Abstract
In this paper we discuss topological problems inspired by robotics. We study in detail the robot motion planning problem. With any path-connected topological space X we associate a numerical invariant TC(X) measuring the “complexity of the problem of navigation in X.„We examine how the number TC(X) determines the structure of motion planning algorithms, both deterministic and random.
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References
Abrams, A. (2002) Configuration spaces of colored graphs, Geom. Dedicata 92, 185–194.
Adem, J., Gitler, S., and James, I. M. (1972) On axial maps of a certain type, Bol. Soc. Mat. Mexicana(2) 17, 59–62.
Connelly, R., Demaine, E. D., and Rote, G. (2003) Straightening polygonal arcs and convexifying polygonal cycles, Discrete Comput. Geom. 30, 205–239.
Cornea, O., Lupton, G., Oprea, J., and Tanré, D. (2003) Lusternik–Schnirelmann category, Vol. 103 of Math. Surveys Monogr., Providence, RI, Amer. Math. Soc.
Davis, D. M. (1993) Immersions of projective spaces: a historical survey, In M. C. Tangora (ed.), Algebraic Topology, Vol. 146 of Contemp. Math., Oaxtepec, 1991, pp. 31–37, Providence, RI, Amer. Math. Soc.
Dold, A. (1972) Lectures on Algebraic Topology, Vol. 200 of Grundlehren Math. Wiss., Berlin, Springer.
Eilenberg, S. and Ganea, T. (1957) On the Lusternik–Schnirelmann category of abstract groups, Ann. of Math. (2) 65, 517–518.
Fadell, E. and Neuwirth, L. (1962) Configuration spaces, Math. Scand. 10, 111–118.
Farber, M. (2003) Topological complexity of motion planning, Discrete Comput. Geom. 29, 211–221.
Farber, M. (2004) Instabilities of robot motion, Topology Appl. 140, 245–266.
Farber, M. (2005) Collision free motion planning on graphs, In Algorithmic Foundations of Robotics.VI, Utrecht/Zeist, 2004, Berlin, Springer, to appear.
Farber, M., Tabachnikov, S., and Yuzvinsky, S. (2003) Topological robotics: motion planning in projective spaces, Int. Math. Res. Not. 34, 1853–1870.
Farber, M. and Yuzvinsky, S. (2004) Topological robotics : subspace arrangements and collision free motion planning, In V. M. Buchstaber and I. M. Krichever (eds.), Geometry, Topology, and Mathematical Physics, Vol. 212 of Amer. Math. Soc. Transl. Ser. 2, Moscow, 2002–2003, pp. 145–156, Providence, RI, Amer. Math. Soc.
Felix, Y. and Halperin, S. (1982) Rational LS category and its applications, Trans. Amer. Math. Soc. 273, 1–38.
Gal,Ś. (2001) Euler characteristic of the configuration space of a complex, Colloq. Math. 89, 61–67.
Ghrist, R. (2001) Configuration spaces and braid groups on graphs in robotics, In J. Gilman, W. W. Menasco, and X.-S. Lin (eds.), Knots, Braids, and Mapping Class Groups—Papers Dedicated to Joan S. Birman, Vol. 24 of AMS/IP Stud. Adv. Math., New York, 1998, pp. 29–40, Providence, RI, Amer. Math. Soc. & Somerville, MA, International Press.
Ghrist, R. W. and Koditschek, D. E. (2002) Safe cooperative robot dynamics on graphs, SIAM J. Control Optim. 40, 1556–1575.
Hausmann, J.-C. and Knutson, A. (1998) Cohomology rings of polygon spaces, Ann. Inst. Fourier(Grenoble) 48, 281–321.
Jordan, D. and Steiner, M. (1999) Configuration spaces of mechamical linkages, Discrete Comput. Geom. 22, 297–315.
Kapovich, M. and Millson, J. J. (1996) The symplectic geometry of polygons in Euclidean space, J. Di.fferential Geom. 44, 479–513.
Kapovich, M. and Millson, J. J. (2002) Universality theorems for configuration spaces of planar linkages, Topology 41, 1051–1107.
Klyachko, A. A. (1994) Spatial polygons and stable configurations of points in the projective line, In Algebraic Geometry and its Applications, Vol. E25 of Aspects Math., Yaroslavl’, 1992, pp. 67–84, Braunschweig, Vieweg.
Lam, K. Y. (1967) Construction of nonsingular bilinear maps, Topology 6, 423–426.
Latombe, J.-C. (1991) Robot Motion Planning, Dordrecht, Kluwer Acad. Publ.
Lebesgue, H. (1950) Leçons sur les constructions géométriques, Paris, Gauthier-Villars.
Milgram, R. J. (1967) Immersing projective spaces, Ann. of Math. (2) 85, 473–482.
Orlik, P. and Terao, H. (1992) Arrangements of Hyperplanes, Vol. 300 of Grundlehren Math. Wiss., Berlin, Springer.
Schwartz, J. T. and Sharir, M. (1983) On the “piano movers” problem. II. General techniques for computing topological properties of real algebraic manifolds, Adv. in Appl. Math. 4, 298–351.
Schwarz, A. S. (1966) The genus of a fiber space, Amer. Math. Soc. Transl. (2) 55, 49–140.
Sharir, M. (1997) Algorithmic motion planning, In J. E. Goodman and J. O’Rourke (eds.), Handbook of Discrete and Computational Geometry, CRC Press Ser. Discrete Math. Appl., Boca Raton, FL, CRC Press, p. 733–754.
Smale, S. (1987) On the topology of algorithms. I, J. Complexity 3, 81–89.
Świątkowski, J. (2001) Estimates for the homological dimension of configuration spaces of graphs, Colloq. Math. 89, 69–79.
Thurston, W. (1987) Shapes of polyhedra, preprint.
Vassil'iev, V. A. (1988) Cohomology of braid groups and complexity of algorithms, Funktsional. Anal. i Prilozhen. 22, 15–24, (Russian) ; English transl. in Funct. Anal. Appl. 22, 182–190.
Walker, K. (1985) Configuration spaces of linkages, Undergraduate thesis, Princeton University.
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FARBER, M. (2006). TOPOLOGY OF ROBOT MOTION PLANNING. In: Biran, P., Cornea, O., Lalonde, F. (eds) Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology. NATO Science Series II: Mathematics, Physics and Chemistry, vol 217. Springer, Dordrecht. https://doi.org/10.1007/1-4020-4266-3_05
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DOI: https://doi.org/10.1007/1-4020-4266-3_05
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