An Unified Approach for the Determination of Instantaneous Screw Axes for Linkages Associated With the Euclidean Group and Its Subgroups

Abstract

Abstract This paper presents a unified approach for the determination of the instantaneous screw axes for one-DOF linkages associated with the different subgroups of the Euclidean group. The theory behind the approach is based on the most general form of the three-axes theorem and the invariant symmetric forms, Killing and Klein, defined on se(3), the Lie algebra of the Euclidean group, SE (3). Using the Killing and Klein forms on the general form of the three-axes theorem provides sufficient equations to find the instantaneous screw axes of any linkage. The contribution provides a unified approach for the determination of the instantaneous screw axis associated with the relative motion of two bodies in general spatial motion. Further, the equations can be simplified when considering the different subalgebras of the Lie algebra of the Euclidean group, in particular, the spherical and planar subalgebras. The contribution includes two subgroups (subalgebras) that have not been previously analyzed. Further, it is proven that in general both the Killing and Klein forms are required to find the instantaneous screw axes of spatial linkages. Several examples for the general spatial case, and the two subal-gebras, that have been not previously analyzed, are presented.