Modeling the boundaries of the working space of a planar three-link manipulator

Abstract

A study of the boundaries of the working space of a three-link planar manipulator, specified by analytical equations, is carried out. A new geometric interpretation of these samples is proposed. On its basis, it is established that outer space consists of two-parameter volumes of eccentric and concentric circles. When transforming such environments into four-dimensional space, two types of hypersurfaces are
obtained, which represent a geometric model of the manipulator's workspace. The discriminants of these hypersurfaces on the hyperplane are two two-dimensional
surfaces. Both an analytical description of these surfaces and their computer models are obtained. As a result, it is established that the boundaries of the working space
on the plane of the mechanism are the discriminants of such surfaces. To confirm the reliability of the results obtained, as an example, an inverse kinematics problem
is solved on discriminant surfaces — the values of generalized coordinates at the boundary points of the manipulator’s workspace are determined for their given
Cartesian coordinates.