## 3 Path Tracking Controller
Figure 9 compares the results of three controllers by tracking a snaking target path. As shown in Figure 9a, a swinging posture of the wheeled robot appeared when tracking the straight-line part for the SNN and PID controllers. The three controllers yielded an abrupt increase in tracking errors in the intersections between a straight line and a sinusoid curve, which is illustrated in Figure 9b. The maximum \(\lvert e_{p}\rvert\) values of the PID and SNN are about 0.18 m and 0.1 m, respectively, and they are much higher than those of the proposed framework. The obviously periodic changes in the three tracking errors of the SNN and PID were caused by slip ratio variations (Figure 5). The slippage disturbance led to degradation in the tracking performance of the SNN and PID controllers. The proposed control framework exhibited the lowest tracking errors and prior anti-disturbance performance when compared with those of the SNN and PID. The robot could track the target path with lower errors and a steady posture. Figure 8: Comparison of results of different control algorithms by tracking a DLC path: (**a**) trajectory and (**b**) tracking errors. Figure 10 compares the accumulate position, orientation, and equivalent errors of three different controllers. The proposed framework yielded the lowest \(S_{qp}\), \(S_{dh}\), and \(S_{qq}\) in both the DLC and snaking path tracking processes, while the PID controller exhibited much higher accumulate errors than those of the other controllers. As shown in Figure 10a, the proposed controller yielded much lower \(S_{qq}\) than \(S_{qp}\) and \(S_{dh}\) when tracking the DLC path. This is because that the equivalent tracking error \(e_{q}\) is defined by taking account for the position error of a future state and the heading error which may lead to approaching the target path, decreases \(e_{q}\). This integrated consideration of tracking error caused a decrease in the accumulate position error. \(S_{dh}\) of the three controllers when tracking the DLC path is much higher than when tracking the snaking path, which is due to a steady heading error in the lane changing process, as shown in Figure 8b. ## 5 Conclusions

An adaptive path-tracking controller is proposed in this study to overcome slippage disturbances. Both a kinematic model and a dynamic model of a wheeled robot with skid-steer characteristics are established for the development of a dual-loop control framework. An RBF neural network is employed to design a yaw rate controller with anti-slippage performance. Simulations to track snaking and DLC paths are conducted under slippage disturbances. The proposed control framework yields much lower accumulate position, orientation, and equivalent tracking errors compared with those of an SNN and a PID controller. It also exhibits steady posture during the tracking process, which indicates its prior anti-slippage performance. The weights of the output layer in the RBF network vary

Figure 10: Comparison of accumulate errors of different control methods by tracking (**a**) a DLC path and (**b**) a snaking path. Figure 9: Comparison of results of different control algorithms in tracking a snaking path: (**a**) trajectory and (**b**) tracking errors. with respect to the disturbances input. The proposed dual-loop controller is demonstrated to adapt to slippage, indicating that it could be employed in future vehicles or robots working in complex conditions. While the model did not consider the effect of vertical direction on path tracking. Our future work will focus on the validation of the proposed control framework in a field environment with complex terrain and take into account the vertical stiffness of the tire model. Conceptualization, Y.K. and R.Z. ; methodology, Y.K. and B.X. ; software, Y.K. ; validation, Y.K., B.X. and R.Z. ; formal analysis, B.X. ; investigation, R.Z. ; resources, Y.K. ; data curation, R.Z. ; writing--original draft preparation, Y.K. and B.X. ; writing--review and editing, Y.K. and B.X. ; visualization, B.X. ; supervision, Y.K.t and R.Z.