# Design Procedure for Motion Profiles with Sinusoidal Jerk for Vibration Reduction
## 3 Parameter Design for Sinusoidal Jerk Profile
### Analysis of Vibration Characteristics  
The algorithm in the last subsection yields the solution that minimizes the execution time with the actuator limits. However, the quantitative consideration of the vibration response relating to the system modes is not incorporated into the motion profile design. The performance improvement in practice therefore appears to be almost exclusively due to smoothing. To investigate the relationship between the profile parameters and the vibration response, consider the simplified model in Figure 4, which includes a base, a mass (\(m\)), a spring with stiffness (\(k\)), and a damper (\(c\)). The dynamics of the system are described by the following expression:  
\[-ky-c\dot{y}=m\big{(}\ddot{x}+\ddot{y}\big{)}, \tag{15}\]  
where \(x\) is the absolute position of the base and \(y\) is the relative position of the mass with respect to the base. In this model, the base stands for the actuation system controlled by an input force, while the mass represents the mechanically linked structure that undergoes vibration. Thus, \(x\) and \(y\) serve as the reference input motion profile and dynamic response, respectively. Equation (15) can be written in Laplace form:  
\[\frac{Y(s)}{X(s)}=\frac{-ms^{2}}{ms^{2}+cs+k}=\frac{-s^{2}}{s^{2}+2\zeta\omega_ {n}s+\omega_{n}{}^{2}}, \tag{16}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Type 1** & **Type 2** & **Type 3** & **Type 4** \\ \hline Critical & \(V_{\text{max}}>\frac{\pi A_{\text{max}}}{2I_{\text{max}}}\) & \(|D|>\frac{\pi A_{\text{max}}}{4I_{\text{max}}}^{3}\) & \(V_{\text{max}}<\frac{\pi A_{\text{max}}}{2I_{\text{max}}}\), & \(|D|<\frac{\pi^{2}A_{\text{max}}}{4I_{\text{max}}}^{3}\), \\ Condition & \(|D|>\frac{V_{\text{max}}}{2I_{\text{max}}}^{2}+\frac{\pi V_{\text{max}}A_{ \text{max}}}{2I_{\text{max}}}\) & \(|D|\leq\frac{V_{\text{max}}}{2I_{\text{max}}}^{2}+\frac{\pi V_{\text{max}}A_{ \text{max}}}{2I_{\text{max}}}\) & \(|D|>\sqrt{\frac{V_{\text{max}}}{2I_{\text{max}}}^{3}}\) & \(|D|<\sqrt{\frac{V_{\text{max}}}{2I_{\text{max}}}^{3}}\) \\ \hline \(T_{1}\) & \(\frac{\pi A_{\text{max}}}{2I_{\text{max}}}\) & \(\frac{\pi A_{\text{max}}}{2I_{\text{max}}}\) & \(\sqrt{\frac{\pi V_{\text{max}}}{2I_{\text{max}}}}\) & \(\sqrt{\frac{\pi|D|}{2I_{\text{max}}}}\) \\ \(T_{2}\) & \(\frac{V_{\text{max}}}{A_{\text{max}}}-T_{1}\) & \(-\frac{3T_{1}}{2}+\sqrt{\frac{T_{1}^{2}}{4}+\frac{|D|}{A_{\text{max}}}}\) & 0 & 0 \\ \(T_{3}\) & \(\left|\frac{D}{V_{\text{max}}}\right|-2T_{1}-T_{2}\) & 0 & \(\left|\frac{D}{V_{\text{max}}}\right|-2T_{1}\) & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Complete minimum-time solutions for parameters of sinusoidal jerk profile. Figure 4: Modeling of flexible-motion system. where \(\omega_{n}=\sqrt{k/m}\) and \(\zeta=c/\sqrt{4 km}\) denote the undamped natural frequency and the damping ratio of the system, respectively. For a given reference input (\(f(t)\)), the dynamic response of the system can be calculated via the convolutional integral \(y(t)=\int_{0}^{\infty}f(\tau)h(t-\tau)d\tau\), where \(h(t)\) is the impulse response of the system, obtained from the inverse Laplace transform of Equation (16). Therefore, the dynamic motion-induced vibration response with the motion profile \(x_{1}(t)\) is derived as follows:  
\[ \begin{array}{l}y(t)=-\int_{0}^{t}\tilde{x}(\tau)\frac{1}{\omega_{n}\sqrt{1- \zeta^{2}}}e^{-\zeta\omega_{n}(t-\tau)}\sin\Bigl{(}\omega_{n}\sqrt{1-\zeta^{2 }}(t-\tau)\Bigr{)}d\tau\\ =-\frac{1}{\omega_{d}}\int_{0}^{t}\tilde{x}(\tau)e^{-\zeta\omega_{n}(t-\tau) }\sin(\omega_{d}(t-\tau))d\tau\end{array} , \tag{17}\]  
where \(\omega_{d}=\omega_{n}\sqrt{1-\zeta^{2}}\) represents the damped natural frequency. For low-damped systems with \(\zeta\approx 0\), the above expression is simplified as follows:  
\[y(t)=-\frac{1}{\omega_{d}}\int_{0}^{t}\tilde{x}(\tau)\sin(\omega_{d}(t-\tau) )d\tau. \tag{18}\]  
By substituting Equation (5) into Equation (18), one can obtain the residual-vibration response after the completion of motion:  
\[