## 5 Initial Alignment Error Compensation for Shearer's SINS
### Fine Alignment Stage Based on Optimized UKF Algorithm  
For SINS, the Kalman filter algorithm is often used to suppress its initial alignment error, but SINS often exhibits different degrees of nonlinearity, and the Kalman filter (KF) algorithm is not good enough to suppress the error of nonlinear systems. Therefore, based on the research of the Kalman filter algorithm, a nonlinear Kalman filter algorithm, namely unscented Kalman filter (UKF) algorithm, is proposed, in which unscented transform (UT transform) is used to determine the sampling point near the estimation point, and the Kalman linear filter framework is used to avoid the degradation defect of particle filter generated by random sampling. Studies show that the filter performance of UKF is much better than that of KF with comparable computational effort. Under the actual conditions, the computational effort and complexity of UKF are much lower than that of the particle filter. In the following section, the UKF optimization algorithm is used to compensate for the fine alignment stage of the initial alignment [29]. We are assuming that after the initial coarse alignment, the error angles \(\varphi_{E}\), \(\varphi_{N}\), and \(\varphi_{U}\) are all small angles. The horizontal error angle is smaller than the orientation error angle in the shearer's SINS, so the influences of the horizontal cross-coupling terms \(\varphi_{N}\omega_{ie}\sin L\), \(\varphi_{E}\omega_{ie}\cos L\), and \(-\varphi_{E}\omega_{ie}\sin L\) can be omitted. While in the fine alignment, the position is known, so the influences of \(\delta L\omega_{ie}\cos L\) and \(-\delta L\omega_{ie}\sin L\) can be omitted. Under these conditions, the error equation related to the initial fine alignment can be obtained as follows:  
\[ \begin{split}\delta\dot{V}_{E}=2\omega_{ie}\sin L\delta V_{N}- \varphi_{N}g+\epsilon_{ax}\\ \delta\dot{V}_{N}=-2\omega_{ie}\sin L\delta V_{E}-\varphi_{E}g+ \epsilon_{ay}\\ \dot{\varphi}_{E}=\varphi_{N}\omega_{ie}\sin L-\varphi_{U}\omega_{ ie}\cos L+\epsilon_{ax}\\ \dot{\varphi}_{N}=-\varphi_{E}\omega_{ie}\sin L+\epsilon_{\omega y} \\ \dot{\varphi}_{U}=\ \theta_{E}\omega_{ie}\cos L+\epsilon_{\omega z} \end{split}  \tag{15}\]  
Due to the short time of initial alignment, the gyroscope drift and the accelerometer zero offset can be considered as random constants, then  
\[\dot{\epsilon}_{w}=0,\dot{\epsilon}_{a}=0. \tag{16}\]  
Therefore, the SINS can be expressed as follows:  
\[\left[\epsilon_{aE}\ \epsilon_{aN}\ \epsilon_{aU}\right]^{T}=C_{b}^{n} \left[\epsilon_{ax}\ \epsilon_{ay}\ \epsilon_{az}\right]^{T} \tag{17}\]\[[\varepsilon_{wE}\ \varepsilon_{wN}\ \varepsilon_{wU}]^{T}=C_{b}^{n}\big{[} \varepsilon_{wx}\ \varepsilon_{wy}\ \varepsilon_{wz}\big{]}^{T} \tag{18}\]  
Based on the model of the inertial sensor, the error state equation of the SINS alignment is expressed as follows:  
\[\dot{X}=AX+W. \tag{19}\]  
In the equation, \(X=\big{[}\delta V_{E}\ \delta V_{N}\ \varphi_{E}\ \varphi_{N}\ \varphi_{U}\ \varepsilon_{ax}\ \varepsilon_{ay}\ \varepsilon_{wx}\ \varepsilon_{uy}\ \varepsilon_{wz}\big{]}^{T}\) is the state variable, where \(\delta V_{E}\) is the velocity error in the east direction, \(\delta V_{N}\) is the velocity error in the north direction, \(\phi_{U}\) is the orientation error angle, \(\phi_{N}\) is the horizontal error angle, \(\varepsilon_{w}\) is the random constant of gyroscope drift, and \(\varepsilon_{a}\) is the random constant zero offset of acceleration. Then, the state transition matrix \(A\) is expressed as follows:  
\[A= \begin{bmatrix}F&T\\ 0_{5\times 5}&0_{5\times 5}\end{bmatrix}  \tag{20}\]  
where  
\[F= \begin{bmatrix}0&2\Omega_{U}&0&-g&0\\ 2\Omega_{U}&0&g&0&0\\ 0&0&0&\Omega_{U}&-\Omega_{U}\\ 0&0&-\Omega_{U}&0&0\\ 0&0&\Omega_{U}&0&0\end{bmatrix}  \tag{21}\]  
\[T= \begin{bmatrix}C_{11}&C_{12}&0&0&0\\ C_{21}&C_{22}&0&0&0\\ 0&0&C_{11}&C_{12}&C_{13}\\ 0&0&C_{21}&C_{22}&C_{23}\\ 0&0&C_{31}&C_{32}&C_{33}.\end{bmatrix} .