Influence of Leg Flexibilities on the Trajectory Planning of a 3-DOF Spherical Parallel Manipulator

Abstract

The paper deals with a spherical 3-DOF parallel manipulator with three linear actuators considered as the linear springs. The spherical motion of the platform was decomposed into major displacement calculated by solving the direct position problem, and the minor displacement resulting from elastic deflections of the legs under a given load applied to the platform. The minor displacement was found by solving the force and moment equilibrium equations for the platform and for the legs with elastic deflections taken into account. The compliance matrix of the mechanism was derived. The simulation software has been implemented to investigate the effect of leg flexilibities on the platform trajectory planning and used to calculate the influence of the manipulator geometry and stiffness coefficients of the legs on the dynamic behavior. An orientation workspace was determined by using the ranges of the Euler angles. The actual workspace is restricted mainly by the limits on the leg lengths. The singular position was determined by det[J] = 0. The inverse dynamics problem was solved for the actuator forces required to produce a given acceleration of the motion platform. The position, velocity and acceleration of the legs were computed by inverse kinematics. The Euler equation for the platform was used to determine the dynamic moment. Numerical example is given.