## 3 Optimization Design
### Reliability-Based Multi-Objective NSGA-II Optimization  
Figure 4 shows the CM configuration corresponding to the PRB model in Figure 3. The designated design space of the mechanism is limited to \((100\times 50\times 30)\) mm\({}^{3}\), which is adequate for the later task of assembly into an available machine tool. The previous PRBM optimization algorithm has constrained this design space in its problem formulation. Therefore, it can be easily changed in appropriately adapting to any other design configurations. The general working principle of the CM is to rely on the deformation of the mechanism to transfer or transmit motion. As a rule of thumb, this deformation must be maintained within the elastic limit of the fabricated material, and prolonging of its fatigue endurance is allowed. Generally, the maximum concentrated stress \((\sigma_{max})\) must be lower than the yield strength \((\sigma_{y})\) when the structure experiences the most significant displacement. For hinge-type CMs, \(\sigma_{max}\) occurs when a semicircular elastic hinge reaches its largest rotation angle \(\theta_{max}\)[40, 41], namely:  
\[\sigma_{max}=\frac{E(1+\eta)}{\eta^{2}f(\eta)}\theta_{max}<\frac{\sigma_{y}}{ SF} \tag{42}\]  
where, \(\eta=T/2R\) is a dimensionless coefficient; \(T\) and \(R\) are the minimum thickness and radius of a hinge, respectively; \(f(\eta)\) is a function of \(\Delta\)[40, 41], and \(SF\) is the safety factor. Equation (5) in Section 2.3 is used in evaluating the displacement vector, including the rotation angle \(\theta_{z}\) of a hinge if given the tensile force \(F_{x}\), transverse force \(F_{y}\), and bending moment \(M_{z}\). This calculation does not consider the effect of out-of-plane loads, since the mechanism has constrained its deformation in the \(xOy\) plane by designing a much larger out-of-plane thickness \(h\) than the minimum in-plane thickness \(T\) of each hinge. The more flexible the CM, the more easily it could deform. Reducing the th \begin{table}
\begin{tabular}{c c c} \hline \hline
1. Objective function: \(Min\|A_{amp}-3|\) & (35) \\ \hline
2. DVs: \(L_{OM},L_{ON},L_{\mathrm{O}_{1}N},L_{\mathrm{O}_{1}P}\) & \\ \hline
3. Subjected to: & \\ \hline (i) Limits of the DVs (mm) & \\ \(10\leq L_{\mathrm{OM}}\leq 20\) & (36) \\ \(15\leq L_{\mathrm{ON}}\leq 30\) & (37) \\ \(10\leq L_{\mathrm{O}_{1}N}\leq 20\) & (38) \\ \(25\leq L_{\mathrm{O}_{1}P}\leq 40\) & (39) \\ (ii) Boundary constraints: & \\ \(L_{\mathrm{ON}}+L_{\mathrm{O}_{1}N}=40\) & (40) \\ \(L_{\mathrm{O}_{1}P}-L_{\mathrm{O}_{1}N}\geq 20\) & (41) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: PRBM optimization formulation. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**DVs** & **Value (mm)** & **DVs** & **Value (mm)** \\ \hline \(L_{\mathrm{OM}}\) & 16.5 & \(L_{\mathrm{O}_{1}N}\) & 17 \\ \hline \(L_{\mathrm{ON}}\) & 23 & \(L_{\mathrm{O}_{1}P}\) & 37 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The optimal results for the DVs. hinge is one of the common alternatives. Therefore, the DVs at this calculation step are the minimum thickness of the hinges (\(T_{1}\), \(T_{2}\), \(T_{3}\)), as shown in Figure 2.  
Before conducting the size optimization using the multi-objective NSGA-II optimization, in this section, the sensitivity of the DVs is analyzed to understand the influence of each variable. The knowledge gained from this could allow eliminating parameters that have little effect, and expanding the boundary of those that have more impact on the desired objective functions. In addition, the convergence speed of the objective functions is enhanced, which reduces the consumption of computer resources and processing time. Figure 2 shows all of the design parameters for the CM model, including the dimensions of the hinges (\(T_{i}\) and \(R_{i}\)) and the length of the rigid links (\(L_{i}\)).