## 1 Introduction  
With the continuous improvement of the Beidou satellite navigation system, navigation and positioning applications based on the Beidou system are also gradually being developed. However, in various highly dynamic applications, such as cruise missiles and high-orbit space, the carrier frequency of the signal received by the receiver will contain a large Doppler shift and rate of change so that the traditional scalar receiver cannot complete the tracking and positioning. At this time, if the correlation between the signal channels can be used to assist the weak signal channel with the tracking result of the strong signal channel, then the loop can continue to track the signal without losing lock. This concept of using the correlation between signal channels to further improve the tracking performance of the receiver is called vector tracking. The concept of vector tracking was proposed as early as 1996 by [PERSON] et al. [1]. After entering the 21 st century, vector tracking technology has been given importance by scholars and developed rapidly. [PERSON] et al., from the University of Munich, Germany, were the first to give a detailed structure of the vector delay locking loop (VDLL), vector frequency locking loop (VFLL), and their software receiver form [2, 3]. Reference [4] introduced a variant of the vector delay locking loop tracking algorithm for tracking L1 civil GPS signals. The VDL architecture uses a single extended Kalman filter to predict the phase of the satellite PRN code and track the user's position, speed, and clock state. In addition, each channel uses a series of independent Kalman filters to track the satellite carrier signal. Then, the advantages of the architecture based on the vector/Kalman filter over traditional methods can be explained. References [5; 6] discussed the continuous tracking of GPS signals by vector tracking loops in the case of weak signals and compared them with the traditional scalar tracking method. Domestic scholars and doctors have also conducted in-depth research on vector tracking rings. [PERSON] from Tsinghua University and [PERSON] from the University of Colorado, USA, together presented the open-source GPS/GNSS vector tracking loop code. Experiments have proven that the proposed VLL structure is superior to SLL in terms of the accuracy of the navigation solution and the ability to cover GPS signal interruptions [7]. In addition, [PERSON] from the National University of Defense Technology gave a nonlinear observation equation of the VDLL, linearized the established model, and realized the VDLL with an extended Kalman filter (EKF). The research results show that the vector code loop can maintain lock even when several channels lose lock momentarily. This advantage makes the vector code loop suitable for highly dynamic and weak signal environments [8]. The abovementioned research on the vector tracking loop provides a new idea for the tracking of highly dynamic weak signals in a GPS system. For the nonlinear model of the vector tracking loop, the more commonly used method is the EKF algorithm. In highly dynamic applications, the high degree of nonlinearity caused by high dynamic stress will incur greater errors if the EKF algorithm is used. In this case, nonlinear filtering algorithms such as unscented Kalman filters (UKF), particle filters (PF) [9], strong tracking Kalman filters [10], and cubature Kalman filters (CKF) are needed. Among them, the cubature Kalman filter algorithm is a newly proposed algorithm [11] that is derived from rigorous theory and can be accurate to the third order of Taylor expansion. Unlike the EKF algorithm, there is no error caused by the linearization process. Compared with the unscented Kalman filter, it reduces one sampling point. However, CKF has the characteristics of numerical instability and a large amount of computation in the recursive process, and once the initialization parameters are set, the parameters cannot be adjusted in real time as the environment changes [12; 13].