# Design Procedure for Motion Profiles with Sinusoidal Jerk for Vibration Reduction
###### Abstract  
High-speed motions performed by industrial machines can induce severe vibrations that degrade the positioning accuracy and efficiency. To address this issue, this paper proposes a novel motion profile design method utilizing a sinusoidal jerk model to generate fast and smooth motions with low vibrations. The expressions for the acceleration, velocity, and displacement were obtained through successive integrations of the continuous jerk profile. A minimum-time solution with actuator limits was formulated based on an analysis of the critical constraint conditions. Differing from previous studies, the current study introduces an analytical optimization procedure for the profile parameters to minimize both the motion duration and excitation frequency contents corresponding to the system pole. By examining the correlation between the input motion profiles and system responses, the conditions for vibration elimination were identified, highlighting the significance of specific time intervals in controlling the vibration amplitude. Numerical and experimental studies were conducted to validate the effectiveness of the proposed method. The comparative results illustrate that this method outperforms existing baseline techniques in terms of smoothness and vibration attenuation. The residual-vibration level and settling time are significantly reduced with the optimized sinusoidal jerk profile, even in the presence of modeling errors, contributing to higher productivity. motion profile; vibration reduction; trajectory optimization; residual vibration; sinusoidal jerk; motion control 202313320133201332013320132013320133201332013320  
Aiming to achieve faster movements, machines may be expected to work within their physical limitations, causing undesired adverse impacts that compromise their performances. It is widely known that, for given limits on the velocity and force/torque, the trapezoidal velocity profile, which consists of constant acceleration and deceleration in the initial and final phases, respectively, while maintaining maximum velocity in the intermediate phase, minimizes the motion completion time [8; 9]. To date, this profile remains a fundamental choice that is broadly employed in production equipment, such as manipulators or machine tools, due to its simplicity. [PERSON] et al. [10] proposed a profile generator based on a cascaded P-PI position controller and dynamic-range limiters to remove the steady-state error. A closed-form parameter-tuning approach for the energy-saving velocity profiles of servomotor systems was investigated in [11] by formulating a quasi-convex optimization problem. The main limitation is that discontinuities in acceleration tend to stimulate resonant modes, leading to mechanical structure damage and additional time consumption for motion-induced vibration dissipation at the desired position [12]. The adoption of more lightweight and flexible structures with low stiffness in modern industry exacerbates this issue [13]. Though some works have attempted to optimize the acceleration time to minimize the residual vibrations [14; 15; 16], the presence of high-frequency components in the trapezoidal profile necessitates a wide servo bandwidth to ensure tracking accuracy. This dilemma has driven researchers to devise more sophisticated and precise motion profiles that satisfy the desired task specifications while respecting the system's limitations and optimization constraints [17]. As a consequence, the S-curve profile has been developed to yield smoother motions that are easy to implement [18; 19]. By replacing the step acceleration pulses with ramps, this profile exhibits a finite distribution of jerk (the time derivative of acceleration) over a period of time. Numerous studies have shown that S-curve profiles offer advantages in minimizing undesirable effects, such as vibrations, overshoot, actuator stress, and component wear, when compared to trapezoidal velocity profiles [20; 21; 22].