Pith. sign in

REVIEW 2 major objections 4 minor 2 cited by

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

T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Solving Poisson's equation on an occupancy map produces a safety function that can certify safe robot motion and drive a CBF-based safety filter.

desk verdict Static Poisson safety-function synthesis is a real contribution, but the paper's dynamic-safety claim rests on an unproven time-varying extension. read the letter →

arxiv 2505.06794 v1 pith:GKB77HWE submitted 2025-05-11 cs.RO cs.SYeess.SY

Gilbert Bahati , Ryan M. Bena , Aaron D. Ames This is my paper · ORCID
classification cs.ROcs.SYeess.SY MSC 35J0593C85
keywords controlbarrierfunctionsPoisson'sequationsafetyfilterssafesetsynthesisoccupancymapsCBFbacksteppingquadrupedrobothumanoid
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper attempts to establish a constructive pipeline from raw perception to verified safety: take a local occupancy map, solve Poisson's equation with zero boundary values on obstacle surfaces and a strictly negative forcing function, and use the unique solution $h$ as the safety constraint in a Control Barrier Function (CBF) safety filter. If successful, the method replaces hand-designed or gradient-discontinuous constraints, such as signed distance functions, with one smooth function that is automatically synthesized from arbitrary obstacle geometry and comes with forward-invariance proofs. The paper proves that $h$ is a safety function, that it renders the safe set forward invariant for first-order models, and that CBF backstepping extends the guarantee to higher-order integrator chains. Hardware demonstrations on a quadruped and a humanoid in static and moving obstacle fields show the computation is fast enough for real-time use, which is why a general reader should care: it is a concrete step toward robots that can be handed a camera feed and a guarantee about not colliding.

What carries the argument

The central object is the Poisson safety function: the solution $h$ of the Dirichlet problem $\Delta h = f$ on the occupancy domain with $h=0$ on obstacle boundaries and $f$ strictly negative. The forcing function is the design knob. The paper's main construction routes through a smooth guidance vector field $\vec{v}$: each component solves Laplace's equation with boundary data $b(y)\,\hat{n}(y)$ encoding the desired flux, and the forcing term is a smooth softplus-style function of $\nabla \cdot \vec{v}$ that stays strictly negative, so the solution is automatically smooth and superharmonic. This also makes $h$ the unique minimizer of the variational problem $\min_h \int_\Omega \left(\tfrac{1}{2}|Dh|^2 - h\, \beta^{-1}\ln(1+e^{-\beta \nabla \cdot \vec{v}})\right) dV$. The companion machinery is CBF backstepping: auxiliary safe controllers $k_i$ are designed recursively, and the function $h_B(\vec{y}) = h(y) - \sum_i \tfrac{1}{2\mu_i}\| y^{(i)} - k_i\|^2$ is shown to define a shrunken invariant safe set, with Lemma 1 supplying the Lipschitz regularity of $D^r h$ that makes the controller locally Lipschitz and well-defined.

What would settle it

Run the hardware pipeline with an obstacle moving through the field faster than the roughly 10 Hz update rate; the claim fails if $h$ becomes negative before the next recomputation, or if one can construct two consecutive updates satisfying (41) and the filter constraint yet with $h$ crossing zero in between. A cheaper falsifier is a simulation of the double integrator with a periodically switching occupancy map that checks the continuous trajectory between safety-function updates.

