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 20: Project Discussion and the Kalman Filter

2. Announcements • Homework 6 Due Friday

3. Project Discussion

4. Project Estimate State Coordinate Frames • Satellite state estimated and propagated in the inertial frame: • Dynamics solve-for parameters are (fundamentally) not tied to a coordinate system: • Ground-station locations are in the Earth-fixed frame:

5. Project Estimate State Coordinate Frames • Since the ground stations are in the Earth-fixed frame, we assume: • Hence, we have:

6. Structure of Project’s STM • The portions of the reference state requiring integration only includes the spacecraft position and velocity • Strictly speaking, we only need to propagate a 6 × 9 matrix!

7. Measurement Modeling • We recommend including this transformation in the measurement model: All of these need to be in the same reference frame!

8. State-Measurement Mapping Matrix • How can we estimate the filter solve-for parameters since the observations do not seem to depend on them? The STM is a function of these values • How/why can we estimate these values? (conceptual and mathematical answers)

9. The (Conventional) Kalman Filter

10. Minimum Variance as a Sequential Processor • Given from a previous filter: • We have new a observation and mapping matrix: • We can update the solution via:

11. Sequential Minimum Variance Measurement Update • Is there a better sequential processing algorithm? • YES! – The equations above may be manipulated to yield the Kalman filter

12. Derivation of the Kalman Filter • Schur Identity (Appendix B, Theorem 4): (Yes, it will simplify things…)

13. Derivation of the Kalman Filter

14. Derivation of the Kalman Filter • Recall that:

15. Derivation of the Kalman Filter Kalman Gain

16. Derivation of the Kalman Filter

17. Derivation of the Kalman Filter

18. Derivation of the Kalman Filter

19. Derivation of the Kalman Filter

20. Kalman Filter Measurement Update • Instead inverting a p×p matrix • Mathematically equivalent to the batch least squares • Also provides a solution to the least squares minimization problem • Yields a new set of problems in filtering (to be covered later)

21. Computational Algorithm Does not map to epoch time! Note the use of Htilde

22. State Transition Matrix Generation • Reinitialize integrator after each observation: • Alternatively, we can use already generated output:

23. Matrix Inversion Elimination • We have to invert a p×p matrix, which is likely more efficient and stable than a n×n matrix inversion • Can we further reduce the computation overhead? • Yes – under certain conditions…

24. Kalman Filter with Scalar Inversion

25. Kalman Filter with Scalar Inversion

26. What if R is not diagonal? • Whitening Transformation Use new values in Kalman filter

27. What if R is not diagonal? • Whitening Transformation