COMPARATIVE ANALYSIS OF SINGULARITIES OF 3-PRS AND 3- PRS PARALLEL MANIPULATORS USING GRASSMANN-CAYLEY

Luc Djimon Clément AKONDE

Abstract: Grassmann-Cayley Algebra (GCA) is a symbolic approach to Plücker coordinates lines. It is used to write expressions for geometric incidences in projective geometry.GCA gives a useful way to analyze mechanism singularity without algebraic coordinate expression. Since it has great powerful tools for coordinate’s free representation, geometric interpretation and singularity analyzing in real time computing, this method is implemented in the present research. The goal of this study is finding the singularities conditions of 3-PRS Parallel Manipulators (PMs) based on GCA when the position of the actuators is firstly on prismatic joints and secondly on revolute joints. Geometric method of reciprocal screw is used to formulate the Global Wrench System (GWS) of the PMs. GWS represents the determinant of Jacobian matrix which rows are Plücker coordinate lines. The degeneracy analysis of determinant of this matrix based on the dependency notion of Plücker coordinate lines is performed in coordinate free expressions and provides double and single singularities conditions, respectively for3-PRS and 3-PRS PMs, which contains all generals and specials cases. The keys elements of transition between the rows of Jacobian matrix and singularity condition are the introduction of Plücker lines, superbrackets, join and meet operators. The vanishing points of these superbrackets decomposition are analyzed to describe the singularity conditions associated with different motions. The results indicate that for the 3-PRS and 3-PRS PMs, a common singularity occurs when the three constraints forces of the mobile platform lie on the common plane and intersect at a unique point. It is suggested that both position and orientation of the actuator have to be integrated in the conceptual design stage in order to optimize the rigidity frame of the mechanism.

Keywords: Singularity, 3-PRS Parallel Manipulators, Grassmann-Cayley Algebra, Projective space, Wrench graph.

DOI: https://doi.org/10.15623/ijret.2013.0212118