The Canonical Form of All Planar Linkage Topologies

Abstract

The current paper shows that all the planar linkages can be constructed from given components called, Assur Graphs, which can be ordered in a table with infinite numbers of rows and columns. In the paper we term this order the canonical form of the planar linkages. This canonical form is proved to be an ordered hierarchy of several levels, enabling systematic generation of all its members. The work has originated from the concept of Assur groups, long known in the field of kinematics, and used to decompose any linkage into basic kinematical atoms. In this paper we introduce a systematic procedure for generating all Assur groups thus finding all the topologies of plane linkages. The work in 2D is based upon known but new mathematical theorems which prove it to be complete and sound. The paper also indicates how this work can be extended into 3D linkages. The mathematical foundation of this work contains several new theorems that have been published by the rigidity theory community during the past six years.