## 6 Conclusions  
This paper proposes a design and control scheme for a track-following servo subsystem of a LGCS suitable for CLRV ground testing:  
1. The track-following servo subsystem consists of a crane, trolley, and servo motors. The crane moves along the bridge, tracking the real-time position of the CLRV, while the trolley moves along the crane girder to follow the real-time position of the CLRV. Three-loop control models are established for both the crane and trolley, including the position loop, motor speed loop, and current loop. Figure 12: The results of tracking desired path four by the crane and trolley are presented. The figures illustrate the following outcomes: a comparison between the trajectory of the upper lifting point and the expected trajectory under three different controllers, the tracking error \(\delta P=\sqrt{\delta P_{x}^{2}+\delta P_{y}^{2}}\) of the track-following servo subsystem, and the movement speed of both the crane and trolley. The driving force of the track-following servo subsystem is provided by a PMSM. Specific parameters for the motor and transmission mechanism are determined based on the subsystem's motion capability. PMSM vector control based on the SVPWM is adopted to achieve fast response and precise control of the motor speed. PI controllers are employed in both the current and speed loops of the motors. 3. In the position loop control of the track-following servo subsystem, explicit MPC control is introduced for multi-parameter constraint control. This paper presents an EMPC controller suitable for the track-following servo subsystem, including the cost function design and offline calculation process. The weighting matrices, prediction horizon, and control horizon are also determined. Especially, the effectiveness of the proposed EMPC controller is demonstrated through a comparison with PI and LQI controllers. The simulation results show that the maximum tracking error \(\delta P\) of the track-following servo subsystem is consistently below 0.2 m. The impact of the errors on the drawbar pull is within 12.5%, and their effect on the compensation force is negligible. The simulation results also suggest that, due to the large inertia of the crane, the CLRV should avoid frequent forward, backward, acceleration, and deceleration in the \(X\) direction to reduce the influence of the crane's large inertia on tracking performance. These simulation results provide valuable theoretical support for designing a track-following servo subsystem that is suitable for CLRV ground testing. In our future work, our objective is to develop a track-following servo subsystem that is suitable for ground testing of the CLRV. The primary contribution of this paper is the development of an EMPC controller for the position control loop, along with PMSM vector control, which has demonstrated excellent tracking performance for the subsystem. However, implementing the EMPC controller in engineering settings may pose significant challenges. In our upcoming research, we plan to refine our work further, including addressing potential deviations in motor load torque, exploring cooperative control of multiple motors, and enhancing the robustness of the EMPC controller. These are essential considerations that must be addressed to fully realize the potential of the proposed approach in practical applications. Conceptualization, M.L. and J.W. ; methodology, Y.Z. ; software, Y.Z. ; validation, Y.Z., H.W. and M.L. ; formal analysis, H.W. ; investigation, H.W. ; resources, Y.Z. ; data curation, Y.Z. ; writing--original draft preparation, Y.Z. ; writing--review and editing, Y.Z. ; visualization, Y.Z. ; supervision, H.W. ; project administration, H.W. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. Not applicable. Not applicable. Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author.