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VertexCBF: Improving Neural Control Barrier Functions via Vertex-Restricted Control Search

Bojan Derajić, Sebastian Bernhard, Wolfgang Hönig

arXiv:2609.12831v2cs.ROcs.LGcs.SYeess.SY

TL;DR

Designing CBFs for constrained systems can be difficult and may produce conservative safe sets. VertexCBF learns a neural approximation to the stationary HJ safety value using physics-informed learning and vertex-restricted search. Across diverse systems it recovers larger, more reliable validated safe sets than the supplied baselines, and a learned CBF is deployed in real time around people.

  • Problem

    Designing CBFs for systems with state and input constraints is difficult and can compromise performance and safety.

    Source: §1 · p. 1

  • Method

    VertexCBF trains a neural approximation of the stationary HJ safety value with physics-informed learning and vertex-restricted finite-horizon search under control-affine dynamics and convex-polytope controls.

    Source: §1 · p. 1, §4.2 · p. 4, §6 · p. 9

  • Results

    VertexCBF recovers larger and more reliable validated safe sets than PDE-only learning and full-control MPPI supervision, and its learned CBF is demonstrated as a real-time CBF-QP filter around people.

    Source: §6 · p. 9

  • Takeaways & Limitations

    The hardware experiment demonstrates deployment of the learned CBF as a real-time CBF-QP safety filter for robots operating around people.

    Source: §6 · p. 9

    The method provides neither formal convergence guarantees nor deterministic error bounds on the learned CBF, so the authors rely on post-training validation with large-scale closed-loop rollouts.

    Source: §7 · p. 9

Abstract

from arXiv · show

As the number of autonomous robots continues to grow, safety becomes increasingly important. Control barrier functions (CBFs) provide a theoretically grounded framework for ensuring safety, but existing design methods often face limitations in effectiveness, scalability, or interpretability, and may result in overly conservative safe sets. In this paper, we propose \emph{VertexCBF}, a framework for learning neural CBFs in a scalable, systematic, and explainable way. We approximate the stationary Hamilton--Jacobi value function using a neural network trained via a combination of physics-informed and sparsely supervised learning. By exploiting control-affine dynamics and a convex polytope control set, under which the Hamiltonian is maximized at the control vertices, we efficiently generate supervision points via GPU-parallel vertex-restricted tree search, while a residual architecture guarantees that the learned CBF is never larger than the specified constraint function. We evaluate the method on 15 systems and compare it against relevant baselines, showing that it reliably recovers large safe sets where the baselines are conservative or fail completely. In addition, we perform a hardware experiment in which a mobile robot safely avoids pedestrians using a neural CBF trained with our method.

1 Introduction

Safe robot deployment needs systematic, explainable control, but designing CBFs under state and input constraints can compromise performance and safety. VertexCBF learns neural CBFs from HJ reachability using vertex-restricted search and is evaluated across systems and on hardware.

1 Introduction

  • Safe and reliable operation remains a key challenge as learning-based robots increasingly interact directly with their environments.The paper frames systematic and explainable safety as important to robot deployment.Source: §1 · p. 1
  • CBFs provide theoretically grounded, explainable safety filters, but designing them for state and input constraints is difficult and can compromise performance and safety.Safety filtering modifies the nominal control input at runtime to preserve safety.Source: §1 · p. 1
  • VertexCBF approximates the HJ safety value function with a neural CBF trained using physics-informed and sparse supervised losses.A residual parametrization constrains the learned function to be no larger than the specified constraint function.Source: §1 · p. 1
  • Control-affine dynamics and a convex polytope control set let the method generate supervision labels through GPU-parallel search restricted to control vertices.Under these assumptions, the continuous-time Hamiltonian is maximized at a control vertex.Source: §1 · p. 2
  • Across 15 systems, VertexCBF recovers large, reliable safe sets where baselines are conservative or fail, and a mobile robot safely navigates among pedestrians.The method is demonstrated in a hardware experiment.Source: §1 · p. 2

2 Related Work

Prior learning-based CBF and HJ reachability methods face limitations involving constrained systems, data generation, or training. VertexCBF combines physics-informed learning and supervision while using stationary reachability and vertex-restricted search under specific system assumptions.

