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ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones

ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 9: Batch Estimator and Weighted LS. Announcements. Homework 3 – Due September 19 When exporting MATLAB figures, please use a high- quality image format, e.g., PNG, EPS, etc.

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ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones

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  1. ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 9: Batch Estimator and Weighted LS

  2. Announcements • Homework 3 – Due September 19 • When exporting MATLAB figures, please use a high-quality image format, e.g., PNG, EPS, etc. • Screen captures and JPEGs are typically not the best option! • Lecture Quiz • Covers Lectures 6-8 • Due Friday at 5pm • Friday and Next Week: • Probability and Statistics • Book Appendix A

  3. Batch Estimator

  4. Linearized Equations

  5. LS with Linearized System • Straightforward way to estimate the state at a time that matches the observations • What about when the observations cover multiple points in time?

  6. Observations at multiple times? • What can we do to estimate the state when we have observations at multiple points in time? • What tool(s) do we have available to alter the formulation? • Given result from above, how might we alter the formulation to use a single relationship of the form:

  7. Reformulation for Epoch State

  8. Reformulation for Epoch State

  9. The Batch Estimator • Process all observations over a given time span in a single batch • The alternative sequential methods will be discussed later • What are the shortcomings of such a formulation?

  10. Example Least Squares Problem

  11. Example Least Squares Problem

  12. Example Least Squares Problem

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

  14. Shortcomings of Basic LS • No weighting of observations • How do we account for different sensors with different accuracies? • No incorporation of previous information • Known a a priori state information • How do we include this in the filter?

  15. Weighted Least Squares Estimation

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

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

  18. Derivation of Weighted LS Estimator

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

  20. Derivation of Weighted LS Estimator

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

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

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

  24. Weighted Least Squares w/ A Priori

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

  26. LS w/ A Priori Solution • As you will show in the homework:

  27. Computation Method

  28. Concept Exercises

  29. Directions • Sent via e-mail shortly before lecture • Take a look between now and Friday • Feel free to work in groups! • Be ready to answer the questions at the start of lecture • The concept quiz will not be turned in for a grade

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