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Probabilistic Robotics

Probabilistic Robotics. Robot Localization. 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.

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Probabilistic Robotics

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  1. Probabilistic Robotics Robot Localization

  2. 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)

  3. Localization

  4. Localization Position tracking

  5. Localization Global localization

  6. Landmark-based Localization

  7. Linearity Assumption Revisited

  8. Non-linear Function

  9. EKF Linearization (1)

  10. EKF Linearization (2)

  11. EKF Linearization (3)

  12. EKF Linearization: First Order Taylor Series Expansion • Prediction: • Correction:

  13. EKF Algorithm • Extended_Kalman_filter( mt-1,St-1, ut, zt): • Prediction: • Correction: • Returnmt,St

  14. EKF_localization ( mt-1,St-1, ut, zt,m):Prediction: Jacobian of g w.r.t location Jacobian of g w.r.t control Motion noise Predicted mean Predicted covariance

  15. EKF_localization ( mt-1,St-1, ut, zt,m):Correction: Predicted measurement mean Jacobian of h w.r.t location Pred. measurement covariance Kalman gain Updated mean Updated covariance

  16. EKF Prediction Step (known correspondences)

  17. EKF Correction Step (known correspondences)

  18. EKF Prediction Step (unknown correspondences)

  19. EKF Correction Step (unknown correspondences)

  20. EKF Prediction Step

  21. EKF Observation Prediction Step

  22. EKF Correction Step

  23. Estimation Sequence (1)

  24. Estimation Sequence (2)

  25. Comparison to GroundTruth

  26. EKF Summary • Highly efficient: Polynomial in measurement dimensionality k and state dimensionality n: O(k2.376 + n2) • Not optimal! • Can diverge if nonlinearities are large! • Works surprisingly well even when all assumptions are violated!

  27. Linearization via Unscented Transform EKF UKF

  28. UKF Sigma-Point Estimate (2) EKF UKF

  29. UKF Sigma-Point Estimate (3) EKF UKF

  30. Unscented Transform Sigma points Weights Pass sigma points through nonlinear function Recover mean and covariance

  31. Motion noise UKF_localization ( mt-1,St-1, ut, zt,m): Prediction: Measurement noise Augmented state mean Augmented covariance Sigma points Prediction of sigma points Predicted mean Predicted covariance

  32. Measurement sigma points UKF_localization ( mt-1,St-1, ut, zt,m): Correction: Predicted measurement mean Pred. measurement covariance Cross-covariance Kalman gain Updated mean Updated covariance

  33. UKF Prediction Step

  34. UKF Observation Prediction Step

  35. UKF Correction Step

  36. EKF Correction Step

  37. Estimation Sequence EKF PF UKF

  38. Estimation Sequence EKF UKF

  39. Prediction Quality EKF UKF

  40. UKF Summary • Highly efficient: Same complexity as EKF, with a constant factor slower in typical practical applications • Better linearization than EKF: Accurate in first two terms of Taylor expansion (EKF only first term) • Derivative-free: No Jacobians needed • Still not optimal!

  41. Kalman Filter-based System • [Arras et al. 98]: • Laser range-finder and vision • High precision (<1cm accuracy) [Courtesy of Kai Arras]

  42. Map-based Localization

  43. Monte Carlo (Particle Filter) Localization

  44. Resampling Algorithm

  45. Monte Carlo (Particle Filter) Localization

  46. Algorithm sample_normal_distribution(b): • return Monte Carlo (Particle Filter) Localization • Algorithm sample_triangular_distribution(b): • return

  47. Monte Carlo (Particle Filter) Localization

  48. Monte Carlo (Particle Filter) Localization

  49. Monte Carlo (Particle Filter) Localization

  50. Monte Carlo (Particle Filter) Localization

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