ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones

1. ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones Professor George H. Born Lecture 32: SNC Example and Dynamic Model Compensation

2. Announcements • Homework 10 due next week • Lecture quiz due by 5pm on Wednesday • Posted over the weekend

3. Application of SNC to Ballistic Trajectory

4. Example – Problem Statement • Ballistic trajectory with unknown start/stop • Red band indicates time with available observations Obs. Stations Start of filter

5. Example – Problem Statement • Object in ballistic trajectory under the influence of drag and gravity • Nonlinear observation model • Two observations stations

6. Example – Problem Statement

7. Changes Since Last Time • Now use an EKF • We will vary the filter model to study the benefits of SNC • Look at two cases: • Run each with and without a process noise model • Error in gravity (g = 9.8 m/s vs. 9.9 m/s) • Error in drag (b = 1e-4 vs. 1.1e-4)

8. Filter Residuals over Time, Gravity Error Station 1 Station 2 • Blue – Range • Green – Range-Rate Edited Points (ignore them)

9. State Error and Uncertainty, Gravity Error

10. SNC Design for Gravity Error • Added SNC to the filter: • Why is the term for x-acceleration smaller?

11. Filter Residuals over Time, Gravity-SNC Station 1 Station 2 • Blue – Range • Green – Range-Rate

12. State Error and Uncertainty, Gravity-SNC • 178.3 vs. 0.8 meters RMS

13. Filter Residuals over Time, Drag Error Station 1 Station 2 • Blue – Range • Green – Range-Rate Edited Points (ignore them)

14. State Error and Uncertainty, Drag Error

15. SNC Design for Drag Error • Added SNC to the filter:

16. Filter Residuals over Time, Drag-SNC Station 1 Station 2 • Blue – Range • Green – Range-Rate

17. State Error and Uncertainty, Gravity-SNC • 27.6 vs. 1.26 meters RMS

18. Relative Performance of Cases • Mitigation of the gravity acceleration error was easier to mitigate than the drag error. Why?

19. Dynamic Model Compensation

20. What is the Markov property? • The Markov property describes a random (stochastic) process where knowledge of the future is only dependent on the present:

21. Do these processes have the Markov property? • An object under deterministic motion? • A satellite in a chaotic orbit? • An object under stochastic motion?

22. Gauss-Markov Processes • Introduction of the random, uncorrelated (in time), Gaussian process noise u(t) makes η a Gauss-Markov process • We will use the GMP to develop another form of process noise

23. GMP Analytic Solution • Stochastic integral cannot be solved analytically, but has a statistical description: Stochastic Deterministic

24. GMP Analytic Solution • Stochastic integral cannot be solved analytically, but has a statistical description: Stochastic Deterministic

25. Correlation of GMP in Time • It may be shown that: • In other words: • The process is exponentially correlated in time • Rate of the correlation fade determined by β • For large β, the faster the decay

26. Equivalent Process • Instead, let’s use the equivalent process • Lk has the same statistical description as the stochastic integral • Hence, it is an equivalent process

27. Alternate Solution for η • Process behavior varies with the equation parameters Add dependence on time to emphasize different realizations for different times

28. GMP Behavior • What happens if β  0?

29. GMP Behavior • What happens if σ 0?

30. Illustration of GMP Evolution

31. Adding the DMC to your Filter • Augment the state vector to include the accelerations • Dubbed Dynamic Model Compensation (DMC) • The random portion determines the process noise matrix Q(t) (see Appendix F, p. 507-508)

32. DMC More Robust than SNC

33. GMP1 vs. GMP2 Image: Leonard, Nievinski, and Born, 2013