cs685-kalman

cs685-kalman - Markov Kalman Filter Localization Markov...

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1 Probabilistic Robotics Bayes Filter Implementations Gaussian filters Markov Kalman Filter Localization • Markov localization localization starting from any unknown position recovers from ambiguous situation. However, to update the probability of all positions within the whole state space at any time requires a discrete representation of the space (grid). The required memory and calculation power can thus become very important if a fine grid is used. Kalman filter localization tracks the robot and is inherently very precise and efficient. However, if the uncertainty of the robot becomes to large (e.g. collision with an object) the Kalman filter will fail and the position is definitively lost. Kalman Filter Localization Prediction Correction Bayes Filter Reminder
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2 Gaussians p ( x ) ~ N ( μ , σ 2 ) : p ( x ) = 1 2 π e 1 2 ( x ) 2 2 - σ σ μ Univariate μ Multivariate Properties of Gaussians Multivariate Univariate We stay in the “Gaussian world” as long as we start with Gaussians and perform only linear transformations 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 8 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 u t changes the state from t to t-1 . Matrix (kxn) that describes how to map the state x t to an observation z t . Random variables representing the process and measurement noise that are assumed to be independent and normally distributed with covariance R t and Q t respectively.
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3 9 Kalman Filter Updates in 1D 10 Kalman Filter Updates in 1D 11 Kalman Filter Updates in 1D 12 Kalman Filter Updates
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4 13 Linear Gaussian Systems: Initialization Initial belief is normally distributed: 14 Dynamics are linear function of state and control
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cs685-kalman - Markov Kalman Filter Localization Markov...

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