## 4 Results
### Algorithm Performance Analysis
#### 4.2.2 Confidence Intervals and Inferential Statistics  
This section confirms the best algorithm behavior to solve each study case considered in this work. The best algorithm behavior is obtained by analyzing the sample (thirty best values of the objective function) obtained in each algorithm execution by 95% confidence intervals [61] and using the non-parametric Friedman test [56]. The 95% Confidence Interval (CI) [61] is a range of values for a selected sample of a study, where, with a 95% confidence, the interval contains the population's true mean. Therefore, the confidence interval values closest to zero indicate the most promising algorithm due to the high probability that the mean behavior of the algorithm is inside such interval. Moreover, in what follows  
The 95%-confidence Friedman test is carried out in those three outstanding algorithms per each trajectory, i.e., in R1B-NCH, R1B-PF, and MUMSA-PF for trajectory 1; and, in R1B-FR, R1B-PF, and R1B-NCH for trajectory 2. According to the Friedman test in those groups of three algorithms, there is no statistically significant difference in the comparisons because the returned \(p\)-value is greater than 0.05. It is also confirmed in the multiple comparisons with the Bonferroni post-hoc error correction method of the Friedman test given in Table 10 that the associated \(p\)-value is greater than 0.05, indicating the no confirmation of the superiority of one algorithm in the comparison. In this paper, a tie is declared in the comparison when that situation occurs. Thus, in all comparisons presented in Table 10, the algorithms draw. In the mechanism synthesis problem, it is common to obtain the optimal solution by setting different (usually thirty) executions to the same problem [47]. However, there may be significant differences among the best solutions obtained through different executions, and the most promising solution obtained by the optimizer in the mechanism synthesis problem is given through the best solution among executions [62]. Figure 7: Graphical representation of confidence intervals for algorithms solving the study case 1 in both trajectories. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**Trajectory 2**} \\ \hline
**Limits** & **R1B-FR** & **R1B-SR** & **R1B-eC** & **R1B-PF** & **R1B-NCH** \\ Low & 0.0180 & 0.0216 & 0.0261 & 0.0205 & 0.0175 \\ Up & 0.0291 & 0.0321 & 0.0330 & 0.0297 & 0.0292 \\ \hline  & **GA-FR** & **GA-SR** & **GA-eC** & **GA-PF** & \\ Low & 0.0313 & 0.0415 & 0.0360 & 0.0295 & \\ Up & 0.0391 & 0.0490 & 0.0434 & 0.0396 & \\ \hline  & **PSO-FR** & **PSO-SR** & **PSO-eC** & **PSO-PF** & \\ Low & 0.0346 & 0.0315 & 0.0324 & 0.0304 & \\ Up & 0.0399 & 0.0391 & 0.0397 & 0.0386 & \\ \hline  & **MUMSA-FR** & **MUMSA-SR** & **MUMSA-eC** & **MUMSA-PF** & \\ Low & 0.0285 & 0.0268 & 0.0220 & 0.0240 & \\ Up & 0.0356 & 0.0345 & 0.0315 & 0.0298 & \\ \hline \hline \end{tabular}
\end{table} 
Table 9: _Cont._
Then, to know whether the proposed R1B-NCH empirically presents a high probability of finding the best results in the kinematic synthesis problem for lower limb rehabilitation, the following is defined: As the confidence interval indicates whether these executions are repeated in several tests, the proportion of calculated 95% confidence intervals that is included the population's true mean value would tend toward 95%; then, the confidence interval closest to zero will indicate the population's true mean value is closer to the reduction of the objective function, and, hence, it is assumed that this interval presents a high probability of granting the best solution (minimum value) through thirty executions of the algorithm (executions commonly used for finding the best solution in the mechanism synthesis problem). In order to verify the previous assumption, the other four sets of thirty executions are performed for the two groups of three algorithms (R1B-NCH, R1B-PF, and MUMSA-PF for trajectory 1; and R1B-FR, R1B-PF, and R1B-NCH for trajectory 2).