## 3 Path Tracking Control Based on States Estimation
### Patch Tracking Control Based on State Estimation  
We define the state of the roadheader as \(\mathbf{S}=(x_{e},y_{e},\theta_{e},\ \omega_{L},\ \omega_{R})\), where \((x_{e},y_{e},\theta_{e})\) is the pose deviation, which is obtained according to the predesigned path and the pose measurements in real time by the UWB pose detection system on-board. The scheduled rotation rates of the left and right driving wheels \((\omega_{L},\ \omega_{R})^{T}\) then are calculated through Equation (8), where the skidding ratio \(i_{L}\) and \(i_{R}\) are uncertain. Notice that the state \(\mathbf{S}\) varies as the recommended heading velocity \(v_{c}\) and turning angular speed \(\omega_{c}\) change as in Equation (4), with the aim that \(\lim\limits_{H\rightarrow\infty}\mathbf{p_{e}}=0\). Apparently, the path tracking control of the  
Figure 8: A prototype of the pose detection system for the roadheader. Figure 9: Error frequency histogram of the ranges within different distance scales. (**a**) The error frequency histogram of the ranges within 20 m; (**b**) The error frequency histogram of the ranges within 35 m; (**c**) The error frequency histogram of the ranges within 57 m; (**d**) The error frequency histogram of the ranges within 90 m.  
roadheader based on the pose deviation model, which is expressed by Equations (1), (2), (4) and (8), is represented by a typical nonlinear system:  
\[ \begin{split}\mathbf{x}_{k+1}&=\mathbf{S}_{k+1}= \mathbf{S}_{k}+\hat{\mathbf{S}}_{k}\cdot\Delta t+\varepsilon_{k}=\text{f}( \mathbf{S}_{k},\ \ \mathbf{u}_{k})+\varepsilon_{k}\\ \mathbf{y}_{k+1}&=\text{h}(\mathbf{S}_{k})+\delta_{k+ 1}\end{split}  \tag{10}\]  
where \(\varepsilon_{k}\) is the process noise and \(\delta_{k+1}\) is the observation error, as introduced in Sections 3.1 and 3.2, respectively. Since the skidding between the tracks and ground results in part of the errors between the real pose of the roadheader and the expected pose, \(\varepsilon_{k}\) is equivalent to the effort of skidding; the observation item, i.e., the pose deviation, is obtained based on measurements; therefore, \(\delta_{k+1}\) is equivalent to measurement noise. We are able to treat the nonlinear system represented by Equation (10) as a joint estimation problem, to which the nonlinear filtering method is widely used for solution. There are many nonlinear filtering methods for state estimation, including the Kalman filter, which is used to estimate linear system and has great applicational value in nonlinear systems; however, the error introduced by linear approximation would lead to divergence [27]. The SVD-Unscented Kalman Filtering is able to suppress divergence while having better estimation accuracy and matrix operations during iterations; therefore, it is applied in this work. The main idea in Unscented Kalman Filtering (UKF) for state estimation is to approximate the probability distribution of the concerned parameters directly by nonlinear transformation, instead of approximating the linearized model of the system [28]. For Unscented Kalman Filtering (UKF), to realize states estimation without derivative operations, the Unscented Transform (UT) is commonly used to calculate the statistical properties of random variables that are formulated by nonlinear expressions. A brief introduction on UT is given in the following section.