Watch

Extended reading notes

Core claim

The paper's central claim is that a single scalar function encoding an entire obstacle field can be manufactured by solving an elliptic boundary value problem. Let $\Omega$ be the unoccupied region bounded by obstacle surfaces $\partial\Omega$, and let $f \in C^{k,\alpha}(\Omega;\mathbb{R}_{<0})$ be H\"older continuous and strictly negative. The unique classical solution $h$ of $\Delta h = f$ in $\Omega$, $h=0$ on $\partial\Omega$, is a safety function of order $2+k$: its zero superlevel set is a compact safe set whose boundary is $\partial\Omega$, and Hopf's lemma guarantees $Dh \cdot \hat{n} < 0$ on the boundary, so the gradient is nonzero exactly where it is needed. Because $\Delta h < 0$, the weak minimum principle keeps $h$ nonnegative inside $\Omega$, so the safe set is the whole free space, and each obstacle interior can be made unsafe by solving with a positive forcing function there. The paper then proves this safety function is a CBF: Proposition 1 gives a Lipschitz controller for $\dot{y} = w$ satisfying $\dot{h} \geq -\gamma h$, and Theorem 6 extends the argument to integrator chains of relative degree $r \geq 2$ using backstepping functions $h_B$ that render a shrunken subset $C_B \subset C \times \mathbb{R}^{3(r-1)}$ forward invariant.

Load-bearing premise

The load-bearing premise is that forward invariance for the static safety function $h$ transfers to the time-varying $h(t,y)$ recomputed online and differentiated by finite differences in (41), an extension the paper assumes but does not prove or cite.

Editorial extensions

If this is right

  • Occupancy maps can be converted directly into safety constraints: the same pipeline yields a smooth CBF for arbitrary obstacle geometry, avoiding the gradient discontinuities of signed distance functions that cause chattering in safety filters.
  • Safety is decoupled from the goal: the synthesized safety function is geometry-only, so a single $h$ serves any nominal controller in a safety-filter quadratic program.
  • The guarantees cover higher-order dynamics, not just kinematic models: for integrator chains of relative degree $r \geq 2$, the backstepped CBF makes a shrunken safe set forward invariant.
  • The regularity of the safety function is tunable through the forcing function: H\"older-continuous forcing already suffices for low relative degree, while smooth forcing gives smooth $h$ for higher relative degree.
  • On hardware, the pipeline runs at about 10 Hz with PDE solve times of 0.2--0.3 ms, which the experiments indicate is fast enough to keep a quadruped safe while obstacles move.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A rigorous dynamic-safety theorem would need to treat $h(t,y)$ as time-varying; one route is to bound $\|\partial h/\partial t\|$ and add it to the CBF condition, turning the current finite-difference approximation (41) into a provable margin.
  • The per-obstacle boundary flux $b$ in (26) is a tuning handle for shaping gradient magnitudes; this suggests a data-driven or optimization-based flux assignment to push the system away from deadlock equilibria, though no such guarantee is given.
  • The same variational formulation could be applied to higher-dimensional state spaces or to directly synthesizing a CLF-CBF pair, since the guidance field construction is independent of the robot's dynamics.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper proposes a method for synthesizing safety functions from occupancy-map perception data by solving Poisson's equation with Dirichlet boundary conditions and a strictly negative forcing function. The resulting solution h is shown to be a safety function (Theorem 1), to define a forward-invariant safe set for first-order integrator systems (Proposition 1), and, via control barrier function backstepping, to yield safety filters for higher-order integrator chains (Theorem 6). The authors demonstrate the approach in simulations and on quadruped and humanoid hardware, including dynamic environments where h is recomputed online at about 10 Hz and its temporal derivative is approximated by finite differences in Eq. (41).

Significance. The static theoretical core is a clean, constructive bridge from geometric perception data to smooth safety functions with classical regularity, avoiding the gradient discontinuities of signed distance functions. The variational characterization and the explicit use of Hopf's lemma and elliptic regularity are credible and useful. If the dynamic-environment claims were backed by a time-varying invariance theorem, the paper would be a strong practical contribution to CBF-based safety filtering. As it stands, the static theory is solid, but the title- and abstract-level claim of 'dynamic safety' rests on an unproven extension to time-varying h, and one hardware experiment shows h becoming negative. The paper should be revised to either provide the missing time-varying guarantee or soften the dynamic-safety claims to heuristic demonstrations.

