Automated Asteroid Selection for a 'Grand Tour' Mission 1 Introduction
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58th International Astronautical Congress , Hyderabad, India, 24 - 28 September 2007. Copyright IAF/IAA. All rights reserved. 57th International Astronautical Congress, Paper IAC-07-C1.7.07 Automated asteroid selection for a 'grand tour' mission Dario Izzoy, Tam¶asVink¶oy, Claudio Bombardelliy, Stefan Brendelbergery, Simone Centuoriy yESA, Advanced Concepts Team, ESTEC Keplerlaan 1, Postbus 299,2200 AG, Noordwijk contact: [email protected] Abstract In designing a mission to visit multiple asteroids and comets ¯nding the optimum choice of the asteroid sequence and computing optimum trajectories linking di®erent small bodies represent a brand new problem in the ¯eld of astrodynamics and trajectory optimisation. After introducing a general description of the multiple asteroid rendezvous problem using low thrust propulsion we propose a possible solution approach divided in three phases: a combinatorial optimisation phase, a global optimisation and local optimisation phase. This solution approach is applied to the particular case of the asteroid grand tour problem assigned at the 2nd edition of the Global Trajectory Optimisation Competition. The results obtained are presented and discussed. 1 Introduction oids were studied by di®erent authors. Brooks and Hampshire [1], for instance, derived multiple-asteroid Missions to asteroids are currently at the centre of the fly-by impulsive trajectories based on geometric dis- attention of national space agencies, academic com- tance between known asteroids and randomly gener- munities and media. Deep Impact, Hayabusa, Dawn ated keplerian trajectories. Bender and Bourke [2] are just famous and successful names among a larger later analysed a similar problem for the case of so- number of ongoing e®orts aimed at improving our lar electric propulsion. Perret et al. [3] performed a understanding of the potential bene¯ts and threats systematic search for multiple asteroid fly-bys using a asteroids o®er to human kind. A number of missions branch-and-prune criterion for limiting the maximum aimed at obtaining the largest possible set of informa- deltaV. These studies did not consider the possibil- tion on asteroids are being studied at di®erent detail ity of randezvous with multiple asteroids due to the levels, from conceptual (e.g. SWARM and ANTS) to much higher deltaV requirements. On the other hand phase-A design (e.g. Don-Quijote) to operations (e.g. later advances of electric propulsion technology made Dawn). From a mission design point of view, asteroid this option feasible and a number of authors started mission design is strongly driven by the initial aster- addressing di®erent aspects of multiple asteroid ran- oid choice, often constrained by scienti¯c rationales, dezvous missions. For example Bender and Friedlan- but otherwise left free to the mission designer. Given der [4]... Morimoto et al. [5] considered a multiple the large number of asteroids and comets known so rendezvous mission in which the sequence of asteroids far when this additional degree of freedom is added to be visited was not selected a priori and employed to the trajectory optimisation process the dimension a genetic algorithm to the trajectory design and opti- of the search space increases considerably. Further- misation. On the other hand the number of candidate more, when the mission objective is to rendezvous asteroids considered was rather low (21 asteroids in with more than one asteroid the problem dimensions total) making it possible, in principle, to analyse all grows enormously. possible sequences in a reasonable amount of time. In Past studies were focused on the problem of multi- this article we deal with the most general case of opti- ple asteroid fly-bys. Missions like Rosetta and Cassini mising a multiple asteroid rendezvous mission (using were designed in order to obtain fast flybys of a either chemical or electric propulsion) in which N as- number of small bodies as secondary science mission teroids have to be visited starting from a relatively goals, while low-cost missions whose primary objec- large sample of n asteroids while maximising a given tive was to fly-by with the highest number of aster- performance index J. For this case the large number 3 The Solution Approach of possible asteroid combinations makes enumerative searches unreasonable and a method has to be de- A solution approach for the asteroid grand tour opti- vised in order to quickly select the most promising misation problem is now presented. We make here sequences. After de¯ning the optimisation problem the assumption that no planetary fly-bys are em- in a rather general way we propose a solution strat- ployed which considerably reduces the complexity of egy aimed at providing a number of optimised chem- an already di±cult problem. The method is gen- ical trajectories in a fully automated way. Thereafter eral enough to accommodate both impulsive and low we describe the process from which low-thrust opti- thrust propulsion. mised