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Probabilistic Robotics

Probabilistic Robotics. Bayes Filter Implementations Gaussian filters. Kalman Filter Localization. Bayes Filter Reminder. Algorithm Bayes_filter ( Bel(x),d ): h = 0 If d is a perceptual data item z then For all x do For all x do

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Probabilistic Robotics

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  1. Probabilistic Robotics Bayes Filter Implementations Gaussian filters

  2. Kalman Filter Localization

  3. Bayes Filter Reminder • Algorithm Bayes_filter( Bel(x),d ): • h=0 • Ifd is a perceptual data item z then • For all x do • For all x do • Else ifd is an action data item uthen • For all x do • ReturnBel’(x)

  4. Bayes Filter Reminder • Prediction • Correction

  5. Kalman Filter • Bayes filter with Gaussians • Developed in the late 1950's • Most relevant Bayes filter variantin practice • Applications range from economics, wheather forecasting, satellite navigation to robotics and many more. • The Kalman filter "algorithm" isa couple of matrix multiplications!

  6. m -s s m Multivariate Gaussians Univariate

  7. Gaussians 1D 3D 2D Video

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

  9. Introduction to Kalman Filter (1) • Two measurements no dynamics • Weighted least-square • Finding minimum error • After some calculation and rearrangements • Another way to look at it – weigthed mean

  10. 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 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.

  11. Kalman Filter Updates in 1D prediction measurement correction It's a weighted mean!

  12. Kalman Filter Updates in 1D

  13. Kalman Filter Updates in 1D

  14. Kalman Filter Updates

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

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

  17. Linear Gaussian Systems: Dynamics

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

  19. Linear Gaussian Systems: Observations

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

  21. Kalman Filter Algorithm

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

  23. Prediction The Prediction-Correction-Cycle

  24. Correction The Prediction-Correction-Cycle

  25. Prediction Correction The Prediction-Correction-Cycle

  26. 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!

  27. Nonlinear Dynamic Systems • Most realistic robotic problems involve nonlinear functions • To be continued

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