Evolving Solutions To Tsp Variants For Active Space Debris-PDF Free Download

Evolving solutions to TSP variants for active space debris
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erally to problems where the travelling salesman needs to number of fragmentation debris. select which cities to visit within a given budget and in a at breakup 1 2015 1 2025. scenario where the cities are moving We create a number of Iridium 33 598 368 191. test instances of these problems considering orbiting debris Cosmos 2251 1603 1128 781. clouds created by past fragmentation events and we solve Fengyun 3312 2651 2363. them applying an evolutionary technique the inver over al. gorithm 7 and the Series Method approach 4 We also Table 1 The cloud size slowly decreases as debris orbit nat. formalize the very same problems as tree search problems urally decay with time. and solve them using a breadth first search also known as. beam search with fixed frontier a popular approach in the. context of interplanetary trajectory design and yet never. used for the design of ADR missions Unlike previous work. 4 5 3 6 we consider large size debris collections, The rest of the paper is organized as follows In section. 2 we discuss the used debris cloud data and their orbital. characteristics Section 3 explains the algorithms used In. section 4 we explain how the problem of finding a trajectory. for an active debris removal mission can be modeled as a. variation of the classic TSP Further we test the performance. of the algorithms described in section 3 on a spacial static. case of the problem In section 5 we consider the debris. pieces as moving objects We analyse this dynamic problem Figure 1 Orbital parameters for an Earth bound orbit The. and again discuss the performance of our algorithms RAAN is denoted by the argument of perigee by and. the orbital inclination by I or i in the text, 2 THE DEBRIS CLOUDS. In this section we introduce the orbital characteristics of putation of orbital transfers A large contribution of such a. three different debris clouds i e sets of orbiting space de cost derives from the relative orbital inclination which can. bris originated from some past fragmentation event We be computed as. will be using these to create our test instances We consider. the database of all orbiting objects visible to radar observa cos i cos i1 cos i2 sin i1 sin i2 cos 1 cos 2. tions1 The orbital elements for each object in the database sin i1 sin i2 sin 1 sin 2 1. in the two line element TLE format are periodically re. leased by NORAD as to maintain a reasonable prediction It is the orbital transfer cost that is the equivalent of the city. capability on the position of all space objects The released distance in classic TSPs debris orbits having a large i are. elements can then be used in a SGP4 orbital propagator 8 thus to be considered as distant while debris orbits having. to predict the future position and velocity In order to build a small i are to be considered as possibly close Since we. some fictitious test scenarios we select three separated sets will eventually want to visit and thus remove the largest. of orbiting objects deriving from distinct recent catastrophic pieces of debris we also show in Figure 2 the dimension of. collision events and containing the resulting fragmentation each fragment of the debris cloud as measured from radar. debris We consider the 2009 Iridium 33 Cosmos 2251 col observations and extracted from the SATCAT catalogue. lision and the 2007 Fengyun 1C event caused by a Chinese. anti satellite missile test These fragmentation events cre 3 METHODOLOGY. ated debris clouds that are a daily threat to all active satel A classic TSP can be formulated as the problem of finding. lites sharing the same orbital environment In Table 1 we a minimum weight Hamiltonian path in a complete weighted. show the number of fragmentation debris belonging to each graph G V W where V is the set containing the ver. cloud at three different epochs at breakup as of 01 2015 texes cities and W V V R is a map that associates. and as of 01 2025 decayed orbits were predicted using the to each ordered vertex pair a positive real number called edge. SGP4 propagator Due to the Earth s gravitational field weight and representing the distance between cities. and the effect of the residual atmospheric drag the orbits Here we define a TSP variant as an optimisation problem. of the debris change with time and some of them are nat concerned to find the best possible Hamiltonian path in a. urally removed by decaying in the atmosphere In Figure graph with respect to some predefined criteria We indicate. 