major comments (2)
  1. [§VI.B.2, Eq. (41)] Proposition 1 and Theorem 6 establish forward invariance only for a time-independent safety function h(y), but the dynamic hardware experiments recompute h from perception at roughly 10 Hz and use the finite-difference term in Eq. (41) to approximate ∂h/∂t. No theorem or prior reference is provided showing that the resulting time-varying filter still enforces the CBF inequality ˙h(t,y) = ∂h/∂t + Dh·u ≥ −γh, and simply applying the static condition at each frozen instant does not preclude a moving obstacle from driving h negative between updates. The paper should either state and prove a time-varying version of the safety-filter guarantee with explicit regularity and rate-of-change assumptions on h(t,y), or explicitly reclassify the dynamic experiments as heuristic demonstrations rather than guaranteed safe.
  2. [§VI.B.3, Fig. 6] The paper's own data show the humanoid's h(t) becoming briefly negative, and the text attributes this to reduced-order-model mismatch. This contradicts the statement that the safety filter renders C forward invariant, since the filter's guarantee applies to the modeled single-integrator state, not the actual robot state; if the claimed dynamic-safety result requires robustness to tracking error, the manuscript needs a formal treatment of that error (e.g., an input-to-state safety argument with a quantified tracking bound) or should present the humanoid result as a limitation rather than as validation of the invariance guarantee.
minor comments (4)
  1. [§II.B] There is a typo: 'aribtrary' should be 'arbitrary'; similarly, the Appendix A heading 'POISSION'S EQUATION' should be 'POISSON'S EQUATION'.
  2. [§V.B] The sentence 'Since hB(y) ≤ h(y) for all y ∈ C' should read 'for all (y, ˙y) ∈ C × R^3', because hB depends on the velocity error as well as the position.
  3. [§VI.B.1, Fig. 5] The claim that 'h remained positive' is based on evaluation at sampled robot states; the paper should explicitly state the sampling rate and avoid implying a continuous-time certificate from discrete evaluations.
  4. [References] References [30] and [43] are the same conference paper; please disambiguate or unify the citation entries.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the Poisson safety functions and CBF filters are derived constructively from standard elliptic PDE theory, and the overlapping-authority backstepping citations are used as published tools rather than as substitutes for the paper's own derivation.

full rationale

The derivation is self-contained in the relevant sense. The safety function h is defined as the unique solution of the Dirichlet problem (16) with a strictly negative Holder forcing function; Theorem 1 derives the safety-function properties (14) and Dh != 0 on dC from the weak minimum principle and Hopf's lemma (Theorem 3, Lemma 2), which are cited to classical external texts, not to the authors' own prior work. The forcing functions in Section IV are user-specified constructions rather than parameters fitted to a desired outcome, and the variational equivalence in Theorem 5 is the standard Dirichlet principle, not a forced identity. Proposition 1 exhibits an explicit locally Lipschitz controller (34) satisfying (33), and Theorem 6/Appendix B invokes the published backstepping result [43, Theorem 5] as an external theorem whose stated assumptions do not include the Poisson-safety-function conclusion; overlapping authorship of [43] is therefore not load-bearing circularity. The only significant weakness is a correctness gap rather than a circularity: the dynamic experiments recompute h at 10 Hz and add a finite-difference dh/dt term (41), but no theorem establishes forward invariance for time-varying h, and the humanoid section admits 'minor safety violations, corresponding to moments when h was briefly negative.' That unsupported time-varying extension does not make any claimed result equivalent to its inputs by construction, so the circularity score remains 0.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central derivation rests on standard elliptic PDE theory and cited backstepping results; the genuinely assumed inputs are the occupancy map fidelity and the accuracy of the single-integrator reduced-order model. Design parameters beta, mu_i, gamma and b are tunable but the theorems hold for any admissible values, so they are not fit-to-target free parameters.

