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Kalman Filter CMPUT 615 Nilanjan Ray

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What is Kalman Filter A sequential state estimator for some special cases Invented in 1960’s Still very much used today in academia, industry, military… Optimal filter if certain conditions hold

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Kalman Filter Model Linear state transition/motion model: Linear measurement model: N t and V t are independent Gaussian random variables, called “noise” Typically N t has a Gaussian distribution with zero mean and R t covariance V t has a Gaussian distribution with zero mean and Q t covariance

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KF as SBE Is KF a type of SBE? Indeed it is: Under the KF model assumptions, all the densities become Gaussian So we need to keep track of the mean and the covariance of the Gaussian densities Smart recursion equations exist for tracking mean and covariance

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KF: From SBE predicted/Filter density: posterior density: where So we need to keep track of μ’s and P’s in a recursion Under the KF model assumptions, we can show that

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KF: Recursion Equations Filter recursions (aka state update) Bayes’ rule recursions (aka measurement update) whereis called Kalman gain

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KF: Schematic

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Extended Kalman Filter Often the assumptions of linear relationships need to be relaxed for applications Extended KF uses the good old Taylor expansion to approximate non-linear relationships linearly

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EKF: Linearization where the Jacobians are defined as

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EFK: Recursion Equations Filter step/ State update Bayes’ rule step/ Measurement update Kalman gain

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Summary KF is optimal when state and measurement relationships are linear and noise model is independent Gaussian EFK can handle non-linear relationships; but we still need Gaussian noise model EKF cannot gracefully handle highly non-linear relationships Particle filter can handle any non-linear relationships and non-Gaussian noise models

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