1. Probabilistic Robotics Bayes Filter Implementations Gaussian filters

2. Prediction Correction Bayes Filter Reminder

3. Gaussians --   Univariate  Multivariate

4. Properties of Gaussians

5. We stay in the “Gaussian world” as long as we start with Gaussians and perform only linear transformations. Multivariate Gaussians

6. 6 Discrete Kalman Filter Estimates the state x of a discrete-time controlled process that is governed by the linear stochastic difference equation with a measurement

7. 7 Components of a Kalman Filter Matrix (nxn) that describes how the state evolves from t to t-1 without controls or noise. Matrix (nxl) that describes how the control u t changes the state from t to t-1. Matrix (kxn) that describes how to map the state x t to an observation z t. Random variables representing the process and measurement noise that are assumed to be independent and normally distributed with covariance R t and Q t respectively.

8. 8 Kalman Filter Updates in 1D

9. 9

10. 10 Kalman Filter Updates in 1D

11. 11 Kalman Filter Updates

12. 12 Linear Gaussian Systems: Initialization Initial belief is normally distributed:

13. 13 Dynamics are linear function of state and control plus additive noise: Linear Gaussian Systems: Dynamics

14. 14 Linear Gaussian Systems: Dynamics

15. 15 Observations are linear function of state plus additive noise: Linear Gaussian Systems: Observations

16. 16 Linear Gaussian Systems: Observations

17. 17 Kalman Filter Algorithm 1. Algorithm Kalman_filter(  t-1,  t-1, u t, z t ): 2. Prediction: 3. 4. 5. Correction: 6. 7. 8. 9. Return  t,  t

18. 18 The Prediction-Correction-Cycle Prediction

19. 19 The Prediction-Correction-Cycle Correction

20. 20 The Prediction-Correction-Cycle Correction Prediction

21. 21 Kalman Filter Summary Highly efficient: Polynomial in measurement dimensionality k and state dimensionality n: O(k 2.376 + n 2 ) Optimal for linear Gaussian systems! Most robotics systems are nonlinear!

22. 22 Nonlinear Dynamic Systems Most realistic robotic problems involve nonlinear functions

23. 23 Linearity Assumption Revisited

24. 24 Non-linear Function

25. 25 EKF Linearization (1)

26. 26 EKF Linearization (2)

27. 27 EKF Linearization (3)

28. 28 EKF Linearization (4)

29. 29 EKF Linearization (5)

30. 30 Prediction: Correction: EKF Linearization: First Order Taylor Series Expansion

31. 31 EKF Algorithm 1.Extended_Kalman_filter (  t-1,  t-1, u t, z t ): 2. Prediction: 3. 4. 5. Correction: 6. 7. 8. 9. Return  t,  t

32. 32 Localization Given Map of the environment. Sequence of sensor measurements. Wanted Estimate of the robot’s position. Problem classes Position tracking Global localization Kidnapped robot problem (recovery) “Using sensory information to locate the robot in its environment is the most fundamental problem to providing a mobile robot with autonomous capabilities.” [Cox ’91]

33. 33 Landmark-based Localization

34. 34 1.EKF_localization (  t-1,  t-1, u t, z t, m): Prediction: 6. 7. 8. Motion noise Jacobian of g w.r.t location Predicted mean Predicted covariance Jacobian of g w.r.t control

35. 35 1.EKF_localization (  t-1,  t-1, u t, z t, m): Correction: 6. 7. 8. 9. 10. Predicted measurement mean Pred. measurement covariance Kalman gain Updated mean Updated covariance Jacobian of h w.r.t location

36. 36 EKF Prediction Step

37. 37 EKF Observation Prediction Step

38. 38 EKF Correction Step

39. 39 Estimation Sequence (1)

40. 40 Estimation Sequence (2)

41. 41 Comparison to GroundTruth

42. 42 EKF Summary Highly efficient: Polynomial in measurement dimensionality k and state dimensionality n: O(k 2.376 + n 2 ) Not optimal! Can diverge if nonlinearities are large! Works surprisingly well even when all assumptions are violated!

43. 43 Linearization via Unscented Transform EKF UKF

44. 44 UKF Sigma-Point Estimate (2) EKF UKF

45. 45 UKF Sigma-Point Estimate (3) EKF UKF

46. 46 Unscented Transform Sigma points Weights Pass sigma points through nonlinear function Recover mean and covariance

47. 47 UKF_localization (  t-1,  t-1, u t, z t, m): Prediction: Motion noise Measurement noise Augmented state mean Augmented covariance Sigma points Prediction of sigma points Predicted mean Predicted covariance

48. 48 UKF_localization (  t-1,  t-1, u t, z t, m): Correction: Measurement sigma points Predicted measurement mean Pred. measurement covariance Cross-covariance Kalman gain Updated mean Updated covariance

49. 49 1.EKF_localization (  t-1,  t-1, u t, z t, m): Correction: 6. 7. 8. 9. 10. Predicted measurement mean Pred. measurement covariance Kalman gain Updated mean Updated covariance Jacobian of h w.r.t location

50. 50 UKF Prediction Step

51. 51 UKF Observation Prediction Step

52. 52 UKF Correction Step

53. 53 EKF Correction Step

54. 54 Estimation Sequence EKF PF UKF

55. 55 Estimation Sequence EKF UKF

56. 56 Prediction Quality EKF UKF

57. 57 UKF Summary Highly efficient: Same complexity as EKF, with a constant factor slower in typical practical applications Better linearization than EKF: Accurate in first two terms of Taylor expansion (EKF only first term) Derivative-free: No Jacobians needed Still not optimal!

58. 58 [Arras et al. 98]: Laser range-finder and vision High precision (<1cm accuracy) Kalman Filter-based System [Courtesy of Kai Arras]

59. 59 Multi- hypothesis Tracking

60. 60 Belief is represented by multiple hypotheses Each hypothesis is tracked by a Kalman filter Additional problems: Data association: Which observation corresponds to which hypothesis? Hypothesis management: When to add / delete hypotheses? Huge body of literature on target tracking, motion correspondence etc. Localization With MHT

61. 61 Hypotheses are extracted from LRF scans Each hypothesis has probability of being the correct one: Hypothesis probability is computed using Bayes’ rule Hypotheses with low probability are deleted. New candidates are extracted from LRF scans. MHT: Implemented System (1) [Jensfelt et al. ’00]

62. 62 MHT: Implemented System (2) Courtesy of P. Jensfelt and S. Kristensen

63. 63 MHT: Implemented System (3) Example run Map and trajectory # hypotheses #hypotheses vs. time P(H best ) Courtesy of P. Jensfelt and S. Kristensen