trajectory can be obtained. Finally we apply The overall optimisation strategy proposed can be our solution strategy to multiple asteroid rendezvous summarised in 3 steps. problem proposed by the Jet Propulsion Laboratories First, a combinatorial optimisation is performed (JPL) for the 2nd edition of the Global Trajectory based on a generalised distance between asteroids or- Optimisation Competition. The method and results bits and providing a number of candidate sequences. obtained are discussed and suggestions are given for Second, for each candidate sequence an optimised im- future improvements. pulsive trajectory is derived based on a global optimi- sation method. Lastly, if electric propulsion is consid- ered, the best trajectory obtained are taken as initial guess for the local optimisation phase. One of the rea- sons to select this particular strategy was to create a tool that can quickly provide a relatively large num- ber of sequences in a (ideally) fully automated way 2 Problem De¯nition without resorting to astrodynamics considerations to direct the search for promising trajectories. The multiple asteroid rendezvous problem (or 1. Combinatorial Optimisation ¶asteroidgrand tour' problem) can be formulated as follows. Given n sets Ai, i = 1; : : : ; n of asteroids not 2. Global Optimisation necessarily disjointed, each of them containing mj as- teroids, consider a generic trajectory which launches 3. Local Optimisation from Earth and subsequently performs a rendezvous of duration Tj with one (not already visited) asteroid During the combinatorial optimisation a number from each of the N sets, in general without a pre- of 'interesting' asteroid sequences are selected. For ferred sequential order. The propulsion system em- each given sequence, a global optimisation problem is ployed (chemical, low-thrust, solar sail, etc.) will be solved in order to locate a good phasing between the speci¯ed by the problem and gravity assist with solar asteroids assuming impulsive deltaV maneuvers. Fi- system planets may or may not be permitted. After nally, if a low-thrust propulsion startegy is requested, de¯ning a suitable performance index J a trajectory a ¯nal optimisation step is performed whereby the is sought which maximises J while subject to a num- most promising solutions found on step 2 are used ber of given constraints (e.g. launch window, maxi- as initial guess to be re¯ned with a local optimisa- mum thrust, etc.). The reason for grouping di®erent tion technique in order to meet the constraints im- asteroids into di®erent sets comes from the need to posed by the problem description. Note that if low- maximise the scienti¯c return of a space mission. For thrust propulsion is considered the global optimisa- example one can choose to place asteroids of di®erent tion phase has to be tailored in order to provide suit- taxonomic classes in di®erent sets so that each class able solutions for carrying out the local optimisation of asteroid is visited. step. 3.1 Combinatorial Optimisation thrust transfer with maximum thrust Fmax). Many choices of d(A ;A ) are possible and generally lead- Let us introduce n sets A , i = 1; : : : ; n containing i j i ing to di®erent sequence ranking and the quest for the the di®erent asteroid groups member and the set A 0 best formulation is the subject of an ongoing research containing only one element, the Earth. Let us then e®ort by the authors. introduce a generalised distance d : Ai £ Aj ! R, i; j = 0; : : : ; n between each pair of asteroids1. At this point the following problem is considered. Choose 3.2 Global Optimisation one element Ai 2 Ai in each set and a permutation s of f1; 2; ::; ng so that The de¯nition of the generalised distance between as- teroid pairs used to select a ¯rst sample of candidate nX¡1 sequences is not at all su±cient to guarantee that the Jcomb := d(A0;As(1)) + d(As(i);As(i+1)) latter will lead to good trajectories. This is because i=1 it cannot account for the phasing between subsequent is minimised. asteroid pairs. As a matter of fact the influence of the This problem is a combinatorial optimisation prob- phasing can have a large impact on the ¯nal merit lem that can be tackled, in general, with a number function as it will be clear from the numerical exam- of di®erent techniques both stochastic and determin- ple reported later on. One possible approach to ad- istic. When the order of visit for the di®erent groups dress the phasing issue is to introduce simpli¯ed dy- is left free the number of all possible combinations is namical models representing impulsive transfer tra- given by: jectories for each asteroid sequence and optimise a Yn given trajectory decision vector using global optimi- N := n! m c j sation techniques. This approach not only returns i=1 asteroid sequences which have good phasing charac- It is easy to see that when the number of aster- teristics but can provide, when properly and care- oids to be visited is ¸ 4 already with a number of fully tuned, good initial guesses for a subsequent low- asteroids of the order of 100 for each group the re- thrust optimisation.