# The assembly modes of rigid 11-bar linkages

2 VEGAS - Effective Geometric Algorithms for Surfaces and Visibility
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
Abstract : Designing an m-bar linkage with a maximal number of assembly modes is important in robot kinematics, and has further applications in structural biology and computational geometry. A related question concerns the number of assembly modes of rigid mechanisms as a function of their nodes n, which is uniquely defined given m. Rigid 11-bar linkages, where n=7, are the simplest planar linkages for which these questions were still open. It will be proven that the maximal number of assembly modes of such linkages is exactly 56. The rigidity of a linkage is captured by a polynomial system derived from distance, or Cayley-Menger, matrices. The upper bound on the number of assembly modes is obtained as the mixed volume of a 5x5 system. An 11-bar linkage admitting 56 configurations is constructed using stochastic optimisation methods. This yields a general lower bound of $\Omega(2.3^n)$ on the number of assembly modes, slightly improving the current record of $\Omega(2.289^n)$, while the best known upper bound is roughly $4^n$. Our methods are straightforward and have been implemented in Maple. They are described in general terms illustrating the fact that they can be readily extended to other planar or spatial linkages. The main results have been reported in conference publication [EM11]. This version (2017) typesets correctly the last Figure 5 so as to include all 28 configurations modulo reflection.
Keywords :
Type de document :
Communication dans un congrès
IFToMM 2011 World Congress, Jun 2011, Guanajuato, Mexico. 2011
Domaine :

https://hal.inria.fr/inria-00530327
Contributeur : Ioannis Emiris <>
Soumis le : vendredi 13 octobre 2017 - 17:31:52
Dernière modification le : mercredi 17 octobre 2018 - 17:02:07
Document(s) archivé(s) le : dimanche 14 janvier 2018 - 12:23:55

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• HAL Id : inria-00530327, version 2
• ARXIV : 1010.6214

### Citation

Ioannis Z. Emiris, Guillaume Moroz. The assembly modes of rigid 11-bar linkages. IFToMM 2011 World Congress, Jun 2011, Guanajuato, Mexico. 2011. 〈inria-00530327v2〉

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