Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

Sequential motion planning of non-colliding particles in Euclidean spaces


Authors: Jesús González and Mark Grant
Journal: Proc. Amer. Math. Soc. 143 (2015), 4503-4512
MSC (2010): Primary 55R80, 55S40; Secondary 55M30, 68T40
DOI: https://doi.org/10.1090/proc/12443
Published electronically: June 5, 2015
MathSciNet review: 3373948
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In terms of Rudyak's generalization of Farber's topological complexity of the path motion planning problem in robotics, we give a complete description of the topological instabilities in any sequential motion planning algorithm for a system consisting of non-colliding autonomous entities performing tasks in space whilst avoiding collisions with several moving obstacles. The Isotopy Extension Theorem from manifold topology implies, somewhat surprisingly, that the complexity of this problem coincides with the complexity of the corresponding problem in which the obstacles are stationary.


References [Enhancements On Off] (What's this?)

  • [1] Ibai Basabe, Jesús González, Yuli B. Rudyak, and Dai Tamaki, Higher topological complexity and its symmetrization, Algebr. Geom. Topol. 14 (2014), no. 4, 2103-2124. MR 3331610, https://doi.org/10.2140/agt.2014.14.2103
  • [2] Robert D. Edwards and Robion C. Kirby, Deformations of spaces of imbeddings, Ann. Math. (2) 93 (1971), 63-88. MR 0283802 (44 #1032)
  • [3] Edward R. Fadell and Sufian Y. Husseini, Geometry and topology of configuration spaces, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2001. MR 1802644 (2002k:55038)
  • [4] Michael Farber, Instabilities of robot motion, Topology Appl. 140 (2004), no. 2-3, 245-266. MR 2074919 (2005g:68166), https://doi.org/10.1016/j.topol.2003.07.011
  • [5] Michael Farber and Mark Grant, Topological complexity of configuration spaces, Proc. Amer. Math. Soc. 137 (2009), no. 5, 1841-1847. MR 2470845 (2009j:55020), https://doi.org/10.1090/S0002-9939-08-09808-0
  • [6] Michael Farber, Mark Grant, and Sergey Yuzvinsky, Topological complexity of collision free motion planning algorithms in the presence of multiple moving obstacles, Topology and robotics, Contemp. Math., vol. 438, Amer. Math. Soc., Providence, RI, 2007, pp. 75-83. MR 2359030 (2008i:55015), https://doi.org/10.1090/conm/438/08446
  • [7] Michael Farber and Sergey Yuzvinsky, Topological robotics: Subspace arrangements and collision free motion planning, Geometry, topology, and mathematical physics, Amer. Math. Soc. Transl. Ser. 2, vol. 212, Amer. Math. Soc., Providence, RI, 2004, pp. 145-156. MR 2070052 (2005i:55019)
  • [8] Jesús González, Bárbara Gutiérrez, and Sergey Yuzvinsky, The higher topological complexity of subcomplexes of spheres - and related polyhedral product spaces. Submitted for publication. arXiv:1501.07474[math.AT]
  • [9] Mark Grant, Gregory Lupton, and John Oprea, Spaces of topological complexity one, Homology Homotopy Appl. 15 (2013), no. 2, 73-81. MR 3117387, https://doi.org/10.4310/HHA.2013.v15.n2.a4
  • [10] Peter Orlik and Hiroaki Terao, Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 300, Springer-Verlag, Berlin, 1992. MR 1217488 (94e:52014)
  • [11] Fridolin Roth, On the category of Euclidean configuration spaces and associated fibrations, Groups, homotopy and configuration spaces, Geom. Topol. Monogr., vol. 13, Geom. Topol. Publ., Coventry, 2008, pp. 447-461. MR 2508218 (2010m:55003), https://doi.org/10.2140/gtm.2008.13.447
  • [12] Yuli B. Rudyak, On higher analogs of topological complexity, Topology Appl. 157 (2010), no. 5, 916-920. MR 2593704 (2011c:55009), https://doi.org/10.1016/j.topol.2009.12.007
  • [13] A. S. Schwarz.
    The genus of a fiber space,
    Amer. Math. Soc. Transl. (2), 55:49-140, 1966.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 55R80, 55S40, 55M30, 68T40

Retrieve articles in all journals with MSC (2010): 55R80, 55S40, 55M30, 68T40


Additional Information

Jesús González
Affiliation: Departamento de Matemáticas, Centro de Investigación y de Estudios Avanzados del IPN, Av. IPN 2508, Zacatenco, México City 07000, México
Email: jesus@math.cinvestav.mx

Mark Grant
Affiliation: School of Mathematics & Statistics, Newcastle University, Herschel Building, Newcastle upon Tyne NE1 7RU, United Kingdom
Email: mark.grant@newcastle.ac.uk

DOI: https://doi.org/10.1090/proc/12443
Keywords: Robot motion planning, higher topological complexity, sectional category, configuration spaces, moving obstacles
Received by editor(s): October 2, 2013
Received by editor(s) in revised form: January 9, 2014
Published electronically: June 5, 2015
Additional Notes: The first author was supported by Conacyt Research Grant 221221.
Communicated by: Michael A. Mandell
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society