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Read about the problem statement, DEIMOS.Space team, asteroid family analysis, and solution steps taken during the 3rd Global Trajectory Optimisation Competition.
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DEIMOS SPACE SOLUTION TO THE 3rd GLOBAL TRAJECTORY OPTIMISATION COMPETITION (GTOC3) Miguel Belló, Juan L. Cano Mariano Sánchez, Francesco Cacciatore DEIMOS Space S.L., Spain
Contents • Problem statement • DEIMOS Space team • Asteroid family analysis • Solution steps: • Step 0: Asteroid Database Pruning • Step 1: Ballistic Global Search • Step 2a: Gradient Restoration Optimisation • Step 2b: Local Direct Optimisation • DEIMOS solution presentation • Conclusions
Problem Statement • Escape from Earth, rendezvous with 3 asteroids and rendezvous with Earth • Depature velocity below 0.5 km/s • Launch between 2016 and 2025 • Total trip time less than 10 years • Minimum stay time of 60 days at each asteroid • Initial spacecraft mass of 2,000 kg • Thrust of 0.15 N and Isp of 3,000 s • Only Earth GAMs allowed (Rmin = 6,871 km) • Minimise following cost function:
DEIMOS Space Team • Miguel Belló Mora, Managing Director of DEIMOS Space, in charge of the systematic analysis of ballistic solutions and the reduction to low-thrust solutions by means of the gradient-restoration algorithm • Juan L. Cano, Senior Engineer, has been in charge of the low-thrust analysis of solution trajectories making use of a local optimiser (direct method implementation) • Francesco Cacciatore, Junior Engineer, has been in charge of the analysis of preliminary low-thrust solutions by means of a shape function optimiser • Mariano Sánchez, Head of Mission Analysis Section, has provided support in a number of issues
Asteroid Family Analysis • Semi-major axis range: [0.9 AU-1.1 AU] • Eccentricity range: [0.0-0.9] • Inclination range: [0º-10º] • Solution makes use of low eccentricity, low inclination asteroids
Step 0: Asteroid Database Pruning • To reduce the size of the problem, a preliminary analysis of earth-asteroid transfer propellant need is done by defining a “distance” between two orbits • This distance is defined as the minimum Delta-V to transfer between Earth and the asteroid orbits • By selecting all asteroids with “distance” to the Earth bellow 2.5 km/s, we get the following list of candidates: • 5, 11, 16, 19, 27, 30, 37, 49, 61, 64, 66, 76, 85, 88, 96, 111, 114, 122 & 129 • In this way, the initial list of 140 asteroids is reduced down to 19 • Among them numbers 37, 49, 76, 85, 88 and 96 shall be the most promising candidates
Step 1: Ballistic Global Search • The first step was based on a Ballistic Scanning Process between two bodies (including Earth swingbys) and saving them into databases of solutions • Assumptions: • Ballistic transfers • Use of powered swingbys • Compliance with the problem constrains • This process was repeated for all the possible phases • As solution space quickly grew to immense numbers, some filtering techniques were used to reduce the space • The scanning procedure used the following search values: • Sequence of asteroids to visit • Event dates for the visits • An effective Lambert solver was used to provide the ballistic solutions between two bodies
Step 1: Ballistic Global Search • Due to the limited time to solve the problem, only transfer options with the scheme were tested: E-E–A1–E–E–A2–E–E–A3–E–E • All possible options with that profile were investigated, including Earth singular transfers of 180º and 360º • The optimum sequence found is: E–49–E–E–37–85–E–E • Cost function in this case is: J = 0.8708 • This step provided the clues to the best families of solutions
Step 2a: Gradient Restoration Optimisation • A tool to translate the best ballistic solutions into low-thrust solutions was used • A further assumption was to use prescribed thrust-coast sequences and fixed event times • The solutions were transcribed to this formulation and solved for a number of promising cases • Optimum thrust directions and event times were obtained in this step • A Local Direct Optimisation Tool was used to validate the solution obtained
Best Solution Found • Final spacecraft mass: 1716.739 kg • Stay time at asteroids: 135.2 / 60.0 / 300.3 days • Minimum stay time at asteroid: 60 days • Cost function • Solution structure: • Mission covers the 10 years of allowed duration • Losses from ballistic case account to a 0.05% E – TCT – 49 – TC – E – C – E – TCT – 37 – TCT – 85 – TC – E – CTCT – E
Best solution: From Earth to asteroid 37 Segment Earth to asteroid 49: • E–TCT–49 • 2½ revolutions about Sun • Duration of 1,047 days • Segment asteroid 49 to 37: • 49-TC-E-C-E-TCT-37 • 2½ revolutions about Sun • Duration of 852 days
Best solution: From asteroid 37 to Earth Segment asteroid 37 to 85: • 37–TCT–85 • 1¼ revolutions about Sun • Duration of 450 days • Segment asteroid 85 to Earth: • 85–TC–E–CTCT–E • 2½ revolutions about Sun • Duration of 836 days
Conclusions • Use of ballistic search algorithms seem to be still applicable to provide good initial guesses to low-thrust trajectories even in these type of problems • Such approach saves a lot of computational time by avoiding the use of other implementations with larger complexity (e.g. shape-based functions) • Transcription of ballistic into low-thrust trajectories by using a GR algorithm has shown to be very efficient • Failure to find a better solution is due to: • The a priori imposed limit in the number of Earth swingbys (best solution shows up to 3 Earth-GAMs) • Non-optimality of the assumed thrust-coast structures between phases