高阶平均场控制障碍函数

中文摘要 AI 辅助

平均场控制障碍函数（MF-CBFs）通过对智能体分布施加约束来确保群体安全。先前的公式，包括应用于随机覆盖和牧群问题，使用一阶微分不等式。这些无法直接强制执行其障碍函数产生无控制一阶导数的约束。因此，我们发展了高阶MF-CBFs的理论，该理论通过高阶微分不等式确保障碍泛函的正性，如同其有限维类似物。更具体地，我们引入了平均场安全泛函的相对阶概念，并开发了一个用于计算其高阶导数的解析框架。此外，对于互相关和自相关泛函的实际例子，我们展示了沿平均场动力学的重复微分是结构保持的，并简化为核递归。作为说明，我们将高阶MF-CBFs应用于双积分器群体跟踪和避障问题，其中仅位置约束导致相对阶为二的MF-CBFs。

英文摘要

Mean-field control barrier functions (MF-CBFs) enforce swarm safety through constraints on the agents' distribution. Previous formulations, including applications to stochastic coverage and shepherding, use first-order differential inequalities. These cannot directly enforce constraints whose barrier functions yield control-free first derivatives. Thus, we develop the theory of higher-order MF-CBFs, which ensures the positivity of the barrier functionals through higher-order differential inequalities, as in their finite-dimensional analogs. More specifically, we introduce the notion of relative degree for mean-field safety functionals and develop an analytic framework for computing their higher-order derivatives. Moreover, for practical examples of cross-correlation and self-correlation functionals, we show that repeated differentiation along mean-field dynamics is structure-preserving and reduces to kernel recursions. As an illustration, we apply higher-order MF-CBFs to double-integrator swarm tracking and avoidance problems, where position-only constraints lead to MF-CBFs of relative degree two.

发表机构

• Emory University（埃默里大学）
• Colorado School of Mines（科罗拉多矿业学院）

机构由 AI 辅助整理，请以论文原文为准。