A Group Theoretic Formalization of Surface Contact

Yanxi Liu, Robin Popplestone

Research output: Contribution to journalArticle

18 Citations (Scopus)

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 symmetry group of the surface. In this article we develop a group theoretic formalization for describing surface contact between solids. In particular we define (1) primitive and compound features of a solid, (2) a topological characterization of these features, and (3) the symmetry groups of primitive and compound features. The symmetry group of a feature is a descriptor of the fea ture 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: 1. When primitive features of a solid are mutually dis tinct, 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. 2. When two solids have surface contact, their rela tive positions can be expressed as a coset of their common symmetry group, which in turn can be ex pressed in terms of the intersections of the symmetry groups of the primitive features involved in this con tact. 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 fea tures. At the end of this article, a geometric representation for symmetry groups and an efficient group intersection algorithm using characteristic invariants are described.

Original languageEnglish (US)
Pages (from-to)148-161
Number of pages14
JournalThe International Journal of Robotics Research
Volume13
Issue number2
DOIs
StatePublished - Jan 1 1994

Fingerprint

Symmetry Group
Formalization
Contact
Group theory
Intersection
Congruent
Group Theory
Symmetry
Geometric Representation
Motion
Coset
Combinatorics
Descriptors
Automation
Robotics
Express
Planning
Acoustic waves
Invariant
Formulation

All Science Journal Classification (ASJC) codes

  • Software
  • Modeling and Simulation
  • Mechanical Engineering
  • Electrical and Electronic Engineering
  • Artificial Intelligence
  • Applied Mathematics

Cite this

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title = "A Group Theoretic Formalization of Surface Contact",
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 symmetry group of the surface. In this article we develop a group theoretic formalization for describing surface contact between solids. In particular we define (1) primitive and compound features of a solid, (2) a topological characterization of these features, and (3) the symmetry groups of primitive and compound features. The symmetry group of a feature is a descriptor of the fea ture 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: 1. When primitive features of a solid are mutually dis tinct, 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. 2. When two solids have surface contact, their rela tive positions can be expressed as a coset of their common symmetry group, which in turn can be ex pressed in terms of the intersections of the symmetry groups of the primitive features involved in this con tact. 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 fea tures. At the end of this article, a geometric representation for symmetry groups and an efficient group intersection algorithm using characteristic invariants are described.",
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A Group Theoretic Formalization of Surface Contact. / Liu, Yanxi; Popplestone, Robin.

In: The International Journal of Robotics Research, Vol. 13, No. 2, 01.01.1994, p. 148-161.

Research output: Contribution to journalArticle

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