On the Solution of Five-Precision-Points Path Synthesis of Planar Four-Bar Linkages Using Algebraic Method

The problem of five precision points path synthesis of planar four-bar linkage can be divided into four types in term of the input parameters. A unified formulation for the four types is built based on the planar displacement matrix. Next, the corresponding resultant matrix is constructed based on Groebner bases generated by applying the new term ordering (the groups graded reverse lexicographic ordering, <_ggrevlex) for four types. Then, a high-degree univariate polynomial equation is accordingly obtained. At last, several examples are provided to validate the algorithm and the solutions are verified in the software SAM. And it is concluded that type I has 36 solutions, type II has 64 including 16 degenerate solutions, type III has 92 solutions and type IV has 82 solutions including 16 degenerate solutions.