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It is known that there are parallel manipulators that can perform nonsingular transitions between different assembly modes. In particular, 3-degree-of-freedom (DOF) manipulators have received primary attention related to this phenomenon. In this paper, the conditions for the existence of special points in the projection of the direct-kinematic-problem-singularity locus onto the joint space for one constant input are obtained. From these conditions, the coordinates of all cusp points can be obtained analytically. Encircling one of these cusp points, it is possible to make a nonsingular transition between two assembly modes of a parallel manipulator. Utilizing these conditions, the range for the existence of cusp points of each input value can be also determined. An extension of the concept of cusp points to the complete joint space is also performed. The procedure is applied to an RPR-2-PRR parallel manipulator that can be solved analytically. Its dimensional variables are parametrized as a 1-D function, and all results are obtained in closed form, which is a benchmark example for other procedures.