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. Importance Sampling with Resampling

6. Weighted samples After resampling

7. Particle Filters

8. Sensor Information: Importance Sampling

9. Robot Motion

10. Sensor Information: Importance Sampling

11. Robot Motion

12. 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

13. 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

14. 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’.

15. 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

16. Start Motion Model Reminder

17. Proximity Sensor Model Reminder Laser sensor Sonar sensor

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37. 36 Sample-based Localization (sonar)

38. 37 Initial Distribution

39. 38 After Incorporating Ten Ultrasound Scans

40. 39 After Incorporating 65 Ultrasound Scans

41. 40 Estimated Path

42. Using Ceiling Maps for Localization

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

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

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

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

47. Global Localization Using Vision

48. 47 Robots in Action: Albert

49. 48 Application: Rhino and Albert Synchronized in Munich and Bonn [Robotics And Automation Magazine, to appear]

50. Localization for AIBO robots

51. 50 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)?

52. 51 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).

53. 52 Random Samples Vision-Based Localization 936 Images, 4MB,.6secs/image Trajectory of the robot:

54. 53 Odometry Information

55. 54 Image Sequence

56. 55 Resulting Trajectories Position tracking:

57. 56 Resulting Trajectories Global localization:

58. 57 Global Localization

59. 58 Kidnapping the Robot

60. Recovery from Failure

61. 60 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.