Topological complexity of rational configuration spaces and applications EACAT4 2011
Topological complexity of rational configuration spaces and applications
Feb 24, 2016
Topological complexity of rational configuration spaces and applications. EACAT4 2011. Configuration space . Robot motion planning. Jean Claude Latombe, Kluwer 1991. Topological robotics in motion plannings . Definition . Topological complexity . Remark. Michael Farber, 2003. - PowerPoint PPT Presentation
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Topological complexity of rational configuration spaces and applications
EACAT4 2011
Robot motion planning
Configuration space
Aniceto Murillo
Robot: palabra introducida por el escritor Karel Capek en 1920 (trabajador forzado del checo)Robotica fue introducida por Isaac Asimov en 1940 en Yo, Robot.
Topological robotics in motion plannings
Jean Claude Latombe, Kluwer 1991
Aniceto Murillo
Latombe: Ingeniero e informatico de StanfordEspacio de configuraciones: Introducidos por Fadell y Neuwirth en 1962
Definition
Aniceto Murillo
An algorithm for a motion planning has as input two states of the cofiguration space and, as output, a motion between these states.
Remark
Michael Farber, 2003Topological complexity
Aniceto Murillo
Existe un entorno de la diagonal (homotopo a ella) en XxX donde existe una seccion continua
Aniceto Murillo
If that is the case, any contraction produces a section, and therefore, an algorithm of the motion planning
Aniceto Murillo
In the same way, a contractible neighborhood, or more general, a neighborhood of the configuration spaces for which there exists a local section of the path fibration produces an algorithmn of the motion as long as the initial and final state are within the neighborhood.
Aniceto Murillo
Hence the samllest number of such neighborhoods needed to cover our configuration space gives a measure of how hard it is to find an algorithm for oour motion planning
A. Schwarz
First properties
Aniceto Murillo
Dada una variedad simplectica simplemente conexa de dimension 2n, existe una fibracion con categoria seccional n
Computing the topological complexity
M. Farber, J. González, S. Tabachnikov, S. Yuzvinsky,…
y
M. Farber, S. Tabachnikov, S. Yuzvinsky, 2003
J. Adem, S. Gitler, I. James, 1972
Corollary
Problem
Michael Farber, 2005
Dennis Sullivan 1977 Daniel Quillen 1969
Rational homotopy theory: the basics
Theorem
Problem’
Theorem
L. Lechuga, A. M.
The answer…
Theorem B. Jessup , A. M., P.E. Parent
Some applications…
Corollary
1. Topological complexity of formal spaces
Examples…
2. Detection of spaces for which TC equals cat
Theorem B. Jessup , A. M., P.E. Parent
3. Reduced topological complexity
Theorem
4. A robot with an extra arm
TheoremB. Jessup , A. M., P.E. Parent
Remark
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