## 2 Scheduling Requirements and Modeling Foundations of Roadheaders
### Basic Path Tracking Control for the Roadheader  
The tracking control of the roadheader was carried out based on a pose deviation model [24], which is demonstrated in Figure 2. An isosceles triangle, which takes the centroid of the roadheader as its apex and the ends of the two tracks as its bottom points, is used to display the pose (position of the centroid, orientation) of the roadheader; the dotted triangle in front represents the scheduled next pose of the roadheader. \((x,y)\) and \((X,Y)\) are the coordinates of the roadheader in the vehicle coordinate system \(xoy\) and geodetic coordinate system \(XOY\), respectively. The letter b presents the distance between the two tracks; \(v\) presents the instant heading velocity; \(\varphi\) presents the turning angle of the roadheader; \(\omega\) presents the turning angular speed, which is also \(\varphi;\theta\) presents the orientation of the roadheader. Taking \(\dot{x}\) and \(\dot{y}\) as the instant velocity of the vehicle along the \(x\) axis and \(y\) axis of the vehicle coordinate system, respectively, the instant heading velocity \(v\) is actually composited by \(\dot{x}\), \(\dot{y}\) and the linear velocity caused by the steering motion of the vehicle. Considering that the turning angular speed is usually very small, the corresponding linear velocity and \(\dot{y}\) are ignorable compared with the \(\dot{x}\). Hence, the value of \(v\) is very close to \(\dot{x}\), namely, \(v\approx\dot{x}\). We take the subscript \(c\) as the current state of a certain parameter, \(d\) as the destined state, and \(e\) as their difference. For instance, we denote the current pose of the centroid as \(\mathbf{P}_{c}=(X_{c},Y_{c},\theta_{c})^{T}\), and the destined pose as \(\mathbf{P}_{d}=(X_{d},Y_{d},\theta_{d})^{T}\); thus, their difference in the vehicle coordinate system is  
\[\mathbf{P_{e}}=\left[ \begin{array}{c}x_{e}\\ y_{e}\\ \theta_{e}\end{array} \right]=\left[ \begin{array}{ccc}\cos\theta_{c}&\sin \theta_{c}&0\\ -\sin\theta_{c}&\cos\theta_{c}&0\\ 0&0&1\end{array} \right]\left[ \begin{array}{c}X_{d}-X_{c}\\ Y_{d}-Y_{c}\\ \theta_{d}-\theta_{c}\end{array} \right] \tag{1}\]  
The differential of \(\mathbf{P_{e}}\) is deduced to be  
\[\dot{\mathbf{P_{e}}}=\left[ \begin{array}{c}\dot{x}_{e}\\ \dot{y}_{e}\\ \dot{\theta}_{e}\end{array} \right]=\left[ \begin{array}{c}v_{d}\cos\theta_{ e}-v_{c}+y_{e}\omega_{c}\\ -x_{c}\omega_{c}+v_{d}\sin\theta_{e}\\ \dot{\theta}_{d}-\dot{\theta}_{c}\end{array} \right]=\left[ \begin{array}{c}v _{d}\cos\theta_{e}-v_{c}+y_{e}\omega_{c}\\ -x_{c}\omega_{c}+v_{d}\sin\theta_{e}\\ \omega_{d}-\omega_{c}\end{array} \right] \tag{2}\]  
Based on Equations (1) and (2), a control law for updating current heading velocity \(v_{c}\) and turning angular speed \(\omega_{c}\) is designed as Equation (3) to realize \(\mathbf{P_{e}}\rightarrow(0,0,0)^{T}\) within limited steps, which is path tracking. Figure 2: Illustration of the position and orientation adjustment for the roadheader. \[\left\{ \begin{array}{c}v_{c}=k_{1}x_{e}+v_{d}\cos\theta_{e}+y_{e}\omega_{c}\\ (1+v_{d}\ x_{e})\omega_{c}=\omega_{d}+{v_{d}}^{2}\sin\theta_{e}+k_{2}\theta_{e}+ k_{2}v_{d}\ y_{e}\end{array} \right. \tag{3}\]  
Equation (3) is obtained by defining the switching function and using the backstepping approach, and it is simplified as Equation (4), considering that (1) \(\omega_{c}\) is very small compared with \(x_{e}\); (2) \(|v_{d}x_{e}|\ll 1\); (3) \(\dot{v}_{d}\approx 0\); (4) \(\sin\theta_{e}\approx\theta_{e}\). \[\mathrm{u}=\left[ \begin{array}{c}v_{c}\\ \omega_{c}\end{array} \right]=\left[ \begin{array}{c}k_{x}x_{e}+v_{d}\cos \theta_{e}\\ \omega_{d}+v_{d}k_{y}y_{e}+k_{\theta}\sin\theta_{e}\end{array} \right] \tag{4}\]  
where \(k_{x}\), \(k_{y}\), and \(k_{\theta}\) are random positive coefficients of the pose errors \(x_{e}\), \(y_{e}\) and \(\theta_{e}\), respectively.