# Residual Vibration Reduction in Flexible Systems Based on Trapezoidal Velocity Profiles
## 5 Experimental Validation
### Experiments on UR5e Robot
#### 5.2.2 Optimal Parameter Computation  
The parameters for the optimized trapezoidal trajectories with different vibration suppression conditions are computed based on the vibration period (0.179 s). To compute these parameters, the minimum time parameters for the original trapezoidal profile are first calculated according to Equations (5) and (6) as follows:  
\[T_{a}=\min\left\{\frac{0.65}{2.4},\sqrt{\frac{0.55}{2.4}}\right\}=0.271\;\text {s},\;T_{v}=\frac{0.55-2.4\times 0.271^{2}}{0.65}=0.575\;\text{s} \tag{22}\]  
Thus, the corresponding switching times are \(t_{1}=T_{a}=0.271\;\text{s},\;t_{2}=T_{a}+T_{v}=0.846\;\text{s}\). Since the acceleration time is determined by the velocity constraint, both the peak velocity and acceleration attain their maximum allowable values, i.e., \(v=0.65\;\text{m/s}\), \(a=2.4\;\text{m/s}^{2}\). Then, the vibration period is taken into account to tune the above original parameters. In particular, if condition C1 is selected, the optimized parameters are calculated using Equations (16)-(18):  
\[\hat{t}_{1}=ceil\left(\frac{0.271}{0.179}\right)\times 0.179=0.358\;\text{s},\; \hat{t}_{2}=\max\{0.358,0.846\}=0.846\;\text{s} \tag{23}\]  
Figure 8: Technical drawing of the first part (unit: mm). Figure 7: Experimental setup. \[\hat{a}=\frac{0.55}{0.358\times 0.846}=1.814\;\mathrm{m/s^{2}} \tag{24}\]  
The peak velocity does not change as \(t_{2}\) remains the same. We can then obtain \(\hat{T}_{a}=\hat{t}_{1}=0.358\;\mathrm{s}\), \(\hat{T}_{v}=\hat{t}_{2}-\hat{t}_{1}=0.488\;\mathrm{s}\). If condition C2 is selected, the optimized parameters are calculated from the formulae in Section 3.2.2 as follows:  
\[\hat{t}_{1}=t_{1}=0.271\;\mathrm{s},\;\hat{t}_{2}=ceil\left(\frac{0.846}{0.179 }\right)\times 0.179=0.896\;\mathrm{s} \tag{25}\]  
\[\hat{v}=\frac{0.55}{0.896}=0.614\;\mathrm{m/s},\;\hat{a}=\frac{0.55}{0.271 \times 0.895}=2.267\;\mathrm{m/s^{2}} \tag{26}\]  
We can then obtain \(\hat{T}_{a}=\hat{t}_{1}=0.271\;\mathrm{s}\), \(\hat{T}_{v}=\hat{t}_{2}-\hat{t}_{1}=0.625\;\mathrm{s}\). Finally, if both C1 and C2 are selected, the optimized parameters are calculated from Equations (18)-(21):  
\[\hat{t}_{2}=ceil\left(\frac{0.846}{0.179}\right)\times 0.179=0.896\;\mathrm{s}, \;\hat{t}_{1}=ceil\left(\frac{0.55}{2.4\times 0.896\times 0.179}\right)\times 0.179=0. 358\;\mathrm{s} \tag{27}\]  
\[\hat{v}=\frac{0.55}{0.896}=0.614\;\mathrm{m/s},\;\hat{a}=\frac{0.55}{0.358 \times 0.896}=1.713\;\mathrm{m/s^{2}} \tag{28}\]  
We can then obtain \(\hat{T}_{a}=\hat{t}_{1}=0.358\;\mathrm{s}\), \(\hat{T}_{v}=\hat{t}_{2}-\hat{t}_{1}=0.538\;\mathrm{s}\). These parameters can be directly set in the robot controller to generate the desired trajectories. Therefore, the implementation can be achieved through standard robot programming interfaces without specialized expertise. The corresponding input panels are depicted in Figure 9. The MoveL command is used to achieve a precise straight-line movement of the robot tool between two path points. The necessary parameters for determining the movement include the reference frame, the poses of the desired path points, and the tool speed and acceleration. The unoptimized trapezoidal profile, which is the robot's default setting, is taken as a baseline for comparison. Table 1 lists all the computed time and kinematic parameters. Note that the original unoptimized trapezoidal profile has a shorter duration than the optimized trajectories. Therefore, to provide a more precise comparison, the unoptimized trapezoidal profile is uniformly scaled to match the duration of each optimized trajectory; namely, the acceleration time and cruise time are scaled with the same ratio. These scaled trajectories are denoted as scaling1, scaling2, and scaling3, whose durations correspond to those of the optimized trajectories with C1, C2, and both C1 and C2, respectively.