Dynamic Safety in Complex Environments: Synthesizing Safety Filters with Poisson's Equation
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918017332412416 |
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| author | Bahati, Gilbert Bena, Ryan M. Ames, Aaron D. |
| author_facet | Bahati, Gilbert Bena, Ryan M. Ames, Aaron D. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_06794 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dynamic Safety in Complex Environments: Synthesizing Safety Filters with Poisson's Equation Bahati, Gilbert Bena, Ryan M. Ames, Aaron D. Robotics Systems and Control 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. |
| title | Dynamic Safety in Complex Environments: Synthesizing Safety Filters with Poisson's Equation |
| topic | Robotics Systems and Control |
| url | https://arxiv.org/abs/2505.06794 |