研究多机器人在变化任务下的路径规划拓扑复杂度，给出抗干扰算法设计的理论极限。
Parametrized Topological Complexity for a Multi-Robot System with Variable Tasks
- 构建参数化纤维化模型，将异构任务路径规划转化为拓扑不变量计算。
- 证明了奇偶维空间下最小算法不稳定性上界，首次覆盖可变目标数场景。
- 适用于需要高鲁棒性的多机器人协同系统，如搜救、巡检等动态环境应用。
我们研究了一个广义的运动规划问题，涉及多个自主机器人在 $d$ 维欧氏空间中导航，存在位置事先未知的障碍物。每个机器人需按序访问一组指定的目标状态，且各机器人的目标数量可不同。该异构设定推广了 Farber 及本文第二作者先前关于序列参数化拓扑复杂度的工作。为确定本问题的拓扑复杂度，我们通过构造适当的纤维化进行数学建模。主要贡献在于在广义设定下确定该不变量，刻画了在参数依赖约束下设计无碰撞运动规划算法所需的最小算法不稳定性。我们对奇偶维环境空间进行了详细分析，包含关键的上同调计算及对应运动规划算法的显式构造。
原文摘要 · Abstract (English)
We study a generalized motion planning problem involving multiple autonomous robots navigating in a $d$-dimensional Euclidean space in the presence of a set of obstacles whose positions are unknown a priori. Each robot is required to visit sequentially a prescribed set of target states, with the number of targets varying between robots. This heterogeneous setting generalizes the framework considered in the prior works on sequential parametrized topological complexity by Farber and the second author of this article. To determine the topological complexity of our problem, we formulate it mathematically by constructing an appropriate fibration. Our main contribution is the determination of this invariant in the generalized setting, which captures the minimal algorithmic instability required for designing collision-free motion planning algorithms under parameter-dependent constraints. We provide a detailed analysis for both odd and even-dimensional ambient spaces, including the essential cohomological computations and explicit constructions of corresponding motion planning algorithms.
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