## 2 Mathematical Modelling for the Micro-Positioning Platform  
The micro-positioning platform in this study consists of a PC, industrial controllers, data acquisition card, linear amplifier and positioning platform, which consists of a voice coil motor, a flexible mechanism and a laser displacement sensor, as shown in Figure 1. In Figure 1, the PC was used to edit the control algorithm and program. We loaded it into the industrial controller, and the control signal given by the industrial controller formed a control voltage through the data acquisition card, which was transferred to the voice coil motor, thus driving the micro-positioning platform to generate position change. Then, the position change of the micro-positioning platform formed a feedback voltage between 0 and 10 V through the sensor, which was transferred to the data acquisition card and inputted the industrial controller to form a feedback signal.  
Figure 1: The micro-positioning platform.  
The micro-positioning platform works on the principle that the controller outputs to the voice coil motor, which drives the displacement of the flexible mechanism. The mechanical equation of the voice coil motor can be obtained from [PERSON]'s second law:  
\[\mathrm{F}{-}\mathrm{F}_{\mathrm{k}}-\mathrm{F}_{\mathrm{c}}{=}\mathrm{m}\mathrm{a}, \tag{1}\]  
where F is the electromagnetic force, \(\mathrm{F}=\mathrm{NBIL}=\mathrm{k_{I}}\mathrm{I}\), N is the number of turns of the coil winding, L is the length of each turn of wire or the circumference of the cross-section of the kinematic cylinder, I is the magnitude of the current, B is the magnetic induction produced by the permanent magnet and \(\mathrm{k_{f}}\) is the force constant of the voice coil motor, which has a size of \(\mathrm{k_{f}}{=}\mathrm{NBIL}\). \(\mathrm{F_{c}}\) is the damping force, which has a size of \(\mathrm{F_{c}}=\mathrm{c\dot{x}}\). \(\mathrm{F_{k}}\) is the spring force, which has a size of \(\mathrm{F_{k}}{=}\mathrm{kx}\). Simplifying the second-order differential equation of the voice coil motor [9]:  
\[\mathrm{F}= \mathrm{m_{e}}\ddot{\mathrm{z}}_{\mathrm{m}}{+}\dot{\mathrm{c}} \mathrm{z_{m}}{+}\mathrm{k}\mathrm{z_{m}}{+}\omega, \tag{2}\]  
where \(\mathrm{m_{e}}=(\mathrm{m_{m}}+\mathrm{m_{c}}+\mathrm{m_{s}})/2\), \(\mathrm{m_{m}}\) is the quality of voice coil electromotor, \(\mathrm{m_{c}}\) and \(\mathrm{m_{s}}\) are the quality of the end of the flexible mechanism and the intermediate platform, \(\mathrm{z_{m}}\) is the displacement of the end platform, \(\mathrm{\omega}\) is the total disturbance and the drive force F of the voice coil motor is proportional to its input current I:  
\[\mathrm{F}= \mathrm{k_{m}}\mathrm{I}, \tag{3}\]  
where \(\mathrm{k_{m}}\) is the force constant of the voice coil motor. The input current I of the voice coil motor is generated by the output voltage \(\mathrm{V_{r}}\) of the data acquisition card via a linear amplifier, and the input current I of the voice coil motor is proportional to the output voltage \(\mathrm{V_{r}}\) of the acquisition card:  
\[\mathrm{I}=\mathrm{\alpha V_{r}}, \tag{4}\]  
where \(\mathrm{\alpha}\) is the amplification factor of the linear amplifier.  
The mathematical model of the micro-positioning platform can be derived as follows:  
\[\mathrm{k_{m}}\mathrm{\alpha V_{r}}=\frac{(\mathrm{m_{m}}+\mathrm{m_{c}}+ \mathrm{m_{s}})\ddot{\mathrm{z}}_{\mathrm{m}}}{2}{+}\mathrm{c_{m}}{+}\mathrm{ k}\mathrm{z_{m}}{+}\omega, \tag{5}\]