  This paper investigates the cuspidal configurations of 3-RPR parallel
manipulators that may appear on their singular surfaces in the joint space.
Cusp points play an important role in the kinematic behavior of parallel
manipulators since they make possible a non-singular change of assembly mode.
In previous works, the cusp points were calculated in sections of the joint
space by solving a 24th-degree polynomial without any proof that this
polynomial was the only one that gives all solutions. The purpose of this study
is to propose a rigorous methodology to determine the cusp points of
3-R\underline{P}R manipulators and to certify that all cusp points are found.
This methodology uses the notion of discriminant varieties and resorts to
Gr\"obner bases for the solutions of systems of equations.
