Poisson Safety Functions

Updated 3 July 2026

Poisson Safety Functions are smooth scalar fields constructed as solutions to Poisson’s equation that define dynamically safe regions in complex robotic environments.
They integrate with control barrier functions by encoding safety constraints into quadratic programs for both first-order and higher relative degree systems.
Extensions for geometry and semantics enable risk-aware navigation and real-time safety filtering in both static and time-varying domains.

A Poisson Safety Function (PSF) is a smooth scalar field constructed as the unique solution to a Dirichlet problem for Poisson’s equation, whose zero-superlevel set certifies a dynamically safe region for a robotic or dynamical system operating in complex, typically perception-derived, environments. The PSF formalism provides a scalable, PDE-based alternative to signed distance and analytic barrier functions, enabling the synthesis of control barrier function (CBF)–compatible safety certificates directly from environmental data, while supporting extensions for geometry-awareness, risk prioritization, semantic adaptation, and real-time filtering in both static and dynamic domains.

1. Mathematical Definition and Construction

The prototypical PSF is defined as a solution to the boundary value problem: $\begin{cases} \Delta h(y) = f(y), & y \in \Omega, \ h(y) = 0, & y \in \partial \Omega, \end{cases}$ where $\Omega \subset \mathbb{R}^p$ is a bounded, connected open domain encoding the perceived free space (e.g., via occupancy mapping), $\partial\Omega$ is the perceived obstacle boundary, and $f: \Omega \rightarrow \mathbb{R}_{<0}$ is a sufficiently regular negative forcing function ensuring $h$ is superharmonic in $\Omega$ (Bahati et al., 11 May 2025, Bahati et al., 29 Oct 2025, Bena et al., 15 Aug 2025, Wilkinson et al., 23 Apr 2026). The choice of $f$ determines the “steepness” and regularity of the safety field; constant $f$ yields maximally smooth $h \in C^\infty(\overline{\Omega})$ .

The associated safety set is characterized as: $C = \{ y \in \Omega \mid h(y) \geq 0 \}, \qquad \partial C = \{ y \mid h(y) = 0 \}$ with $\Omega \subset \mathbb{R}^p$ 0 when $\Omega \subset \mathbb{R}^p$ 1, so the boundary is regular and supports CBF invariance theorems.

The PSF can be synthesized directly (by choosing $\Omega \subset \mathbb{R}^p$ 2), or indirectly by solving a variational problem to best align $\Omega \subset \mathbb{R}^p$ 3 with a user-specified smooth vector field $\Omega \subset \mathbb{R}^p$ 4 (the guidance field), in which case the forcing becomes $\Omega \subset \mathbb{R}^p$ 5, often postprocessed to guarantee negativity—e.g., $\Omega \subset \mathbb{R}^p$ 6 (Bahati et al., 11 May 2025, Bahati et al., 29 Oct 2025, Yang et al., 5 Mar 2026).

2. Integration into Control Barrier Functions and Safety Filtering

Once synthesized, a PSF $\Omega \subset \mathbb{R}^p$ 7 naturally serves as a control barrier function for a class of robotic systems. For first-order integrator dynamics: $\Omega \subset \mathbb{R}^p$ 8 the PSF is incorporated as an inequality constraint for the safe control policy: $\Omega \subset \mathbb{R}^p$ 9 or, employing a risk-modulating guidance field $\partial\Omega$ 0: $\partial\Omega$ 1 which can be encoded in a quadratic program (QP)–based safety filter that minimally modifies a nominal input $\partial\Omega$ 2 (Bahati et al., 11 May 2025, Bahati et al., 29 Oct 2025, Yang et al., 5 Mar 2026). For higher relative degree systems (e.g., $\partial\Omega$ 3), CBF backstepping is applied, and the regularity of the PSF ensures the existence of locally Lipschitz safe controllers as per established backstepping CBF theory (Bahati et al., 11 May 2025, Bahati et al., 29 Oct 2025).

Forward invariance of $\partial\Omega$ 4 (or a backstepped set $\partial\Omega$ 5) is rigorously guaranteed under these constructions provided $\partial\Omega$ 6 and $\partial\Omega$ 7 are sufficiently smooth and $\partial\Omega$ 8 is nonvanishing on the boundary.

3. Geometry-Aware, Dynamic, and Semantic PSF Extensions

PSFs are extended to handle:

Time-dependent environments: The domain $\partial\Omega$ 9 and boundary $f: \Omega \rightarrow \mathbb{R}_{<0}$ 0 are made functions of time, lifted into a space-time domain $f: \Omega \rightarrow \mathbb{R}_{<0}$ 1, and the PSF is obtained by solving parameterized or moving-boundary Poisson problems, often with boundary dynamics estimated via optical flow or level set transport equations (Bena et al., 15 Aug 2025).
Robot geometry: For non-point robots, the safe domain is shrunken via configuration-dependent Minkowski differences: $f: \Omega \rightarrow \mathbb{R}_{<0}$ 2, and the PSF is solved over a product domain $f: \Omega \rightarrow \mathbb{R}_{<0}$ 3 with the Laplacian still acting only on the spatial variable (Wilkinson et al., 23 Apr 2026, Bena et al., 15 Aug 2025).
Semantic and risk awareness: The Laplace guidance field $f: \Omega \rightarrow \mathbb{R}_{<0}$ 4 is constructed with boundary flux $f: \Omega \rightarrow \mathbb{R}_{<0}$ 5 modulated by semantic features or risk (from perception or prior knowledge), and its divergence forms the PSF’s source: $f: \Omega \rightarrow \mathbb{R}_{<0}$ 6. The risk-aware filter then uses $f: \Omega \rightarrow \mathbb{R}_{<0}$ 7 as the safety-gradient surrogate, enabling obstacle-specific repulsion strength and the encoding of social/pass norms (e.g., larger $f: \Omega \rightarrow \mathbb{R}_{<0}$ 8 for humans than for static objects) (Bahati et al., 29 Oct 2025, Yang et al., 5 Mar 2026).

