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 a recent paper, 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.. This algorithm is optimal, it has minimal possible topological complexity for any d odd. We also analyse the parametrized topological complexity of sphere bundles using the Stiefel - Whitney characteristic classes.
翻译:在本文中,我们研究了为各种运动规划问题提供普遍和灵活解决办法的模拟运动规划算法,这些算法意在各种外部条件下运作,这些外部条件被视为参数,并作为算法的一部分投入。我们继续在最近的论文中研究超美化地形复杂程度的概念。我们充分详细地分析在极地空间存在多种障碍的情况下控制一大批机器人的问题,这对我们来说是一个自然激励的例子。我们提出了一个明确的超美化运动规划算法,解决任何数目的机器人和障碍的运动规划问题。这种算法是最佳的,对任何奇异而言,其表面复杂性是最小的。我们还利用斯特里费尔-惠特尼特征类别分析球团的超美化地形复杂程度。