# Residual Vibration Reduction in Flexible Systems Based on Trapezoidal Velocity Profiles
## 1 Introduction  
The advent of modern manufacturing innovations has spurred increased research interest in robotic automation [1, 2]. Industrial applications now employ robot manipulators for various tasks [3]. Achieving precise object manipulation requires minimizing residual vibrations caused by resonance between the robot and the object upon reaching the target [4]. These vibrations can compromise task quality, increase production cost, and extend completion times, as settling periods are necessary post-movement. Recent studies have highlighted the significance of vibration analysis in robotic systems. [PERSON] et al. [5] demonstrated structural vibration phenomena at the robot flange and mounted tools during spot welding applications using triaxial accelerometers and high-speed camera measurements. The dynamic performance of robotic manipulators has been extensively investigated through modal analysis, as evidenced by [PERSON] and [PERSON] [6] in their investigation of flexibility effects on two-link planar manipulators. Furthermore, [PERSON] et al. [7] developed an analytical framework for the modal analysis of flexible-joint robotic manipulators with rigid links, based on linearized dynamic modeling. Although vibration reduction can be achieved through hardware design optimization or introducing damping components [8], changing the system structure for each task in flexible production environments is impractical. Motion control offers an alternative approach to vibration reduction without altering mechanical design [9; 10], which typically employs feedback or feedforward methods [11]. While feedback control measures and compensates for positioning errors, it necessitates additional sensors and modifications to the actuator control scheme. In contrast, feedforward control influences the vibration behavior by manipulating the robot's driving motion profile. This feedforward approach does not require any revision of the actuator control scheme, as profiles are imposed during trajectory generation. Established methods include creating smooth motion profiles [12; 13] and implementing preshaping techniques [14; 15]. Given the same maximum velocity, maximum acceleration, and displacement, the trapezoidal velocity profile (TVP) provides minimum motion duration compared to other profiles [16]. This profile offers a good balance between simplicity and performance for many applications. It is widely used in robots and machine tools due to its easy implementation. A main drawback is its discontinuous acceleration profile, which may cause undesirable vibrations for flexible systems. Previous research has explored various smooth velocity profiles to reduce robot vibrations, including polynomial and S-curve profiles [17; 18]. [PERSON] et al. [19] planned smooth trajectories using the asymmetric higher-degree polynomial profiles under kinematic constraints, guaranteeing continuity of displacement, velocity, acceleration, and jerk. [PERSON] and [PERSON] [20] carried out collision-free trajectory planning using the fifth-order polynomial to ensure continuous position, speed, and acceleration between the starting and target points. [PERSON] [21] developed an optimal trajectory planning technique to achieve the minimum vibration in two-link rigid-flexible manipulators undergoing large deformations based on a cubic spline function and the particle swarm optimization algorithm, taking into account the effects of geometric nonlinearities. [PERSON] et al. [22] utilized quintic polynomials to design joint trajectory functions and optimized the trajectory with particle swarm optimization for suppressing residual vibration while satisfying the acceleration limits of flexible manipulators. However, these approaches require additional computation due to increased polynomial coefficients. They also demand larger acceleration and torque compared to trapezoidal profiles for equivalent displacements. S-curve motion profiles offer another approach to ensure motion smoothness [23].