal low-thrust and two-impulse solutions for all Earth – Asteroid 1 – Asteroid 2 – Asteroid 3 sequences. The figures above indicate that as calculated, the two-impulse solutions do not provide a reliable upper bound. However, if the two-impulse solutions are shifted slightly, then they could provide an upper bound for the low-thrust solutions. For the sample problem, this would require a multiplication factor of 1.09, 1.14, and 1.21 for Leg 1, Leg 1 + Leg 2, and Leg 1 + Leg 2 + Leg 3, respectively. Of course, on a larger problem, this multiplication factor is not known a priori. Choosing a value of this parameter that is too small will result in a number of good sequences being pruned out during the branch-and-bound procedure. Choosing a value that is too large, however, will result in an unnecessarily large number of low-thrust trajectory optimizations to be carried out because very few asteroid sequences will be pruned. The solution is to iterate on the best value of this multiplication factor during the branch-and-bound procedure, as will be illustrated on the sample problem. Another important aspect of the branch-and-bound method is the order in which the branches are evaluated. Choosing this order in an intelligent fashion can significantly reduce the number of asteroid sequences that require low-thrust trajectory optimization. If an asteroid sequence with a high final mass is evaluated early in the process, then more branches will be pruned out than if all low mass sequences are evaluated initially. One way to choose the order in which to evaluate the various branches is based on the pruning 79
metrics which were calculated in the previous phase of the methodology. For each full asteroid sequence, the pruning metrics can be combined as follows, where the two pruning metrics are weighted equally: ∑ wedge,Legi ⎞ ⎛ ∆ ⎟ ( ) ⎟ ⎞ ⎟ ⎜ ⎜ ⎜ ∑ VLegi i ⎟ + 0.5 ⋅ θwedge ⎟ max( ∆V) ⎟ ⎝ ⎠ ⎛ θ ⎜ i Wi = 0.5⋅ ⎜ (18) ⎜ max ⎝ ⎠ For each asteroid sequence i, W i will fall between 0 and 1, with smaller numbers being better. The first sequence evaluated, which is used to set the initial lower bound, is chosen based on the results of Equation 18. The branches are then evaluated in sequential order based on the weighted combination of the pruning metrics. The first example applies the branch-and-bound method without any multiplier on the two-impulse solutions. For the sample problem, the first asteroid sequence evaluated is Earth – 2006 QQ56 – Medusa – Pandarus. The resulting optimal low-thrust solution is 638 kg. This becomes the current lower bound. The next asteroid sequence, based on the pruning metric rank, is Earth – 2006 QQ56 – Chicago – Pandarus. First, the optimal twoimpulse solution is calculated for the first leg: Earth – 2006 QQ56, yielding an optimal final mass of 1371 kg. Because this is greater than the current lower bound, the next level down must be calculated. The resulting two-impulse optimal solution for Earth – 2006 QQ56 – Chicago is 824 kg. Again, this branch can not be pruned. Finally, the twoimpulse optimal for the entire sequence, Earth – 2006 QQ56 – Chicago – Pandarus, is 736 kg. Because this is still greater than the current lower bound, the optimal low-thrust solution for this asteroid sequence must be calculated. This sequence does not yield a feasible solution (MBfB < 500 kg), and therefore, the previous lower bound remains the best known solution thus far. The third ranking asteroid sequence is Earth – 2006 QQ56 – Medusa – Kostinsky. The same process is carried out, and again the optimal two-impulse 80
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 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
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: is calculated using the same specif
Page 101 and 102: sorted by final mass. All of the se
Page 103 and 104: impulse optima. Figure 35 plots the
Page 105 and 106: CHAPTER III OVERVIEW OF METHODOLOGY
Page 107 and 108: (1) All asteroid sequences are rank
Page 109 and 110: two-impulse optimum solutions. Ther
Page 111 and 112: identifying another metric that cou
Page 113 and 114: CHAPTER IV VALIDATION OF METHODOLOG
Page 115 and 116: Table 8 lists the 10 best asteroid
Page 117 and 118: In order to further validate the pr
Page 119 and 120: impulse ∆V). The first iteration
Page 121 and 122: Because the impulsive multiplier ha
Page 123 and 124: Table 12: Effectiveness of the meth
Page 125 and 126: function calls to MALTO were requir
Page 127 and 128: order of 0.8 (assuming a final mass
Page 129 and 130: Table 16: Design variables for gene
Page 131 and 132: known solution to 1621 kg. Based on
Page 133 and 134: MBfB (kg) #1, #63, and #16, respect
Page 135 and 136: problem (Table 17). Additionally, t
Page 137 and 138: ≤ Inclination (deg) 60 50 40 30 2
Page 139 and 140: order to have a benchmark with whic
Page 141 and 142: Of the remaining sequences, the 1 s
Page 143 and 144: Table 22: Settings for the genetic
Page 145 and 146: 120 Impulsive Solutions Low-Thrust
Page 147 and 148: of exactly four weeks. As a benchma
Page 149 and 150:
Objective Function (kg/yr) 120 100
Page 151 and 152:
these two problems, the best known
Page 153 and 154:
asteroid sequences would be require
Page 155 and 156:
Table 27: Best known solutions rema
Page 157 and 158:
Table 29: Best known solutions rema
Page 159 and 160:
good solutions exist in the design
Page 161 and 162:
heuristics chosen were based on the
Page 163 and 164:
problem, the best set of solutions
Page 165 and 166:
Finally, for each set of inner loop
Page 167 and 168:
solutions from the GTOC2 competitio
Page 169 and 170:
sequences were optimized in low-thr
Page 171 and 172:
application is for the conceptual d
Page 173 and 174:
types of the TSP, they have not bee
Page 175 and 176:
methodology would be applied in the
Page 177 and 178:
APPENDIX A SET OF GTOC2 ASTEROIDS T
Page 179 and 180:
3054373 "2000 UK11" 0.88325596 0.24
Page 181 and 182:
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
Page 187 and 188:
2000496 Gryphia 2.1987751 0.079568
Page 189 and 190:
2001216 Askania 2.2322234 0.1793551
Page 191 and 192:
2000134 Sophrosyne 2.5632069 0.1166
Page 193 and 194:
2000712 Boliviana 2.5738464 0.18812
Page 195 and 196:
2005209 "1989 CW1" 5.1533221 0.0495
Page 197 and 198:
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
show all

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

Create successful ePaper yourself

Delete template?

Save as template?