## 6 Installation Error Compensation for the Shearer's SINS
### Lever Arm Effect Error and Compensation Algorithm  
In actual engineering application, the installation position of the inertial measurement unit will not coincide completely with the centroid of the moving target, which will not affect the output of the gyroscope, but the tangential and centripetal accelerations caused by the noncoincidence of the centroids will cause the measurement error of accelerometer which is called \"lever arm effect\" [30]. The tangential acceleration is proportional to the product of the lever arm length and the carrier swing angular acceleration, and the centripetal acceleration is proportional to the product of the lever arm length and the squared carrier swing angular velocity. They are equivalent to the accelerometer error [31]. The presence of the lever arm effect will also lead to a reduction in the positioning accuracy of the shearer's SINS, so it needs to be compensated. We establish an inertial coordinate system (\(i\) system), and the orientation relationship between \(i\) system and \(b\) system is shown in Figure 11. Then, the accelerometer error caused by the lever arm effect can be expressed by the following equation:  
\[\delta a=\left.\frac{d\omega_{ib}}{dt}\right|_{i}\times r_{p}+\omega_{ib}\times \left(\omega_{ib}\times r_{p}\right), \tag{35}\]  
where \(\delta a\) is the acceleration output error, \(\omega_{ib}\) is the angular velocity of the \(b\) system relative to the \(i\) system, and \(r_{p}\) is the lever arm length. The lever arm effect affects the shearer positioning accuracy, so it is necessary to eliminate or reduce the lever arm effect with error compensation. Let the SINS installation point be \(P\), the position vector of point \(P\) relative to \(O_{i}\) be \(R_{ip}\), the position vector of point \(P\) relative to \(O_{b}\) be \(r_{p}\), and the position vector from \(O_{i}\) to \(O_{b}\) be \(R_{i\omega_{b}}\). Then,  
\[R_{ip}=R_{i\omega_{b}}+r_{p} \tag{36}\]  
then  
\[\left.\frac{dR_{ip}}{dt}\right|_{i}=\left.\frac{dR_{i\omega_{b}}}{dt}\right|_ {i}+\left.\frac{dr_{p}}{dt}\right|_{i}=\left.\frac{dR_{i\omega_{b}}}{dt} \right|_{i}+\left.\frac{dr_{p}}{dt}\right|_{b}+\omega_{ib}\times r_{p} \tag{37}\]  
\[\left.\frac{d^{2}R_{ip}}{dt^{2}}\right|_{i}=\left.\frac{d^{2}R_{i\omega_{b}}}{ dt^{2}}\right|_{i}+\left.\frac{d}{dt}\right(\left.\frac{dr_{p}}{dt}\right|_{b} \right)\right|_{i}+\left.\frac{d}{dt}\right(\omega_{ib}\times r_{p})\right|_ {i}. \tag{38}\]  
Figure 11: Schematic diagram of the lever arm effect. The installation point \(P\) of SINS is a fixed point in the carrier coordinate system, so  
\[\left.\frac{dr_{p}}{dt}\right|_{b}=\left.\frac{d^{2}r_{p}}{dt^{2}}\right|_{b}=0, \tag{39}\]  
then  
\[\left.\frac{d^{2}R_{ip}}{dt^{2}}\right|_{i}=\left.\frac{d^{2}R_{i_{0}}}{dt^{2}} \right|_{i}+\left.\frac{d\omega_{ib}}{dt}\right|\times r_{p}+\omega_{ib}\times \big{(}\omega_{ib}\times r_{p}\big{)}. \tag{40}\]  
Ideally, \(r_{p}=0\), and there is no error at this moment. However, the installation point \(P\) of SINS is not the centroid of the shearer in practice, i.e., \(r_{p}\
eq 0\); therefore, the lever arm effect error introduced by \(r_{p}\) is  
\[\delta a=a_{p}-a_{b}=\left.\frac{d^{2}R_{ip}}{dt^{2}}\right|_{i}-\left.\frac{d ^{2}R_{i_{0}}}{dt^{2}}\right|_{i}=\left.\frac{d\omega_{ib}}{dt}\right|_{i} \times r_{p}+\omega_{ib}\times\big{(}\omega_{ib}\times r_{p}\big{)} \tag{41}\]  
where \(a_{p}\) is the specific force at the installation point of the SINS, and \(a_{b}\) is the specific force at the centroid of the shearer. The difference between them is the lever arm error \(\delta a\), and \(\delta a\) is positively proportional to \(r_{p}\). To improve the attitude accuracy of the shearer, error compensation for the lever arm effect is required. Generally, the mechanical theory is used to compensate the lever arm error.