1. University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 10: Weighted LS and A Priori

2. University of Colorado Boulder  Homework 4 – Due September 26  Exam 1 – October 10 ◦ Open book, open notes ◦ Calculator, but no computer 2

3. University of Colorado Boulder  Lecture 10 – WLS w/ a priori and OD Process  Lecture 11 – Probability and Statistics I ◦ This afternoon @ 3pm (ECCS 1B14)  Lecture 12 – Probability and Statistics II ◦ Monday morning  Friday and Next Week: ◦ Probability and Statistics ◦ Book Appendix A 3

4. University of Colorado Boulder 4 Weighted Least Squares Estimation

5. University of Colorado Boulder  Process all observations over a given time span in a single batch 5

6. University of Colorado Boulder  For each y i, we have some weight w i 6

7. University of Colorado Boulder 7  Consider the case with two observations (m=2)  If w 2 > w 1, which ε i will have a larger influence on J(x) ? Why? 2.0

8. University of Colorado Boulder  For the weighted LS estimator: 8  How do we find W ?

9. University of Colorado Boulder 9

10. University of Colorado Boulder 10

11. University of Colorado Boulder 11 Weighted Least Squares w/ A Priori

12. University of Colorado Boulder  A priori ◦ Relating to or denoting reasoning or knowledge that proceeds from theoretical deduction rather than from observation or experience  We have: 12

13. University of Colorado Boulder  As you showed in the homework: 13

14. University of Colorado Boulder 14

15. University of Colorado Boulder 15 Orbit Determination Algorithm (so far)

16. University of Colorado Boulder  We want to get the best estimate of X possible  What would we consider when deciding if we should include a solve-for parameter? 16

17. University of Colorado Boulder  Truth  Reference  Best Estimate (goal)  Observations are functions of state parameters, but usually NOT state parameters themselves.  Mismodeled dynamics  Underdetermined system ◦ l*(n+p) 17

18. University of Colorado Boulder  We have noisy observations of certain aspects of the system.  We need some way to relate each observation to the trajectory that we’re estimating. 18 Observed Range Computed Range True Range = ??? X*X*

19. University of Colorado Boulder  Assumptions: ◦ The reference/nominal trajectory is near the truth trajectory.  Why do we introduce this assumption? ◦ Force models are good approximations for the duration of the measurement arc.  Why does this matter? ◦ The filter that we are using is unbiased:  The filter’s best estimate is consistent with the true trajectory. 19

20. University of Colorado Boulder  Linearization  Introduce the state deviation vector  If the reference/nominal trajectory is close to the truth trajectory, then a linear approximation is reasonable.  If they are not, then higher order terms are no longer negligible! 20

21. University of Colorado Boulder  Goal of the Stat OD process:  Find a new state/trajectory that best fits the observations:  If the reference is near the truth, then we can assume: 21

22. University of Colorado Boulder  Goal of the Stat OD process:  The best fit trajectory is represented by 22 This is what we want

23. University of Colorado Boulder  How do we map the state deviation vector from one time to another? 23 X*X*

24. University of Colorado Boulder  How do we map the state deviation vector from one time to another?  The state transition matrix.  It permits: 24

25. University of Colorado Boulder  Now we can relate an observation to the state at an epoch. 25 Observed Range Computed Range X*X*

26. University of Colorado Boulder  Still need to know how to map measurements from one time to a state at another time! 26

27. University of Colorado Boulder 27  Since we linearized the formulation, we can still improve accuracy through iteration (more on this in a future lecture)  How do we get the weights? Probability and Statistics

28. University of Colorado Boulder 28 Concept Exercises

29. University of Colorado Boulder  Work in groups  At 11:40am, we will reassemble and discuss  The concept quiz will not be turned in for a grade 29