  Distributed, scalable, and safe control of large-scale multi-agent systems
(MAS) 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 (CBF) 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 (GNNs) to parameterize
a candidate GCBF and a distributed control policy. The proposed framework is
distributed and is capable of directly 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
nonlinear agents (e.g., Crazyflie drones) GCBF+ outperforms the handcrafted
CBF-based method with the best performance by up to 20% for relatively
small-scale MAS for 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.
