1 st ACT Competition on Global Trajectory Optimisation Solution of Team #7

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1 1 st ACT Competition on Global Trajectory Optimisation Solution of Team #7 Régis Bertrand Thierry Ceolin Richard Epenoy CNES-DCT/SB/MO CS-ESPACE/FDS CNES-DCT/SB/MO

2 Contents Presentation of team #7 Available tools Methodology used Trajectory found Lessons learnt and conclusions International Workshop on Global Trajectory Optimisation ESTEC 2 nd of February

3 Presentation of Team 7 Composition Dr. Régis Bertrand CNES DCT/SB/MO Dr. Thierry Ceolin CS ESPACE/FDS Dr. Richard Epenoy CNES DCT/SB/MO CNES/CS: long time working relations since several decades Mars Sample Return mission analysis (Netlanders) development of the Rosetta descending phase calculation software, Rosetta mission analysis (phase A design + re-design), development of a low thrust trajectory optimisation software. International Workshop on Global Trajectory Optimisation ESTEC 2 nd of February

4 Presentation of Team 7 (cont d) CNES is involved in several trajectory optimisation thesis F. Dufour, Optimisation of the space shuttle s phasing manoeuvres, Ph.D. thesis, Jun. 1993, UPS Toulouse III, (in French). S. Geffroy, Generalisation of average techniques in optimal control: Application to lowthrust orbital transfers and rendezvous, Ph.D. thesis, Oct. 1997, INPT Toulouse, (in French). T. Ceolin, Optimisation of Impulsive Thrusts for the Calculation of Heliocentric Trajectories and Planetary Station Acquisition, Ph.D. thesis, Jan. 1998, UPS Toulouse III, (in French). R. Bertrand, Optimisation of Low-Thrust Interplanetary Trajectories, Ph.D. thesis, Jun. 2001, UPS Toulouse III, (in French). International Workshop on Global Trajectory Optimisation ESTEC 2 nd of February

5 CNES trajectory optimisation tools LOTH Interplanetary Trajectory Optimisation Software ETOPH Electric Transfer Optimisation with Planetocentric and Heliocentric phases MIPELEC Electric Station Acquisition Other tools: LOGARITHME, DRAGON, OPTTNS International Workshop on Global Trajectory Optimisation ESTEC 2 nd of February

6 LOTH Goal: Optimise an interplanetary trajectory with D.S.M., Manoeuvres are Impulsive Principle: Non Linear Parametric optimisation problem with constraints Algorithms and techniques used: patched conics formulation, Nelder-Mead simplex method, multi-start technique, exact penalties, (for mathematic programming sense). Criteria: Final mass or global velocity increment 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). International Workshop on Global Trajectory Optimisation ESTEC 2 nd of February

7 LOTH (cont d) Modelling: a trajectory is made of several arcs, each arc has a duration and is solution of a Lambert s problem, each manoeuvre is determined by the departure and arrival velocities, body parameters Unknown variables: Fly-by dates & dates and positions of D.S.M Optimisation formulation: min J nbody+ 1 = Vi ( X ) + V j ( X ) + i= 1 ndsm j= 1 c. P( X ) nconstra int max min P( X ) = max[ max (0,( K K ),( K K r r= 1 r r r ))] Optimisation algorithm: + direct search method, no gradient calculation - no proof of optimality + local method with some global properties - local method + very robust method - less efficiency for great dimension + constraints are treated by penalty functions problems International Workshop on Global Trajectory Optimisation ESTEC 2 nd of February

8 ETOPH Goal: Optimise a low-thrust interplanetary trajectory Principle: (Bang-Bang) optimal control problems 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 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 , International Workshop on Global Trajectory Optimisation ESTEC 2 nd of February

9 ETOPH (cont d) Modelling: the mission scenario is fixed (in terms of visited bodies), swing-by manoeuvres: instantaneous direction change on the S/C relative velocity. Unknown variables: dates, thrusting periods, swing-by parameters Optimisation formulation: PMP drives to MPBVP Solution algorithms: smoothing methods to overcome issues due to the bang-bang control structure decomposition-coordination techniques: solve separately each trajectory arc, link all the arcs to build the overall trajectory in an optimal way (coordination parameters). International Workshop on Global Trajectory Optimisation ESTEC 2 nd of February

10 Software validation LOTH ETOPH Validation and Reference Trajectories - Mars98, 01 and 2003, - Cassini (~65m/s ; error 1.3%), - Rosetta (~8 m/s ; error 0.5%), - Voyager II Internal studies of SOLO and BepiColombo trajectories Mission Analysis Used for Fire mission preliminary studies (Earth-Jupiter-Sun), mission to observe the Poles of the Sun Mars Sample Return Mission, NEO Phase 0 study International Workshop on Global Trajectory Optimisation ESTEC 2 nd of February

