Closed-form displacement analysis of 8, 9 and 10-link mechanisms: Part I: 8-link 1-DOF mechanisms

This paper presents closed-form solutions to the displacement analysis problem of planar 8-link mechanisms with 1 degree-of-freedom (DOF). Using the successive elimination procedure presented herein, the degrees of the input–output (I/O) polynomials as well as the number of assembly configurations for all 71 mechanisms resulting from 16 8-link kinematic chains are presented. It is shown that the displacement analysis problems for these mechanisms can be classified into nine distinct structures each of which can be reduced into a univariate polynomial devoid of any extraneous roots. This univariate polynomial corresponds to the I/O polynomial of the mechanism. Three numerical examples illustrating the applicability of the successive elimination procedure to the displacement analysis of 8-link mechanisms are presented. The first example deals with the determination of I/O polynomial for an 8-link mechanism which does not contain any 4-link loops. The second and third examples address in detail some of the problems associated with the conversion of transcendental loop closure equations into an algebraic form using tangent half-angle substitutions. An application of the proposed approach to the displacement analysis of spherical 8-link mechanisms is also presented.