## 1 Introduction  
The state estimation of nonlinear systems is widely used in engineering applications, such as radar target tracking, complex image processing, highly dynamical navigation and positioning, and signal processing. In order to obtain the optimal estimation of a nonlinear system, the posterior probability distribution of the system state needs to be obtained. However, a complete description of the posterior probability distribution can be accurately known only in a few specific cases. In recent years, driven by engineering applications, a large number of suboptimal filtering methods have been proposed, which can be classified into two main categories: The first one is the linearization method such as the extended Kalman filter (EKF). The second one is the sampling method, i.e., the unscented Kalman filter (UKF) and particle filter (PF). EKF is a traditional approach to solving nonlinear problems, such as navigation, target tracking, information fusion, monitoring state estimation [1, 2], etc. However, in a system with a higher degree of nonlinearity, the EKF will have a larger truncation error under a low carrier to noise ratio, which can easily cause the filter to diverge. In addition, the EKF implementation needs to calculate the Jacobian matrix, which also limits its scope of applications. The second method directly uses nonlinear filtering algorithms to process observation information, which can improve the accuracy of state estimation. Among them, UKF approximates the distribution function of random variables through a set of determined weighted sampling points [3]. When the sampling points are propagated through a nonlinear function, the statistical properties of the non-linear functions are captured, and the precision of UKF can reach the third-order [4; 5]. However, UKF still has some limitations. In high-dimensional systems (usually higher than three-dimensions), in order to avoid the propagation of non-positive definite covariance matrices, it should choose the parameters of UKF very carefully [6]. At this time, the UKF is prone to numerical instability, which can lead to dimensional disasters or divergence, and limited their application in complex systems, such as high dynamics, weak signals, and strong non-linearity. The PF is proposed based on the idea of the Monte Carlo method [7; 8], which uses a large number of randomly-generated particles to approximate the posterior probability density. The PF is used to solve signal processing problems in nonlinear and non-Gaussian systems. In recent years, it has been widely used for target tracking, state estimation under low-dynamical conditions, and modality detection [9]. However, as the number of iterations increases, there will be particle degradation and depletion when using PF [10]. In order to solve the problem of particle degradation, many improved particle algorithms have emerged. However, the generation of a large number of particles will still increase the computational burden, and the reduction of degradation and depletion is at the cost of increasing the complexity of the PF algorithm. Therefore, it is difficult for PF to meet real-time requirements, especially for state estimation under highly dynamical conditions. In [11], the authors proposed a new non-linear filtering method based on the cubature transformation: the cubature Kalman filter (CKF). Once proposed, the method was used for navigation [12; 13; 14], attitude estimation [15], continuous system [16], hybrid filtering [17], and so on. The CKF is similar to the UKF filtering process, but its theoretical derivation is more rigorous [18]. It generates new points by converting 2n equal-valued cubature points through nonlinear functions and is used to predict the system state at the next moment. The CKF also does not need to linearize the non-linear dynamics and has all the features of the UKF. CKF uses a symmetrical sampling strategy and has fewer sampling points than the UKF does.