  This paper investigates the conditions in the design parameter space for the
existence and distribution of the cusp locus for planar parallel manipulators.
Cusp points make possible non-singular assembly-mode changing motion, which
increases the maximum singularity-free workspace. An accurate algorithm for the
determination is proposed amending some imprecisions done by previous existing
algorithms. This is combined with methods of Cylindric Algebraic Decomposition,
Gr\"obner bases and Discriminant Varieties in order to partition the parameter
space into cells with constant number of cusp points. These algorithms will
allow us to classify a family of degenerate 3-RPR manipulators.
