1. Manifold Filtering Problem Lockheed Martin Jarett Hailes Jonathan Wiersma Richard VanWeelden July 21, 2003

2. Outline Pod problem description Signal / Observation Model SERP Simulation Parameter Details Results IDEX Implementation Sketch of progress towards proof

3. Problem Description Pod Sensor Second Sensor

4. Signal Description 233 33 : : x h    Process WienerD2:  t W Stratonovich SDE 3 Dimensional State

5. Signal Constraints

6. Signal Implementation Stratonovich SDEIto SDE

7. Pod Observation Model : basis of plane normal to r r : resting position of pod sensor

8. 2 4 3 1) Resample Particles 2) Evolve Particles 3) Update Weights given Y k If W(ξ t i ) < ρ W(ξ t j ): )()( )( j t i t j t WW W    Prob: 22 )()( )( j t i t i t WW W    )( j t W  )( i t W  2 )()( i t j t WW  Filtering Using SERP

9. Simulation Parameters Error Function: r = 1 e Signal Estimate

10. Simulation

11. Future Directions - Workable explicit solution -Eliminate approximation errors in SERP particle evolve - Use IDEX as filter - More realistic manifolds - Cantilever equations - Enhance signal motion - Damped Harmonic Motion

12. θ φ Other Observation Model

13. Filtering with IDEX Goal: prove explicit solutions exist IDEX provides: Faster computation No inherent approximation error

14. Background From Kouritzin and Remillard (2000): are 1-step nilpotent, h constraint holds

15. Problem Description Two dimensional manifold in three space

16. Conditions

17. Equivalency By Ito’s formula and martingale theory: (1)

18. Results Conjecture: Φ exists iff (2) where (2)

19. Sketch of Proof Assuming that Φ exists, then (2) is equivalent to two-step nilpotentcy Apply chain rule Simplify equation

20. Sketch of Proof Cont. If (2) holds, and all σ m are two step nilpotent, then Φ exists Idea: find satisfying (3) (4) (5) (6) (7)

21. Sketch of Proof Cont. Let Φ be such that Get dΦ is exact by (3-7) Future: construct (4-6), show they converge