## 5 Conclusions  
In the field of compliant mechanisms, this study has proposed a proper size optimization method based on two steps: pseudo-rigid-body model optimization, and reliability-based multi-objective optimization. The benchmarking for the optimization results between the proposed RBDO and DO shows that all candidates in the Pareto front of the RBDO satisfy the expected safety level, \(R_{0}\). Meanwhile, not all optimum designs of the DO meet the requirement. This optimization algorithm has been applied to design a new CFDM with the reliability of \(R_{0}=99.87\%\) for two objective functions of natural frequency and displacement amplification. It considers the uncertainty of different input/output parameters of the problem, such as cutting forces, fabrication variations, material properties, and input loads from the PZT actuator. This CFDM has been fabricated, tested, and controlled using the GA-PID algorithm. The experimental results show that the discrepancies between theory, simulation, and experiment are less than 5% for the first natural frequency and displacement amplification ratio. Due to the symmetrical design of the mechanism, the parasitic motion is decoupled from the instrument's end-effector. In addition, the error of displacement closed-loop control is 0.025 \(\upmu\)m. This CFDM is also being used to investigate the machining efficiency of microfabrication. The experimental results will be presented in another published work. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Performance Parameters** & **Analytical** & **Simulation** & **Error (\%)** & **Experiment** & **Error (\%)** \\ \hline Amplification ratio & 3 & 3.104 & 3.46\% & 2.93 & 2.33\% \\ Natural frequency (Hz) & 1519.2 & 1500 & 1.26\% & 1529.4 & 0.67\% \\ Travel range (\(\upmu\)m) & 40 & 39.65 & 0.87\% & 39.16 & 2.1\% \\ Input stiffness\(K_{in}\) (N/\(\upmu\)m) & 65.43 & 59.26 & 9.42\% & 62.35 & 4.70\% \\ Output stiffness \(K_{out}\) (N/\(\upmu\)m) & 5.21 & 5.23 & 0.38\% & 5.42 & 4.03\% \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of simulation and experiment results. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Studies** &
\begin{tabular}{l} **Natural Frequency** \\ **(Hz)** \\ \end{tabular} & **Amplification Ratio** & **Stress (MPa)** \\ \hline [PERSON] [42] & 398.8 & 1.21 & 53 \\ [PERSON] [43] & 156.77 & unknown & 94.636 \\ [PERSON] [44] & 480 & 3 & unknown \\ [PERSON] [45] and [PERSON] [4] & 200 & 2.8725 & 260.65 \\ This study & 1500 & 3.104 & 73.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison with the results of previous studies. **Author Contributions:** Conceptualization, V.-K.N. and H.-T.P. ; methodology, V.-K.N. and H.-H.P. ; software, H.-T.P. ; validation, V.-K.N. ; formal analysis, V.-K.N. and H.-T.P. ; investigation, V.-K.N. ; resources, P.S.M. and Q.-K.D. ; data curation, V.-K.N. and Q.-K.D. ; writing--original draft preparation, V.-K.N. ; writing--review and editing, H.-T.P. and H.-H.P. ; visualization, H.-T.P. ; supervision, H.-T.P. ; project administration, H.-T.P. ; funding acquisition, V.-K.N. and H.-T.P. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was supported by the Ho Chi Minh City University of Technology and Education, project grant number B2021-SPK-02, funded by the Ministry of Education and Training, and hosted by the Ho Chi Minh City University of Technology and Education, Vietnam. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Informed consent was obtained from all subjects involved in the study. **Data Availability Statement:** Not applicable. **Conflicts of Interest:** The authors declare no conflict of interest.