Organizations: Indian Institute of Information Technology and Management, Gwalior · California Institute of Technology · The University of Texas at Austin · University of Virginia
Control barrier functions (CBFs) are a standard way to keep a planner inside a safe set, but they assume control affine dynamics, a barrier function given in closed form, and a known nominal controller. Integral control barrier functions (ICBFs) remove the first assumption and absorb input constraints, yet they still assume a given nominal controller and a barrier function. In this work, we present an approach to learn all three from data, using neural ordinary differential equations (ODEs) for the dynamics, imitation for the nominal controller, and supervised regression for the barrier function. The learned dynamics, nominal controller, and the barrier function are used to build the ICBF filter constraint analytically. Furthermore, the barrier is trained on viability rather than geometry, meaning that a state counts as safe only if some emergency maneuver can still avoid the constraint under the actuator limits, a correction that geometric labels miss entirely and that grows with speed. Actuator limits then enter the online program as hard bounds while the learned barrier rows may be relaxed, which keeps the program solvable at every state and the actuator limits exact, and we bound how far the input it returns can drift from the one the true models would give. On agile racing with a 6-D dynamic bicycle, the filter removes all 12 unsafe episodes from 200 held out layouts at an unchanged success rate and above 100 Hz on one CPU core, holds the input constraint on every episode of a benchmark where a jointly trained baseline holds it on 53.38%, and keeps that advantage under large errors in the plant friction.
Figures & tables
Method
Safety
Goal
Input compliance
Cost
Ours
100%
71%
100%
16758
SABLAS
98.49%
89%
53.38%
11474
Table 1: Double integrator benchmark of Qin et al. (2022) , 100 random test episodes. Input compliance is the fraction of episodes on which the actuator constraint held throughout.
Figure 1: What the viability labeled barrier represents. (left, middle) The learned obstacle barrier hθ in the body frame at two speeds, with its zero level set and the geometric obstacle. At 1 m/s the two nearly coincide, whereas at 5 m/s the unsafe set extends roughly 1.5 m along the approach direction, covering states clear of the obstacle that can no longer avoid it under the actuator limits. (c) Fraction of geometrically safe states that viability relabels unsafe, by speed: the signature of a viability set rather than a smoothed obstacle indicator.
v (m/s)
Method
Success
Collision
Out of bounds
Unsafe
Stall
Timeout
3.5
MPPI
172
0
1
1
27
0
HOCBF-QP
122
58
0
58
14
6
Ours, πθ
162
3
0
3
35
0
Ours, MPPI
166
0
0
0
34
0
4.0
MPPI
168
7
5
12
20
0
HOCBF-QP
55
69
1
70
18
57
Table 2: Agile racing on 200 held out layouts, in episode counts, with every row summing to 200. MPPI is the planner alone, HOCBF-QP is the analytical filter of Xiao and Belta (2022) applied to it, and Ours is the Neural ICBF with either πθ or MPPI as the nominal controller. Unsafe is the sum of the collision and out of bounds columns.
Appendix figures & tables2 assets
Supplementary material from the paper’s appendix.
Appendix
Vehicle
Quadrotor-pendulum
Method
Time (s)
Cost
Time (s)
Cost
NMPC
5.562
4.583
10.242
15.232
ORCA-MAPF
1.373
6.839
2.328
21.281
CADRL
0.837
5.683
2.181
19.283
Ours
0.736
4.631
1.550
15.672
Appendix
Table 3: Time to goal and average closed loop cost on the two low dimensional systems. ORCA-MAPF and CADRL are runtime reference points rather than safety filters, so the cost column is not a like for like comparison against them.
Configuration
Success
Unsafe
minhθ
QP feasible
Slack active
Interventions
Default ( η=0.2 , α=5 , grip row)
8/12
0/12
+0.29
100%
0.16%
44%
No enforcement margin ( η=0 )
9/12
0/12
+0.20
100%
0.13%
45%
Barrier gain α=10
9/12
0/12
+0.14
100%
0.15%
45%
No grip row
7/12
0/12
−0.05
100%
2.77%
31%
Appendix
Table 4: Ablations on the 12 held out layouts at 4.0 m/s on which MPPI alone is unsafe, with MPPI as the nominal controller. minhθ is the smallest obstacle barrier value reached in closed loop, before the enforcement margin is subtracted, so a negative entry means the learned safe set was left. Interventions is the fraction of steps at which v⋆=0 .
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 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.
Bojan Derajić, Sebastian Bernhard, Wolfgang Hönig
Technical University of Berlin, Germany · AUMOVIO, Germany · Technical University of Applied Sciences Augsburg, Germany +1
Safety filters are an effective tool for enforcing constraints in safety-critical systems, but most existing methods assume perfect state information, which is rarely available in practice. Recent work has begun to close this gap by developing filtering mechanisms that are robust to state estimation error, but these methods can still exhibit safety violations or overly conservative behavior as estimation error grows. Focusing on obstacle avoidance, we develop two new control barrier function (CBF) formulations: drift-measurement-robust (DMR)-CBFs and neural measurement-robust (NMR)-CBFs. The DMR-CBF augments the standard CBF condition with an inner optimization over the worst-case uncertainty in the drift dynamics, improving robustness to estimation error. This DMR-CBF then supervises a pretraining phase for the NMR-CBF, which replaces the inner optimization with a learned term. The NMR-CBF is subsequently finetuned through differentiable trajectory rollouts, yielding a filter that achieves empirical safety comparable to the DMR-CBF while reducing both conservativeness and computational cost. We provide theoretical analysis of the DMR-CBF along with numerical results on a planar double integrator and a 12D quadrotor, where both proposed approaches prevent collisions while other robust methods either fail or are overly conservative. Finally, we deployed the NMR-CBF on a Unitree Go2, enabling successful navigation of an obstacle field under odometry errors that caused a standard CBF to collide.
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.
Mattias Trende, Petter Ögren
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