Four Gaussian Filter Algorithms for Nonlinear State Estimation

Gaussian filtering algorithms serve as essential tools for nonlinear system state estimation, widely applied in target tracking, robot localization, and related fields. The four common Gaussian filter algorithms each possess distinct characteristics: Extended Kalman Filter (EKF) EKF linearizes nonlinear systems through first-order Taylor series expansion, offering relatively high computational efficiency suitable for weakly nonlinear systems. However, linearization errors may lead to increased estimation bias in strongly nonlinear scenarios. In code implementation, EKF typically requires analytical derivation of Jacobian matrices for state transition and observation models. Unscented Kalman Filter (UKF) UKF employs Unscented Transform (UT) to capture statistical properties of nonlinear systems, eliminating the need for Jacobian matrix calculations. Compared to EKF, UKF demonstrates more stable performance under strong nonlinear conditions, though with slightly higher computational overhead. The algorithm implements sigma point propagation through nonlinear functions to approximate posterior distributions. Quadrature Kalman Filter (QKF) QKF approximates nonlinear distributions using numerical integration rules (e.g., Gauss-Hermite quadrature), achieving higher computational accuracy than UKF, particularly suitable for high-dimensional state estimation. However, its computational complexity increases significantly with dimensionality. Implementation involves weighted summation of function evaluations at quadrature points. Central Difference Kalman Filter (CKF) CKF utilizes polynomial interpolation to approximate nonlinear functions, featuring good stability and straightforward implementation. It provides balanced performance for moderately nonlinear systems, though may be less accurate than UKF or QKF in extreme nonlinear cases. The algorithm employs finite difference approximations for derivative calculations. Summary: EKF suits scenarios with limited computational resources and weak nonlinearity. UKF offers greater robustness in strongly nonlinear systems as a general-purpose choice. QKF applies to high-dimensional problems requiring high precision. CKF provides a balanced solution for medium-complexity problems.