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Point Charges and Polygonal Linkages

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Abstract

We investigate the critical points of Coulomb potential of point charges placed at the vertices of a planar polygonal linkage. It is shown that, for a collection of positive charges on a pentagonal linkage, there is a unique critical point in the set of convex configurations which is the point of absolute minimum. This enables us to prove that two controlling charges are sufficient to navigate between any two convex configurations of a pentagonal linkage.

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References

  1. Agrachev A. Homology of intersections of real quadrics. Sov Math Dokl 1988;37: 493–6.

    MathSciNet  MATH  Google Scholar 

  2. Agrachev A, Lerano L. Systems of quadratic inequalities. J London Math Soc 2012;105(3):622–60.

    Article  MathSciNet  MATH  Google Scholar 

  3. Connelly R, Demaine E. Geometry and topology of polygonal linkages, Handbook of discrete and computational geometry, 2nd ed. Boca Raton: CRC Press; 2004, pp. 197–218.

    Google Scholar 

  4. Cors J, Roberts G. Four-body co-circular central configurations. Nonlinearity 2012;25(2):343–8.

    Article  MathSciNet  MATH  Google Scholar 

  5. Dziobek O. Uber einen Merkwurdigen Fall von Vierkoerper problem. Astron Nachrichten 1900;152:33–46.

    Article  Google Scholar 

  6. Farber M. Invitation to topological robotics. Eur Math Soc 2008.

  7. Hampton M. Concave central configurations in the four-body problem, Doctoral Thesis, University of Washington, Seattle; 2002.

  8. Jing H, Chuanjiang L, Guangfu M, Gang L. Coulomb control of a triangular three-body satellite formation using nonlinear model predictive method. Proc 33rd Chinese Control Conference (CCC) 2014.

  9. Khimshiashvili G, Panina G. Cyclic polygons are critical points of area. Zap Nauchn Sem S-Peterburg Otdel Mat Inst Steklov (POMI) 2008;360(8):238–45.

    MathSciNet  MATH  Google Scholar 

  10. Khimshiashvili G, Panina G, Siersma D. Coulomb control of polygonal linkages. J Dyn Contr Syst 2014;14(4):491–501.

    Article  MathSciNet  MATH  Google Scholar 

  11. Khimshiashvili G, Panina G, Siersma D, Zhukova A. Critical configurations of planar robot arms. Centr Europ J Math 2013;11(3):519–29.

    MathSciNet  MATH  Google Scholar 

  12. 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.

    Article  Google Scholar 

  13. Wang S, Schaub H. Coulomb control of nonequilibrium fixed shape triangular three-vehicle cluster. J Guid Control Dyn 2011;34(1):259–70.

    Article  Google Scholar 

Download references

Acknowledgments

The present paper was completed during a “Research in Pairs” session in CIRM (Luminy) in January of 2015. The authors acknowledge the hospitality and excellent working conditions at CIRM. We also thank Pavel Galashin for useful and inspiring discussions. G. Panina and V. Zolotov were supported by RFBR, research project No. 15-01-02021.

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Authors and Affiliations

  1. Ilia State University, Tbilisi, Georgia

    Giorgi Khimshiashvili

  2. Institute for Informatics and Automation, St. Petersburg, Russia

    Gaiane Panina

  3. Saint-Petersburg State University, St. Petersburg, Russia

    Gaiane Panina & Vladimir Zolotov

  4. University of Utrecht, Utrecht, The Netherlands

    Dirk Siersma

Authors
  1. Giorgi Khimshiashvili
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  2. Gaiane Panina
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  3. Dirk Siersma
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  4. Vladimir Zolotov
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Corresponding author

Correspondence to Gaiane Panina.

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Khimshiashvili, G., Panina, G., Siersma, D. et al. Point Charges and Polygonal Linkages. J Dyn Control Syst 23, 1–17 (2017). https://doi.org/10.1007/s10883-015-9286-3

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  • DOI: https://doi.org/10.1007/s10883-015-9286-3

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