GTOC 7 Team 12 Solution

Transcription

1 GTOC 7 Team 12 Solution Telespazio Vega Deutschland GmbH (Germany) Holger Becker, Gianni Casonato, Bernard Godard, Olympia Kyriopoulos, Ganesh Lalgudi, Matteo Renesto Contact: bernard <dot> godard <at> gmail <dot> com This document describes the optimization strategy employed by team 12 in the 7th Global Trajectory Optimization Competition (GTOC7). We start with presenting the assumptions we have made to prune the search space. This is followed by an outline of the global and local optimisation methods leading to the final proposed trajectory. Finally a description of the best trajectory found by the team is given. 1 Search space pruning Probes are assumed to be in rendez vous conditions with an asteroid at release and capture by the mothership. Due to the limited number of impulses and the low efficiency of the mothership engine, it is not possible to have all probes be released and captured at different asteroids, which means that some probe(s) will begin or end their trip in rendez vous at an asteroid which is also visited by another probe (of course only one probe is requested to stay 30 days in rendez vous conditions with that asteroid). To prune the search space, initially a graph was generated with all asteroids as nodes. A path was added between any two distinct asteroid pairs if the asteroid orbital plane relative inclination and semi major axis differences were below a certain threshold (1 degree, 0.1 AU). The graph was then pruned of all nodes with less than a certain amount of neighbours (100). This left only approximately 1000 asteroids with orbital periods of about 5 years. In retrospect, it seems this pruning was too strict, since allowing only small semi major axis differences between start and end of of each transfer arc makes it difficult to adjust orbital phase quickly enough. Then all paths in the pruned graph were removed and regenerated using a different approach: a path was added between any two distinct asteroid pairs if a ballistic transfer in one year with a total delta velocity cost (from Lambert solution) below a certain threshold could be found between those 2 asteroids during the mission lifetime. These paths were later used during the asteroid sequence generation as candidates for next asteroids given a current asteroid to avoid looping over the larger list of 1000 pruned asteroids.

2 The Mothership transfer to the first asteroid was computed with a Lambert solver with 2 impulses. One impulse is fired at launch since the launcher does not provide enough delta velocity. Thus the additional required delta velocity is obtained by a Mothership engine burn. The other impulse is fired at the first asteroid rendezvous. 2 Asteroid sequence search 2.1 Probes solution using RNNA Repetitive Nearest Neighbour Algorithm (RNNA) is a greedy algorithm able to find a sub optimal solution of a Traveling Salesman Problem (TSP). The idea is to use all nodes as starting points and to compute from each of them the sequence of asteroids choosing at each step the shortest node/asteroid, based on a problem specific distance criteria. The initial time for each starting asteroid is selected based on a mothership launch to asteroid rendezvous optimization. For GTOC, the TSP has the characteristic to be time dependent, as asteroids distances vary with time. The RNNA adaptation to GTOC performs then the search starting from a predefined set of asteroids and trying to find at each step the closest asteroid in term of delta velocity for the transfer. Additionally initial and final asteroids can be imposed so as to come back to the mothership at the end. The core of the RNNA is the identification, at each search step, of the closest asteroid starting from the current one. To do that taking into account the delta velocity and the flight time limits, the concept of distance is translated into a combined index calculated aggregating both of those state variables. The minimisation performed on such index guarantee a trade off in the selection of the next asteroids, trying in this way to balance the consumptions of both of them. 2.2 Probes solutions refinement using Simulated Annealing The results obtained with the RNNA method is used as input to an heuristic algorithm based on simulated annealing technique for trying to refine and improve the initial solution. Simulated annealing is a random search technique which exploits an analogy between the way in which a metal cools and freezes into a minimum energy crystalline structure (the annealing process) and the search for a minimum in a more general system; it forms the basis of an optimisation technique for combinatorial and other problems. Simulated annealing approaches the global maximisation problem using a controlled iteration loop where at each stage a certain set of bounces in the solution are allowed, exploring in that way alternative solutions not directly reachable from the initial one. The loop control is based on the solution temperature concept, beginning at a high "temperature" which

3 enables the ball to make very high bounces, and decreasing to lower temperatures, with less probability to make solution bounces. The possible changes in the solution are produced by a certain generating distribution typically built using problem specific knowledge. An acceptance distribution is defined as well, for deciding on the basis of probability if a new solution shall be accepted. Both the generating and acceptance distributions depend on the temperature. For the GTOC7 TSP, the generating distribution introduced try both to modify a certain asteroids sequence without adding new ones, and to completely scratch a part of it and replace it with a new asteroid sequence. More specifically the change trials which are performed are: swap position for a pair of asteroids and recalculate the asteroids arrival and departure times according to the new flight time values replace an asteroid in a solution sequence with another one picked up from the search basket, and update the overall solution, trying also to append further asteroids scratch a solution starting from a certain asteroid and start creating a completely new one from that point on using RNNA, adding other asteroids picked up from the search basket 2 Low thrust trajectory generation The low thrust trajectory were generated arc by arc using an indirect optimization code. 3 Software used: The low thrust optimiser was written in Java for GTOC5. It is an adaptation of an algorithm from Bryson s book Dynamic Optimization. The implementation uses the open source mathematics library Apache Commons Math. In general, a lot of open source software was used such as PyGMO, PyKEP, Scipy and their many dependencies. PyGMO and PyKEP were used for example in finding a 3 impulses transfer for mothership from initial asteroid (where probes are released) to final asteroid (where probes are captured). 4 The final solution Our final solution starts with a ballistic transfer to asteroid from Earth using one engine burn at Earth to add to the launcher velocity. Then a burn is made at asteroid to rendezvous with the asteroid where the probes are released. Then a 3 impulses transfer brings the mothership to rendezvous with asteroid 7393 where it will later collect the probes. Only probe 1 stays 30 days at asteroids and 7393.

4 The solution is summarized in the table and figures below. The symbol in the plots represents a probe release while the symbol represents a probe capture. The times when either mothership or probes are in rendezvous conditions with an asteroid are marked with thicker lines. launch date (Modified Julian Date) escape velocity (launcher only not including main engine simultaneous burn) in meters per seconds J2000 ecliptic probe 1 release date (Modified Julian Date) probe 2 release date (Modified Julian Date) probe 3 release date (Modified Julian Date) probe 1 capture date (Modified Julian Date) probe 2 capture date (Modified Julian Date) probe 3 capture date (Modified Julian Date) probe 1 visited asteroids (minimum stay of 30 days) probe 2 visited asteroids (minimum stay of 30 days) probe 3 visited asteroids (minimum stay of 30 days) 10945, 5033, 11092, 1042, 1883, , 10975, 14256, , 2090, 9727, 2087, 9712 probe 1 final mass (kg) probe 2 final mass (kg) probe 3 final mass (kg) mother ship final mass (kg) with recovered probes primary performance index 15 secondary performance index (kg)

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