Skewed Unscented Kalman Filter Using Gaussian Sum

Abstract

The unscented Kalman filter (UKF) finds extensive application in the state estimation of systems characterized by significant nonlinearity. A constraint of the UKF is its presumption that the probability density function (PDF) of the states maintains Gaussian distribution throughout the filter recursion, which restricts the estimation accuracy even though the UKF may still be applicable when the Gaussian assumption is not strictly met. To overcome this problem, a skewed unscented Kalman filter (SUKF) based on Gaussian sum is presented in this article. First, as the theoretical basis, the conventional UKF is reviewed. Then, the SUKF algorithm is presented in a manner analogous to the UKF, following a two-step process. In the time-update step, an approximation method, employing no additional sigma points compared to the UKF algorithm, has been proposed to obtain the skewness of the random variable after nonlinear transformation, which approximates the Taylor series of skewness up to fourth-order terms. In the measurement-update step, a Gaussian sum PDF matching the known mean, covariance, and skewness is constructed to represent the non-Gaussian joint PDF of states and measurements. Finally, taking one nonlinear transformation and three nonlinear systems with different dimensions as examples, the effectiveness of the proposed SUKF is verified through numerical simulations. The results demonstrate that the SUKF algorithm offers higher estimation accuracy compared to the conventional UKF while requiring similar computational time, which can provide a practical option for state estimation for highly nonlinear systems.