Kinematic Dexterity of Robotic Mechanisms

In this article we develop a mathematical theory for optimizing the kinematic dexterity of robotic mechanisms and obtain a collection of analytical tools for robot design. The performance criteria we consider are workspace volume and dexterity; by the latter we mean the ability to move and apply forces in arbitrary directions as easily as possible. Clearly, dexterity and workspace volume are intrinsic to a mechanism, so that any mathematical formulation of these properties must necessarily be independent of the particular coordinate representation of the kinematics. By regarding the forward kinematics of a mechanism as defining a mapping between Riemannian manifolds, we apply the coordinate-free language of differential geometry to define natural measures of kinematic dexterity and workspace volume. This approach takes into account the geometric and topolog ical structures of the joint and workspaces. We show that the functional associated with harmonic mapping theory provides a natural measure of the kinematic dexterity of a mechan ism. Optimal designs among the basic classes of mechanisms are determined as extrema of this measure. We also examine the qualitative connections between kinematic dexterity and workspace volume.