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Kinematic Isotropy and Optimal Kinematic Design of Planar Manipulators and a 3-DOF Spatial Manipulator

Robot operation near isotropic configurations, in which the condition number of the Jacobian matrix reaches unity, is desirable from several points of view. However, determination of all such configurations, given arbitrary robot geometry, is a rather complex problem. In this article all the isotropic configurations of planar manipulators with two, three, and four degrees of freedom are determined, together with that of a 3- DOF spatial manipulator. The solutions are obtained in the form of a fourth-order polynomial for the 3-DOF robot and an eighth-order polynomial for the 4-DOF planar manipulator, resulting in a maximum of eight sets of solutions. The condition numbers are obtained as explicit analytic functions of joint coordinates and link lengths ratios. The optimal length of the links is determined by minimizing the criterion that the condition number increase most slowly with joint angles in the vicinity of isotropic configurations.