## 2 The Proposed Synchronous Control Method for the Dual-Motor System  
### Kinetic Model of the Dual-Motor System  
A schematic of the redundant direct-drive gantry platform is shown in Figure 1. A cross beam was installed on two linear guides arranged in parallel and jointly driven by two PMLSMs. Motors Y\({}_{1}\) and Y\({}_{2}\) were rigidly connected to the beam, and the stators were installed on the base where the two linear guides were located. For the sake of simplicity, this study only focused on the synchronous control of the Y-axis, and it is assumed that the motor X is rigidly connected to the cross-beam. Therefore, from Figure 1 and the dynamic analysis in literature [27], the kinetic model of the dual-motor system can be obtained as follows:  
\[\begin{cases}M_{1}\ddot{y}_{1}+c_{1}\dot{y}_{1}+k_{1}\xi_{1}=F_{m1}-F_{r1}-F_{ c1}\\ M_{2}\ddot{y}_{2}+c_{2}\dot{y}_{2}+k_{2}\xi_{2}=F_{m2}-F_{r2}+F_{c2}\end{cases}, \tag{1}\]  
where \(F_{ci}\) = \(f\) (\(y_{1}\), \(y_{2}\)) is the coupling force, which is related to the displacements of motors Y\({}_{1}\) and Y\({}_{2}\), and \(i\) = 1, 2. \(F_{mi}\) is the thrust force, \(F_{ri}\) is the friction, \(M_{i}\) is the mass, \(c_{i}\) is the viscosity coefficient, \(k_{i}\) is the elasticity coefficient, \(y_{i}\) is the displacement of the motor, and \(\xi_{i}\) is the deformation between the mover and slider. The control method typically adopted by a dual-motor system is also shown in Figure 1. Each motor has an independent feedback controller (FBC); by designing the reasonable friction compensator (FC) and the synchronous control method, the synchronous motion performance of the dual-motor system is guaranteed.  
Figure 1: Schematic of the structure and control strategy of redundant direct-drive gantry platform.  
The relationship between the motor thrust force and control system characteristics can be obtained as follows:  
\[\begin{cases}F_{m1}=K_{u1}u_{1}=K_{u1}sat\{u_{FC1}+u_{FBC1}\},\\ F_{m2}=K_{u2}u_{2}=K_{u2}sat\{u_{FC2}+u_{FBC2}\},\end{cases} \tag{2}\]  
where \(K_{ui}\) is the thrust force constant of the motor, \(sat\) {} is the saturation function, and \(u_{FCI}\) and \(u_{FBCi}\) are the control voltages of FC and FBC, respectively. Because the coupling force is related to the displacements of motors Y\({}_{1}\) and Y\({}_{2}\), it appears with the generation of the synchronization error. When the synchronization error is eliminated, the influence of the coupling force on the motor movement will also be negligible, so the kinetic model can be re-expressed as follows:  
\[\begin{cases}M_{1}\dot{y}_{1}+c_{1}\dot{y}_{1}+k_{1}\dot{\xi}_{1}=F_{m1}\\ M_{2}\dot{y}_{2}+c_{2}\dot{y}_{2}+k_{2}\dot{\xi}_{2}=F_{m2}.\end{cases} \tag{3}\]  
We can establish the following transfer function of each motor system according to Equation (3):  
\[g_{yi}(s)=\frac{b_{0i}s^{m_{i}}+b_{1i}s^{(m-1)_{i}}+\cdots+b_{mi}}{s^{n_{i}}+a _{1i}s^{(n-1)_{i}}+\cdots+a_{ni}}, \tag{4}\]  
where \(a_{ni}\) and \(b_{mi}\) are the parameters of the denominator and numerator polynomials, respectively, and \(n_{i}\) and \(m_{i}\) are the orders of the denominator and numerator polynomials, respectively. By designing a proper feedback controller \(c_{i}\)(s), the transfer function between the input signal \(r\) and output signal \(y_{i}\) is as follows:  
\[G_{i}(s)=\frac{y_{i}(s)}{r(s)}=\frac{c_{i}(s)g_{yi}(s)}{1+c_{i}(s)g_{yi}(s)}. \tag{5}\]  
From Equation (5), we can calculate the amplitude and phase characteristics of each closed-loop feedback control loop in the dual-motor system, and these characteristics determine the dynamic response characteristics of the motor system. Therefore, we can know when the dual-motor system is in a synchronous-state, and the motion of the motor system is mainly affected by its electric characteristic, structure, and control system, leading to the differences in the dynamic response characteristics between the motor systems.