free parameters (4)
  • beta
    Smoothing temperature in the softplus forcing (29); chosen by hand, not fitted. The theorem holds for any beta>0.
  • mu_i (backstepping margins)
    Weights in the backstepping CBF (36), (60); chosen by hand. Guarantees hold for any mu_i>0.
  • gamma (class K gain)
    Gain in CBF condition (5), (33); chosen by hand; any gamma>0 preserves the invariant-set property.
  • boundary flux b(y)
    User-assigned boundary flux in (26) to encode repulsive strength per obstacle; a design input, not fitted to data.
assumptions (5)
  • standard math Classical elliptic regularity and existence for the Dirichlet problem (Theorem 4)
    Invoked in Theorem 1 to conclude smoothness of h from Holder-continuous f; standard Schauder theory [14].
  • standard math Weak maximum/minimum principle and Hopf's lemma (Theorem 3, Lemma 2)
    Used to prove h>0 in Omega and Dh.n<0 on the boundary; standard results [51].
  • standard math Nagumo's theorem and CBF backstepping theorem [43, Thm 4/5]
    Used to render C and CB forward invariant; cited published results.
  • domain assumption Perception pipeline produces a faithful occupancy map and boundary ∂Omega
    The whole method starts from 'given an occupancy map' (Section III); errors in SAM2 segmentation or buffering are not modeled. This is an assumption about the environment representation.
  • domain assumption Single-integrator ROM (32) accurately represents the robot for filtering
    Hardware filter commands velocity for a single integrator; the paper notes the humanoid tracks velocity poorly, causing safety violations (Section VI.B.3).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Dynamic Safety in Complex Environments: Synthesizing Safety Filters with Poisson's Equation." pith.science (2026). https://pith.science/paper/GKB77HWE

@misc{pith2026250506794,
  author       = {Pith},
  title        = {Pith review of: Dynamic Safety in Complex Environments: Synthesizing Safety Filters with Poisson's Equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GKB77HWE}},
  note         = {Machine review of arXiv:2505.06794}
}
read the original 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

Figures reproduced from arXiv: 2505.06794 by the authors.

Figure 1
Figure 1. Safe set synthesis from perception data via Poisson’s equation. [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. [From left to right] Solutions to Poisson’s equation (16) with the following forcing functions: [left] a Holder continuous function ¨ (19) with α = 0.1; [mid-left] an average flux forcing function with ¯b = −1 in (23); [mid-right] a smooth forcing function (29) constructed using the guidance field (26) under a uniform boundary flux b(y) = −1 for all y ∈ ∂Ω; and [right] the same forcing function with a non-uniform bo… view at source ↗
Figure 3
Figure 3. Smooth guidance field generation via Laplace’s equation (26). [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Double integrator simulations using safety filters synthesized from: [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. 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 …
Figure 6
Figure 6. Figure 6: Hardware experiments demonstrating dynamic behavior on quadrupedal and humanoid robots. [Top Left] A timelapse showing the [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: Interior sphere condition for [left] smooth boundary and [right] Lipschitz boundary. Lemma 2. (Hopf’s Lemma [51]) Consider an open, bounded and connected set Ω ⊂ R 3 with smooth boundary ∂Ω, and let h ∈ C 2 (Ω) such that ∆h ≥ 0 (≤ 0) in Ω. If y0 ∈ ∂Ω is an extremal poi…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Grasp Execution Without a Planner: Configuration-Space Grasp Distance Fields with Certified Safety & Guaranteed Quality

    cs.RO 2026-08 reject novelty 6.0 of 10

    Grasp execution via a softmin field over grasp configurations with CBF-QP safety filtering, eliminating trajectory replanning, with a force-closure margin guarantee that fails in one reported trial.

  2. Towards General Language-Conditioned Latent Safety Filters

    cs.RO 2026-07 conditional novelty 6.0 of 10

    A single Hamilton-Jacobi safety filter conditioned on language constraints reduces violations in simulated pick-and-place, wiping, and stacking, with partial transfer to unseen constraint instances.

