## 5 Initial Alignment Error Compensation for Shearer's SINS
### Initial Alignment of SINS  
Initial alignment refers to the determination of the attitude angle, position, and velocity of the carrier at the initial time point. The initial alignment process includes two stages, i.e., coarse alignment and fine alignment. The coarse alignment stage is to calculate the initial attitude matrix of the system directly using the output information of the gyroscope and accelerometer, and this alignment stage is not accurate enough and ignores the influences of such factors as the possible vibration of the carrier itself and temperature change, so this stage only provides the initial attitude matrix that meets the requirements of the next fine alignment stage. The fine alignment stage occurs after the coarse alignment stage, and the inertial sensor is generally required to be in a stationary state to estimate the attitude error angle of the system. After the inertial sensor reaches a stable working state, the initial attitude matrix of the system is corrected to finally complete the fine alignment stage. The specific process is as follows:  
The initial outputs of the accelerometer and gyroscope in the \(b\) system are  
\[\left\{ \begin{array}{l}g^{0}=\left(a_{x0},a_{y0},a_{z0}\right)^{T}\\ \omega^{0}=\left(a_{x0},a_{y0},a_{z0}\right)^{T}\end{array} \right.. \tag{6}\]  
According to the transformation relationship between the \(b\) system and the \(n\) system, we can obtain  
\[ \begin{pmatrix}a_{x0}&a_{y0}&a_{z0}\\ \omega_{x0}&\omega_{y0}&\omega_{z0}\\ B_{x0}&B_{y0}&B_{z0}\end{pmatrix} = \begin{pmatrix}0&0&-g\\ 0&\omega_{ie}\cos L&\omega_{ie}\sin L\\ \omega_{ie}\mathcal{G}\cos L&0&0\end{pmatrix} C_{0}^{n^{\prime}}, \tag{7}\]  
that is  
\[C_{0}^{n^{\prime}}= \begin{pmatrix}0&0&\frac{1}{\omega_{ie}\mathcal{G}\cos L} \\ \frac{\sin L}{\mathcal{G}\cos L}&\frac{1}{\omega_{ie}\cos L}&0\\ -\frac{1}{g}&0&0\end{pmatrix}\begin{pmatrix}a_{x0}&a_{y0}&a_{z0}\\ \omega_{x0}&\omega_{y0}&\omega_{z0}\\ B_{x0}&B_{y0}&B_{z0}\end{pmatrix} . \tag{8}\]  
\(C_{0}^{n^{\prime}}\) is the coarse alignment attitude matrix, and then it is corrected with the error model in the next step to complete the fine alignment stage. A transition navigation coordinate system is obtained by transforming the carrier coordinates with \(C_{0}^{n^{\prime}}\). Assuming that there is a small attitude error angle between the transition navigation coordinate system and the \(n\) system, the antisymmetric matrix \(C_{0}^{n^{\prime}}\) of the attitude error angle between the transition navigation coordinate system and the \(n\) system is  
\[C_{0}^{n}=C_{n^{\prime}}^{n}C_{0}^{n^{\prime}}, \tag{9}\]  
where  
\[C_{n}^{n^{\prime}}= \begin{pmatrix}1&-\varphi_{2}&-\varphi_{y}\\ \varphi_{z}&1&-\varphi_{x}\\ -\varphi_{y}&\varphi_{x}&1\end{pmatrix} , \tag{10}\]  
where \(\varphi_{x}\), \(\varphi_{y}\), and \(\varphi_{z}\) are the attitude error angles between the transition navigation coordinate system and the \(n\) system, so it is necessary to solve \(C_{n^{\prime}}^{n}\) to correct \(C_{0}^{n^{\prime}}\), and then an accurate initial attitude matrix \(C_{0}^{n}\) can be obtained. Let the change in velocity in one solving cycle be \(\Delta v^{n}\) and the change in angle be \(\Delta\theta^{n}\), then  
\[\left\{ \begin{array}{l}\Delta v^{n}=\left(0,0,-gt_{0}\right)^{T}\\ \Delta\theta^{n}=\left(0,\omega_{ie}\cos Lt_{0},\omega_{ie}\sin Lt_{0}\right) ^{T}\end{array} \right.. \tag{11}\]  
For accelerometer and gyroscope outputs, it follows that  
\[\left\{ \begin{array}{l}\Delta v^{0}=C_{0}^{n}\Delta v^{n}+\Delta v_{r}{}^{0 }\\ \Delta\theta^{0}=C_{n}^{0}\Delta\theta^{n}+\Delta\theta_{r}{}^{0}\end{array} \right.. \tag{12}\]  
The velocity increment error and angle increment error thus can be obtained as follows:  
\[\left\{ \begin{array}{l}\delta v^{n}=C_{0}^{n^{\prime}}\Delta v^{0}-\Delta v ^{n}\\ \delta\theta^{n}=C_{0}^{n^{\prime}}\Delta\theta^{0}-\Delta\theta^{n}\end{array}  \right.. \tag{13}\]Since \(C_{n}^{n^{\prime}}=\left(C_{n^{\prime}}^{n}\right)^{T}\), it gives  
\[\left( \begin{array}{c}\varphi_{x}\\ \varphi_{y}\\ \varphi_{z}\end{array}