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arXiv math/0210018math.ATcs.ROmath.DG
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拓扑机器人:射影空间中的运动规划

Topological robotics: motion planning in projective spaces

Michael Farber, Serge Tabachnikov, Sergey Yuzvinsky

摘要:

我们研究拓扑机器人的一个基本问题:旋转的一条线,这是固定的旋转关节在一个基点:一个人想把线从其初始位置到最终位置的空间中的连续运动。最终的目标是构造一个算法,一旦给定初始和最终位置,该算法将执行此任务。任何这样的运动规划算法都会存在不稳定性,这是由拓扑原因造成的。第一作者最近提出了研究机器人运动不稳定性的一般方法。对于任何路连通的拓扑空间X,我们都可以将一个数TC(X)联系起来,称为X的拓扑复杂度。这个数字对于运动规划问题是非常重要的:TC(X)决定了X中所有运动规划算法的不稳定性特征。本文研究了实射影空间的拓扑复杂性。特别地,我们计算所有n<24的TC(RP^n)。我们的主要结果是:(对于不同于1,3,7的n)TC(RP^n)的计算问题等价于寻找最小的k,使得RP^n可以浸入欧几里得空间R^{k-1}。

英文摘要:

We study an elementary problem of topological robotics: rotation of a line, which is fixed by a revolving joint at a base point: one wants to bring the line from its initial position to a final position by a continuous motion in the space. The final goal is to construct an algorithm which will perform this task once the initial and final positions are given. Any such motion planning algorithm will have instabilities, which are caused by topological reasons. A general approach to study instabilities of robot motion was suggested recently by the first named author. With any path-connected topological space X one associates a number TC(X), called the topological complexity of X. This number is of fundamental importance for the motion planning problem: TC(X) determines character of instabilities which have all motion planning algorithms in X. In the present paper we study the topological complexity of real projective spaces. In particular we compute TC(RP^n) for all n<24. Our main result is that (for n distinct from 1, 3, 7) the problem of calculating of TC(RP^n) is equivalent to finding the smallest k such that RP^n can be immersed into the Euclidean space R^{k-1}.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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