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

arXiv: https://arxiv.org/abs/2505.06794 · HTML: https://mlsanity.com/paper/2505.06794/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 May 2025.

## Built on by (most cited first, 1 of 1)

- [VLPSA: Vision-Language-Poisson-Safe Actions for Full-Body Safety of Learned Policies](https://mlsanity.com/paper/2609.22462/) (2026)

## Abstract (arXiv)

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.