Results found by the CNES team (team #4)

Transcription

1 3 rd Global Trajectory Optimisation Competition (GTOC3) organized by the Aerospace Propulsion Group of the Dipartimento di Energetica at Politecnico di Torino Results found by the CNES team (team #4) Presented by Régis Bertrand Centre National d Etudes Spatiales (CNES) 18 av. Edouard Belin, F Toulouse, France Regis.Bertrand@cnes.fr

2 Conclusions Global Optimization = Flight Dynamics Considerations + Local Optimization Flight dynamics considerations for pruning the search space Efficient local optimization methods: indirect methods One selection rule was too much restrictive GTOC ranking of the CNES team: #6 in 2005, #4 in 2006, Thank you to the GTOC2 organizers for this very interesting problem and for the invitation to the workshop R. Bertrand, CNES, GTOC3 Workshop, June 27, 2008, Turin. 2

3 Outline The CNES team: Who are we? What are we working on? The GTOC3 problem Solution strategy and numerical methods Solution found Local optimization: a still open problem Conclusions R. Bertrand, CNES, GTOC3 Workshop, June 27, 2008, Turin. 3

4 CNES Toulouse Space Center Projects and engineering activities Earth observation satellites scientific missions telecommunication satellites ATV control center Photo credit: CNES Orbital maneuvers office flight dynamics activities orbit control and formation flying ATV mission analysis and operations studies on interplanetary missions Team members for GTOC3: Dr. Régis Bertrand Dr. Richard Epenoy Mr. Benoît Meyssignac Dr. Jean-Paul Berthias Mr. Jacques Foliard Mr. Flavien Mercier the place where the solution was found Photo credit: CNES R. Bertrand, CNES, GTOC3 Workshop, June 27, 2008, Turin. 4

5 Low-thrust trajectory optimization in CNES Since 1994 mission analysis team is involved in this topic (initiated by Dr. R. Epenoy) Collaboration with engineering schools (ENSEEIHT), universities (UPS-Toulouse) and private space companies (Thalès Alenia Space and Astrium) Several Ph.D. theses has been supported by CNES: minimum time trajectories around the Earth how to tackle constraints on the thrust direction low-thrust interplanetary trajectories homotopy techniques for bang-bang optimal control problems optimizing gravity assist sequences for interplanetary trajectories Post-doc grants supported by CNES: minimum time trajectories around the Earth with eclipse constraints minimum fuel deployment of satellites formation flying Internal studies are conducted: J2 perturbations Methods are applied to phase 0/A studies: Marco R. Bertrand, CNES, GTOC3 Workshop, June 27, 2008, Turin. 5

6 GTOC3 problem Objective function: Four rendezvous are mandatory, 3 with asteroids and one with the Earth Departure date: Maximum duration, τ max = 10 years Stay-times τ i=1,..,3 60 days Departure hyperbolic excess velocity less or equal than 0.5 km/s Gravity assists from the Earth are permitted, r p 6871 km K = 0.2 S/C characteristics: initial mass m i = 2000 kg, m f final mass I sp = 2000 s T = 0.15 N a 114-months thrust phase is possible (m f = 470 kg, J = ) Asteroids set 140 asteroids 66 Atens (a 1 AU) 74 Apollos (a 1 AU, q AU) 0.9 a 1.1 AU, uniformly spread 0 e 0.4, 80 % uniformly spread 0 i 10 deg, uniformly spread R. Bertrand, CNES, GTOC3 Workshop, June 27, 2008, Turin. 6

7 GTOC3 problem (cont d) Preliminary analysis before optimization Problem sensitivity J = m f m i = 5.10 J τ A 0.02 increase of J can be obtained by: a 40 kg increase of m f (i.e. a 594 m/s-bof Delta-V decrease) a 1 year increase of τ i (i.e. 1/3 of the mission duration is stay-time) The maximization of m f is crucial for this problem Optimization of stay-times will be done at a second step i = K τ max = 2.10 R. Bertrand, CNES, GTOC3 Workshop, June 27, 2008, Turin. 7

8 GTOC3 problem (end) Number of combinations (except Earth flyby): = Duration of the competition: 4 weeks = s For scanning the overall search space: 1 s of CPU time per case impossible with our methods (and computers!) Global search on a simplified problem R. Bertrand, CNES, GTOC3 Workshop, June 27, 2008, Turin. 8

9 How to find out this asteroid sequence? orbit of asteroid 49 orbit of asteroid 37 orbit of asteroid 85 Earth s orbit R. Bertrand, CNES, GTOC3 Workshop, June 27, 2008, Turin. 9