Reference graph

Works this paper leans on

58 extracted references · 54 canonical work pages · cited by 2 Pith papers

  1. [1]

    Hamilton-Jacobi reachability: A brief overview and re- cent advances,

    S. Bansal, M. Chen, S. Herbert, and C. J. Tomlin, “Hamilton-Jacobi reachability: A brief overview and re- cent advances,” in Proc. Conf. Decis. Control, pp. 2242– 2253, 2017

    work page 2017

  2. [2]

    Fastrack: A modular framework for real-time motion planning and guaranteed safe tracking,

    M. Chen, S. L. Herbert, H. Hu, Y . Pu, J. F. Fisac, S. Bansal, S. Han, and C. J. Tomlin, “Fastrack: A modular framework for real-time motion planning and guaranteed safe tracking,” IEEE Trans. Autom. Control , vol. 66, no. 12, pp. 5861–5876, 2021

    2021

  3. [3]

    Data- driven safety filters: Hamilton-jacobi reachability, control barrier functions, and predictive methods for uncertain systems,

    K. P. Wabersich, A. J. Taylor, J. J. Choi, K. Sreenath, C. J. Tomlin, A. D. Ames, and M. N. Zeilinger, “Data- driven safety filters: Hamilton-jacobi reachability, control barrier functions, and predictive methods for uncertain systems,” IEEE Control Systems Magazine, vol. 43, no. 5, pp. 137–177, 2023

    work page 2023

  4. [4]

    Borrelli, A

    F. Borrelli, A. Bemporad, and M. Morari, Predictive con- trol for linear and hybrid systems. Cambridge University Press, 2017

    work page 2017

  5. [5]

    Predictive con- trol barrier functions: Enhanced safety mechanisms for learning-based control,

    K. P. Wabersich and M. N. Zeilinger, “Predictive con- trol barrier functions: Enhanced safety mechanisms for learning-based control,” IEEE Transactions on Automatic Control, vol. 68, no. 5, pp. 2638–2651, 2023

    work page 2023

  6. [6]

    Real-time obstacle avoidance for manipu- lators and mobile robots,

    O. Khatib, “Real-time obstacle avoidance for manipu- lators and mobile robots,” The international journal of robotics research, vol. 5, no. 1, pp. 90–98, 1986

    work page 1986

  7. [7]

    Comparative analysis of control barrier functions and artificial potential fields for obstacle avoidance,

    A. Singletary, K. Klingebiel, J. Bourne, A. Browning, P. Tokumaru, and A. Ames, “Comparative analysis of control barrier functions and artificial potential fields for obstacle avoidance,” in 2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) , pp. 8129–8136, IEEE, 2021

    work page 2021

  8. [8]

    Control barrier function based quadratic programs for safety critical systems,

    A. D. Ames, X. Xu, J. W. Grizzle, and P. Tabuada, “Control barrier function based quadratic programs for safety critical systems,” IEEE Trans. Autom. Control , vol. 62, no. 8, pp. 3861–3876, 2017

    work page 2017

Show all 58 references
  1. [9]

    Nonsmooth barrier functions with applications to multi-robot sys- tems,

    P. Glotfelter, J. Cort ´es, and M. Egerstedt, “Nonsmooth barrier functions with applications to multi-robot sys- tems,” IEEE Control Systems Letters , vol. 1, no. 2, pp. 310–315, 2017

    2017

  2. [10]

    Composing control barrier functions for complex safety specifications,

    T. G. Molnar and A. D. Ames, “Composing control barrier functions for complex safety specifications,”IEEE Control Systems Letters , 2023

    2023

  3. [11]

    Safety-critical manipulation for collision- free food preparation,

    A. Singletary, W. Guffey, T. G. Molnar, R. Sinnet, and A. D. Ames, “Safety-critical manipulation for collision- free food preparation,” IEEE Robotics and Automation Letters, vol. 7, no. 4, pp. 10954–10961, 2022

    2022

  4. [12]

    V oxblox: Incremental 3d euclidean signed distance fields for on-board mav planning,

    H. Oleynikova, Z. Taylor, M. Fehr, R. Siegwart, and J. Nieto, “V oxblox: Incremental 3d euclidean signed distance fields for on-board mav planning,” in IEEE/RSJ International Conference on Intelligent Robots and Sys- tems (IROS), pp. 1366–1373, IEEE, 2017

