# Design Procedure for Motion Profiles with Sinusoidal Jerk for Vibration Reduction
###### Abstract
An algorithm to generate a seven-segment velocity profile is developed in [23] for motion control in a hybrid electronic platform. [PERSON] et al. [24] optimized the ramp-up time of S-curve profiles based on the frequency content of the force input to minimize the response time and residual vibrations. [PERSON] et al. [25] proposed an optimization model to achieve the time-optimal asymmetrical S-curve velocity scheduling of a specified end-effector path with kinematic constraints. However, the limited jerk employed in these profiles is accompanied by sudden transitions at the switching instances of segments, which are associated with high-frequency content. To further dampen vibrations, considerable research efforts have been devoted to attaining smooth changes in jerk. A representative idea of defining such a motion law is based on higher-order polynomials and trigonometric functions [26; 27; 28; 29]. [PERSON] et al. [30] suggested a fourth-order trajectory-planning algorithm with feedforward control under given physical bounds for electromechanical motion systems. [PERSON] et al. [31] developed a smooth, real-time, snap-bounded profile generation technique based on an embedded-system platform. The experimental results showed that the snap-constrained method effectively reduces the tracking error when compared to the seven-segment and trapezoidal velocity methods. [PERSON] et al. [32] employed the concatenation of quintic polynomials to provide a bounded and continuous jerk point-to-point trajectory for the vibration control of a flexible-joint manipulator with parametric uncertainties. However, the complexity becomes significant as the number of profile segments increases, presenting a challenge for real-time applications. In some cases, the utilization of a numerical method is necessary, which may fail to obtain a feasible solution or become stuck in local convergence due to the selection of an inappropriate initial solution. To avoid the excessive computational burden for the controller brought on by high-degree polynomials, [PERSON] et al. [33] present a low-vibration motion profile generation scheme using a level-shifted sinusoidal waveform to design the acceleration profile for high-speed positioning. Simulation and experimental studies have demonstrated that their method achieves remarkably lower residual vibrations relative to the conventional trapezoidal profile and S-curve profile. The main limitation of this scheme is its incompleteness in addressing special cases, rendering it inapplicable under certain physical constraints. [PERSON] et al. [34] studied a jerk-bounded trigonometric S-curve trajectory planner based on a three-phase sine jerk motion profile for the pick-and-place operations of a six-DOF robotic manipulator. However, the study does not guarantee the fulfillment of specified kinematic constraints and time optimality. More recently, [PERSON] et al. [35], using the same profile, adopted a multivariable time optimization method to expand the range of feasible solutions for achieving the shortest running time. [PERSON] et al. [36] utilized a locally asymmetrical jerk profile to establish smooth and time-optimal point-to-point trajectories for industrial robots. In [37], a trigonometric feed-rate-scheduling algorithm is introduced to enable the continuous velocity, acceleration, and jerk control of the parametric interpolation, which reduces undesirable vibrations in high-speed and high-accuracy machining. However, most of these studies mainly dealt with the generation of trajectories with time optimality under physical limits, while comprehensive quantitative analyses of the influence of the profile parameters on the vibration responses were neglected. To ensure that machines produce rapid and precise movements, the optimization of the profile parameters adapted to the vibration behavior is of significance.