A New Framework for Nonlinear Kalman Filters

TL;DR

This work identifies a fundamental overconfidence problem in nonlinear Kalman filters when using the conventional Predict/Update framework with nonlinear measurements. It introduces a recalibration framework that re-approximates measurement statistics after the state update, plus a back-out mechanism to withdraw unhelpful updates, preserving robustness without iterative gains as in IKF. Theoretical analysis and simulations show that the approach reduces state-estimation errors by orders of magnitude in low-noise regimes and yields covariance estimates that better match actual errors, across EKF, EKF2, UKF, CKF, and IEKF variants. While incurring modest additional computation, the framework offers substantial improvements in accuracy and convergence, suggesting broad practical impact for discrete-time nonlinear KF applications as sensor quality improves. The method is demonstrated on five tasks and is readily combined with existing nonlinear KF architectures, with codes available at the provided repository.

Abstract

The Kalman filter (KF) is a state estimation algorithm that optimally combines system knowledge and measurements to minimize the mean squared error of the estimated states. While KF was initially designed for linear systems, numerous extensions of it, such as extended Kalman filter (EKF), unscented Kalman filter (UKF), cubature Kalman filter (CKF), etc., have been proposed for nonlinear systems over the last sixty years. Although different types of nonlinear KFs have different pros and cons, they all use the same framework of linear KF. Yet, according to our theoretical and empirical analysis, the framework tends to give overconfident and less accurate state estimations when the measurement functions are nonlinear. Therefore, in this study, we designed a new framework that can be combined with any existing type of nonlinear KFs and showed theoretically and empirically that the new framework estimates the states and covariance more accurately than the old one. The new framework was tested on four different nonlinear KFs and five different tasks, showcasing its ability to reduce estimation errors by several orders of magnitude in low-measurement-noise conditions. The codes are available at https://github.com/Shida-Jiang/A-new-framework-for-nonlinear-Kalman-filters

Figures (22)

• Figure 1: The basic idea of the new framework for nonlinear KFs. The additional step we propose re-approximates the nonlinear measurement function around the updated states, making the final covariance estimation more accurate.
• Figure 2: The comparison of the variance estimation after the state update and recalibration using different frameworks and different types of nonlinear Kalman filters. The recalibrated variance of CKF (new) is approximately $144^2$, almost 10,000 times larger than the predicted variance, which underscores the importance of the "back out" step.
• Figure 3: The root mean squared error of the state estimations under different measurement noise setups (3D target tracking).
• Figure 4: The root mean squared error of the state estimations under different measurement noise setups (terrain-referenced navigation).
• Figure 5: The root mean squared error of the state estimations under different measurement noise setups (synchronous generator state estimation).