## 5 Results and Discussion
### Analysis of Results among DE Variants
This may be caused by the fact that designs are generally aimed to meet in the best possible way some performance criteria, such as accuracy in developing a specific trajectory without limiting the possible solutions. At the same time, the use of prescribed time restricts the alternatives that can be generated, and it also requires a priori knowledge of the synthesized mechanism that is expected to be obtained. * It is possible to verify in Figures 2-5 that the best mechanisms synthesized with the most promising DE variant rand/1/exp could generate trajectories that were accurate to the precision points with or without prescribed time. Some comments are given for each specific case study: For C1, it is easy to observe in Figure 2 that the best trajectories generated by the DE variants showed remarkable differences. This is because the precision points for this case did not form a closed trajectory, and there was no prescribed time, so different mechanisms could generate trajectories with segments that were similar to the one described with these points. In the case C2, Figure 3 illustrates that the mechanisms synthesized with all the DE variants followed trajectories with imperceptible differences, except for current-to-best/1, whose trajectory hardly differed from the rest. With respect to C3, Figure 4 indicates that only the variants rand/1/exp and best/1/bin could generate trajectories that passed close to all precision points. The other variants deviated from the desired trajectory at some points. In turn, Figure 5 shows that the variants rand/1/exp, best/1/bin, best/1/exp, and rand/2/dir developed similar trajectories that passed near most of the precision points. In contrast, the rest of the variants generated trajectories that passed near some of them. * Regarding the execution time, all the DE variants completed the mechanism synthesis process in relatively short and affordable times (on average, for the most challenging case, the execution time was 5.4437 (s)), as shown in Tables 8-11, which highlights the efficiency of the algorithm in solving problems of this type. The execution times in these tables show little differences between the variants, which was partly due to the processes carried out in the operative system while the mechanism optimization process was executed. Furthermore, based on the information in these tables, it is possible to observe that the execution time of the algorithms was proportional to the number of precision points considered in each case study (see Table 2). * Regarding the precision of the mechanisms generated with DE, Table 12 indicates that the error between the precision points and the trajectories developed by the best mechanisms, obtained with the most promising DE variants, did not exceed the value of 6.1321 \(\times\) 10\({}^{-1}\) (mm). This highlights the high performance of the alternatives synthesized by DE. Overall Performance of the DE Variants among the Studied Cases  
At this point, it is essential to emphasize that DE is a stochastic optimization technique, so each independent execution of its variants may generate different results with a probability distribution that is different from the normal one. To ensure that the results obtained so far were significant, they were subjected to the Wilcoxon signed-rank test [47]. This test allowed us to detect significant differences between the distributions of the results obtained by each pair of DE variants. For this purpose, a statistical significance of \(\alpha=5\%\) was established. Table 13 shows the Wilcoxon test results for the samples of the best values of the objective function \(J(\vec{p})\) obtained from the 30 independent runs of all the DE variants for each case study. This table presents the probability values (\(p\)-value) obtained as a result of each pairwise test, which were labeled with (\(+\)) when the first variant was better than the second, with (\(-\)) when the second outperformed the first, and with (\(\approx\)) when there was no significant difference between the two.