    2017

  5. [13]

    Learning barrier functions with memory for robust safe naviga- tion,

    K. Long, C. Qian, J. Cort ´es, and N. Atanasov, “Learning barrier functions with memory for robust safe naviga- tion,” IEEE Robotics and Automation Letters , vol. 6, no. 3, pp. 4931–4938, 2021

    2021

  6. [14]

    Gilbarg and N

    D. Gilbarg and N. S. Trudinger, Elliptic partial differen- tial equations of second order , vol. 224. Springer, 1977

    1977

  7. [15]

    Jost, Partial Differential Equations, vol

    J. Jost, Partial Differential Equations, vol. 214. Springer Science & Business Media, 3rd ed., 2012

    2012

  8. [16]

    Path plan- ning using laplace’s equation,

    C. I. Connolly, J. B. Burns, and R. Weiss, “Path plan- ning using laplace’s equation,” in Proceedings., IEEE International Conference on Robotics and Automation , pp. 2102–2106, IEEE, 1990

    1990

  9. [17]

    The applications of harmonic functions to robotics,

    C. I. Connolly and R. A. Grupen, “The applications of harmonic functions to robotics,” Journal of robotic Systems, vol. 10, no. 7, pp. 931–946, 1993

    1993

  10. [18]

    Real-time obstacle avoidance using harmonic potential functions,

    J.-O. Kim and P. Khosla, “Real-time obstacle avoidance using harmonic potential functions,” IEEE Transactions on Robotics and Automation , vol. 8, no. 3, pp. 338–349, 1992

    1992

  11. [19]

    Path planning based on fluid mechanics for mobile robots using unstructured terrain models,

    D. Gingras, E. Dupuis, G. Payre, and J. de Lafontaine, “Path planning based on fluid mechanics for mobile robots using unstructured terrain models,” in 2010 IEEE International Conference on Robotics and Automation , pp. 1978–1984, 2010

    2010

  12. [20]

    Control bar- rier function based quadratic programs with application to adaptive cruise control,

    A. D. Ames, J. W. Grizzle, and P. Tabuada, “Control bar- rier function based quadratic programs with application to adaptive cruise control,” in Proc. Conf. Decis. Control, (Los Angeles, CA), pp. 6271–6278, Dec. 2014

    2014

  13. [21]

    Towards a framework for realizable safety critical control through active set invariance,

    T. Gurriet, A. Singletary, J. Reher, L. Ciarletta, E. Feron, and A. Ames, “Towards a framework for realizable safety critical control through active set invariance,” in 2018 ACM/IEEE 9th International Conference on Cyber- Physical Systems (ICCPS) , pp. 98–106, IEEE, 2018

    2018

  14. [22]

    Control barrier functions: theory and applications,

    A. D. Ames, S. Coogan, M. Egerstedt, G. Notomista, K. Sreenath, and P. Tabuada, “Control barrier functions: theory and applications,” in Proc. Eur. Control Conf. , pp. 3420–3431, 2019

    2019

  15. [23]

    Model-free safety-critical con- trol for robotic systems,

    T. G. Molnar, R. K. Cosner, A. W. Singletary, W. Ubel- lacker, and A. D. Ames, “Model-free safety-critical con- trol for robotic systems,” IEEE Robot. Aut. Lett. , vol. 7, no. 2, pp. 944–951, 2022

    2022

  16. [24]

    Construc- tive safety-critical control: Synthesizing control barrier functions for partially feedback linearizable systems,

    M. H. Cohen, R. K. Cosner, and A. D. Ames, “Construc- tive safety-critical control: Synthesizing control barrier functions for partially feedback linearizable systems,” IEEE Control Systems Letters , 2024

    2024

  17. [25]

    Control barrier function synthesis for nonlinear systems with dual relative degree,

    G. Bahati, R. K. Cosner, M. H. Cohen, R. M. Bena, and A. D. Ames, “Control barrier function synthesis for nonlinear systems with dual relative degree,” arXiv preprint arXiv:2504.00397, 2025

