Algorithm of Extended Kalman Filter

The Extended Kalman Filter (EKF) is a state estimation algorithm designed for nonlinear systems. By performing local linearization of nonlinear dynamic and observation models, EKF plays a vital role in practical applications such as robotic localization, autonomous driving, and sensor fusion.

The core concept of EKF involves first-order Taylor expansion of nonlinear models around the current state estimate, effectively approximating them as linear models. This adaptation allows the standard Kalman filter framework to remain applicable. While first-order EKF considers only first-order derivatives (Jacobian matrices), second-order EKF incorporates additional second-order terms (Hessian matrices) to improve estimation accuracy at the cost of increased computational complexity.

The algorithm typically consists of two main phases: Prediction and Update. Prediction Phase: Based on the system dynamics model, this phase predicts the next state and covariance matrix using current state estimates and control inputs. Update Phase: This phase integrates sensor measurements to refine predictions and obtain more accurate state estimates.

Although EKF demonstrates broad applicability, its performance depends heavily on model accuracy and linearization quality. In strongly nonlinear scenarios, alternative filtering methods such as Unscented Kalman Filter (UKF) or Particle Filter (PF) may be preferable.

For developers implementing EKF, understanding mathematical derivations and linearization processes is crucial. During code implementation, it's recommended to first validate the correctness of first and second derivative calculations to ensure filter stability and convergence. Key implementation aspects include proper Jacobian matrix computation using numerical differentiation or symbolic math tools, and maintaining numerical stability through covariance matrix regularization techniques.