Dual number method, rank of a screw system and generation of Lie sub-algebras

Abstract

The classical method used to estimate the rank of a set χ of m screws, that is to calculate the rank of matrix of dimension m×6, is not often practical especially in singular cases. The aim of this paper is to explain a general method to evaluate the rank of such a set by the determination of the rank of a three-dimensional system with the help of dual numbers and of the representation of screws by elements of the Lie algebra of skew symmetric vector fields. With this method, we propose a new method of classification of screw sets based on the concept of free maximal list of χ. Moreover, we also study the subspaces generated by the set χ using the Lie brackets of ith order. We prove that the subspaces generated by the 4th order bracket is always a Lie subalgebra. Finally, we give a complete demonstration of the mobility conditions of the Bennett mechanisms without using assumptions used in the previous proof by Ogino in [9].