2 we show for the three debris clouds the distribution of with n the cardinality of V i e the number of cities and we. the orbital parameters see Figure 1 for their definition as label each vertex in V with an increasing natural number. of 01 2015 On the left column we show the Gabbard dia i e 1 2 n A Hamiltonian path is then to be encoded. gram widely used to characterize fragmentation events 9 by a permutation of the set containing the first n natural. and showing the perigee and apogee of all fragments against numbers Note that in a classic TSP cycling a chosen per. their orbital period On the right column we show the dis mutation does not change the fitness function i e the path. tribution of inclinations i versus the Right Ascension of length but in the more complex variants introduced here. the Ascending Node RAAN The importance of the this no longer needs to be the case For example 1 2 3 can. and i distributions derive from their importance in the com have a different fitness with respect to say 3 1 2. The SATCAT catalogue can be downloaded from http, celestrak com pub satcat txt. Figure 3 Best Hamiltonian tours found for the three debris clouds in the classic TSP variant Each node represents an orbital. debris Node positioning given by a spring model see text for details As the problems are all non metric edges can cross. The color of each node is related to the RAAN of the debris it is representing. without mutation the algorithm constructs an offspring by. only using edges from the previous generation Consider. as an example the first iteration of the for loop given the. following fictitious population of a five city problem. S1 1 5 2 3 6 4 7 S2 3 6 1 5 7 4 2, S3 2 5 4 7 6 3 1 S4 7 2 3 6 1 4 5. S5 4 1 3 2 7 6 5, Since we look at the first iteration of the for loop.
Stmp S1 1 5 2 3 6 4 7, We select randomly a city from Stmp lets say c1 5 Next. assume our sampled uniformly distributed random number. is larger than the mutation probability ri Therefore we. randomly select an individual say S5 4 1 3 2 7 6 5 and. look which city is following 5 In our case that would be. c2 4 since in the case of a closed TSP each individual is. represented as a circular array Since 5 is not an adjacent. city of 4 in Stmp we invert the segment from the city next. to 5 namely 2 to city 4 We obtain, Figure 2 Distribution of the orbital parameters for the three. debris clouds as of 01 2015 2009 Iridium 33 top Cosmos Stmp 1 5 4 6 3 2 7. 2251 mid and Fengyun 1C event bottom The marker We proceed by setting c1 to the city that followed c2 4 in. size is proportional to the debris size as measured by radar Stmp before we inverted namely 7 and by going back to the. observations from SATCAT catalogue beginning of the repeat loop Assume that we again do not. chose c2 randomly Therefore we select a random individual. say S3 2 5 4 7 6 3 1 The city following c1 7 is 6, 3 1 Inver over algorithm We set c2 6 and observe that c2 6 occurs before c1 7. The Inver over operator is a well established evolutionary in Stmp In contrast to the previous repeat loop iteration. technique that provides a powerful genetic operator for clas where c2 was in front of c1 in Stmp we invert the section. sic TSPs 7 We used a slightly different version of the al from c2 to the city coming before c1 namely 2 We get. gorithm with respect to the originally proposed Inver over Stmp 1 5 4 2 3 6 7. The version we implemented takes care of not making in. version of sections including the edge from the last city to For the following repeat loop iteration we obtain c1 4. the first one a modification that improves considerably per That is the city that comes previous to 6 before the inversion. formances in the TSP variant that are not invariant to cy Assume our random generator returns a number larger than. cling The algorithm s pseudo code is sketched in Algorithm ri We randomly select an individual from the population. 1 The termination condition could be a maximum amount say S4 7 2 3 6 1 4 5 and check which city comes next. of generations and ri denotes the chance of a random in to c1 4 It is city 5 We set c2 5 and after observing. version and is thus called mutation probability Note that that c2 is already next to c1 in Stmp we exit the repeat loop. Initialization of the population P 3 3 A Tree Search approach. while termination condition is not satisfied do The design of a mission for the removal of multiple or. for each individual Si P do biting debris is remarkably similar from a trajectory design. Stmp Si point of view to the design of missions involving multiple. select randomly a city c1 from Stmp asteroids rendezvous or fly bys These missions have been. repeat the subject of extensive research in