## 2 Integrated Navigation System Model
### Combined Navigation Model
#### 2.2.3. Observation Equation  
There are two types of model observations: one is the horizontal attitude angle provided by the VG algorithm, and the other is the dual-antenna heading angle, pseudo-range, and pseudo-range rate from the GNSS. When using the horizontal attitude angle provided by the VG and the dual-antenna heading angle provided by GNSS as observations, the observation equation can be obtained as  
\[\mathbf{Z}_{\text{I}}(t)=\mathbf{H}_{\text{I}}(t)\mathbf{X}(t)+\mathbf{V}_{ \text{I}}(t) \tag{8}\]  
Observation \(\mathbf{Z}_{\text{I}}\) and observation matrix \(\mathbf{H}_{\text{I}}\) can, in turn, be expressed in detail as  
\[\mathbf{Z}_{\text{I}}(t)= \begin{bmatrix}\theta_{ins}-\theta_{accel}\\ \phi_{ins}-\phi_{accel}\\ \psi_{ins}-\psi_{ant}\end{bmatrix} , \begin{bmatrix}\theta_{accel}=\arctan(f,u) \\ \phi_{accel}=-\arctan(r,u)\end{bmatrix}  \tag{9}\]  
\[\mathbf{H}_{\text{I}}(t)= \begin{bmatrix}\frac{-\epsilon_{11}}{\epsilon_{20}+ \epsilon_{22}^{2}}&\frac{\epsilon_{01}}{\epsilon_{20}+\epsilon_{22}^{2}}&0& \mathbf{0}_{1\times 18}\\ \frac{\epsilon_{01}}{\epsilon_{11}+\epsilon_{11}^{2}}&\frac{\epsilon_{01}}{ \epsilon_{21}+\epsilon_{11}^{2}}&0&\mathbf{0}_{1\times 18}\\ \frac{\epsilon_{01}\epsilon_{21}}{\epsilon_{01}+\epsilon_{11}^{2}}&\frac{ \epsilon_{11}\epsilon_{21}}{\epsilon_{01}+\epsilon_{11}^{2}}&-1&\mathbf{0}_{1 \times 18}\end{bmatrix}  \tag{10}\]  
In observation vector \(\mathbf{Z}_{\text{I}}\), \(\theta_{ins}\), \(\phi_{ins}\), and \(\psi_{ins}\) are the pitch, roll, and yaw angles calculated by the inertial navigation system; \(\theta_{accel}\) and \(\phi_{accel}\) represent the pitch and roll angles calculated from the vertical gyro observation data, respectively; and the formula is Equation (9), where \(r\) is the rightward acceleration, \(f\) is the forward acceleration, \(u\) is the upward acceleration, and \(\psi_{ant}\) is the dual-antenna heading angle provided by the GNSS. In observation matrix \(\mathbf{H}_{\text{I}}\), element \(e\) is from the rotation matrix \(\mathbf{C}_{b}^{n}\), whose subscript represents the position of element \(e\) in the rotation matrix \(\mathbf{C}_{b}^{n}\). Since the accuracy of the observations obtained by the VG is affected by the harmful acceleration of the carrier, the practical use of the algorithm stipulates that VG observations be considered valid when the three-directional acceleration paradigm operations satisfy the inequality in Equation (11), with \(g_{local}\) representing the local gravitational acceleration. \[0.9*g_{local}<\parallel\text{Accel}\ _{r,f,u}\parallel<1.1*g_{local} \tag{11}\]  
When using the observations provided by the GNSS, the observation equation is obtained by considering the pseudo-range and pseudo-range rate observations as  
\[\mathbf{Z}_{G}(t)=\mathbf{H}_{G}(t)\mathbf{X}(t)+\mathbf{V}_{G}(t) \tag{12}\]  
Observation \(\mathbf{Z}_{G}\) and observation matrix \(\mathbf{H}_{G}\) can, in turn, be expressed in detail as  
\[\mathbf{Z}_{G}(t)= \begin{bmatrix}\mathbf{P}_{G}-\mathbf{P}_{\text{I}}\\ \dot{\mathbf{P}}_{\text{G}}-\dot{\mathbf{P}}_{\text{I}}\end{bmatrix} \cdot \begin{cases}\mathbf{P}_{G}-\mathbf{P}_{\text{I}}=\begin{bmatrix}\rho_{\text{G }1}-\rho_{\text{I}1}&\rho_{\text{G}2}-\rho_{\text{I}2}&\cdots&\rho_{\text{G}n} -\rho_{\text{Im}}\end{bmatrix}^{\mathsf{T}}\\ \dot{\mathbf{P}}_{\text{G}}-\dot{\mathbf{P}}_{\text{I}}=\begin{bmatrix}\rho_{ \text{G}1}-\dot{\rho}_{\text{I}1}&\dot{\rho}_{\text{G}2}-\dot{\rho}_{\text{I} 2}&\cdots&\dot{\rho}_{\text{G}n}-\dot{\rho}_{\text{Im}}\end{bmatrix}^{ \mathsf{T}}\end{cases}  \tag{13}\]  
\[\mathbf{H}_{G}(t)=\begin{bmatrix}\mathbf{H}_{\rho}(t)\\ \mathbf{H}_{\rho}(t)\end{cases} \tag{14}\]  
\[