Cubature Kalman Filter vs Extended Kalman Filter: Implementation and Algorithm Comparison

In this article, we explore two variants of Kalman filters: the Cubature Kalman Filter and the Extended Kalman Filter. Both filters serve as tools for state estimation and uncertainty analysis, but they employ different implementation approaches and algorithmic strategies. The Cubature Kalman Filter is a nonlinear filtering technique that extends the standard Gaussian-based Kalman filter to handle non-Gaussian distributions. This filter maintains the fundamental Kalman filter equations and covariance matrix derivations but introduces a novel approach for state estimation and uncertainty propagation. The core innovation lies in using cubature rules (typically through spherical-radial transformations) to numerically approximate Gaussian-weighted integrals. In code implementation, this involves generating cubature points (2n points for n-dimensional states) using deterministic sampling, propagating these points through nonlinear system models, and reconstructing the mean and covariance. Key functions would include point generation via Cholesky decomposition of the covariance matrix and numerical integration using spherical-radial cubature rules. In contrast, the Extended Kalman Filter also addresses nonlinear problems but employs a different methodology based on linearization. It performs state estimation and uncertainty analysis by linearizing nonlinear system models around the current state estimate using first-order Taylor series expansion. While retaining the basic Kalman filter equations, EKF uses Jacobian matrices (partial derivatives of system and measurement models) to approximate the covariance propagation. Code implementation typically requires analytical or numerical computation of Jacobians at each time step, which can be computationally intensive and prone to linearization errors. The algorithm structure follows the predict-update cycle but with linearized transformations of covariance matrices. Both Kalman filter variants serve as valuable tools for various applications including robotics, navigation systems, and control engineering. Understanding their implementation nuances, algorithmic strengths (CKF's better numerical stability for highly nonlinear systems vs EKF's computational efficiency for mildly nonlinear cases), and limitations enables developers to select the appropriate filter for specific application requirements.