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Journal of Dynamical and Control Systems

, Volume 20, Issue 4, pp 491–501 | Cite as

Coulomb Control of Polygonal Linkages

  • G. Khimshiashvili
  • G. Panina
  • D. Siersma
Article

Abstract

Equilibria of polygonal linkage with respect to Coulomb potential of point charges placed at the vertices of linkage are considered. It is proved that any convex configuration of a quadrilateral linkage is the point of global minimum of Coulomb potential for appropriate values of charges of vertices. Similar problems are treated for the equilateral pentagonal linkage. Some corollaries and applications in the spirit of control theory are also presented.

Keywords

Polygonal linkage Planar configuration Point charge Coulomb potential Critical point Equilibrium 

References

  1. 1.
    Connelly R., Demaine E. Geometry and topology of plygonal linkages, Handbook of discrete and computational geometry, 2nd ed. Boca Raton: CRC Press; 2004, pp. 197–218.Google Scholar
  2. 2.
    Farber M. Invitation to topological robotics, Zuerich Lectures in Advanced Mathematics. Zuerich: European Mathematical Society (EMS); 2008.CrossRefGoogle Scholar
  3. 3.
    Gabrielov A., Novikov D., Shapiro B. Mystery of point charges. Proc. Lond. Math. Soc. 2007;95:443–472.CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Khimshiashvili G., Panina G. Cyclic polygons are critical points of area. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 2008;360(8):238–245.Google Scholar
  5. 5.
    Khimshiashvili G., Siersma D. Cyclic configurations of planar multiple penduli. ICTP Preprint IC/2009/047. p. 11.Google Scholar
  6. 6.
    Khimshiashvili G., Panina G., Siersma D., Zhukova A. Critical configurations of planar robot arms. Centr. Europ. J. Math. 2013;11(3):519–529.CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Kudernac T., Ruangsupapichat N., Parschau M., Mac B., Katsonis N., Harutyunyan S., Ernst K-H., Feringa B. Electrically driven directional motion of a four-wheeled molecule on a metal surface. Nature 2011;479(7372):208–11.CrossRefGoogle Scholar
  8. 8.
    Maxwell J.C. A treatise on electricity and magnetism: London; 1853.Google Scholar
  9. 9.
    Panina G. Moduli space of a planar polygonal linkage: a combinatorial description. arXiv:1209.3241.

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Ilia State UniversityTbilisiGeorgia
  2. 2.Institute for Informatics and AutomationSaint-Petersburg State UniversitySt. PetersburgRussia
  3. 3.University of UtrechtUtrechtThe Netherlands

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