1. ASEN 5070: Statistical Orbit Determination I Fall 2014
Professor Brandon A. Jones
Lecture 27: Kalman Filter Case Study

2. Announcements Homework 8 Due Friday Lecture Quiz
Due by 5pm on Friday
Exam 2 – Friday, November 7
Covers material through the end of this week
Office Hours Thursday
This week: 11:30-12:30pm

3. Sample Biases GPS receiver solutions for Jason-2
Antenna is offset ~1.4 meters from COM
What could be causing the bias change after 80 hours?

4. Solution Sensitivity
The numeric results are sensitive to the implementation
For example, these two methods of computing the input yield different results:

5. Sample Solution Plots Online
Includes both the semilogx() and loglog() solutions (either one will be accepted)

6. Kalman Filter Case Study - Introduction

7. Example – Problem Statement
Ballistic trajectory with unknown start/stop
Red band indicates time with available observations
Obs. Stations
Start of filter

8. Example – Problem Statement
Object in ballistic trajectory under the influence of drag and gravity
Nonlinear observation model
Two observation stations

9. Example – Problem Statement

10. Filter Characterization
What should we look at to characterize the filter performance?
Residuals (pre-/post-fit)
Covariance
State Estimate
There are different ways to visualize these
We will consider the case where we have a known truth for comparison

11. Filter Residuals over Time
Station 1
Station 2
Blue – Range
Green – Range-Rate

12. Observation Residual Histograms
Prefits
Postfits

13. State Error and Uncertainty
Position
Velocity

14. What are some of the things we may want to consider adding to our filter?

15. Kalman Filter Case Study – Filter Saturation

16. Process Noise
To prevent filter saturation, we add a constant term to the covariance time update to set a minimum value:
This is usually referred to as process noise
More typically based on stochastic acceleration (more on this in November)

17. State Estimate with Process Noise
Position
Velocity

18. Residuals with Process Noise
Station 1
Station 2

19. Residual Histogram with Process Noise
Prefits
Postfits

20. Kalman Filter Case Study – Observation Editing

21. Process Noise Compute the prefit residual variance via
An observation is not processed in the filter if:

22. Predicted Residual Editing
Compute the prefit residual variance via
An observation may be ignored in the filter if (for example):

23. Kalman Filter with Scalar Inversion

24. Kalman Filter with Scalar Inversion

25. Residuals with Observation Editing
Station 1
Station 2

26. Residual Histograms w/ Editing
Prefits
Postfits

27. Filter Accuracy w/ Editing
Position
Velocity

28. Kalman Filter Case Study – Bias Estimation

29. Bias Estimation
To estimate the bias, we add it to the estimated state vector

30. Residual Histograms w/o Bias Estimation
Prefits
Postfits

31. Residual Histograms w/ Bias Estimation
Prefits
Postfits

32. Residuals without Bias Estimation
Station 1
Station 2

33. Residuals w/ Bias Estimation
Station 1
Station 2

34. Accuracy w/o Bias Estimation
Position
Velocity

35. Accuracy w/ Bias Estimation
Position
Velocity

36. Filter Estimated State Correlation
No Augmentation
Proc. Noise, Editing, Bias Est.