In this article we develop a mathematical theory for optimizing the ki
nematic dexterity of robotic mechanisms and obtain a collection of ana
lytical tools for robot design. The performance criteria we consider a
re workspace volume and dexterity; by the latter we mean the ability t
o 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 necessar
ily be independent of the particular coordinate representation of the
kinematics. By regarding the forward kinematics of a mechanism as defi
ning a mapping between Riemannian manifolds, we apply the coordinate-f
ree language of differential geometry to define natural measures of ki
nematic dexterity and workspace volume. This approach takes into accou
nt the geometric and topological structures of the joint and workspace
s. We show that the functional associated with harmonic mapping theory
provides a natural measure of the kinematic dexterity of a mechanism.
Optimal designs among the basic classes of mechanisms are determined
as extrema of this measure. We also examine the qualitative connection
s between kinematic dexterity and workspace volume.