    2025 arXiv

  18. [26]

    Poisson image editing,

    P. P ´erez, M. Gangnet, and A. Blake, “Poisson image editing,” ACM Trans. Graph. , vol. 22, p. 313–318, July 2003

    2003

  19. [27]

    Perko, Differential equations and dynamical systems , vol

    L. Perko, Differential equations and dynamical systems , vol. 7. Springer Science & Business Media, 2013

    2013

  20. [28]

    Set invariance in control,

    F. Blanchini, “Set invariance in control,” Automatica, vol. 35, no. 11, pp. 1747–1767, 1999

    1999

  21. [29]

    Isidori, Nonlinear control systems: an introduction

    A. Isidori, Nonlinear control systems: an introduction . Springer, 1985

    1985

  22. [30]

    Safe backstepping with control barrier functions,

    A. J. Taylor, P. Ong, T. G. Molnar, and A. D. Ames, “Safe backstepping with control barrier functions,” in Proc. Conf. Decis. Control , pp. 5775–5782, 2022

    2022

  23. [31]

    Invariant sets for integrators and quadrotor obstacle avoidance,

    L. Doeser, P. Nilsson, A. D. Ames, and R. M. Murray, “Invariant sets for integrators and quadrotor obstacle avoidance,” in Proceedings of the American Control Conference, pp. 3814–3821, 2020

    2020

  24. [32]

    Generalizing Hopf’s boundary point Lemma,

    L. Rosales, “Generalizing Hopf’s boundary point Lemma,” Canadian Mathematical Bulletin, vol. 62, no. 1, pp. 183–197, 2019

    2019

  25. [33]

    Around the nor- mal derivative lemma,

    D. E. Apushkinskaya and A. I. Nazarov, “Around the nor- mal derivative lemma,”arXiv preprint arXiv:2206.08043, 2022

    2022 arXiv

  26. [34]

    Grisvard, Elliptic problems in nonsmooth domains

    P. Grisvard, Elliptic problems in nonsmooth domains . SIAM, 2011

    2011

  27. [35]

    Borsuk and V

    M. Borsuk and V . Kondratiev, Elliptic boundary value problems of second order in piecewise smooth domains , vol. 69. Elsevier, 2006

    2006

  28. [36]

    Lectures on elliptic and parabolic equa- tions in H ¨older spaces,

    N. V . Krylov, “Lectures on elliptic and parabolic equa- tions in H ¨older spaces,” American Mathematical Society, 1996

    1996

  29. [37]

    Fiorenza, H¨older and locally H ¨older continuous func- tions, and open sets of class Ck,Ck,λ

    R. Fiorenza, H¨older and locally H ¨older continuous func- tions, and open sets of class Ck,Ck,λ. Birkh ¨auser, 2017

    2017

  30. [38]

    L. C. Evans, Partial differential equations . American Mathematical Society, 1998

    1998

  31. [39]

    J. E. Marsden and A. Tromba, Vector calculus. Macmil- lan, 2003

    2003

  32. [40]

    O. D. Kellogg, Foundations of potential theory , vol. 31. Springer Science & Business Media, 2012

    2012

  33. [41]

    Char- acterizing smooth safety filters via the implicit function theorem,

    M. H. Cohen, P. Ong, G. Bahati, and A. D. Ames, “Char- acterizing smooth safety filters via the implicit function theorem,” IEEE Contr. Syst. Lett., vol. 7, pp. 3890–3895, 2023

    2023

  34. [42]

    A Universal Construction of Artstein’s Theorem on Nonlinear Stabilization,

    E. D. Sontag, “A Universal Construction of Artstein’s Theorem on Nonlinear Stabilization,” Systems & control letters, vol. 13, no. 2, pp. 117–123, 1989

    1989

  35. [43]

    Safe backstepping with control barrier functions,

    A. J. Taylor, P. Ong, T. G. Molnar, and A. D. Ames, “Safe backstepping with control barrier functions,” in 2022 IEEE 61st Conference on Decision and Control (CDC), pp. 5775–5782, IEEE, 2022

