# 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
This test is conducted using the following steps:

* A target point is fixed by the user, it mimics the simulated random point test, then the proposed metaheuristic solver generates an online inverse solution and generates the joints control using the Arduino pulse wave modulation outputs, PWM. * The test is video reported, and the Kinovea video analysis software is used to gather the practical joints motions as well as the real target point, as in Figure 9.
* This test is repeated 20 times and the position errors are reported to evaluate the repeatability precision. Figure 9 shows the target point, which is \(X_{t}=[18,10,25]\), in centimeters, and the obtained solution with Kinovea annotations for analysis. Table 7 presents the mean, minimum, maximum, and standard deviation errors towards the base frame \([X_{0},Y_{0},Z_{0}]\). Results showed that the computational results of the proposed method are very precise in regards to the experimental results. The limitations of the 3D-printed robotic arm are clear, essentially when based on maximum and minimum real positions. The real target points position errors were of \([-0.7,+0.5]\) toward \(X\), it ranges among \([-0.6,+0.5]\) toward \(Y\), and among \([-0.7,+0.4]\) toward \(Z\). Standard deviations were around 0.3. The precision limitations are essentially due to the limited quality of the 3D-printing mechanical frame, making it very limited for industrial use while being acceptable for educational use, and may help in explaining the gaps that may occur between the computational models and the mechanical system limitations. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Target Point** & **Mean** &
\begin{tabular}{c} **Standard** \\ **Deviation** \\ \end{tabular} & **Minimum** & **Maximum** \\ \hline \(X=18\) & 17.98 & 0.37360266 & 17.3 & 18.5 \\ \(Y=10\) & 9.885 & 0.32650461 & 9.4 & 10.5 \\ \(Z=25\) & 24.85 & 0.29824045 & 24.3 & 25.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Experimental repeatability test for the target point [18; 10; 25] statistical analysis. Figure 9: (a) 3D arm setup (b) Solution simulated in the control interface. ## 7 Conclusions and Future Works

This paper addresses the inverse kinematics problems of a 3D-printed 5-DOF robotics arm for educational use. The paper proposed a behavior-modified PSO, m-PSO, and its multi-objective version (MO-m-PSO) to solve the inverse kinematics problem of a 3D-printed 5-DOF robotic arm, with a focus on the multi-objective approach. The proposed approach adapts the particles' behavior rather than the PSO parameters, and particles evolve differently considering their level of optimality. Mo-m-PSO for inverse kinematics was tested using test scenarios which mimic the industrial needs of robotics arms:

* The simple target point test, which assesses the precision capacities of the inverse kinematics solver as well as its convergence. * The random target point test, which allows to prove the robustness of the proposal and its capacity to correctly address any inverse solution within its work space. * The path planning task, which allows to confirm that the proposed method can be used to plan specific inverse solutions belonging to a specific geometric shape; here, a circle. Results demonstrated that the mono-objective m-PSO as well as the MO-m-PSO are good potential solvers for inverse kinematics; however, the multi-objective Pareto strategy reduced the position precision while satisfying the needed orientation. Acceptable trade-offs were correctly returned with solutions which are mechanically up to the precision and resolution of the system. Errors were on the order of millimeters for a system acting within the centimeter range. Nonparametric statistical comparatives demonstrate that the proposed methods are very competitive toward existing state-of-the-art methods. Future investigations may concern the impact of the heuristic parameters on the level of performances such as in [39].