Topological complexity of rational configuration spaces and applications

Transcript of 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

Topological complexity of rational configuration spaces and applications

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Transcript of Topological complexity of rational configuration spaces and applications

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