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Extended Kalman Filter
The extended Kalman filter (EKF) is a recursive Bayesian state-estimation algorithm that extends the linear Kalman filter to systems governed by nonlinear state-transition or measurement functions [1]. Like the linear Kalman filter, the EKF operates in two alternating steps: a time update that propagates the state estimate and its associated error covariance forward through the nonlinear process model, and a measurement update that corrects the prediction using an available observation [2]. The nonlinear functions are linearized about the current state estimate by a first-order Taylor expansion, with the Jacobian matrices of the process and observation functions evaluated at the current estimate replacing the constant state-transition and observation matrices of the linear filter [1]. The EKF assumes additive Gaussian process and measurement noise and Gaussian initial uncertainty, and it propagates only the first two moments—mean and covariance—of the state distribution [3]. Because the linearization is performed around the current estimate rather than a fixed trajectory, the EKF is a nonlinear filter in which the gain and covariance depend on the actual measurements received [2]. It is distinguished from the linear Kalman filter by its handling of nonlinearity, and from second-order or sigma-point filters by its first-order Taylor approximation [4].
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