Using the kinematics of the RC linkage to find the degree of the adjoint representation of SE(3)

Abstract

This work studies the projective algebraic variety formed from the closure of the adjoint representation of the group of rigid-body displacements, $SE\left(3\right)$. This is motivated by asking how many assembly configurations a mechanism would have in general, if it was designed to keep six given lines in six linear line complexes.

The main result is to find the degree of the variety defined by the adjoint representation and hence answer the motivating question. A simple special case is discussed, a mechanism that maintains a single given line reciprocal to three fixed lines from the regulus of a cylindrical hyperboloid of one sheet. The three dimensional variety defined in this way can be realised by an RC linkage. More specifically, the variety splits into two components each of which can be realised by an RC linkage. The homology of these 3-dimensional varieties, as subvarieties of the Study quadric, is found and used to determine the degree of the adjoint representation as an algebraic variety.

The possible equations defining the variety determined by the adjoint representation of $SE\left(3\right)$, are also discussed but no definitive result is found.