10 Solution strategy No selection with respect to the orbital elements! (lesson learnt from GTOC2) Two-step strategy Global search for the asteroid selection phasing is omitted almost ballistic transfers: limited number of impulsive maneuvers only one Earth flyby between two bodies Delta-V as optimization criterion Low-thrust optimization S/C characteristics and problem assumptions are taken into account key dates (departure, flyby and rendezvous) and stay-times are optimized a second Earth flyby is introduced into the mission scenario (if it is suitable) R. Bertrand, CNES, GTOC3 Workshop, June 27, 2008, Turin. 10

11 Global search Each leg was optimized separately (phasing is omitted) The following cases were inspected: Earth asteroid trajectories (140 cases) Earth Delta-VEGA asteroid trajectories (140 cases) asteroid asteroid (1 or 2 loops around the Sun) trajectories (38920 cases) asteroid Delta-VEGA asteroid trajectories (19460 cases) asteroid Earth trajectories (140 cases) Sequences were built Ranking with respect to the Delta-V value Phasing analysis: numerous sequences were rejected CNES software LOTH was used R. Bertrand, CNES, GTOC3 Workshop, June 27, 2008, Turin. 11

12 CNES software LOTH Goal: Optimize an interplanetary trajectory with Impulsive Deep Space Maneuvers Principle: Non Linear Parametric optimization problem with constraints Algorithms and techniques used: patched conics formulation Nelder-Mead simplex method multi-start technique exact penalties (in terms of mathematical programming) Criteria: Final Mass or Global Velocity Increment Delta-V Main Reference: T. Ceolin, J.M. Garcia, J.M Enjalbert, C. Brochet and J. Bernard, A General Method for Interplanetary Trajectory Optimization, 11 th IAS, May 1996, Gifu (Japan). R. Bertrand, CNES, GTOC3 Workshop, June 27, 2008, Turin. 12

13 Global search results sequence rejected due to bad phasing selected sequence Sequence E-E-19-E-37-E-49-E-E E-E-49-E E-E E-E E-37-E-E E-E-90-E-37-E-49-E-E E-E-96-E E-E E-E-5-E E-E E-E E-49-E-E E-E E-49-E-E E-E-122-E-37-E-49-E-E E-E-49-E-16-E-96-E-E E-E E-11-E-E E-E-37-E E-E E-E-19-E E-E E-E E-96-E-E E-E-49-E-37-E-90-E-E E-E-114-E-37-E-49-E-E E-E-49-E-37-E-96-E-E E-E E-37-E-E... Delta-V [km/s] R. Bertrand, CNES, GTOC3 Workshop, June 27, 2008, Turin. 13

14 Low-thrust optimization Each leg of the selected sequence is optimized by taking into account problem assumptions: S/C characteristics, departure constraints A second Earth flyby is introduced into the mission scenario in particular for the departure and arrival legs The overall trajectory is optimized by adjusting key dates (Earth flyby and rendezvous) and stay-times at each asteroid. This modifies the propellant consumption related to each leg Main issues: the optimal control is characterized by a large number of switches. This makes the numerical integration very sensitive the accuracy required is very high: 1000 km on position and 1 m/s on velocity CNES software ETOPH is used R. Bertrand, CNES, GTOC3 Workshop, June 27, 2008, Turin. 14

15 CNES software ETOPH Goal: Optimize a low-thrust interplanetary trajectory Principle: (Bang-Bang) optimal control problem with constraints Algorithms and techniques used: patched conics formulation Pontryagin Maximum Principle (PMP) and shooting methods continuation-smoothing techniques decomposition-coordination methods Criteria: Final mass or mission duration or a new criterion Main Reference: R. Bertrand and R. Epenoy, New Smoothing techniques for solving Bang-Bang Optimal Control Problems Numerical Results and Statistical Interpretation, Optimal Control Applications and Methods, Vol. 23, pp , R. Bertrand, CNES, GTOC3 Workshop, June 27, 2008, Turin. 15

16 Smoothing methods: example on an Earth to Venus transfer Source: R. Bertrand and R. Epenoy, New Smoothing techniques for solving Bang-Bang Optimal Control Problems Numerical Results and Statistical Interpretation, Optimal Control Applications and Methods, Vol. 23, pp , R. Bertrand, CNES, GTOC3 Workshop, June 27, 2008, Turin. 16

17 The GTOC3 solution found in details Total flight time [years] from Oct. 20, 2025 to Oct. 20, 2035 Total duration of thrust [year] 17% of cruise duration Final S/C mass [kg] 86.7% of launch mass Delta-V [m/s] from 423 to 1645 m/s (per arc) Stay-times [day] τ 1 = τ 2 = τ 3 60 Value of objective function R. Bertrand, CNES, GTOC3 Workshop, June 27, 2008, Turin. 17

