## 2 Shearer Positioning System Based on SINS  
### Principle of the Shearer Positioning System Based on SINS  
SINS is applied to the shearer to detect the position and attitude of the shearer. The angular velocity and acceleration of the shearer are also detected by using an inertial measurement unit (IMU). Then, the position and attitude information of the shearer can be obtained by calculations [32]. As shown in Figure 2, because the position and attitude of the shearer are three-dimensional, the IMU of the shearer positioning system is not a single-axis gyroscope and an accelerometer, but a three-axis gyroscope and a three-axis accelerometer, whose axes are orthogonal. Shearer positioning entails actually determining the coordinates of the shearer relative to a certain reference coordinate system at a certain time [33]. However, the angular velocity and acceleration measured by the shearer positioning system are in the carrier coordinate system and cannot be directly  
Figure 2: Shearer positioning system based on strapdown inertial navigation system (SINS). used for the calculation of the position and attitude of the shearer. To solve the position and attitude of the shearer, it is necessary to convert the carrier coordinate system to the navigation coordinate system. Most mine maps are based on the geographic coordinate system. The geographic coordinate system is used as the navigation coordinate system; this will alleviate the need to convert the mine map coordinate information in the navigation coordinate system, reducing the calculation burden of the positioning system [34; 35; 36; 37; 38]. The attitude of the shearer is the orientation relationship between the carrier coordinate system and the geographic coordinate system. If we let the heading angle be \(\varphi\), the pitch angle be \(\theta\), and the roll angle be \(\gamma\), then the orientation relationship between the carrier coordinate system and the geographic coordinate system is as shown in Figure 3.  
The following is the transformation matrix from the carrier coordinate system to the geographic coordinate system [39]:  
\[C_{t}^{b}=C_{\gamma}C_{\theta}C_{\varphi}=\left( \begin{array}{ccc}\cos\gamma& 0&-\sin\gamma\\ 0&1&0\\ \sin\gamma&0&\cos\gamma\end{array} \right)\left( \begin{array}{ccc}1&0&0\\ 0&\cos\theta&\sin\theta\\ 0&-\sin\theta&\cos\theta\end{array} \right)\left( \begin{array}{ccc}\cos \varphi&-\sin\varphi&0\\ \sin\varphi&\cos\varphi&0\\ 0&0&1\end{array} \right)=\]  
\[\left( \begin{array}{ccc}\cos\gamma\cos\varphi-\sin\gamma\sin\theta\sin \varphi&-\cos\gamma\sin\varphi+\sin\gamma\sin\theta\cos\varphi&-\sin\gamma \cos\theta\\ \sin\varphi\cos\theta&\cos\varphi\cos\theta&\sin\theta\\ \sin\gamma\cos\varphi-\cos\gamma\sin\varphi\sin\theta&-\sin\gamma\sin\varphi- \cos\gamma\sin\varphi\cos\theta&\cos\gamma\cos\theta\end{array} \right) \tag{1}\]  
Obviously, matrix \(C_{t}^{b}\) is an orthogonal matrix, then,  
\[C_{b}^{n}=C_{b}^{t}=\left(C_{t}^{b}\right)^{T}=\left( \begin{array}{ccc}\cos \gamma\cos\varphi-\sin\gamma\sin\theta\sin\varphi&\sin\varphi\cos\theta&\sin \gamma\cos\varphi-\cos\gamma\sin\varphi\sin\theta\\ -\cos\gamma\sin\varphi+\sin\gamma\sin\theta\cos\varphi&\cos\varphi\cos\theta&- \sin\gamma\sin\varphi-\cos\gamma\sin\varphi\cos\theta\\ -\sin\gamma\cos\theta&\sin\theta&\cos\gamma\cos\theta\end{array} \right) \tag{2}\]  
Matrix \(C_{b}^{n}\) is called an attitude matrix or a strapdown matrix. The angular velocity measured by the gyroscope can update the attitude matrix in real time, and the attitude matrix can be solved to obtain the information of the shearer attitude angle. The acceleration measured by the accelerometer also uses the attitude matrix to convert the system from a shearer coordinate system to a navigation coordinate system and calculate the velocity and position of the shearer. In brief, the shearer positioning system based on SINS is shown in Figure 4.  
Figure 3: Orientation relationship between the carrier coordinate system and the geographic coordinate system.