Autonomous Navigation in Libration Point Orbits

1. Autonomous Navigation in Libration Point Orbits Keric A. Hill Thesis Committee: George H. Born, chair R. Steven Nerem Penina Axelrad Peter L. Bender Rodney Anderson 27 April 2007

2. Why Do We Need Autonomy? Image credit: http://solarsystem.nasa.gov/multimedia/gallery/

3. Measurement Type Accuracy Horizon Scanner angles to Earth Stellar Refraction angles to Earth Landmark Tracker angles to Landmark km Space Sextant scalar to the Moon km Sun sensors angles to the Sun Star trackers angles to stars Magnetic field sensors angles to Earth km Optical Navigation angles to s/c or bodies X-ray Navigation scalar to barycenter km Forward Link Doppler scalar to groundstation km DIODE (near Earth) scalar to DORIS stations m GPS (near Earth) 3D position, time cm Crosslinks (LiAISON) scalar to other s/c m Measurement Types

4. Crosslinks • Scalar measurements (range or range-rate) • Estimate size, shape of orbits • Estimate relative orientation of the orbits. SST picture Image credit: http://www.centennialofflight.gov/essay/Dictionary/TDRSS/

5. Crosslinks • Scalar measurements (range or range-rate) • Estimate size, shape of orbits • Estimate relative orientation of the orbits. Image credit: http://www.centennialofflight.gov/essay/Dictionary/TDRSS/

6. Two-body Problem SST

7. Two-body Problem SST

8. Two-body Problem SST

9. Two-Body Symmetry The vector field of accelerations in the x-y plane for the two-body problem.

10. Two-Body Solutions Initial Conditions

11. Two-Body Solutions • NOT all observable: • Ω1, Ω2, i1, i2, ω1, ω2 Radius Initial Conditions • All observable: • a1, a2, e1, e2, v1, v2

12. J2 Symmetry The vector field of accelerations in the x-z plane for two-body and J2.

13. J2 Solutions Initial Conditions

14. J2 Solutions Radius: Height: Initial Conditions • Observable: • a1, a2, e1, e2, v1, v2, • ΔΩ, i1, i2, ω1, ω2 • NOT observable: • Ω1, Ω2

15. Circular Restricted Three-body Problem y spacecraft z r1 r2 P1 P2 Barycenter x μ 1-μ

16. Three-body Symmetry The vector field of accelerations in the x-z plane for the three-body problem.

17. Lagrange Points y L4 P1 P2 x L2 L1 L3 L5

18. Three-body Solutions

19. Proving Observability • Orbit determination with two spacecraft. • One spacecraft is in a lunar halo orbit. • Observation type: Crosslink range. • Gaussian noise 1 σ = 1.0 m. • Batch processor : • Householder transformation. • Fit span = 1.5 halo orbit periods (~18 days). • Infinite a priori covariance. • Observations every ~ 6 minutes. • LOS checks.

20. OD Accuracy Metric

21. Position Along the Halo

22. Initial Positions Sat 1

23. Spacecraft Separation

24. Out of Plane Component LL1 Halo2 constellations

25. LL3 Results: Weak

26. Halo-Moon

27. Monte Carlo Analysis

28. Constellation Design Principles • At least one spacecraft should be in a libration orbit. • Spacecraft should be widely separated. • Orbits should not be coplanar. • Shorter period orbits lead to better results. • More spacecraft lead to better results.

29. Some Interesting Questions • How does orbit determination work for unstable orbits? • Why do the phase angles of the spacecraft affect the orbit determination so much?

30. Observation Effectiveness Accumulating the Information Matrix: The effectiveness of the observation at time ti:

31. Observation Effectiveness for Two-body Orbits

32. Observation Effectiveness for Three-body Orbits

33. Two-body Orbits, by Components

34. Three-body Orbits, by Components

35. Observation Effectiveness Dissected Observation Geometry Uncertainty Growth

36. Instability and Aspect Ratio Smaller Aspect Ratio Larger Aspect Ratio

37. Uncertainty Growth

38. Observation Geometry Axis of Most Uncertainty Axis of Least Uncertainty Least Effective Observation Vector Most Effective Observation Vector

39. Local Unstable Manifolds

40. Realistic Simulations • Truth Model: • DE403 lunar and planetary ephemeris • DE403 lunar librations • Solar Radiation Pressure (SRP) • LP100K Lunar Gravity Model • 7th-8th order Runge-Kutta Integrator • Stationkeeping maneuvers with execution errors • Orbit Determination Model: • Extended Kalman Filter with process noise • SRP error ~10-9 m/s2 • LP100K statistical clone • Stationkeeping maneuvers without execution errors

41. Snoopy-Woodstock Simulation Propagation: RK78 with JPL DE405 ephemeris, SRP, LP100K Lunar Gravity (20x20) Orbit Determination: Extended Kalman Filte Observations: Crosslink range with 1 m noise every 60 seconds Moon Halo Orbiter: 4 Δv’s per period 5% Δv errors cR error -> 1 x 10-9 m/s2 position error RSS ≈ 80 m Lunar Orbiter: 50x 95 km, polar orbit cR error -> 1 x 10-9 m/s2 5% Δv errors 1σ gravity field clone position error RSS ≈7 m The lunar orbiter could hold science instruments and be tracked to estimate the far side gravity field. Earth

42. Snoopy L2 halo orbiter EKF position error

43. Woodstock Lunar orbiter EKF position error

44. L2-Frozen Orbit Simulation L2 halo orbiter EKF position error

45. L2-Frozen Orbit Simulation Frozen orbiter EKF position error

46. Frozen Orbit Constellation Frozen orbiter EKF position error

47. L1-LEO L1 halo orbiter EKF position error

48. Application: Comm/Nav for the Moon Earth Far Side Moon L1 L2 6 out of 10 of the lunar landing sites mentioned in ESAS require a communication relay. South Pole/ Aitken Basin Image credit: http://photojournal.jpl.nasa.gov

49. Future Work • Perform navigation simulations using independently validated software (GEONS was not quite ready). • Compare ground-based navigation with space-based navigation at the Moon. • Obtain and process crosslink measurements for any of the following situations: • Halo Orbiter – Halo Orbiter • Halo Orbiter – Lunar Orbiter • Lunar Orbiter – Earth Orbiter

50. Acknowledgements • This material is based upon work supported under a National Science Foundation Graduate Research Fellowship. Any opinions, findings, conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the National Science Foundation. • The idea for this research came from Him for whom all orbits are known.