11 Global optimisation / Globalisation of local methods Multi-Start Technique try of several initial values of the unknown variables in their research domain advantages: possibility of parallelisation, possible use with any local search techniques. drawbacks: no proof of global optimum, need several draws, may be long. International Workshop on Global Trajectory Optimisation ESTEC 2 nd of February

12 Methodology for finding a solution Step one: Software customisation implementation of the ACT criterion, setting of ACT model parameters (gravitational constants, planet s radius, ) integration of the asteroid ephemeris. Step two: Intuitive understanding of the ACT problem lowest consumption trajectory, reach the asteroid at its perihelion, S/C relative velocity co-linear to asteroid velocity, highest relative S/C velocity, (on retrograde orbit). Step three: Validation of the customisations on direct Earth-2001TW229 trajectories. max J = m f r. V ast r. U rel r.cos( V ast r. U rel ) Step four: Improving the cost function value with several swing-bys enumerative method (flyby sequence), problems of convergence. International Workshop on Global Trajectory Optimisation ESTEC 2 nd of February

13 Methodology for finding a solution (cont d) Step five: How to reach a retrograde orbit? use of giant s planets, close flyby of the Sun (forbidden). Step six: Split of the trajectory into two parts Earth to Jupiter, several inner planets swing-bys final mass maximisation, search of a ballistic trajectory, provide an initial guess for low-thrust optimisation trajectory tool. Jupiter to 2001TW229, with Saturn and Jupiter swing-bys ACT criteria optimisation Step seven: Coordination of the two parts International Workshop on Global Trajectory Optimisation ESTEC 2 nd of February

14 A direct Earth-2001TW229 trajectory Impact relative velocity: 43.50km.s -1 Velocities angle: 157deg (cos ~ -0.92) Mean anomaly at encounter: 36deg Final S/C mass: m f = 441.7kg t 0 = 2012/01/04 t f = 2039/11/04 Criterion value: J = kg.km 2.s -2 International Workshop on Global Trajectory Optimisation ESTEC 2 nd of February

15 Our EEVEEJSJA trajectory, arc by arc 1 st Earth swing-by Periaster radius = km Thrust durations = 73.5d, 52d, 26d Venus swing-by Periaster radius = km Thrust duration = 0.3d International Workshop on Global Trajectory Optimisation ESTEC 2 nd of February

16 Our EEVEEJSJA trajectory, arc by arc 2 nd Earth swing-by Periaster radius = 6 683km Thrust duration = 22d 3 rd Earth swing-by Periaster radius = 6 688km Thrust duration = 76d International Workshop on Global Trajectory Optimisation ESTEC 2 nd of February

17 Our EEVEEJSJA trajectory, arc by arc 1 st Jupiter swing-by Periaster radius ~ km Thrust duration = 74.5d Mass at Jupiter = 1453kg Saturn swing-by Periaster radius ~ km Thrust duration = 3 190d International Workshop on Global Trajectory Optimisation ESTEC 2 nd of February

18 Our EEVEEJSJA trajectory, arc by arc 2 nd Jupiter swingby Periaster radius ~ km Ballistic arc Encounter with 2001TW229 Ballistic arc International Workshop on Global Trajectory Optimisation ESTEC 2 nd of February

19 Our EEVEEJSJA trajectory: summary Impact relative velocity: 50.15km.s -1 Velocities angle: 166deg (cos ~ -0.97) Mean anomaly of asteroid at encounter: M = 1.91deg Final S/C mass: m f = kg Criterion value: J = kg.km 2.s -2 (movie) International Workshop on Global Trajectory Optimisation ESTEC 2 nd of February

20 Lessons learnt Our tools are efficient but can (have to) be improved, The search of a trajectory including outer and inner planets may certainly be split into two sub-problems, (high level of decomposition-coordination algorithm), Maximisation of the ACT criterion implies maximisation of the final mass, so the optimal trajectory has to be as ballistic as possible, then parametric optimisation tool is most suitable, Experience (training) of this kind of studies is fundamental. International Workshop on Global Trajectory Optimisation ESTEC 2 nd of February

21 Conclusions Lack of experience: loss time, Initialisation of such problem: very tricky, well known for all classical methods, Having a set of trajectories fulfilling the mission constraints, Global optimisation: The multi-start technique is not efficient enough for such open problem, Optimisation of the mission scenario (flyby sequence) work in progress: Ph. D Thesis CNES/AAS/CMA International Workshop on Global Trajectory Optimisation ESTEC 2 nd of February