In this paper we study paramertized motion planning algorithms which provide universal and flexible solutions to diverse motion planning problems. Such algorithms are intended to function under a variety of external conditions which are viewed as parameters and serve as part of the input of the algorithm. Continuing the recent paper [2], we study further the concept of parametrized topological complexity. We analyse in full detail the problem of controlling a swarm of robots in the presence of multiple obstacles in Euclidean space which served for us a natural motivating example. We present an explicit parametrized motion planning algorithm solving the motion planning problem for any number of robots and obstacles in R d \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb R}^d$$\end{document}. This algorithm is optimal, it has minimal possible topological complexity for any d 3 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\ge 3 $$\end{document} odd. Besides, we describe a modification of this algorithm which is optimal for d 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\ge 2$$\end{document} even. We also analyse the parametrized topological complexity of sphere bundles using the Stiefel - Whitney characteristic classes.


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    Title :

    Parametrized Motion Planning and Topological Complexity


    Additional title:

    Springer Proceedings in Advanced Robotics


    Contributors:

    Conference:

    International Workshop on the Algorithmic Foundations of Robotics ; 2022 ; , MD, USA June 22, 2022 - June 24, 2022



    Publication date :

    2022-12-15


    Size :

    17 pages





    Type of media :

    Article/Chapter (Book)


    Type of material :

    Electronic Resource


    Language :

    English




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