Higher-Order Unscented Estimator

Abstract

This paper introduces a new extension of the unscented Kalman filter with asymmetric sample points and weights chosen to match third-and fourth-order moments in addition to the mean and covariance. Explicit solutions are obtained for sample points and weights, making their evaluation efficient and robust, and rigorous constraints are derived for their applicability. The use of the new filter is demonstrated with three dynamic systems (an aircraft coordinated turn model, a rotating rigid body, and a projectile with drag), and filter performance is compared with that of the conventional unscented Kalman filter and conjugate unscented transform filters. The new filter is found to be more robust in most cases where the initial distribution, process noise, and measurement noise have a high kurtosis, in that it does not generate extreme outliers in the estimation error. Also, execution times for the new filter are found to be only slightly longer than for the conventional unscented Kalman filter and significantly shorter than for the conjugate unscented transform filters.

The Nonsingular Estimator for Exoplanet Orbits: An Unscented Batch Estimation Method for Direct Imaging Measurements

“…We call this method the Nonsingular Estimator for Exoplanet Orbits (NEXO). This paper is a continuation of our previous work (Stojanovski & Savransky 2022;Stojanovski 2023).…”

“…In our previous work (Stojanovski & Savransky 2022), we proposed a new set of orbital elements specifically for the purposes of exoplanet orbit fitting, combining features of both the Thiele-Innes parameters and the Cohen-Hubbard elements. These elements are defined as…”

“…These sigma points and their corresponding weights are chosen to match the mean and covariance of the distribution; this process is referred to as the unscented transform. In order to obtain more accurate estimates, particularly for non-Gaussian distributions, the UKF has been further extended to account for higher-order moments-for example, in the cubature KF (Singh & Bhaumik 2015), the conjugate unscented transform filter (Adurthi et al 2018), and the higher-order unscented estimator (Stojanovski & Savransky 2021). Yet another variant of the UKF is the square root SPF (SRSPF), which replaces the covariance matrix with its square root (specifically, its Cholesky decomposition) to improve numerical stability (Brunke & Campbell 2004).…”

A Recursive Non-Uniform Sampling Estimator for Asynchronous Nonlinear Systems

“…The UKF algorithm obtains the sigma points and corresponding weights through unscented transformation. The weights are generally negative in high-dimensional systems, which introduces highorder truncation error terms and reduces the accuracy of the algorithm [33]. The CKF algorithm acquires cubature points and propagates them using nonlinear equations.…”

A Novel Extended Unscented Kalman Filter Is Designed Using the Higher-Order Statistical Property of the Approximate Error of the System Model

“…For an exploratory study of Unscented Kalman filtering methods, Stojanovski et al [19] proposed an extended UKF based on Unscented Kalman filtering, which considers asymmetric sample points and weights to match third-order and fourth-order moments in addition to the mean and covariance to improve the performance. Cui et al [8] proposed a sampling design method based on the advantages of statistical sampling of the UKF and random sampling of the EnKF to overcome the shortcomings of both.…”