EKF

Extended Kalman Filter (EKF) is an algorithm for estimating the state of a nonlinear dynamic system from noisy measurements. It extends the linear Kalman filter by linearizing nonlinear process and observation models about the current state estimate. The process model is x_k = f(x_{k-1}, u_{k-1}) + w_{k-1}, and the observation model is z_k = h(x_k) + v_k, where w_k and v_k are zero-mean Gaussian noises with covariances Q_k and R_k, respectively.

Prediction step: x_hat_{k|k-1} = f(x_hat_{k-1|k-1}, u_{k-1}); P_{k|k-1} = F_{k-1} P_{k-1|k-1} F_{k-1}^T + Q_{k-1}, where F_{k-1} is the Jacobian of f with respect to x, evaluated at x_hat_{k-1|k-1}.

Update step: y_k = z_k - h(x_hat_{k|k-1}); H_k = ∂h/∂x evaluated at x_hat_{k|k-1}; S_k = H_k P_{k|k-1} H_k^T + R_k; K_k = P_{k|k-1} H_k^T S_k^{-1}; x_hat_{k|k} = x_hat_{k|k-1} + K_k y_k; P_{k|k} = (I - K_k H_k) P_{k|k-1}.

Assumptions and limitations: The EKF relies on first-order linearization, which is accurate only for mildly nonlinear systems and when the estimate is close to the true state. It assumes Gaussian noise and may diverge if the initial estimate is poor or nonlinearities are strong. Numerical stability and Jacobian computations are practical considerations.

Applications: EKF is widely used in robotics, navigation, aerospace, land surveying, and computer vision for tasks such as pose estimation, sensor fusion, and tracking.

Variations and related methods: Methods such as the iterated EKF, square-root EKF, and constrained EKF modify the update to improve stability or enforce constraints. The unscented Kalman filter (UKF) and other nonlinear filters provide alternatives that can handle larger nonlinearities without explicit Jacobians.