用拓扑距离量化轨迹群随时间的一致性，助力机器人行为分析。
The Induced Matching Distance: A Novel Topological Metric with Applications in Robotics
- 基于动态时间规整与0维持久同调识别相似轨迹群。
- 通过诱导匹配距离追踪群体动态，生成1维一致性信号。
- 适合研究多智能体行为演化，尤其在复杂运动模式区分上表现突出。
本文提出一种新型拓扑度量——诱导匹配距离，用于比较由对称非负函数表示的离散结构。该方法应用于分析随时间演化的智能体轨迹：首先使用动态时间规整（DTW）衡量轨迹相似性，再计算0维持久同调以识别相关连通分支，这些分支在上下文中对应于相似轨迹群。为追踪这些分支随时间的演化，我们计算诱导匹配距离，以保持其动态行为的一致性。最终获得一个1维信号，量化轨迹群在时间上的稳定性。实验表明，该方法能有效区分不同智能体行为，展现出在机器人学及相关领域进行鲁棒拓扑分析的巨大潜力。
原文摘要 · Abstract (English)
This paper introduces the induced matching distance, a novel topological metric designed to compare discrete structures represented by a symmetric non-negative function. We apply this notion to analyze agent trajectories over time. We use dynamic time warping to measure trajectory similarity and compute the 0-dimensional persistent homology to identify relevant connected components, which, in our context, correspond to groups of similar trajectories. To track the evolution of these components across time, we compute induced matching distances, which preserve the coherence of their dynamic behavior. We then obtain a 1-dimensional signal that quantifies the consistency of trajectory groups over time. Our experiments demonstrate that our approach effectively differentiates between various agent behaviors, highlighting its potential as a robust tool for topological analysis in robotics and related fields.
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