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.

翻译：卡尔曼滤波（KF）是一种状态估计算法，其通过最优结合系统知识与测量值来最小化估计状态的均方误差。虽然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（EKF）、无迹卡尔曼滤波（UKF）、容积卡尔曼滤波（CKF）等。尽管不同类型的非线性KF各有优劣，但它们均沿用线性KF的同一框架。然而，根据本文的研究发现，当测量函数为非线性时，该框架往往会产生过度自信且精度较低的状态估计。因此，本研究设计了一种可与任何现有非线性KF类型结合的新框架，并从理论与实验两方面证明，新框架比原有框架能更准确地估计状态与协方差。该新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，其能将估计误差降低数个数量级。