Critical configurations of planar robot arms

• Autorzy: Giorgi Khimshiashvili , Gaiane Panina , Dirk Siersma , Alena Zhukova
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Mechanical linkage; Robot arm; Configuration space; Moduli space; Oriented area; Morse function; Morse index; Cyclic polygon

Kategorie tematyczne

• 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel\cprime man) theory, etc.)
• 52Cxx: Discrete geometry

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2013
• Tom: 11
• Numer: 3
• Strony: 519-529

Daty

wydano

2013-03-01

online

2012-12-22

Twórcy

autor

Giorgi Khimshiashvili

• Institute for Fundamental and Interdisciplinary Mathematical Studies, Ilia State University, K. Cholokashvili Ave. 3/5, Tbilisi, 0132, Georgia

autor

Gaiane Panina

autor

Dirk Siersma

• Mathematisch Instituut, Universiteit Utrecht, P.O.Box 80010, 3508 TA, Utrecht, The Netherlands

autor

Alena Zhukova

• St. Petersburg State University, Universitetskaya emb. 7-9, St. Petersburg, 199034, Russia

Bibliografia

• [1] Kapovich M., Millson J., On the moduli space of polygons in the Euclidean plane, J. Differential Geom., 1995, 42(2), 430–464
• [2] Khimshiashvili G., Cyclic polygons as critical points, Proc. I. Vekua Inst. Appl. Math., 2008, 58, 74–83
• [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
• [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
• [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]
• [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]
• [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]

Identyfikatory

DOI

10.2478/s11533-012-0147-y