1. University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 41: Information Filter

2. University of Colorado Boulder  Exam 3 ◦ In-class Students: Due Friday by 5pm ◦ CAETE Students: Due 11:59pm (Mountain) on 12/13  Final Project Due December 14 by noon 2

3. University of Colorado Boulder 3 Project Q&A

4. University of Colorado Boulder 4 Information Filter

5. University of Colorado Boulder 5  Well, we know that the CKF has problems… Negative Values

6. University of Colorado Boulder 6  How about the Joseph formulation of the measurement update? Negative Values

7. University of Colorado Boulder  How about the EKF?  How about the Potter square-root filter? 7

8. University of Colorado Boulder  Time Update 8  Measurement Update:

9. University of Colorado Boulder 9  What if we go back to the minimum variance?

10. University of Colorado Boulder 10  If I don’t want to invert the information matrix, do I have another option?

11. University of Colorado Boulder  Well, that was easy.  What about the time update? 11

12. University of Colorado Boulder  What can we do to simplify this? 12 (Assume Q k-1 non-singular)

13. University of Colorado Boulder  Require that Q k-1 be non-singular  Do not need to invert the n×n information matrix 13 Still need to maintain information matrix separate from D !

14. University of Colorado Boulder  From the time update of the information matrix: 14

15. University of Colorado Boulder 15  Can I initialize the filter with an infinite a priori state covariance matrix?  What happens if we have very accurate measurements?

16. University of Colorado Boulder  Once the information matrix has a sufficiently small condition number: 16

17. University of Colorado Boulder 17  Provides a more numerically stable solution  Stability equals that of the Batch, but in a sequential implementation  Don’t need to generate state/covariance until needed  Square-root information filter (SRIF) ◦ Refined through extensive use in POD ◦ Includes smoothing capabilities

18. University of Colorado Boulder 18 Information Filter with Bierman’s Problem

19. University of Colorado Boulder 19

20. University of Colorado Boulder 20

21. University of Colorado Boulder 21

22. University of Colorado Boulder 22 Monte Carlo Analysis

23. University of Colorado Boulder  There are many unknowns in orbit determination. What are some? 23

24. University of Colorado Boulder  There are many unknowns in orbit determination ◦ Dynamics Model ◦ Dynamics Errors (systematic and stochastic) ◦ Measurement Model ◦ Measurement Noise  Many of these may be characterized using covariance analysis (CH. 6, StatOD 2)  Given the large number of random inputs, how would we characterize the possible OD performance when covariance analysis is limited? 24

25. University of Colorado Boulder  Consider many different types of models and model errors  What about the accuracy of input models? ◦ Example: Gravity Field ◦ Our best estimate of the gravity field still has a variance.  How do we consider the filter performance with such errors? 25

26. University of Colorado Boulder 26

27. University of Colorado Boulder 27

28. University of Colorado Boulder 28