2 Related Work

  • Learning-based CBF methods with control limits can still yield conservative or suboptimal safe sets, while distance-based formulations can be inadequate for higher-order systems and limited actuation.Earlier approaches include support vector machines, neural networks, and Gaussian splats.Source: §2 · p. 2
  • HJ-value learning methods relying on numerically generated training data face scalability limits, while DeepReach-style methods use curriculum procedures that can be difficult to tune and unstable.DeepReach and later refinements are discussed in this context.Source: §2 · p. 2
  • A stationary HJB-VI loss alone admits trivial or suboptimal solutions in practice, motivating additional supervision to guide learning.The paper reports experimental demonstrations of this issue for.Source: §2 · p. 2
  • Compared with, VertexCBF targets stationary rather than time-dependent HJ reachability and replaces full-control sampling-based MPC supervision with vertex-restricted search under control-affine, polytope-control assumptions.The paper also states that this formulation eliminates curriculum learning and makes pretraining unnecessary.Source: §2 · p. 2

3 Preliminaries and Problem Formulation

The paper seeks a CBF whose safe set approximates the maximal controlled invariant subset of the constraint set. It formulates this target through a safety value function and runtime CBF-QP filtering.

3 Preliminaries and Problem Formulation

  • The system is required to remain in the constraint set C = {x : c(x) ≥ 0}, with controls drawn from a compact convex polytope whose vertices form V.The dynamics are control-affine.Source: §3 · pp. 2–3
  • The infinite-horizon safety value V gives the largest worst-case constraint value maintainable from a state, and its nonnegative set is the maximal controlled invariant subset of C.The paper assumes the supremum defining V is finite for every state.Source: §3 · p. 3
  • A CBF supports a runtime quadratic-program safety filter that modifies a nominal control input when the program is feasible.By Nagumo’s theorem [22], the filter renders the CBF’s nonnegative set forward invariant.Source: §3 · p. 3
  • The central design problem is to construct a CBF h whose nonnegative set approximates the maximal safe set S.Source: §3 · p. 3

4 Methodology

VertexCBF approximates the stationary HJ safety value function by combining a PDE-based objective with sparse tree-search labels. Its discretization, finite-horizon search, and residual model make the approach systematic while introducing approximation conditions and boundaries.

4 Methodology

  • Sparse supervised state-value labels complement the stationary HJB-VI loss because the PDE objective admits multiple solutions and gradient optimization can encounter local optima.VertexCBF uses a neural approximation of the safety value function as its CBF.Source: §4 · p. 3

4.1 Time Discretization of the Vertex-Restricted Problem

  • The method first restricts the continuous-time Hamiltonian maximization to vertices, then applies forward Euler; its Bellman equation is exact for the Euler-discretized vertex-restricted system but only approximates the original continuous-time problem.The vertex-restricted Euler operator is a first-order consistent local approximation of the continuous-time HJB-VI.Source: §4.1 · p. 4

4.2 Finite-Horizon Search over Vertex-Restricted Controls

  • Vertex-restricted finite-horizon search reduces the trajectory search to M^K possibilities, with complete search exact for low-dimensional systems and approximate search variants used when M or K is large.Here M is the number of control vertices and K is the horizon length; listed approximate variants include beam search, stochastic beam search, and branch-and-bound.Source: §4.2 · p. 4
  • Finite-horizon labels assume the attained safety level can be maintained after the horizon; that assumption can fail for short horizons, causing labels to overestimate the true value.Width-limited search returns lower bounds on the full vertex-restricted search, shrinking rather than enlarging the predicted safe set.Source: §4.2 · p. 5
  • Vertex restriction is effective under control-affine dynamics, polytope controls, and sufficiently small ∆t, but large ∆t, short horizons, or narrow beams can make results conservative or suboptimal.The paper does not claim vertex-restricted search always outperforms full-control optimization.Source: §4.2 · p. 5

4.3 Model Architecture and Training Procedure

  • The residual architecture sets VΘ = c − rΘ with a nonnegative residual, ensuring the learned safe set stays within the constraint set and making constant solutions less natural.The residual uses a smooth softplus output; periodic inputs are represented with cosine and sine features.Source: §4.3 · p. 5
  • Training combines a physics-informed PDE loss and a data loss: the PDE term enforces local self-consistency, while search-generated labels anchor nontrivial values and steer optimization away from spurious PDE-only minima.The PDE term uses collocation states, and the data term is mean-squared error against vertex-restricted tree-search labels.Source: §4.3 · pp. 5–6

5 Experimental Results

The section evaluates VertexCBF against PDE-only and full-control MPPI supervision on 15 systems using rollout validation, then reports a pedestrian-avoidance hardware test.

