# Design Procedure for Motion Profiles with Sinusoidal Jerk for Vibration Reduction
## 3 Parameter Design for Sinusoidal Jerk Profile
### Analysis of Vibration Characteristics
Particularly, if two of the vibration suppression conditions are satisfied for a given value of \(\omega_{d}\), the following holds:  
\[y_{res}(\omega_{d})=\frac{\partial y_{res}(\omega_{d})}{\partial\omega_{d}}=0. \tag{29}\]  
Thus, for a small estimation error (\(\Delta\omega_{d}\)), we have the following:  
\[y_{res}(\omega_{d}+\Delta\omega_{d})\cong y_{res}(\omega_{d})+\frac{\partial y _{res}(\omega_{d})}{\partial\omega_{d}}\cdot\Delta\omega_{d}\cong 0. \tag{30}\]  
Equation (30) reveals that the residual-vibration amplitude is not only zero at \(\omega_{d}\) but is also approximately zero near \(\omega_{d}\), which implies that the vibration is robustly mitigated even with a parameter estimation error to some extent. Similarly, if all three vibration suppression conditions are satisfied, then the insensitivity performance is further enhanced, as the following holds:  
\[\frac{\partial^{2}y_{res}(\omega_{d})}{\partial^{2}\omega_{d}}=0. \tag{31}\]  
In fact, the first and second derivatives with respect to the nominal natural frequency represent, respectively, the gradient and curvature of the vibration amplitude evolution curve versus the estimated frequency. If both of these derivatives are equal to zero, then the vibrations remain limited within a broader frequency band, enabling the motion profile to effectively handle the system uncertainties typically encountered in practice. In what follows, the motion profiles that satisfy one, two, and three conditions are referred to as onefold, twofold, and threefold vibration mitigation, respectively. The robustness degrees for the three cases are defined as \(r=1\), \(2\), and \(3\), respectively. This degree can be specified by the user according to actual demands in applications. From a spectral point of view, tuning the profile parameters essentially removes the frequency content of the input motion profile that is related to the resonances. When the vibration suppression conditions are met, the frequency spectrum of the reference acceleration profile exhibits zero magnitude at the resonant frequency, aligning with the system's pole. As a matter of fact, a mechanical system functions as a filter that amplifies or diminishes the magnitudes of different harmonics based on its frequency response. The frequency contents near the resonances in the input are significantly amplified. To more clearly illustrate this point, we consider the expression of the acceleration profile \(a(t)\) in (5) in the Laplace domain:  
\[A(s)=\frac{A_{peak}}{2}\bigg{(}\frac{\pi}{t_{1}}\bigg{)}^{2}\frac{\big{(}1+e^{-st _{1}}\big{)}\big{(}1-e^{-st_{2}}\big{)}\big{(}1-e^{-st_{4}}\big{)}}{\bigg{[}s^{ 2}+\Big{(}\frac{\pi}{t_{1}}\Big{)}^{2}\bigg{]}s}. \tag{32}\]  
Thus, the frequency spectrum of the acceleration profile can be deduced as follows:  
\[|A(j\omega)|=\left|\frac{A_{peak}}{2}\bigg{(}\frac{\pi}{t_{1}}\bigg{)}^{2} \frac{\big{(}1+e^{-j\omega t_{1}}\big{)}\big{(}1-e^{-j\omega t_{2}}\big{)} \big{(}1-e^{-j\omega t_{4}}\big{)}}{\bigg{[}-\omega^{2}+\Big{(}\frac{\pi}{t_{1 }}\Big{)}^{2}\bigg{]}j\omega}\right|=\frac{\bigg{|}4D\cos\Big{(}\frac{\omega t _{1}}{2}\Big{)}\sin\Big{(}\frac{\omega t_{2}}{2}\Big{)}\sin\Big{(}\frac{\omega t _{3}}{2}\Big{)}\bigg{|}}{\bigg{|}\bigg{(}1-\Big{(}\frac{\omega t_{1}}{\pi} \Big{)}^{2}\bigg{)}\omega t_{2}t_{4}\bigg{|}}. \tag{33}\]  
Note that if any one of the conditions C1, C2, or C3 is satisfied, \(|A(j\omega)|\) is equal to zero at \(\omega=\omega_{d}\). Figure 6 gives an example of the acceleration spectra before and after parameter optimization. It can be observed that the contributions of the proposed sinusoidal jerk profile to vibration reduction are twofold: On the one hand, the smoothness of the profile induces a low-pass filtering effect, causing high-frequency components to attenuate rapidly. On the other hand, the optimization of the profile parameters further eliminates frequency contents in the resonant mode that can excite vibrations.