  This paper investigates two situations in which the forward kinematics of
planar 3-RPR parallel manipulators degenerates. These situations have not been
addressed before. The first degeneracy arises when the three input joint
variables r1, r2 and r3 satisfy a certain relationship. This degeneracy yields
a double root of the characteristic polynomial in t, which could be erroneously
interpreted as two coalesce assembly modes. But, unlike what arises in
non-degenerate cases, this double root yields two sets of solutions for the
position coordinates (x, y) of the platform. In the second situation, we show
that the forward kinematics degenerates over the whole joint space if the base
and platform triangles are congruent and the platform triangle is rotated by
180 deg about one of its sides. For these "degenerate" manipulators, which are
defined here for the first time, the forward kinematics is reduced to the
solution of a 3rd-degree polynomial and a quadratics in sequence. Such
manipulators constitute, in turn, a new family of analytic planar manipulators
that would be more suitable for industrial applications.
