# Performance and Evaluation of GNSS Receiver Vector Tracking Loop Based on Adaptive Cascade Filter
## 2 Design of Cascaded Adaptive Vector Tracking Loop
### Extended Kalman Navigation Filter Design  
This section adopts the receiver position error \(\Delta p_{x},\Delta p_{y},\Delta p_{z}\), receiver velocity error \(\Delta v_{x},\Delta v_{y},\Delta v_{z}\), receiver clock drift, and clock drift rate \(\Delta b,\Delta d\) as the state variables of the EKF navigation processor to establish the state and measurement equations. \[X=\left[ \begin{array}{cccc}\Delta p_{x}&\Delta p_{y}&\Delta p_{z}&\Delta v _{x}&\Delta v_{y}&\Delta v_{z}&\Delta b&\Delta d\end{array} \right]^{T} \tag{13}\]  
\[\hat{X}_{k}=\Phi_{k-1}\hat{X}_{k-1}+\omega_{k} \tag{14}\]  
\[\Phi_{k-1}=\left[ \begin{array}{cccc}I_{3\times 3}&T_{3\times 3}&0_{3\times 2} \\ 0_{3\times 3}&I_{3\times 3}&0_{3\times 2}\\ 0_{2\times 3}&0_{2\times 3}&K\end{array} \right]_{8\times 8},\;\;\;K=\left[ \begin{array}{cccc}1&T\\ 0&1\end{array} \right] \tag{15}\]  
where \(\Phi_{k-1}\) is the discretized state transition matrix, \(\omega_{k}\) is the state noise matrix, and the statistical characteristics can be expressed as:  
\[\left\{ \begin{array}{l}E(W_{k})=0_{8\times 1}\\ E\big{(}W_{k}W_{k}^{T}\big{)}=Q_{k}=\left[\begin{array}{cccc}Q_{dyn,1}&Q_{ dyn,2}&0_{3\times 2}\\ Q_{dyn,2}&TQ_{dyn}&0_{3\times 2}\\ 0_{2\times 3}&0_{2\times 3}&Q_{clk}\end{array}\right]\end{array} \right. \tag{16}\]\[Q_{dyn}=\left[ \begin{array}{ccc}\sigma_{x}^{2}&0&0\\ 0&\sigma_{y}^{2}&0\\ 0&0&\sigma_{z}^{2}\end{array} \right],Q_{dyn,1}=\frac{T^{3}}{3}Q_{dyn},Q_{dyn,2}= \frac{T^{2}}{2}Q_{dyn} \tag{17}\]  
\[Q_{clk}=\left[ \begin{array}{ccc}\sigma_{b}^{2}T+\sigma_{d}^{2}\frac{T^{3}}{ 3}&\sigma_{d}^{2}\frac{T^{2}}{2}\\ \sigma_{d}^{2}\frac{T^{2}}{2}&\sigma_{d}^{2}\end{array} \right] \tag{18}\]  
where \(\sigma_{i}^{2}(i=x,y,z),\sigma_{b}^{2},\sigma_{d}^{2}\) are the variances of the receiver position error, the clock drift, and the clock drift rate, respectively. The discretized measurement equation can be expressed as:  
\[Z=\left[\Delta\rho^{1},\Delta\rho^{2},\cdots,\Delta\rho^{m},\Delta\dot{\rho}^ {1},\Delta\dot{\rho}^{2},\cdots,\Delta\dot{\rho}^{m}\right]^{T} \tag{19}\]  
\[Z_{k}=H_{k}\cdot X_{k}+\
u_{k} \tag{20}\]  
\[H=\left[ \begin{array}{cccccc}-\alpha_{x}^{1}&-\alpha_{y}^{1}&-\alpha_{z}^{ 1}&0&0&0&1&0\\ -\alpha_{x}^{2}&-\alpha_{y}^{2}&-\alpha_{z}^{2}&0&0&0&1&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&1&\vdots\\ -\alpha_{x}^{m}&-\alpha_{y}^{m}&-\alpha_{z}^{m}&0&0&0&1&0\\ 0&0&0&-\alpha_{x}^{1}&-\alpha_{y}^{1}&-\alpha_{z}^{1}&0&1\\ 0&0&0&-\alpha_{x}^{2}&-\alpha_{y}^{2}&-\alpha_{z}^{2}&0&1\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\ 0&0&0&-\alpha_{x}^{m}&-\alpha_{y}^{m}&-\alpha_{z}^{m}&0&0\end{array} \right] \tag{21}\]  
where \(\Delta\rho^{i},\Delta\dot{\rho}^{i}(i=1,2\cdots m)\) represent the pseudo-range error and pseudo-range rate error, \(m\) represents the number of satellites, \(\alpha_{x}^{i},\alpha_{y}^{i},\alpha_{z}^{i}\) represent the line-of-sight distance vector between the \(i\)-th satellite and the receiver, and \(\
u_{k}\) is the measurement noise. Under the premise of satisfying Gaussian white noise:  
\[\left\{ \begin{array}{ccc}E(\
u_{k})=0_{2n\times 1}\\ E(\
u_{k}\
u_{k}^{T})=R_{k}=\left[\begin{array}{cc}R_{\Delta\rho,k}&0_{n \times n}\\ 0_{n\times n}&R_{\Delta\dot{\rho},k}\end{array}\right]_{2n\times 2n}\end{array} \right. \tag{22}\]  
\[R_{\Delta\rho,k}=\left[ \begin{array}{ccc}R_{\Delta\rho,k}^{1}&\cdots&0\\ \vdots&\ddots&\vdots\\ 0&\cdots&R_{\Delta\rho,k}^{n}\end{array} \right]_{n\times n},R_{\Delta\dot{ \rho},k}=\left[ \begin{array}{ccc}R_{\Delta\dot{\rho},k}^{1}&\cdots&0\\ \vdots&\ddots&\vdots\\ 0&\cdots&R_{\Delta\dot{\rho},k}^{n}\end{array}