## 2 Mathematical Model of the Horizontal System  
### Modeling of Horizontal Thrust-Current Dynamics  
The \"moving armature\" method is adopted, with the horizontal control coils wound around the MRC [2]. The Ampere force generated in the horizontal direction by the energized horizontal control coils in the air-gap magnetic field drives the MRC to move horizontally. The winding configuration of the horizontal control coils is depicted in Figure 1.  
The MAR system is equipped with six stator yokes in the Y-direction. The air-gap magnetic fields between the two sets of stator yokes on both sides and the MRC share the same direction. However, the air-gap magnetic field between the middle stator yoke and the MRC has an opposite direction to that of the side stator yokes. If a coil is wound around the entire MRC along its Y-axis, the Ampere forces acting on the MRC in the three magnetic fields will not be in the same direction. The Ampere forces at the two-side force-application points of the MRC are in the same direction, which is contrary to the Ampere force at the middle force-application point. Since the MRC is driven by the Ampere forces on both sides, the Ampere force in the middle interferes with the movement of the MRC. Consequently, no coil is wound in the middle part of the MRC. Instead, four coils, Co1, Co2, Co3, and Co4, are wound on both sides of the MRC. As illustrated in Figure 1, Co1 and Co2 are fastened to one side of the MRC from the front and rear, while Co3 and Co4 are attached to the other  
Figure 1: Schematic diagram of the winding method of horizontal control coils. side of the MRC from the front and rear. This setup ensures that the wires of the horizontal control coils in the air-gap magnetic field are perpendicular to the Y-direction. In this paper, the permanent magnet of the MAR system is designated as the front side of the system, and the suspension control coils are considered the rear side. After energization, Co1 can generate a forward-driving force, propelling one end of the MRC forward. Similarly, Co3 can drive the other end of the MRC forward. Co2 and Co4 generate backward-driving forces to move the MRC backward. As there are no wires in the air-gap magnetic field at the middle of the MRC, no reverse interference force is generated. Figure 2 presents the schematic diagram of the horizontal driving force. In the figure, \(F_{t1}\) represents the Ampere force generated by the energization of coil Co1, \(F_{t2}\) is the Ampere force generated by the energization of coil Co2, \(F_{t3}\) is the Ampere force generated by the energization of coil Co3, and \(F_{t4}\) is the Ampere force generated by the energization of coil Co4. Therefore, the thrust on the right-hand side of the MRC is as follows:  
\[F_{rt}=F_{t1}-F_{t2} \tag{1}\]  
Based on the Ampere force calculation formula [2], the formula for calculating the right-hand side thrust \(F_{rt}\) is as follows:  
\[F_{rt}=nB_{r}Li_{cr} \tag{2}\]  
In this formula, \(n\) is the sum of the turns of the right-front horizontal control coil and the right-rear horizontal control coil. Each group of coils has 15 turns, amounting to a total of 30 turns. \(B_{r}\) represents the sum of the right-hand side air-gap magnetic fields. \(L\) is the length of a single-turn horizontal control coil in the air-gap magnetic field. \(i_{cr}\) is the current of the horizontal control coils on the right-hand side of the MRC, which is composed of the current \(i_{c1}\) of the right-front horizontal control coil Co1 and the current \(i_{c2}\) of the right-rear horizontal control coil Co2. \(i_{c1}\) and \(i_{c2}\) flow in opposite directions. When observing the two horizontal control coils Co1 and Co2 on the right-hand side from the front side of the MAR system, if the current \(i_{c1}\) in coil Co1 flows counter-clockwise, it is considered positive, ensuring that the horizontal control coil Co1 generates a forward thrust \(F_{t1}\).