# A Multi-Objective Modified PSO for Inverse Kinematics of a 5-DOF Robotic Arm
## 3 Multi-Objective Inverse Kinematics Problem Statement
### Inverse Kinematics as a Multi-Objective Problem
Figure 6: (a) Position error of each randomly generated point, (b) orientation error of each randomly generated point. ## 5 Nonparametric Statistical Comparatives

To compare the relative performances of the proposed m-PSO to classical PSO and also to related and close techniques, the nonparametric Wilcoxon statistical analysis was used based on the single target point, and using the same configuration for all methods in terms of number of search agents, which were 50, and in terms of maximum iteration, which was fixed at 500. Tests were based on Matlab (2019) simulations, with an AMD A6 processor and 4 GB of RAM. Wilcoxon nonparametric analysis was conducted based on 30 independent executions of the compared metaheuristics. ### Mono-Objective m-PSO Comparative Analysis

The mean fitness evolutions on the compared mono-objective techniques for 500 iterations are visible in Figure 8a. Results evidently show a different evolution dynamic of the m-PSO metaheuristic, essentially, when compared to classical PSO, SSA, and CSA. The obtained position errors for the 30 tests appear in Figure 8b, and show clearly that the m-PSO is returning a more precise inverse kinematics solution than its concurrent. Figure 7: Circular path tracking of 50 points: (**a**) The input trajectory and the tracked trajectory by robotic arm, (**b**) Angle values resulted by MO-m-PSO for each point of the trajectory, (**c**) 3D representation of the robotic arm and the circular path. These results were confirmed by the Wilcoxon signed-rank test reported in Table 5, where we can see that, globally, the position error obtained based on the m-PSO is better than the classical PSO such as in [32; 33], salp swarm algorithm, SSA [34], and crow search algorithm [35]. ### Multi-Objective m-PSO, MO-m-PSO, Comparative Analysis

In order to benchmark the performance of the proposed method compared to existing multi-objective approaches used for inverse kinematics, MO-m-PSO was compared to multi-objective PSO for inverse kinematics, MO-PSO, and in [36], multi-objective CSA [37] and multi-objective JAYA algorithm (MO-JAYA) [38]. Comparatives are based on 30 random points RUNS and using the Wilcoxon nonparametric method. That target positions were set randomly, while the orientation was fixed to 90\({}^{\circ}\) for the end-effector. Comparisons showed that the proposal ranking is better than the MO-PSO and the MO-JAYA, while very close to MO-CSA, the worst. Detailed results appear in Table 6, and all tests satisfied the null hypothesis condition with \(\rho<0.05\). ## 6 Experimental Validation on a 3D-Printed ARM

The experimental investigation consists of printing, assembling, and controlling a 3D-printed arm using inverse solutions generated by the m-PSO and M0-m-PSO. The 3D-printed arm is for educational use only and serves as an integration prototype. The experimental setup is shown in Figure 9. It includes the following:

* Matlab software;
* An Arduino Uno board;
* The 3D-printed arm. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Data** & \(R^{+}\) & \(R^{-}\) & \(\rho\) & **Best** \\ \hline MO-m-PSO & 486 & 778 & \(2.31\times 10^{-3}\) & MO-m-PSO \\ MO-m-PSO & 195 & 1069 & \(6.87\times 10^{-4}\) & MO-m-PSO \\ MO-m-PSO & 663 & 661 & \(1.02\times 10^{-4}\) & MO-m-PSO \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Statistical results of MO-M-PSO for IK problem. Figure 8: (**a**) Comparative mean fitness evolutions of m-PSO, PSO, CSA, and SSA. (**b**) Comparative position errors over 30 tests of m-PSO, PSO, CSA, and SSA. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Data 1** & **Data 2** & \(R^{+}\) & \(R^{-}\) & \(\rho\) & **Best** \\ \hline m-PSO & PSO & 0 & 465 & \(3.01\times 10^{-11}\) & m-PSO \\ m-PSO & SSA & 30 & 435 & \(5.57\times 10^{-10}\) & m-PSO \\ m-PSO & CSA & 0 & 465 & \(3.01\times 10^{-11}\) & m-PSO \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Nonparametric comparatives with PSO, SSA, and CSA. The inverse kinematics solver runs on Matlab software with a PC powered by i7 Intel processor and a RAM of 12 GB.