## 5 Results of Research  
Generally, to certify the correctness and fitness of our approach, a series of numerical simulation tests, such as tracking circular reference path, linear trajectory under varying loads, and S-curve shape of working space, has been completed in this section. According to previous research, our model has been built with physical parameters as in Table 1. In the computer's environment, the design was modified so that each computational time did not have much cost and step sizes were all set to 0,1. We considered that the initial reference location of the AGR was \((x_{r}(0),y_{r}(0),\theta_{r}(0))=(2,1,0.25\pi)\). In  
Figure 5: Result of linear velocity (**a**) and angular velocity (**b**) in tracking circular trajectory under varying loads. Figure 6: _Cont._  
In the effort to reveal the advantages of our approach, the tracking performance in a linear reference path under load variations is shown in Figure 7. Additionally, the test information including linear velocity and circular velocity is noticed in Figure 8. In a straight line trajectory, the autonomous robot seemed, with no trouble, to track the desired velocity in any situation. The output tracking resulted in errors under various loads as depicted in Figure 9. In a short time, the adaptation of the \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(\beta_{1}=\beta_{2}\) & 70 & \(k_{1}\) & 15 \\ \(\varepsilon_{1}=\varepsilon_{3}\) & 5 & \(k_{2}\) & 3 \\ \(\varepsilon_{2}=\varepsilon_{4}\) & 3 & \(\varepsilon\) & 0.005 \\ \(\sigma_{1}=\sigma_{2}\) & 15 & \(\widetilde{\phi}_{1}(0)\) & 0.1 \\ \(p\) & \(\frac{5}{7}\) & \(\widetilde{\chi}_{1}(0)\) & 0.2 \\ \(\widetilde{\phi}_{2}(0)\) & 0.3 & \(\widetilde{\chi}_{2}(0)\) & 0.15 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: List of tuning parameters for AGR controller. Figure 6: Result of linear errors \(e_{x}\) (**a**), \(e_{y}\) (**b**), and angular error \(e_{\theta}\) (**c**) in tracking circular trajectory under varying loads. proposed scheme impacted on system state to drive to reduce the differences in the two axes \((x,y)\) and \(\theta\). It is admitted that the output results were reasonable while the convergence time was still ensured. Figure 8: Result of linear velocity (**a**) and angular velocity (**b**) in tracking linear trajectory under varying loads. Figure 7: Result of tracking linear trajectory under varying loads. Figure 9: Result of linear errors \(e_{x}\) (**a**), \(e_{y}\) (**b**), and angular error \(e_{0}\) (**c**) in tracking linear trajectory under varying loads. More specifically, the real S-curve of the working map as in Figure 10a is suggested for the AGR to confirm the robustness and correctness of the proposed algorithm. For continuous trajectories as in Figure 10b, the robot must follow the alterations of the reference path and decide to drive the whole system immediately. In this test, the tracking performance in the circle path was not as good as the one in the line path. It can be explained that the AGR changed both linear and angular velocity as in Figure 11 to adapt with the unexpected fluctuations in trajectory when it entered a corner. In detail, the tracking errors in linear direction and angular movement for the S-curve reference path are demonstrated in Figure 12. The errors still existed in tracking the desired map when the robot executed its task in the warehouse. Under the effective driving of the proposed controller, the existing errors tended to zero in finite time. Figure 11: Performance of linear velocity (**a**) and angular velocity (**b**) in tracking S-curve of working map. Figure 10: Real S-curve of working map (**a**) and tracking result with continuous linear and circular trajectory (**b**). Competitive performance of tracking trajectory for the autonomous robot should be required. Typical robust nonlinear control (TRNC) [23] was chosen since it is popular and proper for industrial applications.