A study of the singularity locus in the joint space of planar parallel manipulators: special focus on cusps and nodes

Cusps and nodes on plane sections of the singularity locus in the joint space of parallel manipulators play an important role in nonsingular assembly-mode changing motions. This paper analyses in detail such points, both in the joint space and in the workspace. It is shown that a cusp (resp. a node) defines a point of tangency (resp. a crossing point) in the workspace between the singular curves and the curves associated with the so-called characteristics surfaces. The study is conducted on a planar 3-RPR manipulator for illustrative purposes.Comment: 4th International Congress Design and Modeling of Mechanical Systems, Sousse : Tunisia (2011

Kinematic analysis of a class of analytic planar 3-RPR parallel manipulators

A class of analytic planar 3-RPR manipulators is analyzed in this paper. These manipulators have congruent base and moving platforms and the moving platform is rotated of 180 deg about an axis in the plane. The forward kinematics is reduced to the solution of a 3rd-degree polynomial and a quadratic equation in sequence. The singularities are calculated and plotted in the joint space. The second-order singularities (cups points), which play an important role in non-singular change of assembly-mode motions, are also analyzed

Nonsingular change of assembly mode without any cusp

International audienceThis paper shows for the first time a parallel manipulator that can execute nonsingular changes of assembly modes while its joint space is free of cusp points and cuspidal edges. The manipulator at hand has two degrees of freedom and is derived from a 3-RPR manipulator; the shape of its joint space is a thickening of a figure-eight curve. A few explanations concerning the relationship between cusps and alpha curves are given

Hidden cusps

International audienceThis paper investigates a situation pointed out in a recent paper, in which a non-singular change of assembly mode of a planar 2-RPR-PR parallel manipulator was realized by encircling a point of multiplicity 4. It is shown that this situation is, in fact, a non-generic one and gives rise to cusps under a small perturbation. Furthermore , we show that, for a large class of singularities of multiplicity 4, there are only two types of stable singularities occurring in a small perturbation: these two types are given by the complex square mapping and the quarto mapping. Incidentally , this paper confirms the fact that, generically, a local non-singular change of solution must be accomplished by encircling a cusp point