## 2 Vector Tracking Loop Operation Mechanism
If the output of the code phase discriminator and carrier frequency discriminator is selected as the observation, the system measurement equation is as follows [7]:  
\[[\textbf{Z}_{k}]^{2}=[h(\Delta\textbf{X}_{k})]^{2}+[\textbf{U}_{k}]^{2} \tag{2}\]  
where  
\[\textbf{Z}=\left[z_{code,1,k}\ z_{carrier,1,k}\ z_{code,2,k}\ z_{carrier,2,k}\ \cdots\ z_{code,n,k}\ z_{carrier,n,k}\right]^{T} \tag{3}\]  
where  
\[z_{code,j,k}=e_{x,j,k}\Delta x_{k}\ +e_{y,j,k}\Delta y_{k}+e_{z,j,k}\Delta z _{k}+\Delta t_{b,k}+v_{code,j,k} \tag{4}\]  
\[z_{carrier,j,k}=e_{x,j,k}\Delta v_{x,k}\ +e_{y,j,k}\Delta v_{y,k}+e_{z,j,k} \Delta v_{z,k}+\Delta t_{d,k}+v_{carrier,j,k} \tag{5}\]  
In the above expression, subscript '\(j\)' represents the \(j\)th satellite, '\(k\)' represents the time epoch of the data, and '\(a_{j}\)' is the line-of-sight direction unit vector between the receiver and the \(j\)th satellite. It is assumed that the position coordinates of the user at time \(k\) are \((x_{k},y_{k},z_{k})\). The position of the \(j\)th satellite at time \(k\) is \((x_{j,k},y_{j,k},z_{j,k})\)[7], then  
\[e_{x,j,k}=\frac{x_{j,k}-x_{k}}{q_{j,k}} \tag{6}\]  
\[e_{y,j,k}=\frac{y_{j,k}-y_{k}}{q_{j,k}} \tag{7}\]  
\[e_{z,j,k}=\frac{z_{j,k}-z_{k}}{q_{j,k}} \tag{8}\]  
where, \(q_{j,k}\) is  
\[q_{j,k}=\sqrt{\left(x_{j,k}-x_{k}\right)^{2}+\left(y_{j,k}-y_{k}\right)^{2}+ \left(z_{j,k}-z_{k}\right)^{2}}=\sqrt{\left(x_{j,k}-x_{k-1}-\Delta x_{k} \right)^{2}+\left(y_{j,k}-y_{k-1}-\Delta y_{k}\right)^{2}+\left(z_{j,k}-z_{k- 1}-\Delta z_{k}\right)^{2}} \tag{9}\]  
From expressions (2), (9), it can be seen that there is a nonlinear relationship between measurements \(\textbf{Z}_{k}\) and states \(\Delta\textbf{X}_{k}\). In the highly dynamic state of the receiver, the vector tracking loop has better tracking performance than scalar tracking. For example, it can be assisted by strong signals when tracking weak signals, and fast reacquisition can be achieved when the signals are blocked. However, it can be seen from the established highly dynamic vector tracking model that there is a highly nonlinear relationship between the measurements and the states. At this time, it is necessary to use not only a high-performance nonlinear filtering algorithm to estimate the states as accurately as possible but also real-time processing of the measurement noise to adapt to the high-frequency real-time changes of various parameters in a highly dynamic environment. Therefore, this paper adopts a new nonlinear filtering algorithm--cubature Kalman filtering (CKF)--proposed in recent years, which uses a set of sampling points to approximate the state value and covariance of the nonlinear system based on the third-order spherical radial criterion, has a strict and complete theoretical basis of numerical integration, and is superior to nonlinear filtering algorithms such as UKF and PF in terms of computational complexity and reliability. At the same time, to avoid filtering divergence caused by rounding errors in practical applications, the square root filtering idea is applied to CKF and further combined with adaptive filtering; an adaptive square root cubature Kalman filtering algorithm is proposed to realize real-time signal tracking.