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

TL;DR

The paper addresses risk-aware safety for robots in real-world settings by encoding environment geometry with Poisson safety functions and obstacle-driven risk via Laplace guidance fields. By solving a Dirichlet Poisson problem for $h$ and a vector-Laplace problem for $\vec{v}$, it constructs a safety function and a guidance field that together enable a risk-aware safety filter capable of online synthesis and obstacle-prioritized conservatism. The key contributions are (i) a novel PSF-Laplace framework for spatially varying conservatism, (ii) analysis of how the guidance field affects activation zones and safety activation, and (iii) case studies across geometric, dynamic, and semantic features demonstrating how a priori risk can shape safe behaviors. This approach provides a practical, perception-driven method for real-time, risk-aware navigation in complex environments, with potential extensions to broader nonlinear systems and hardware deployment.

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.

Figures (5)

• Figure 1: The proposed risk-aware control synthesis method. By incorporating user-assigned risk values in the guidance field, our approach generates controllers and closed-loop behaviors exhibiting conservatism aligned with the prescribed risk levels.
• Figure 2: Comparison of activation zones \ref{['eq: activation']} for varying boundary flux with, decreasing the flux for the center and bottom right obstacles. Reducing $\lVert b(\mathbf{y}) \rVert$ yields shallower boundary gradients and smaller activation zones, allowing trajectories closer to the obstacle, whereas larger $\lVert b(\mathbf{y}) \rVert$ produces steeper gradients, larger activation zones and more conservative behavior. Top row: Activation zones \ref{['eq: activation set']} with $\mathbf{k}_{\mathrm{nom}}(\mathbf{y})\!=\!-\mu Dh(\mathbf{y}), \mu\!>\!0$, driving in the direction of steepest decrease of $h$ (i.e., the worst-case, adversarial direction). Bottom row: Closed-loop trajectories for \ref{['eq: single integrator']} with \ref{['eq: k_qp']} and $\mathbf{k}_{\mathrm{nom}}(\mathbf{y})\!=\!-\mu(\mathbf{y}-\mathbf{y}_{\mathrm d})$, driving towards the goal $\mathbf{y}_{\mathrm d}$.
• Figure 3: Activation zones based on obstacle uncertainty. Left: Without smoothing, boundary fluxes vary irregularly. Right: Smoothing forces spatial regularity. High confidence regions (yellow) yield tighter activation zones, while low confidence regions (blue) yield expanded activation zones, reflecting the risk prioritization.
• Figure 4: Time-lapse of the activation zones with a moving object in the environment, defined by the zero level set of the function \ref{['eq: activation dynamic']}. The object's velocity increases to a maximum before decreasing to a minimum. Larger velocities yield higher boundary fluxes and hence larger activation zones. These zones extend in the direction of motion and are affected by the object's acceleration; they expand during acceleration and contract during deceleration, reflecting the rate of change of the safe set.
• Figure 5: An occupancy map is obtained from semantic segmentation of perception data using a pretrained YOLOv11 model Jocher_Ultralytics_YOLO_2023 with user designated risk values, assigned according to label classes as by Algorithm \ref{['alg:main']}: $b_{\text{wall}} = 1, b_{\text{chair}} = 3$ and $b_{\text{person}} = 6$. The occupancy map and flux information are then used to generate the guidance field and derive activation zones.

Theorems & Definitions (10)

• Definition 1
• Definition 2: Relative Degree $r$ isidori1985nonlinear
• Definition 3
• Definition 4: Guidance Field
• Proposition 1
• proof
• Definition 5: Activation Zone
• Corollary 1
• proof
• Remark 1: Combining features