1. University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 20: Project Discussion and the Kalman Filter

2. University of Colorado Boulder  Homework 6 Due Friday 2

3. University of Colorado Boulder 3 Project/Homework Discussion

4. University of Colorado Boulder  Satellite state estimated and propagated in the inertial frame: 4  Dynamics solve-for parameters are (fundamentally) not tied to a coordinate system:  Ground-station locations are in the Earth-fixed frame:

5. University of Colorado Boulder 5  Since the ground stations are in the Earth-fixed frame, we assume:  Hence, we have:

6. University of Colorado Boulder 6  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. University of Colorado Boulder  We recommend including this transformation in the measurement model: 7 All of these need to be in the same reference frame!

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

9. University of Colorado Boulder  Compare to solution online  Results available as.txt and.mat ◦ Results generated for the.txt files did not use ode45()! ◦ Results in.mat file appear to have used Rel/Abs tolerances of 1e-11  Note: some elements of the project website need to be updated (suggestions and rubric) 9

10. University of Colorado Boulder  We ask for relative differences to quickly identify differences between your result and the one online:  Example: 10

11. University of Colorado Boulder 11 Conventional Kalman Filter (CKF)

12. University of Colorado Boulder  Given from a previous filter: 12  We have new a observation and mapping matrix:  We can update the solution via:

13. University of Colorado Boulder 13  Is there a better sequential processing algorithm? ◦ YES! – The equations above may be manipulated to yield the Kalman filter

14. University of Colorado Boulder  Today – Outline derivation from minimum variance estimator ◦ Demonstrates mathematical equivalence of CKF and Batch  Wednesday – Derivation as a solution to Bayes theorem ◦ Demonstrates strengths of Kalman filter in context of probability/statistics ◦ Also helps to understand impacts of assumptions 14

15. University of Colorado Boulder 15  Schur Identity (Appendix B, Theorem 4): (Yes, it will simplify things…)

16. University of Colorado Boulder 16 Kalman Gain

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18. University of Colorado Boulder 18  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)

19. University of Colorado Boulder 19 Note the use of Htilde Does not map to epoch time!

20. University of Colorado Boulder  Reinitialize integrator after each observation: 20  Alternatively, we can use already generated output:

21. University of Colorado Boulder  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… 21

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24. University of Colorado Boulder  Whitening Transformation 24 Use new values in Kalman filter

25. University of Colorado Boulder  Whitening Transformation 25

26. University of Colorado Boulder 26 The Kalman Filter – Prediction Residuals

27. University of Colorado Boulder  Previously, we have discussed the pre-fit and post-fit residuals:  How can this change in the context of the CKF? 27

28. University of Colorado Boulder  At each measurement time in the CKF, we can take a look at the prediction residual:  Covariance of the prediction residual: 28

29. University of Colorado Boulder  How might we use the prediction residual PDF? 29