Position analysis in polynomial form of planar mechanism with an Assur group of class 4 including one prismatic joint

This paper presents the solution in polynomial form to the position analysis of the Assur group of class 4 (tetrad) with four links, one prismatic and five revolute joints. The aim of this position analysis is to determine all possible configurations of the Assur 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, a final polynomial equation of sixth order in one unknown is obtained. The roots of the polynomial equation lead to, at most, six different configurations of the Assur group in the complex field. Two kinds of the Assur group of class 4, the first with one external and the second with one internal prismatic joint, are investigated. In both cases, the numerical examples lead to two real solutions, which correspond to assembly modes of the Assur group. An application of the proposed method is also given for planar mechanism including such groups.