Axis Constraint Analysis and Its Resultant 6R Double-Centered Overconstrained Mechanisms

Abstract

This paper investigates the 6R overconstrained mechanisms by looking at an arrangement that axes intersect at two centers with arbitrary intersection-angles. From the close-loop matrix equation of the mechanism, the paper develops a set of geometric constraint equations of the 6R double-centered overconstrained mechanisms. This leads to the axis constraint equation after applying the Sylvester’s dialytic elimination method. The equation reveals the geometric constraint of link and axis parameters and identifies three categories of the 6R double-centered overconstrained mechanisms with arbitrary axis intersection-angles. The first two categories present two 6R double-centered overconstrained mechanisms and a 6R spherical mechanism. The last category evolves into the 6R double-spherical overconstrained mechanism with arbitrary axis intersection-angles at each spherical center. This further evolves into Baker’s double-Hooke mechanism and his derivative double-spherical mechanism with orthogonal axis intersection. The paper further develops the joint-space solution of the 6R double-centered overconstrained mechanisms based on the geometric constraint equation and verifies the result with a numerical example.