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 27: Orthogonal Transformations

2. Announcements • No homework • No lecture quiz • Exam 2 – Friday, November 8 • E-mail Marco with questions for Wednesday’s lecture

3. Exam 2 Topics • Minimum Variance • Conventional Kalman Filter • Extended Kalman Filter • Prediction Residual • Handling Observation Biases • Numeric Considerations in the Kalman • Batch vs. CKF vs. EKF • Effects of Uncertainties on Estimation • Potter Square-Root Filter • Cholesky Decomposition w/ Forward and Backward substitution • Singular Value Decomposition Methods

4. End of Semester Schedule • I need to turn grades in by Thursday December 19 • Exam 3 will be take home • Expect to post it on Friday Dec. 6 • In-class students: Due by noon, Monday Dec. 16 • CAETE students: TBD • -10 pts for each 24 hours late • Final Projects • In-class students: Due by noon, Monday Dec. 16 • CAETE students: TBD • -10 pts for each 24 hours late • “Freebies” not applicable to exam or project • We are happy to accept your exam and project earlier!

5. Update to Online Requirements • See: http://ccar.colorado.edu/ASEN5070/odproject/reportSuggestions.htm

6. Lecture Quiz 5

7. Question 1 • Percent Correct: 96.97%

8. Question 2 • Percent Correct: 79.67% • Residuals • Covariance Matrix • State

9. Question 3 • Percent Correct: 82.58%

10. Question 4 • Percent Correct: 33.33%

11. Question 5 • Percent Correct: 90.15%

12. Least Squares via Orthogonal Transformations

13. Properties of an Orthogonal Matrix

14. Orthogonal Transformation of LS Cost Fcn. • Recall the least squares cost function: • By property 4 on the previous slide and Q an orthogonal matrix:

15. Derivation of LS Solution via Orthogonal Transformation

16. Derivation of LS Solution via Orthogonal Transformation

17. Derivation of LS Solution via Orthogonal Transformation

18. How do we select R ? • The method for selecting R defines a particular algorithm • Givens Transformations (Section 5.4) • Householder Transformation (Section 5.5) • Gram-Schmidt Orthogonalization • Not in the book and we won’t cover it

19. LS Solution via Givens Transformations

20. Givens Rotation

21. Givens Rotation

22. Givens Transformation for n>1 • Consider the desired result • To achieve this, we select the Givens matrix such that • We then use this transformation in top equation

23. Givens Transformation for m>2, n>1 • We do not want to non-zero terms to the previously altered rows, so we use identity matrix except in the rows of interest:

24. Givens Transformation • After applying the transformation, we get: • Repeat for all remaining non-zero elements in the third column • What if the term is already 0 ?

25. Givens Transformation Algorithm (Outline)

26. Batch vs. Givens – An Example

27. Problem Statement • Consider the case where: • The exact solution is: • After truncation:

28. Solution via Cholesky • Well, the Batch can’t handle it. What about Cholesky decomposition? • Darn, that’s singular too. • Let’s give Givens a shot!

29. Givens Solution

30. Givens Solution

31. Givens Solution

32. Givens Solution • Hence, Givens transformations give us a solution for the state Home Exercise: Why is this true? Note: R is not equal to H ! Still a problem w/ P !

33. Givens vs. Householder • Givens uses a sequence of rotation to generate the R matrix • Instead, Householder transformations use s sequence of reflections to generate R