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:

3 Measurement Types Measurement TypeAccuracy Horizon Scannerangles to Earth Stellar Refraction angles to Earth Landmark Trackerangles to Landmark km Space Sextantscalar to the Moon km Sun sensorsangles to the Sun Star trackersangles to stars Magnetic field sensorsangles to Earth km Optical Navigationangles to s/c or bodies X-ray Navigationscalar to barycenter km Forward Link Dopplerscalar 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

4 Crosslinks SST picture Image credit: Scalar measurements (range or range-rate) Estimate size, shape of orbits Estimate relative orientation of the orbits.

5 Crosslinks Scalar measurements (range or range-rate) Estimate size, shape of orbits Estimate relative orientation of the orbits. Image credit:

6 Two-body Problem SST

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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 All observable: – a 1, a 2, e 1, e 2, v 1, v 2 NOT all observable: – Ω 1, Ω 2, i 1, i 2, ω 1, ω 2 Initial Conditions Radius

12 J 2 Symmetry The vector field of accelerations in the x-z plane for two-body and J 2.

13 J 2 Solutions Initial Conditions

14 J 2 Solutions Observable: – a 1, a 2, e 1, e 2, v 1, v 2, – ΔΩ, i 1, i 2, ω 1, ω 2 NOT observable: – Ω 1, Ω 2 Initial Conditions Radius: Height:

15 Circular Restricted Three-body Problem P1P1 P2P2 x y Barycenter z μ 1-μ r1r1 r2r2 spacecraft

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

17 Lagrange Points x y L1L1 L2L2 L4L4 L5L5 L3L3 P1P1 P2P2

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 LL 1 Halo 2 constellations

25 LL 3 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 t i :

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 Uncertainty Growth Observation Geometry

36 Instability and Aspect Ratio Larger Aspect Ratio Smaller Aspect Ratio

37 Uncertainty Growth

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

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 – 7 th -8 th order Runge-Kutta Integrator – Stationkeeping maneuvers with execution errors Orbit Determination Model: – Extended Kalman Filter with process noise – SRP error ~10 -9 m/s 2 – LP100K statistical clone – Stationkeeping maneuvers without execution errors

41 Halo Orbiter: 4 Δv’s per period 5% Δv errors c R error -> 1 x m/s 2 position error RSS ≈ 80 m Snoopy-Woodstock Simulation Lunar Orbiter: 50x 95 km, polar orbit c R error -> 1 x m/s 2 5% Δv errors 1σ gravity field clone position error RSS ≈ 7 m 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 Earth The lunar orbiter could hold science instruments and be tracked to estimate the far side gravity field.

42 Snoopy L 2 halo orbiter EKF position error

43 Woodstock Lunar orbiter EKF position error

44 L 2 -Frozen Orbit Simulation L 2 halo orbiter EKF position error

45 L 2 -Frozen Orbit Simulation Frozen orbiter EKF position error

46 Frozen Orbit Constellation Frozen orbiter EKF position error

47 L 1 -LEO L 1 halo orbiter EKF position error

48 Application: Comm/Nav for the Moon Image credit: L1L1 L2L2 South Pole/ Aitken Basin Far Side Earth Moon 6 out of 10 of the lunar landing sites mentioned in ESAS require a communication relay.

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 – 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.