  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.
