Safety requirements in dynamical systems are commonly enforced with set invariance constraints over a safe region of the state space. Control barrier functions, which are Lyapunov-like functions for guaranteeing set invariance, are an effective tool to enforce such constraints and guarantee safety when the system is represented as a point in the state space. In this paper, we introduce extent-compatible control barrier functions as a tool to enforce safety for the system including its volume (extent) in the physical world. In order to implement the extent-compatible control barrier functions framework, a sum-of-squares based optimization program is proposed. Since sum-of-squares programs can be computationally prohibitive, we additionally introduce a sampling based method in order to retain the computational advantage of a traditional barrier function based quadratic program controller. We prove that the proposed sampling based controller retains the guarantee for safety. Simulation and robotic implementation results are also provided.

中文翻译：

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范围兼容的控制屏障函数

动态系统中的安全要求通常通过在状态空间的安全区域上设置不变性约束来强制执行。控制屏障函数是用于保证集合不变性的类李雅普诺夫函数，当系统表示为状态空间中的一个点时，它是强制执行此类约束并保证安全性的有效工具。在本文中，我们引入了范围兼容的控制屏障功能作为一种工具，以加强系统的安全性，包括其在物理世界中的体积（范围）。为了实现范围兼容的控制屏障函数框架，提出了一种基于平方和的优化程序。由于平方和程序在计算上可能会令人望而却步，我们还引入了一种基于采样的方法，以保留基于传统障碍函数的二次程序控制器的计算优势。我们证明所提出的基于采样的控制器保留了安全保证。还提供了仿真和机器人实施结果。