1. Probabilistic Robotics Bayes Filter Implementations Particle filters

2. Sample-based Localization (sonar)

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

4. Distributions

5. Distributions Wanted: samples distributed according to p(x| z1, z2, z3)

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

7. Importance Sampling with Resampling

8. Importance Sampling with Resampling Weighted samples After resampling

9. Particle Filters

10. Sensor Information: Importance Sampling

11. Robot Motion

12. Sensor Information: Importance Sampling

13. Robot Motion

14. Particle Filter Algorithm • Algorithm particle_filter( St-1, ut-1 zt): • For Generate new samples • Sample index j(i) from the discrete distribution given by wt-1 • Sample from using and • Compute importance weight • Update normalization factor • Insert • For • Normalize weights

15. draw xit-1from Bel(xt-1) draw xitfrom p(xt | xit-1,ut-1) Importance factor for xit: Particle Filter Algorithm

16. Resampling • Given: Set S of weighted samples. • Wanted : Random sample, where the probability of drawing xi is given by wi. • Typically done n times with replacement to generate new sample set S’.

17. w1 wn w2 Wn-1 w1 wn w2 Wn-1 w3 w3 Resampling Implementation • Stochastic universal sampling • Systematic resampling • Linear time complexity • Easy to implement, low variance • Roulette wheel • Binary search, n log n

18. Resampling Algorithm • Algorithm systematic_resampling(S,n): • For Generate cdf • Initialize threshold • For Draw samples … • While ( ) Skip until next threshold reached • Insert • Increment threshold • ReturnS’ Also called stochastic universal sampling

19. Motion Model Reminder Start

20. Proximity Sensor Model Reminder Sonar sensor Laser sensor

39. Sample-based Localization (sonar)

40. Initial Distribution

41. After Incorporating Ten Ultrasound Scans

42. After Incorporating 65 Ultrasound Scans

43. Estimated Path

44. Using Ceiling Maps for Localization

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

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

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

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

49. Global Localization Using Vision

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