1. Slam is a State Estimation Problem

3. Predicted belief corrected belief

4. Bayes Filter Reminder

5. Gaussians

6. Standard deviation Covariance matrix

7. Gaussians in one and two dimensions One standard deviation two standard deviations

8. Gaussians in three dimensions Multivariate probability

9. Properties of Gaussians for Univariate case Linear system Standard deviation on output of linear system Mean on output of linear system For two-dimensional system:

10. Properties of Gaussians Properties of Gaussians for Multivariate case From previous slide

11. Properties of Gaussians Important Property of all these methods

12. Discrete Kalman Filters

13. Kalman Filter background 1.Kalman Filter is a Bayes Filter 2.Kalman Filter uses Gaussians 3.Estimator for the linear Gaussian case 4.Optimal solution for linear models and Gaussian distributions 5.Developed in late 1950’s 6.Most relevant Bayes filter variant in practice 7.Applications in econcomics, weather forecasting, satellite navigations, GPS, robotics, robot vision and many other 8.Kalman filter is just few matrix operations such as multiplication.

14. Discrete Kalman Filter

15. Components of a Kalman Filter

16. Example of Example of Kalman Filter Updates in one dimension Kalman Filter calculates a weighted mean value!

17. Kalman Filter Updates in 1D: PREDICTION Single dimension Matrices in multi-dimensions Again generalization to many dimensions here

18. CORRECTION Kalman Filter Updates in 1D: CORRECTION Variant single variable Generalization: Generalization: Variant of multiple variables matrix

19. Kalman Filter Updates

20. Linear Gaussian Systems

21. Initialization Linear Gaussian Systems: Initialization Initial belief has a normal distribution:

22. Dynamics Linear Gaussian Systems: Dynamics Gaussian

23. Linear Gaussian Systems: Dynamics From previous slide Linear, gaussian

24. Linear Gaussian Systems: Observations R = correction

25. Linear Gaussian Systems: Observations

26. : Marginalization and Conditioning Properties: Marginalization and Conditioning Notation for Gaussians All are Gaussian

27. Kalman Filter assumes linearity Zero-mean Gaussian Noise

28. Linear Motion Model We want to calculate this probability variable

29. Theorem 1

30. We want to calculate this probability variable

31. Theorem 2

32. the belief is Gaussian! Everything stays Gaussian: the belief is Gaussian! Probabilistic Robotics Proofs of these theorems and properties are not trivial and can be found in the book by ‘three Germans” called Probabilistic Robotics. Theorem 3

33. Kalman Filter Algorithm

34. The Kalman Filter Assumptions are: 1.Gaussian distributions 2.Gaussian noise 3.Linear motion 4.Linear observation model Discuss later

35. Calculates multi- dimensional mean and covariance matrix Prediction phase Correction phase R for motion Q for measurement Prediction of multi-dimensional mean Prediction of multi-dimensional covariance matrix Calculates corrected multi- dimensional mean and covariance matrix Kalman

36. Kalman Filter Algorithm Different notation to previous slide Measurement noise

37. Kalman Filter Algorithm: navigation using odometry and measurement to landmark Predicted and corrected position of the ship

38. The Prediction-Correction-Cycle The phase of Prediction

39. The Prediction-Correction-Cycle The phase of Correction

40. The Prediction-Correction-Cycle

41. The general Optimal State Estimation Problem

42. Diagram of general State Estimation 123123 2 or 3 !

43. Discrete Kalman Filter This is what we discussed

44. Linear-Optimal State Estimation Compare with this Change with time derivative

45. Linear-Optimal State Estimation (Kalman-Bucy Filter) Similar to before Kalman

46. Estimation Gain for the Kalman-Bucy Filter Same equations as those that define control gain, except – solution matrix, P, propagated forward in time – Matrices and matrix sequences are different

48. Second-Order Example of Kalman- Bucy Filter

50. Kalman-Bucy Filter with Two Measurements

51. State Estimate with Angle Measurement Only

52. Kalman Filter Summary

53. Non-Linear Dynamic Systems

54. Sources Wolfram Burgard Cyrill Stachniss, Maren Bennewitz Kal Arras