# 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
 \tag{18}\]  
Now, differentiating both sides of the coordinate transformation formula yields the following transformation relationship:  
\[\delta x =\cos L\cos\lambda\delta h-\left(R_{n}+h\right)\sin L\cos \lambda\delta L-\left(R_{n}+h\right)\cos L\sin\lambda\delta\lambda\] \[\delta y =\cos L\sin\lambda\delta h-\left(R_{n}+h\right)\sin L\sin\lambda \delta L+\left(R_{n}+h\right)\cos L\cos\lambda\delta\lambda \tag{19}\] \[\delta z =\sin L\delta h+\left[R_{n}\big{(}1-e^{2}\big{)}+h\big{]}\delta L\]  
Hence, the following observation equation can be derived:  
\[Z_{\rho}(t)=H_{\rho}(t)X(t)+V_{\rho}(t) \tag{20}\]  
where  
\[H_{\rho}=\left[O_{4\times 6}\vdots H_{\rho 1}\vdots O_{4\times 6}\vdots H_{ \rho 2}\right]\]  
\[H_{\rho 1}=\left[ \begin{array}{cccc}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\\ a_{41}&a_{42}&a_{43}\end{array} \right],H_{\rho 2}=\left[ \begin{array}{cccc}1&0\\ 1&0\\ 1&0\\ 1&0\end{array} \right] \tag{21}\]  
The coefficient calculation formula is as follows:  
\[Z_{\rho}(t)=H_{\rho}(t)X(t)+V_{\rho}(t) \tag{22}\]  
(2)  
Observation Equation for the System Pseudorange Rate Composition  
The pseudorange rate between the vehicle output by the INS and the \(j\)-th satellite can be expressed as follows:  
\[\rho_{1j}^{\bullet}=e_{j1}\left(\overset{\bullet}{x_{1}}-\overset{\bullet}{x _{sj}}\right)+e_{j2}\left(\overset{\bullet}{y_{1}}-\overset{\bullet}{y_{sj}} \right)+e_{j3}\left(\overset{\bullet}{z_{1}}-\overset{\bullet}{z_{sj}}\right) +e_{j1}\delta\overset{\bullet}{x}+e_{j1}\delta\overset{\bullet}{y}+e_{j1} \delta\overset{\bullet}{z} \tag{23}\]  
The pseudorange rate measured by the GNSS receiver is given by  
\[\rho_{Gj}^{\bullet}=e_{j1}\left(\overset{\bullet}{x_{1}}-\overset{\bullet}{x _{sj}}\right)+e_{j2}\left(\overset{\bullet}{y_{1}}-\overset{\bullet}{y_{sj}} \right)+e_{j3}\left(\overset{\bullet}{z_{1}}-\overset{\bullet}{z_{sj}}\right) +\delta t_{ru}+v_{pj} \tag{24}\]  
Therefore, the observation equation for pseudorange rate error can be derived. \[\rho_{1j}^{\bullet}-\rho_{Gj}^{\bullet}=e_{j1}\delta\overset{\bullet}{x}+e_{ j2}\delta\overset{\bullet}{y}+e_{j3}\delta\overset{\bullet}{z}-\delta t_{ru}-v_{pj} \tag{25}\]  
In the case of four observed satellites, it has  
\[\delta\overset{\bullet}{\rho}= \begin{bmatrix}e_{11}&e_{12}&e_{13}&-1\\ e_{21}&e_{22}&e_{23}&-1\\ e_{31}&e_{32}&e_{33}&-1\\ e_{41}&e_{42}&e_{43}&-1\end{bmatrix}\begin{bmatrix}\delta\overset{\bullet}{x }\\ \delta\overset{\bullet}{y}\\ \delta\overset{\bullet}{z}\\ \delta t_{ru}\end{bmatrix} - \begin{bmatrix}v_{\rho 1}\\ v_{\rho 2}\\ v_{\rho 3}\\ v_{\rho 4}^{\bullet}\end{bmatrix}  \tag{26}\]As the velocity information outputted by the INS is represented in the geodetic coordinate system (i.e., the East-North-Up coordinate system), it is necessary to convert it through the coordinate transformation matrix from the geodetic coordinate system to the ECEF coordinate system, as follows:  
\[\delta\lx@note{footnote}{$\star$} =-\delta V_{e}\sin\lambda-\delta V_{n}\sin L\cos\lambda+\delta V_{u }\cos L\cos\lambda\] \[\delta\lx@note{footnote}{$\star$} =\delta V_{e}\cos\lambda-\delta V_{n}\sin L\sin\lambda+\delta V_{u }\cos L\sin\lambda \tag{27}\] \[\delta\lx@note{footnote}{$\star$} =\delta V_{n}\cos L+\delta V_{n}\sin L\]  
Through the above analysis, the observation equation for the pseudorange rate is derived as follows:  
\[Z_{\lx@paragraphsign}(t)=H_{\lx@paragraphsign}(t)X(t)+V_{\lx@paragraphsign}(t) \tag{28}\]  
where  
\[H_{\lx@paragraphsign}=\left[ \begin{array}{ccc}b_{11}&b_{12}&b_{13}\\ b_{21}&b_{22}&b_{23}\\ b_{31}&b_{32}&b_{33}\\ b_{41}&b_{42}&b_{43}\end{array} \right],H_{\lx@paragraphsign}=\left[ \begin{array} []{ccc}0&1\\ 0&1\\ 0&1\\ 0&1\end{array}