## 6 Installation Error Compensation for the Shearer's SINS
### Sculling Error and Compensation Algorithm  
When the shearer is working underground in a coal mine, the angular vibration and linear vibration of SINS will be probably caused by such factors as eccentricity of the traveling wheel, gear meshing clearance, and the undulation of the coal seam floor. When the in-phase angular and linear vibrations occur simultaneously on the two orthogonal axes of the shearer, a sculling motion will be formed on the third axis perpendicular to the above two axes, with a constant velocity generated.  
Assuming that there are angular vibration and linear vibration with the same frequency along the X axis and Y axis of the carrier, respectively, the angular vibration  
\[\alpha(t)=\begin{bmatrix}A_{\alpha}\cos\Omega t\\ 0\\ 0\end{bmatrix}, \tag{62}\]  
the linear vibration (displacement)  
\[P(t)=\begin{bmatrix}0\\ A_{p}\cos(\Omega t+\varphi)\\ 0\end{bmatrix}, \tag{63}\]  
Figure 16: Contrast with or without coning error compensation.  
Figure 15: Shearer SINS coning error.  
and the angular velocity  
\[\omega(t)=\dot{\alpha}(t)=\begin{bmatrix}-A_{a}\sin\Omega t\\ 0\\ 0\end{bmatrix}, \tag{64}\]  
the velocity  
\[v(t)=\dot{P}(t)=\begin{bmatrix}0\\ -A_{p}\Omega\sin(\Omega t+\varphi)\\ 0\end{bmatrix}, \tag{65}\]  
and the acceleration  
\[f(t)=\dot{v}(t)=\begin{bmatrix}0\\ -A_{p}\Omega^{2}\cos(\Omega t+\varphi)\\ 0\end{bmatrix}, \tag{66}\]  
then there will be a velocity full flow along the third axis  
\[\Delta v_{scultm}=\frac{1}{2}\int\limits_{t_{m-1}}^{t_{m}}[\alpha(t)\times f( t)+v(t)\times\omega(t)]dt=\begin{bmatrix}0\\ 0\\ -(1/2)A_{a}A_{p}\Omega^{2}T\cos\varphi\end{bmatrix}. \tag{67}\]  
When the carrier makes a sculling motion, a direct component of the velocity \(-(1/2)A_{a}A_{p}\Omega^{2}T\cos\varphi\), that is, a sculling error, is generated on the third orthogonal axis. Under the condition of high-frequency motion, the SINS will produce sculling error in the velocity calculation. To improve the position and attitude monitoring accuracy of the SINS, the sculling error must be compensated in the velocity calculation.  
Assuming that the information on \(\Delta\theta_{i}\) and \(\Delta v_{i}\) can be obtained by sampling \(\omega^{b}\) and \(f^{b}\) for \(N\) times at equal intervals in the time range of \([t_{m-1},\ t_{m}]\), and that \(\omega\) and \(f\) can be fitted with an \(N-1\)-degree polynomial about \(t\), then the expression of sculling error compensation algorithm is obtained as follows:  
\[\Delta v_{scullm}=\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}K_{ij}\big{[}\Delta\theta_{i }\times\Delta v_{j}+\Delta v_{i}\times\Delta\theta_{j}\big{]}. \tag{68}\]  
Similar to the rotation vector algorithm, the above equation is simplified as follows:  
\[\Delta v_{scullm}=\sum_{i=1}^{N-1}K_{i}[\Delta\theta_{i}\times\Delta v_{N}+ \Delta v_{i}\times\Delta\theta_{N}]. \tag{69}\]  
With the minimum drift error of the algorithm under the sculling motion condition as the accuracy criterion, the four-sample optimal compensation algorithm is obtained as follows:  
\[\Delta v_{scullm}=\begin{bmatrix}\frac{54}{105}\theta_{1}+\frac{92}{105} \theta_{2}+\frac{214}{105}\theta_{3}\end{bmatrix}\times\Delta v_{4}+\begin{bmatrix} \frac{54}{105}\Delta v_{1}+\frac{29}{105}\Delta v_{2}+\frac{214}{105}\Delta v _{3}\end{bmatrix}\times\theta_{4}. \tag{70}\]