# Iterative Learning Control for Vibration Suppression of a Robotic Arm
## 2 Control of Robotic Arm with Elastic Joints
### Vibration Signal Analysis and Feature Extraction  
The vibration signals from the manipulator contain important information about the operation state of the arm. Usually, the manipulator vibrates during motion due to its structure. The vibration information produced during motion can be measured by sensors, such as the accelerometer, and analyzed through the torque signal feedback from the motor side. Therefore, the accelerometer and torque signal contain vibrational signals. The vibrations of the manipulator are low-frequency due to its structure, but some high-frequency noises will influence the measurement. Therefore, it is improper to directly analyze the information from the accelerometer or torque sensors. The HHT and VMD can be used to extract the characteristics of the vibration signal before analysis. A recursive mode decomposition technique can quickly produce mode mixing problems, resulting in distortion of the IMFs after decomposition and thus leading to hindered analysis. However, the VMD method must determine the mode number, \(v\), required for decomposition in advance. Because the vibration signal is random, it is impossible to determine the number of mode numbers in advance. The proposed method is to observe whether the decomposition is over-decomposed or under-decomposed to judge the number of mode numbers that need to decompose. The VMD-HHT method combines the HHT and VMD to extract the features of mechanical robotic arms and vibration signals for analysis. The procedure of the VMD-HHT analysis method is as follows:  
1. Based on the observation of the mode number using the HHT, the value of the mode number, \(V\), is determined to decompose the IMFs. The HHT method is applied to the accelerometer and torque signals to obtain multiple IMFs. 2. The IMFs obtained via the decomposition of the accelerometer signal are compared to the torque signal in a pairwise manner. IMFs similar in terms of the time domain signal waveform are identified to determine the two groups of IMFs related to the vibration. 3. After extracting the IMFs with the vibration components, the marginal spectrum is used to analyze the frequency distribution. If the results of the two are similar, this group of IMFs is related to the vibration of the arm. In the signal analysis, VMD decomposes the vibration signal into a series of IMFs containing information regarding the vibration signal for characteristic information extraction. Then, the proposed method is applied to calculate the vibration signal and remove trivial information for quick compensation for the error caused by the vibration. However, the number of modes must be determined before utilizing the VMD algorithm. If the value of the preset mode number is less than the mode number of the original signal, the unique frequency component of the original signal is lost. When the preset value of the mode number is greater than the original mode number of the signal, the decomposed high frequency is very close. Therefore, the proximity of the center frequencies obtained by the decomposition is used to obtain the proper mode number for decomposition. A search method to automatically determine the appropriate mode number is listed in Algorithm 1. Correctly decomposed center frequencies are spaced apart by a certain distance. This distance is defined as the proximity of the two center frequencies:  
\[d_{V,\ m,\ n}=\frac{\left|H_{V,\ m}-H_{V+1,\ n}\right|^{2}}{H_{V,\ m}^{2}} \tag{9}\]  
where \(H_{V,m}\) represents the \(m\)th center frequency when the mode number for decomposition is \(V\). \(H_{V+1,n}\) represents the \(n\)th center frequency for decomposition when the mode number is \(V+1\), and \(d_{V,m,n}\) represents the proximity of the \(m\)th center frequency when the mode number is \(V\) and the \(n\)th center frequency when the mode number is \(V+1\). First, the center frequency when \(V=1\) is used as a reference.