## 4 PMSM Driving Control Design  
A PMSM mainly actuates feed drive axes; thus, the PMSM driving control design can significantly affect the motion performance of each feed drive axis and further affects the operating performance of the PCB-ILCC. Considering the dynamic equation of the feed drive axis actuated by the PMSM as Equation (15):  
\[\tau_{m}=\dot{J}\dot{\theta}_{m}+B\dot{\theta}_{m}+\tau^{sc}+\tau^{c}, \tag{15}\]  
where \(J\) and \(B\) denote the equivalent moment of inertia and equivalent viscous friction coefficient, respectively. \(\theta_{m}\) denotes the rotating angular position of the PMSM, \(\tau_{m}\) and \(\tau^{c}\), respectively, denote the driving torque and external disturbance torque of the PMSM, and \(\tau^{sc}\) as Equation (16) denotes the compound friction torque composed of the static friction torque and Coulomb friction torque:  
\[\tau^{sc}=\left(\tau^{c}+(\tau^{s}-\tau^{c})e^{-(|\frac{\dot{\theta}_{m}}{ \omega_{s}}|)^{\delta}}\right)\cdot sgn\left(\dot{\theta}_{m}\right), \tag{16}\]  
where \(\tau^{s}\) and \(\tau^{c}\) denote the static friction torque and Coulomb friction torque, respectively. \(\omega_{s}\) denotes the Stribeck velocity, and \(\delta\) denotes the Stribeck shape factor. Supposing that the feed drive axis would not be influenced by the external disturbance torque (i.e., \(\tau^{c}=0\)), this study designed the driving control as Equation (17):  
\[\tau_{m}=\dot{J}a_{m}+\dot{B}\dot{\theta}_{m}+\tau^{sc}, \tag{17}\]  
where \(\dot{J}\) denotes the estimated equivalent moment of inertia, \(\dot{B}\) denotes the estimated equivalent viscous friction coefficient, \(\tau^{sc}\) denotes the estimated compound friction torque, and \(a_{m}\) denotes the control term required to be further designed. Therefore, by substituting Equation (17) into Equation (15), the dynamic equation can be rewritten as Equation (18):  
\[\dot{J}\ddot{\theta}_{m}=\dot{J}a_{m}+\big{(}\dot{B}-B\big{)}\dot{\theta}_{m} +(\tau^{sc}-\tau^{sc})=\dot{J}a_{m}+\widetilde{B}\dot{\theta}_{m}+\widetilde{ \tau}^{sc}, \tag{18}\]  
where \(\widetilde{B}=\dot{B}-B\) and \(\widetilde{\tau}^{sc}=\dot{\tau}^{sc}-\tau^{sc}\) denote the estimation errors of the equivalent viscous friction coefficient and compound friction torque, respectively. The estimation error of the equivalent moment of inertia can also be defined as \(\widetilde{J}=\dot{J}-J\), and Equation (18) can be rewritten as Equation (19):  
\[\ddot{\theta}_{m}=\dot{J}^{-1}\left(\dot{J}a_{m}+\widetilde{B}\dot{\theta}_{m }+\widetilde{\tau}^{sc}\right)=a_{m}+\dot{J}^{-1}\left(\widetilde{J}a_{m}+ \widetilde{B}\dot{\theta}_{m}+\widetilde{\tau}^{sc}\right) \tag{19}\]  
The uncertainty \(\eta\) can be defined as Equation (20):  
\[\eta=\dot{J}^{-1}\left(\widetilde{J}a_{m}+\widetilde{B}\dot{\theta}_{m}+ \widetilde{\tau}^{sc}\right) \tag{20}\]  
Then, Equation (19) can be rewritten as Equation (21):  
\[\ddot{\theta}_{m}=a_{m}+\eta \tag{21}\]  
To overcome the influence of the uncertainty \(\eta\), this study designed the control term \(a_{m}\) as shown in Equation (22):  
\[a_{m}=\ddot{\theta}_{d}-K_{1}\Big{(}\dot{\theta}_{m}-\dot{\theta}_{d}\Big{)}- K_{0}(\theta_{m}-\theta_{d})+\delta a_{m}, \tag{22}\]  
where \(\ddot{\theta}_{d}\), \(\dot{\theta}_{d}\), and \(\theta_{d}\) denote the commands for the rotating angular acceleration, angular velocity, and angular position of the PMSM, respectively. \(K_{1}>0\) and \(K_{0}>0\) denote the control gains; \(\delta a_{m}\) denotes the additional term and is used to attenuate the influence ofthe uncertainty.