1. Particle Filters Pieter Abbeel UC Berkeley EECS Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAA

2. 2  For continuous spaces: often no analytical formulas for Bayes filter updates  Solution 1: Histogram Filters: (not studied in this lecture)  Partition the state space  Keep track of probability for each partition  Challenges:  What is the dynamics for the partitioned model?  What is the measurement model?  Often very fine resolution required to get reasonable results  Solution 2: Particle Filters:  Represent belief by random samples  Can use actual dynamics and measurement models  Naturally allocates computational resources where required (~ adaptive resolution)  Aka Monte Carlo filter, Survival of the fittest, Condensation, Bootstrap filter Motivation

3. Sample-based Localization (sonar)

4. Given a sample-based representation of Bel (x t ) = P(x t | z 1, …, z t, u 1, …, u t ) Find a sample-based representation of Bel (x t+1 ) = P(x t+1 | z 1, …, z t, z t+1, u 1, …, u t+1 ) Problem to be Solved

5. Given a sample-based representation of Bel (x t ) = P(x t | z 1, …, z t, u 1, …, u t ) Find a sample-based representation of P(x t+1 | z 1, …, z t, u 1, …, u t+1 ) Solution: For i=1, 2, …, N Sample x i t+1 from P(X t+1 | X t = x i t ) Dynamics Update

6. Sampling Intermezzo

7. Given a sample-based representation of P(x t+1 | z 1, …, z t ) Find a sample-based representation of P(x t+1 | z 1, …, z t, z t+1 ) = C * P(x t+1 | z 1, …, z t ) * P(z t+1 | x t+1 ) Solution: For i=1, 2, …, N w (i) t+1 = w (i) t * P(z t+1 | X t+1 = x (i) t+1 ) the distribution is represented by the weighted set of samples Observation update

8. Sample x 1 1, x 2 1, …, x N 1 from P(X 1 ) Set w i 1 = 1 for all i=1,…,N For t=1, 2, … Dynamics update: For i=1, 2, …, N Sample x i t+1 from P(X t+1 | X t = x i t ) Observation update: For i=1, 2, …, N w i t+1 = w i t * P(z t+1 | X t+1 = x i t+1 ) At any time t, the distribution is represented by the weighted set of samples { ; i=1,…,N} Sequential Importance Sampling (SIS) Particle Filter

9. SIS Demo

10. The resulting samples are only weighted by the evidence The samples themselves are never affected by the evidence  Fails to concentrate particles/computation in the high probability areas of the distribution P(x t | z 1, …, z t ) SIS particle filter major issue

11. At any time t, the distribution is represented by the weighted set of samples { ; i=1,…,N}  Sample N times from the set of particles  The probability of drawing each particle is given by its importance weight  More particles/computation focused on the parts of the state space with high probability mass Sequential Importance Resampling (SIR)

12. 1. Algorithm particle_filter( S t-1, u t, 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

13. Particle Filters

14. Sensor Information: Importance Sampling

15. Robot Motion

16. Sensor Information: Importance Sampling

17. Robot Motion

18. SIR Demo

19. SIR demo with random

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38. 38 Initial Distribution

39. 39 After Incorporating Ten Ultrasound Scans

40. 40 After Incorporating 65 Ultrasound Scans

41. 41 Estimated Path

42. 42 Summary – Particle Filters  Particle filters are an implementation of recursive Bayesian filtering  They represent the posterior by a set of weighted samples  They can model non-Gaussian distributions  Proposal to draw new samples  Weight to account for the differences between the proposal and the target

43. 43 Summary – PF Localization  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.