A fully analytical method for coupler-curve synthesis of planar four-bar linkages

Coupler-curve synthesis is a classical problem in linkage kinematics. A recent study shows that the solutions to the coupler-curve synthesis problems of four-bar linkages can be obtained based on a combination of analytical and graphical methods. In this work, a fully analytical method is developed to find the exact solutions of coupler-curve equations for the planar four-bar linkages. The developed method is based on a novel algebraic formulation of the system of coefficient equations, which yields a small system of four equations and four variables about the positions of pivoting joints. With any given algebraic coupler curve of a planar four-bar linkage, the position parameters of pivoting joints can be easily solved, while the other parameters can be calculated by the remaining equations. Two cases are provided to demonstrate and validate the fully analytical method.