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Probabilistic Robotics: Kalman Filters

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68 diapositives · 968 Views · Jul 25, 2014

Probabilistic Robotics: Kalman Filters. Sebastian Thrun & Alex Teichman Stanford Artificial Intelligence Lab.

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  1. Probabilistic Robotics: Kalman Filters Sebastian Thrun & Alex Teichman Stanford Artificial Intelligence Lab Slide credits: Wolfram Burgard, Dieter Fox, Cyrill Stachniss, Giorgio Grisetti, Maren Bennewitz, Christian Plagemann, Dirk Haehnel, Mike Montemerlo, Nick Roy, Kai Arras, Patrick Pfaff and others

  2. Bayes Filters in Localization

  3. Bayes Filter Reminder • Prediction • Measurement Update

  4. m Univariate -s s m Multivariate Gaussians

  5. Properties of Gaussians

  6. Multivariate Gaussians • We stay in the “Gaussian world” as long as we start with Gaussians and perform only linear transformations.

  7. Discrete Kalman Filter Estimates the state x of a discrete-time controlled process that is governed by the linear stochastic difference equation with a measurement

  8. Components of a Kalman Filter Matrix (nxn) that describes how the state evolves from t to t-1 without controls or noise. Matrix (nxl) that describes how the control ut changes the state from t to t-1. Matrix (kxn) that describes how to map the state xt to an observation zt. Random variables representing the process and measurement noise that are assumed to be independent and normally distributed with covariance Rt and Qt respectively.

  9. Kalman Filter Updates in 1D

  10. Kalman Filter Updates in 1D

  11. Kalman Filter Updates in 1D

  12. Kalman Filter Updates

  13. Linear Gaussian Systems: Initialization • Initial belief is normally distributed:

  14. Linear Gaussian Systems: Dynamics • Dynamics are linear function of state and control plus additive noise:

  15. Linear Gaussian Systems: Dynamics

  16. Linear Gaussian Systems: Observations • Observations are linear function of state plus additive noise:

  17. Linear Gaussian Systems: Observations

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

  19. Kalman Filter Algorithm

  20. Prediction • Observation • Correction • Matching Kalman Filter Algorithm

  21. Prediction The Prediction-Correction-Cycle

  22. Correction The Prediction-Correction-Cycle

  23. Prediction Correction The Prediction-Correction-Cycle

  24. Kalman Filter Summary • Highly efficient: Polynomial in measurement dimensionality k and state dimensionality n: O(k2.376 + n2) • Optimal for linear Gaussian systems! • Most robotics systems are nonlinear!

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

  26. Linearity Assumption Revisited

  27. Non-linear Function

  28. EKF Linearization (1)

  29. EKF Linearization (2)

  30. EKF Linearization (3)

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

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

  33. 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)

  34. Landmark-based Localization

  35. 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

  36. 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

  37. EKF Prediction Step

  38. EKF Observation Prediction Step

  39. EKF Correction Step

  40. Estimation Sequence (1)

  41. Estimation Sequence (2)

  42. Comparison to GroundTruth

  43. 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!

  44. EKF Localization Example • Line and point landmarks

  45. EKF Localization Example • Line and point landmarks

  46. EKF Localization Example • Lines only (Swiss National Exhibition Expo.02)

  47. Linearization via Unscented Transform EKF UKF

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

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

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