Online Synthesis of Control Barrier Functions with Local Occupancy Grid Maps for Safe Navigation in Unknown Environments

Shaoyuan Li; Xiang Yin; Yu Chen; Yuda Li; Yuepeng Zhang

REVIEW 4 major objections 4 minor 37 references

Online Synthesis of Control Barrier Functions with Local Occupancy Grid Maps for Safe Navigation in Unknown Environments

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

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Pith's one-line read This paper claims that solving Laplace's equation over a local occupancy grid map produces a control barrier function that keeps a robot safe in unknown environments and can be recomputed in about 9 milliseconds.

desk verdict A useful engineering paper for real-time CBF synthesis from local grids, but the proof covers an exact global harmonic function rather than the truncated local map actually used, and the unicycle transfer is asserted, not proved. read the letter →

arxiv 2505.11941 v1 pith:RVQAESY4 submitted 2025-05-17 cs.RO

Yuepeng Zhang , Yu Chen , Yuda Li , Shaoyuan Li , Xiang Yin This is my paper · ORCID

classification cs.RO

keywords controlbarrierfunctionoccupancygridmapLaplace'sequationsafenavigationonlinesynthesisharmonicsparselinearsolvermobilerobotsafety

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 claims that a safe navigation command can be synthesized directly from a local occupancy grid map by treating the map as a boundary-value problem for steady-state heat conduction. The safety function h(p) is the solution of Laplace's equation, with obstacles held at temperature -a and safe boundaries at temperature b, and the paper argues that this h(p) is a valid control barrier function for single-integrator dynamics. Because the resulting linear system is sparse, the field can be recomputed online in about 9 ms for a 200 by 200 grid, fast enough to track a moving robot's changing sensor view. A single scalar constraint then keeps the robot inside the safe set regardless of how many obstacles or what shapes appear in the map. The payoff is a perception-driven safety filter that needs no pretraining and no per-obstacle modeling.

What carries the argument

The load-bearing object is the discrete Laplace equation with a five-point stencil: each transition cell's safety value equals the average of its four neighbors, assembled into the sparse linear system Ah = b, where A has diagonal 4 and off-diagonal entries -1 or 0. Boundary cells are fixed by the Dirichlet condition (10), and interior cells in the safety and obstacle regions are fixed at b and -a. The sparsity lets the system be solved iteratively by GMRES or BiCGSTAB, which is what makes online recomputation at millisecond scale possible. The harmonic nature of the solution supplies the smoothness, maximum principle, and isolated-critical-point properties on which the safety proof rests.

What would settle it

Run the controller in an arena with a thin pole placed just outside the robot's 200 by 200 local map in the direction of travel, so it is not yet in the occupancy grid; if the robot does not brake or deviate before the map expands to reveal the pole, the online safety claim is disproved.

Watch

Extended reading notes

Core claim

The central claim is that the harmonic function obtained from Laplace's equation, with Dirichlet conditions h = -a on obstacle boundaries and h = b on the safety-region and workspace boundaries, is a control barrier function for the single-integrator model, and that its discrete version over a local occupancy grid map can be solved in real time. The paper defines this as the steady-state thermal field-inspired CBF (SSTF-CBF). Propositions 1 and 3 argue that h is continuously differentiable, that its critical points are isolated, and that even if a robot crosses the h = 0 boundary at a point where the gradient vanishes, it cannot reach an obstacle before leaving the unsafe sublevel set. Experiments with a TurtleBot in simulation and in the lab show the field synthesized in average times around 9 ms on 200 by 200 maps, with safety values remaining positive during navigation.

Load-bearing premise

The safety guarantee rests on the assumption that the local map's outer boundary really is a safe boundary, so anything outside the current sensor range is treated as safe; an obstacle lurking just beyond the map edge is invisible to the computed field.

Editorial extensions

If this is right

Only one CBF constraint is needed in the safety QP, independent of obstacle count or shape, so the optimization stays small even in cluttered maps.
The per-cycle synthesis time of about 9 ms on a 200 by 200 grid makes it practical to refresh the safety field as fast as sensor maps are produced, including with dynamic obstacles.
Because the field is harmonic in the transition region, there is no need for the interpolation or smoothing that signed-distance-function-based CBFs require.
A robot that momentarily crosses into the h < 0 region is still protected from collisions by Proposition 3, so the filter tolerates temporary violations of the nominal safe set.
The same construction works with local maps that treat other robots as obstacles, as demonstrated by the two-robot real-world experiment.

