1. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 1 Chapter 8 State Estimation

2. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 2 Why estimate states? State information needed to control aircraft We measure many things –accelerations, angular rates, pressure altitude, dynamic pressure (airspeed), magnetic heading (sometimes), GPS position Don’t have measurement of everything we need

3. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 3 Benchmark Maneuver

4. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 4 Inverting Sensor Measurement Model Several states can be estimated by inverting sensor measurement model –Angular rates, altitude, airspeed –LPF denotes low pass filter

5. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 5

6. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 6

7. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 7 Inverting Sensor Measurement Model

8. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 8

9. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 9 Inverting Sensor Measurement Model

10. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 10

11. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 11 Dynamic Observer Theory

12. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 12 Dynamic Observer Theory, cont.

13. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 13 Dynamic Observer Theory, cont.

14. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 14 Kalman Filter

15. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 15 Quadratic Forms

16. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 16 Multivariate Gaussian

17. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 17 Kalman Filter, cont.

18. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 18 Kalman Filter, cont.

19. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 19 EKF Algorithm

20. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 20 Attitude Estimation Using EKF

21. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 21 Attitude Estimation Using EKF

22. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 22 Attitude Estimation Using EKF

23. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 23 Attitude Estimation Using EKF

24. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 24 Attitude Estimation Results Not perfect, but significantly better!

25. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 25 GPS Smoothing

26. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 26 GPS Smoothing

27. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 27 GPS Smoothing

28. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 28 GPS Smoothing

29. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 29 GPS Smoothing

30. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 30 GPS Smoothing

31. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 31 GPS Smoothing Results

32. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 32 Matlab Simulation

33. Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 8, Slide 33 Suggestions for Tuning Your EKF Implement EKF in steps –Attitude estimation –GPS smoother Test and tune each component independently and thoroughly –Attitude estimator first –GPS smoother second As a first step, make sure your filter estimates track the states when sensors are perfect (no sensor error). This will expose coding errors and show the limits of performance of your filter. Keep the wind set to zero until you are confident that your filter is well tuned. We know R pretty well typically. Tune filter by changing Q. Tune filter by focusing on individual states. Give inputs to those states while keeping other states “quiet”. Adjust Q value corresponding to state of interest. Tune by trial and error (use your brain to make a good guess!). Don’t put extreme inputs into the system when tuning the filter (e.g., large steps). Your filter will have its own dynamic limits – don’t make the problem impossibly difficult. Tune your controllers so that the response of the aircraft to typical inputs is smooth and graceful. Abrupt and jerky state dynamics are difficult to estimate. Check your equations AGAIN!