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Machine Theory
Mechanism and Machine Theory 43 (2008) 1401–1415
ate/mechmt
Position analysis in polynomial form of class-three Assur
groups with two or three prismatic joints
S. Mitsi a,*, K.-D. Bouzakis a, G. Mansour a, I. Popescu b
a Laboratory for Machine Tools and Manufacturing Engineering, Aristoteles University, 54124 Thessaloniki, Greece
b Faculty of Mechanics, University of Craiova, Romania
Received 14 February 2006; received in revised form 28 November 2007; accepted 2 December 2007
Available online 14 January 2008
Abstract
This paper presents some new results on the position analysis in polynomial form of the class-three Assur group. The
aim of this position analysis is to determine all possible configurations of the Assur group, for a given position of its exter-
nal joints. Five new kinds of the class-three Assur group with two or three prismatic joints are investigated. The analysis
leads to a non-linear system of equations. Using a essive elimination procedure, a final polynomial equation in one
unknown is obtained. The real solutions of this polynomial equation correspond to assembly modes of the Assur group.
Finally, five numerical applications of the proposed methods are presented.
Ó 2007 Elsevier Ltd. All rights reserved.
Keywords: Class-three Assur group; Position analysis; Polynomial form
1. Introduction
The position analysis of a mechanism is considered as the most important task of the kinematic analysis. In
the case of the multiloop planar mechanisms with decoupled structure the position analysis is performed in a
hierarchical order defined by the mechanism structure, where the constraint equations can be written and
solved separately for each module [1–7]. The modules are the input groups and the well knows Assur groups.
The position analysis for an Assur group is to find all possible configurations, when the position of the exter-
nal joints and dimensions of the links are give