## 4 Numerical Simulations  
The finite element analysis of the stage was carried out using ABAQUS (Dassault Systemes, version 2017). Considering the simulation's computational efficiency, the simulation model was simplified by only including all motion links and the output stage. The virtual prototype was created using SOLIDWORKS (Dassault Systemes, version 2018), as shown in Figure 10a. The analysis process was divided into two steps: (1) static analysis and (2) modal analysis. Regarding the simulation, the material parameters are shown in Table 2. The ends of all the links were fixed. Loads were applied at the surface in the \(x\)-axis. First, given an input force of 60 N in a single direction (simulating only one PZT working), the output stage displacement was extracted. The displacements in the X- and Y-directions are shown in Figure 11a,b, respectively. The simulation analysis results are shown in Table 3. The displacements in the X- and Y-directions were 76.3 \(\upmu\)m and 24.9 \(\upmu\)m, respectively. According to Equation (13), the theoretical values were 78 \(\upmu\)m and 26.3 \(\upmu\)m, respectively. The error between the theoretical calculation and the simulation analysis was within 5.3%. The displacement ratio between the X- and Y-directions were 3.06 (simulation) and 2.96 (theoretical). The X-direction stiffness was approximately 3 times that of the Y-direction. The simulation results verified the reliability of the theory. The stress figure of the proposed stage under the input limit state of the piezoelectric actuator is shown in Appendix D, (Figures A9-A12). Figure 10: Virtual prototype. (**a**) Modeled micropositioning stage using SOLIDWORKS (Dassault Systemes, version 2018), and the (**b**) finite element model. Figure 9: Parameters of each part. The frequency results for the stage are shown in Figure 12 and Table 4. The first to third natural frequencies were 1088.9 Hz, 2240.1 Hz, and 5989.9 Hz, respectively. The third natural frequency was much larger than the first and second, which illustrates that the vibration had little effect on working conditions. Considering the characteristics of the selected piezo-actuator for verifying the performance of the structure, the range of input displacements was assumed to be 0-20 \(\upmu\)m in the simulations. The performance was assessed in terms of the corresponding output displacements. (1) Given a \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model Analysis** & **First** & **Second** & **Third** & **Fourth** & **Fifth** \\ \hline Frequency (Hz) & 1088.9 & 2240.1 & 5989.9 & 7866.9 & 10,739 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Natural frequencies. Figure 11: Numerical simulation analysis results: (**a**) \(X\)-axis and (**b**) \(Y\)-axis. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Axis** & \begin{tabular}{c} **Theory** \\ **(\(\upmu\)m)** \\ \end{tabular} & \
\begin{tabular}{c} **Simulation** \\ **(\(\upmu\)m)** \\ \end{tabular} &
\begin{tabular}{c} **Error** \\ **(\%)** \\ \end{tabular} \\ \hline \(x\) & 78 & 76.3 & 2.2 \\ \(y\) & 26.3 & 24.9 & 5.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison between the theory and simulation output displacements. Figure 12: Modal analysis: (**a**) first-order mode, (**b**) second-order mode, (**c**) third-order mode, (**d**) fourth-order mode, and (**e**) fifth-order mode. single-direction displacement input, the output displacement of the output stage was extracted to get the displacement relationship as shown in Figures 13 and 14. In the \(X\)-axis direction, the ratio was close to 1 between the input and output, and the average error was 3.83%. The material was considered linear elastic in the simulation. The result showed that there was also a slight displacement in the \(Y\)-axis, as shown in Figure 14. Regarding the \(X\)-axis direction, the displacement ratio of the X- to Y-axes was 10.07%.