Robotic systems navigating in real-world settings require a semantic understanding of their environment to properly determine safe actions. This work aims to develop the mathematical underpinnings of such a representation--specifically, the goal is to develop safety filters that are risk-aware. To this end, we take a two step approach: encoding an understanding of the environment via Poisson's equation, and associated risk via Laplace guidance fields. That is, we first solve a Dirichlet problem for Poisson's equation to generate a safety function that encodes system safety as its 0-superlevel set. We then separately solve a Dirichlet problem for Laplace's equation to synthesize a safe \textit{guidance field} that encodes variable levels of caution around obstacles -- by enforcing a tunable flux boundary condition. The safety function and guidance fields are then combined to define a safety constraint and used to synthesize a risk-aware safety filter which, given a semantic understanding of an environment with associated risk levels of environmental features, guarantees safety while prioritizing avoidance of higher risk obstacles. We demonstrate this method in simulation and discuss how \textit{a priori} understandings of obstacle risk can be directly incorporated into the safety filter to generate safe behaviors that are risk-aware.

翻译：在现实环境中导航的机器人系统需要对其环境进行语义理解，以准确确定安全行动。本研究旨在为此类表示建立数学基础——具体而言，目标是开发具备风险感知能力的安全滤波器。为此，我们采用两步方法：通过泊松方程编码对环境的风险理解，并通过拉普拉斯引导场关联风险。即，首先求解泊松方程的狄利克雷问题，生成以0-超水平集编码系统安全性的安全函数；随后单独求解拉普拉斯方程的狄利克雷问题，通过可调通量边界条件合成编码障碍物周围可变警戒级别的安全引导场。安全函数与引导场结合后定义安全约束，并用于合成风险感知安全滤波器。该滤波器在给定环境语义理解及环境特征关联风险等级的前提下，能保证安全性并优先规避高风险障碍物。我们在仿真中验证了该方法，并探讨如何将障碍物风险的先验理解直接融入安全滤波器，以生成具备风险感知的安全行为。