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This paper investigates the singular curves in two-dimensional slices of thejoint space of a family of planar parallel manipulators. It focuses on specialpoints, referred to as cusp points, which may appear on these curves. Cusppoints play an important role in the kinematic behavior of parallelmanipulators since they make possible a nonsingular change of assembly mode.The purpose of this study is twofold. First, it reviews an important previouswork, which, to the authors' knowledge, has never been exploited yet. Second,it determines the cusp points in any two-dimensional slice of the joint space.First results show that the number of cusp points may vary from zero to eight.This work finds applications in both design and trajectory planning.

Singular curves and cusp points in the joint space of 3-RPR parallel manipulators