18 Characteristics of Earth flybys Earth flyby v-inf [km/s] Perigee radius [km] Equivalent Delta-V V S/C (t + ) - V S/C (t - ) [m/s] #1 Dec. 6, #2 May 9, #3 Oct. 2, #4 Jul. 4, greater than the Delta-V performed by the S/C engine 5625 R. Bertrand, CNES, GTOC3 Workshop, June 27, 2008, Turin. 18

19 Earth to asteroid 49: departure v-inf = 0.5 km/s cruise duration 1099 days Delta-V = 423 m/s Delta-m = kg two Earth flybys asteroid 49 to asteroid 37: cruise duration 860 days Delta-V = 759 m/s Delta-m = kg one Earth flyby 1 2 asteroid 37 to asteroid 85: cruise duration 428 days Delta-V = 1381 m/s Delta-m = kg 3 4 asteroid 37 to Earth: cruise duration 1086 days Delta-V = 1645 m/s Delta-m = kg one Earth flyby R. Bertrand, CNES, GTOC3 Workshop, June 27, 2008, Turin. 19

20 Engine thrust vs. time of flight 17 different thrusting phases minimum duration 1 day maximum duration 140 days Total thrust duration 1.65 years (17 % of the cruise duration) mission around asteroid 49 mission around asteroid 37 mission around asteroid 85 R. Bertrand, CNES, GTOC3 Workshop, June 27, 2008, Turin. 20

21 Characteristics of the end-to-end trajectory Sun distance [AU] S/C velocity [km/s] mission around asteroid 49 mission around asteroid 37 mission around asteroid 85 R. Bertrand, CNES, GTOC3 Workshop, June 27, 2008, Turin. 21

22 Orbital elements Earth flyby #1 Earth flyby #4 Earth flyby #1 Earth flyby #4 Earth flyby #3 Earth flyby #2 Earth flyby #2 Earth flyby #3 semi-major axis [AU] eccentricity mission around asteroid 49 mission around asteroid 37 mission around asteroid 85 R. Bertrand, CNES, GTOC3 Workshop, June 27, 2008, Turin. 22

23 Orbital elements (cont d) Earth flyby #3 Earth flyby #4 Earth flyby #2 Earth flyby #4 Earth flyby #2 inclination [deg] LAN [deg] mission around asteroid 49 mission around asteroid 37 mission around asteroid 85 R. Bertrand, CNES, GTOC3 Workshop, June 27, 2008, Turin. 23

24 Orbital elements (end) Earth flyby #1 Earth flyby #3 Earth flyby #4 Earth flyby #2 argument of perihelion [deg] mission around asteroid 49 mission around asteroid 37 mission around asteroid 85 R. Bertrand, CNES, GTOC3 Workshop, June 27, 2008, Turin. 24

25 Global and local optimization GTOC3 problem is characterized by: a large number of combinations possibility to treat the combinatorial problem by solving a simplified problem close to the real one For a given trajectory leg (departure and arrival conditions are fixed) several local solutions exist and can be slightly or highly different Some solutions are not achievable by means of direct optimization methods because of the required accuracy Optimization of the overall trajectory is fundamental (optimization of the trajectory leg by leg is not sufficient) Local optimization is a still open problem R. Bertrand, CNES, GTOC3 Workshop, June 27, 2008, Turin. 25

26 4 groups of solutions same asteroid sequence Difference between the best and the worst solutions (value of objective function J) for each group group 1: group 2: 0.02 group 3: group 4: Differences due to: rendezvous order (TAC/TAS) number of Earth flybys (JPL/Georgia) control law (CNES/Deimos) R. Bertrand, CNES, GTOC3 Workshop, June 27, 2008, Turin. 26

27 consumption = 28.6 ; 50.2 ; 88.1 ; kg consumption = 33.2 ; 48.1 ; 88.1 ; kg δm = ; ; 0 ; kg CNES solution Better solution? JPL solution different trajectories and different control laws (leg #1) same trajectories but different control laws (legs #2 and #4) same trajectories and same control laws (leg #3) R. Bertrand, CNES, GTOC3 Workshop, June 27, 2008, Turin. 27

28 Conclusions Indirect methods (combined with smoothing methods) are very efficient to find out complex thrust laws and to achieve fine accuracy. Nevertheless they still remain difficult to initialize Earth flybys are extremely important to reduce the propellant consumption even if the hyperbolic excess velocities are low Up to now the choice of Earth flybys is guided by experience but promising studies are in progress What about the global optimality of the winning solution? R. Bertrand, CNES, GTOC3 Workshop, June 27, 2008, Turin. 28

29 Thank you very much to the GTOC3 organizers for this very interesting problem R. Bertrand, CNES, GTOC3 Workshop, June 27, 2008, Turin. 29