# Design Procedure for Motion Profiles with Sinusoidal Jerk for Vibration Reduction
## 4 Numerical Results
### Kinematic Comparative Study  
To verify the feasibility and versatility of the method, the study was conducted with various constraint conditions, as listed in Table 2. In addition, it was assumed that the natural frequency of the system is 8 Hz, with a damping ratio of 0.01 (\(T_{d}=0.125\) s). Both the minimum-time and time-frequency-optimal sinusoidal jerk profiles adapted to the vibration characteristics were generated. Note that for the latter, the robustness degree specified by the user affects the resulting profile. In the first study, this degree was set to \(r=1\). The parameters of the minimum-time sinusoidal jerk profiles and the optimized sinusoidal jerk profiles were solved according to the algorithms in Sections 3.1 and 3.3, respectively. These optimization algorithms were implemented in the Matlab language, and the conditions listed in Table 2 were used as the input arguments. Tables 3 and 4 contain the values of the obtained profile parameters, along with the total durations of the resulting profiles. Then, the corresponding displacement, velocity, acceleration, and jerk profiles were obtained by applying the formulations in Section 2.2. The results under different constraints are illustrated in Figures 7-10. It is recognized that the yielded motion profiles for all the test cases successfully arrive at the desired positions while strictly complying with the actuator limits, demonstrating the high reliability of the method. Moreover, within each segment of the minimum-time sinusoidal jerk profiles, the peak value of a derivative equals the corresponding bound. As a consequence, the duration of this motion profile cannot be further shortened without violating any of the imposed constraints. In contrast, the peak kinematic values of the time-frequency-optimal profiles are slightly smaller than their bound values due to the fine tuning of the time parameter for vibration mitigation. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Case** & **Selection** & \(\mathbf{T_{1}}\) **(s)** & \(\mathbf{T_{2}}\) **(s)** & \(\mathbf{T_{3}}\) **(s)** & \(\mathbf{T_{f}}\) **(s)** \\ \hline
1 & C2 & 0.1047 & 0.1453 & 0.5828 & 1.2922 \\
2 & C3 & 0.0589 & 0.3745 & 0.0078 & 0.9923 \\
3 & C2 & 0.1144 & 0.0106 & 1.0406 & 1.5194 \\
4 & C3 & 0.1279 & 0 & 0.1191 & 0.6309 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters of optimized sinusoidal motion profiles (\(r=1\)). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Case** & \(\mathbf{D}\) **(m)** & \(\mathbf{V_{\text{max}}}\) **(m/s)** & \(\mathbf{A_{\text{max}}}\) **(m/s\({}^{2}\))** & \(\mathbf{J_{\text{max}}}\) **(m/s\({}^{3}\))** \\ \hline
1 & 0.75 & 0.8 & 4 & 60 \\
2 & 0.32 & 1 & 1.5 & 40 \\
3 & 0.32 & 0.25 & 2.4 & 30 \\
4 & 0.08 & 0.5 & 3 & 30 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Constraint conditions for computation of motion profiles. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Case** & \(\mathbf{T_{1}}\) **(s)** & \(\mathbf{T_{2}}\) **(s)** & \(\mathbf{T_{3}}\) **(s)** & \(\mathbf{T_{f}}\) **(s)** \\ \hline
1 & 0.1047 & 0.0953 & 0.6328 & 1.2422 \\
2 & 0.0589 & 0.3745 & 0 & 0.9845 \\
3 & 0.1144 & 0 & 1.0512 & 1.5088 \\
4 & 0.1279 & 0 & 0 & 0.5118 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of minimum-time sinusoidal motion profiles. Figure 7: Motion profiles resulting from different techniques in Case 1: **(a)** displacement; **(b)** velocity; **(c)** acceleration; **(d)** jerk. Figure 8: Motion profiles resulting from different techniques in Case 2: (**a**) displacement; (**b**) velocity; (**c**) acceleration; (**d**) jerk. Figure 9: Motion profiles resulting from different techniques in Case 3: (**a**) displacement; (**b**) velocity; (**c**) acceleration; (**d**) jerk. For Case 1 shown in Figure 7, all the kinematic limit values are reached in the minimum-time sinusoidal jerk profile, which is a standard seven-segment profile. Because the \(r\) is set to 1, we can choose one condition from C1, C2, and C3.