## 3 Numerical Simulation Processes  
Assume that the spring-mass-damper system of the PA and stage has a resonant frequency of 1 kHz and a mechanical quality factor Q of 50. Through the flexure hinge amplifier based on the principle of triangle amplification, the final output displacement of one PA can be amplified to 60 \(\upmu\)m with a driving voltage of 100 V. The driving force of the PA can be measured by the theory of spring deformation force; thus, the conversion coefficient \(\delta\) can be calculated to \(-0.6\) N\(\cdot\)V\({}^{-1}\). The resonant frequency and damper ratio can be calculated according to the formulas based on free vibration as follows:  
\[f_{r}=\frac{1}{2\pi}\sqrt{\frac{k}{m_{p}}}, \tag{6}\]  
Figure 3: Karnopp friction model.  
\[\xi=\frac{1}{2Q}=\frac{c}{2\sqrt{km_{p}}} \tag{7}\]  
where \(f_{r}\) is the resonant frequency, \(k\) is the stiffness of the PA and stage, \(m_{p}\) is the equivalent mass of the PA and stage, \(c\) is the damper of the PA and stage, and \(\xi\) is the damping ratio.  
Structural parameters of the PIDM were assumed equal in the \(x\)-axis and \(y\)-axis. By choosing an appropriate slider, the parameters for numerical simulation were calculated and are listed in Table 1.  
With these simulation parameters, the simulation blocks were established (Figure 4) according to the abovementioned kinematics equations [36]. The whole simulation block used a subsystem module, which included a relay module, a saturation module, and a logical operator module. Thus, it could distinguish the friction force when the system was driving under different speeds. The relay module was used to judge whether the slider was sliding on the contact surface between the slider and stage. Because the stage and PA were tightly adhered, the speed of the stage and PA were assumed to be equal. If the relative speed between the slider and PA was beyond the velocity range of \(a\) in the Karnopp friction model, the sliding friction was chosen in the kinematics equations. The sliding friction was usually equal to Coulomb friction, and if the relative speed between the slider and PA was in the velocity range of \(a\), the static friction was chosen and used with the kinematics equations according to the logical operator module. The saturation module was used to judge whether the calculated static friction exceeded the maximum static friction. If the calculated static friction was greater than the maximum static friction, the maximum static friction was chosen. If the calculated static friction was less than the maximum static friction, the calculated static friction was chosen and used with the kinematics equations.  
\begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Value** & **Unit** \\ \hline \(m_{px}\) (\(m_{py}\)) & \(2.5\times 10^{-3}\) & kg \\ \(m_{xx}\) (\(m_{sy}\)) & \(5.0\times 10^{-3}\) & kg \\ \(k_{x}\) (\(k_{y}\)) & \(1.0\times 10^{6}\) & N\(\cdot\)m\({}^{-1}\) \\ \(c_{x}\) (\(c_{y}\)) & 3.2 & N\(\cdot\)s\(\cdot\)m\({}^{-1}\) \\ \(\delta_{x}\) (\(\delta_{y}\)) & \(-0.6\) & N\(\cdot\)V\({}^{-1}\) \\ \(f_{s}\) & 1.2 & N \\ \(f_{c}\) & 1.0 & N \\ \(a\) & \(1.0\times 10^{-6}\) & m\(\cdot\)s\({}^{-1}\) \\ \hline \end{tabular}
\end{table}
Table 1: Parameters of the parallel 2-DOF linear PIDM.