Grassmann?Cayley algebra and robotics

Abstract

Abstract. PlUcker coordinates are a suitable way to represent lines in space for purposes of studying instantaneous motions of robot arms. Similarly, we can represent any affine subspace of Euclidean space of any dimension. This can be useful for studying motion spaces of robot arms. In this paper, after we introduce Pltieker coordinates and develop a few of their basic properties, we present them in a somewhat more abstract setting, called the Grassmann-Cayley algebra. Several other applications of this algebra are also described.

A Proof of the Molecular Conjecture

“…Refer to [3,12,26,27] for more detailed descriptions. Throughout the paper, we simply refer to a d-dimensional affine space as a d-affine space for any nonnegative integer d.…”

“…[26]) treats a Plücker coordinate vector at a symbolic level, that is, no coordinate basis is specified, and the symbolic version of a Plücker coordinate vector is referred to as a k-extensor, which is denoted by p 1 ∨ p 2 ∨ · · · ∨ p k . Although we will work on the coordinatized version, we would like to exploit this terminology to follow the conventional notation.…”

“…The Plücker coordinate vector of U is defined as the d+1 k -dimensional vector obtained by writing down all of possible Plücker coordinates of U in some predetermined order, say, the lexicographic order of the indices (see e.g. [12,26] for more details).…”

Straightening-Free Algorithm for the Singularity Analysis of Stewart-Gough Platforms with Collinear/Coplanar Attachments

“…Obtaining the shortest form for each case may need applying syzygies (see [4] for details). There is a direct connection between the exterior calculus (Grassmann-Cayley algebra [5]) and bracket algebra. For instance, the exterior product of four points in space in Grassmann-Cayley algebra translates directly to a bracket.…”

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.…”