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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access December 22, 2012

Critical configurations of planar robot arms

  • Giorgi Khimshiashvili EMAIL logo , Gaiane Panina , Dirk Siersma and Alena Zhukova
From the journal Open Mathematics


It is known that a closed polygon P is a critical point of the oriented area function if and only if P is a cyclic polygon, that is, P can be inscribed in a circle. Moreover, there is a short formula for the Morse index. Going further in this direction, we extend these results to the case of open polygonal chains, or robot arms. We introduce the notion of the oriented area for an open polygonal chain, prove that critical points are exactly the cyclic configurations with antipodal endpoints and derive a formula for the Morse index of a critical configuration.

MSC: 52Cxx; 58E05

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Published Online: 2012-12-22
Published in Print: 2013-3-1

© 2013 Versita Warsaw

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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