The Kalman Filter: A Study of Covariances

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

2. Kalman Overview: Predict(P) Forward One Step P P P P P P P P U U U • Common Applications1: • 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 Update (U) Use Measurements If Available

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

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

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

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

7. Kalman Intuition: Summary Predict Step Predict state forward one step. Predict covariance forward one step. Kalman Gain, Update Step Determine Kalman Gain (optimal weighting between and ). Update state using Kalman gain and residual. Update state covariance . 1 2 3 4 5 ?

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

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

10. Results: Linear Example • Ten Steps • “Predict" Only: • 500 runs • 10 time steps

11. Results: Linear Example • Experimental covariance (Cyan dots) MATLAB cov command • Analytical covariance (Red solid line) • Individual runs (Magenta dots)(Dark blue dots)

12. Results: Linear Example • Update Step: • 500 runs • 0.01

13. Results: Linear Example • Experimental covariance(Green dots)MATLAB cov command • Analytical covariance(Magenta solid line) • Individual runs(Dark blue dots)

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

15. Linear Example: Convergence of Covariances

16. Non-Linear Example • Process Example 2

17. Results: Non-linear Example • 30 Time Steps • 500 runs • Input: • Input Noise is Gaussian, ±5% • (known to start at origin) • Analytical Covariance(Cyan Ellipse) • Beacon Location(Red Circle)

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)

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!

20. Conclusion 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

21. Questions & Discussion