# 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 GPS-only system uses an Earth-centered Earth-fixed (ECEF) frame of reference as the state vector reflects whereas the complete (integrated) system state vector is in Local Level Frame (LLF). Thus, we need to project this into that then we can connect the measurements with the system states for the system measurement update step. The errors in the pseudorange measurements are connected to the errors in the position through the following relation [6]:  
\[\delta Z_{p}=\left[ \begin{array}{c}\delta\rho^{1}\\ \vdots\\ \delta\rho^{N}\end{array} \right]=\left[ \begin{array}{ccc}L_{x}^{1}&L_{y}^{1 }&L_{z}^{1}\\ L_{x}^{2}&L_{y}^{2}&L_{z}^{2}\\ \vdots&\vdots&\vdots\\ L_{x}^{N}&L_{y}^{N}&L_{z}^{N}\end{array} \right]\left[ \begin{array}{c}\delta x \\ \delta y\\ \delta z\end{array} \right]+\delta b_{r}+\left[ \begin{array}{c}\varepsilon_{ p}^{1}\\ \vdots\\ \varepsilon_{p}^{N}\end{array} \right] \tag{28}\]  
\[\delta\rho^{j}=\rho_{GPS}^{j}-\rho_{est}^{j} \tag{29}\]  
where \(\rho_{GPS}^{j}\) is the pseudorange measured by the GPS receiver to the \(j\)th satellite, \(\rho_{est}^{j}\) is the estimated pseudorange to the \(j\)th satellite, \((L_{x}^{j},\ L_{y}^{j},\ L_{z}^{j})\) is the line-of-sight vector from receiver to the \(j\)th satellite, \(\left[\varepsilon_{p}\right]_{N\times 1}\) is the pseudorange measurement noise vector. The transformation matrix above, also called the geometry matrix, will henceforth be denoted as \(G\). We use the following relation to transform the ECEF Cartesian coordinates into ECEF geodetic coordinates:  
\[x=(R_{N}+h)\cos\varphi\cos\lambda \tag{30}\]  
\[y=(R_{N}+h)\cos\varphi\sin\lambda \tag{31}\]  
\[z=\left[R_{N}\Big{(}1-e^{2}\Big{)}+h\right]\sin\varphi \tag{32}\]  
However, since we are interested in the error terms of these coordinates, again we linearize this relation and then get:  
\[\left[ \begin{array}{c}\delta x\\ \delta y\\ \delta z\end{array} \right]=\left[ \begin{array}{ccc}-(R_{N}+h)\sin\varphi \cos\lambda&-(R_{N}+h)\cos\varphi\sin\lambda&\cos\varphi\cos\lambda\\ -(R_{N}+h)\sin\varphi\sin\lambda&+(R_{N}+h)\cos\varphi\cos\lambda&\cos\varphi \sin\lambda\\ \left[R_{N}\Big{(}1-e^{2}\Big{)}+h\right]\cos\varphi&0&\sin\varphi&\sin\varphi \end{array} \right] \tag{33}\]  
For simplicity, we will re-write the above equation on the form:  
\[\delta r^{e}=T\delta r^{J} \tag{34}\]  
where \(\delta r^{e}\) represents the Cartesian position error vector, T is the transformation matrix, and \(\delta r^{J}\) is the geodetic position error vector. Substituting Equation (34) into Equation (28), we get:  
\[\delta Z_{p}=GT\delta r^{J}+\delta b_{r}+\varepsilon_{p} \tag{35}\]  
We will join the G and T matrices in one whole matrix for better readability:  
\[H_{N\times 3}^{p}=G_{N\times 3}T_{3\times 3} \tag{36}\]  
Thus, the ultimate pseudorange measurement model becomes:  
\[\delta Z_{pN\times 1}=H_{N\times 3}^{p}\delta r^{J}+\delta b_{r}+\varepsilon_{pN \times 1} \tag{37}\]We deal with the pseudorange rate measurements in a similar way. The errors in the pseudorange rate measurements can be connected to the errors in velocity as:  
\[\delta Z_{\dot{p}}=\left[ \begin{array}{c}\delta\dot{\rho}^{1}\\ \vdots\\ \delta\dot{\rho}^{N}\end{array} \right]=\left[ \begin{array}{ccc}L_{x}^{1}&L_{y} ^{1}&L_{z}^{1}\\ L_{x}^{2}&L_{y}^{2}&L_{z}^{2}\\ \vdots&\vdots&\vdots\\ L_{x}^{N}&L_{y}^{N}&L_{z}^{N}\end{array} \right]\left[ \begin{array}{c}\delta v _{x}\\ \delta v_{y}\\ \delta v_{z}\end{array} \right]+\delta d_{r}+\left[ \begin{array}{c}\varepsilon _{\dot{p}}^{1}\\ \vdots\\ \varepsilon_{\dot{p}}^{N}\end{array} \right] \tag{38}\]  
\[\delta\dot{\rho}^{j}=\dot{\rho}^{j}_{GPS}-\dot{\rho}^{j}_{est} \tag{39}\]  
where \(\dot{\rho}^{j}_{GPS}\) is the pseudorange rate measured by the GPS receiver to the \(j\)th satellite, \(\dot{\rho}^{j}_{est}\) is the estimated pseudorange rate to the \(j\)th satellite, \(\left[\varepsilon_{\dot{p}}\right]_{N\times 1}\) is the pseudorange rate measurement noise vector.