## 2 Scheduling Requirements and Modeling Foundations of Roadheaders
### Optimization of Weights Based on BP Neural Network  
According to the deduction of Equation (4), as long as \(k_{x}\), \(k_{y}\), and \(k_{\theta}\) are positive, the position and orientation errors would converge to 0 within a limited adjustment period. To verify this, two groups of simple pre-designed trajectories, i.e., two series of planned positions together with orientations are given for the roadheader at certain initial deviation situations to the track. Since the centreline of the roadway is usually straight, the two target centrelines of the roadway should be along the lines \(Y=2\) and \(X=2\), respectively, in the geodetic coordinate \(XOY\). Correspondingly, the initial poses of the roadheader are \((0,\ 1,\pi/6)\) and \((1,\ 0,\pi/3)\), respectively. Here, \((0,1)\) and \((1,0)\) are coordinates, and \(\pi/6\) and \(\pi/3\) are orientations. As shown in Figure 2(a), path #1 leads the roadheader to move from \((0,1,\pi/6)\) onto the line \(Y=2\), while path #2 leads the roadheader onto the line \(X=2\) from \((1,0,\pi/3)\). The expected orientations for the roadheader along the paths are also presented in Figure 2(b), and their variances, i.e., the steering angular velocities, are shown in Figure 2(c). To simulate the path tracking process using Equation (4), we used the parameters as shown in Table 1. Note that the values of \(k_{x}\), \(k_{y}\), \(k_{\theta}\) are randomly taken to be 0.6, 0.7, and 0.2. The simulation results are shown in Figure 4, and for brief illustration, only the coincidences between the planned and tracked paths are shown here. It can be seen that both sets of adjustments achieve the purpose of path tracking, i.e., the roadheader moves close to the planned path and finally moves onto the target centerline within a certain number of adjustment steps in the two cases of simulation. However, it is obvious that the tracking effect following path #2 is not as good as the case of path #1 in the early stage of tracking. Commonly, changing the values of \(k_{x}\), \(k_{y}\), \(k_{\theta}\) for simulation is carried out repeatedly for trials. A set of values at 0.1, 0.5, and 0.5 for \(k_{x}\), \(k_{y}\), \(k_{\theta}\), respectively, results in more satisfying tracking performances for both cases, as shown in Figure 5.  
Inclatently, some other sets of \(k_{x}\), \(k_{y}\), \(k_{\theta}\), for instance, 0.5, 0.2 and 1.8, also give good tracking results. To summarize, first, the path control using Equation (4) will not fail if the values of \(k_{x}\), \(k_{y}\), \(k_{\theta}\) are suitable. Second, different sets of \(k_{x}\), \(k_{y}\), \(k_{\theta}\) would give different tracking effects in the aspects of convergence speed of the position and orientation deviations, or the overlap ratio of the tracked path over the pre-planned path. Lastly, \(k_{x}\), \(k_{y}\), \(k_{\theta}\) giving satisfying tracking results is not unique in terms of value, and these different sets of values apparently show no patterns. The BP neural network might be appropriate for the situation presented above, to obtain ideal values for \(k_{x}\), \(k_{y}\), \(k_{\theta}\) automatically while eliminating the pose deviation effectively in the process of tracking. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Parameters**} & **Value or** & \multirow{2}{*}{**Parameters**} & \multirow{2}{*}{**Value or Description**} \\  & **Description** & & \\ \hline Radius of driving wheel \(r\) & 0.5 m & & \(k_{x}\) & 0.6 \\ Distance between the & 1 m & & coefficients & \(k_{y}\) & 0.7 \\ \multirow{2}{*}{Width of track \(B\)} & 320 mm & & \(k_{\theta}\) & 0.2 \\ Length of track where & & Maximum rotating & & \\ touching the ground \(l\) & 3 m & speed of the & 4 rad/s \\ \multirow{2}{*}{Maximum travelling speed \(v_{\max}\)} & \multirow{2}{*}{\(v_{\max}\) = 0.8 m/s} & \multirow{2}{*}{Range of angular acceleration} & \multirow{2}{*}{(0, 0.2)} \\ Referred average speed & \multirow{2}{*}{\(v_{ref}\) = 0.6 m/s} & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The data used in simulations.