1. Colorado Center for Astrodynamics Research The University of Colorado STATISTICAL ORBIT DETERMINATION Project Report Unscented kalman Filter Information Filter Combining Estimates Concept Exam ASEN 5070 LECTURE 38, 39 12/07, 09/09

2. 2 Copyright 2006 Requirements/Suggestions for Term Project Report **1. General description of the OD problem and the batch and sequential algorithms **2. Discussion of the results - contrasting the batch processor, and sequential filter. Discuss the relative advantages, shortcomings, applications, etc. of the algorithms. **3. Show plots of residuals for all algorithms. Plot the trace of the covariance for position and velocity for the sequential filter for the first iteration. You may want to use a log scale. **4. When plotting the trace of P for the position and velocity, do any numerical problems show up? If so discuss briefly how they may be avoided. **5. Contrast the relative strengths of the range and range rate data. Generate solutions with both data types alone for the batch and discuss the solutions. How do the final covariances differ? You could plot the two error ellipsoids for position. What does this tell you about the solutions and the relative data strength? **6. Why did you fix one of the stations? Would the same result be obtained by not solving for one of the stations i.e., leaving it out of the solution list? Does it matter which station is fixed? **7. A discussion of what you learned from the term project and suggestions for improving it. **Required items for the final report

3. 3 Copyright 2006 Requirements/Suggestions for Term Project Report Extra Credit Items 8. Code the Extended Kalman Filter 9. How does varying the a priori covariance and data noise covariance affect the solution? What would happen if we used an a priori more compatible with the actual errors in the initial conditions, i. e. a few meters in position etc. 10. Do an overlap study (see lecture 34). 11. Code the Potter algorithm and compare results to the conventional Kalman filter. 12. Solve for the state deviation vector using the Givens square root free algorithm. Compare solution and RMS residuals for range and range rate from Givens solution with results from conventional Kalman and Potter filters. 13. Plot pre and post fit residuals for the Kalman filter. Include the 1 sigma pre-fit standard deviations on the plot (See Eqn. 4.7.34 of text). 14. Convert the estimation error covariance matrix into classical orbit element and nonsingular orbit element space. What elements are most in error? Can you see a reason for this. The coordinate transformation code is being mailed to the class. 15. Examine the effects of fixing various parameters such as J2 and seeing the results on the solution and residuals. They will no longer be Gaussian.

4. 4 Copyright 2006 4 Unscented Kalman Filter Introduced by Julier & Uhlmann circa 1997 Minimal set of samples (sigma points) are deterministically drawn from the square root decomposition of prior covariance, using a specific algorithm –Monte Carlo draws samples randomly (based upon a priori statistics) Sigma points are propagated through the modeled non-linear function –No Jacobians need be computed State estimate and covariance are accurate to 2 nd order for all non-linearities –3 rd order accuracy is expected for Gaussian distributions

5. 5 Copyright 2006 5 Unscented Kalman Filter (1) Compute weights: where: and (2) Initialize the UKF given:(3) Compute Sigma Points (matrix) for t-1 (4) Propagate Sigma Points through non-linear system note: this implies integrating 2L+1 state vectors at each time step (i.e. each column) = 3 - L Use =1 Use matlab fn. Sqrtm to get square Root of P

6. 6 Copyright 2006 6 Unscented Kalman Filter and (5) Perform a state and covariance time update where: Q is the process noise covariance (6) Re-compute Sigma Points to incorporate effects of process noise (7) Compute the measurements associated with each Sigma Point vector and their weighted average

7. 7 Copyright 2006 7 Unscented Kalman Filter and (8) Compute the innovation and cross-correlation covariances (9) Compute the Kalman Gain, the best estimate of the state and the a posteriori state covariance where: (10) Replace t with t-1 and return to (3) until all the observations have been processed. are the observations

8. Information Filter

9. Statistical Orbit Determination Copyright 2006 University of Colorado at Boulder

10. Statistical Orbit Determination Copyright 2006 University of Colorado at Boulder Recall that the equations for the standard sequential algorithm (also called the covariance filter) are:

11. Statistical Orbit Determination Copyright 2006 University of Colorado at Boulder The equations for the information filter are as follows.

12. Statistical Orbit Determination Copyright 2006 University of Colorado at Boulder

13. Statistical Orbit Determination Copyright 2006 University of Colorado at Boulder

14. Statistical Orbit Determination Copyright 2006 University of Colorado at Boulder

15. Statistical Orbit Determination Copyright 2006 University of Colorado at Boulder

16. Statistical Orbit Determination Copyright 2006 University of Colorado at Boulder

17. Combining Estimates

18. Statistical Orbit Determination Copyright 2006 University of Colorado at Boulder

19. Statistical Orbit Determination Copyright 2006 University of Colorado at Boulder Treat estimates as observations. Write observation equations. Then use Linear Unbiased Minimum Variance Estimate Equation, as follows.

20. Statistical Orbit Determination Copyright 2006 University of Colorado at Boulder

21. Statistical Orbit Determination Copyright 2006 University of Colorado at Boulder :

22. Statistical Orbit Determination Copyright 2006 University of Colorado at Boulder