Extended Kalman Filter: Theory and Implementation for Nonlinear Systems

In this article, we introduce the Extended Kalman Filter (EKF), a sophisticated filtering technique designed for nonlinear systems. Unlike the traditional Kalman Filter, which is limited to linear systems, the EKF handles nonlinear dynamics by employing first-order Taylor series approximations to linearize nonlinear functions around the current state estimate. This linearization enables the filter to operate effectively in nonlinear environments while maintaining the core Kalman Filter structure. The implementation typically involves two main steps: prediction (using the nonlinear state transition function) and update (using linearized measurement models). Key functions include computing Jacobian matrices for system dynamics and measurement models, which are crucial for the linearization process. We will provide detailed explanations of the EKF's mathematical foundation, practical applications in areas like robotics and navigation, and code implementation strategies using numerical computation libraries. This comprehensive coverage will equip readers with both theoretical understanding and practical skills to effectively implement and apply this powerful filtering algorithm.