## 4 Optimization of Path Generating Four-Bar Linkage
### Five Problems of the Dimensional Synthesis of the Path-Generating Four-Bar Linkages
The comparison results revealed that the optimal value (7.42E\(-\)07) of L-EPSDE was the same as those of the EPSDE and DE-gr (Table 2) and HLIDE methods (Table 3), slightly superior to those of the BAS, ODSRA + CP, and REA methods (Table 3), and superior to those of the PSO, HPSO (Table 2) and GA-KK, Exact gradient, GA-CSP, MUMSA, IOAS-at, and HCDI) methods (Table 3). However, based on the SD and mean value in Table 2, both the L-EPSDE and EPSDE were superior to DE-gr. The trajectories of the coupler point obtained by the five metaheuristic algorithms in Case 1, simulated by a MATLAB coded program, are shown in Figure 8, in which the curves of HPSO, EPSDE, and L-EPSDE coincide with one another, as their optimal solutions were the same. The trajectory of the coupler point and the optimal mechanism obtained by L-EPSDE method are shown in Figure 9. The convergence rates of the five metaheuristic methods are shown in Figure 10. The HPSO method exhibited the best convergence rate to the optimal solution and almost achieved an optimal fitness value at the eleventh generation, and the convergence rate of L-EPSDE was slightly slower than PSO, HPSO, EPSDE, and DE-gr. However, all five metaheuristic optimization methods converged rapidly (under 20 iterations) because the number of design variables was six, which is easier than four other cases. _Case 2:_ Irregular closed curve (18 target points, 10 design variables). This is a representative problem for the synthesis of path generation with prescribed timing, where 18 target points are specified in an irregular closed curve. This well-known problem was originally proposed by [PERSON] and [PERSON] [39], and has been solved by numerous researchers. The problem is defined as follows:  
The coordinates of eighteen target points (unit: mm):  
\[\{C_{d}^{i}\}= \{(0.5,\,1.1)\,,\,(0.4,\,1.1)\,,\,(0.3,\,1.1)\,,(0.2,\,1.0)\,,\,(0.1, 0.9)\,,\,(0.05,\,0.75),(0.02,\,0.6)\,,\,(0,\,0.5)\,,\,(0,\,0.4)\,,\] \[(0.03,\,0.3)\,,(0.1,\,0.25)\,,\,(0.15,\,0.2)\,,\,(0.2,\,0.3)\,,\,(0,\,0.4)\,,\,(0.4,\,0.5)\,,(0.5,\,0.7)\,,\,(0.6,\,0.9)\,,\,(0,\,0.6,\,1.0)\}\]  
The first input crank position, \(\theta_{2}^{1}\), is a design parameter and prescribed timing with:  
\[\theta_{2}^{i}=\theta_{2}^{1}+\frac{\pi}{9}(i-1),\,\,i=1\,\,\text{to}\,\,18.\]  
Figure 10: Convergence rate of the five metaheuristic methods for Case 1. Figure 9: Trajectory of the coupler point of Case 1. The vector of ten design variables:  
\[\mathbf{X}=\left[r_{1},r_{2},r_{3},r_{4},r_{cx},r_{cy},x_{0},y_{0},\theta_{0}, \ \theta_{2}^{1}\right]\]  
The limits of the design variables:  
\[r_{1},r_{2},r_{3},r_{4}\in[0,\ 12];\ \ x_{0},y_{0},r_{cx},r_{cy}\in[-15,\ 15],\ \theta_{0},\ \theta_{2}^{1}\in[0,\ 2\pi]. \tag{32}\]  
In this study, the design domain exhibits a large size with 18 target points. There are slight differences in the limit of design parameters in various studies, such as the GA-CSP [41], GA-DE [47], MUMSA [48], IOA\({}^{\text{S-at}}\)[53], HCDJ [77] methods, \(r_{1},r_{2},r_{3},r_{4}\in[0,\ 50]\); \(x_{0},y_{0},r_{cx},r_{cy}\in[-50,50]\), DE-SRT [56] and BAS [37] methods, \(r_{1},r_{2},r_{3},r_{4}\in[0,5]\); \(r_{cx},r_{cy}\), \(x_{0},y_{0}\in[-5,5]\). Hereafter, in the following cases, the existence of difference in the limits of design variables will not be mentioned again. The comparison of the optimal solution of L-EPSDE for Case 2 to those of the PSO, HPSO, DE-gr, and EPSDE algorithms and twenty-two other methods are shown in Tables 2 and 4a,b, respectively.