## 5 Results and Discussion
The tracking error \(\delta P=\delta P_{x}\) and the tracking error of the EMPC controller for path one is within 0.2 m.  
Figure 10 depicts the simulation results of the trolley tracking path two, which involves frequent acceleration, deceleration, and forward and reverse movements in the \(Y\) direction. Figure 9: The simulation results’ comparison of tracking path one used the PI, LQI, and EMPC controllers. The figures present the following results: the results of the crane tracking desired path one in the \(X\) direction; tracking errors \(\delta P_{x}\) between the crane’s position and the desired position along path one in the \(X\) direction; the crane speed \(v_{ux}\) of tracking path one. Figure 10 shows the simulation results of the trolley tracking path two in the \(Y\) direction. The tracking error is mainly caused by the trolley's inertia. Comparing it with Figure 9, it can be observed that as the crane's inertia is much greater than that of the trolley, the error will be even greater when the crane tracks the expected path with rapid changes in the \(X\)-directional speed. The results shown in Figure 10 show the trolley's speed \(\
u_{\text{up}}\) of tracking path two. From the tracking error plot and trolley speed plot, it can be seen that large errors usually occur during the time periods when the trolley undergoes acceleration and deceleration switching. To minimize the impact of the crane's large inertia on tracking performance, it is recommended to avoid frequent acceleration and deceleration of the lunar rover in the \(X\) direction. Instead, it is advisable to conduct forward and reverse movement performance testing solely in the \(Y\) direction to achieve optimal results. In the tracking servo of path two, there is only an error in the \(Y\) direction. The tracking error \(\delta P=\delta P_{y}\) and the tracking error of the EMPC controller for path two is within 0.2 m.  
Figure 11 depicts the simulation results of the crane and trolley tracking desired path three, which follows a turning motion trajectory of the CLRV with a radius of 10 m, and operates at a speed of 3 m/s. On the other hand, Figure 12 illustrates the crane and trolley tracking path four, which represents the driving trajectory of the CLRV when encountering continuous turns during obstacle avoidance. The CLRV operates at a speed of around 3 m/s while following this path. Figure 10: Results of the trolley tracking desired path two. The figures present the following results: the results of trolley tracking desired path one in the \(Y\) direction; tracking errors \(\delta P_{y}\) between the trolley’s position and the desired position along path two in the \(Y\) direction; the trolley speed \(v_{uy}\) of tracking path two. Figures 11 and 12 illustrate that the EMPC controller results in the smallest tracking error, followed by LQI and PI controllers. Furthermore, the results suggest that the LQI controller exhibits excellent stability, as the movement speed of the crane and trolley remains relatively stable in response to dynamic disturbances. On the other hand, when using a PI controller for a large inertia system, the overshoot can be relatively large, and it can be challenging to balance the response speed and stability. Figures 11 and 12 displays the simulation results of crane and trolley tracking for desired paths three and four, utilizing the EMPC controller.