# Residual Vibration Reduction in Flexible Systems Based on Trapezoidal Velocity Profiles
## 5 Experimental Validation
### Experiments on AUBO-i5 Robot
#### 5.3.2 Experiment 2: Results  
The generated position, velocity, and acceleration profiles in the Cartesian coordinate system are depicted in Figure 18. Note that the seventh-order polynomial trajectory proposed in [17] results in a peak velocity higher than its limit value, as this method only considers the acceleration limit for parameter design. The measured vibration responses are presented in Figure 19, and the results are summarized in Table 4. The colored vertical dashed lines indicate the motion completion time for each trajectory. Figure 17: Technical drawing of the second part (unit: mm). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Trajectory** & \(\mathbf{T_{a}}\) **(s)** & \(\mathbf{T_{v}}\) **(s)** & \(\mathbf{T_{tot}}\) **(s)** & \(\mathbf{v}\) **(m/s)** & \(\mathbf{a}\) **(m/s)** \\ \hline Unoptimized trapezoidal velocity & 0.220 & 0.507 & 0.947 & 0.55 & 2.5 \\ Optimized & C1 & 0.233 & 0.495 & 0.960 & 0.55 & 2.365 \\ trapezoidal velocity & C1, C2 & 0.233 & 0.581 & 1.047 & 0.491 & 2.113 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Computed trajectory parameters for experiment 2. Figure 19: Measured vibration responses with different techniques. Figure 18: (**a**) Position; (**b**) velocity and (**c**) acceleration profiles generated by the unoptimized trapezoidal trajectory, optimized S-curve, seventh-order polynomial, and optimized trapezoidal trajectory planning techniques (TVP: trapezoidal velocity profile). As expected, the optimized trapezoidal trajectory achieves significantly better performance than the unoptimized one. The optimized trapezoidal trajectory with C1 results in an 81.5% attenuation of residual vibration, while the optimized trapezoidal trajectory with both C1 and C2 results in a 91.8% attenuation. In this experiment, the remaining residual vibration is smaller than in the previous experiment, as the part is characterized by a relatively higher mode, thus being less influenced by parameter estimation errors. Notably, the residual vibration amplitude caused by the optimized trapezoidal trajectory with double conditions is comparable to that from the optimized S-curve profile and slightly smaller than that from the seventh-order polynomial profile while leading to shorter time delay. The optimized trapezoidal trajectory with C1 yields the shortest settling time, as the resulting residual vibration is marginally larger but has a remarkably shorter motion duration than the other optimized trajectories. In practice, choosing a suitable profile could help minimize the settling time. A limitation of the proposed method is that the vibration during the acceleration and deceleration phases of the movement is higher than that of the S-curve and polynomial profiles. If the vibration during the entire motion is also required to be attenuated, smoother profiles may be preferred. Since the proposed method operates at the motion planning level using standard programming, it demonstrates good adaptability to various load conditions and robot types. In this study, the two robots possess similar link dimensions. If the robot has flexible links with a significantly greater length or a considerably larger payload mass, its dynamic performance may be substantially affected by the payload. Longer links lead to lower natural frequencies and increased dynamic coupling, resulting in higher deflection amplitudes. The flexible element type impacts mass distribution during different robot poses and inertial forces and may result in multiple vibration modes and complex mode shapes. In this case, the influence of the payload on dynamic performance needs to be considered. However, within the scope of this study, the robots maintain good dynamic performance as long as the payload is within the allowable range. Therefore, given the same robot type and payload conditions, the proposed method maintains consistent vibration reduction performance across the payload range relative to benchmark techniques.