# Design Procedure for Motion Profiles with Sinusoidal Jerk for Vibration Reduction
## 3 Parameter Design for Sinusoidal Jerk Profile  
### Minimum-Time Solution with Actuator Limits  
Before addressing the optimization of the residual vibrations, we first consider the minimum-time solution with only the actuator limits. The motion profile design should ensure that the terminal displacement at the end of the movement is equal to the target moving distance. Moreover, the kinematic values during motion should stay within the physical limits. Given the desired displacement (\(D\)) and the maximum allowable values of the velocity (\(V_{\text{max}}\)), acceleration (\(A_{\text{max}}\)), and jerk (\(J_{\text{max}}\)) imposed by the physical limitations of the actuators, the design constraint conditions are as follows: \(d_{end}=D\), \(\left|J_{peak}\right|\leq J_{\text{max}}\), \(\left|A_{peak}\right|\leq A_{\text{max}}\), \(\left|V_{peak}\right|\leq V_{\text{max}}\). To minimize the motion duration, it is necessary for machines to take full advantage of the kinematic limits and maintain the saturated state for as long as possible. However, there are mutual restrictions among the constraint conditions so that it is not always possible for the motion profile to reach all the limit values. Specifically, the acceleration limit may not be achieved due to the tight limit on the velocity or a short moving distance, while the velocity limit may be unreachable due to the short distance traveled. Consequently, the corresponding constant acceleration and velocity segments may disappear, resulting in a motion profile with fewer than seven segments under different constraint conditions. According to the included segments, the profile model can be classified into four specific types, as illustrated in Figures 2 and 3. By substituting the expressions of the peak kinematic values, the design constraints can be written as follows:  
\[\left\{ \begin{array}{l}\left|J_{peak}\right|\leq J_{\text{max}}\\ \left|\frac{2I_{\text{peak}}T_{1}}{\pi}\right|\leq A_{\text{max}}\\ \left|\frac{2I_{\text{peak}}T_{1}}{\pi}(T_{1}+T_{2})\right|\leq V_{\text{max}} \\ \frac{2I_{\text{peak}}T_{1}}{\pi}(T_{1}+T_{2})(2T_{1}+T_{2}+T_{3})=D\end{array} .\right. \tag{10}\]Note that the sinusoidal jerk segments must exist in the profile, enabling the jerk to consistently reach its limit value, namely, \(J_{peak}=\text{sgn}(D)J_{\text{max}}\) and \(T_{1}>0\), \(T_{2}\geq 0\), \(T_{3}\geq 0\). Hence, the optimal value of \(T_{1}\) that does not violate any constraints is as follows:  
\[T_{1}=\text{min}\left\{\frac{\pi A_{\text{max}}}{2J_{\text{max}}},\sqrt{\frac{ \pi V_{\text{max}}}{2J_{\text{max}}}},\sqrt{\frac{\pi|D|}{2J_{\text{max}}}} \right\}. \tag{11}\]  
If \(T_{1}=\frac{\pi A_{\text{max}}}{2J_{\text{max}}}\), then \(T_{1}\) is determined via the acceleration limit. The acceleration limit (\(A_{\text{max}}\)) can be reached, namely, \(A_{peak}=\text{sgn}(D)A_{\text{max}}\); thus, we can continue to calculate \(T_{2}\). The design constraints for \(T_{2}\) are as follows:  
\[\left\{ \begin{array}{l}|A_{\text{max}}(T_{1}+T_{2})|\leq V_{\text{max}}\\ A_{\text{max}}(T_{1}+T_{2})(2T_{1}+T_{2}+T_{3})=|D|\end{array} \right.. \tag{12}\]  
Therefore, the optimal value of \(T_{2}\) that does not violate any constraints is as follows:  
\[T_{2}=\text{min}\left\{\frac{V_{\text{max}}}{A_{\text{max}}}-T_{1},\ -\frac{3T_{1}}{2}+\sqrt{\frac{T_{1}^{2}}{4}+\frac{|D|}{A_{\text{max}}}} \right\}. \tag{13}\]  
Two cases are distinguished here: If \(T_{2}=\frac{V_{\text{max}}}{A_{\text{max}}}-T_{1}\), then the moving distance is long enough for the motion profile to attain the velocity limit (\(V_{\text{max}}\)). The motion profile belongs to Type 1 in Figure 2, with all seven segments present. Then, the time period for the constant velocity is calculated as follows:  
\[T_{3}=\frac{|D|}{V_{\text{max}}}-2T_{1}-T_{2}; \tag{14}\]  
otherwise, if \(T_{2}=-\frac{3T_{1}}{2}+\sqrt{\frac{T_{1}^{2}}{4}+\frac{|D|}{A_{\text{max}}}}\), then the moving distance falls short of reaching the velocity limit (\(V_{\text{max}}\)) within the motion profile.