## 3 Laser Processing Control Method for MCR Frequency Modulation
### Workflow of Laser Processing Control System for MCR Frequency Modulation
The d-q axis mathematical model is used to analyze motor performance, and the voltage equation of the permanent magnet synchronous linear motor is established [21]. \[\left\{ \begin{array}{l}\mathrm{u_{q}}=\mathrm{Ri_{q}}+\mathrm{\psi_{q}}+ \mathrm{\omega_{r}}\mathrm{\psi_{d}}\\ \mathrm{u_{d}}=\mathrm{Ri_{d}}+\mathrm{\psi_{d}}+\mathrm{\omega_{r}}\mathrm{ \psi_{q}}\end{array} \right. \tag{2}\]  
Figure 3: Automatic operation flow chart of laser process for MCR frequency modulation. The flux linkage equation of linear motor is as follows. \[\left\{ \begin{array}{l}\psi_{\rm q}={\rm L_{q}}{\rm i_{q}}\\ \psi_{\rm d}={\rm L_{d}}{\rm i_{d}}+\psi_{\rm f}\end{array} \right. \tag{3}\]  
where \(u_{d}\) and \(u_{q}\) are the voltage components on the d-q axis, \(V\); \(i_{d}\) and \(i_{q}\) are the current components on the d-q axis, \(A\); \(L_{d}\) and \(L_{q}\) are the inductance components on the d-q axis, \(mH\); \(\psi_{d}\) and \(\psi_{q}\) are the flux linkage vectors on the d-q axis, \(Wb\); \(\psi_{f}\) is the coupling flux linkage of the stator permanent magnet in the winding, \(Wb\); \(R\) is the stator winding resistance, \(\Omega\), and \(\omega_{r}\) is the rotor angular speed, \(rad/s\). The electromagnetic thrust formula is as follows [21]. \[{\rm F_{e}}=\frac{3\pi}{2\tau_{\rm n}}[\psi_{\rm f}{\rm i_{q}}+({\rm L_{d}}-{ \rm L_{q}}){\rm i_{d}}{\rm i_{d}}] \tag{4}\]  
The electromagnetic thrust of the linear motor is controlled by the q-axis current component, that is, when the excitation component current is 0, the electromagnetic thrust can be expressed as the following formula. \[{\rm F_{e}}=\frac{3\pi}{2\tau_{\rm n}}\psi_{\rm f}{\rm i_{q}}={\rm K_{m}}{\rm i _{q}} \tag{5}\]  
Figure 4: Frequency modulation flow chart. The mechanical motion equation of linear motor is the following formula. \[\mathrm{F_{e}=K_{m}i_{q}=Mv^{\prime}+Bv+F} \tag{6}\]  
where \(M\) is the motor mover mass, \(kg\); \(B\) is motor viscous friction coefficient, \(N\cdot s/m\); \(F\) is the disturbance of thrust fluctuation caused by load resistance, friction and end effect, \(N\); \(K_{m}\) is the motor force constant, \(N/A\); \(v\) is the operating velocity, \(m/s\), and \(\tau\) is the polar distance, \(mm\). The voltage balance equation is as follows. \[\mathrm{L}\frac{\mathrm{di_{q}}}{\mathrm{dt}}+\mathrm{Ri_{q}=u_{q}-\varphi_{I }Pv=u_{q}-K_{e}v} \tag{7}\]  
where \(P\) is the pole-pair number and \(K_{e}=\phi_{f}P\) is the back EMF coefficient, \(V/(m/s)\). Ignoring the effect of load disturbance during simulation, the linear motor model can be simplified, and the simulation model is depicted in Figure 5, where s is the Laplace operator. Since the main travel direction of the platform is the X direction, the linear motor used in the X direction is designed and analyzed, and the motor parameters are listed in Table 2.  
2.2 Carrier Platform Control Method of Laser Processing for MCR Frequency Modulation Based on Fuzzy PID  
Since the linear motor current controller and speed controller are set in the driver, the three closed-loop control mode based on position control mode is selected to control the linear motor. In order to facilitate the analysis, the current loop is regarded as a linear gain, \(\mathrm{K_{c}}\), and the velocity loop controller is a Proportion-Integral (PI) controller with certain parameters. The velocity loop PI controller transfer function of the linear motor is shown in formula 8 [21]. \[\mathrm{G_{v}(s)=K_{v}(1+\frac{1}{T_{v}\cdot s})=\frac{K_{v}(1+T_{v}\cdot s)}{T _{v}\cdot s}} \tag{8}\]  
where \(T_{v}\) is the integral time constant of the velocity loop and \(K_{v}\) is the proportional gain of the velocity loop.