## 5 Results and Discussion  
In this system, the impact of the tracking error of the track-following system mainly includes the impact on the drawbar pull force \(F_{DP}\)[33] and gravity compensation force \(F_{C}\) of the CLRV. Supposing that the upper lifting point is positioned 10 m above the CLRV, the drawbar pull force error in the horizontal direction caused by the tracking error denoted as \(\Delta F_{h}\) can be expressed as:  
\[\frac{\Delta F_{h}}{\frac{5}{6}G_{c}}=\frac{\delta P}{10} \tag{35}\]  
where \(G_{c}\) is the weight of the CLRV and \(\delta P\) is the tracking error. The gravity compensation force error \(\Delta F_{v}\) in the vertical direction caused by the position tracking error is:  
\[\Delta F_{v}=F_{C}\bigg{[}1-cos\bigg{(}arctan\frac{\delta P}{10}\bigg{)} \bigg{]} \tag{36}\]  
Supposing that the drawbar pull factor of the CLRV's wheel with a radius of 0.4 m on loose soil is \(u_{DP}=0.8\), the maximum drawbar pull force and gravity compensation force can be roughly expressed as:  
\[F_{DP}=u_{DP}W_{N} \tag{37}\]  
\[F_{C}=\frac{5G_{c}}{6} \tag{38}\]  
where \(W_{N}\) is the wheel vertical load. Hence, the maximum drawbar pull of the entire rover can be estimated by setting \(W_{N}\) as \(1/6G_{c}\). Combining Equations (35)-(38) gives:  
\[\epsilon_{h}=\frac{\Delta F_{h}}{F_{DP}} \tag{39}\]  
\[\epsilon_{v}=\frac{\Delta F_{v}}{F_{C}} \tag{40}\]  
where \(\epsilon_{h}\) and \(\epsilon_{v}\) are the error impact factors to measure the influence of tracking errors. This section presents and discusses the simulation results of the track-following servo control for paths one, two, three, and four using the PI, LQI, and EMPC controllers. Figure 9 presents a comparison of simulation results for track-following servo control along path one, using three different controllers: PI, LQI, and EMPC. The \"Reference\" curve shows the real-time position of the CLRV. The figure displays the tracking results, tracking errors, and the crane speed of the track-following servo for each controller. It can be observed from Figure 9 that the EMPC controller has the best performance and the smallest tracking error against the disturbance caused by the rapid change in the \(X\)-direction speed of the CLRV, followed by the LQI and PI controllers. Moreover, by examining the motion speed diagram of the crane, it can be inferred that the tracking stability of LQI is highest. As shown in Figure 9, the positional deviation resulting from the track-following servo of the PI controller is significantly greater than those of the EMPC and LQI controllers. This is primarily due to the high inertia of the crane, which poses challenges in tracking the desired path. To overcome this, a simple PI controller with higher proportional coefficients is employed, leading to improved response speed but degraded steady-state performance. Interestingly, LQI has a lower settling time than PI and EMPC, and its stability is best. This can be attributed to its very high weight in the state of the tracking error, which helps it reach the steady state earlier. However, LQI suffers from delay errors between the desired path and the crane position, resulting from its reliance on the output value of the error integrator. In contrast, the EMPC controller enhances control performance by generating optimal inputs while satisfying constraints. The model predictive control algorithms used in EMPC are more complex, taking into account various parameters and predicting the most optimal path for possible trajectories. As a result, the EMPC controller performs better at minimizing errors compared to controllers that use a simple cost function or a set of gains to correct deviations. Figure 9 shows the tracking error of tracking desired path one using the EMPC controller. \(P_{x}\) refers to the real-time error of the crane and lunar rover in the \(X\) direction, while \(P_{y}\) refers to the real-time error of the trolley and lunar rover in the \(Y\) direction. In the tracking servo of path one, there is only an error in the \(X\) direction.