The Geometric Unscented Kalman Filter

Abstract

Many filters have been proposed in recent decades for the nonlinear state estimation problem. The linearization-based extended Kalman filter (EKF) is widely applied to nonlinear industrial systems. As EKF is limited in accuracy and reliability, sequential Monte-Carlo methods or particle filters (PF) can obtain superior accuracy at the cost of a huge number of random samples. The unscented Kalman filter (UKF) can achieve adequate accuracy more efficiently by using deterministic samples, but its weights may be negative, which might cause instability problem. For Gaussian filters, the cubature Kalman filter (CKF) and Gauss Hermit filter (GHF) employ cubature and respectively Gauss-Hermite rules to approximate statistic information of random variables and exhibit impressive performances in practical problems. Inspired by this work, this paper presents a new nonlinear estimation scheme named after geometric unscented Kalman filter (GUF). The GUF chooses the filtering framework of CKF for updating data and develops a geometric unscented sampling (GUS) strategy for approximating random variables. The main feature of GUS is selecting uniformly distributed samples according to the probability and geometric location similar to UKF and CKF, and having positive weights like PF. Through such way, GUF can maintain adequate accuracy as GHF with reasonable efficiency and good stability. The GUF does not suffer from the exponential increase of sample size as for PF or failure to converge resulted from non-positive weights as for high order CKF and UKF.