## 3 Path Tracking Controller
The reference snaking path was composed of two straight lines and a sinusoid, and abrupt changes in the reference orientation are occurred the intersections of the straight line and curve. Although the tracking error increased to about 0.2 m when changing from a straight line to a sinusoid, it rapidly converged to zero. Figures 6c and 7c illustrate the angular velocities of the left and right driving wheels generated by the proposed control framework. Their periodic variations are due to slippage disturbance, as shown in Figure 5. The weighting coefficients of the output layer from neurons No. 1, 3, 5, 7, and 9 were selected and shown in Figures 6d and 7d. These coefficients also varied with the slippage to achieve a lower evaluation indicator, while they showed abrupt increases and decreases in the discontinuous position of the reference paths. The results indicate that the proposed control framework can adapt to slippage disturbance in terms of adjusting the weights of the network. It can also adapt to the abrupt change in reference orientation. The adaptiveness and effectiveness of the proposed control framework were verified. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline \(r\) & 0.21 & m \\ \(B\) & 0.67 & m \\ \(m\) & 115 & kg \\ \(J\) & 20.59 & kgm\({}^{2}\) \\ \(J_{L}\) & 3.29 & kgm\({}^{2}\) \\ \(J_{R}\) & 3.29 & kgm\({}^{2}\) \\ \(C_{\pm L}\) & 10 & kN \\ \(C_{xR}\) & 10 & kN \\ \(C_{y}\) & 240 & N (\({}^{\circ}\))\({}^{-1}\) \\ \(D\) & 0.5335 & Nm (A)\({}^{-1}\) \\ \(Z\) & 0.005 & Nms (rad)\({}^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the robot model. ### Comparison with Other Control Algorithms

In this paper, a single neuron network (SNN) [27] and a PID control algorithms were selected for comparisons with the proposed control framework in order to demonstrate

Figure 6: Tracking results of the proposed framework under a double-lane-change maneuver: (**a**) trajectory, (**b**) tracking errors, (**c**) angular velocity of the left and right driving wheels, and (**d**) weights of the RBF network. Figure 7: Tracking results of the proposed framework under a snaking path maneuver: (**a**) trajectory, (**b**) tracking errors, (**c**) angular velocity of the left and right driving wheels and (**d**) weights of the RBF network. the tracking performance. Equivalent error \(e_{q}\) was also employed as the control target for the two controllers. The results of the three control algorithms to in tracking a DLC and a snaking reference path under slippage disturbance are shown in Figure 8.

A quantification method was employed to evaluate the path tracking performance of the different algorithms. Accumulate position, orientation and equivalent errors during the tracking process, denoted by \(S_{ep}\), \(S_{ch}\), and \(S_{eq}\), respectively, are defined as

\[\left\{ \begin{array}{l}S_{ep}=\int_{0}^{T}\lvert e_{p}\rvert dt\\ S_{eh}=\int_{0}^{T}\lvert e_{h}\rvert dt\\ S_{eq}=\int_{0}^{T}\lvert e_{q}\rvert dt\\ \end{array} \right. \tag{24}\]

where \(T\) is the total time of path tracking. The trajectories and variations in the tracking errors with the time of the proposed control framework, SNN, and PID when tracking a DLC path are illustrated in Figure 8. The DLC path could be tracked by all of the afore-mentioned controllers. The variations in \(e_{p}\) and \(e_{q}\) of the PID and SNN controllers occurred in a larger range when compared with those of the proposed control framework. The maximum \(\lvert e_{q}\rvert\) values of the PID and SNN controllers are nearly 0.15 m and 0.08 m, respectively, and they are much higher than those of the RBF. The swinging postures of the robot are exhibited in trajectories of the SNN and PID (Figure 8a), and this is due to the oscillating variations in \(e_{h}\) of the two controllers in the tracking process (Figure 8b). Although the heading error remained nearly constant in the lane-changing process, the proposed framework yielded a steady posture.