## 4 Optimization of Path Generating Four-Bar Linkage
### Five Problems of the Dimensional Synthesis of the Path-Generating Four-Bar Linkages
The problem can be defined as follows:  
The coordinates of five target points:  
\[\left\{C_{d}^{i}\right\}=\left\{\left(3.000,\ 3.000\right)\,,\ \left(2.759,\ 3. 363\right)\,,\ \left(2.372,\ 3.663\right)\,,\left(1.890,\ 3.862\right)\,,\ \left(1.355,\ 3.943\right)\right\}\]  
The input crank positions for prescribed timing:  
\[\left\{\theta_{2}^{i}\right\}=\left\{\pi/6,\pi/4,\pi/3,5\pi/12,\ \pi/2\right\}\, \left(i=1,\ 2,..,\ 5\right)\]  
The vector of six design variables:  
\[\mathbf{X}=\left[r_{1},r_{2},r_{3},r_{4},r_{cx},r_{cy}\right]\]  
The limits of the design variables:  
\[r_{1},r_{2},r_{3},r_{4}\ \in\ \left[1/20,\ 12\right],\ \ r_{cx},r_{cy}\ \in\ \left[-15,\ 1 5\right].\]  
The synthesized optimal parameters and the corresponding values of the objective function for each method are listed in Table 3. In this work, the design domain was medium-sized with five target points. It is important to note that there was a slight difference in the limits of design variables in previous literature, such as GA-CSP [41] and IOAS\({}^{\text{Sat}}\)[53] methods, \(r_{1}\), \(r_{2}\), \(r_{3}\), \(r_{4}\ \in\ \left[0,\ 50\right]\), MUMSA [48] and HLIDE [66] methods, \(r_{1}\), \(r_{2}\), \(r_{3}\), \(r_{4}\ \in\ \left[0,\ 5\right]\), \(r_{cx}\), \(r_{cy}\ \in\ \left[-5,\ 5\right]\). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Case 1: 5 Target Points, Non-Aligned Point Curve**} & \multicolumn{4}{c}{**Case 2: 18 Target Points, Irregular Closed Curve**} \\ \hline  & **Best** & **Worst** & **Median** & **Mean** & **SD** & **Best** & **Worst** & **Median** & **Mean** & **SD** \\ \hline EPSDE & **7.42E\(-\)07** & **7.42E\(-\)07** & **7.42E\(-\)07** & **7.42E\(-\)07** & **9.64E\(-\)20** & **9.91E\(-\)03** & **2.60E\(-\)01** & 1.14E\(-\)01 & 1.18E\(-\)01 & **5.88E\(-\)02** \\ \hline L-EPSDE & **7.42E\(-\)07** & **7.42E\(-\)07** & **7.42E\(-\)07** & **7.42E\(-\)07** & 1.08E\(-\)19 & **9.91E\(-\)03** & 4.53E\(-\)01 & **4.53E\(-\)02** & **7.77E\(-\)02** & 8.08E\(-\)02 \\ \hline \multicolumn{10}{c}{**Case 3: 6 target points, semi-circular arc curve**} & \multicolumn{4}{c}{**Case 4: 6 target points, straight-line segment curve**} \\ \hline  & Best & Worst & Median & Mean & SD & Best & Worst & Median & Mean & SD \\ \hline PSO & 3.26E+00 & 1.91E+02 & 7.55E+00 & 1.21E+01 & 2.16E+01 & 5.74E\(-\)03 & 1.68E+01 & 2.99E+00 & 4.00E+00 & 3.38E+00 \\ \hline HPSO & 2.95E+00 & 1.40E+02 & 8.27E+00 & 1.55E+01 & 1.89E+01 & 4.54E\(-\)03 & 1.43E+01 & 6.32E+00 & 6.12E+00 & 3.09E+00 \\ \hline DE-gr & **2.58E+00** & 7.16E+01 & 4.67E+00 & 1.02E+01 & 1.62E+01 & **2.32E\(-\)04** & 2.50E+01 & 2.36E+00 & 3.53E+00 & 4.10E+00 \\ \hline EPSDE & **2.58E+00** & **1.37E+01** & **2.58E+00** & 3.47E+00 & **1.90E+00** & 2.59E\(-\)04 & 8.84E+00 & **1.00E\(-\)01** & 1.17E+00 & 2.43E+00 \\ \hline L-EPSDE & **2.58E+00** & 7.16E+01 & **2.58E+00** & **2.91E+00** & 5.26E+00 & 4.71E\(-\)04 & **7.12E+00** & 1.09E\(-\)01 & **3.24E\(-\)01** & **8.15E\(-\)01** \\ \hline \multicolumn{10}{c}{**Case 5: 10 target points, elliptical path curve**} \\ \hline  & & & & Best & Worst & Median & Mean & SD \\ \hline  & & & & PSO & 2.30E\(-\)01 & 3.44E+02 & 9.95E+01 & 1.21E+02 & 9.81E+01 \\ \cline{2-6}  & & & & HPSO & **5.42E\(-\)03** & 2.92E+02 & 1.71E+01 & 4.96E+02 & 7.35E+01 \\ \cline{2-6}  & & & & & & & & \\ \cline{2-6}  & & & & & & & & & \\ \cline{2-6}  & & & & & & & & & \\ \cline{2-6}  & & & & & & & & & \\ \cline{2-6}  & & & & & & & & & \\ \cline{2-6}  & & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Cont. The comparison of the optimal solutions of EPSDE and L-EPSDE for Case 1 to those of the three other metaheuristic optimization algorithms and ten other methods are shown in Tables 2 and 3, respectively.