## 2 Vector-Tracking Loops
\begin{array}{cccc}\mathbf{I}_{3\times 3}&\mathbf{T}_{3} \mathbf{I}_{3\times 3}&\mathbf{T}_{s}^{2}\mathbf{I}_{3\times 3}/2&\mathbf{0}_{3\times 2}\\ \mathbf{0}_{3\times 3}&\mathbf{I}_{3\times 3}&\mathbf{T}_{s}\mathbf{I}_{3\times 3}&\mathbf{0}_{3 \times 2}\\ \mathbf{0}_{2\times 3}&\mathbf{0}_{2\times 3}&\mathbf{I}_{3\times 3}&\mathbf{0}_{3 \times 2}\\ \mathbf{0}_{2\times 3}&\mathbf{0}_{2\times 3}&\mathbf{0}_{2\times 3}&\mathbf{L}\end{array} \right] \tag{3}\]  
\[\mathbf{L}=\left[ \begin{array}{cc}1&\mathbf{T}_{s}\\ 0&1\end{array} \right]\]  
where \(T_{s}\) is the discretization interval and \(\mathbf{I}_{3\times 3}\) denotes a third-order unit matrix. In Equation (2), \(\mathbf{\eta}_{k-1}\) is the process noise. The variance matrix is:  
\[\mathbf{Q}_{k}=\left[ \begin{array}{cccc}\mathbf{U}\cdot\mathbf{T}_{s}^{5}/20&\mathbf{U} \cdot\mathbf{T}_{s}^{4}/8&\mathbf{U}\cdot\mathbf{T}_{s}^{3}/6&\mathbf{0}_{3\times 2}\\ \mathbf{U}\cdot\mathbf{T}_{s}^{4}/8&\mathbf{U}\cdot\mathbf{T}_{s}^{3}/3&\mathbf{U}\cdot\mathbf{T}_{s}^ {2}/2&\mathbf{0}_{3\times 2}\\ \mathbf{U}\cdot\mathbf{T}_{s}^{3}/6&\mathbf{U}\cdot\mathbf{T}_{s}^{2}/2&\mathbf{T}_{s}\cdot\mathbf{U}& \mathbf{0}_{3\times 2}\\ \mathbf{0}_{2\times 3}&\mathbf{0}_{2\times 3}&\mathbf{0}_{2\times 3}&\mathbf{V}\end{array} \right] \tag{4}\]  
\[\mathbf{U}=\left[ \begin{array}{cccc}q_{x}^{2}&0&0\\ 0&q_{y}^{2}&0\\ 0&0&q_{z}^{2}\end{array} \right]\ \mathbf{V}=\left[ \begin{array}{cccc}q_{b}^{2}T_{s}+q_{d}^{2} \frac{T_{s}^{3}}{3}&q_{d}^{2}\frac{T_{s}^{2}}{2}\\ q_{d}^{2}\frac{T_{s}^{2}}{2}&q_{d}^{2}T_{s}\end{array} \right]\]  
where \(q_{x}^{2}\), \(q_{y}^{2}\) and \(q_{z}^{2}\) denote the power spectral density (PSD) of the acceleration disturbance noise in three directions and \(q_{b}^{2}\) and \(q_{d}^{2}\) are the PSDs of the receiver clock phase noise and the frequency noise, respectively. The measurement equation of the navigation Kalman filter is as follows:  
\[\left[ \begin{array}{cccc}\Delta\rho_{1}\\ \Delta\dot{\rho}_{1}\\ \vdots\\ \Delta\rho_{M}\\ \Delta\dot{\rho}_{M}\end{array} \right]=\left[ \begin{array}{cccc}\mathbf{u}_{ los,1}&\mathbf{0}_{1\times 3}&\mathbf{0}_{1\times 3}&-1&0\\ \mathbf{0}_{1\times 3}&\mathbf{u}_{los,1}&\mathbf{0}_{1\times 3}&0&-1\\ \vdots&&\vdots&&\\ \mathbf{u}_{los,M}&\mathbf{0}_{1\times 3}&\mathbf{0}_{1\times 3}&-1&0\\ \mathbf{0}_{1\times 3}&\mathbf{u}_{los,M}&\mathbf{0}_{1\times 3}&0&-1\\ \end{array} \right]\left[ \begin{array}{c}\delta P_{3\times 1}\\ \delta P_{3\times 1}\\ \delta A_{3\times 1}\\ \delta B_{2\times 1}\end{array} \right] \tag{5}\]where \(\Delta\rho\) and \(\Delta\dot{\rho}\) are the pseudorange residuals and the pseudorange rate residuals, respectively, that are output by the discriminators; \(\mathbf{u}_{los}\) denotes the unit vector of the receiver and satellite line of sight; and \(M\) is the number of visible satellites. The symbols \(IE_{k}\), \(QE_{k}\), \(IL_{k}\) and \(QL_{k}\) represent the early and late complex correlator output samples at time \(k\). The correlator spacing is 1/2 chip. \(IP_{k,1}\), \(IP_{k,2}\), \(QP_{k,1}\) and \(QP_{k,2}\) represent prompt complex correlator output samples (subscripts 1 and 2 denote the sample outputs in the first half and the second half, respectively, of the integration period). As shown in Figure 1, each tracking channel uses a code discriminator and a carrier frequency discriminator to process the correlator outputs and to estimate the code and carrier tracking errors. The tracking errors of the discriminator output are converted into pseudorange and pseudorange rate residuals as Kalman filter measurements. The forms of the code phase and carrier frequency discriminator are as follows [15]. 1.