Motion Generation Synthesis for Four and Five Multiply Separated Position Problems Using Monad Auxiliary Equations

Abstract

The need for a linkage exactly to generate desired motions such as passing through some existing finitely, infinitesimally separated positions or their mixed separated positions is an important kinematic synthesis problem. In accordance with the closed vector loops in a linkage, the relative motion differences of monads were used to build a synthesis equation system, which contained some independent unknown angular variables and hence was a mixed trigonometric polynomial system. By adding the component variables of the monad vectors and some auxiliary equations, the mixed trigonometric polynomial system was transformed skillfully to a general polynomial system, and then the total degree of the synthesis system of five positions for thousands, even tens of thousands, dropped sharply to 16. The complete possible combinations for four- and five- multiply separated positions were also presented. Examples were given to demonstrate modelling the synthesis system with the monad method, calculating their total degrees and mixed volumes, and then solving it with Homotopy method. Comparing the complexities in derivation, the simplicity, and the total degrees of the synthesis equations building with different approaches shows that our method seems to be more easily used to model such equations with best modularity and lower total degrees or mixed volumes.