Coupler-curve synthesis of four-bar linkages via a novel formulation

The coupler-curve synthesis of four-bar linkages is a fundamental problem in kinematics. According to the Roberts-Chebyshev theorem, three cognate linkages can generate the same coupler curve. While the problem of linkage synthesis for coupler-curve generation is determined, it has been regarded as overdetermined, given that the number of coefficients in an algebraic coupler-curve equation exceeds that of linkage parameters available. In this paper, we develop a new formulation of the synthesis problem, whereby the linkage parameters are determined "exactly", within unavoidable roundoff error. A system of coupler-curve coefficient equations is derived, with as many equations as unknowns. The system is thus determined, which leads to exact solutions for the linkage parameters. A method of linkage synthesis from a known coupler-curve equation is further developed to find the three cognate mechanisms predicted by the Roberts-Chebyshev theorem. An example is included to demonstrate the method.