# Design Procedure for Motion Profiles with Sinusoidal Jerk for Vibration Reduction
## 4 Numerical Results
### Comparative Dynamic Study  
To assess the influence of the motion laws on residual vibrations, the dynamic model of a flexible-motion system (Figure 4) was taken into account for comparative analysis. This simplified model can effectively represent a wide range of industrial equipment, such as transfer robots, inspection machines, and cranes. The motion profiles obtained in the last subsection were utilized as the reference input to the system in order to evaluate the response of the mass following the previous settings: \(\omega_{n}\) = 50.27 rad/s and \(\zeta\) = 0.01. The simulation was performed in the Matlab/Simulink environment, and Figure 11 depicts the model of the dynamic response of the mass. The time step was set to 0.5 ms. Figures 12-15 depict the vibration responses with different profiles for Cases 1-4. The vertical dashed lines marked in corresponding colors represent the motion completion times for the profiles. The same applies to the subsequent figures. Table 5 reports the comparison results in terms of the peak-to-peak value of the residual vibration and settling time, with the values in bold indicating the best results achieved among these profiles. The settling time is defined as the time at which the induced vibration decays and remains within a specified error band (\(\pm\)0.2 mm). It can be observed that the vibration level decreased remarkably with an increase in the continuity of the motion profiles, albeit at the expense of a slight prolongation of the duration. Figure 11: Simulation model for motion-induced vibration. Figure 12: Comparison of vibration responses induced by different motion profiles in Case 1. Figure 14: Comparison of vibration responses induced by different motion profiles in Case 3. Figure 15: Comparison of vibration responses induced by different motion profiles in Case 4. As expected, the trapezoidal profile, characterized by acceleration discontinuities, induced the most severe residual vibrations in all the cases, whereas the proposed optimized sinusoidal jerk profile exhibited the best performance in vibration mitigation. In addition, the minimum-time sinusoidal jerk outperformed the trapezoidal velocity and S-curve profiles, which is consistent with observations in previous literature. Specifically, the residual-vibration level generated by the optimized sinusoidal jerk profile decreased by an average of 95.2% relative to the trapezoidal velocity profile, and by 89.9% relative to the S-curve profile. Meanwhile, the settling time, which serves as a reliable productivity metric, was significantly shortened when using the optimized profile for Cases 1 and 2. In all the test cases, the settling times were identical to the motion durations (\(T_{f}\)), as the vibration magnitudes fell below the specified error after reaching the target positions. Hence, in terms of the settling time, the optimized sinusoidal jerk profile achieves the most efficient movement, despite having a slightly longer duration than the other profiles. To further evaluate the robustness of the method, we performed experiments considering a 10% estimation error in the natural frequency (the actual natural frequency varies from 8 Hz to 7.2 Hz). As mentioned earlier, setting a higher robustness degree (\(r\)) enhances the insensitivity of the generated sinusoidal jerk profile to parameter uncertainties. Tables 6 and 7 aggregate the computed parameters for the optimized profiles with the \(r\) set to 2 and 3, respectively. Note that these profiles were tuned according to the nominal natural frequency. The vibration responses for the optimized sinusoidal jerk profiles with different robustness degrees in the absence and presence of a modeling error are plotted in Figures 16-19, and the comparison results are summarized in Table 8.  
Figure 16: Vibration responses for optimized sinusoidal jerk profiles with different robustness degrees in Case 1. Figure 17: Vibration responses for optimized sinusoidal jerk profiles with different robustness degrees in Case 2.