1. David Wheeler Kyle Ingersoll EcEn 670 December 5, 2013 A Comparison between Analytical and Simulated Results The Kalman Filter: A Study of Covariances

2. Kalman Overview:  Common Applications 1 : Inertial Navigation (IMU + GPS) Global Navigation Satellite Systems Estimating Constants in the Presence of Noise Simultaneous Localization and Mapping (SLAM) Object Tracking In Computer Vision Economics Predict (P) Forward One Step Update (U) Use Measurements If Available PPPPPPPP UU U 2

3. Kalman Intuition: Predict Using Underlying Model 1 2 3 4 5 ? 3

4. 1 2 3 4 5 ? 4

5. Kalman Intuition: Update by Weighing Measurement and Model 1 2 3 4 5 ? Residual 5

6. Kalman Intuition: Update by Weighing Measurement and Model 1 2 3 4 5 ? 6

7. Kalman Intuition: Summary 1 2 3 4 5 ? 7

8.  Prediction Derivation: Prediction Step: Linear Example Current State Recent State Recent Input k=1 k=2 Example 1 8 = = = = = = = = = = = =

9. Update Step: Linear Example Measurement Model’s Guess for Measurement Residual Weighting 9

10. Results: Linear Example 10

11. Results: Linear Example 11

12. Results: Linear Example 12

13. Results: Linear Example 13

14. Linear Example: Comparing Covariance Trends Experimental Covariance (Blue) Analytical Covariance (Red) 14

15. Linear Example: Convergence of Covariances 15

16. Non-Linear Example Example 2 16

17. Results: Non-linear Example 17

18. Results: Non-linear Example  Beacon Location (Red Circle)  Measurement (7/500) (Green Lines)  Gaussian Noise on Measurement (Red Xs)  Covariance (before update)  Analytical (Thin Cyan)  Experimental (Thick Cyan) 18

19. Results: Non-linear Example  Covariance  Before update  Analytical (Thin Cyan)  Experimental (Thick Cyan)  After update  Analytical (Thin Magenta)  Experimental (Thick Magenta) Note – the update step reduces the uncertainty in the direction of the measurement only! 19

20.  Under certain conditions, a Kalman filter causes the covariance to converge  Analytical and simulated covariances match closely  Analytical and simulated covariances converge quickly if seeded with different values  Individual measurements can significantly reduce the covariance of the state estimate Conclusion 20

21. Questions & Discussion 21