Reading between the lines

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

A natural next step is to derive a quantitative bound on the error between the local-window harmonic solution and the global harmonic solution; without such a bound the safety certificate formally covers the closed workspace, not a moving sensor window.
Because the map-to-CBF mapping is linear, the fixed-stencil computation could be shifted almost entirely to a GPU or embedded accelerator, potentially reducing the 9 ms figure further.
The same harmonic construction could serve as a navigation function, with gradient descent on h used for planning as well as for safety filtering, uniting the two layers in one scalar field.
For multi-robot operation, the experiments show mutual avoidance, but the proof treats other agents as static obstacles in each local map; a formal treatment of concurrent updates would be needed before claiming pairwise guarantees between moving robots.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Know this paper for two things: a clean way to turn a local occupancy grid into a single smooth safety function by solving Laplace's equation, and a formal guarantee that stops short of the actual algorithm. The experiments are real and the timings are credible.

What's genuinely new is the combination: solve the heat equation on a 200x200 local OGM with a sparse iterative solver, then use that as the CBF for the whole workspace. That gives you one smooth safety function for arbitrary obstacle shapes, which is a real advantage over ellipse-based or SDF-based constructions. The authors know the related literature and the writing is clear. The experiments include both Gazebo and real TurtleBots, and showing ~10ms synthesis online is a legitimate contribution.

The soft spots are concentrated in the gap between the math and the implementation. The main theorem proves safety for the harmonic function on the full workspace with true obstacle boundaries. Algorithm 1, however, solves the same PDE on a truncated local map whose outer edge is artificially assigned the safe value b. Obstacles just outside the sensor range are simply absent from that boundary-value problem, and there is no error bound linking the local solution to the global one. In the worst case, the artificial boundary condition can actually create a gradient that steers the robot toward an obstacle that hasn't entered the map yet. So the 'safe in unknown environments' claim is wider than what the mathematics supports. That's the kind of thing a serious revision can fix—e.g., by adding a pessimistic buffer at the map edge or by analyzing the harmonic extension—but the current paper does not address it.

Second, the unicycle mapping in Eq. (19) is not a near-identity diffeomorphism in the usual sense: the commanded [vx,vy] gets projected onto the heading direction, so the single-integrator CBF guarantee does not automatically transfer. The experiments work, but that is empirical evidence, not a proof. Proposition 3 is also written in a way that is hard to follow; the vmax machinery seems unnecessary and it quietly assumes the path encounters no critical points. I think the conclusion can be recovered more directly—inside the unsafe set, the barrier constraint forces h nondecreasing—so this is a presentation problem more than a fatal flaw.

Third, the experimental section has no baselines and no error bars. The timing numbers are nice, but without comparison to SDF or ellipse methods the practical claims stay suggestive. No code is released either.

The paper deserves a serious referee. It is an honest engineering effort with a promising idea and a gap that can be closed. I would bring it to a reading group and would cite it if I were working on sensor-based CBF synthesis, but I would cite it with a caveat about the truncation issue.

Referee Report

4 major / 4 minor

Summary. The paper proposes SSTF-CBF, a method for online synthesis of control barrier functions from local occupancy grid maps. The safety function h is obtained by solving Laplace's equation over the grid with Dirichlet boundary conditions: h=-a on obstacle boundaries and h=b on the safety-region and workspace boundaries. The resulting h is used to filter a nominal velocity controller by a quadratic program with the single-integrator CBF constraint. The authors claim that this construction yields a valid CBF (Proposition 1), that critical points of h are isolated (Proposition 2), and that safety is maintained even if the robot enters the zero-sublevel set through a critical boundary point (Proposition 3). Experiments in Gazebo and with two real TurtleBots show millisecond-level synthesis times on 200x200 local OGMs. The computational claim is supported by the sparse linear-system formulation and by Table I.

Significance. The core idea of using the steady-state heat equation to turn a local occupancy map into a smooth safety field is genuinely attractive, and the sparse-matrix formulation is a sensible alternative to SDF-based smoothing or per-obstacle ellipsoidal CBFs. The experiments are informative and the reported computation times support the real-time feasibility claim. I also credit the authors for a construction that is not fitted to make the experiments pass; the parameters a, b, delta, and alpha are user-chosen design parameters. However, the formal safety guarantees stated in Propositions 1 and 3 are proved for an idealized global harmonic solution, not for the truncated local-grid function actually used in Algorithm 1, and the transfer from the single-integrator model to the unicycle experiments is asserted rather than proved. These gaps are load-bearing because the title and abstract promise safe navigation in unknown environments. The underlying approach is promising and the gaps appear addressable in principle, but the manuscript in its current form does not establish its central formal claim.

major comments (4)

