  This paper deals with the in-depth kinematic analysis of a special spherical
parallel wrist, called the Agile Eye. The Agile Eye is a three-legged spherical
parallel robot with revolute joints in which all pairs of adjacent joint axes
are orthogonal. Its most peculiar feature, demonstrated in this paper for the
first time, is that its (orientation) workspace is unlimited and flawed only by
six singularity curves (rather than surfaces). Furthermore, these curves
correspond to self-motions of the mobile platform. This paper also demonstrates
that, unlike for any other such complex spatial robots, the four solutions to
the direct kinematics of the Agile Eye (assembly modes) have a simple geometric
relationship with the eight solutions to the inverse kinematics (working
modes).
