Modeling the workspace of a three-link planar manipulator

Abstract

A study of the working space of a three-link planar manipulator was carried out. The basis is taken from analytical dependencies that allow solving the direct problem of kinematics, i.e. determine the coordinates of the gripper center point using three generalized coordinates of the manipulator. The analysis performed made it possible to give a geo-metric interpretation of the dependencies. It has been established that the workspace of the manipulator is a three-parameter set of points. On a plane, this set of points is represented in the form of two disks consisting of ring cells, for which the corresponding analytical dependencies are obtained. The geometric image of this set is a three-dimensional torus. The resulting models are visualized, which facilitates the solution of this problem. To determine the values of the generalized coordinates of the gripper center points, a mapping was carried out by orthogonal projection of the families of circles obtained in the work into four-dimensional space. As a result, three-dimensional hypersurfaces in four-dimensional space were obtained. It is proposed to study them by constructing hypersurface sections models by hyper-planes. Such models in visualization mode allow solving direct and inverse kinematics problems of the manipulator under study.