Workspace analytic determination of two similar translational parallel manipulators

Abstract

Summary The design of a parallel manipulator for a given workspace would be greatly facilitated, if the analytic expressions of the hypersurfaces bounding the workspace were available. In translational parallel manipulators (TPMs), the hypersurfaces bounding the workspace are actually threedimensional surfaces, which are the geometric locus of all the positions assumed by an end-effector point when the TPM reaches the workspace borders. These surfaces can be represented in a Cartesian reference system fixed to the frame. This paper studies the workspace of two TPMs, that have the same closure equations and workspace when a few geometric conditions are satisfied. The two TPMs are the translational 3-RUU and the DELTA robot. The kinematic analyses of these two TPMs are different from one another only when velocity and accelerations are considered. The result of this study is that the analytic expression of the surfaces bounding their workspace is a fourth degree polynomial equation, in the coordinates of a plattorm point, which contains all the manipulator geometric parameters. This analytic expression is given in an explicit form. The use of this expression is illustrated through a numerical example.