# Residual Vibration Reduction in Flexible Systems Based on Trapezoidal Velocity Profiles
## 1 Introduction
By minimizing or bounding jerk, researchers have demonstrated vibration reduction. However, determining appropriate trajectory parameters for robot actuators remains challenging. [PERSON] et al. proposed a trajectory planning particle swarm optimization algorithm for optimizing the joint angles or paths of robot movements [24]. [PERSON] et al. [25] investigated the systematic design of high-order S-curve trajectories adapted to vibratory behavior under derivative bounds for high-speed operations. [PERSON] et al. [26] presented an angular velocity trajectory generation scheme based on a two-stage S-curve profile to control the load sway for a rotary crane boom in a minimum motion time while considering the crane constraints. In addition, some innovative motion laws based on radial basis, trigonometric, and elliptic functions were also investigated in recent studies [27; 28; 29]. [PERSON] and [PERSON] [30] applied an elliptic jerk profile to the Cartesian space position control of serial robots. [PERSON] et al. [31] defined an acceleration reference profile based on a Gaussian function to ensure a continuous jerk contour for industrial machines, which reduced detrimental vibration effects and improved the machine stability. A general framework for robust vibration mitigation under model parametric uncertainties with three motion profile primitives was introduced in [32] for underactuated mechanical systems. However, commercial industrial robots and machines often feature closed architecture controllers, preventing users from integrating custom profile algorithms or preshaping methods [33; 34]. For safety reasons, motion parameter adjustments are restricted to skilled personnel using the provided trajectory planning method. Most commercial robots employ trapezoidal velocity profiles for trajectory planning [35]. For instance, major robot manufacturers allow operators to input maximum velocity and acceleration values to configure trapezoidal velocity profiles in their industrial robots. Recently, some studies attempted to optimize the vibration level by trial and error [36; 37; 38], suggesting that trapezoidal profiles can achieve good performance by properly tuning the time parameters [39; 40]. Nevertheless, a systematic approach for designing trapezoidal trajectories with actuator limits and vibration suppression constraints is still lacking. This study proposes an analytical approach for designing trapezoidal velocity profiles under actuator limits to reduce residual vibration in flexible motion systems. The method focuses on determining the optimal acceleration/deceleration time by analyzing the correlation between the vibration response characteristics and the natural frequency at which resonance may occur during high-speed movements, such that the trajectory duration and vibration can be minimized. Unlike previous trajectory planning methods that demand extensive computational resources, the proposed approach leverages a simple trapezoidal profile, thus avoiding numerical complexity, and does not require additional modification to the control architecture. It offers a practical solution for engineers and designers to enable vibration suppression in robotic systems, potentially improving precision and efficiency in various industrial applications. The paper is structured as follows: Section 2 gives the formulation of the trapezoidal velocity profile and the time-optimal solution. Section 3 presents a theoretical analysis for vibration behavior and a tuning strategy for determining the optimal time parameters to minimize residual vibration with actuator limits based on the modal frequency. Then, parameter sensitivity evaluation is reported in Section 4 to show the influence of variations in natural frequency and damping ratio on the trajectories with different vibration suppression conditions. Section 5 details the experiments conducted on two industrial robots to validate the proposed method. Finally, Section 6 gives the conclusions.