5 Experimental Results

  • Validation balances predicted-safe and complementary initial states, then counts a state as validated safe only if its closed-loop rollout stays within C.The reported metrics are false-safe rate, false-unsafe rate, and effective safe volume.Source: §5 · pp. 6–7
  • Across 15 systems, VRCD generally achieves low false-safe rates and competitive false-unsafe rates while attaining the highest or tied-highest effective safe volume (Table 1).Table 1 reports means ± standard deviations over five seeds; dashes mark systems whose predicted safe set is empty for every seed.Source: Table 1 · p. 7, §5 · p. 7
  • PDE-only learning is unreliable and can collapse to an empty safe set, FCD supervision is often conservative, and VRCD approximately recovers the maximal safe set.For the inverted pendulum, Fig. 2 contrasts these learned neural CBFs.Source: §5 · p. 7, Figure 2 · p. 8
  • For the inverted pendulum, vertex-restricted beam search keeps trajectories inside the constraint set for all sampled true-safe initial states, while MPPI succeeds for only a few.Fig. 1 shows trajectories from the same 160 random initial states.Source: §5 · p. 7, Figure 1 · p. 8

5.1 Hardware Experiments

  • In 10 trials with one or two pedestrians intercepting the robot, the CBF-QP-filtered mobile robot avoided collision in every trial (Fig. 3).The filter modifies a nominal MPC controller's input using pedestrian detections.Source: §5.1 · p. 8, Figure 3 · p. 9

6 Conclusion

VertexCBF approximates stationary HJ safety value functions using physics-informed learning, vertex-restricted search, and a residual architecture. The conclusion reports larger, more reliable validated safe sets than PDE-only learning and full-control MPPI supervision, while noting that guarantees remain future work.

6 Conclusion

  • VertexCBF approximates stationary HJ safety value functions using vertex-restricted finite-horizon search, physics-informed loss, and a residual architecture bounded by the constraint function.The search exploits control-affine dynamics and convex-polytope control constraints.Source: §6 · p. 9
  • Experiments show larger and more reliable validated safe sets than PDE-only learning and full-control MPPI supervision, with hardware deployment as a real-time CBF-QP safety filter around people.The conclusion attributes the supervision signal to vertex-restricted control.Source: §6 · p. 9
  • The learned CBFs lack formal convergence guarantees and deterministic error bounds; future work includes stronger guarantees, scalability to many control vertices, and extensions to uncertain dynamics and perception-based constraints.Source: §6 · p. 9

7 Limitations

VertexCBF’s limitations include the absence of formal convergence guarantees and rapidly growing tree-search costs. The appendix establishes local first-order consistency and describes three width-bounded search strategies, but local consistency is not a full convergence proof.

7 Limitations

  • The learned CBF has no formal convergence guarantees or deterministic error bounds; approximation error comes from discretization, finite horizon, width-limited search, and neural approximation.The authors rely on large-scale closed-loop validation and identify sample-based certification as possible future work.Source: §7 · p. 9
  • Tree search grows rapidly with horizon and control-vertex count; although tested up to 13 states and 6 controls, direct use for very high-dimensional humanoids may be computationally prohibitive.Reduced-order models or safety-critical subsystems may be more suitable in such settings.Source: §7 · p. 9

A First-order consistency of the vertex-restricted discretization

  • The vertex-restricted Euler Bellman operator is locally first-order consistent with the continuous stationary HJB-VI, not a complete convergence proof for the infinite-horizon value function.A full proof would also require well-posedness conditions, including a comparison principle and appropriate boundary or state-constraint conditions.Source: A First-order consistency of the… · p. 12

B Overview of the Tree Search Methods

  • The three GPU-parallel search methods approximate the same vertex-control optimization; width-bounded heuristic searches replace intractable enumeration of the M^K-leaf tree.They differ in how candidates are selected for retention.Source: B Overview of the Tree Search Methods · pp. 12–13

B.1 Beam Search (BS)

