Trajectory Planning Method for Flexible Arm Mounted on Flexible Base

In this study, we consider a point-to-point motion of a flexible arm mounted on a flexible base, composed of a slide rail and helical extension springs, and then propose a trajectory planning method for reducing residual vibrations. The Lagrangian approach, in conjunction with the assumed mode method, is applied to derive the equations of motion of the system. In order to accurately construct the mathematical model, the values of the coefficients in the equations of motion are determined by an identification experiment, where Coulomb friction is considered in the dynamics of the flexible base. In the proposed method, the joint angle of the flexible arm is expressed by a combination of polynomial and cycloidal functions. In this case, the trajectory profile of the joint angle depends on the coefficients in the polynomial function. In order to minimize the vibrations of the flexible arm and base after the positioning, the coefficients are tuned by an optimization technique based on metaheuristic algorithms, and then the optimal trajectory is obtained. By rotating the joint angle along the obtained optimal trajectory, the residual vibration of the flexible arm and base can be suppressed. Therefore, the proposed trajectory planning method is classified as a feedforward vibration control technique. Finally, the experiment is performed, to demonstrate the validity and effectiveness of the optimal trajectory obtained from the simulation.