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M onte C arlo L ocalization for Mobile Robots

M onte C arlo L ocalization for Mobile Robots. Frank Dellaert 1 , Dieter Fox 2 , Wolfram Burgard 3 , Sebastian Thrun 4 1 Georgia Institute of Technology 2 University of Washington 3 University of Bonn 4 Carnegie Mellon University. Take Home Message.

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M onte C arlo L ocalization for Mobile Robots

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  1. Monte Carlo Localizationfor Mobile Robots Frank Dellaert1, Dieter Fox2,Wolfram Burgard3, Sebastian Thrun4 1Georgia Institute of Technology 2University of Washington 3University of Bonn 4Carnegie Mellon University

  2. Take Home Message Representing uncertainty using samples is powerful, fast, and simple !

  3. Outline Robot Localization Sensor ? Density Representation ? Monte Carlo Localization Results

  4. Outline Robot Localization Sensor ? Density Representation ? Monte Carlo Localization Results

  5. Minerva

  6. Motivation • Crowded public spaces • Unmodified environments

  7. Museum Application Desired Location Exhibit

  8. Global Localization • Where in the world is Minerva the Robot ? • Vague initial estimate • Noisy and ambiguous sensors

  9. Local Tracking • Sharp initial estimate • Noisy and ambiguous sensors

  10. The Bayesian Paradigm • Knowledge as a probability distribution 60% Rain 40% dry

  11. Probability of Robot Location P(Robot Location) Y State space = 2D, infinite #states X

  12. Bayesian Filtering • Two phases: 1. Prediction Phase2. Measurement Phase

  13. 1. Prediction Phase u xt-1 xt P(xt) =  P(xt|xt-1,u) P(xt-1) Motion Model

  14. 2. Measurement Phase z xt P(xt|z) = k P(z|xt) P(xt) Sensor Model

  15. Outline Robot Localization Sensor ? Density Representation ? Monte Carlo Localization Results

  16. What sensor ? • Sonar ? • Laser ? • Vision ?

  17. Problem: Large Open Spaces • Walls and obstacles out of range • Sonar and laser have problems • One solution: Coastal Navigation

  18. Problem: Large Crowds • Horizontally mounted sensors have problems • One solution: Robust filtering

  19. Solution: Ceiling Camera • Upward looking camera • Model of the world = Ceiling Mosaic

  20. Global Alignment (other talk)

  21. Ceiling Mosaic

  22. Large FOV Problems • 3D ceiling -> 3D Model ? • Matching whole images slow

  23. Small FOV Solution • Model = orthographic mosaic • No 3D Effects • Very fast

  24. P(z|x) z h(x) Vision based Sensor

  25. Outline Robot Localization Sensor ? Density Representation ? Monte Carlo Localization Results

  26. Hidden Markov Models A B C D E A B C D E

  27. Kalman Filter • Very powerful • Gaussian, unimodal sensor motion motion

  28. Under Light

  29. Next to Light

  30. Elsewhere

  31. Markov Localization • Fine discretization over {x,y,theta} • Very successful: Rhino, Minerva, Xavier…

  32. Dynamic Markov Localization • Burgard et al., IROS 98 • Idea: use Oct-trees

  33. Sampling as Representation P(Robot Location) Y X

  34. Samples <=> Densities • Density => samplesObvious • Samples => densityHistogram, Kernel Density Estimation

  35. Sampling Advantages • Arbitrary densities • Memory = O(#samples) • Only in “Typical Set” • Great visualization tool ! • minus: Approximate

  36. Outline Robot Localization Sensor ? Density Representation ? Monte Carlo Localization Results

  37. Disclaimer • Handschin 1970 (!)lacked computing power • Bootstrap filter 1993 Gordon et al. • Monte Carlo filter 1996 Kitagawa • Condensation 1996 Isard & Blake

  38. Added Twists • Camera moves, not object • Global localization

  39. Monte Carlo Localization Sk-1 weighted S’k Sk S’k Predict Weight Resample

  40. 1. Prediction Phase u P(xt| ,u) Motion Model

  41. 2. Measurement Phase P(z|xt) Sensor Model

  42. 3. Resampling Step O(N)

  43. A more in depth look

  44. Bayes Law, new look • Densities:update prior p(x) to p(x|z) via l(x;z) • Samplesupdate a sample from p(x) to a sample from the posterior p(x|z) through l(x;z)

  45. Bayes Law Problem • We really want p(x|z) samples • But we only have p(x) samples ! • How can we upgrade p(x) to p(x|z) ?

  46. More General Problem • We really want h(x) samples • But we only have g(x) samples ! • How can we upgrade g(x) to h(x) ?

  47. Solution = Importance Sampling • 1. generate xi from g(x) • 2. calculate wi = h(xi)/g(xi) • 3. assign weight qi = wi/ wi • Still works if h(x) only known up to normalization factor

  48. Mean and Weighted Mean • Fair sample:obtain samples xi from p(x|z)E[m(x)|z] ~ m(xi)/N • Weighted sample: obtain weighted samples (xi,qi) from p(x|z) E[m(x)|z] ~ qi m(xi)

  49. Bayes Law using Samples • 1. generate xi from p(x) • 2. calculate wi = l(xi;z) • 3. assign weight qi = wi/  wi • Indeed: wi=p(x|z)/p(x) = l(x;z) p(x) /p(x) = l(x;z) • 4. if you want, resample from (xi,qi)

  50. Monte Carlo Localization Sk-1 weighted S’k Sk S’k Predict Weight Resample

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