On the complete synthesis of finite positions with constraint decomposition via kinematic mapping

In this paper, we revisit the classical Burmester problem of the exact synthesis of a planar four-bar mechanism with up to five task positions. A novel algorithm is presented that uses prescribed task positions to obtain "candidate" manifolds and then find feasible constraint manifolds among them. The first part is solved by null space analysis,and the second part is reduced to finding the solution of two quadratic equations. Five-position synthesis could be solved exactly with up to four resulting dyads. For four-position synthesis, a limited number of solutions could be selected from the ∞¹ many through adding an additional linear constraint equation without increasing the computational complexity. This linear constraint equation could be obtained either by defining one of the coordinates of the center/circle points, by picking the ground line/coupler line, or by adding one additional task position, all of which are proved to be able to convert into the same form as in (23). For three-position synthesis, two additional constraints could be imposed in the same way to select from the ∞² many solutions. The result is a novel algorithm that is simple and efficient, which allows for task driven design of four-bar linkages with both revolute and prismatic joints, as well as handling of different kinds of additional constraint conditions in the same way.