Design Parameter Space of Spherical Four-Bar Linkages

Abstract

Four link twist angles are the design parameters for spherical 4R linkages: changing the magnitudes of the twist angles changes the motion characteristics of the linkage. A new quartic algebraic input-output equation for spherical four-bar linkages, obtained in another paper, contains four terms which each factor into pairs of distinct cubics in the link twist parameters. These eight cubic factors possess a symmetry that suggest they combine to form a shape that, at least locally, bears a remarkable resemblance to a pair of dual tetrahedra in the design parameter space of the link twists. In this paper we show that the location of points relative to the eight distinct cubic surfaces implies a complete classification scheme for all possible spherical 4R linkages. Moreover, we show that the design parameter spaces of both the spherical and planar 4R linkages, with suitable scaling, intersect in 12 lines which form the 12 edges of a pair of dual tetrahedra.