# Design Procedure for Motion Profiles with Sinusoidal Jerk for Vibration Reduction
## 4 Numerical Results
### Comparative Dynamic Study
Figure 18: Vibration responses for optimized sinusoidal jerk profiles with different robustness degrees in Case 3. As shown in Table 8, under nominal conditions, all the optimized sinusoidal jerk profiles effectively attenuated the residual vibrations. However, when an estimation error was introduced, the residual-vibration levels increased in all the test cases. It is worth noting that the optimized profiles with onefold vibration mitigation produced noticeably larger vibrations as the resonant frequency deviated, resulting in a certain degree of performance degradation. In contrast, the optimized profile with threefold vibration mitigation achieved a superior robustness performance under the frequency perturbation, leading to the smallest increment in the residual vibrations. The profile with twofold vibration mitigation was characterized by an intermediate robustness between the profiles with twofold and twofold vibration mitigation. This difference can also be observed with ease by measuring the setting times. Table 8 reports the percentage increments in the settling time due to the modeling error for each motion profile. On average, the increments in the settling time for the profiles with onefold and twofold vibration mitigation are 167% and 145%, respectively, while the settling times for the profiles with threefold vibration mitigation remain the same as the moving time (\(T_{f}\)). To better quantify the influence of the parameter perturbations on the results, the magnitude evolution curves of the residual-vibration envelope with respect to a \(\pm\) 10% variation in the actual natural frequency in the four test cases are plotted in Figure 20. The results confirm the consistency of the sensitivity reduction brought on by the optimized profiles with threefold vibration mitigation. The vibration level remains limited within the allowed tolerance despite minor parameter perturbations, as the harmonic contents are eliminated in a wider neighborhood of the resonant frequency with the increase in \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multicolumn{3}{c}{**\(r=1\)**} & \multicolumn{2}{c}{\(r=2\)} & \multicolumn{2}{c}{\(r=3\)} \\ \cline{3-8}  & & **w/o Modeling** & **w/ Modeling** & **w/o Modeling** & **w/ Modeling** & **w/o Modeling** & **w/ Modeling** \\  & & **Error** & **Error** & **Error** & **Error** & **Error** & **Error** \\ \hline \multirow{2}{*}{1} & Residual vibration (mm) & 0.224 & 2 201 & 0.051 & 1.768 & **0.001 *** & **0.221 * \\  & Settling time (s) & **1.292 * (ref)** & 5.102 (+295\%) & 1.355 (ref) & 4.637 (+242\%) & 1.438 (ref) & **1.438 * (+0\%)** \\ \hline \multirow{2}{*}{2} & Residual vibration (mm) & 0.359 & 1.362 & 0.037 & 2.418 & **0 *** & **0.156 * \\  & Settling time (s) & **0.992 * (ref)** & 3.760 (+279\%) & 1.184 (ref) & 5.182 (+338\%) & 1.438 (ref) & **1.438 * (+0\%)** \\ \hline \multirow{2}{*}{3} & Residual vibration (mm) & 0.041 & 0.564 & 0.018 & 0.332 & **0 *** & **0.045 *** \\  & Settling time (s) & **1.520 * (ref)** & 2.356 (+55\%) & 1.615 (ref) & **1.615 * (+0\%)** & 1.813 (ref) & 1.813 (+0\%) \\ \hline \multirow{2}{*}{4} & Residual vibration (mm) & 0.010 & 0.445 & 0.002 & 0.190 & **0 *** & **0.081 *** \\  & Settling time (s) & **0.631 * (ref)** & 0.881 (+39.6\%) & 0.750 (ref) & **0.750 * (+0\%)** & 0.938 (ref) & 0.938 (+0\%) \\ \hline \hline \end{tabular}  
* The values in bold indicate the best results achieved among the tested profiles.  
\end{table} 
Table 8: Comparative results of optimized sinusoidal jerk profiles with different robustness degrees with/without modeling errors. Figure 19: Vibration responses for optimized sinusoidal jerk profiles with different robustness degrees in Case 4.  
the \(r\). Therefore, the motion profile with threefold vibration reduction is preferable in the presence of system modeling errors.