## 3 Dynamic Model of the Dual Linear-Motor Differential-Drive System
### Mechanical Model
#### 3.1.2 Dual Linear-Motor Differential-Drive System Analysis  
As shown in Figure 3, the mechanical model of the dual linear-motor differential-drive servo-feed system is presented. In order to ensure that the transmission parameters of the two single-drive systems are consistent, two linear motors with identical parameters are used in the design, and it is ensured that the equivalent mass and viscous friction coefficient of the upper linear motor actuator during the use of the upper linear-motor single drive are equal to the equivalent mass and viscous friction coefficient of the under linear motor actuator during the use of the under linear motor single drive, respectively, namely, \(m_{a}=m_{a1}=m_{a2}\), \(B_{s}=B_{s1}=B_{s2}\). At the same time, it is ensured that the equivalent mass and viscous friction coefficient of the upper table during the use of the upper linear-motor single drive are equal to the equivalent mass and viscous friction coefficient of the under table during the use of the under linear-motor single drive, respectively, that is \(m_{t}=m_{t1}=m_{t2}\), \(B_{t}=B_{t1}=B_{t2}\). In addition, it is required that the comprehensive transmission stiffness of the two single-drive systems and the differential-drive system be consistent, that is \(K_{eq}=K_{eq1}=K_{eq2}\).  
The dual linear-motor differential-drive system is achieved by comparing the difference values in force, speed, and displacement between the upper linear-motor single drive and the under linear-motor single drive, respectively.  
\[\Delta F_{s} =F_{s1}-F_{s2} \tag{6}\] \[\Delta F_{a} =F_{a1}-F_{a2}\] (7) \[\Delta F_{d} =F_{d1}-F_{d2}\] (8) \[\Delta F_{sf} =F_{sf1}-F_{sf2}\] (9) \[\Delta F_{f} =F_{f1}-F_{f2}\] (10) \[\Delta v_{t} =v_{t1}-v_{t2}\] (11) \[\Delta x_{a} =x_{a1}-x_{a2}\] (12) \[\Delta x_{t} =x_{t1}-x_{t2} \tag{13}\]  
where \(F_{s1},F_{s2}\) are the servo driving forces generated by the upper linear motor and under linear motor, respectively. \(F_{a1},F_{a2}\) are the driving forces exerted by the upper linear motor and under linear motor on the actuator, respectively, and generate driving forces \(F_{d1}\), \(F_{d2}\) acting on the upper table and the under table, respectively. \(F_{sf1},F_{sf2}\) are the electromagnetic resistance generated on the upper linear motor and under linear motor, respectively. \(F_{f1},F_{f2}\) are the equivalent coulomb friction forces acting on the table when the upper linear motor and under linear motor are driven separately. \(v_{t1},v_{t2}\) respectively represent the actual speed of the table when the upper linear motor and under linear motor are driven individually. \(x_{a1},x_{a2}\) represent the theoretical displacement generated by the upper linear motor actuator and under linear motor actuator, respectively. \(x_{t1},x_{t2}\) respectively represent the actual displacement generated by the table when the upper linear motor and under linear motor are driven individually.  
The equations of motion for the dual linear-motor differential-drive servo-feed system are Equations (14)-(18), then  
\[\Delta F_{s}=m_{d}\Delta\ddot{x_{a}}+B_{s}\Delta\dot{x_{a}}+\Delta F_{sf}+\Delta F _{a} \tag{14}\]  
\[\Delta F_{a}=\frac{\Delta F_{d}}{\eta} \tag{15}\]  
\[\Delta F_{d}=m_{t}\Delta\ddot{x}_{t}+B_{t}\Delta\dot{x}_{t}+\Delta F_{f} \tag{16}\]  
\[\Delta x_{t}=\Delta x_{a}-\frac{\Delta F_{d}}{K_{eq}} \tag{17}\]  
\[K_{eq}=\frac{K_{s1}K_{a1}}{K_{s1}+K_{a1}}=\frac{K_{s2}K_{a2}}{K_{s2}+K_{a2}} \tag{18}\]  
where \(K_{s1},K_{s2}\) represent the servo stiffness acting on the upper linear motor actuator and under linear motor actuator, respectively. \(K_{a1}\), \(K_{a2}\) respectively represent the contact stiffness between the upper linear motor actuator and the upper table, and the contact stiffness between the under linear motor actuator and the under table.