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, which, according to what we found in this paper, tends to give overconfident and less accurate state estimations when the measurement functions are nonlinear. Therefore, in this study, we designed a new framework for nonlinear KFs and showed theoretically and empirically that the new framework estimates the states and covariance matrix 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 the estimation errors by several orders of magnitude in low-measurement-noise conditions, with only about a 10 to 90% increase in computational time. All types of nonlinear KFs can benefit from the new framework, and the benefit will increase as the sensors become more and more accurate in the future. As an example, EKF, the simplest nonlinear KF that was previously believed to work poorly for strongly nonlinear systems, can now provide fast and fairly accurate state estimations with the help of the new framework. The codes are available at https://github.com/Shida-Jiang/A-new-framework-for-nonlinear-Kalman-filters.

翻译：卡尔曼滤波器（KF）是一种状态估计算法，它通过最优结合系统知识与测量数据来最小化估计状态的均方误差。尽管KF最初是为线性系统设计的，但针对非线性系统已提出了多种扩展形式，例如扩展卡尔曼滤波器（EKF）、无迹卡尔曼滤波器（UKF）、容积卡尔曼滤波器（CKF）等。虽然不同类型的非线性KF各有优劣，但它们均沿用线性KF的相同框架。本文研究发现，当测量函数为非线性时，该框架易导致状态估计过度自信且精度下降。因此，本研究设计了一种适用于非线性KF的新框架，并从理论与实验两方面证明：相较于旧框架，新框架能更精确地估计状态及协方差矩阵。该框架在四种不同的非线性KF及五项不同任务中进行了测试，结果表明在低测量噪声条件下，其能够将估计误差降低数个数量级，而计算时间仅增加约10%至90%。所有类型的非线性KF均可受益于该新框架，且随着未来传感器精度不断提升，其优势将愈加显著。以最简单的非线性KF——EKF为例，以往认为其在强非线性系统中表现不佳，而借助新框架，EKF现在能够实现快速且相当精确的状态估计。相关代码已发布于 https://github.com/Shida-Jiang/A-new-framework-for-nonlinear-Kalman-filters。