## 3 Path Tracking Controller
When the value of \(\eta\) is too low, the speed of learning decreases. On the contrary, when the value of \(\eta\) is too high, the change in the weight is unstable. Therefore, the momentum term is added to Equation (22). If the gradient direction of the current time is similar to that of the historical time, the gradient direction of the current time is strengthened; if it

Figure 4: Scheme of the yaw rate controller based on an RBF neural network. is different, the gradient direction at the current time weakens. The weight of the output layer, the width, and center vectors of the Gaussian function can be updated by

\[\left\{ \begin{array}{c}\omega_{j}(t)=\omega_{j}(t-1)+\Delta\omega_{j}(t)+\alpha \big{(}\omega_{j}(t-1)-\omega_{j}(t-2)\big{)}\\ b_{j}(t)=b_{j}(t-1)+\Delta b_{j}(t)+\alpha\big{(}b_{j}(t-1)-b_{j}(t-2)\big{)}\\ c_{jk}(t)=c_{jk}(t-1)+\Delta c_{jk}(t)+\alpha\Big{(}c_{jk}(t-1)-c_{jk}(t-2) \Big{)}\end{array} \right. \tag{23}\]

where \(\alpha\) is factor of momentum, and the value is between 0 and 1. A PID tracking controller is also formulated for the wheeled robot to obtain the central vector values of the RBF network. Path tracking simulations are conducted by employing the PID controller, and both position error and orientation error are collected to determine the central vector of the RBF network by the K-means clustering method. In the simulation employing the PID controller, the slippage disturbances are not taken into account. The purpose of such simulation is to identify a suitable central vector for the RBF network rather than a random approach. ## 4 Results and Discussion

The proposed path tracking algorithm is verified by tracking two different reference paths, namely, a snaking path and a double lane change (DLC) path. As shown in Figure 5, the robot tracks the path on a separation surface, and the slip ratios of the left and right wheels are taken as disturbances input to the kinematic model. The effectiveness and adaptability of the proposed control framework are demonstrated under tracking simulations. ### Algorithm Verification

The parameters of the wheeled robot for algorithm verification are listed in Table 1. The learning rate \(\eta\) and factor of momentum \(\alpha\) in the RBF network are set to 0.3 and 0.35, respectively. The weight coefficients \(s_{1}\) and \(s_{2}\) are 0.6 and 0.004, respectively. The preview distance \(l_{p}\) is 0.01 m in this paper. The initial values of neural network weights (the weighting vector and the width vector) are set as follows: \(\mathbf{w}=[0.05\,0.05\cdots 0.05]^{T}\) and \(\mathbf{b}=[0.5\,0.5\cdots 0.5]^{T}\). Simulations to track both a snaking path and a DLC path are conducted by employing the PID controller, and the central vector of the RBF network is determined by the K-means clustering method. Unlike general vehicles operating on prepared roads, wheeled mobile robots are used to accomplish special tasks in specific occasions. Usually, the speed of a wheeled mobile robot is not fast. Therefore, in this paper, the desired speed of the robot in the simulation is set to 1 m/s. Figure 5: Slip ratio variations of the left and right tracks. Figures 6 and 7 show the results of the proposed control framework by tracking a DLC and a snaking path, respectively. The wheeled robot controlled by the proposed framework can track both reference paths precisely without any significant offset from the \"reference\", and these are shown in Figures 6a and 7a. Position and orientation errors, as well as the equivalent tracking error \(e_{q}\), when tracking the reference paths, are illustrated in Figures 6b and 7b. The maximum position error of the proposed framework was less than 0.08 m when tracking a DLC path, and it occurred in the lane-change part. The equivalent error remained at nearly zero during the tracking process, which indicates the effectiveness of the RBF network in reducing error indicators (Equation (20)). In contrast, sharp increases in errors occurred when tracking a snaking path.