1. Graphical Models for Mobile Robot Localization Shuang Wu

2. Global Localization In an occupancy map, estimate the pose of the robot X = (0,0) θ occupied free

3. The inputs Laser readings z t = {r 1,r 2 … r 180 } Odometer reading a t = {d,theta} Prior sample set, Bel(X 0 ) =, n possible pose

4. Could use Bayes Filter prediction update

5. Recursive Bayes Filter sensor model action model normalizing constant

6. New Positions: Generate X ’ according to P(X ’ |X,a) t odometer reading t+1 a t = (1, 45˚) p(X’|X,a) ~ X’ + N(0,Σd) + N(0, Σθ)

7. Sensor model weight w ’ by p(z|X ’ ) z t = (1, 1.41,…,3) w, t+1 p(z|X’)*w, t+1 p(z|X’) = N(d,Σs), where Σs is the measure of noise in the laser readings d is the distance to the closest obstacle p(z|X’) =

8. Sampling Likelihood weighted sampling Samples are drawn from the target distribution Importance Sampling Samples are drawn from a proposal distribution(g) Re-weight samples to account for the difference between the proposal and target distribution proposal target Key: represent belief states by set of weighted samples g 0, f 0

9. Resampling Reason: Waste of CPU time if we keep propagating particles that have 0 weight. Goal: Minimize variance of the importance weights

10. Rao-Blackwell Theorem Particle Filters approximate any distribution independent of size of state space. The complexity of the standard PF algorithm is O(N). Motivation: In very high-dimensional spaces, a large number of particles is needed to represent the posterior Solution: Reduce size of state space by marginalizing out some of the variables.

11. Rao-Blackwellised Step1. Divide the set of variables into sets R and L where R = set of sampled variables and L = remaining variables in the DBN at time t. Choice of RB variables: nodes that have no parents in the current time slice or have parents that are already in R. Keep growing R until the remaining variables can be updated exactly. Step2. Sample the set of variables R t from R t-1 using standard PF Step3. Compute the remain variables analytically.

12. RBPF Now the particle set is represented by Prediction: Update: Proposal distribution: Importance weights

13. Implementation Implementation Summary: Used Rao-Blackwellised particle filtering to solve localization problem. Able to localize within about 4cm (using a 20 x 20 x 10 resolution grid). Implementation is written in Java. Takes 16 seconds on my 2.4 GHz 512MB Windows XP machine for data set 1 (not counting loading data and ray tracing pre-computations). Ray tracing takes 8 seconds, plus 14 seconds for pre-computing grid ranges.

14. Results(Implemented in Java)

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18. End of Forward Run

19. The End