Sequential motion planning of non-colliding particles in Euclidean spaces

González, Jesús; Grant, Mark

doi:10.1090/proc/12443

Sequential motion planning of non-colliding particles in Euclidean spaces
HTML articles powered by AMS MathViewer

by Jesús González and Mark Grant PDF

Proc. Amer. Math. Soc. 143 (2015), 4503-4512 Request permission

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

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, DOI 10.2140/agt.2014.14.2103
Robert D. Edwards and Robion C. Kirby, Deformations of spaces of imbeddings, Ann. of Math. (2) 93 (1971), 63–88. MR 283802, DOI 10.2307/1970753
Edward R. Fadell and Sufian Y. Husseini, Geometry and topology of configuration spaces, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2001. MR 1802644, DOI 10.1007/978-3-642-56446-8
Michael Farber, Instabilities of robot motion, Topology Appl. 140 (2004), no. 2-3, 245–266. MR 2074919, DOI 10.1016/j.topol.2003.07.011
Michael Farber and Mark Grant, Topological complexity of configuration spaces, Proc. Amer. Math. Soc. 137 (2009), no. 5, 1841–1847. MR 2470845, DOI 10.1090/S0002-9939-08-09808-0
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, DOI 10.1090/conm/438/08446
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, DOI 10.1090/trans2/212/07
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]
Mark Grant, Gregory Lupton, and John Oprea, Spaces of topological complexity one, Homology Homotopy Appl. 15 (2013), no. 2, 73–81. MR 3117387, DOI 10.4310/HHA.2013.v15.n2.a4
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, DOI 10.1007/978-3-662-02772-1
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, DOI 10.2140/gtm.2008.13.447
Yuli B. Rudyak, On higher analogs of topological complexity, Topology Appl. 157 (2010), no. 5, 916–920. MR 2593704, DOI 10.1016/j.topol.2009.12.007
A. S. Schwarz. The genus of a fiber space, Amer. Math. Soc. Transl. (2), 55:49–140, 1966.