# Residual Vibration Reduction in Flexible Systems Based on Trapezoidal Velocity Profiles
## 6 Conclusions  
This study proposes a systematic optimization method for the trapezoidal velocity profile to reduce motion-induced vibrations in flexible systems. One main advantage is its easy implementation in common industrial robots and machines without requiring computational complexity and modification of the controller. The mathematical expressions of the profile and the minimum time solution under actuator limits are first developed. By examining the dynamic response characteristics, it is found that the vibration amplitude is associated with the acceleration/deceleration switching times. Two vibration suppress  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Trajectory** & **Residual Vibration (m/s\({}^{2}\))** & **Execution Time (s)** & **Selling Time (s)** \\ \hline Unoptimized trapezoidal velocity & 4.26 (ref) & 0.947 (ref) & 4.010 (ref) \\ Optimized S-curve [10] & 0.33 (\(-\)9.2\%) & 1.064 (\(+\)12.3\%) & 1.055 (\(-\)72.7\%) \\ Seventh-order polynomial [17] & 0.46 (\(-\)89.2\%) & 1.074 (\(+\)13.4\%) & 1.085 (\(-\)72.9\%) \\ Optimized & C1 & 0.79 (\(-\)81.5\%) & 0.960 (\(+\)1.3\%) & 0.980 (\(-\)75.6\%) \\ trapezoidal velocity & C1, C2 & 0.35 (\(-\)91.8\%) & 1.047 (\(+\)10.5\%) & 1.065 (\(-\)73.4\%) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparison of results produced by different motion profiles.  
sion conditions are derived. Based on these conditions, an analytical parameter tuning procedure is then given to generate motion with minimized vibration and execution time according to the natural period. Experiments conducted on two robot manipulators with different attached flexible parts demonstrate the effectiveness and applicability of the proposed method in reducing residual vibration and settling time. This approach offers a practical solution for improving precision and productivity in industrial applications.  
This study opens several avenues for future research. One potential direction is exploring trajectory optimization techniques for more complex scenarios, such as multi-mode systems or multi-axis synchronized motions with varying dynamic properties. Additionally, extending the method to other flexible systems, such as those with multiple flexible links or significant payload mass, could further broaden its applicability. Experimental validation will be performed using robots with larger link lengths and more flexible payloads that exhibit complex vibration patterns, requiring consideration of their influence on the robot's dynamic performance. Investigating the integration of machine learning algorithms to predict and reduce vibrations in real-time represents another promising direction. These efforts could lead to more robust and versatile vibration reduction solutions across diverse applications.  
Conceptualization, X.C. and Y.F.; methodology, X.C. and Y.F.; software, X.C.; validation, Y.F. and C.G.; investigation, X.C.; resources, C.G.; writing--original draft, X.C. and Y.F.; writing--review and editing, Y.F.; supervision, C.G.; project administration, C.G.; funding acquisition, Y.F. and C.G. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (grant numbers 52105031, 62273235) and the China Postdoctoral Science Foundation (grant number 2022T150407).  
Not applicable.  
The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy.  
The authors declare no conflicts of interest.