Risk-Aware Safety Filters with Poisson Safety Functions and Laplace Guidance Fields

2025-10-29

AI Summary

Robots navigating real-world environments require integrated semantic understanding and risk awareness to achieve formal safety guarantees. This paper proposes a risk-aware safety filter framework: first, a safety function encoding global safety is constructed via the Poisson equation with Dirichlet boundary conditions; second, a Laplacian-guided field with tunable flux boundary conditions is introduced to explicitly embed semantic risk levels of obstacles, enabling proactive avoidance of high-risk regions. This two-stage mechanism is the first to directly incorporate environmental semantic risk information into certified control design—safety constraints are formally defined via superlevel sets and embedded within an optimization-based safety filter. Simulation results demonstrate that the proposed approach significantly enhances risk-sensitive behavioral decision-making while maintaining strict formal safety guarantees.

Technology Category

• Natural Language Processing: Safety and Robustness
• Intelligent Robots: Behavior Learning & Control
• Philosophy and Ethics of AI: Safety, Robustness & Trustworthiness

Application Category

• Semantics and Knowledge: Provenance, trust, security and privacy, and ethical issues in managing semantic data
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Security and Privacy: Security and privacy of machine learning and AI applications

Abstract

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 extit{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 extit{a priori} understandings of obstacle risk can be directly incorporated into the safety filter to generate safe behaviors that are risk-aware.

Problem

• Developing risk-aware safety filters for robotic navigation systems
• Encoding environmental safety through Poisson equation solutions
• Creating guidance fields with adjustable risk levels around obstacles

Innovation

• Poisson equation encodes safety as 0-superlevel set
• Laplace equation generates tunable risk guidance fields
• Combined safety functions guarantee risk-aware obstacle avoidance