Yanxi Liu, and Robin J. Popplesone
"A Group Theoretic Formalization of Surface Contact"
International Journal of Robotics Research,
13-2, pp. 148-161, The MIT Press. April 1994.


## Abstract

The surface contacts between solids are always associated with a set of symmetries of the contacting surfaces. These symmetries form a group, known as the {\em symmetry group} of the surface.

In this article we develop a group theoretic formalization for describing surface contact between solids. In particular we define: \begin{itemize} \item primitive and compound features of a solid; \item a topological characterization of these features; and \item the symmetry groups of primitive and compound features. \end{itemize}

The symmetry group of a feature is a descriptor of the feature that is at once abstract and quantitative. We show how to use group theory concepts to describe the exact relative motion (position) of solids under surface contacts, which can be either rigid or articulated. The central result of this article is to prove the following:

\begin{itemize} \item when primitive features of a solid are mutually distinct, 1-congruent or 2-congruent, the symmetry group of a compound feature can be expressed in terms of the intersection of the symmetry groups of its primitive features; \item when two solids have surface contact, their relative positions can be expressed as a coset of their common symmetry group, which in turn can be expressed in terms of the intersections of the symmetry groups of the primitive features involved in this contact. \end{itemize}

These results show that using group theory to formalize surface contacts is a general approach for specifying spatial relationships and forms a sound basis for the automation of robotic task planning. One advantage of this formulation is its ability to express continuous motions between two surface-contacting solids in a computational manner, and to avoid combinatorics arising from multiple relationships, especially from discrete symmetries in the assembly parts and their features. At the end of this article a geometric representation for symmetry groups and an efficient group intersection algorithm using characteristic invariants are described.