## 2 Integrated Navigation System Model
### Combined Navigation Model
#### 2.2.2 State Equation  
To clearly explain the state equations, the errors of the INS and the GNSS will be described separately.  
For the INS error state equation, only the random errors of the INS's three-axis gyroscopes and accelerometers, as well as the errors of the INS itself, are considered. The specific equations are  
\[\dot{\mathbf{x}_{\mathsf{I}}}\left(t\right)=\mathbf{F}_{\mathsf{I}}(t) \mathbf{X}_{\mathsf{I}}(t)+\mathbf{G}_{\mathsf{I}}(t)\mathbf{W}_{\mathsf{I}}(t) \tag{2}\]  
where the individual matrices are represented as  
\[\mathbf{X}_{\mathsf{I}}=\begin{bmatrix}\phi^{\mathsf{T}}&(\delta v)^{\mathsf{ T}}&(\delta p)^{\mathsf{T}}&\left(\varepsilon^{b}\right)^{\mathsf{T}}&\left( \
abla^{b}\right)^{\mathsf{T}}&\left(\delta l^{b}\right)^{\mathsf{T}}&\delta t \end{bmatrix}^{\mathsf{T}} \tag{3}\]  
Figure 1: A schematic diagram of the ECEF navigation coordinate frame and body coordinate frame. The X-, Y-, and Z-axes and text within the same framework are represented in the same color.  
\[\begin{split}\mathbf{F}_{\text{I}}=\begin{bmatrix}\mathbf{M}_{aa}& \mathbf{M}_{av}&\mathbf{M}_{ap}&-\mathbf{C}_{b}^{n}&\mathbf{0}_{3\times 3}& \mathbf{0}_{3\times 4}\\ \mathbf{M}_{va}&\mathbf{M}_{vv}&\mathbf{M}_{vp}&\mathbf{0}_{3\times 3}& \mathbf{C}_{b}^{n}&\mathbf{0}_{3\times 4}\\ \mathbf{0}_{3\times 3}&\mathbf{M}_{pv}&\mathbf{M}_{pp}&\mathbf{0}_{3\times 3}& \mathbf{0}_{3\times 3}&\mathbf{0}_{3\times 4}\\ &\qquad\qquad\qquad\qquad\qquad\mathbf{0}_{10\times 19}&\end{bmatrix},\\ \mathbf{G}_{\text{I}}=\begin{bmatrix}-\mathbf{C}_{b}^{n}&\mathbf{0}_{3\times 3 }\\ \mathbf{0}_{3\times 3}&\mathbf{C}_{b}^{n}\\ \mathbf{0}_{13\times 6}\end{bmatrix},\mathbf{W}_{\text{I}}=\begin{bmatrix}w_{b} \\ w_{a}\end{bmatrix}\end{split}\end{split} \tag{4}\]  
detailed definitions of the individual submatrices in the matrix \(\mathbf{F}_{\text{I}}\) can be found in the literature [23]. \(\mathbf{C}_{b}^{n}\) is the rotation matrix from the b-frame to the n-frame. \(w_{b}\) and \(w_{a}\) represent the measured white noise of the gyroscope's angular velocity and the accelerometer's specific force, respectively.  
For the GNSS error equation of state, only the receiver clock error is considered. The specific equation is  
\[\dot{\mathbf{X}}_{\text{G}}\left(t\right)=\mathbf{F}_{\text{G}}(t)\mathbf{X}_ {\text{G}}(t)+\mathbf{G}_{\text{G}}(t)\mathbf{W}_{\text{G}}(t) \tag{5}\]  
where the individual matrices are represented as  
\[\begin{split}\mathbf{X}_{\text{G}}=\begin{bmatrix}\delta t_{u}& \delta t_{ru}\end{bmatrix}^{\top},\mathbf{F}_{\text{G}}=\begin{bmatrix}0&1 \\ 0&-\frac{1}{\tau_{ru}}\end{bmatrix},\\ \mathbf{G}_{\text{G}}=\begin{bmatrix}1&0\\ 0&1\end{bmatrix},\mathbf{W}_{\text{G}}=\begin{bmatrix}w_{tu}\\ w_{tu}\end{bmatrix}\end{split} \tag{6}\]  
where \(\tau_{tru}\) is the time-dependent constant, and \(w_{tu}\) and \(w_{tu}\) are white noise.  
Equations (3) and (5) can be combined to obtain the system state equation in this paper:  
\[\dot{\mathbf{X}}\left(t\right)=\begin{bmatrix}\dot{\mathbf{X}}_{\text{I}}(t) \\ \dot{\mathbf{X}}_{\text{G}}(t)\end{bmatrix}=\begin{bmatrix}\mathbf{F}_{\text{I} }(t)&\mathbf{O}\\ \mathbf{O}&\mathbf{F}_{\text{G}}(t)\end{bmatrix}\begin{bmatrix}\mathbf{X}_{ \text{I}}(t)\\ \mathbf{X}_{\text{G}}(t)\end{bmatrix}+\begin{bmatrix}\mathbf{G}_{\text{I}}(t)& \mathbf{O}\\ \mathbf{O}&\mathbf{G}_{\text{G}}(t)\end{bmatrix}\begin{bmatrix}\mathbf{W}_{ \text{I}}(t)\\ \mathbf{W}_{\text{G}}(t)\end{bmatrix} \tag{7}\]  
i.e., \(\dot{\mathbf{X}}\left(t\right)=\mathbf{F}(t)\mathbf{X}(t)+\mathbf{G}(t) \mathbf{W}(t)\)