# Design Procedure for Motion Profiles with Sinusoidal Jerk for Vibration Reduction
## 6 Conclusions  
This paper proposes a novel motion profile design methodology based on a piecewise trigonometric jerk function to address the vibration excitation problem of high-speed machines. The complete analytical mathematical expressions of the motion profile are formulated from the jerk profile. Based on the obtained results, the following conclusions can be drawn:  
1. The motion profile features good smoothness and ensures continuity up to the jerk level. The possibility of introducing physical limits on the velocity, acceleration, and jerk enables the adaptation of the profile to different machines. The critical constraint conditions corresponding to specific cases are derived to obtain the minimum-time solution under physical limitations;
2. Unlike most previous investigations that focus solely on time optimality as the design metric, this study examined the characteristics of the motion profiles from the perspective of dynamic response in order to take full advantage of the vibration elimination capability inherent in the profiles. The findings highlight the significance of the time durations of specific profile segments, rather than lower jerk values, as the usual manner for motion-induced vibration control;
3. Based on the established vibration-free conditions, an optimization algorithm is suggested to generate the time-frequency-optimal motion profile such that both the execution time and residual vibration are minimized while respecting the given constraints on the actuator capacities and robustness. The determination of the optimal profile parameters is achieved in an analytical way and does not pose computational difficulties;
4. A sequence of simulation and experimental tests on a motion stage were carried out to validate the effectiveness of the proposed technique. The results show that the generated velocity, acceleration, and jerk profiles are all bounded and continuous for various given constraint conditions. The comparison studies with other baseline methods demonstrate that the optimized sinusoidal jerk profile exhibits a better performance in terms of vibration reduction and robustness. In particular, it achieves a more than 90% suppression ratio in the residual-vibration amplitude compared to the classical trapezoidal velocity profile. Consequently, the settling time is remarkably shortened, which is beneficial to prevent unnecessary standstill in manufacturing processes. The PSD curves indicate that the high-frequency harmonics are effectively suppressed due to the low-pass filtering effect of the profile;
5. By adjusting the specified robustness degree, the proposed profiles exhibit high insensitivity to system modeling errors. The optimal profile with threefold vibration mitigation maintains a constant settling time, even if the actual natural frequency deviates by over 10% from its nominal value, for which the motion profile is optimized. This feature makes it suitable for cases in which accurate system parameters are not available. The application of this newly elaborated motion profile design allows industrial machines to achieve fast and high-precision movements, thereby enhancing both the productivity and task quality. In future work, this profile design will be improved by a complex domain transformation to address damped cases, and the optimization of more types of motion laws will be investigated through an analysis of the vibration characteristics. Conceptualization, Y.F. ; methodology, Y.F. and Y.Z. ; software, Y.F. ; validation, G.-N.Z. and Y.Z. ; investigation, Y.F. and Y.Z. ; resources, C.G. ; writing--original draft, Y.F. ; writing--review and editing, G.-N.Z. and Y.Z. ; supervision, G.-N.Z. ; project administration, C.G. ; funding acquisition, G.-N.Z. and C.G. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was funded by the National Natural Science Foundation of China (grant numbers 62273235, 52105031) and the China Postdoctoral Science Foundation (grant numbers 2022T150407, 2022 TQ0209).