## 5 Simulations of the Path Tracking Control for the Roadheader  
To evaluate the performance of the proposed path tracking model, a group of tracking simulations are carried out in this section. A transition path heading to the aimed roadway was designed previously for the yawed roadheader, which is shown in Figure 11a with a series of positions. The corresponding expected orientation of the roadheader during tracking is shown in Figure 11b, and the required steering angular velocity in the process is shown in Figure 11c. Other relevant parameters used in the simulations are specified in Table 1, but not the randomly chosen \(k_{x}\), \(k_{y}\), \(k_{\theta}\), as BP neural network optimization is applied. Instead, they would update regularly until the pose deviations become zero. To explain how the strategy of path tracking combining state estimation and BP neural network is carried out, the tracking simulations consider three scenarios for comparison. The first scenario is as specified in Section 2.2: \(k_{x}\), \(k_{y}\), \(k_{\theta}\) is not optimized and the noises are not considered. The second scenario is to obtain a set of \(k_{x}\), \(k_{y}\), \(k_{\theta}\) through the BP neural network but without using state estimation when noises are considered. The third scenario is to use both \(k_{x}\), \(k_{y}\), \(k_{\theta}\) optimization and state estimation as the proposed path tracking, combining state estimation and BP neural network to deal with the uncertainty during the real path tracking for the roadheader. The path tracking under the first scenario is successful when using the basic tracking control based on the pose derivation model. The results are close to the situation presented by Figure 5, which is achieved by choosing suitable values for \(k_{x}\), \(k_{y}\), \(k_{\theta}\). Although this is rarely the case in a real situation, it is another example to verify that the basic path tracking control based on the pose derivation model is theoretically reasonable for the roadheader. The simulation results of path tracking under the second and third scenarios are shown in Figures 12 and 13, respectively. Figure 12a shows the coincidence between the planned  
Figure 10: Framework of the proposed path tracking combining state estimation and BP neural network. path and centroid trajectory during the scheduling process. Figure 12b shows how the position and orientation errors change in the process; these errors are expected to converge to zero as soon as possible. Figure 12c shows the variance of the angular speeds of the driving wheels for the tracks on the left and right sides, respectively. Similar explanations are suitable for the illustrations in Figure 13a-c. Comparing Figures 12a and 13a, it can be seen that the simulated tracking path and expected path are generally coincident in both scenarios. However, the situation in Figure 13a applying states estimation in addition is better, namely, the differences between expected and tracking paths are smaller, and the obtained whole tracking path is visually smoother. Comparing Figures 12b and 13b, it can be seen that  
(1) The position and orientation errors \(y_{e}\) and \(\theta_{e}\) of the roadheader converge to a small range around 0 within a limited number of tracking steps in both scenarios. The convergence rate in Figure 13b is faster than that in Figure 12b. (2) The position error \(x_{e}\) of the roadheader also converges as tracking takes place in both scenarios; however, it does not end at zero but at an average slightly higher than zero. According to the illustration in Figure 1 and Equation (1), it is acceptable that \(x_{e}\) is larger than \(y_{e}\), as long as the errors are not absolutely zero. A reasonable explanation is as  
Figure 11: The pre-designed transition path for the yawed roadheader.