## 4 Optimization of Path Generating Four-Bar Linkage
### Five Problems of the Dimensional Synthesis of the Path-Generating Four-Bar Linkages  
Several previous researchers have investigated these five representative synthesis problems using different algorithms. The comparative study of the results of the metaheuristic optimization methods (DE-gr, PSO, HPSO, EPSDE, and L-EPSDE) employed in this study to those of more than ten other algorithms utilized in previous studies will be discussed in this section. The statistical results (Best, Worst, Median, and Mean values, and SD) of the five metaheuristic methods applied to solve the five selected problems undergoing 100 experimental trials are shown in Table 2. The accuracy and efficiency (effectiveness) of those metaheuristic methods were compared using the SD, mean value, and convergence rate. In Table 2, the bold numbers on each column for all cases are the minimum values of the statistical results of 100 optimal solutions obtained using the five metaheuristic methods. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Optimization Method** & **PSO** & **HPSO** & **DE-gr** & **EPSDE** & **L-EPSDE** \\ \hline Number of particles or & \multirow{2}{*}{100} & \multirow{2}{*}{100} & \multirow{2}{*}{100} & \multirow{2}{*}{100} & \multirow{2}{*}{100} & \multirow{2}{*}{100} & \multirow{2}{*}{100} \\ population (\(N_{P}\)) & & & & & & \\ \hline Number of iterations (\(N_{linear}\)) & 1000 & 1000 & 1000 & 1000 & 1000 \\ \hline Minimum group (\(NP_{min}\)) & – & – & – & – & – & 5 \\ \hline Crossover method & – & \
\begin{tabular}{c} Single point- \\ uniform \\ \end{tabular} & Multi-point & Multi-point & Multi-point \\ \hline Crossover rate (\(C\tau\)) & – & – & 0.618 & 0.1–0.9 & 0.1–0.9 \\ \hline Mutation method & – & – & DE/best/1 &
\begin{tabular}{c} DE/best/2 \\ DE/rand/1 \\ DE/current-to- \\ rand/1 \\ \end{tabular} & DE/rand/1 \\ \hline Mutation factor (\(F\)) & – & – & 0.382 & 0.4–0.9 & 0.4–0.9 \\ \hline Selection method & – & – & Competition law & Competition law \\ \hline Learning factor (\(c_{1},c_{2}\)) & 2.05 & 2.05 & – & – & – \\ \hline Inertia weight (\(\omega\)) & 0–1 & 0–1 & – & – & – \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of five optimization methods. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{**Case 1: 5 Target Points, Non-Aligned Point Curve**} & \multicolumn{5}{c}{**Case 2: 18 Target Points, Irregular Closed Curve**} \\ \hline  & **Best** & **Worst** & **Median** & **Mean** & **SD** & **Best** & **Worst** & **Median** & **Mean** & **SD** \\ \hline PSO & 1.16E–05 & 1.92E–03 & 4.21E–05 & 1.12E–04 & 2.83E–04 & 1.92E–02 & 2.53E+00 & 4.67E–02 & 2.43E–01 & 6.33E–01 \\ \hline HPSO & 8.01E–06 & 6.37E–05 & 4.21E–05 & 3.95E–05 & 1.32E–05 & 1.09E–02 & 8.43E–01 & 4.64E–02 & 1.03E–01 & 1.50E–01 \\ \hline DE-gr & **7.42E–07** & 2.36E–05 & 7.42E–07 & 2.36E–06 & 2.35E–05 & 1.09E–02 & 2.53E+00 & 4.53E–02 & 2.32E–01 & 5.93E–01 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results of optimized values and statistical analyses of five representative cases. _Case 1:_ Five non-aligned points (5 target points, 6 design variables). This is a path generation problem with prescribed timing, whose coupler point must trace a path with five non-aligned points. This problem was originally proposed by [PERSON] and [PERSON] [39]. The four-bar mechanism exhibits a crank-fixed point in the coordinate system, and the direction of fixed link is collinear to the \(X\)-axis, that is \(\theta_{0}=0\), x\({}_{0}=0\), y\({}_{0}=0\).