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Top1. Introduction
The problem of nonlinear filtering is common in many areas such as integrated navigation system, geodetic positioning, automatic control, information fusion and signal processing. The extended Kalman filtering is a commonly used filtering method to nonlinear systems (Julier, Uhlmann & Durrant-Whyte, 2000; Lefebvre, Bruyninckx & Schutter, 2004). This is an approximation method, in which nonlinear system equations are linearized by the Taylor expansion and the linearized states are assumed to obey the Gaussian distribution. The linearization stage of the state equations may lead to a problem of divergence or instability (Simon, 2006). Especially, when the practical probability function has multiple peak values, the estimated state error is very large or even divergent (Grewal & Andrews, 2008). The unscented Kalman filtering (UKF) method combines the concept of unscented transform with the Kalman filtering (Julier & Uhlmann, 2004; Wan & van der Merwe, 2000). This method inherits the linear update structure of the Kalman filtering. It uses only the second-order system moments, which may not be sufficient for some nonlinear systems.
The particle filtering (PF) is an optimal recursive Bayesian filtering method based on Monte Carlo simulation (Doucet, Godsill & Andrieu, 2000; Rawlings & Bakshi, 2006). This method aims to produce a sample of independent random variables distributed according to the conditional probability distribution. It is not limited by the linearized errors and the assumption of Gaussian noise, and thus can deal with nonlinear system models and non-Gaussian noise (Rawlings & Bakshi, 2006). It is also easier to implement, even for high dimensional problems. Therefore, particle filtering has been widely used in the fields of navigation, target tracking, fault detection, robotic control and computer vision (Zhang, Chen, Zhou & Li, 2007; Oppenheim & Philippe & de Rigal, 2008). However, the accuracy of the PF method largely depends on the choice of importance sampling density and resampling scheme (Yang, Tian, Jin & Zhang, 2005; Arulampalam, Maskell, Gordon & Clapp, 2002; Watzenig, Brandner & Steiner, 2007).
Recently, various methods have been proposed to design a good importance sampling density or modify the resampling scheme (Yang, Tian & Jin, 2006; Zhang, Tian & Jin, 2006; Budhiraja, Chen & Lee, 2007; Ning & Fang, 2008). The unscented particle filtering (UPF) method uses unscented transformation to get a better importance sampling density (van der Merwe, Doucet, Freitas & Wan, 2000; Liang, Ma & Dai, 2008; Ning & Fang, 2008; Ali & Fang, 2009). The unscented transformation calculates the statistics of a random variable that undergoes a nonlinear transform (Julier & Uhlmann, 2004). It enables the estimation of state mean and variance to achieve the third-order accuracy, thus providing higher accuracy for filtering. However, the particle degeneracy phenomenon could occur if a dynamic system has very small noise or the observational noise has very small variance (Oppenheim & Philippe & de Rigal, 2008; Ning & Fang, 2008; Ali & Fang, 2009). In fact, it is unavoidable in practical engineering applications that a dynamic system has small systematic noise due to the disturbances caused by singular observations and uncertain factors.