Planar kinematic mapping is applied to the five-position Burmester problem for planar four-bar mechanism synthesis. The problem formulation takes the five distinct rigid body poses directly as inputs to generate five quadratic constraint equations. The five poses are on the fourth order curve of intersection of up to four hyperboloids of one sheet in the image space. Moreover, the five poses uniquely specify these two hyperboloids. So, given five positions of any reference point on the coupler and five corresponding orientations, we get the fixed revolute centres, the link lengths, crank angles, and the locations of the coupler attachment points by solving a system of five quadratics in five variables that always factor in such a way as to give two pairs of solutions for the five variables (when they exist).

27th Biennial Mechanisms and Robotics Conference
Department of Mechanical and Aerospace Engineering

Hayes, M.J.D, & Zsombor-Murray, P.J. (P. J.). (2002). Solving the burmester problem using kinematic mapping. In Proceedings of the ASME Design Engineering Technical Conference (pp. 1439–1446).