# An Efficient Ultra-Tight GPS/RISS Integrated System for Challenging Navigation Environments
## 3 The Proposed Method
### GPS/SD RISS Integration
#### 3.1.3 The System Measurement Model
The velocity error vector will be annotated as \(\delta v^{e}\), and hence, Equation (36) becomes:  
\[\delta Z_{\dot{p}}=G\delta v^{e}+\delta d_{r}+\varepsilon_{\dot{p}} \tag{40}\]  
The LLF-calculated velocity can be transformed into Earth Centred Earth Fixed velocity using:  
\[\left[ \begin{array}{c}\delta v_{x}\\ \delta v_{y}\\ \delta v_{z}\end{array} \right]=\left[ \begin{array}{ccc}-\sin\lambda&-\sin \varphi\ cos\lambda&\cos\varphi\cos\lambda\\ \cos\lambda&-\sin\varphi\ sin\lambda&\cos\varphi\sin\lambda\\ 0&\cos\varphi&\sin\varphi\end{array} \right]\left[ \begin{array}{c}\delta v_{ e}\\ \delta v_{n}\\ \delta v_{u}\end{array} \right] \tag{41}\]  
This LLF to ECEF transformation matrix will be denoted as \(R_{l}^{e}\). Furthermore, the velocity error vector in the LLF will be denoted as \(\delta v^{l}\). Therefore, the last equation becomes:  
\[\delta v^{e}=R_{l}^{e}\delta v^{l} \tag{42}\]  
Substituting this into Equation (40) we obtain:  
\[\delta Z_{\dot{p}}=GR_{l}^{e}\delta v^{l}+\delta d_{r}+\varepsilon_{\dot{p}} \tag{43}\]  
We will combine the geometry matrix \(G\) and the transformation matrix \(R_{l}^{e}\) into one matrix:  
\[H_{N\times 3}^{\dot{p}}=G_{N\times 3}R_{l3\times 3}^{e} \tag{44}\]  
Thus, the final pseudorange rate measurement model could be expressed as:  
\[\delta Z_{\dot{p}N\times 1}=H_{N\times 3}^{\dot{p}}\delta v^{l}+\delta d_{r}+ \varepsilon_{\dot{p}N\times 1} \tag{45}\]  
Now the complete system measurement model could be formed by combining the pseudorange errors measurement model as in Equation (37) and pseudorange rate measurement errors as in Equation (45):  
\[\delta Z_{2N\times 1}=H_{2N\times 11}\delta x_{11\times 1}+\left[ \begin{array}{ c c}\delta b_{rN\times 1}\\ \delta d_{rN\times 1}\end{array} \right]+\left[ \begin{array}{c}\varepsilon _{\dot{p}N\times 1}\\ \varepsilon_{\dot{p}N\times 1}\end{array} \right] \tag{46}\]  
where  
\[H_{2N\times 11}=\left[ \begin{array}{ccc}H_{N\times 3}^{p}&0_{N\times 3}&0_{N \times 3}&1_{N\times 1}&0_{N\times 1}\\ 0_{N\times 3}&H_{N\times 3}^{\dot{p}}&0_{N\times 3}&0_{N\times 1}&1_{N\times 1} \end{array} \right] \tag{47}\]As discussed in Section 3, the pseudorange errors and pseudorange rate errors are the scaled code phase errors and frequency errors, as generated by the code and carrier frequency discriminators, respectively. These error terms can be expressed as:  
\[\delta\rho_{k}^{j}=\frac{c\Delta\tau_{k}^{j}}{f_{code}} \tag{48}\]  
\[\delta\dot{\rho}_{k}^{j}=\frac{c\Delta f_{k}^{j}}{f_{carrier}} \tag{49}\]  
where c is the speed of light, \(\Delta\tau\) is the code phase error, \(f_{code}\) is the code frequency = \(1.023\times 10^{6}\) Hz, \(\Delta f\) is the carrier frequency error, \(f_{carrier}\) is the nominal L1 frequency = \(1575.42\) MHz, \(k\) represents the epoch number, and \(j\) is the corresponding satellite number. Every time a cycle of the tracking/navigation process is completed, the carrier frequency and code frequency are estimated and fed back into the tracking loops to control the code and carrier NCOs inputs to align the locally generated signal with the incoming signal.