## 4 Optimization of Path Generating Four-Bar Linkage
### Five Problems of the Dimensional Synthesis of the Path-Generating Four-Bar Linkages
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**0\(\sharp\)**} \\ \hline  & **CMDE** & **CS** & **DE** & **ADEL3** & **Multi-Scen** & **Gene** & **BAS** & **And-SC** & **CSA** & **ODSEA** & **HCDI** & **L-EPSDE** \\  & [141] & [33] & [63] & [167] & **Gene** & [163] & [167] & [92] & [71] & **CP** [73] & **Trivial Work** \\ \hline \(r_{0}\) & 0.20766 & 0.20766 & 0.3030 & 1.20044 & 0.309593 & 1.803078 & 0.25700 & 0.2409 & 0.2560 & 0.2044 & -14.86045 & 0.20444 \\ \hline \(r_{0}\) & 0.15845 & 0.15845 & -0.0874 & -0.01802 & -0.107000 & 0.4912575 & 0.154427 & 0.1429 & 0.2214 & 0.1480 & 20.1402 & 0.1406 \\ \hline \(\theta_{0}\) & 0.28833 & 0.28802 & 0.1110 & 6.22754 & 0.528027 & 3.40143 & 0.256040 & 0.2878 & 0.2371 & 0.3366 & 2.52861 & 0.33656 \\ \hline \(\theta_{2}\) & 0.09155 & 0.09156 & 0.8721 & 1.189200 & – & 3.90796 & 1.176411 & 0.8822 & 1.1692 & 0.8377 & 5.04422 & 0.83707 \\ \hline Error & **9.0882-03** & **9.0882-03** & 1.8841-02 & 2.7786-02 & 8.0771E-02 & 1.21E-02 & **9.0882-03** & 1.1962-02 & **4.082-03** & **9.082-03** & **9.082-03** \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._  
Figure 13: Convergence rate of five metaheuristic methods for Case 2. HTRCA methods (Table 5), and inferior to those of the DE (reduced number of design parameters) [63], IOA-Sat, BAS, and HCDJ methods (Table 5) and the ATLBO-DA method ([PERSON] et al. [73]). The trajectories of the coupler point obtained by five metaheuristic algorithms in Case 3 are shown in Figure 14, in which the curves of DE-gr, EPSDE, and L-EPSDE coincide one another, as their optimal solutions were same. The trajectory of the coupler point and optimal mechanism obtained by the L-EPSDE method is shown in Figure 15. The convergence rates of the five metaheuristic methods are shown in Figure 16, which indicated that the convergence rate of L-EPSDE was faster than those of PSO, EPSDE, and HPSO, but was slightly slower than DE-gr. _Case 4:_ Straight-line segment curve (6 target points, 15 design variables)  
This is a synthesis problem for the path generation without prescribed timing where six target points are specified in a vertical line segment. This problem was first presented by [PERSON] [41] and has been investigated by several researchers. The problem can be defined as follows:  
The coordinates of six target coupler points:  
\[\left\{C_{d}^{i}\right\}=\left\{\text{(20, 20), (20, 25), (20, 30), (20, 35), (20, 40), (20, 45)}\right\}\]  
The vector of fifteen design variables:  
\[\mathbf{X}=\left[r_{1},r_{2},r_{3},r_{4},r_{cx},r_{cy},\ x_{0},\ y_{0},\ \theta_{0},\theta_{2}^{1},\theta_{2}^{2},\theta_{2}^{3},\theta_{2}^{4},\theta_ {2}^{5},\theta_{2}^{6}\right]\,.\]  
The limits of the variables:  
\[r_{1},r_{2},r_{3},r_{4}\ \in\ [5,\ 60];\ \ r_{cx},\ r_{cy},\ x_{0},\ y_{0}\in[-60, \ 60];\theta_{0},\ \theta_{2}^{i}(i=1,\ 2,\ \ldots\,6)\ \in\ [0,\ 2\pi].\]  
The comparison of the optimal synthesis results of the DE-gr and L-EPSDE methods for Case 4 to those of the PSO, HPSO, and EPSDE methods and sixteen other methods are shown in Tables 2 and 6a,b, respectively. The optimal value (4.71E\(-\)04) of L-EPSDE was superior to those of the PSO and HPSO methods (Table 2), the GA-CSP, DE, GA-DE, HPSO, ICA, and HTRCA methods (Table 6a), the SAP-TLBO, TLBO, ImHS, GSEF-IAA, and RBDO-MCS methods (Table 6b), and the ATLBO-DA method ([PERSON] et al. [73]), but was inferior to those of the MUMSA, IOA\({}^{\text{S-at}}\), and MKH methods (Table 6a), and the HLIDE, ODSRA + CP, and HCDJ methods (Table 6b). In addition, although the optimal solutions (2.32E\(-\)04 and 2.59E\(-\)04) of both the DE-gr and EPSDE methods were slightly superior to that of the L-EPSDE method, the mean value (0.324) and SD (0.815) of L-EPSDE were the lowest, indicating that L-EPSDE exhibited the best accuracy and efficiency.