# Research on Kalman Filter Fusion Navigation Algorithm Assisted by CNN-LSTM Neural Network
## 2 Kalman Filtering GNSS/INS Fusion Navigation Basic Principles
### Mathematical Model of GNSS/INS Fusion Navigation
#### 2.2.2 Observation Equations  
(1)  
Observation Equation of the System Pseudorange Composition  
Since the position information output by the strapdown inertial navigation solution is typically represented in the Earth-centered Earth-fixed (ECEF) coordinate system, which is the longitude, latitude, and altitude coordinate system, it needs to be converted to the ECEF coordinate system. The conversion relationship is as follows:  
\[ \begin{split} x_{I}&=(R_{n}+h)\cos L\cos\lambda\\ y_{I}&=(R_{n}+h)\cos L\cos\lambda\\ z_{I}&=\begin{bmatrix}R_{n}\big{(}1-e^{2}\big{)}+h\end{bmatrix}\sin L\end{split}  \tag{12}\]  
Assuming the position of the \(j\)-th satellite in the ECEF coordinate system is \(\left(x_{s}^{j},y_{s}^{j},z_{s}^{j}\right)\), then the pseudorange from the vehicle to the \(j\)-th satellite can be obtained using the position of the vehicle \((x_{I},y_{I},z_{I})\) calculated by the inertial navigation solution:  
\[\rho_{I}^{j}=\sqrt{\left(x_{I}-x_{s}^{j}\right)^{2}+\left(y_{I}-y_{s}^{j} \right)^{2}+\left(z_{I}-z_{s}^{j}\right)^{2}} \tag{13}\]  
Expanding Equation (14) in a Taylor series around the true position of the vehicle in the ECEF coordinate system (\(x,y,z\)) and retaining terms up to the first order, it can be obtained as follows:  
\[\rho_{I}^{j}=\left[\left(x_{I}-x_{s}^{j}\right)^{2}+\left(y_{I}-y_{s}^{j} \right)^{2}+\left(z_{I}-z_{s}^{j}\right)^{2}\right]^{\frac{1}{2}}+\frac{ \partial\rho_{I}^{j}}{\partial x}\delta x+\frac{\partial\rho_{I}^{j}}{ \partial y}\delta y+\frac{\partial\rho_{I}^{j}}{\partial z}\delta z \tag{14}\]  
Then, it has the following relationship:  
\[ \begin{split}\frac{\partial\rho_{I}^{j}}{\partial x}& =\frac{x-x_{s}^{j}}{\left[\left(x-x_{s}^{j}\right)^{2}+\left(y-y_{s}^{j} \right)^{2}+\left(z-z_{s}^{j}\right)^{2}\right]^{\frac{1}{2}}}=e_{j1}\\ \frac{\partial\rho_{I}^{j}}{\partial y}&= \frac{y-y_{s}^{j}}{\left[\left(x-x_{s}^{j}\right)^{2}+\left(y-y_{s}^{j} \right)^{2}+\left(z-z_{s}^{j}\right)^{2}\right]^{\frac{1}{2}}}=e_{j2}\\ \frac{\partial\rho_{I}^{j}}{\partial z}&= \frac{z-z_{s}^{j}}{\left[\left(x-x_{s}^{j}\right)^{2}+\left(y-y_{s}^{j} \right)^{2}+\left(z-z_{s}^{j}\right)^{2}\right]^{\frac{1}{2}}}=e_{j3}\end{split}  \tag{15}\]  
The expression for the pseudorange measured by the GNSS receiver between the vehicle and the \(j\)-th GNSS satellite is given by  
\[\rho_{G}^{j}=\left[\left(x_{I}-x_{s}^{j}\right)^{2}+\left(y_{I}-y_{s}^{j} \right)^{2}+\left(z_{I}-z_{s}^{j}\right)^{2}\right]^{\frac{1}{2}}-\delta t_{u} -v_{\rho}^{j} \tag{16}\]  
where \(\delta t_{u}\) represents the distance corresponding to the equivalent clock error, and \(v_{\rho}^{j}\) denotes the pseudorange measurement noise, primarily stemming from effects such as multipath, tropospheric delay errors, and ionospheric errors. Thus, it can be obtained the observation equation for pseudorange error as follows:  
\[\delta\rho_{j}=\rho_{I}^{j}-\rho_{G}^{j}=e_{j1}\delta x+e_{j2}\delta y+e_{j3} \delta z+\delta t_{u}+v_{\rho}^{j} \tag{17}\]  
In general, the observation equation for pseudorange is established by selecting the best four satellites as observation satellites. \[\delta\rho= \begin{bmatrix}\delta\rho_{1}\\ \delta\rho_{2}\\ \delta\rho_{3}\\ \delta\rho_{4}\end{bmatrix} = \begin{bmatrix}e_{11}&e_{12}&e_{13}&1\\ e_{21}&e_{22}&e_{23}&1\\ e_{31}&e_{32}&e_{33}&1\\ e_{41}&e_{42}&e_{43}&1\end{bmatrix}\begin{bmatrix}\delta x\\ \delta y\\ \delta z\\ \delta t_{u}\end{bmatrix} + \begin{bmatrix}v_{1}^{1}\\ v_{2}^{2}\\ v_{\rho}^{3}\\ v_{\rho}^{4}\end{bmatrix}