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Bayesian Filtering

Bayesian Filtering. Dieter Fox. Probabilistic Robotics. Key idea: Explicit representation of uncertainty (using the calculus of probability theory) Perception = state estimation Control = utility optimization. Bayes Filters: Framework. Given:

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Bayesian Filtering

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  1. Bayesian Filtering Dieter Fox

  2. Probabilistic Robotics Key idea: Explicit representation of uncertainty (using the calculus of probability theory) • Perception = state estimation • Control = utility optimization

  3. Bayes Filters: Framework • Given: • Stream of observations z and action data u: • Sensor modelP(z|x). • Action modelP(x|u,x’). • Prior probability of the system state P(x). • Wanted: • Estimate of the state X of a dynamical system. • The posterior of the state is also called Belief:

  4. Markov Assumption Underlying Assumptions • Static world • Independent noise • Perfect model, no approximation errors

  5. Bayes Markov Total prob. Markov z = observation u = action x = state Bayes Filters

  6. Bayes Filters are Familiar! • Kalman filters • Particle filters • Hidden Markov models • Dynamic Bayesian networks • Partially Observable Markov Decision Processes (POMDPs)

  7. Localization “Using sensory information to locate the robot in its environment is the most fundamental problem to providing a mobile robot with autonomous capabilities.” [Cox ’91] • Given • Map of the environment. • Sequence of sensor measurements. • Wanted • Estimate of the robot’s position. • Problem classes • Position tracking • Global localization • Kidnapped robot problem (recovery)

  8. Bayes Filters for Robot Localization

  9. Probabilistic Kinematics • Odometry information is inherently noisy. p(x|u,x’) x’ x’ u u

  10. Proximity Measurement • Measurement can be caused by … • a known obstacle. • cross-talk. • an unexpected obstacle (people, furniture, …). • missing all obstacles (total reflection, glass, …). • Noise is due to uncertainty … • in measuring distance to known obstacle. • in position of known obstacles. • in position of additional obstacles. • whether obstacle is missed.

  11. Mixture Density How can we determine the model parameters?

  12. Raw Sensor Data Measured distances for expected distance of 300 cm. Sonar Laser

  13. Approximation Results Laser Sonar 300cm 400cm

  14. Representations for Bayesian Robot Localization • Kalman filters (late-80s?) • Gaussians • approximately linear models • position tracking • Discrete approaches (’95) • Topological representation (’95) • uncertainty handling (POMDPs) • occas. global localization, recovery • Grid-based, metric representation (’96) • global localization, recovery Robotics AI • Particle filters (’99) • sample-based representation • global localization, recovery • Multi-hypothesis (’00) • multiple Kalman filters • global localization, recovery

  15. Discrete Grid Filters

  16. Piecewise Constant Representation

  17. Grid-based Localization

  18. Sonars and Occupancy Grid Map

  19. Tree-based Representation Idea: Represent density using a variant of Octrees

  20. Tree-based Representations • Efficient in space and time • Multi-resolution

  21. Particle Filters

  22. 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

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

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

  25. Particle Filters

  26. Sensor Information: Importance Sampling

  27. Robot Motion

  28. Sensor Information: Importance Sampling

  29. Robot Motion

  30. Sample-based Localization (sonar)

  31. Using Ceiling Maps for Localization [Dellaert et al. 99]

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

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

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

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

  36. Global Localization Using Vision

  37. Localization for AIBO robots

  38. Adaptive Sampling

  39. KLD-sampling • Idea: • Assume we know the true belief. • Represent this belief as a multinomial distribution. • Determine number of samples such that we can guarantee that, with probability (1- d), the KL-distance between the true posterior and the sample-based approximation is less than e. • Observation: • For fixed d and e, number of samples only depends on number k of bins with support:

  40. Example Run Sonar

  41. Example Run Laser

  42. Kalman Filters

  43. Bayes Filter Reminder • Prediction • Correction

  44. m Univariate -s s m Multivariate Gaussians

  45. Gaussians and Linear Functions

  46. Kalman Filter Updates in 1D

  47. Kalman Filter Algorithm • Algorithm Kalman_filter( mt-1,St-1, ut, zt): • Prediction: • Correction: • Returnmt,St

  48. Nonlinear Dynamic Systems • Most realistic robotic problems involve nonlinear functions

  49. Linearity Assumption Revisited

  50. Non-linear Function

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