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arXiv · 2505.06794

Dynamic Safety in Complex Environments: Synthesizing Safety Filters with Poisson's Equation

Gilbert Bahati·Ryan M. Bena·Aaron D. Ames
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Abstract

Synthesizing safe sets for robotic systems operating in complex and dynamically changing environments is a challenging problem. Solving this problem can enable the construction of safety filters that guarantee safe control actions -- most notably by employing Control Barrier Functions (CBFs). This paper presents an algorithm for generating safe sets from perception data by leveraging elliptic partial differential equations, specifically Poisson's equation. Given a local occupancy map, we solve Poisson's equation subject to Dirichlet boundary conditions, with a novel forcing function. Specifically, we design a smooth guidance vector field, which encodes gradient information required for safety. The result is a variational problem for which the unique minimizer -- a safety function -- characterizes the safe set. After establishing our theoretical result, we illustrate how safety functions can be used in CBF-based safety filtering. The real-time utility of our synthesis method is highlighted through hardware demonstrations on quadruped and humanoid robots navigating dynamically changing obstacle-filled environments.

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Gilbert Bahati, Ryan M. Bena, Aaron D. Ames. 2025-05-11. Dynamic Safety in Complex Environments: Synthesizing Safety Filters with Poisson's Equation. https://arxiv.org/abs/2505.06794

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