Closed-form displacement analysis of 10-link 1-DOF mechanisms: Part 2 – polynomial solutions

This paper presents closed-form polynomial solutions to the displacement analysis problem of planar 10-link mechanisms with 1 degree of freedom (DOF). Using the successive elimination procedure, the input–output (I/O) polynomials as well as the number of assembly configurations for five mechanisms resulting from two 10-link kinematic chains are presented. It is shown that the displacement analysis problems for all five mechanisms can be reduced to a univariate polynomial devoid of any extraneous roots. This univariate polynomial is the I/O polynomial of the mechanism. In addition, one of the examples also illustrates how trigonometric manipulations in conjunction with tangent half-angle substitutions can lead to non-trivial extraneous roots in the solution process. Theoretical conditions for identifying and eliminating these extraneous roots are also presented.