Five precision point-path synthesis of planar four-bar linkage using algebraic method

The problem of synthesizing a planar four-bar linkage with two given fixed pivots such that the coupler curve passes through five given points is considered with the Groebner-Sylvester hybrid approach.First,closed-form equations of a single point are constructed.The reduced Groebner basis in degree lexicographic ordering for the closed-form equations is then obtained using com-puter algebra.A 23623 Sylvester’s matrix can be con-structed by selecting 23 out of 89 Groebner bases.A 36th degree univariate equation is obtained directly from the determinate of the matrix.The same result can be obtained with a continuation method.A numerical example is given and verifies that the problem has at most 36 solutions in the complex field.