Global Trajectory Optimization 3 How to fail 3

Global Trajectory Optimization 3 (How to fail 3 times in a row) Massimiliano Vasile Space Advanced Research Team Department of Aerospace Engineering Space • ART space advanced research team Dept. of Aerospace Engineering University of Glasgow www. gla. ac. uk/Research/Space. Art

1. The Team • Global Trajectory Optimization in Glasgow 2. Lessons Learnt from Previous GTOCs • • What went wrong To do list 3. GTOC 3 • • What we did What went wrong 4. New Year Resolutions Agenda Space • ART space advanced research team Dept. of Aerospace Engineering University of Glasgow www. gla. ac. uk/Research/Space. Art 2

Space. ART 1. The Team 2. Global Trajectory Optimisation in Glasgow Space • ART space advanced research team Dept. of Aerospace Engineering University of Glasgow www. gla. ac. uk/Research/Space. Art

• Space. ART and the Global Optimization Laboratory The whole Space. ART participated in different measure to the competition. The hard coding and computation involved mainly the following people in Glasgow: – – – – • Massimiliano Vasile, Edmondo Minisci Camilla Colombo Pau Sanchez Matteo Ceriotti Christie Maddock Daniel Novak And the following people in Turin and Florence: – – Bernadetta Addis Andrea Cassioli Marco Locatelli Fabio Schoen The Team Space • ART space advanced research team Dept. of Aerospace Engineering University of Glasgow www. gla. ac. uk/Research/Space. Art 4

Global Trajectory Optimization in Glasgow • One of the main streams of research in Glasgow is the analysis and development of methods for global single and multiobjective optimization. • We tested and developed a whole range of optimization methods both stochastic and deterministic, from pure EA to hybrid approaches. • In particular the relation between search heuristics and problem characteristics is of our interest. • The hope is that GTOCs are a way to quickly assess what works best. Why participating in GTOCs Space • ART space advanced research team Dept. of Aerospace Engineering University of Glasgow www. gla. ac. uk/Research/Space. Art 5

Lessons Learnt from Previous GTOCs 1. GTOC-1 2. GTOC-2 Space • ART space advanced research team Dept. of Aerospace Engineering University of Glasgow www. gla. ac. uk/Research/Space. Art

GTOC-1 • The first competition was proposing a particular instance of a multi-gravity assist trajectory problem in which the objective function was a combination of the impact velocity with an asteroid and of the mass of the impacting spacecraft. • Due to the objective function and of the limited number of celestial bodies involved, the identification of a potentially optimal strategy was relatively easy. • The challenging part was to find the optimal trajectory for each sequence of swing-bys • All the top 5 teams used essentially two methods to identify the optimal solution: systematic search and experience. • The trajectory model for the first estimation was essentially multi-impulsive • We classified first among the solutions with no retrograde orbit and among the solution that were identified with a stochastic global optimisation algorithm Problem and Solution Space • ART space advanced research team Dept. of Aerospace Engineering University of Glasgow www. gla. ac. uk/Research/Space. Art 7

GTOC-1 Main Points • The correct modelling of the problem was essential. We had a bug in the Lambert’s solver and all the retrograde orbits were wrong • A brute-force systematic search coupled with experience was the best optimization approach. No sophisticated global optimization heuristics, human experience more important than the algorithm. • Effective multi-impulsive first estimation for LT trajectories Minor Points • Always remember to say that you worked only a little, during the weed-end. • In the following the week-end will be the man-time unit: 1 w/e=1 man-week Lessons Learnt Space • ART space advanced research team Dept. of Aerospace Engineering University of Glasgow www. gla. ac. uk/Research/Space. Art 8

GTOC-2 • The problem was to randezvous with multiple asteroids minimising the following quantity: • The problem was conceptually different from GTOC-1: a very challenging combinatorial part and a relatively simple search for solutions for each combination. • The winning team used a simplified model and experience to identify the optimal sequence. Again the search was rather systematic and simple. • Some of the top teams reached good results with the use of EA of some sort but the use of a deterministic pruning based on a simplified model was essential. Problem and Solution Space • ART space advanced research team Dept. of Aerospace Engineering University of Glasgow www. gla. ac. uk/Research/Space. Art 9

