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

TL;DR

The paper addresses dynamic safety in complex robotic environments by formulating safe-set generation as a Dirichlet problem for Poisson's equation: solve $\Delta h(\mathbf{y}) = f(\mathbf{y})$ in a perception-defined open domain $\Omega$ with $h(\mathbf{y}) = 0$ on $\partial\Omega$, producing a smooth safety function $h$ whose 0-superlevel set defines the safe region. It then shows how $h$ can be used as a Control Barrier Function to yield safety filters for first- and high-order systems, via backstepping and related techniques, ensuring forward invariance of the safe set. The main contributions include a constructive Poisson-based method to synthesize safe sets from perception data, a formal link to CBF-based safety filters, and real-time hardware demonstrations on quadruped and humanoid robots in static and dynamic environments. The approach leverages elliptic PDE regularity to provide differentiable safety functions, enabling stable, real-time safety filtering in highly dynamic scenarios and offering flexible forcing-function design to encode obstacle geometry through boundary flux.

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.

Figures (7)

• Figure 1: Safe set synthesis from perception data via Poisson's equation. Hardware experimental footage: https://youtu.be/fBRdkAJGixI.
• Figure 2: [From left to right] Solutions to Poisson’s equation \ref{["eq: poisson's eq"]} with the following forcing functions: [left] a Hölder continuous function \ref{['eq: holder continous f']} with $\alpha = 0.1$; [mid-left] an average flux forcing function with $\bar{b} = -1$ in \ref{['eq: forcing function average flux']}; [mid-right] a smooth forcing function \ref{['eq: softplus forcing function']} constructed using the guidance field \ref{['eq: guidance field']} under a uniform boundary flux $b(\mathbf{y}) = -1$ for all $\mathbf{y} \in \partial \Omega$; and [right] the same forcing function with a non-uniform boundary flux $b: \partial \Omega \to \mathbb{R}_{<0}$, allowing different flux values across regions of the boundary, corresponding to different obstacles.
• Figure 3: Smooth guidance field generation via Laplace's equation \ref{['eq: guidance field']}. [left] Boundary conditions $\vec{\mathbf{v}} = b \hat{\mathbf{n}}$ encoding the desired negative flux on obstacle surfaces; and [right] solution to Laplace's equation for each component in $\vec{\mathbf{v}} = (v_x, v_y)$.
• Figure 4: Double integrator simulations using safety filters synthesized from: [left] Signed Distance Function \ref{['eq: SDF']}; and [middle and right] the Poisson Safety Function, constructed with the forcing function \ref{['eq: softplus forcing function']} with the guidance field \ref{['eq: guidance field']} where the boundary conditions use [middle] a uniform boundary flux $b(\mathbf{y}) =-1$ for all $\mathbf{y} \in \partial \Omega$ and [right] a non-uniform boundary flux $b: \partial \Omega \to \mathbb{R}_{<0}$, assigning different flux values across regions of the boundary associated with different obstacles. Sharp ridges in the SDF surface introduce gradient discontinuities, which lead to oscillations in the resulting trajectories---an issue which does not arise with Poisson safety functions due to their classical regularity (i.e., differentiability) properties. Furthermore, in contrast to SDFs with fixed gradients, the guidance field promotes the manipulation of boundary flux values via the assignment of $b$ in \ref{['eq: guidance field']}, enabling the ability to encode gradients customized to specific obstacles, which we are unable to do with traditional SDFs. This flexibility helps yield trajectories that avoid undesired equilibria i.e., “deadlocks”, in [right].
• Figure 5: Hardware experiments demonstrating Poisson safety functions for safety filtering. [Top Left] A timelapse showing the motion of the Go2 quadruped during the 25-second tracking experiment, starting from three difference initial conditions (ICs). [Top Middle] The Poisson safety function constructed from real-time segmented image data. [Top Right] The resultant safe trajectories for each IC. [Bottom Left/Middle] The nominal (orange) and CBF safety-filtered (blue) velocity commands, in m/s, sent to the robot. [Bottom Right] The evaluated value of the Poisson safety function over the course of each experiment. This value remains above zero, confirming that safety was maintained.

Theorems & Definitions (25)

• Definition 1
• Definition 2
• Definition 3: Relative Degree $r$ isidori1985nonlinear
• Definition 4
• Theorem 1
• proof
• Remark 1
• Proposition 1
• proof
• Lemma 1