# Iterative Learning Control for Vibration Suppression of a Robotic Arm
## 4 Experimental Results
### Multiple Joint Vibration Suppression  
Figure 11 demonstrates the experimental results of applying the VH-ILC-EJ method to the laboratory-developed 6-DOF robotic arm with \(K_{p}=1.2\), \(K_{d}=0.3\), \(K_{a}=1\), and \(\lambda_{k}=10\). Figure 11a shows two screenshots of a scenario where the robotic arm moves a cup filled with liquid. Figure 11b,c are screenshots of the movements of iterations 1 and 20 for the experiment to validate the effects of VMD-HHT. Iteration 1, considered the ILC, is not implemented, while iteration 20, as the VH-ILC-ET method, is implemented. The \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Joint 1** & \multicolumn{3}{c}{**Acceleration Signal (Hz)**} & \multicolumn{3}{c}{**Feedback Torque Signal (Hz)**} \\ \hline
**No. of Times** & **1** & **2** & **3** & **5** & **5** & **5** \\ \hline IMF 1 & 1.156 & 1.66 & 1.163 & 1.551 & 1.473 & 1.523 \\ \hline IMF 2 & 6.370 & 6.354 & 6.147 & 7.155 & 6.841 & 6.968 \\ \hline \end{tabular}
\end{table} 
Table 4: Vibration characteristics of the VMD-HHT analysis of joint 1. Figure 10: Signals of joint 1. (**a**) VMD of the torque signal. (**b**) VMD of the acceleration signal. (**c**) Acceleration marginal spectrum of IMF 1 and 2. (**d**) Torque marginal spectrum of IMF 1 and 2.  
experimental results show that the liquid in the container is more stable when implementing the VH-ILC-EJ method, demonstrating a vibration suppression ability. Figure 11d,e are screenshots of the movements of iterations 1 and 20 for the experiment to validate the effects of ILC. Iteration 1, considered the ILC, is not implemented, while iteration 20, as the ILC is implemented. The liquid in the container is more stable when the ILC is implemented. From Figure 11c,e, the VH-ILC-EJ method performs better than ILC-EJ method. Figure 12a,b shows the link-side angular displacements of the first and second joints, and Figure 12c shows the acceleration data of the second joint with and without implementing ILC. In multi-joint experiments, the vibrations of the first joint are obvious. Similarly, in the iterative process, the error caused by vibration gradually decreases, and the conver  
Figure 11: Experiments of multiple joint vibration suppression. (**a**) Screenshots of the movement. (**b**) Without ILC-VMD-HHT. (**c**) With ILC-VMD-HHT. (**d**) Without ILC. (**e**) With ILC. gence rate of ILC-VMD-HHT is faster than that of ILC. Because there is no obvious vibration in the measurement of second joint, the performance of vibration suppression from the acceleration in the relevant direction is discussed. Similarly, in the iterative process, the error caused by vibration gradually decreases, and the convergence rate of ILC-VMD-HHT is faster than that of ILC. Because there is no obvious vibration in the measurement of the second joint, we discuss the performance of vibration suppression from the acceleration in the relevant direction. The two curves coincide in Figure 12b. The angular displacements with and without ILC are close. Therefore, the vibration suppression performance from the acceleration in the relevant direction is discussed in Figure 12c. The result shows that the residual vibration still exists at the fixed position. The statistics data for the maximum and 2-norm errors at \(k=1\) and \(k=20\) are summarized in Table 4. The experimental results validate the ILC for vibration suppression. Figure 12c,d compares the L2 norm of the error plots for each iteration with and without the VMD-HHT. The experimental results show that the VH-ILC-EJ controller has a small L2 norm error and converges faster than without the application of the VMH-HHT. The maximum and L2 norm errors are listed in Table 5. The results show that the ILC reduces the maximum and L2 norm errors. After ILC-VMD-HHT is applied, the residual vibration is suppressed. As the number of iterations increases, the error and vibration of the actual output of the link side are reduced. Therefore, the proposed control method can also be applied to multi-joint vibration control.