  This paper presents an algorithm for detecting and computing the cusp points
in the joint space of 3-RPR planar parallel manipulators. In manipulator
kinematics, cusp points are special points, which appear on the singular curves
of the manipulators. The nonsingular change of assembly mode of 3-RPR parallel
manipulators was shown to be associated with the existence of cusp points. At
each of these points, three direct kinematic solutions coincide. In the
literature, a condition for the existence of three coincident direct kinematic
solutions was established, but has never been exploited, because the algebra
involved was too complicated to be solved. The algorithm presented in this
paper solves this equation and detects all the cusp points in the joint space
of these manipulators.
