## 3 Shearer Positioning Technology
### The Backward Calibration of Shearer Trajectory  
According to the measured retreat and creep displacements outlined in Section 3.2, the start and end points of each cutting cycle can be determined. In the LASC, the minimum-variance fixed-interval smoothing (MFS) algorithm was applied to correct the shearer trajectory backwardly when the shearer remains stationary at the end of the longwall face [33]. In fixed-interval smoothing, the estimation of the current state is based on the utilization of the measurement data from both before and after the current time within the interval. Smoothing is an off-line processing algorithm. Compared with filtering, smoothing can provide better estimation accuracy with the double computation load. The minimum-variance fixed-interval smoothing algorithm is briefly expressed as follows. Suppose that the system satisfies the state equation and observation equation as Equations (1) and (2). \[\mathbf{X}_{\mathbf{k}+1}=\mathbf{F}_{\mathbf{k}}\mathbf{X}_{\mathbf{k}}+\mathbf{B}_{\mathbf{k}}\mathbf{W}_{\mathbf{k}} \tag{1}\]  
where \(\mathbf{X}_{\mathbf{k}}\) is the state vector, \(\mathbf{F}_{\mathbf{k}}\) is the state transition matrix, \(\mathbf{B}_{\mathbf{k}}\) is the noise transition matrix, and \(\mathbf{W}_{\mathbf{k}}\) is the state noise whose covariance is \(\mathbf{Q}_{\mathbf{k}}\). \[\mathbf{Z}_{\mathbf{k}}=\mathbf{H}_{\mathbf{k}}\mathbf{X}_{\mathbf{k}}+\mathbf{V}_{\mathbf{k}} \tag{2}\]  
where \(\mathbf{Z}_{\mathbf{k}}\) is the observation vector, \(\mathbf{H}_{\mathbf{k}}\) is the observation transition matrix, and \(\mathbf{V}_{\mathbf{k}}\) is the observation noise whose covariance is \(\mathbf{R}_{\mathbf{k}}\). Figure 14: The diagram of (**a**) the measurement principle, (**b**) the measurement device, and (**c**) a field test in an underground coal mine [32]. There are \(N\) sample data in the system. The smoothing algorithm has two steps. Step 1: calculating (3) from \(k=0\) to \(k=N\) to obtain \(\mathbf{\alpha}_{k}\). Step 2: calculating (4) from \(k=N\) to \(k=0\) to acquire \(\mathbf{\beta}_{k}\) under the known \(\mathbf{\alpha}_{k}\) from (3). \[ \begin{bmatrix}\mathbf{X}_{k+1}\\ \mathbf{\alpha}_{k}\end{bmatrix} = \begin{bmatrix}\mathbf{F}_{k}-\mathbf{K}_{k}\mathbf{H}_{k}& \mathbf{K}_{k}\\ -\mathbf{H}_{k}(\mathbf{H}_{k}\mathbf{P}_{k-1}\mathbf{H}_{k}{}^{T}+\mathbf{R}_{k})^{-1/2}&(\mathbf{H}_ {k}\mathbf{P}_{k-1}\mathbf{H}_{k}{}^{T}+\mathbf{R}_{k})^{-1/2}\end{bmatrix}\begin{bmatrix} \mathbf{X}_{k}\\ \mathbf{Z}_{k}\end{bmatrix} ,\ \mathbf{X}_{0}=0 \tag{3}\]  
\[ \begin{bmatrix}\mathbf{Y}_{k-1}\\ \mathbf{\beta}_{k}\end{bmatrix} = \begin{bmatrix}\mathbf{F}_{k}{}^{T}-\mathbf{H}_{k}{}^{T} \mathbf{K}_{k}{}^{T}&\mathbf{H}_{k}{}^{T}(\mathbf{H}_{k}\mathbf{P}_{k-1}\mathbf{H}_{k}{}^{T}+\mathbf{ R}_{k})^{-1/2}\\ -\mathbf{K}_{k}{}^{T}&(\mathbf{H}_{k}\mathbf{P}_{k-1}\mathbf{H}_{k}{}^{T}+\mathbf{R}_{k})^{-1/2} \end{bmatrix}\begin{bmatrix}\mathbf{Y}_{k}\\ \mathbf{\alpha}_{k}\end{bmatrix} ,\ \mathbf{Y}_{N}=0 \tag{4}\]  
where \(\mathbf{K}_{k}\) is the gain, which can be given as (5), and \(\mathbf{P}_{k}\) is the mean square of the system, which is updated as (6). \[\mathbf{K}_{k}=\mathbf{F}_{k}\mathbf{P}_{k-1}\mathbf{H}_{k}{}^{T}(\mathbf{H}_{k}\mathbf{P}_{k-1}\mathbf{H} _{k}{}^{T}+\mathbf{R}_{k})^{-1} \tag{5}\]  
\[\mathbf{P}_{k}=\mathbf{H}_{k}\mathbf{P}_{k-1}\mathbf{H}_{k}{}^{T}-\mathbf{K}_{k}(\mathbf{H}_{k}\mathbf{P} _{k-1}\mathbf{H}_{k}{}^{T}+\mathbf{R}_{k})\mathbf{K}_{k}{}^{T}+\mathbf{B}_{k}\mathbf{Q}_{k}\mathbf{B}_ {k}{}^{T} \tag{6}\]  
Therefore, the estimated output is expressed as  
\[\hat{\mathbf{Z}}_{k}=\mathbf{Z}_{k}-\mathbf{R}_{k}\mathbf{\beta}_{k} \tag{7}\]  
In CUMT, the authors of this paper proposed the global optimization model (GOM) for shearer trajectory through borrowing the simultaneous localization and mapping (SLAM) technology. The SLAM frame is divided into front-end and back-end procedures [34,35]. The front-end process acquires the environment information using a camera or lidar to realize the map construction.