1. University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 18: Minimum Variance Estimator and the Term Project

2. University of Colorado Boulder  Exam 1 – Friday, October 10 2

3. University of Colorado Boulder  Minimum Variance w/ A Priori  Sequential Processing w/ Minimum Variance  Term Project 3

4. University of Colorado Boulder 4 Minimum Variance w/ A Priori

5. University of Colorado Boulder  With the least squares solution, we minimized the square of the residuals  Instead, what if we want the estimate that gives us the highest confidence in the solution: ◦ What is the linear, unbiased, minimum variance estimate of the state x? 5

6. University of Colorado Boulder  What is the linear, unbiased, minimum variance estimate of the state x ? ◦ This encompasses three elements  Linear  Unbiased, and  Minimum Variance  We consider each of these to formulate a solution 6

7. University of Colorado Boulder 7  Put into the context of scalars:

8. University of Colorado Boulder  Turns out, we get the weighted, linear least squares!  Hence, the linear least squares gives us the minimum variance solution ◦ Of course, this is predicated on all of our statistical/linearization assumptions 8

9. University of Colorado Boulder  To add a priori in the least squares, we augment the cost function J(x) to include the minimization of the a priori error.  How do we control the weighting of the a priori solution and the observations in the cost function? 9

10. University of Colorado Boulder  This is analogous to treating the a priori information as an observation of the estimated state at the epoch time 10

11. University of Colorado Boulder 11

12. University of Colorado Boulder 12

13. University of Colorado Boulder  Like the previous case, the statistical least squares w/ a priori is equivalent to the minimum variance estimator  Do I have to use a statistical description of the observation/state errors to estimate the state? 13

14. University of Colorado Boulder  Least squares does not require a probabilistic definition of the weights/state  The minimum variance estimator demonstrates that, for a Gaussian definition of the observation and state errors, the LS is the best solution  Also know as the Best Linear Unbiased Estimator (BLUE)  Now, we can use the minimum variance estimator as a sequential estimator… 14

15. University of Colorado Boulder 15 Minimum Variance and Sequential Processing

16. University of Colorado Boulder 16 X*X*  Batch – process all observations in a single run of the filter  Sequential – process each observations individually (usually as they become available over time)

17. University of Colorado Boulder  Recall how to map the state deviation and covariance matrix (previous lecture) 17  Can we leverage this information to sequentially process measurements in the minimum variance / least squares algorithm?

18. University of Colorado Boulder  Given from a previous filter run: 18  We have new a observation and mapping matrix:  We can update the solution via:

19. University of Colorado Boulder  Two principle phases in any sequential estimator ◦ Time Update  Map previous state deviation and covariance matrix to the current time of interest ◦ Measurement Update  Update the state deviation and covariance matrix given the new observations at the time of interest  Jargon can change with communities ◦ Forecast and analysis ◦ Prediction and fusion ◦ others… 19

20. University of Colorado Boulder 20  No assumptions on the number of observations at t k.  Wait, but what if we have fewer observations than unknowns at t k ? ◦ Do we have an underdetermined system?

21. University of Colorado Boulder 21  The a priori may be based on independent analysis or a previous estimate ◦ Independent analysis could be a product of:  Expected launch vehicle performance  Previous analysis of system (a priori gravity field)  Initial orbit determination solution

22. University of Colorado Boulder 22  We still have to invert a n × n matrix  Can be computationally expensive for large n ◦ Gravity field estimation: ~n 2 +2n-3 coefficients!  May become sensitive to numeric issues

23. University of Colorado Boulder 23  Is there a better sequential processing algorithm? ◦ YES! – This equations above may be manipulated to yield the Kalman filter (next week)

24. University of Colorado Boulder 24 Term Project Introduction

25. University of Colorado Boulder  The OD project consists of ◦ Deriving the algorithms for estimating the state of a spacecraft ◦ Implementing the algorithms ◦ Processing range and range-rate observations from three ground-stations 25

26. University of Colorado Boulder  3 Ground-based tracking stations ◦ Equatorial station’s position fixed in filter  Dynamics Model ◦ Two-body, unknown μ ◦ Also estimating J 2 ◦ Drag with unknown C D ◦ Already completed! ◦ Simple satellite, spherical with known mass and area 26

27. University of Colorado Boulder  Estimate satellite position and velocity in ECI  Constants governing satellite dynamics  Station positions on Earth (ECEF)  18x1 state vector 27

28. University of Colorado Boulder  Unknown ground tracking station positions ◦ Boat in Pacific at equator  Equatorial station still estimated, but a priori covariance is very small ◦ Turkey ◦ Greenland 28

29. University of Colorado Boulder  Range (m) and range-rate (m/s) ◦ Range: Zero mean, σ = 1 cm ◦ Range-Rate: Zero mean, σ = 1 mm/s  There is an observation at t=0 !  Observation intervals do not overlap ◦ Number of observations at a given time always equals 2 ◦ Would add more complexity to the filter 29

30. University of Colorado Boulder  Derive A and H_tilde and implement the partials in software ◦ Use the symbolic toolbox with the jacobian() command ◦ Output results to a file to later copy/paste  Add numeric integration of STM ◦ Check intermediate results with the website 30

31. University of Colorado Boulder  Implement a batch least squares filter to estimate the state and error covariance ◦ First iteration implemented at the end of HW 9  Implement a Kalman/Sequential filter to estimate the state and error covariance ◦ Completed with HW 10 31

32. University of Colorado Boulder  Discussion of OD problem and batch processor in the report  Discuss the results ◦ Compare results from the Kalman and LS filters  Plots: ◦ Residuals ◦ Covariance ◦ Error ellipsoids (more in future lecture) 32

33. University of Colorado Boulder  Process one data type at a time ◦ Range-only processing ◦ Range-rate only ◦ Compare results (which does better?)  Discussion of results 33

34. University of Colorado Boulder  The previous tasks get you in the high-B range  Additional elements boost your grade from there ◦ Extended Kalman filter (EKF needed for StatOD 2 anyway) ◦ Potter Filter ◦ Givens transformations ◦ Smoothing ◦ Interval processing of data ◦ Others…  The additional elements is your chance to further explore a topic of interest covered in class 34