Generic global rigidity of body-bar frameworks

R. Connelly, T. Jordán, W. Whiteley

Research output: Contribution to journalArticle

15 Citations (Scopus)

Abstract

A basic geometric question is to determine when a given framework G(p) is globally rigid in Euclidean space Rd, where G is a finite graph and p is a configuration of points corresponding to the vertices of G. G(p) is globally rigid in Rd if for any other configuration q for G such that the edge lengths of G(q) are the same as the corresponding edge lengths of G(p), then p is congruent to q. A framework G(p) is redundantly rigid, if it is rigid and it remains rigid after the removal of any edge of G.When the configuration p is generic, redundant rigidity and ( d+ 1)-connectivity are both necessary conditions for global rigidity. Recent results have shown that for d= 2 and for generic configurations redundant rigidity and 3-connectivity are also sufficient. This gives a good combinatorial characterization in the two-dimensional case that only depends on G and can be checked in polynomial time. It appears that a similar result for d≥. 3 is beyond the scope of present techniques and there are examples showing that the above necessary conditions are not always sufficient.However, there is a special class of generic frameworks that have polynomial time algorithms for their generic rigidity (and redundant rigidity) in Rd for any d≥ 1, namely generic body-and-bar frameworks. Such frameworks are constructed from a finite number of rigid bodies that are connected by bars generically placed with respect to each body. We show that a body-and-bar framework is generically globally rigid in Rd, for any d≥ 1, if and only if it is redundantly rigid. As a consequence there is a deterministic polynomial time combinatorial algorithm to determine the generic global rigidity of body-and-bar frameworks in any dimension.

Original languageEnglish
Pages (from-to)689-705
Number of pages17
JournalJournal of Combinatorial Theory. Series B
Volume103
Issue number6
DOIs
Publication statusPublished - Nov 2013

Fingerprint

Rigidity
Polynomials
Configuration
Polynomial-time Algorithm
Connectivity
Sufficient
Necessary Conditions
Combinatorial Algorithms
Congruent
Finite Graph
Framework
Rigid Body
Euclidean space
Polynomial time
If and only if

Keywords

  • Body-bar framework
  • Global rigidity
  • Redundant rigidity
  • Rigid graph
  • Stress matrix

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Theoretical Computer Science
  • Computational Theory and Mathematics

Cite this

Generic global rigidity of body-bar frameworks. / Connelly, R.; Jordán, T.; Whiteley, W.

In: Journal of Combinatorial Theory. Series B, Vol. 103, No. 6, 11.2013, p. 689-705.

Research output: Contribution to journalArticle

Connelly, R. ; Jordán, T. ; Whiteley, W. / Generic global rigidity of body-bar frameworks. In: Journal of Combinatorial Theory. Series B. 2013 ; Vol. 103, No. 6. pp. 689-705.
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