Slides for the book: Probabilistic Robotics

1. Slides for the book:Probabilistic Robotics • Authors: • Sebastian Thrun • Wolfram Burgard • Dieter Fox • Publisher: • MIT Press, 2005. • Web site for the book & more slides: http://www.probabilistic-robotics.org/

2. Probabilistic Robotics Bayes Filter Implementations Particle filters

3. Sample-based Localization (sonar)

4. Particle Filters • 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

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

6. Importance Sampling with Resampling:Landmark Detection Example

7. Distributions

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

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

10. Importance Sampling with Resampling

11. Importance Sampling with Resampling Weighted samples After resampling

12. Particle Filters

13. Sensor Information: Importance Sampling

14. Robot Motion

15. Sensor Information: Importance Sampling

16. Robot Motion

17. 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

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

19. 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’.

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

21. 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

22. Motion Model Reminder Start

23. Proximity Sensor Model Reminder Sonar sensor Laser sensor

42. Sample-based Localization (sonar)

43. Initial Distribution

44. After Incorporating Ten Ultrasound Scans

45. After Incorporating 65 Ultrasound Scans

46. Estimated Path

47. Using Ceiling Maps for Localization

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

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

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