用对称动力系统分析机器人集群的集体行为演化规律
Analyzing Symmetries of Swarms of Mobile Robots Using Equivariant Dynamical Systems
- 基于对称性理论，构建机器人集群动态演化的数学框架
- 发现集群对称性可沿特定层级递增，且与连接图自同构相关
- 适用于状态依赖线性动力学的群集协议，适合分布式系统研究者
本文研究移动机器人集群在分布式系统中的对称性特征。考虑在OBLOT模型下的n个机器人集群，利用等变动力系统理论分析其集体同步（Fsync）动态。证明其演化函数与ℝ²中的旋转和反射变换保持交换关系，形成同构于O(2) × Sₙ的群结构。通过将LCM周期中的感知阶段与计算-移动阶段解耦，该对称性增长层次可由连接图的自同构刻画。特别地，在计算-移动阶段可逆时，可确定所有可能的对称性增强类型。最后，将结果应用于诱导状态依赖线性动力学的协议，其中仅包含计算-移动阶段的简化系统为线性系统。
原文摘要 · Abstract (English)
In this article, we investigate symmetry properties of distributed systems of mobile robots. We consider a swarm of $n\in\mathbb{N}$ robots in the $\mathcal{OBLOT}$ model and analyze their collective $\mathcal{F}$sync dynamics using of equivariant dynamical systems theory. To this end, we show that the corresponding evolution function commutes with rotational and reflective transformations of $\mathbb{R}^2$. These form a group that is isomorphic to $\mathbf{O}(2) \times S_n$, the product group of the orthogonal group and the permutation on $n$ elements. The theory of equivariant dynamical systems is used to deduce a hierarchy along which symmetries of a robot swarm can potentially increase following an arbitrary protocol. By decoupling the Look phase from the Compute and Move phases in the mathematical description of an LCM cycle, this hierarchy can be characterized in terms of automorphisms of connectivity graphs. In particular, we find all possible types of symmetry increase, if the decoupled Compute and Move phase is invertible. Finally, we apply our results to protocols which induce state-dependent linear dynamics, where the reduced system consisting of only the Compute and Move phase is linear.
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