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 computer algebra. A 23 × 23 Sylvester’s matrix can be constructed 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.