  • Beam search deterministically retains the B partial trajectories with the largest running-minimum score after each expansion.At each depth it expands the current beam across all control vertices, then keeps the top B children.Source: §B.1 · p. 13

B.2 Stochastic Beam Search (SBS)

  • Stochastic beam search replaces deterministic top-B selection with score-proportional sampling without replacement to preserve search diversity.The appendix lists softmax, Gumbel top-k, rank-based, and ε-greedy sampling strategies.Source: §B.2 · p. 13

B.3 Branch and Bound (BnB)

  • Branch and bound discards children whose upper-bound score cannot exceed the best known leaf, and returns a value at least as large as beam search for the same B.It runs multiple passes, trading additional runtime for the opportunity to explore previously unseen subtrees.Source: §B.3 · pp. 13–14

C Influence of Beam-Search Hyperparameters

Beam-search label quality depends on beam width, prediction horizon, and step size. Longer horizons require wider beams to maintain low error, while runtime scales with beam width and depth rather than step size.

C Influence of Beam-Search Hyperparameters

  • If K∆t is too short to detect future violations, the value is systematically overestimated; increasing ∆t can reduce the needed number of steps, but excessive ∆t makes discretization inaccurate.Source: C Influence of Beam-Search… · p. 14
  • As K increases, B must also increase to retain near-optimal trajectories and maintain low MSE against the ground-truth value.Figure 4 reports this error as a function of B and K for different ∆t values on the inverted pendulum.Source: Figure 4 · p. 14, C Influence of Beam-Search… · p. 14
  • Beam-search runtime grows approximately linearly with B and K and is independent of ∆t; its complexity is O(NKBM), apart from transition simulation and objective evaluation.Source: C Influence of Beam-Search… · p. 14

D Influence of the Search Step Size

Step size trades off discretization accuracy against how far into the future the search can detect violations. For the underwater vehicle, intermediate step sizes performed stably, while the smallest and largest tested values underperformed for different reasons.

D Influence of the Search Step Size

  • For ∆t = 0.05–0.2 s, false-safe rate was at most 1.18% and effective safe volume ranged from 0.348 to 0.356; choose an accurate Euler step, then K so K∆t covers stopping time.With K = 40, look-ahead ranged from 0.8 s to 16 s, and search cost did not depend on ∆t.Source: D Influence of the Search Step Size · p. 15
  • At ∆t = 0.02 s, the 0.8 s look-ahead is shorter than braking time and false-safe rate reaches 3.94%; at ∆t = 0.4 s, false-unsafe rate reaches 7.02% and safe volume falls to 0.317.The largest step also makes the discretization inaccurate and visibly shrinks the predicted safe set.Source: D Influence of the Search Step Size · p. 15, Figure 6 · p. 15
  • For ∆t between 0.05 and 0.2 s, false-safe rate stays at or below 1.18% and effective safe volume ranges from 0.348 to 0.356; ∆t = 0.1 s gives the best result.The spread across three seeds is small throughout this interval.Source: D Influence of the Search Step Size · p. 15

E Ablation on the Residual Architecture

The residual parametrization prevents trivial PDE-only solutions and guarantees that the learned safe set stays within the constraint set. Supervision helps guide the learned set toward the maximal safe set, though removing the residual still degrades many supervised results.

E Ablation on the Residual Architecture

  • The ablation replaces the residual softplus output with direct value-function prediction using a linear output, while keeping other settings unchanged.The architecture, hyperparameters, supervision data, and validation protocol remain unchanged.Source: E Ablation on the Residual Architecture · pp. 15–16
  • Without the residual parametrization, PDE-only learning produced an empty predicted safe set on 14 of 15 systems.Only the 3-DOF Manipulator retained a non-empty safe set, with a high and strongly seed-dependent false-safe rate.Source: E Ablation on the Residual Architecture · p. 16
  • For VRCD, removing the residual increased false-safe rates on 13 of 15 systems and reduced effective safe volume on 11.The largest losses were on Cart-Pole and Quadrotor; supervised methods were less sensitive overall, but most entries still degraded.Source: E Ablation on the Residual Architecture · p. 16
  • The residual parametrization guarantees the predicted safe set is contained in the constraint set, while supervision guides it toward the maximal safe set.The two components are complementary: the residual reduces trivial-solution risk, and supervision determines how closely the learned safe set approaches the maximal one.Source: E Ablation on the Residual Architecture · p. 16

F Sensitivity to the PDE-Loss Weight

The PDE-loss weight has a broad stable range on the inverted pendulum, but larger weights can yield trivial predictions. Against grid-based ground truth, VRCD performs best or nearly best on all seven systems with available references.

