1. EE 495 Modern Navigation Systems Kalman Filtering – Part II Mon, April 4 EE 495 Modern Navigation Systems Slide 1 of 23

2. Kalman Filtering – Part II Mon, April 4 EE 495 Modern Navigation Systems Slide 2 of 23

3. Kalman Filtering – Part II Mon, April 4 EE 495 Modern Navigation Systems CASE 1: A Fixed Constant  Simulation results Slide 3 of 23

4. Kalman Filtering – Part II Mon, April 4 EE 495 Modern Navigation Systems CASE 2: Measuring only position at 100 Hz estimate the velocity  Direct differentiation of noisy meas would be very bad  Kalman Filter: Let’s assume that the velocity is ~constant o State model: Slide 4 of 23

5. Kalman Filtering – Part II Mon, April 4 EE 495 Modern Navigation Systems A guess? From pos sensor specs Slide 5 of 23

6. Kalman Filtering – Part II Mon, April 4 EE 495 Modern Navigation Systems CASE 2: Measuring only position estimate velocity  Simulation results Slide 6 of 23

7. Kalman Filtering – Part II Mon, April 4 EE 495 Modern Navigation Systems CASE 2: Measuring only position estimate velocity  Simulation results o What if we tried to directly generate a velocity measurement? Slide 7 of 23

8. Kalman Filtering – Part II Mon, April 4 EE 495 Modern Navigation Systems CASE 2: Measuring only position estimate velocity  Simulation results: A Comparison Slide 8 of 23

9. Kalman Filtering – Part II Mon, April 4 EE 495 Modern Navigation Systems CASE 3: Estimate 1D Position and Velocity  The Holloman AFB High-Speed Test Track Slide 9 of 23

10. Kalman Filtering – Part II Mon, April 4 EE 495 Modern Navigation Systems Slide 10 of 23

11. Kalman Filtering – Part II Mon, April 4 EE 495 Modern Navigation Systems Accelerometer measurement  Bias instability + accel VRW type noise o where o and the bias instability can be modeled as o with  Accelerometer model Slide 11 of 23

12. Kalman Filtering – Part II Mon, April 4 EE 495 Modern Navigation Systems Accelerometer measurement Slide 12 of 23

13. Kalman Filtering – Part II Mon, April 4 EE 495 Modern Navigation Systems What if we simply integrated the accelerometer measurements (twice) to estimate position? IMU mechanization!! Slide 13 of 23

14. Kalman Filtering – Part II Mon, April 4 EE 495 Modern Navigation Systems GPS position measurement Slide 14 of 23

15. Kalman Filtering – Part II Mon, April 4 EE 495 Modern Navigation Systems Solution Approach #1:  Estimate the pos, vel, and accel o Will need a dynamic model for the “sled” – The dynamics can get complex (i.e., A & B)!! » Mass, friction, … Solution Approach #2:  Let’s estimate the error in the accel derived position estimate o Need only a model of the error dynamics – Do NOT need the dynamic model of the system (i.e., sled)!! IMU Slide 15 of 23

16. Kalman Filtering – Part II Mon, April 4 EE 495 Modern Navigation Systems Solution Approach #2:  Modeling the error dynamics o The velocity error dynamics o The position error dynamics white Non-white Augment the state vector!! Slide 16 of 23

17. Kalman Filtering – Part II Mon, April 4 EE 495 Modern Navigation Systems Solution Approach #2:  Modeling the error dynamics (summary)  Modeling the measurement equation Est = Truth -  Meas = Truth +  GPS measurement Slide 17 of 23

18. Kalman Filtering – Part II Mon, April 4 EE 495 Modern Navigation Systems Implementing the Kalman Filter: Slide 18 of 23

19. Kalman Filtering – Part II Mon, April 4 EE 495 Modern Navigation Systems Kalman Filter Results:  Position  Remember that we are estimating the error in the accel only derived position estimate!! Slide 19 of 23

20. Kalman Filtering – Part II Mon, April 4 EE 495 Modern Navigation Systems Kalman Filter Results:  Velocity Slide 20 of 23

21. Kalman Filtering – Part II Mon, April 4 EE 495 Modern Navigation Systems Kalman Filter Results: Bias Instability Slide 21 of 23

22. Kalman Filtering – Part II Mon, April 4 EE 495 Modern Navigation Systems Slide 22 of 23

23. Kalman Filtering – Part II Mon, April 4 EE 495 Modern Navigation Systems Overall architecture  Note that we are estimating the “error in the inertial-only estimate” !! o Then correcting the inertial-only estimate by subtracting this error!! Slide 23 of 23