DEIMOS SPACE SOLUTION TO THE 3 rd GLOBAL TRAJECTORY OPTIMISATION COMPETITION (GTOC3)

1. 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

2. 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

3. 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:

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. Best Solution Found

12. Best solution: Full trajectory

13. Best solution: Distances

14. Best solution: Mass

15. Best solution: Thrust components

16. 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

17. 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

18. 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