F Sensitivity to the PDE-Loss Weight

  • The data term keeps optimization away from trivial PDE-loss solutions, while a moderate PDE weight enforces local consistency without dominating the objective.For the inverted pendulum, λ = 0.2 lies within the stable region; the authors selected per-system weights from flat validation ranges.Source: F Sensitivity to the PDE-Loss Weight · pp. 16–17
  • From λ = 0 to 0.5, effective safe volume stayed between 20.2% and 22.1%, false-safe rate below 0.3%, and ground-truth error between 1.8·10^-4 and 7.8·10^-4.Above λ = 0.5, results changed abruptly: the predicted set was empty at λ = 0.7, while at λ = 0.9 and 1 every predicted-safe state was false-safe.Source: Figure 7 · p. 17, F Sensitivity to the PDE-Loss Weight · p. 17

G Comparison with the Ground Truth

  • Grid-based reference value functions were available for seven of the 15 systems, all with at most four states, because grid-solver memory and runtime grow exponentially with dimension.The remaining systems could not be solved on the authors’ hardware with a grid-based solver.Source: G Comparison with the Ground Truth · p. 17
  • On seven systems with grid references, VRCD achieved the highest IoU on five and stayed within 0.3 percentage points of the best on the other two.IoU penalizes predicted safe sets that are too large or too small; the three methods were nearly identical on systems they all solved well, while differences were large on the inverted pendulum, vertical drone, and cart-pole.Source: G Comparison with the Ground Truth · p. 17, Table 4 · p. 18

H Details on Hardware Experiments

This section presents hardware pedestrian-avoidance trials using a learned neural CBF and describes the modeled systems, their safety constraints, and learned-CBF slices. It also discusses deployment details and limitations relevant to the trials.

H Details on Hardware Experiments

  • Across 10 pedestrian-avoidance trials, the mobile robot avoided collisions using the neural CBF trained with this approach and deployed as in [13] (α(h(x)) = kh(x), k = 2.8).The CBF was evaluated relative to each tracked pedestrian.Source: H Details on Hardware Experiments · p. 18
  • Pedestrian-specific CBF constraints are aggregated into one QP constraint and relaxed with penalized slack, which preserves QP feasibility but removes the forward-invariance guarantee when active.Slack may activate when a newly detected pedestrian makes the CBF constraint unsatisfiable under any admissible control.Source: H Details on Hardware Experiments · p. 18
  • Brief negative CBF values reflect model mismatch and 5 cm inflated training radii, which make the learned safe set smaller than the real collision-free set and absorb deployment effects.The mismatch includes skid-steer wheel slip, velocity-controller dynamics, noisy and delayed pedestrian estimates, approximate constant-velocity dynamics, and finite-rate QP updates; the robot still completed every trial without collision.Source: H Details on Hardware Experiments · p. 18

I.13 Quadruped Trunk

  • The quadruped trunk uses a smooth box constraint on height, roll, and pitch to prevent falling, while retaining the uncontrolled yaw-rate state for gyroscopic coupling.Figure 21 shows a learned-CBF slice in the (ϕ, θ) plane at fixed trunk height and velocities.Source: §I.13 · p. 34, Figure 21 · p. 36

I.14 6-DoF Underwater Vehicle

  • The 6-DoF underwater vehicle is modeled as neutrally buoyant with rigid-body dynamics, and its constraint confines it to a tunnel in the world (py, pz) plane.Figure 22 shows a learned-CBF slice in (py, pz) at fixed position, orientation, and body-frame velocities.Source: §I.14 · pp. 34–35, Figure 22 · p. 36

I.15 Quadrotor

  • The quadrotor’s constraint is a vertical cylindrical obstacle of radius r centered on the world z-axis; Figure 23 shows a learned-CBF slice in (px, py).The slice is shown at fixed altitude, orientation, linear velocity, and angular velocity.Source: §I.15 · p. 37, Figure 23 · p. 38
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