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

TL;DR

This paper develops risk-aware safety filters for robotic navigation by combining Poisson and Laplace equations to encode environmental safety and caution levels, ensuring safe and risk-sensitive robot behaviors.

Contribution

It introduces a novel mathematical framework using Poisson and Laplace equations to create safety functions and guidance fields for risk-aware robotic safety filtering.

Findings

Successfully demonstrated in simulation

Guarantees safety while prioritizing high-risk obstacle avoidance

Integrates semantic risk understanding into safety filters

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 \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.