1. University of Paderborn Applied Mathematics Michael Dellnitz Albert Seifried Applied Mathematics University of Paderborn Energetically efficient formation flight of spacecraft: numerical experiments Jens Levenhagen Astrium GmbH Oliver Junge Applied Mathematics University of Paderborn

2. Applied Mathematics Motivation NASA: Terrestrial Planet Finder ESA: Darwin

3. University of Paderborn Applied Mathematics Table of Contents 1.Computing the deformation of a spacecraft formation under the natural dynamics (globally) 2.Computing the deformation along Halo orbits 3.Controlling the formation on a Halo orbit

4. University of Paderborn Applied Mathematics circular restricted three body problem  rotating coordinates  3D flow:   L4L4 L3L3 L5L5 L1L1 L2L2 The model

5. University of Paderborn Applied Mathematics Computing the deformation* Compute the deformation of tetrahedra under evolution of their vertices: *related approach for near earth formations: W. Koon, J. E. Marsden, J. Masdemont, and R. M. Murray. J2 dynamics and formation flight, Proceedings of AIAA Guidance, Navigation, and Control Conference, Montreal, Canada, August, AIAA 2001 deformation globally deformation along Halos controlling the formation

6. University of Paderborn Applied Mathematics Computing the deformation Consider a partition of part of the phase space into boxes: L2L2 In every box B compute the average over the deformation of n tetrahedra: deformation globally deformation along Halos controlling the formation

7. University of Paderborn Applied Mathematics Deformation globally time of integration: ca. 8 months length of the edges: 150m – 1.5 km deformation globally deformation along Halos controlling the formation

8. University of Paderborn Applied Mathematics Deformation near the Earth deformation globally deformation along Halos controlling the formation

9. University of Paderborn Applied Mathematics Deformation near L 2 L2L2 deformation globally deformation along Halos controlling the formation

10. University of Paderborn Applied Mathematics Deformation near L 2 in dependence of T deformation globally deformation along Halos controlling the formation

11. University of Paderborn Applied Mathematics Observations there is no region where the deformation always vanishes; there are regions where the deformation regularily grows and shrinks again; typically in these regions the deformation is high only for a short period of time („peaks“). T Def(.,T) deformation globally deformation along Halos controlling the formation

12. University of Paderborn Applied Mathematics Deformation averaged over time deformation globally deformation along Halos controlling the formation

13. University of Paderborn Applied Mathematics Trajectory with low time-averaged deformation L2L2 L2L2 L2L2 deformation globally deformation along Halos controlling the formation

14. University of Paderborn Applied Mathematics Maximal deformation within [0,T] deformation globally deformation along Halos controlling the formation

15. University of Paderborn Applied Mathematics Solution with low maximal deformation L2L2 deformation globally deformation along Halos controlling the formation

16. University of Paderborn Applied Mathematics Halo-Orbits L2L2 Sun deformation globally deformation along Halos controlling the formation

17. University of Paderborn Applied Mathematics Deformation along Halo-Orbits deformation globally deformation along Halos controlling the formation

18. University of Paderborn Applied Mathematics Deformation along Halo-Orbits (adapted initial velocity) deformation globally deformation along Halos controlling the formation

19. University of Paderborn Applied Mathematics Controlled formation flight Model: additive force in each space direction. Control for each spacecraft (j=1,...,4): from current point q j (t i ) (configuration space) reach prescribed point q j (t i+1 ), i.e. solve h : stepsize, u i  R 3 : control. deformation globally deformation along Halos controlling the formation

20. University of Paderborn Applied Mathematics Results for a Halo orbit near L 2 distances between actual and target vertices: meter months deformation globally deformation along Halos controlling the formation

21. University of Paderborn Applied Mathematics Results for a Halo orbit near L 2 norm of controls: N/kg months deformation globally deformation along Halos controlling the formation

22. University of Paderborn Applied Mathematics Results for a Halo orbit near Earth distances between actual and target vertices: meter months deformation globally deformation along Halos controlling the formation

23. University of Paderborn Applied Mathematics Results for a Halo orbit near Earth norm of controls: N/kg months deformation globally deformation along Halos controlling the formation

24. University of Paderborn Applied Mathematics Transition to a local coordinate system Observation: scales differ by a factor of 10 13 : distance Sun-Earth: 1.5·10 11 m maximal error for distances between spacecraft: 0.01 m rounding effects start to play a role But: machine precision („ double “): 10 -16 Idea: transition to a local coordinate system origin: center of mass of the formation origin evolves on Halo-orbit dynamics of the formation according to the linearized dynamics (variational equation along Halo-orbit) distances between actual and target vertices: meter months deformation globally deformation along Halos controlling the formation

25. University of Paderborn Applied Mathematics Results for the linearized model distances between actual and target vertices: meter months deformation globally deformation along Halos controlling the formation

26. University of Paderborn Applied Mathematics Results for the linearized model norm of controls: N/kg months deformation globally deformation along Halos controlling the formation

27. University of Paderborn Applied Mathematics Outlook replace „reach exact target vertex“ by a more flexible target configuration: allow for rotation around normal allow for maximal error in distances between the spacecraft locally optimal control of formation compute globally optimal control using Bellman´s principle application of NTG?