the past decade and in. if rand ri then particular the design of their trajectories defined most of. select randomly a city c2 6 c1 from Stmp the problems released to the international community dur. else ing the Global Trajectory Optimisation Competitions2 All. select randomly an individual S 6 Si of the solution methods of the most successful teams in. from P these competitions were based on standard tree searches. select the city that follows c1 in S and breadth first with fixed frontier depth first with branch. name it c2 pruning etc We thus implemented a simple breadth first. end tree search with fixed frontier size F S i e only the best F S. if c2 is adjacent c1 in Stmp then solutions found at each depth are kept for further branch. exit from repeat loop ing to search for solutions to our TSP variants While in. else the classic TSP case this algorithm is not likely to be compa. if c2 comes after c1 in Stmp then rable to known approaches in the TSP variants introduced. invert section from the city that here where not all cities are scored it actually may pro. comes after c1 until c2 vide an efficient mechanism to select which cities to visit A. c1 gets assigned the city that was frontier size F S 10 is considered for all tests performed. moved to the position of c2 by the, previous inversion. else 4 STATIC CASE, invert section from c2 until the city How is an active debris removal ADR mission related to.
that is previous to c1 the TSP In this section we formalize such a relation starting. c1 gets assigned the city that comes from the classic TSP and then introducing the City Selection. previous to c2 before the inversion TSP variant TSP CS Given a set V of orbiting debris any. was performed ADR mission will consist of one or more spacecraft that will. end have to visit all or part of the considered debris in order to. end de orbit them Such spacecraft will thus have to rendezvous. until and de orbit each selected debris in a given order When. if objfun Stmp objfun S then these orbits are considered as Keplerian ignoring all pertur. S Stmp bations acting upon them we speak of a static case as the. end orbital parameters a e i are considered as constants. end 4 1 ADR as a classic TSP, We assume that we can model the propellant cost to trans. Algorithm 1 Inver over algorithm used, fer between orbits as a function W V V R We con. sider one spacecraft with unlimited fuel budget this very. restrictive hypothesis will be discarded later when we will. and check if our newly generated Stmp represents a better introduce variants to the classic TSP and we ask ourselves. solution than S1 with respect to our objective function If what would the optimal visiting order be in order to min. so we replace S1 by Stmp The algorithm will proceed with imize the cumulative transfer cost Clearly this creates a. a new iteration of the for loop this time trying to evolve classic TSP where G V W The function W can be. S2 In our experiments we use a population size of 100 and computed with the aid of basic spacecraft mechanics compu. a mutation probability ri 0 05 We stop the algorithm tations and with different degrees of approximations Low. when no improvements are made to the objective function thrust transfers can be modelled using the Edelbaum ap. for more than 20000 iterations proximation 10 or equivalents while chemically propelled. 3 2 Nearest Neighbour Search transfers can be modelled at a preliminary mission design. phase using multiple impulse approximations In this pa. The nearest neighbour NN algorithm is well known in per we use the widely used three impulse approximation see. the context of classic TSPs It returns solutions using small 3 for example where the first and last impulses are used. computational effort and that are quite close to the optimal to match the apogee and perigee of the orbits and the mid. solution typically within some 20 Interestingly a sub impulse is used to cancel the orbits relative inclination i e. stantially equivalent version of this algorithm is called Series matching both i and Note though that our method. Method and has been used for the design of ADR missions ology can easily accommodate different choices The classic. 4 In trajectory design problems the computation of the TSP thus created is symmetric and . Evolving solutions to TSP variants for active space debris removal Dario Izzo Advanced Concepts Team European Space Agency Noordwijk The Netherlands Dario Izzo esa int Ingmar Getzner Advanced Concepts Team European Space Agency Noordwijk The Netherlands Ingmar Getzner esa int Daniel Hennes Advanced Concepts Team European Space Agency

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