## 5 Comparative Experimental Study
### Statistical Analysis of the CHT Behavior through Metrics
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**CHT** & \multicolumn{4}{c}{**Metric FP**} & \multicolumn{4}{c}{**Metric P**} \\ \hline  & \multicolumn{4}{c}{**Study Case**} & \multicolumn{4}{c}{**Study Case**} \\  & **1** & **2** & **3** & **1** & **2** & **3** \\ \hline FR & **1** & **1** & 0.9 & **0.053** & 0.7 & **0.12** \\ SR & 0.78 & 0.99 & 0.52 & 0.01 & 0.48 & 0.023 \\ \(\epsilon\)C & **1** & **1** & **0.98** & 0.04 & 0.08 & 0.047 \\ PF & **1** & **1** & 0.7 & 0.043 & **0.71** & 0.09 \\ \hline
**CHT** & \multicolumn{4}{c}{**Metric AFES**} & \multicolumn{4}{c}{**Metric SP**} \\ \hline  & \multicolumn{4}{c}{**Study Case**} & \multicolumn{4}{c}{**Study Case**} \\  & **1** & **2** & **3** & **1** & **2** & **3** \\ \hline FR / NSE & \(\mathbf{4.8\times 10^{6}/5}\) & \(2.2\times 10^{6}/2\) & \(2.391\times 10^{7}/5\) & \(\mathbf{6.1\times 10^{7}/5}\) & \(2.7\times 10^{6}/2\) & \(\mathbf{1.3\times 10^{8}/5}\) \\ SR / NSE & \(5.9\times 10^{5}/9\) & \(1.4\times 10^{6}/2\) & \(1.8\times 10^{7}/7\) & \(5.9\times 10^{6}/9\) & \(3\times 10^{6}/2\) & \(3.4\times 10^{8}/7\) \\ \(\epsilon\)C / NSE & \(4.5\times 10^{6}/7\) & \(1.9\times 10^{5}/6\) & \(\mathbf{2.386\times 10^{7}/5}\) & \(6.7\times 10^{7}/7\) & \(2.1\times 10^{6}/6\) & \(3.4\times 10^{8}/5\) \\ PF / NSE & \(4.9\times 10^{6}/6\) & \(\mathbf{2.3\times 10^{6}/1}\) & \(2.381\times 10^{7}/7\) & \(4.9\times 10^{7}/6\) & \(\mathbf{3.5\times 10^{6}/1}\) & \(1.3\times 10^{8}/7\) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Average metric values _FP_, \(P\), _AFES_, and _SP_ for each study case and for each constraint-handling technique included in the metaheuristic algorithms. executions required at least 200 evaluations to find a feasible solution (the feasible region). For this reason, the samples were not significant to determine a winner in the CHT in such a metric. In study case 2, the \(FR\) confirmed the superior performance in the \(EVALS\) metric. In study case 3, the \(\epsilon\)-constraint method improved the convergence to the feasible region in fewer generations. Finally, in the \(PR\) metric, it was confirmed in two study cases that the \(\epsilon\)-constraint method can improve the solutions through the feasible region when the generations occur, followed by the SR that won in only one study case. According to the results obtained by the previous analysis in the performance metrics, the following findings of the CHTs in the optimal synthesis of the lower limb rehabilitation mechanism synthesis were observed:  
* The FR had one of the best performance and the best one of the search for feasible solutions and satisfactory regions for the three study cases, respectively, because this constraint-handling technique has a high probability value in the FP and P metrics. The relationship between the AFES and P gave by SP is the best among the CHTs. Therefore, among the studied CHTs, the Feasibility Rules technique is the most reliable one for the synthesis of mechanisms for rehabilitation. * The PF can be considered the second option for solving mechanism synthesis for rehabilitation problems because it showed a suitable performance to find satisfactory solutions (P and SP metrics) compared to Stochastic-Ranking and \(\epsilon\)-constraint. This establishes acceptable reliability to find feasible solutions (FP metric). * \(\epsilon\)C method presented the best performance in the search for feasible regions as shown in FP. Inside the feasible region, this CHT produces improved solutions based on the PR metric. Nevertheless, the probability of this CHT to search for satisfactory solutions was low (P and SP metrics). This behavior is attributed to the gradient-based mutation where fast convergence to feasible local regions is promoted. * SR is the less reliable CHT because some algorithm executions can not find feasible solutions (FP metric). This presents a low performance in the search for suitable solutions (P metric).