Least Restrictive Hyperplane Control Barrier Functions

Mattias Trende; Petter Ögren

cs.ROOct 21, 2025

Least Restrictive Hyperplane Control Barrier Functions

Authors: Mattias Trende, Petter Ögren

Organizations: The authors are with the Robotics, Perception and Learning Lab., School of Electrical Engineering and Computer Science, Royal Institute of Technology (KTH), SE-100 44 Stockholm, Sweden

Abstract

Control Barrier Functions (CBFs) can provide provable safety guarantees for dynamic systems. However, finding a valid CBF for a system of interest is often non-trivial, especially for systems with low computational resources, higher-order dynamics, and moving close to obstacles of complex shape. A common solution to this problem is to use a purely distance-based CBF. In this paper, we study Hyperplane CBFs (H-CBFs), where a hyperplane separates the agent from the obstacle. First, we note that the common distance-based CBF is a special case of an H-CBF where the hyperplane is a supporting hyperplane of the obstacle that is orthogonal to a line between the agent and the closest point of the obstacle. We then show that a less conservative CBF can be found by optimising over the orientation of the supporting hyperplane, in order to find the Least Restrictive Hyperplane CBF. This enables us to maintain the safety guarantees while allowing controls that are closer to the desired ones, especially when moving fast and passing close to obstacles. We illustrate the approach on a double integrator dynamical system with acceleration constraints, moving through a group of arbitrarily shaped static and moving obstacles, and show that the proposed approach reduces the average gap between desired and safe controls by an order of magnitude.

Figures & tables

Fig. 1: An agent

A

with velocity

\mathbf{v}_{A}

in front of an obstacle

O

. Five of the infinitely many supporting hyperplanes, parameterised by some

\theta

, are shown (

P_{1}-P_{5}

). A common choice used in [ 6 , 7 , 8 , 9 , 10 ] is

P_{2}

(denoted MDH-CBF) going through the closest point on the obstacle. Note, however, that if the goal of the agent is at

G_{1}

, an H-CBF based on

P_{4}

would make more sense (allowing the agent to pass the obstacle at full speed), whereas if the goal is at

G_{2}

, an H-CBF based on

P_{1}

might be better (allowing a sharp left turn). The key idea in this paper is that, given the desired action of the agent

\mathbf{u}_{\text{des}}

, we optimise over a family of CBFs using

\theta

to find the least restrictive H-CBF, allowing a safe control

\mathbf{u}

that is as close as possible to the desired

\mathbf{u}_{\text{des}}

Fig. 2: The geometry of a general 2D obstacle. The red area is the obstacle

O

, and combined with the light grey areas, we get

\text{conv}(O)

. Notice how the braking distance

b

added to the supporting hyperplane defines the boundary of the set of safe positions, given the current velocity

\mathbf{v}_{A}

Fig. 3: The mean computational time and the aggregate control intervention

||\mathbf{u}-\mathbf{u}_{\text{des}}||

, over 10000 random states in the proximity of a circular obstacle, as a function of

N_{\theta}

in Equation ( 22 ). The C-LRH-CBF is used as a baseline and does not depend on the resolution

N_{\theta}

. Upper: the linear relationship between

N_{\theta}

and mean computational time, giving us the expected time

t(N_{\theta})=6.8N_{\theta}

\mu

s. The mean time to solve a step with the CasADi solver is 7.2 ms, too long to be shown in the same plot. Middle: the full range in

||\mathbf{u}-\mathbf{u}_{\text{des}}||

. Lower: a zoomed-in version. Note that no improvement is gained when going from

N_{\theta}=1

N_{\theta}=3

for

\Theta^{0.5\pi}

, which is an expected result from Section IV-G . Increasing the resolution from

N_{\theta}=1

quickly decreases

||\mathbf{u}-\mathbf{u}_{\text{des}}||

. At resolutions

N_{\theta}<9

, a smaller spread of

\Theta^{0.3\pi}

performs best, while at higher resolutions the wider spread

\Theta^{0.4\pi}

performs best.

Fig. 4 : The setup of Example 3 . The agent moves on a straight line at constant velocity (

\mathbf{v}=[2,0]^{T}

m/s,

\mathbf{u}_{\text{des}}=0

), safely passing an obstacle (the dotted rounded triangle corresponds to the configuration space of the obstacle).

Fig. 5 : The

h

-value for the fixed trajectory of the agent in Example 3 . Note that this safe trajectory would not be allowed by the MDH-CBF of [ 6 ] (dashed), as the

h

-value is negative for

t\in[1.4,2.0]

, right before passing the obstacle. This is independent of the choice of

\alpha

. However, since the

h

-value is positive during the entire trajectory for the other methods, there exists an

\alpha

that allows their trajectories.

Method	Time to Goal [s]	Path Length [m]	Control Intervention [m/s 2 ]	Min Clearance [m]
MDH-CBF	13.820 $\pm$ 1.681	22.658 $\pm$ 1.835	6.754 $\pm$ 1.456	0.239 $\pm$ 0.124
MHH-CBF	11.134 $\pm$ 1.112	22.196 $\pm$ 1.010	2.739 $\pm$ 0.899	0.161 $\pm$ 0.079
C-LRH-CBF	10.944 $\pm$ 1.066	22.209 $\pm$ 0.933	2.509 $\pm$ 0.848	0.048 $\pm$ 0.047
VO-MDH-CBF	12.490 $\pm$ 0.695	21.579 $\pm$ 0.177	5.444 $\pm$ 0.974	0.623 $\pm$ 0.200
VO-LRH-CBF	10.301 $\pm$ 0.223	21.679 $\pm$ 0.124	2.084 $\pm$ 0.466	0.163 $\pm$ 0.089

TABLE I : The statistics after 500 runs of Example 4 . Bold marks the best performance for each metric.

Fig. 6 : The trajectories of one instance of Example 4 . Concurrent snapshots of the agent and the moving obstacle are shown, with older ones increasingly opaque. The agent is driven towards the goal with a PD controller, using the five different types of H-CBFs. Note how the C-LRH-CBF is allowed to move closer to the obstacle than the other methods, enabling it to reach the goal first.

Fig. 7 : Control deviation and distance to goal for one simulation of Example 4 . Upper: The difference between the desired control and the safe control

||\mathbf{u}-\mathbf{u}_{\text{des}}||

over time. The C-LRH-CBF starts to intervene last and restricts less than the other methods. Lower: The distance to the goal as a function of time. The C-LRH-CBF is first, with the MHH-CBF and VO-LRH-CBF close behind.

Fig. 8 : The trajectories in Example 5 , featuring moving and irregularly shaped obstacles. Concurrent snapshots of the agent and the moving obstacle are shown, with older ones increasingly opaque. Notice how the C-LRH-CBF finds a longer but faster trajectory.

Fig. 9 : The clearance to the closest obstacle (in configuration space) and distance to goal corresponding to Example 5 . Upper: note that methods ensure safety. Lower: notice how the three optimising methods perform better than the fixed hyperplane ones.

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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.

Abstract

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