# Iterative Learning Control for Vibration Suppression of a Robotic Arm
## 3 Numerical Validation  
The proposed VH-ILC-EJ controller and its effects for controlling a robotic arm with vibration suppression are illustrated numerically and validated by an example of vibration suppression for a laboratory-developed 6-DOF robotic arm in Figure 1b. The mechanical  
Figure 4: Block diagram of the input-based iterative learning control. parameters of the robotic arm and the control parameters are listed in Tables 1 and 2, respectively. The common motor feedback control has good tracking performance in the simulated environment. Still, the vibration caused by the structure at the endpoint will cause position error, as shown in Figure 5. The solid line is the output of the link side, and the dotted line is the output of the motor side. Figure 5 shows the simulation results of applying the elastic-joint feedback controller to manipulate a 6-DOF robotic arm on the desired trajectory. The vibration caused by the structure at the endpoint causes a position error. \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline d & m & 1.22 & 0 & 0.371 & 0 & 0.28 & 0 \\ \hline a & m & 0 & 0 & 0.01 & \(-\)0.01 & 0 & 0.1298 \\ \hline \(\alpha\) & rad & 0.5\(\pi\) & 0.5\(\pi\) & 0.5\(\pi\) & \(-\)0.5\(\pi\) & 0.5\(\pi\) & 0.5\(\pi\) \\ \hline m & kg & 1.529 & 0.344 & 2.034 & 0.236 & 1.246 & 0.274 \\ \hline C\({}_{\rm x}\) & m & \(-\)0.0036 & 0.0005 & \(-\)0.0072 & 0.0031 & \(-\)0.0023 & 0.0002 \\ \hline C\({}_{\rm y}\) & m & \(-\)0.0342 & 0.0068 & \(-\)0.1382 & \(-\)0.0003 & \(-\)0.1303 & 0.0371 \\ \hline C\({}_{\rm z}\) & m & \(-\)0.012 & 0.02 & 0.0024 & 0.0347 & 0.0004 & \(-\)0.0012 \\ \hline I\({}_{\rm xx}\) & kg m\({}^{2}\) & 0.0035 & 0.0002 & 0.0328 & 0.0003 & 0.0073 & 0.0001 \\ \hline I\({}_{\rm yy}\) & kg m\({}^{2}\) & 0.0022 & 0.0002 & 0.0152 & 0.0002 & 0.0008 & 0.0001 \\ \hline I\({}_{\rm zz}\) & kg m\({}^{2}\) & 0.0025 & 0.0003 & 0.0327 & 0.0002 & 0.0074 & 0.0001 \\ \hline G\({}_{\rm r}\) & kg m\({}^{2}\) & 0.005 & 0.004 & 0.005 & 0.004 & 0.005 & 0.0033 \\ \hline K & Nm/rad & 1000 & 1000 & 1000 & 1000 & 1000 & 1000 \\ \hline \end{tabular}
\end{table} 
Table 1: Dynamic parameters of the 6-DOF robotic arm. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Joint** & **1** & **2** & **3** & **4** & **5** & **6** & **Unit** \\ \hline \(\mathbf{K_{P}}\) & 5 & 5 & 5 & 5 & 5 & 5 & None \\ \hline \(\mathbf{K_{v}}\) & 35 & 35 & 25 & 75 & 65 & 25 & None \\ \hline \(\mathbf{K_{a}}\) & 10 & 10 & 10 & 35 & 65 & 25 & None \\ \hline \(\mathbf{K_{mp}}\) & 70 & 70 & 70 & 70 & 50 & 50 & None \\ \hline \(\mathbf{K_{mv}}\) & 30 & 30 & 30 & 30 & 15 & 15 & None \\ \hlineFigure 6 shows a simulation using the proposed VH-ILC-EJ controller at the 400 th iteration. The results show that most of the vibration at the link-side end is suppressed, but some residual vibrations at the fixed point of the link-side arm exist after compensating for the elastic torque. However, the residual vibration is eliminated by using the ILC repeatedly. The effects of using the previous information are shown in Figure 7. The simulation results show that the convergence rate increases and the tracking errors are reduced after using the previous information. Figure 5: Motor- and link-side angular displacements with an elastic-torque feedback controller. (solid line: link side, dotted line: motor side). Figure 6: Motor- and link-side angular displacements using the VH-ILC-ET method. (solid line: link side, dotted line: motor side). After 400 iterations, a new input is produced to effectively suppress the residual vibration, so there is no residual vibration on the six-axis of the manipulator at the point of arrival. The position 2 norm error is shown in Figure 6. Using the previous information, a new input is produced to suppress the residual vibration effectively. After using the previous information, the rate of convergence increases, and the tracking error is reduced. At the same time, the input changes in the initial iteration are also relatively stable.