# Residual Vibration Reduction in Flexible Systems Based on Trapezoidal Velocity Profiles
## 5 Experimental Validation
### Experiments on UR5e Robot
#### 5.2.3 Experiment 1: Results  
The generated position, velocity, and acceleration profiles in the Cartesian coordinate system are depicted in Figures 10 and 11. It is shown that all the motion trajectories are bounded within the kinematic constraints. The measured vibration responses obtained from different trajectories are presented in Figures 12-15. The vertical dashed red lines indicate the completion time of the motion, and the light-blue area between the horizontal gray dashed lines represents the allowable error band. The comparison results in terms of the peak-to-peak residual vibration amplitude and settling time are summarized in Table 2.  
Figure 9: User input panels of the robot controller: (**a**) initial waypoint setting; (**b**) final waypoint setting; (**c**) movement parameter setting. It can be seen that the unoptimized trapezoidal profile produces significant residual vibration close to 12 m/s\({}^{2}\), requiring a considerably longer period for the vibration to dissipate to acceptable levels. This would delay subsequent operations after the motion is completed. In contrast, while the optimized trapezoidal profile proposed in this work slightly increases the motion completion time relative to the unoptimized one, this trade-off leads to a substantial reduction in the vibration level. As a result, the system's response reaches the required tolerance more quickly with the optimized profile, leading to a shortened overall settling time. A slight vibration still remains, even when the trajectory parameters are properly tuned, due to the tracking error and estimation error in the modal properties. Different selections of vibration suppression conditions produce different trajectory durations and vibration amplitudes. When condition C2 is chosen, the trajectory duration is the shortest, only 0.05 s longer than that of the unoptimized one. The optimized trajectories reduce the residual vibration by 71.6% under condition C1 and by 60.2% under condition C2, compared to the unoptimized trajectory. Notably, the selection of both conditions C1 and C2 produces the best performance in terms of vibration mitigation, as the vibration level is reduced by 76.2%. The settling time is identical to the trajectory duration, as the vibration amplitude settles within the desired bounds almost immediately after motion completion. Therefore, the proposed approach provides several benefits, including faster settling, which enables higher production rates in pick-and-place operations, and improved positioning accuracy, which leads to enhanced product quality. Figure 10: (**a**) Position; (**b**) velocity and (**c**) acceleration profiles generated by the unoptimized and optimized trapezoidal trajectory planning techniques (TVP: trapezoidal velocity profile). Figure 11: (**a**) Position; (**b**) velocity and (**c**) acceleration profiles generated by the unoptimized trapezoidal trajectory planning technique after scaling (TVP: trapezoidal velocity profile). Figure 12: Vibration response with the unoptimized trapezoidal velocity trajectory. Figure 14: Vibration responses with the unoptimized trapezoidal velocity trajectory after scaling and optimized trapezoidal velocity trajectory (condition C2 is chosen). Figure 13: Vibration responses with the unoptimized trapezoidal velocity trajectory after scaling and optimized trapezoidal velocity trajectory (condition C1 is chosen). Figure 15: Vibration responses with the unoptimized trapezoidal velocity trajectory after scaling and optimized trapezoidal velocity trajectory (conditions C1 and C2 are chosen). This method is particularly cost-effective because it leverages existing robot controllers and does not require any additional hardware or software modification. The only cost incurred is the slightly longer motion duration. However, this delay is compensated for by the significant reduction in settling time.