# Residual Vibration Reduction in Flexible Systems Based on Trapezoidal Velocity Profiles
## 5 Experimental Validation
### Experiments on AUBO-i5 Robot  
#### 5.3.1 Experiment 2: Setup  
To validate the applicability of the proposed method across different robotic platforms and payload conditions, the second experiment was conducted on an AUBO-i5 robot with a heavier transferred part attached to the end-effector, as shown in Figure 16. The part is made of aluminum alloy, and the technical drawing in Figure 17 depicts its principal dimensions. The flexible part is approximately characterized by a natural frequency of 8.6 Hz and a damping ratio of 0.011. The weight of the part is about 2.2 kg. Similarly, the robot performed a straight-line motion with a moving distance of 0.4 m from (\(-50\), \(-620\), 100) mm to (\(-50\), \(-620\), 500) mm with the orientation fixed to (0, 0, \(-\pi/2\)) rad using RPY representation. The velocity and acceleration constraints are defined as: \(V_{\text{max}}=0.55\) m/s, \(A_{\text{max}}=2.5\) m/s\({}^{2}\). A wireless accelerometer was mounted on the part to measure the vibration response along the \(z\)-axis direction with a sampling rate of 200 Hz. The settling time is defined as the duration from when the movement starts until the part's residual vibration remains consistently within \(\pm\) 0.5 m/s\({}^{2}\).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Trajectory** & \multicolumn{2}{c}{**Residual Vibration (m/s\({}^{2}\))**} & **Settling Time (s)** \\ \hline \multirow{3}{*}{Unoptimized trapezoidal velocity} & Original & 11.92 & 3.444 \\  & Scaling1 & 8.86 (red) & 2.664 (red) \\  & Scaling2 & 8.70 (red2) & 2.794 (red2) \\  & Scaling3 & 6.74 (red3) & 2.376 (red3) \\ \hline \multirow{3}{*}{Optimized trapezoidal velocity} & C1 & 3.42 (\(-51.8\)) & 1.232 (\(-53.8\)) \\  & C2 & 4.71 (\(-44.4\)) & 1.454 (\(-48.0\)) \\ \cline{1-1}  & C1, C2 & 2.83 (\(-60.8\)) & 1.255 (\(-47.2\)) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparative results of residual vibrations and settling times induced by different motion profiles.  
Figure 16: Experimental setup of AUBO-i5 robot.  
The unoptimized trapezoidal profile is taken as a baseline. The parameters for the optimized trapezoidal trajectories with single and double conditions are computed based on the vibration period (0.116 s). Table 3 lists all the computed time and kinematic parameters. Additionally, the optimized S-curve [10] and seventh-order polynomial [17] profiles introduced in previous studies were also tested for comparison. As mentioned earlier, an important advantage of the proposed method is that its implementation does not require any additional resources. Therefore, the optimized trapezoidal trajectories can be generated directly using built-in commands provided by the robot control panel, while the S-curve and polynomial profiles were implemented on a host computer with the Ubuntu system that served as the primary controller. A Python program was developed to incorporate the algorithms for computing the position data. The generated data are then converted into specific commands using MoveIt within the ROS environment and sent to the robot control cabinet, thereby driving the robot to track the trajectory.