    2022

  36. [44]

    Br ´ezis, Functional analysis, Sobolev spaces and par- tial differential equations , vol

    H. Br ´ezis, Functional analysis, Sobolev spaces and par- tial differential equations , vol. 2. Springer, 2011

    2011

  37. [45]

    J. C. Strikwerda, Finite difference schemes and partial differential equations. SIAM, 2004

    2004

  38. [46]

    On variational and pde-based distance function approximations,

    A. G. Belyaev and P.-A. Fayolle, “On variational and pde-based distance function approximations,” in Com- puter Graphics Forum , vol. 34, pp. 104–118, Wiley Online Library, 2015

    2015

  39. [47]

    On the behavior of the fundamental solution of the heat equation with variable coefficients,

    S. R. S. Varadhan, “On the behavior of the fundamental solution of the heat equation with variable coefficients,” Communications on Pure and Applied Mathematics , vol. 20, no. 2, pp. 431–455, 1967

    1967

  40. [48]

    Safety- critical control for autonomous systems: Control barrier functions via reduced order models,

    M. H. Cohen, T. G. Molnar, and A. D. Ames, “Safety- critical control for autonomous systems: Control barrier functions via reduced order models,” Annual Reviews in Control, vol. 57, p. 100947, 2024

    2024

  41. [49]

    Sam 2: Segment any- thing in images and videos,

    N. Ravi, V . Gabeur, Y .-T. Hu, R. Hu, C. Ryali, T. Ma, H. Khedr, R. R¨adle, C. Rolland, L. Gustafson, E. Mintun, J. Pan, K. V . Alwala, N. Carion, C.-Y . Wu, R. Girshick, P. Doll´ar, and C. Feichtenhofer, “Sam 2: Segment any- thing in images and videos,” 2024

    2024

  42. [50]

    Coron, Control and Nonlinearity

    J.-M. Coron, Control and Nonlinearity. American Math- ematical Society, 2007

    2007

  43. [51]

    M. H. Protter and H. F. Weinberger, Maximum principles in differential equations . Springer Science & Business Media, 2012

    2012

  44. [52]

    Fern ´andez-Real and X

    X. Fern ´andez-Real and X. Ros-Oton, Regularity Theory for Elliptic PDE . Zurich Lectures in Advanced Mathe- matics, European Mathematical Society, 2022

    2022

  45. [53]

    Dacorogna, Introduction to the Calculus of Variations

    B. Dacorogna, Introduction to the Calculus of Variations. World Scientific, 2004

    2004

  46. [54]

    L. A. Caffarelli and X. Cabr ´e, Fully nonlinear elliptic equations, vol. 43. American Mathematical Society, 1995

    1995

  47. [55]

    Dacorogna, Direct methods in the Calculus of Varia- tions, vol

    B. Dacorogna, Direct methods in the Calculus of Varia- tions, vol. 78. Springer Science & Business Media, 2007

    2007

  48. [56]

    High-order control barrier func- tions,

    W. Xiao and C. Belta, “High-order control barrier func- tions,” IEEE Transactions on Automatic Control, vol. 67, no. 7, pp. 3655–3662, 2021

    2021

  49. [57]

    High-order barrier functions: Robustness, safety, and performance-critical control,

    X. Tan, W. S. Cortez, and D. V . Dimarogonas, “High-order barrier functions: Robustness, safety, and performance-critical control,” IEEE Transactions on Au- tomatic Control, vol. 67, no. 6, pp. 3021–3028, 2022

    2022

  50. [58]

    Rectified control barrier functions for high-order safety constraints,

    P. Ong, M. H. Cohen, T. G. Molnar, and A. D. Ames, “Rectified control barrier functions for high-order safety constraints,” IEEE Control Systems Letters , 2024. APPENDIX A POISSON ’S EQUATION This appendix reviews key concepts from second-order lin- ear elliptic PDEs, emphasiz...

    2024

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.