## 5 Comparative Experimental Study
### Evaluation of the Obtained Mechanisms  
In this section, the best solutions obtained by the feasibility rules (according to the overall performance presented in Table 8), and the worst constraint-handling technique are presented. In study case 1, the best constraint-handling technique was the feasibility rules, and the worst was stochastic-ranking. The Computer-Aided Design (CAD) and the generated trajectory of the best mechanisms obtained by both constraint-handling techniques are presented in Figure 8, where the objective function values of the mechanisms obtained by the feasibility rules (with the algorithm DECR) is \(J=0.002098\), and by stochastic-ranking (with the algorithm DECR1E) is \(J=0.003726\). Those values are given in Table A7 of Appendix C.  
According to the design parameters obtained in both CHTs (see Table A26 of Appendix F), the feasibility rules present a Pearson correlation coefficient of 0.9939 with respect to the stochastic-ranking. This indicates a high degree of similarity (see Figure 8). On the other hand, the performance of the design goals is presented in Figure 8,d. Figure 8 shows the trajectory of the mechanisms, the current, and the desired precision points. The average error between the desired precision points and the current ones is \(J_{1}=4.7095\times 10^{-4}\) [m] by Feasibility Rules (FR) and \(J_{1}=4.6009\times 10^{-4}\) [m] by Stochastic-Ranking (SR). Figure 8 shows the angles obtained by the crank and the desired angles, where the average angular error is \(J_{2}=1.6270\) [rad] by feasibility rules, and \(J_{2}=3.2663\) [rad] by stochastic-ranking. We observed that the design goals in both mechanisms related  
to \(f_{1}\) and \(J_{2}\), present a trade-off, i.e., in one goal, it is better but in the other not. Then, solutions obtained by the feasible rules and the stochastic-ranking are suitable for the rehabilitation mechanism. Nevertheless, the feasible rules provide more reliable results through different algorithm executions and with a better overall performance (related to the objective function \(J\)). In study case 2, the feasibility rule was the best constraint-handling technique, and the worst was the epsilon-constraint. The CAD and the generated trajectory of the best mechanisms obtained by both constraint-handling techniques are presented in Figure 9. The objective function values given by the mechanisms obtained by the feasibility rules (DEB1B) is \(J=0.6557\) and by epsilon-constraint (DEB1E) is \(J=0.7588\). Those values are given in Table 10 of Appendix C.  
According to the design parameters obtained in both CHTs (see Table 11 of Appendix F), the feasibility rules present a Pearson correlation coefficient of \(0.9991\) with respect to the epsilon-constraint. This correlation is larger than the obtained mechanisms in the previous study case 1, which implies a fairly strong relationship. In order to highlight the difference, the performance of the design goals is presented in Figure 9c,d for the cam shapes and the angles obtained by the crank, respectively. Figure 8: CAD representation and trajectory generated of the mechanisms (design goals) given by the best and the worst constraint-handling technique for solving study case 1. The normalized average error between the desired crank angles and the current ones is \(\hat{f}_{1}=1.2217\) by feasibility rules and \(\hat{f}_{1}=1.7122\) by epsilon-constraint. The normalized base radius of the cam is \(\hat{f}_{2}=0.4137\) by FR, and \(\hat{f}_{2}=0.3502\) by epsilon-constraint. It is observed again that the design goals present a trade-off, such that both mechanisms are suitable for the rehabilitation routine, and it depends on the application that one selects. However, the feasible rules can search for solutions with a similar performance through different executions and with better quality in the overall performance in the weighted objective function \(J\). Finally, for study case 3, the best constraint-handling technique was the feasibility rules, and the worst one was stochastic-ranking.