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 over the last sixty years. Although different types of nonlinear KFs have different pros and cons, they all use the same framework of linear KF. Yet, according to our theoretical and empirical analysis, 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. 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，它通过最优地结合系统知识与测量数据，以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，但在过去六十年中，针对非线性系统已提出了多种扩展形式，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。尽管不同类型的非线性KF各有优缺点，但它们均沿用线性KF的相同框架。然而，根据我们的理论与实证分析，当测量函数为非线性时，该框架往往会导致状态估计过于自信且准确性降低。因此，在本研究中，我们设计了一个新框架，该框架可与任何现有类型的非线性KF结合使用，并从理论和实证上证明，新框架在状态与协方差估计上比旧框架更为准确。新框架在四种不同的非线性KF和五项不同任务上进行了测试，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。代码可在 https://github.com/Shida-Jiang/A-new-framework-for-nonlinear-Kalman-filters 获取。