2. Probabilistic Robotics Bayes Filter Implementations Particle filters

3. Sample-based Localization (sonar)

4.  Represent belief by random samples  Estimation of non-Gaussian, nonlinear processes  Monte Carlo filter, Survival of the fittest, Condensation, Bootstrap filter, Particle filter  Filtering: [Rubin, 88], [Gordon et al., 93], [Kitagawa 96]  Computer vision: [Isard and Blake 96, 98]  Dynamic Bayesian Networks: [Kanazawa et al., 95]d Particle Filters

5. Weight samples: w = f / g Importance Sampling

6. Importance Sampling with Resampling: Landmark Detection Example

7. Distributions

8. 7 Wanted: samples distributed according to p(x| z 1, z 2, z 3 )

9. This is Easy! We can draw samples from p(x|z l ) by adding noise to the detection parameters.

10. Importance Sampling with Resampling

11. Weighted samples After resampling

12. Particle Filters

13. Sensor Information: Importance Sampling

14. Robot Motion

15. Sensor Information: Importance Sampling

16. Robot Motion

17. 1. Algorithm particle_filter( S t-1, u t-1 z t ): 2. 3. For Generate new samples 4. Sample index j(i) from the discrete distribution given by w t-1 5. Sample from using and 6. Compute importance weight 7. Update normalization factor 8. Insert 9. For 10. Normalize weights Particle Filter Algorithm

18. draw x i t  1 from Bel (x t  1 ) draw x i t from p ( x t | x i t  1,u t  1 ) Importance factor for x i t : Particle Filter Algorithm

19. Resampling Given: Set S of weighted samples. Wanted : Random sample, where the probability of drawing x i is given by w i. Typically done n times with replacement to generate new sample set S’.

20. w2w2 w3w3 w1w1 wnwn W n-1 Resampling w2w2 w3w3 w1w1 wnwn W n-1 Roulette wheel Binary search, n log n Stochastic universal sampling Systematic resampling Linear time complexity Easy to implement, low variance

21. 1. Algorithm systematic_resampling(S,n): 2. 3. ForGenerate cdf 4. 5. Initialize threshold 6. ForDraw samples … 7. While ( )Skip until next threshold reached 8. 9. Insert 10. Increment threshold 11. Return S’ Resampling Algorithm Also called stochastic universal sampling

22. Start Motion Model Reminder

23. Proximity Sensor Model Reminder Laser sensor Sonar sensor

24. 23

25. 24

26. 25

27. 26

28. 27

29. 28

30. 29

31. 30

32. 31

33. 32

34. 33

35. 34

36. 35

37. 36

38. 37

39. 38

40. 39

41. 40

42. 41

43. 42 Sample-based Localization (sonar)

44. 43 Initial Distribution

45. 44 After Incorporating Ten Ultrasound Scans

46. 45 After Incorporating 65 Ultrasound Scans

47. 46 Estimated Path

48. Using Ceiling Maps for Localization

49. Vision-based Localization P(z|x) h(x) z

50. Under a Light Measurement z:P(z|x):

51. Next to a Light Measurement z:P(z|x):

52. Elsewhere Measurement z:P(z|x):

53. Global Localization Using Vision

54. 53 Robots in Action: Albert

55. 54 Application: Rhino and Albert Synchronized in Munich and Bonn [Robotics And Automation Magazine, to appear]

56. Localization for AIBO robots

57. 56 Limitations The approach described so far is able to track the pose of a mobile robot and to globally localize the robot. How can we deal with localization errors (i.e., the kidnapped robot problem)?

58. 57 Approaches Randomly insert samples (the robot can be teleported at any point in time). Insert random samples proportional to the average likelihood of the particles (the robot has been teleported with higher probability when the likelihood of its observations drops).

59. 58 Random Samples Vision-Based Localization 936 Images, 4MB,.6secs/image Trajectory of the robot:

60. 59 Odometry Information

61. 60 Image Sequence

62. 61 Resulting Trajectories Position tracking:

63. 62 Resulting Trajectories Global localization:

64. 63 Global Localization

65. 64 Kidnapping the Robot

66. Recovery from Failure

67. 66 Summary Particle filters are an implementation of recursive Bayesian filtering They represent the posterior by a set of weighted samples. In the context of localization, the particles are propagated according to the motion model. They are then weighted according to the likelihood of the observations. In a re-sampling step, new particles are drawn with a probability proportional to the likelihood of the observation.