These extensions support both model predictive control (MPC) and low-level CBF filtering with real-time updates as environmental or configuration data changes.

4. Computational and Practical Properties

PSF synthesis via numerical solution of Poisson’s equation is computationally efficient (sub-millisecond on modern GPUs), and admits warm starts for dynamic scenes(Bahati et al., 11 May 2025, Bena et al., 15 Aug 2025). The guidance field construction is fully compatible with discrete occupancy maps from perception, and boundary flux modulation is directly tied to semantic, probabilistic, or risk-based labels from multi-modal segmentation and sensor fusion (Yang et al., 5 Mar 2026). Experiments on quadruped, humanoid, and manipulator platforms demonstrate real-time operation and improved safety behavior over signed-distance–based CBFs, notably in deadlock-prone, cluttered, or socially structured environments (Bahati et al., 11 May 2025, Bena et al., 15 Aug 2025, Yang et al., 5 Mar 2026, Wilkinson et al., 23 Apr 2026).

The PSF framework is distinct from several other “Poisson safety” notions:

Poisson kernels in harmonic analysis: These serve in the context of potential theory, maximal function estimates, and differentiability theorems in boundary value problems (Krantz, 2010), but are not safety certificates for dynamical systems.
Poisson-based risk/scale functions: In stochastic processes and risk theory (e.g., $f: \Omega \rightarrow \mathbb{R}_{<0}$ 9-scale functions for spectrally negative compound Poisson processes), the Poisson equation arises as an expected drift condition in continuous-time Markov or Lévy processes to compute ruin probabilities and related risk metrics (Behme et al., 2020). These are analytically and application-wise disconnected from the geometric/PDE-based PSF framework for robotics.
Piecewise stochastic barrier functions: While stochastic barrier functions for finite-horizon safety probabilities in controlled Markov models share the “certificate” conceptual space, their synthesis is based on robust drift inequalities and combinatorial optimization, not Poisson PDEs (Mazouz et al., 2024).

A distinctive feature of modern PSFs is the use of Poisson's equation as a geometric and semantic “glue” between perception, risk modeling, and control, rather than as an analytic tool for boundary convergence or expectation calculation.

6. Applications and Demonstrated Impact

PSFs enable:

Obstacle avoidance for mobile robots and manipulators by mapping perceptual geometry into smooth, CBF-compatible safety sets that handle arbitrary nonconvex obstacles in real time (Bahati et al., 11 May 2025, Wilkinson et al., 23 Apr 2026).
Risk and context-aware navigation, allowing different margins for semantic classes (e.g., humans, robots, static objects) and dynamic adaptation to uncertain or moving obstacles (Bahati et al., 29 Oct 2025, Yang et al., 5 Mar 2026).
Full-body safety guarantees for manipulators via coverage-based sampling and buffered domains, achieving true surface-level collision avoidance with tractable CBF synthesis (Wilkinson et al., 23 Apr 2026).
Predictive safety filtering in MPC architectures for legged robots and humanoids, incorporating both geometric asymmetries and evolving environments (Bena et al., 15 Aug 2025).

Experimental evaluations confirm that PSFs enable qualitative gains such as reduced conservatism, context-adaptive social navigation, and smoother motion near complex obstacle boundaries as compared to traditional SDF or hand-crafted CBFs.

7. Theoretical Guarantees and Limitations

Poisson safety functions enjoy strong guarantees:

Existence, uniqueness, and $h$ 0 smoothness under standard elliptic PDE theory for smooth source and boundary data.
Nonvanishing boundary gradient (by Hopf’s lemma), ensuring regularity of safe set boundaries.
Forward invariance: For any locally Lipschitz feedback satisfying the CBF condition (using PSF and, if needed, Laplace guidance field), the safe set defined by $h$ 1 is forward invariant; analogous guarantees hold for high-order and time-varying systems.
Generalizability: Geometry- and semantics-aware PSFs can be constructed for arbitrary segmented environments, are robust to sensor noise, and can be solved iteratively or warm-started.

Notable limitations include possible deadlocks or spurious equilibria for purely reactive (non predictive) filters and loss of safety if perception fails or the occupancy map is substantially incorrect. The buffering required for sampling and geometry awareness may conservatively shrink feasible safe sets in high-dimensional or dense-clutter scenarios (Wilkinson et al., 23 Apr 2026, Bena et al., 15 Aug 2025).

References

(Bahati et al., 11 May 2025) Dynamic Safety in Complex Environments: Synthesizing Safety Filters with Poisson's Equation
(Bahati et al., 29 Oct 2025) Risk-Aware Safety Filters with Poisson Safety Functions and Laplace Guidance Fields
(Bena et al., 15 Aug 2025) Geometry-Aware Predictive Safety Filters on Humanoids: From Poisson Safety Functions to CBF Constrained MPC
(Wilkinson et al., 23 Apr 2026) Full-Body Dynamic Safety for Robot Manipulators: 3D Poisson Safety Functions for CBF-Based Safety Filters
(Yang et al., 5 Mar 2026) Safe-SAGE: Social-Semantic Adaptive Guidance for Safe Engagement through Laplace-Modulated Poisson Safety Functions