# Design Procedure for Motion Profiles with Sinusoidal Jerk for Vibration Reduction
## 3 Parameter Design for Sinusoidal Jerk Profile
### Analysis of Vibration Characteristics
\begin{array}{rl}y(t|t_{5}\leq t<t_{6})=-\frac{1}{\omega_{d}}\left[\begin{array} []{c}\int_{t_{0}}^{t_{1}}\frac{A_{peak}}{2}\Big{(}1-\cos\frac{\pi}{T_{1}}( \tau-t_{0})\Big{)}\sin(\omega_{d}(t-\tau))d\tau+\int_{t_{1}}^{t_{2}}A_{peak} \sin(\omega_{d}(t-\tau))d\tau\\ +\int_{t_{2}}^{t_{3}}\frac{A_{peak}}{2}\Big{(}1+\cos\frac{\pi}{T_{1}}(\tau-t_ {2})\Big{)}\sin(\omega_{d}(t-\tau))d\tau\\ -\int_{t_{4}}^{t_{5}}\frac{A_{peak}}{2}\Big{(}1-\cos\frac{\pi}{T_{1}}(\tau-t_ {4})\Big{)}\sin(\omega_{d}(t-\tau))d\tau-\int_{t_{5}}^{t}A_{peak}\sin(\omega_{ d}(t-\tau))d\tau\end{array}\right]\\ =\frac{A_{peak}}{\omega_{d}^{2}}\left[1-\frac{\pi^{2}}{\pi^{2}-(\omega_{d}T_{1 })^{2}}\cos\Big{(}\frac{\omega_{d}T_{1}}{2}\Big{)}\left[\cos\Big{(}\omega_{d} \Big{(}(t-t_{4})-\frac{T_{1}}{2}\Big{)}\Big{)}+2\sin\Big{(}\frac{\omega_{d}(T _{1}+T_{2})}{2}\Big{)}\sin\Big{(}\omega_{d}\Big{(}(t-t_{0})-T_{1}-\frac{T_{2}} {2}\Big{)}\Big{)}\right]\right]\end{array} , \tag{26}\]  
and the dynamic response during the constant velocity segment \([t_{3},\ t_{4}]\) can be derived as follows:  
\[ \begin{array}{rl}y(t|t_{3}\leq t<t_{4})=-\frac{1}{\omega_{d}}\left[\begin{array} []{c}\int_{t_{0}}^{t_{1}}\frac{A_{peak}}{2}\Big{(}1-\cos\frac{\pi}{T_{1}}( \tau-t_{0})\Big{)}\sin(\omega_{d}(t-\tau))d\tau+\int_{t_{1}}^{t_{2}}A_{peak} \sin(\omega_{d}(t-\tau))d\tau\\ +\int_{t_{2}}^{t_{3}}\frac{A_{peak}}{2}\Big{(}1+\cos\frac{\pi}{T_{1}}(\tau-t_ {2})\Big{)}\sin(\omega_{d}(t-\tau))d\tau\end{array}\right]\.\end{array}  \tag{27}\]  
Substituting (22) into (25)-(27) yields the following:  
\[y(t|t_{1}\leq t<t_{2})=-\frac{A_{peak}}{\omega_{d}^{2}},\ y(t|t_{5}\leq t<t_{6 })=\frac{A_{peak}}{\omega_{d}^{2}},\ y(t|t_{3}\leq t<t_{4})=0. \tag{28}\]  
Figure 5: Vibration responses of sinusoidal jerk profile optimized under different suppression conditions: (**a**) only condition C1 is satisfied; (**b**) only condition C2 is satisfied; (**c**) only condition C3 is satisfied. Equation (28) implies that satisfying C1 leads to zero vibrations during the constant acceleration segments and constant velocity segment (see Figure 5a). The physical meaning of this condition is that the vibrations induced in the sinusoidal jerk segments are offset at the end of these segments (\(t=t_{1}\), \(t_{3}\), \(t_{5}\), \(t_{7}\)). Likewise, it can be observed that if C2 is satisfied, then the vibrations induced in the positive sinusoidal jerk segment \([t_{0},\ t_{1}]\) and the negative sinusoidal jerk segment \([t_{2},\ t_{3}]\) cancel each other out at the end of the acceleration phase (\(t=t_{3}\)), ensuring no vibrations during the constant velocity segment (see Figure 5b). In contrast, these properties are not present for C3. The vibration induced in the acceleration phase counteracts that in the deceleration phase, and only the residual vibration after the motion stops is suppressed (see Figure 5c). Therefore, when a smooth transfer is required in the movement, condition C1 should be preferentially selected for parameter tuning. Unless otherwise specified, the scheme will choose the one that produces a shorter moving time. Note that these conditions depend on the system parameter \(\omega_{d}\) and are thus influenced by the accuracy of the parameter estimation. In cases in which parameter estimation errors occur or vibration modes vary during motion, it is expected that the motion profile will remain robust against parameter variations. One solution is to prolong the motion time length, as it reduces both the frequency range and amplitudes of the acceleration profile. However, this would lead to much slower movement, which is undesirable in production. Alternatively, the robustness against system parameter uncertainties can be improved when the profile parameters satisfy multiple conditions from C1 to C3, which can be evaluated by taking partial derivatives of the vibration amplitude with respect to the estimated natural frequency.