## 3 Parameter Modeling and Error Analysis of the INS and GNSS Systems
### GNSS Error Analysis and Modeling
#### 3.2.4 Multipath Effect Error
\tag{35}\]  
According to the above analysis, the state equation of the INS is as follows:  
\[\dot{X}_{I}(t)=F(t)X(t)+G(t)W(t), \tag{36}\]  
where \(X=X_{I}\), \(F=F_{I}\), \(G=G_{I}\) and \(W=W_{I}\). Generally, the loose combination observation equation is based on the position and velocity combination, and the close combination observation equation is based on pseudorange and pseudorange rates. For the aircraft navigation and positioning in the incomplete observation mode of GNSS, the loose combination mode is not applicable. Therefore, the single-satellite integrated navigation and positioning algorithm based on broadband LEO constellation communication links adopts the tight combination mode, and the observation equation is introduced below. In the ECEF coordinate system, the carrier position calculated by INS is assumed to be \(\left(x_{I},y_{I},z_{I}\right)^{T}\), the satellite position calculated by LEO satellite ephemeris is \(\left(x_{s},y_{s},z_{s}\right)^{T}\), and the real position of the carrier is \(\left(x_{u},y_{u},z_{u}\right)^{T}\). The pseudorange from the carrier to the GNSS satellite obtained by INS is as follows:  
\[\dot{\rho_{I}^{i}}=\sqrt{\left(x_{I}-x_{s}^{i}\right)^{2}+\left(y_{I}-y_{s}^{i }\right)^{2}+\left(z_{I}-z_{s}^{i}\right)^{2}}, \tag{37}\]  
where the superscript \(i\) denotes the \(i\)th satellite. Let Equation (37) be expanded by the Taylor series at the true value. Taking the first and second order derivatives, we can obtain  
\[\dot{\rho_{I}^{i}}=\rho_{i}+\frac{x_{u}-x_{s}^{i}}{\rho_{i}}\delta x_{u}+\frac {y_{u}-y_{s}^{i}}{\rho_{i}}\delta y_{u}+\frac{z_{u}-z_{s}^{i}}{\rho_{i}}\delta z _{u}, \tag{38}\]  
where \(\rho_{i}\) is the true distance from the carrier to satellite \(S_{i}\). Let \(e_{ix_{u}}=\frac{x_{u}-x_{s}^{i}}{\rho_{i}},e_{iy_{u}}=\frac{y_{u}-y_{s}^{i}} {\rho_{i}},e_{iz_{u}}=\frac{z_{u}-z_{s}^{i}}{\rho_{i}}\). Then, Equation (38) can be expressed as follows:  
\[\rho_{I}^{i}=\rho_{i}+e_{ix_{u}}\delta x_{u}+e_{iy_{u}}\delta y_{u}+e_{iz_{u} }\delta z_{u}. \tag{39}\]  
The pseudorange measured by the GNSS receiver is as follows:  
\[\rho_{G}^{i}=\rho_{i}+b_{cl}+v_{pi}, \tag{40}\]  
where \(v_{pi}\) is the pseudorange measurement white noise of the \(i\)th satellite \(S_{i}\). According to Equation (39) and Equation (40), the pseudorange difference observation equation of INS and GNSS can be calculated:  
\[\delta\rho_{i}=\rho_{I}^{i}-\rho_{G}^{i}=e_{ix_{u}}\delta x_{u}+e_{iy_{u}} \delta y_{u}+e_{iz_{u}}\delta z_{u}-b_{cl}-v_{pi}. \tag{41}\]  
According to the previous discussion, the position state variables of the geographical coordinate system are \(L\), \(\lambda\), and \(h\), while the position state variables of the ECEF coordinate system are \(x_{u}\), \(y_{u}\), and \(z_{u}\). Through the coordinate transformation matrix \(\left(C_{e}^{n}\right)^{T}\), the transformation of position state variables can be completed. The position expression of the transformed ECEF coordinate system is as follows:  
\[\left\{ \begin{array}{l}x_{u}=-L\sin\lambda-\lambda\sin L\cos\lambda+\left(R_ {n}+h\right)\cos\lambda\cos L\\ y_{u}=L\cos\lambda-\lambda\sin L\sin\lambda+\left(R_{n}+h\right)\sin\lambda \cos L\\ z_{u}=\lambda\cos L+\left[R_{n}(1-e^{2})+h\right]\sin L\end{array} \right.. \tag{42}\]Then, after omitting the infinitesimal in Equation (42), the two sides of the equation take the differential at the same time, and the state transfer matrix of the error from the geographical system to the ECEF coordinate system is obtained:  
\[\delta C_{n}^{e}=\left[ \begin{array}{ccc}-(R_{n}+h)\cos\lambda\sin L&(R_{n}+h) \cos L\sin\lambda&\cos L\cos\lambda\\ -(R_{n}+h)\sin\lambda\sin L&-(R_{n}+h)\cos\lambda\cos L&\cos L\sin\lambda\\ [R_{n}(1-e^{2})+h]\cos L&0&\sin L\end{array} \right].