## 2 Train SINS/GNSS Integrated Navigation System Model
### SINS State Equation
\tag{5}\]  
where \(\delta p\) represents position, and \(\delta v\) represents velocity. For the velocity error differential equation, after expanding the three-dimensional attitude error equation, there are more than ten error terms, making it difficult to identify error propagation patterns. However, in the context of train applications, assuming a two-dimensional plane navigation, the velocity error differential equation becomes relatively simple, with a clearer physical meaning. Based on the specific force differential Equation (1) and neglecting the effect of Earth's curvature, i.e., \(\omega_{en}^{n}\)= 0, and considering that the Earth's rotational speed is 10\({}^{-5}\) rad/s in magnitude, while train speeds are typically lower than 100 m/s, in the derivation of the error equation for short-term plane navigation, we can simplify the specific force differential Equations (6) and (7). \[\dot{v}^{n}=C_{b}^{n}f_{st}^{b}-\left(2\omega_{ie}^{n}+\omega_{en}^{n}\right) \times v^{n}+g^{n} \tag{6}\]\[\dot{v}^{n}=C_{b}^{n}f_{st}^{b}+g^{n} \tag{7}\]  
In the two-dimensional plane assumption, according to the definition of the heading angle in Figure 1, we obtain the following equation:  
\[\mathrm{C}_{b}^{n}= \begin{bmatrix}\cos\psi&-\sin\psi\\ \sin\psi&\cos\psi\end{bmatrix}  \tag{8}\]  
where \(\psi\) represents the heading angle. To simplify the specific force equation, we use the following equation:  
\[\left\{ \begin{array}{l}\dot{v}^{n}=C_{b}^{n}f_{sf}^{b}+g^{n}\\ \dot{\widetilde{v}}^{n}_{b}=\widetilde{C}_{b,sf}^{n}f_{sf}^{b}+g^{n}\\ \dot{v}^{n}+\delta\dot{v}^{n}=[I-(\phi)\times]C_{b}^{n}(f^{b}+\delta f^{b}+n_{ a}^{b})+g^{n}\\ \delta\dot{v}^{n}=C_{b}^{n}\delta f^{b}+C_{b}^{n}n_{a}^{b}-(\phi)\times C_{b}^{ n}f^{b}\end{array} \right. \tag{9}\]  
Similar to the anti-matrix of a three-dimensional vector, here, we define a two-dimensional antisymmetric matrix and is as follows:  
\[-(\phi)\times=- \begin{bmatrix}0&\phi\\ -\phi&0\end{bmatrix}  \tag{10}\]  
Then, we obtain the following:  
\[ \begin{array}{l}-(\phi)\times\mathrm{C}_{b}^{n}f^{b}=-\begin{bmatrix}0&\phi \\ -\phi&0\end{bmatrix}\begin{bmatrix}\cos\psi&-\sin\psi\\ \sin\psi&\cos\psi\end{bmatrix}\begin{bmatrix}\begin{bmatrix}f_{x}^{b}\\ f_{y}^{b}\end{bmatrix}=-\begin{bmatrix}0&1\\ -1&0\end{bmatrix}\begin{bmatrix}\cos\psi&-\sin\psi\\ \sin\psi&\cos\psi\end{bmatrix}\begin{bmatrix}\begin{bmatrix}f_{x}^{b}\\ f_{y}^{b}\end{bmatrix}\phi\\ -\cos\psi\end{bmatrix}\begin{bmatrix}\begin{bmatrix}f_{x}^{b}\\ f_{y}^{b}\end{bmatrix}\phi\end{bmatrix}\\ \end{array}  \tag{11}\]  
Substituting Equation (11) into Equation (9) and simplifying, we obtain the following:  
\[\left\{ \begin{array}{l}\dot{\delta\mathbf{V}}_{N}^{n}=-(\sin\psi f_{x}^{b}+ \cos\psi f_{y}^{b})\phi+\cos\psi\delta f_{x}^{b}-\sin\psi\delta f_{y}^{b}+ \cos\psi n_{\mathrm{ex}}^{b}-\sin\psi n_{ay}^{b}\\ \dot{\delta\mathbf{V}}_{E}^{n}=(\cos\psi f_{x}^{b}-\sin\psi f_{y}^{b})\phi+ \sin\psi\delta f_{x}^{b}+\cos\psi\delta f_{y}^{b}+\sin\psi n_{ax}^{b}+\cos \psi n_{ay}^{b}\end{array} \right. \tag{12}\]  
According to the theory of linear systems, when a coefficient of the system has a linearly unrelated change to other coefficients under certain conditions, and both the measurement and state coefficients are known, then that state is considered to have an analytical solution and is observable. As the corresponding coefficient of that state changes more significantly, its susceptibility to noise decreases, leading to a higher observability. However, when the train is in a state of constant velocity and linear motion, the train's acceleration output is equal to zero; thus, the coefficient in front of the heading misalignment angle error is zero. Therefore, the heading is unobservable when the train is in a state of constant velocity and linear motion. From the velocity calculated in Equation (1), we obtain the following:  
\[