1. University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 22: Further Discussions of the CKF

2. University of Colorado Boulder  Homework 7 Due Friday  Lecture Quiz ◦ Due by 5pm on Friday 10/23 2

3. University of Colorado Boulder 3 The Kalman Filter – Implementation Discussion

4. University of Colorado Boulder 4 Note the use of Htilde Does not map to epoch time!

5. University of Colorado Boulder  Like the batch processor, we need to use linearization and a reference trajectory ◦ This gives us the STM and we use it to evaluate Htilde  At any point in time, we have an estimate of the state: 5

6. University of Colorado Boulder  Reinitialize integrator after each observation: 6  Alternatively, if we want to use one call to the integrator, we can use already generated output:

7. University of Colorado Boulder  In the CKF presented, we have to invert a p×p matrix, which is more efficient and (likely) stable than the n×n matrix inversion for the batch  Can we further reduce the computation overhead?  Yes – under certain conditions… 7

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9. University of Colorado Boulder 9 Home Exercise: Prove to yourself that the scalar update is equivalent to the original form if R k is diagonal.

10. University of Colorado Boulder  Whitening Transformation 10 Use new values in Kalman filter

11. University of Colorado Boulder  Whitening Transformation 11

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13. University of Colorado Boulder 13 The Kalman Filter – Prediction Residuals

14. University of Colorado Boulder  Previously, we have discussed the pre-fit and post-fit residuals:  What else might we consider in the context of the CKF? 14

15. University of Colorado Boulder  At each measurement time in the CKF, we can take a look at the prediction residual (sometime called innovation):  Covariance of the prediction residual: 15

16. University of Colorado Boulder  What would the predicted residual PDF be useful for? 16

17. University of Colorado Boulder  If we take another look at the Kalman gain equation: 17

18. University of Colorado Boulder  If we take a closer look, the CKF is using the predicted residual PDF at each time to update the state:  In other words, the CKF estimate of the state is a weighted sum of the a priori and a correction due to the predicted residual and its statistics 18

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20. University of Colorado Boulder 20 Comparison of Kalman and Batch

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26. University of Colorado Boulder  What are the similarities between the batch and the sequential processor (as discussed up until now)?  What are the differences between the batch and the sequential processor (as discussed up until now)? 26