GTOC-2 Main Points • As for GTOC-1 the use of a correct and appropriate model was essential. • No sophisticated global optimization heuristics • Human experience more important than the algorithm • Reduced model for preliminary estimation What went wrong • We were more than 20 in our team distributed in 3 groups • We spent almost three weeks to develop the model for us and for the other groups participating with us in our team • At 4 days from the deadline we found a bug in the ephemeris routine! • We imposed an excessively stringed limit on the maximum stay time at each asteroid. Minor Points • To win a global trajectory optimization, remember to have always a good local optimization tool to produce a solution with a 1 day step size. • A lot of people in the team could mean a big mess!!!! Lessons Learnt Space • ART space advanced research team Dept. of Aerospace Engineering University of Glasgow www. gla. ac. uk/Research/Space. Art 10

GTOC-3 1. The Problem 2. What we did 3. What went wrong Space • ART space advanced research team Dept. of Aerospace Engineering University of Glasgow www. gla. ac. uk/Research/Space. Art

GTOC-3 • The problem was to rendezvous with 4 asteroids minimizing the following quantity: • Where t was the stay time at each asteroid. Only swing-bys of the Earth were allowed. j • The problem was conceptually similar to GTOC-2: a challenging combinatorial part but with the added difficulty of the insertion of a swing-by of the Earth. The Problem Space • ART space advanced research team Dept. of Aerospace Engineering University of Glasgow www. gla. ac. uk/Research/Space. Art 12

Preliminary Estimation of the Global Optimum • In the ideal case, the maximum stay time can be obtained when the spacecraft stops over for 1/3 of the total admissible duration of the mission. • In this ideal case the contribution of the time component of the objective function would be 0. 0666(6). • On the other hand for a zero- v mission, the mass contribution to the objective function would be 1, which gives a total maximum value of 1. 066(6). • By computing the average value of the v required to reach the asteroids in the list, a lower bound on the objective function was estimated to be about 0. 7. • Given that a zero- v mission would be unrealistic, we expected an upper bound close to 1 which would correspond to less than 3 km/s for the whole round trip. WHAT WE DID Space • ART space advanced research team Dept. of Aerospace Engineering University of Glasgow www. gla. ac. uk/Research/Space. Art 13

Optimization Approach • Two different approaches were used to look for a solution: a systematic search and a stochastic based search. In both cases a simple trajectory model based on impulsive manoeuvres was used. • The total v for a generic transfer arc from a celestial body A to a celestial body B had to fulfil the following constraint: • where m 0 is the initial mass of the spacecraft at the beginning of the arc and TTOF is the time of flight of the transfer leg. • Although the problem allows for the use of Earth gravity assists, the search was mainly focused on direct transfers. WHAT Space. ART DID Space • ART space advanced research team Dept. of Aerospace Engineering University of Glasgow www. gla. ac. uk/Research/Space. Art 14

Systematic Search • Parallel exhaustive search for optimal and feasible bi-impulsive transfers between pairs of celestial bodies (E-A, A-E). – The feasible pairs were stored in a database. • Incremental composition of the whole roundtrip starting from the Earth and adding one transfer arc from the database at the time. – A constraint on the starting time of the added transfer arc was imposed, such that one transfer leg was starting between 0. 5 and 3. 5 years after the end of the previous transfer leg. • All the complete and feasible roundtrips were ranked according to the objective function of the competition and re-optimised with a low-thrust model. • The local optimizer for trajectory design called DITAN, WHAT Space. ART DID Space • ART space advanced research team Dept. of Aerospace Engineering University of Glasgow www. gla. ac. uk/Research/Space. Art 15

