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Probabilistic Robotics

Probabilistic Robotics. Localization KF, EKF. Bayes Filter Reminder. Prediction Correction. Localization Taxonomy. Local vs. Global localization. Static vs. Dynamic environments. Passive vs. Active approaches. Single-robot vs. Multirobot localization. Review topics….

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Probabilistic Robotics

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  1. Probabilistic Robotics Localization KF, EKF

  2. Bayes Filter Reminder • Prediction • Correction

  3. Localization Taxonomy • Local vs. Global localization. • Static vs. Dynamic environments. • Passive vs. Active approaches. • Single-robot vs. Multirobot localization.

  4. Review topics… • KF-EKF tutorial on Welch-Bishop website. • Table 7.2: EKF with known correspondences. • Table 7.3: EKF with unknown correspondences. • Table 7.4: Unscented Kalman Filter. • Read Chapter 7, 8 very carefully! • Need volunteers to discuss topics in class…

  5. 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

  6. 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.

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

  8. Prediction Correction The Prediction-Correction-Cycle

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

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

  11. 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)

  12. Landmark-based Localization

  13. 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

  14. 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

  15. 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

  16. 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

  17. Prediction Quality EKF UKF

  18. Summary • Gaussian filters. • Kalman filter: linearity assumption. • Robot systems non-linear. • Works well in practice. • Extended Kalman filters: linearization. • Tangent at the mean. • Unscented Kalman filters: better linearization. • Sigma control points. • ReBEL Matlab software. • Information filter (Section 3.5).

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