# Hierarchical Synchronization Control Strategy of Active Rear Axle Independent Steering System
## 4 Experiments and Analysis
### Static Steering Test  
In order to test the steering angle tracking performance under the proposed control strategy, the static experiment was studied for the ARIS system in three cases. The prototype vehicle mass was 95 kg. The reference steering angle is a sinusoidal sweep signal with a time-varying amplitude and frequency. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Value** & **Parameters** & **Value** \\ \hline \(m\) & 95 kg & \(b_{r}\) & 0.16 m \\ \(a\) & 0.65 m & \(g_{r}\) & 0.03 m \\ \(b\) & 0.7 m & \(c_{r}\) & 120 N/deg/s \\ \(w\) & 1.2 m & \(k_{r}\) & 460 N/deg \\ \(N\) & 10 & \(m_{r}\) & 12 kg \\ \(J_{1},J_{2}\) & 1.37 kg.m\({}^{2}\) & \(\omega_{o}\) & 80 Hz \\ \(k_{r1},k_{2}\) & 2.2 Nm/deg & \(\omega_{c}\) & 20 Hz \\ \(c_{r1},c_{r2}\) & 4.2 Nm/deg/s & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the prototype vehicle and ARIS system. Figure 6: (a) The prototype vehicle with the ARIS system; (b) the structure of the ARIS system. Figure 7: The control structure of the experiment. Case 1: The master-slave method (MS) [16] was adopted for the synchronization controller, while the PID method was adopted for the tracking controller. Case 2: The MS method was adopted for the synchronization controller, while the LADRC method was adopted for the tracking controller. Case 3: The VRSM method was adopted for the synchronization controller, while the LADRC method was adopted for the tracking controller. In order to evaluate the effectiveness of the proposed hierarchical synchronization control strategy, the root mean square error (RSME) was defined to examine control performance as follows. \[RSME_{t}=\sqrt{\frac{1}{2n}\sum_{k=1}^{n}\left(\left|e_{t1}(k)\right|^{2}+ \left|e_{t2}(k)\right|^{2}\right)} \tag{20}\]  
\[RSME_{s}=\sqrt{\frac{1}{n}\sum_{k=1}^{n}\left|e_{s}(k)\right|^{2}} \tag{21}\]  
where \(RSME_{t}\) is the root mean square error of left and right steering wheel angle tracking error; \(e_{ti}(k)=\delta_{ri}(k)-\delta_{ri}^{*}(k)\),\(i=1,2\); \(RSME_{s}\) is the root mean square error of the synchronization error; \(RSME_{c}\) is defined to evaluate comprehensive performance; \(\lambda\) is the weight coefficient. Synchronization error is more important than tracking error for tire wear, so we assign \(\lambda\) = 0.6. \[RSME_{c}=(1-\lambda)RSME_{t}+\lambda RSME_{s} \tag{22}\]  
First of all, case 1 (Figures 8 and 9) and case 2 (Figures 10 and 11) are compared. Figures 8 and 10 show the rear wheel steering angle tracking profiles and errors under case 1 and case 2, respectively. Although the tracking errors of both controllers increase as the frequency of the steering angle command grows. Specifically, comparing Figure 8b,d with Figure 10b,d, it can be seen that the maximum tracking error of PID increases to 0.665 deg, which is much larger compared to that of LADRC (0.485 deg) and the peak error occurs when the steering direction changes. Case 1 and case 2 do not adopt the synchronization control strategy, and the synchronization errors are shown in Figures 9 and 11, respectively. The synchronization error is reduced in case 2 compared with case 1 under static and basically undisturbed conditions. This is because the tracking angle accuracy of LADRC method is improved. Figure 8: The steering angle of rear axle wheel profiles and tracking errors in case 1. L2 and R2 are marked as the left rear wheel and right rear wheel, respectively. **(a)** The steering angle of left rear wheel profiles; **(b)** the tracking error of left rear wheel profiles; **(c)** the steering angle of right rear wheel profiles; **(d)** the tracking error of right rear wheel profiles. Compared with case 1 and case 2, we can conclude that at either large amplitudes or high frequencies, the control accuracy of LADRC is better than that of PID, with lower tracking error and synchronization error. It is because that LESO in LADRC can observe the external interference and compensate it to the control law.