Singularity configurations analysis of a 4-DOF parallel mechanisms using Grassmann-Cayley Algebra

Abstract

This article investigates the regularity of the inverse Jacobian matrices of a 4-DOF parallel manipulators performing 3R1T motion pattern, based on Grassmann-Cayley Algebra. The screw theory is borrowed to obtain the Jacobian matrix structure. As the next step, Bracket Ring helps to formulate the Jacobian matrix in a new language, i.e., the brackets. The synergy of brackets and Grassmann-Cayley Algebra enables one to obtain the singularity conditions at the symbolic level. Finally, the Grassmann Geometry (GG) approach paves the way to correspond geometrical configurations of linear varieties to the algebraic expressions which were computed in the previous stage. The last step will reveal the undesired geometrical configurations of limbs that cause the singularity of the inverse Jacobian matrix. Avoiding these configurations will guarantee the proper functionality of parallel mechanisms. Moreover, this paper by touching upon fundamental concepts can be regarded as the reference for further use of Grassmann-Cayley Algebra on obtaining singularity configurations of parallel mechanisms.