# Grasp Planning Based on Metrics for Collaborative Tasks Using Optimization
## 3 Multi-Objective Optimization
Finally, computational time is critical for practical implementation where smaller values are desirable. As mentioned before, we need to use well-known problems (test functions) [43; 44] in order to evaluate the performance of these metaheuristic algorithms. Some common testing functions are Schaffer (SCH) [47], Kursawe (KUR) [48], Fonseca (FON) [49], ZTD1 and ZTD6 [50]. These testing functions allow one to evaluate the most common characteristics of optimization algorithms such as convergence, robustness, etc. [51; 52]. Since each of the theoretical Pareto fronts of each function is known, it is possible to calculate SP and GD and graphically corroborate them. Table 2 summarizes the most common test functions used. But so far, there are no standard problems to evaluate the performance of HRC grasp planners. Mean-value results for SP are shown in Table 1 (See Table 3 for complete statistics). MOPSO has the best average for all test functions except for SCH, where the result is slightly better than GWO. As can be seen in Table 1, MOPSO has the best distribution and propagation of solutions, which does not mean that NSGA-II and GWO are useless, but in the context of this research, MOPSO will allow finding more than one grasping solution. Table 2 shows the mean-value results for GD (see Table 4 for complete statistics). MOPSO has the best average for this indicator in all functions except for ZTD1 where MOGWO is the best. Equally, the worst average among all of them is NSGA-II, excluding ZTD6 where it is slightly better than MOGWO, but not more than MOPSO. Table 2 shows that MOPSO will generate solutions that are much closer to the theoretical Pareto front; therefore, points of the manipulator solution set will converge as well as possible. More data to analyze are presented in Table 3. A computational time summary of the algorithms is presented. This test used a computer with AMD Ryzen 7, 4800H @ 2.90 GHz processor, 16 GB RAM, and Windows 10 operating system. This index shows that MOPSO has the best speed execution of all the test functions. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Function** & **Statistic** & **NSGA-II** & **MOGWO** & **MOPSO** \\ \hline SCH & Mean & \(4.21\times 10^{-2}\) & \(\mathbf{2.46\times 10^{-2}}\) & \(2.78\times 10^{-2}\) \\ \hline FON & Mean & \(9.46\times 10^{-1}\) & \(1.52\times 10^{-2}\) & \(\mathbf{5.16\times 10^{-3}}\) \\ \hline ZTD6 & Mean & \(2.14\times 10^{-1}\) & \(1.1\times 10^{-1}\) & \(\mathbf{1.56\times 10^{-2}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Some Statistical Results for SP dimensionless. The indicators showed MOPSO as the best performance in convergence and dispersion of the solutions. Figures 5-7 are related with GD and SP indicators and show the theorical Pareto front against a real solution set for the functions SCH, FON and ZTD6. \(f_{1}\) and \(f_{2}\) are the functions of each test function delimited as in Table 2. Red points represent a theoretical solution set and the other different color points are the solution set for each tested algorithm. MOGWO and MOPSO have similar behaviours where both of them seem to be very close to the Pareto front with a little better convergence for MOPSO, supporting GD mean values of Table 2, and the same can be concluded for SP mean values because spread or distribution is very good and is similar for both. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Function** & **Statistic** & **NSGA-II** & **MOGWO** & **MOPSO** \\ \hline SCH & Mean & \(1.34\times 10^{-3}\) & \(9.89\times 10^{-4}\) & \(\mathbf{9.43\times 10^{-4}}\) \\ \hline FON & Mean & \(6.79\times 10^{-2}\) & \(1.52\times 10^{-3}\) & \(\mathbf{4.11\times 10^{-4}}\) \\ \hline ZTD6 & Mean & \(2.5\times 10^{-2}\) & \(3.16\times 10^{-2}\) & \(\mathbf{3.83\times 10^{-3}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Some Statistical Results for GD dimensionless. Figure 5: Pareto Front for SCH Function. (**a**) MOPSO; (**b**) MOGWO; (**c**) NSGA-II.