A New Framework for Nonlinear Kalman Filters

2024-07-08

arXiv.org

AI Summary

Traditional Kalman filters for nonlinear systems—such as the Extended, Unscented, and Cubature Kalman Filters—suffer from overconfident state estimates, underestimated covariances, and degraded accuracy due to nonlinear measurement functions. This paper identifies, for the first time, the intrinsic bias in covariance propagation underlying these deficiencies and proposes a general, plug-and-play correction framework. Grounded in Bayesian estimation reformulation and explicit covariance propagation calibration, the framework is compatible with mainstream nonlinear Kalman filter variants. Theoretical analysis and extensive experiments across five canonical tasks demonstrate that the method reduces state estimation error by one to three orders of magnitude—particularly under low measurement noise—while significantly improving covariance fidelity and effectively mitigating overconfidence.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Probabilistic Inference

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

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. Although different types of nonlinear KFs have different pros and cons, they all use the same framework of linear KF. Yet, according to what we found in this paper, 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.

Problem

• Kalman Filter
• Nonlinear Systems
• Prediction Accuracy

Innovation

• Nonlinear Kalman Filter
• Prediction Accuracy
• Noise Robustness