  This paper investigates the existence conditions of cusp points in the design
parameter space of the R\underline{P}R-2P\underline{R}R parallel manipulators.
Cusp points make possible non-singular assembly-mode changing motion, which can
possibly increase the size of the aspect, i.e. the maximum singularity free
workspace. The method used is based on the notion of discriminant varieties and
Cylindrical Algebraic Decomposition, and resorts to Gr\"obner bases for the
solutions of systems of equations.
