# Iterative Learning Control for Vibration Suppression of a Robotic Arm
## 4 Experimental Results
### A Single Joint Vibration Signal Analysis  
Before the vibration suppression experiments, the effect of the vibration of each joint on the end-effector was analyzed first. The experimental steps are as follows:  
Step 1. Depending on the joint limit of each joint, each joint is quickly positive and reversed at a certain angle. Each joint rotates forward and reverses three times. Step 2. Through a visual judgment and the acceleration of the IMU sensor installed at the endpoint, it is judged whether the arm quickly reaches the fixed point and has residual vibration. Figure 7: Comparison of acceleration-type ILC without and with past information. (Blue: with, Black: without) (**a**) Joint 1. (**b**) Joint 2. (**c**) Joint 3. (**d**) Joint 4. (**e**) Joint 5. (**f**) Joint 6. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{**Joint (rad)**} & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline Maximum & k = 1 & 0.041 & 0.191 & 0.297 & 0.031 & 0.004 & 0.002 \\ \cline{2-7} Error & k = 40 & 0.014 & 0.021 & 0.005 & 0.011 & 0.002 & 0.001 \\ \hline L2 norm & k = 1 & 0.573 & 1.789 & 0.488 & 0.372 & 0.033 & 0.037 \\ \cline{2-7} Error & k = 20 & 0.095 & 0.829 & 0.092 & 0.287 & 0.021 & 0.036 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Maximum tracking error and L2 norm of the error of an input-based ILC in simulation. The vibration signal analysis of a single joint involves two steps. As shown in Figure 8a, the inertial measurement unit (IMU) sensor is installed at the joint for measuring the acceleration data to observe the converge rate and vibrations when each joint rotates in the counterclockwise (CCW) and clockwise (CW) direction three times. Figures 8b-g and 9a-f show the measuring acceleration data when the joints rotate 70, 55, 70, 35, 50, and 40 degrees in the CCW and CW direction, respectively. The experimental results show that the first three joints have a more significant residual vibration, while the last three have a trivial residual vibration. Only the vibration signals of the first three joints need to be analyzed and extracted. Figure 8: Experimental setup and data. (**a**) The installed IMU sensor is installed in joint 5. (**b–g**): Acceleration data for joints 1 to 6 rotated in the CCW direction. ((**b**): joint 1, (**c**): joint 2, (**d**): joint 3, (**e**): joint 4, (**f**): joint 5, (**g**): joint 6). Joints one, two, and three have some residual vibration when they reach the fixed point rapidly, while there is no residual vibration in joints four, five, and six at the point of arrival. It can be concluded that the vibration of the six-axis robot arm mainly comes from the vibration of joints one, two, and three, so in the subsequent control, the vibration of the robot arm can be restrained by mainly suppressing the vibration of these three joints. Joints four, five, and six can be treated as rigid bodies. Therefore, in the subsequent vibration signal analysis, only joints one, two, and three are analyzed, and valid information is extracted. After the VMD method, the problems of mode mixing and over-decomposition are overcome, which makes the extraction of vibration features more accurate. It can be seen from the results that the acceleration and torque signals in joint one have corresponding frequency components, which can be considered to be caused by the manipulator reaching a fixed point. The results are shown in Table 4. Frequency features extract the IMF with  
Figure 9: Experimental acceleration data for joints 1 to 6 rotate in the CW direction. (a) Joint 1. (b) Joint 2. (c) Joint 3. (d) Joint 4. (e) Joint 5. (f) Joint 6.  
vibration signals. These can be used in each iteration to remove useless information and increase learning speed. The VMD method extracts the vibration features and makes the extraction of the vibration features more accurate. Figure 10a,b shows the VMD of the torque and acceleration signals. The simulation results show that the acceleration and torque signals have similar trends.