Position analysis in polynomial form of planar mechanisms with a closed chain of the Assur group of class 4 L’analyse de position sous forme polynomiale des mecanismes plans avec une chaine fermee du groupe d'Assur de 4-eme classe

Abstract

This paper presents a symbolic solution for the position analysis of the Assur group with four links and six revolute joints (tetrad). The aim of this position analysis is to determine all possible configurations of the group, for a given position of its external joints. The analysis leads to a non-linear system of three equations with three unknown parameters. Using a successive elimination procedure of the unknown parameters, a final polynomial equation of sixth order in only one unknown is derived. The six roots of the polynomial equation lead to, at most, six different configurations of the group in the complex field. Numerical examples lead to four real solutions, which correspond to the assembly modes of the tetrad. Finally, application of the proposed method is given for planar mechanisms including such groups.