Critical configurations of planar robot arms

Giorgi Khimshiashvili; Gaiane Panina; Dirk Siersma; Alena Zhukova

Open Mathematics (2013)

  • Volume: 11, Issue: 3, page 519-529
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Giorgi Khimshiashvili, et al. "Critical configurations of planar robot arms." Open Mathematics 11.3 (2013): 519-529. <http://eudml.org/doc/269625>.

@article{GiorgiKhimshiashvili2013,
abstract = {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.},
author = {Giorgi Khimshiashvili, Gaiane Panina, Dirk Siersma, Alena Zhukova},
journal = {Open Mathematics},
keywords = {Mechanical linkage; Robot arm; Configuration space; Moduli space; Oriented area; Morse function; Morse index; Cyclic polygon; mechanical linkage; robot arm; configuration space; moduli space; oriented area; cyclic polygon},
language = {eng},
number = {3},
pages = {519-529},
title = {Critical configurations of planar robot arms},
url = {http://eudml.org/doc/269625},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Giorgi Khimshiashvili
AU - Gaiane Panina
AU - Dirk Siersma
AU - Alena Zhukova
TI - Critical configurations of planar robot arms
JO - Open Mathematics
PY - 2013
VL - 11
IS - 3
SP - 519
EP - 529
AB - 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.
LA - eng
KW - Mechanical linkage; Robot arm; Configuration space; Moduli space; Oriented area; Morse function; Morse index; Cyclic polygon; mechanical linkage; robot arm; configuration space; moduli space; oriented area; cyclic polygon
UR - http://eudml.org/doc/269625
ER -

References

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  1. [1] Kapovich M., Millson J., On the moduli space of polygons in the Euclidean plane, J. Differential Geom., 1995, 42(2), 430–464 Zbl0854.51016
  2. [2] Khimshiashvili G., Cyclic polygons as critical points, Proc. I. Vekua Inst. Appl. Math., 2008, 58, 74–83 Zbl1222.52010
  3. [3] Khimshiashvili G., Panina G., Siersma D., Zhukova A., Extremal Configurations of Polygonal Linkages, Oberwolfach Preprints, 24, Mathematisches Forschungsinstitut, Oberwolfach, 2011, available at http://www.mfo.de/scientificprogramme/publications/owp/2011/OWP2011_24.pdf Zbl1319.51017
  4. [4] Khimshiashvili G., Siersma D., Cyclic configurations of planar multiply penduli, preprint available at http://users.ictp.it/~pub_off/preprints-sources/2009/IC2009047P.pdf Zbl06269927
  5. [5] Panina G., Khimshiashvili G., Cyclic polygons are critical points of area, J. Math. Sci. (N.Y.), 2009, 158(6), 899–903 http://dx.doi.org/10.1007/s10958-009-9417-z[Crossref] Zbl1193.52015
  6. [6] Panina G., Zhukova A., Morse index of a cyclic polygon, Cent. Eur. J. Math., 2011, 9(2), 364–377 http://dx.doi.org/10.2478/s11533-011-0011-5[Crossref][WoS] Zbl1242.52018
  7. [7] Takens F., The minimal number of critical points of a function on a compact manifold and the Lusternik-Schnirelman cathegory, Invent. Math., 1968, 6, 197–244 http://dx.doi.org/10.1007/BF01404825[Crossref] Zbl0198.56603

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