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Probabilistic Robotics

Probabilistic Robotics. Bayes Filter Implementations Particle filters. Sample-based Localization (sonar). Particle Filters. Represent belief by random samples Estimation of non-Gaussian, nonlinear processes

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Probabilistic Robotics

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  1. Probabilistic Robotics Bayes Filter Implementations Particle filters

  2. Sample-based Localization (sonar)

  3. Particle Filters • 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

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

  5. Importance Sampling with Resampling:Landmark Detection Example

  6. Distributions

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

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

  9. Importance Sampling with Resampling

  10. Importance Sampling with Resampling Weighted samples After resampling

  11. Particle Filters

  12. Sensor Information: Importance Sampling

  13. Robot Motion

  14. Sensor Information: Importance Sampling

  15. Robot Motion

  16. 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

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

  18. 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’.

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

  20. 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

  21. Motion Model Reminder Start

  22. Proximity Sensor Model Reminder Sonar sensor Laser sensor

  23. Sample-based Localization (sonar)

  24. Initial Distribution

  25. After Incorporating Ten Ultrasound Scans

  26. After Incorporating 65 Ultrasound Scans

  27. Estimated Path

  28. Using Ceiling Maps for Localization

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

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

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

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

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