In this article we survey some recent developments in optimal robot de
sign, and collect some of the differential geometric approaches into a
general mathematical framework for robot design. The geometric framew
ork permits a set of coordinate-free definitions of robot performance
that can be optimized for designing both open- and closed-chain roboti
c mechanisms. In particular, workspace volume is precisely defined by
regarding the rigid body motions as a Riemannian manifold, and various
features of actuators, as well as inertial characteristics of the rob
ot, can be captured by the suitable selection of a Riemannian metric i
n configuration space. The integral functional of harmonic mapping the
ory also provides a simple and elegant global description of dexterity
.