GCBF+: A Neural Graph Control Barrier Function Framework for Distributed Safe Multi-Agent Control

TL;DR

This work tackles scalable safe control for large-scale multi-agent systems by introducing Graph Control Barrier Functions (GCBF) and the GCBF+ learning framework. GCBF encodes safety on a graph so that a single barrier can certify safety for arbitrary numbers of agents, with local neighbor information and attention-based aggregations. GCBF+ jointly learns a neural GCBF $h_\theta$ and a distributed control policy $\pi_\phi$ via Graph Neural Networks, enabling LiDAR-based obstacle handling and scalable execution without centralized coordination. The approach is validated through extensive numerical experiments and hardware demonstrations on Crazyflie drones, showing superior safety and competitive goal-reaching compared to MARL, MPC, and hand-crafted CBF baselines, including scenarios with up to 1024 agents and 128 obstacles. Overall, GCBF+ advances safe, scalable, and perception-rich MAS control with theoretical safety guarantees for any $N$ and practical real-world applicability.

Abstract

Distributed, scalable, and safe control of large-scale multi-agent systems is a challenging problem. In this paper, we design a distributed framework for safe multi-agent control in large-scale environments with obstacles, where a large number of agents are required to maintain safety using only local information and reach their goal locations. We introduce a new class of certificates, termed graph control barrier function (GCBF), which are based on the well-established control barrier function theory for safety guarantees and utilize a graph structure for scalable and generalizable distributed control of MAS. We develop a novel theoretical framework to prove the safety of an arbitrary-sized MAS with a single GCBF. We propose a new training framework GCBF+ that uses graph neural networks to parameterize a candidate GCBF and a distributed control policy. The proposed framework is distributed and is capable of taking point clouds from LiDAR, instead of actual state information, for real-world robotic applications. We illustrate the efficacy of the proposed method through various hardware experiments on a swarm of drones with objectives ranging from exchanging positions to docking on a moving target without collision. Additionally, we perform extensive numerical experiments, where the number and density of agents, as well as the number of obstacles, increase. Empirical results show that in complex environments with agents with nonlinear dynamics (e.g., Crazyflie drones), GCBF+ outperforms the hand-crafted CBF-based method with the best performance by up to 20% for relatively small-scale MAS with up to 256 agents, and leading reinforcement learning (RL) methods by up to 40% for MAS with 1024 agents. Furthermore, the proposed method does not compromise on the performance, in terms of goal reaching, for achieving high safety rates, which is a common trade-off in RL-based methods.

Figures (15)

• Figure 1: 8-Crazyflie swapping with GCBF: We learn a distributed graph control barrier function (GCBF) for an 8-agent swapping task on the Crazyflie hardware platform. We visualize the learned GCBF for the blue agent and draw the edges with its neighboring agents in grey. The learned GCBF can handle arbitrary graph topologies and hence can scale to an arbitrary number of agents without retraining.
• Figure 2: GCBF contours: For a fixed neighborhood $\mathcal{N}_0$ of agent $0$, we plot the contours of the learned GCBF $h_\theta$, projected on $xy$-plane, by varying the position $p_0$ of agent $0$. Since agent $3$ is outside of agent $0$'s sensing radius, i.e., not a neighbor of agent $0$, it does not contribute to the value of $h(\bar{x}_{\mathcal{N}_0})$.
• Figure 3: GCBF+ training architecture: The sampled input features are labeled as safe control invariant $\mathcal{D}_{\mathcal{C}}$ and unsafe $\mathcal{D}_{\mathcal{A}}$ using the previous step learned control policy $\pi_\phi$. A nominal control policy $\pi_{\mathrm{nom}}$ for goal reaching is used in a CBF-QP with the previously learned GCBF $h_\theta$ to generate $\pi_{\mathrm{QP}}$. Finally, the QP policy and the GCBF conditions are used to define the loss $\mathcal{L}$.
• Figure 4: Satisfaction of \ref{['def: gcbf']} in practice: The attention weights $w_{ij}$ in \ref{['eq:attention_aggregate']} are plotted against inter-agent distances $d_{ij}$ for sensing radius $R=0.5$ from multiple trajectories. The weight $w_{ij}$ approaches $0$ as the inter-agent distance $d_{ij}$ approaches $R$ without explicit supervision, showing that GCBF+ automatically learns to satisfy conditions 1) and 2) in \ref{['def: gcbf']} via the approach outlined in \ref{['rmk:attn']}.
• Figure 5: Comparison of the choice of control loss: The learned policy $\boldsymbol{\textcolor{pol_color}{\pi_\phi}}$ is sensitive to the choice of $\eta_{\mathrm{ctrl}}$ when using $\pi_{\mathrm{nom}}$ as in GCBFv0 zhang2023distributed (top) since a learned policy $\boldsymbol{\textcolor{pol_color}{\pi_\phi}}$ that is close to $\pi_{\mathrm{nom}}$ may be unsafe and not satisfy the GCBF conditions \ref{['eq:graph CBF']}. Consequently, choosing $\eta_{\mathrm{ctrl}}$, which controls the relative weight between $\mathcal{L}_{\mathrm{CBF}}$ and $\mathcal{L}_{{ctrl}}$, becomes a balancing act of staying close to $\pi_{\mathrm{nom}}$ while remaining safe. In contrast, by definition of $\pi_{\mathrm{QP}}$\ref{['eq:opt control policy']}, the control $\boldsymbol{\textcolor{pol_color}{\pi_\phi}}$ is already safe. Hence, the safety of $\boldsymbol{\textcolor{pol_color}{\pi_\phi}}$ is not sensitive to $\eta_{\mathrm{ctrl}}$ when using $\pi_{\mathrm{QP}}$ for the control loss, as in GCBF+.

Theorems & Definitions (11)

• Remark 1
• Definition 1: GCBF
• Lemma 1
• Remark 2
• Theorem 1
• Remark 3
• Remark 4
• Remark 5
• Remark 6
• proof