[§IV-A and Algorithm 1, Eq. (10)] The theoretical results in Propositions 1 and 3 concern the harmonic function defined over the full workspace W with Dirichlet data on the true obstacle boundary ∂O and the true outer boundary ∂W. Algorithm 1, however, constructs the same boundary-value problem on a 200x200 local occupancy grid and, per lines 9-13, assigns the artificial map edge the 'safe' value b. Obstacles beyond the current sensor range simply do not appear in the boundary data, and no theorem or error bound relates the local finite-difference solution to the global harmonic solution. Consequently, the formal guarantee established in Propositions 1 and 3 does not cover the online implementation. This is not merely a theoretical nicety: in the no-obstacle branch (Algorithm 1, lines 5-7) the nominal command is executed unfiltered, so a goal-directed controller can drive the robot toward an obstacle that has not yet entered the sensing range. None of the experiments appears to place obstacles at the sensing boundary, so this regime is untested.
[§V-A, Eq. (19)] The safety filter is designed for the single-integrator model (15), but the experiments use the unicycle model (18). The near-identity transformation (19) is applied to velocity commands, not to the state, and with p defined as the robot center the resulting position dynamics are \dot p = [cos θ; sin θ](cos θ v_x + sin θ v_y), which is not \dot p = [v_x; v_y]ᵀ. Therefore the single-integrator condition L_f h + L_g h u ≥ -α(h) does not automatically imply the corresponding condition for the actual unicycle center dynamics. If p is instead intended to be the offset point used in the Robotarium reference [37], then h is evaluated at that offset point and additional reasoning is needed to conclude safety of the robot center. The paper asserts that safety transfers to the unicycle model but provides no proof; this is a load-bearing gap because the experiments and the claimed experimental safety rely on the unicycle implementation.
[§IV-B, Proposition 1] The proof of Proposition 1 reduces to the observation that u=0 is admissible in the single-integrator model. For any continuously differentiable function h with 0 in the input constraint set U, condition (6) is satisfied on the safe set C because 0 ≥ -α(h(p)) whenever h(p) ≥ 0. Thus Proposition 1 does not distinguish the proposed SSTF-CBF from any smooth function; the Laplace construction is not needed for the CBF condition itself. The meaningful safety-relevant ingredient is the direction and magnitude of ∇h in the QP constraint (16), and the paper should present the non-vacuous invariance argument for that constraint directly. As written, the theoretical contribution of Section IV-B is overstated.
[§IV-B, Proposition 3] The proof of Proposition 3 is incomplete. The quantity vmax is defined as the maximum safety value among critical points in Int(I), but no argument is given that such critical points exist; if there are none, the proof cannot proceed. More seriously, the QP (16) is infeasible at any state with h(p)<0 and ∂h/∂p=0, because the constraint becomes 0 ≥ -α(h(p)) with -α(h(p))>0. Proposition 3 does not rule out reaching such a state, and the statement that the robot encounters no critical points between p0 and p(t2) is asserted without proof. The claim that the robot 'will not enter the obstacle region' before exiting Uc is therefore not established by the given argument.

minor comments (4)

[§IV-A, Eq. (12)] The notation for grid values is inconsistent: the text uses both h_i,j and hij for the same quantity; please unify.
[Abstract / Introduction] Several typos should be corrected, including 'yeilds' in §IV-A, 'simlified' in §III, and 'the CBFs needs' in the Introduction.
[§V-B] The text says a target is considered reached when the robot is within 0.5 cm of the destination; given the 3 m x 3 m arena this is likely a typo for 0.5 m. Please clarify.
[§IV-A, Fig. 3] The caption states 'region O and ∂O are identical,' which is confusing because O is a set of obstacle cells and ∂O is its boundary; please rephrase or explain the discretized convention.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the safety function is constructed from declared boundary data, and the CBF property follows from harmonic-function facts plus the zero-control option, not from a fitted or self-cited input.

full rationale

The paper's derivation chain is self-contained and non-circular. The safety function h(p) is defined as the solution of Laplace's equation (11) with explicit Dirichlet boundary conditions (10): h = -a on obstacle boundaries and h = b on the safety-region and workspace boundaries. Proposition 1 then proves the CBF property by invoking harmonic-function smoothness and the fact that for single-integrator dynamics the control input can be chosen as zero, which trivially satisfies the lower bound in Definition 2 because h(p) >= 0 on the safe set. Propositions 2 and 3 use standard harmonic-function facts (isolated critical points, the maximum principle) to handle the zero-gradient case; these are external mathematical facts, not assumptions equivalent to the conclusion. The online algorithm solves the same discrete Laplace system (12)-(13) on the occupancy grid, and Table I reports measured computation times rather than predicted quantities obtained from fitted parameters. The design parameters a, b, delta, and alpha are stated user choices, not calibrated to force the experimental outcomes. The only potentially overlapping citation, reference [25], appears in a related-work comparison and is not load-bearing for any formal claim. The gap between the local truncated occupancy grid and the global workspace boundary is a real correctness limitation of the safety guarantee, but it is not a circularity: no equation or fitted value is used as its own input, and the local map is not used to derive the theorem that is then claimed as a prediction. Therefore the paper should receive a circularity score of 0.

