Design Guidelines for Nonlinear Kalman Filters via Covariance Compensation

TL;DR

This paper introduces a theoretical framework for nonlinear Kalman filters, emphasizing covariance compensation to enhance robustness and accuracy, supported by theoretical analysis and empirical validation.

Contribution

It establishes new design guidelines for nonlinear Kalman filters based on covariance compensation, improving their robustness and estimation accuracy.

Findings

Adherence to the guidelines improves estimation accuracy.

Covariance compensation beyond EKF enhances filter robustness.

Fixed parameter choices are often suboptimal.

Abstract

Nonlinear extensions of the Kalman filter (KF), such as the extended Kalman filter (EKF) and the unscented Kalman filter (UKF), are indispensable for state estimation in complex dynamical systems, yet the conditions for a nonlinear KF to provide robust and accurate estimations remain poorly understood. This work proposes a theoretical framework that identifies the causes of failure and success in certain nonlinear KFs and establishes guidelines for their improvement. Central to our framework is the concept of covariance compensation: the deviation between the covariance predicted by a nonlinear KF and that of the EKF. With this definition and detailed theoretical analysis, we derive three design guidelines for nonlinear KFs: (i) invariance under orthogonal transformations, (ii) sufficient covariance compensation beyond the EKF baseline, and (iii) selection of compensation magnitude that favors underconfidence. Both theoretical analysis and empirical validation confirm that adherence to these principles significantly improves estimation accuracy, whereas fixed parameter choices commonly adopted in the literature are often suboptimal. The codes and the proofs for all the theorems in this paper are available at https://github.com/Shida-Jiang/Guidelines-for-Nonlinear-Kalman-Filters.