## 5 Simulation and Experimental Results
### Simulation Experiments
Boxplot graphics are used as a visual aid to identify the algorithm with the lowest variability, characterized by lower dispersion and lower fitness function values. A lower fitness value corresponds to smaller errors in both position and orientation. Moreover, lower fitness values indicate that joint values are feasible, as no penalties are applied. The results are presented using statistical measures such as mean, standard deviation, and the best and worst outcomes of each algorithm, all of which are summarized in tables. In the following paragraphs, we consider the term'standard algorithm' for algorithms that have been applied as described by their author, while the term 'hybrid algorithm' is used for algorithms that include the Jacobian pseudoinverse operations. Figure 5 presents a comparison of data dispersion between standard and hybrid (H) algorithms using a boxplot. It is evident that the standard algorithms, Figure 4(a), exhibit higher dispersion compared to the hybrid algorithms, Figure 4(b). The DE and IWO hybrid algorithms demonstrate similar performance. Table 5 provides a comparison of the statistical measures obtained by the algorithms, showing that the DE hybrid algorithm (DE-H) achieves the best performance in solving the inverse kinematics of the Puma-560 robot. Figure 6 illustrates the simulation results for the 7 DOF Baxter robot. Figure 5(a) shows the data dispersion of standard algorithms, while Figure 5(b) demonstrates that the hybrid DE algorithm achieved the best result. Table 6 compares the statistical measures obtained by the algorithms in solving the inverse kinematics of the Baxter robot. Similar to the previous case, the hybrid DE algorithm (DE-H) exhibits the best performance. Figure 4: Randomly generated set of values. \begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline  & **BA** & **BA-H** & **DE** & **DE-H** & **PSO** & **PSO-H** & **ICA** & **ICA-H** & **IWO** & **IWO-H** \\ \hline mean & 0.05588 & 0.06406 & 0.30928 & **5.8349 \(\times 10^{-4}\)** & 0.59901 & 0.10483 & 0.21649 & 0.21025 & 0.03125 & 0.02216 \\ std & 0.06336 & 0.08  
Finally, Figure 7 presents the simulation results of KUKA iwa robot. Figure 7 displays a boxplot of the error obtained by the standard algorithms, while Figure 7b shows a boxplot of the performance of the hybrid algorithms. The distribution comparison indicates that hybrid algorithms perform better, with DE and IWO demonstrating similar performance. Table 7 reveals that DE hybrid algorithm (DE-H) achieves the most significant improvement for the KUKA iwa Robot. \begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline  & **BA** & **BA-H** & **DE** & **DE-H** & **PSO** & **PSO-H** & **ICA** & **ICA-H** & **IWO** & **IWO-H** \\ \hline mean & 0.10814 & 0.08995 & 0.37702 & **2.2135 \(\times\)10\({}^{-3}\)** & 0.50789 & 0.08282 & 0.29066 & 0.28297 & 0.01056 & 0.01515 \\ std & 0.07306 & 0.09374 & 0.19066 & **8.3733 \(\times\)10\({}^{-3}\)** & 0.09158 & 0.16858 & 0.17908 & 0.18431 & 0.02845 & 0.04382 \\ best & 6.227 \(\times\)10\({}^{-5}\) & 2.023 \(\times\)10\({}^{-16}\) & 0.00146 & 2.0507 \(\times\)10\({}^{-15}\) & 0.28868 & 1.182 \(\times\)10\({}^{-16}\) & 0.00925 & 0.01281 & 0.00052 & **1.034**\(\times\)10\({}^{-16}\) \\ worst & 0.32282 & 0.34745 & 0.71676 & **0.05046** & 0.71212 & 0.68462 & 0.88735 & 0.7619 & 0.21314 & 0.31733 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of the statistical performance of standard and hybrid algorithms in solving the inverse kinematics of the Baxter robot. Bold text indicates better fitness values. Figure 5: Comparison of standard and hybrid algorithms in solving inverse kinematics of the Puma-560 robot. (**a**) Standard algorithms. (**b**) Hybrid algorithms. Figure 6: Comparison of standard and hybrid algorithms in solving inverse kinematics of the Baxter robot. (**a**) Standard algorithms. (**b**) Hybrid algorithms.