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

The method introduces no new physical entities. It relies on standard harmonic function theory and on several design parameters chosen by hand (a, b, delta, alpha). The main unstated assumptions are that the OGM and its boundaries are accurate, that the discrete finite-difference solution is close enough to the continuous harmonic solution to preserve the CBF condition, and that safety for the single-integrator model transfers to the unicycle model through the diffeomorphism in Eq. (19).

free parameters (4)

a = 1
Boundary safety value on obstacle boundaries; user-chosen in experiments (Section V-B). Scales h and affects where the zero level set sits relative to obstacles.
b = 1
Boundary safety value on safety-region and workspace boundaries; user-chosen in experiments (Section V-B).
delta = 0.15 m
Safety margin defining S; chosen based on robot radius and dilation. It sets the distance between the obstacle boundary and the hot boundary, hence the position of the zero contour.
alpha = 0.15 h
Extended class K function in the CBF constraint (16b); user-chosen in experiments, controls how aggressively safety is enforced.

assumptions (5)

standard math Laplace's equation solutions are infinitely differentiable in the interior (standard elliptic regularity).
Used in Section IV-A and Proposition 1 to claim smoothness of h.
standard math A non-constant harmonic function on a compact set attains extrema on the boundary and has finitely many isolated critical points in compact subsets.
Used in Propositions 2 and 3 to analyze critical points and extremal values inside unsafe regions.
domain assumption The occupancy grid map Mt accurately represents the obstacle set O, including its boundaries, and the local map boundary can be treated as a safe Dirichlet boundary.
The algorithm extracts ∂O and ∂S from the OGM; errors in perception or the artificial outer boundary are not modeled.
domain assumption The discrete finite-difference solution of Laplace's equation on the grid is sufficiently close to the continuous harmonic function for the gradient-based CBF constraint to remain valid.
The paper solves (13) on a grid and computes gradients by finite differences, but does not provide error bounds; this is an unstated approximation.
domain assumption Safety of the single-integrator model transfers to the unicycle model through the transformation in Eq. (19).
The formal propositions are proved for x-dot = u, while experiments use the TurtleBot unicycle model; the mapping is called near identity but no proof is given.

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Pith. "Pith review of Online Synthesis of Control Barrier Functions with Local Occupancy Grid Maps for Safe Navigation in Unknown Environments." pith.science (2026). https://pith.science/paper/RVQAESY4

@misc{pith2026250511941,
  author       = {Pith},
  title        = {Pith review of: Online Synthesis of Control Barrier Functions with Local Occupancy Grid Maps for Safe Navigation in Unknown Environments},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RVQAESY4}},
  note         = {Machine review of arXiv:2505.11941}
}

read the original abstract

Control Barrier Functions (CBFs) have emerged as an effective and non-invasive safety filter for ensuring the safety of autonomous systems in dynamic environments with formal guarantees. However, most existing works on CBF synthesis focus on fully known settings. Synthesizing CBFs online based on perception data in unknown environments poses particular challenges. Specifically, this requires the construction of CBFs from high-dimensional data efficiently in real time. This paper proposes a new approach for online synthesis of CBFs directly from local Occupancy Grid Maps (OGMs). Inspired by steady-state thermal fields, we show that the smoothness requirement of CBFs corresponds to the solution of the steady-state heat conduction equation with suitably chosen boundary conditions. By leveraging the sparsity of the coefficient matrix in Laplace's equation, our approach allows for efficient computation of safety values for each grid cell in the map. Simulation and real-world experiments demonstrate the effectiveness of our approach. Specifically, the results show that our CBFs can be synthesized in an average of milliseconds on a 200 * 200 grid map, highlighting its real-time applicability.

Figures

Figures reproduced from arXiv: 2505.11941 by the authors.

**Figure 1.** (Left) The robot’s navigation environment, where orange regions denote obstacles and the blue region represents the robot’s local sensing range. (Right) The real-time occupancy grid map generated during navigation, where black grids indicate obstacle-occupied areas while blue grids correspond to free space. • Human-Robot Collaborations: The obstacles, such as humans, are dynamic rather than fixed. As a result, the … view at source ↗

**Figure 3.** A simple example of the domain definition for our SSTF-CBF is [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** A rough schematic illustrating the robot entering an unsafe region [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: (Left) Obstacle distribution and Turtlebot’s trajectory during sequential navigation. (Right) Color map of SSTF-CBF generated by the entire map. The three target points are consistently represented by five-pointed stars in different colors across both figures [PITH_F…

**Figure 6.** Figure 6: The safety values of the TurtleBot 3 during navigation. These values [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

**Figure 7.** Figure 7: (a-c) illustrate the trajectories of two robots during navigation. (d-i) present robot 1’s local occupancy grid maps (before dilation) along with the [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]

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