Systematic Search E E A 1 • • A 2 A 3 A 4 The value of each connected pair of nodes was computed in parallel as an instance of a constrained bi-impulsive transfer. Unpromising branches were pruned while building the tree. The source and sink of the graph was the Earth A complete path along the graph was a complete trajectory WHAT Space. ART DID Space • ART space advanced research team Dept. of Aerospace Engineering University of Glasgow www. gla. ac. uk/Research/Space. Art 16

Stochastic Search • • • Stochastic search of the entire roundtrip based on Monotonic Basin Hopping technique. Constraint on the maximum stay time of 60 days. Outer loop with combinatorial generation of sequences and inner loop with global search via MBH. COMB S xi MBH WHAT GOL DID Space • ART space advanced research team Dept. of Aerospace Engineering University of Glasgow www. gla. ac. uk/Research/Space. Art 17

Description of the Solution The selected sequence emerged initially from the systematic search and later on appeared also in the list of the stochastic method: Earth – 88 – 19 – 49 – Earth Launch date (MJD) 58812. 7187071516 Launch, hyperbolic excess velocity, v∞ (km/s) 0. 5 Total mission time (days) 3551. 814905518397 Value of the objective function, J (kg/year) 0. 8066214725096204 Results Space • ART space advanced research team Dept. of Aerospace Engineering University of Glasgow www. gla. ac. uk/Research/Space. Art 18

Our Pride The solution we selected has no swing -bys. The systematic search for solutions with swing-by was limited, however many direct transfers suggested the insertion of a flyby of the Earth. None of these solution was refined with low-thrust for lack of time Results Space • ART space advanced research team Dept. of Aerospace Engineering University of Glasgow www. gla. ac. uk/Research/Space. Art 19

First Leg WHAT WE DID Space • ART space advanced research team Dept. of Aerospace Engineering University of Glasgow www. gla. ac. uk/Research/Space. Art 20

Second Leg WHAT WE DID Space • ART space advanced research team Dept. of Aerospace Engineering University of Glasgow www. gla. ac. uk/Research/Space. Art 21

Third Leg WHAT WE DID Space • ART space advanced research team Dept. of Aerospace Engineering University of Glasgow www. gla. ac. uk/Research/Space. Art 22

Fourth Leg WHAT WE DID Space • ART space advanced research team Dept. of Aerospace Engineering University of Glasgow www. gla. ac. uk/Research/Space. Art 23

One Remark 1 Week-end Time to analyse and prepare the problem 1 Week-end 2 Week-ends Time to prepare Time to refine our solutions for the other with a time step of 1 day group Time for global optimization: 3 days 3 e 6 solutions • We found a bug in the ephemeredes during the second week: the bug was fixed • Problematic translation of the impulsive transfer solution to the LT model wasted the time required to re-optimise transfer with swing-bys. WHAT WENT WRONG Space • ART space advanced research team Dept. of Aerospace Engineering University of Glasgow www. gla. ac. uk/Research/Space. Art 24

New Year Resolutions Space • ART space advanced research team Dept. of Aerospace Engineering University of Glasgow www. gla. ac. uk/Research/Space. Art

HOW THE GTOCs ANSWER TO OUR QUESTIONS Major Points • The systematic approach was competitive against the more sophisticated stochastic method • The reduced model is still an essential component • A multi-impulse model can be effectivelly used for LT transfers Minor Points • Having a good local optimisation tool is essential to win a global optimization competition!!!! • The number of week-ends spent to coordinate with your mates should be minimal if not zero Space • ART space advanced research team Dept. of Aerospace Engineering University of Glasgow www. gla. ac. uk/Research/Space. Art 26

“When I was Research Head of General Motors and wanted a problem solved, I'd place a table outside the meeting room with a sign: LEAVE SLIDE RULES HERE! If I didn't do that, I'd find some engineer reaching for his slide rule. Then he'd be on his feet saying, ‘Boss you can't do that!’” - Charles F. Kettering (American engineer, Inventor of the electric starter, 1876 -1958) Questions? Space • ART space advanced research team Dept. of Aerospace Engineering University of Glasgow www. gla. ac. uk/Research/Space. Art