2. Particle filters for Robot Localization An implementation of Bayes Filtering Markov Localization

3. Monte Carlo Localization Repeated Random Sampling

4.  Idea: Represent belief by random samples  Estimation of non-Gaussian, nonlinear processes  Monte Carlo filter, Survival of the fittest, Condensation, Bootstrap filter, Particle filter Particle Filters

5. MC Localization Start with lots of random samples (aka particles) Weight each with probability based on sensor reading.

6. MC Localization Given an action a, Use importance weighting to select from the old particles Probabilistically create new particle given action and motion model

7. Adaptive Monte Carlo Localization often abbreviated AMCL

8. Particle Filters

9. Sensor Information: Importance Sampling

10. Robot Motion

11. Sensor Information: Importance Sampling

12. Robot Motion

13. Particle Filter Algorithm Sample the next generation for particles using the proposal distribution Compute the importance weights : weight = target distribution / proposal distribution Resampling: "Replace unlikely samples by more likely ones"

14. 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

15. 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

16. 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’.

17. 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

18. 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

19. Start Motion Model Reminder

20. 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. Mobile Robot Localization Each particle is a potential pose of the robot Proposal distribution is the motion model of the robot (prediction step) The observation model is used to compute the importance weight (correction step)