Extended Kalman Filter

In this article, we introduce the Kalman filter, an efficient recursive filter (autoregressive filter) that estimates the state of dynamic systems from incomplete, noisy measurement sequences. Named after its inventor Rudolf E. Kalman, the standard Kalman filter requires systems with Gaussian distributions. Subsequent research yielded significant improvements, notably the Extended Kalman Filter (EKF) designed for time-nonlinear dynamic systems. The EKF is a widely-used nonlinear filtering approach that linearizes nonlinear systems through first-order Taylor series expansions, effectively transforming nonlinear problems into linear ones. This implementation typically involves calculating Jacobian matrices of system models and measurement functions at each time step. Compared to traditional Kalman filters, EKF demonstrates superior adaptability and accuracy in nonlinear scenarios. Key implementation steps include: 1) Predicting state estimates using nonlinear system models 2) Linearizing system dynamics and measurement equations 3) Updating error covariance matrices 4) Correcting state estimates with new measurements Due to these advantages, EKF finds extensive practical applications in robotics navigation, aerospace systems, autonomous vehicles, and other domains requiring real-time state estimation in nonlinear environments.