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Probabilistic Robotics Bayes Filter Implementations Particle filters

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Sample-based Localization (sonar)

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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 Particle Filters

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Importance Sampling with Resampling

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Weighted samples After resampling

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Particle Filters

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Sensor Information: Importance Sampling

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Robot Motion

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Sensor Information: Importance Sampling

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Robot Motion

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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

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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

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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’.

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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

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Start Motion Model Reminder

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Proximity Sensor Model Reminder Laser sensor Sonar sensor

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36 Sample-based Localization (sonar)

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37 Initial Distribution

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38 After Incorporating Ten Ultrasound Scans

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39 After Incorporating 65 Ultrasound Scans

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40 Estimated Path

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Using Ceiling Maps for Localization

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

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

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

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Elsewhere Measurement z:P(z|x):

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Global Localization Using Vision

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47 Robots in Action: Albert

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

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Localization for AIBO robots

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50 Limitations The approach described so far is able to track the pose of a mobile robot and to globally localize the robot. How can we deal with localization errors (i.e., the kidnapped robot problem)?

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51 Approaches Randomly insert samples (the robot can be teleported at any point in time). Insert random samples proportional to the average likelihood of the particles (the robot has been teleported with higher probability when the likelihood of its observations drops).

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52 Random Samples Vision-Based Localization 936 Images, 4MB,.6secs/image Trajectory of the robot:

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53 Odometry Information

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54 Image Sequence

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55 Resulting Trajectories Position tracking:

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56 Resulting Trajectories Global localization:

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57 Global Localization

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58 Kidnapping the Robot

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Recovery from Failure

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60 Summary Particle filters are an implementation of recursive Bayesian filtering They represent the posterior by a set of weighted samples. 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.

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