# Design Procedure for Motion Profiles with Sinusoidal Jerk for Vibration Reduction
## 2 General Formulation of Motion Profiles
### Mathematical Expressions of Sinusoidal Jerk Profile  
To mitigate the rate of change in acceleration, the rectangular jerk pulses in segments 1, 3, 5, and 7 of the S-curve profile are replaced by sinusoidal waveforms to ensure jerk  
Figure 1: Seven-segment S-curve motion profile. continuity. Similarly, the sinusoidal jerk profile consists of the acceleration phase, cruise phase, and deceleration phase, which can be further divided into seven segments, as shown in Figure 2. \(T_{1}\), \(T_{2}\), and \(T_{3}\) represent the time periods for the sinusoidal jerk segment, constant acceleration segment, and constant velocity segment, respectively. Under the assumption that the profile has a symmetrical shape with the same acceleration and deceleration time, the time periods are defined as follows: \(T_{1}=t_{1}-t_{0}=t_{3}-t_{2}=t_{5}-t_{4}=t_{7}-t_{6}\), \(T_{2}=t_{2}-t_{1}=t_{6}-t_{5}\), and \(T_{3}=t_{4}-t_{0}\). Consequently, the motion profile is fully defined by these three time parameters and the peak jerk parameter. The total duration of the motion profile is calculated as \(T_{f}=4T_{1}+2T_{2}+T_{3}\). The expression of the jerk profile is described as follows:  
\[j(t)= \begin{cases}J_{peak}\sin\Big{(}\frac{\pi}{T_{1}}(t-t_{0})\Big{)},&t_{0} \leq t<t_{1}\\ 0,&t_{1}\leq t<t_{2}\\ -J_{peak}\sin\Big{(}\frac{\pi}{T_{1}}(t-t_{2})\Big{)},&t_{2}\leq t<t_{3}\\ 0,&t_{3}\leq t<t_{4}\,,\\ -J_{peak}\sin\Big{(}\frac{\pi}{T_{1}}(t-t_{4})\Big{)},&t_{4}\leq t<t_{5}\\ 0,&t_{5}\leq t<t_{6}\\ J_{peak}\sin\Big{(}\frac{\pi}{T_{1}}(t-t_{6})\Big{)},&t_{6}\leq t<t_{7}\\ \end{cases}  \tag{2}\]  
Figure 2: General sinusoidal jerk motion profile (Type 1). where \(J_{peak}\) stands for the peak jerk value during the sinusoidal jerk segment. The corresponding mathematical expressions for the acceleration, velocity, and position profiles can then be derived by sequentially integrating Equation (2):  
\[\left\{ \begin{array}{l}a(t)=a(t_{1})+\int_{t_{1}}^{t}j(t)dt\\ v(t)=v(t_{1})+\int_{t_{1}}^{t}a(t)dt\\ d(t)=d(t_{1})+\int_{t_{1}}^{t}v(t)dt\end{array} \right.. \tag{3}\]  
Note that the peak acceleration is attained at the end of the sinusoidal jerk segment and is maintained during the segment \([t_{1},\ t_{2}]\):  
\[A_{peak}=a(t_{1})=\int_{t_{0}}^{t_{1}}J_{peak}\sin(\frac{\pi}{T_{1}}(t-t_{0}) )dt=\frac{2J_{peak}T_{1}}{\pi}. \tag{4}\]  
Consequently, the expression of the acceleration profile can be written as follows:  
\[a(t)=\left\{ \begin{array}{ll}\frac{A_{peak}}{2}\Big{(}1-\cos\frac{\pi}{T_{1} }(t-t_{0})\Big{)},&t_{0}\leq t<t_{1}\\ A_{peak},&t_{1}\leq t<t_{2}\\ \frac{A_{peak}}{2}\Big{(}1+\cos\frac{\pi}{T_{1}}(t-t_{2})\Big{)},&t_{2}\leq t <t_{3}\\ 0,&t_{3}\leq t<t_{4}\\ -\frac{A_{peak}}{2}\Big{(}1-\cos\frac{\pi}{T_{1}}(t-t_{4})\Big{)},&t_{4}\leq t <t_{5}\\ -A_{peak},&t_{5}\leq t<t_{6}\\ -\frac{A_{peak}}{2}\Big{(}1+\cos\frac{\pi}{T_{1}}(t-t_{6})\Big{)},&t_{6}\leq t <t_{7}\end{array} \right.. \tag{5}\]  
Likewise, the peak velocity is reached at the end of the acceleration phase and remains constant during the segment \([t_{3},\ t_{4}]\), which can be readily determined using geometric principles due to symmetry:  
\[V_{peak}=v(t_{3})=A_{peak}(T_{1}+T_{2})=\frac{2J_{peak}T_{1}}{\pi}(T_{1}+T_{2 }). \tag{6}\]  
The expression of the velocity profile is given as follows:  
\[v(t)=\left\{ \begin{array}{ll}\frac{A_{peak}}{2}\Big{(}(t-t_{0})-\frac{T_{1} }{\pi}\sin\frac{\pi}{T_{1}}(t-t_{0})\Big{)},&t_{0}\leq t<t_{1}\\ \frac{A_{peak}T_{1}}{2}+A_{peak}(t-t_{1}),&t_{1}\leq t<t_{2}\\ A_{peak}\Big{(}\frac{T_{1}}{2}+T_{2}\Big{)}+\frac{A_{peak}}{2}(t-t_{2})+\frac{A _{peak}T_{1}}{2\pi}\sin\frac{\pi}{T_{1}}(t-t_{2}),&t_{2}\leq t<t_{3}\\ V_{peak},&t_{3}\leq t<t_{4}.\\ V_{peak}-\frac{A_{peak}}{2}(t-t_{4})+\frac{A_{peak}T_{1}}{2\pi}\sin\frac{\pi} {T_{1}}(t-t_{4}),&t_{4}\leq t<t_{5}\\ V_{peak}-\frac{A_{peak}T_{1}}{2}-A_{peak}(t-t_{5}),&t_{5}\leq t<t_{6}\\ \frac{A_{peak}T_{1}}{2}-\frac{A_{peak}}{2}(t-t_{6})-\frac{A_{peak}T_{1}}{2\pi} \sin\frac{\pi}{T_{1}}(t-t_{6}),&t_{6}\leq t<t_{7}\end{array} \right.