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Probabilistic Robotics

Probabilistic Robotics. Bayes Filter Implementations Particle filters. Sample-based Localization (sonar). Importance Sampling. Weight samples: w = f / g. Distributions. Distributions. Wanted: samples distributed according to p(x| z 1 , z 2 , z 3 ). This is Easy!.

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Probabilistic Robotics

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  1. Probabilistic Robotics Bayes Filter Implementations Particle filters

  2. Sample-based Localization (sonar)

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

  4. Distributions

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

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

  7. Importance Sampling with Resampling

  8. Importance Sampling with Resampling Weighted samples After resampling

  9. Particle Filters

  10. Sensor Information: Importance Sampling

  11. Robot Motion

  12. Sensor Information: Importance Sampling

  13. Robot Motion

  14. 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

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

  16. 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’.

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

  18. 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

  19. Motion Model Reminder Start

  20. Proximity Sensor Model Reminder Sonar sensor Laser sensor

  21. Sample-based Localization (sonar)

  22. Initial Distribution

  23. After Incorporating Ten Ultrasound Scans

  24. After Incorporating 65 Ultrasound Scans

  25. Estimated Path

  26. Using Ceiling Maps for Localization

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

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

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

  30. Elsewhere Measurement z: P(z|x):

  31. Global Localization Using Vision

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

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