Singularity Analysis of Lower Mobility Parallel Manipulators Using Grassmann–Cayley Algebra

Abstract

This paper introduces a methodology to analyze geometrically the singularities of manipulators, of which legs apply both actuation forces and constraint moments to their moving platform. Lower mobility parallel manipulators and parallel manipulators, of which some legs have no spherical joint, are such manipulators. The geometric conditions associated with the dependency of six Plücker vectors of finite lines or lines at infinity constituting the rows of the inverse Jacobian matrix are formulated using Grassmann-Cayley algebra (GCA). Accordingly, the singularity conditions are obtained in vector form. This study is illustrated with the singularity analysis of four manipulators.Index Terms-Grassmann-Cayley algebra (GCA), infinite elements, parallel manipulator, screw theory, singularity.

On the Primal and Dual Forms of the Stewart Platform Pure Condition

“…Finally, it is worth realizing that the pure condition, either in its primal or dual forms, can be applied to any parallel manipulator with line-based singularities, a type of manipulators first characterized in [36]. This fact has been used in [9] to analyze lower mobility platforms with three legs, first by using screw algebra to obtain the governing lines, and then applying the pure condition in its primal form to the result. The interest of using the dual form of the pure condition instead is certainly a point that deserves further attention.…”

“…Then, a generic Stewart platform is usually referred to as a 6-6 Stewart platform while, on the other side of the spectrum, a 3-3 platform refers to any of the three possible topologies in which a Stewart platform has three attachments both in the base and in the platform. Although coalescing attachments leads to multiple spherical joints whose implementation is difficult [6], studying the singularities for all these possible topologies is of great practical relevance because the singularities of 6-6 platforms with some particular arrangements of attachments [7], and the singularities of some 3-legged robots can be reduced to the study of the singularities of some of these topologies [8], [9].…”

Conceptual Design of Schoenflies Motion Generators Based on the Wrench Graph

“…Points at infinity were used initially in a superbracket to examine the singularity of 3-UPU by Kanaan et al [19,20]. This improvement enhanced the application of Grassmann-Cayley Algebra for limited dof parallel manipulators.…”

Singularity Condition of Wrist-Partitioned 6-R Serial Manipulator Based on Grassmann-Cayley Algebra

“…Among these approaches Grassmann-Cayley Algebra ( ) GCA is probably one of the most efficient since it provides sufficient tools to properly analyze geometrically the singularity condition without coordinate expression. GCA approach is suitable for analyzing the rigidity of the framework of the architecture and for scene analysis [1][2][3][4][5][6][7][8].It has a powerful tools for geometric interpretation of coordinate free representation and singularity analyzing in real time computing .The solution provided in GCA language by vanishing the superbrackets decomposition is a single condition which contains all general and particular cases [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17].To prevent the clash of serial robot's actuators which are in singularity configuration, we firstly determined t J related to its twist and secondly calculate the dependency condition of the det( ) t J which rows are Plücker coordinate lines.…”