Mitigating Overconfidence in Nonlinear Kalman Filters via Covariance Recalibration

TL;DR

This paper identifies a systematic covariance underestimation in nonlinear Kalman filters and introduces a covariance recalibration method that significantly improves estimation accuracy across various filters and applications.

Contribution

It provides the first mathematical proof of covariance underestimation in nonlinear KFs and proposes a general recalibration framework that enhances their accuracy.

Findings

Covariance underestimation occurs systematically in nonlinear KFs.

Recalibration improves state and covariance estimation accuracy.

Errors are reduced by several orders of magnitude in simulations.

Abstract

The Kalman filter (KF) is an optimal linear state estimator for linear systems, and numerous extensions, including the extended Kalman filter (EKF), unscented Kalman filter (UKF), and cubature Kalman filter (CKF), have been developed for nonlinear systems. Although these nonlinear KFs differ in how they approximate nonlinear transformations, they all retain the same update framework as the linear KF. In this paper, we show that, under nonlinear measurements, this conventional framework inherently tends to underestimate the true posterior covariance, leading to overconfident covariance estimates. To the best of our knowledge, this is the first work to provide a mathematical proof of this systematic covariance underestimation in a general nonlinear KF framework. Motivated by this analysis, we propose a covariance-recalibrated framework that re-approximates the measurement model after the state update to better capture the actual effect of the Kalman gain on the posterior covariance; when recalibration indicates that an update is harmful, the update can be withdrawn. The proposed framework can be combined with essentially any existing nonlinear KF, and simulations across four nonlinear KFs and five applications show that it substantially improves both state and covariance estimation accuracy, often reducing errors by several orders of magnitude. The code and supplementary material are available at https://github.com/Shida-Jiang/A-new-framework-for-nonlinear-Kalman-filters.