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

Cusp points in the parameter space of RPR-2PRR parallel manipulator