Iterative unscented statistically linearized filter for nonlinear Gaussian observation models

State filtering for nonlinear Gaussian observation models still remains as one of the challenging problems in the estimation and control research field. It is mainly due to the non-existence of the realizable optimal filter and for addressing to this problem, the following two approaches are often taken. The first approach is based on the Gaussian assumed density filter (Gaussian filter) and the most well-known realization is the Unscented Kalman filter. The second approach is based on the series expansion based filter and the most widely-used algorithm is the extended Kalman filter or iterative extended Kalman filter. Note that both approaches are the approximation to the optimal filter and thus a room is still left for the further exploration into the effective filters in terms of both filtering accuracy and speed. In this paper, we focus on the unscented statistical linearization (USL) which is a realization method of the statistical linearization whose linearization accuracy is theoretically better than that of the truncated Taylor series. The filter employing the USL is called the unscented statistically linearized filter (USLF). We newly propose the iterative type algorithm to tackle the filtering problem of the nonlinear Gaussian observation models and numerically show the superior state filtering performance over the aforementioned state-of-the-art filtering approaches.