1. ASEN 5070: Statistical Orbit Determination I Fall 2015
Professor Brandon A. Jones
Lecture 10: Weighted LS and A Priori

2. Announcements Lecture Quiz 5 Due September 18 @ 5pm
Lecture Quiz 4 posted over weekend
Will cover lectures 9 & 10
Homework 3 Due September 9am
Homework 4 Due September 25
Next Week:
Probability and Statistics
Book Appendix A

3. Weighted Least Squares Estimation

4. The Batch Estimator
Process all observations over a given time span in a single batch

5. We define a set of weights
For each yi, we have some weight wi

6. Effects of Weights in J(x)
Consider the case with two observations (m=2)
If w2 > w1, which εi will have a larger influence on J(x) ? Why?

7. Derivation of Weighted LS Estimator

8. Weighted Least Squares Estimator
For the weighted LS estimator:
How do we find W ?

9. What is the effect of W on the solution?

10. What is the effect of W on the solution?

11. Weighted Least Squares w/ A Priori

12. LS w/ A Priori Formulation
Relating to or denoting reasoning or knowledge that proceeds from theoretical deduction rather than from observation or experience
We have:

13. LS w/ A Priori Solution
You will show in the homework:

14. Computation Method

15. Orbit Determination Algorithm (so far)

16. Estimated State Vector
We want to get the best estimate of X possible
What would we consider when deciding if we should include a solve-for parameter?

17. Stat OD in a Nutshell Truth Reference Best Estimate (goal)
Observations are functions of state parameters, but usually NOT state parameters themselves.
Mismodeled dynamics
Underdetermined system
l*(n+p)

18. Stat OD in a Nutshell
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. X*
Observed Range
Computed Range
True Range = ???

19. Stat OD in a Nutshell 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.

20. State Deviation and 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!

21. State Deviation and Linearization
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:

22. State Deviation and Linearization
Goal of the Stat OD process:
The best fit trajectory is represented by
This is what we want

23. State Deviation Mapping
How do we map the state deviation vector from one time to another? X*

24. State Deviation Mapping
How do we map the state deviation vector from one time to another?
The state transition matrix.
It permits:

25. Mapping an observation
Now we can relate an observation to the state at an epoch. X*
Observed Range
Computed Range

26. Measurement Mapping
Still need to know how to map measurements from one time to a state at another time!

27. State Update
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. Lecture Topics Next Week
Probability and Statistics
Appendix A of the book