[Paper Review] Multi-agent systems with CBF-based controllers -- collision avoidance and liveness from instability

Mrdjan Janković, Mario Santillo+1 more|arXiv (Cornell University)|Jul 11, 2022

Traffic control and management4 citations

TL;DR

This paper proposes a novel decentralized control policy, Predictor-Corrector for Collision Avoidance (PCCA), using Control Barrier Functions (CBF) to achieve both safety and liveness in multi-agent systems. Unlike traditional decentralized policies that suffer from gridlock due to stable equilibria, PCCA induces unstable equilibria, enabling fast convergence and near-zero gridlock—matching centralized performance while operating with local information.

ABSTRACT

Assuring system stability is typically a major control design objective. In this paper, we present a system where instability provides a crucial benefit. We consider multi-agent collision avoidance using Control Barrier Functions (CBF) and study trade-offs between safety and liveness -- the ability to reach a destination without large detours or gridlock. We compare two standard decentralized policies, with only the local (host) control available, to co-optimization policies (PCCA and CCS) where everyone's (virtual) control action is available. The co-optimization policies compute control for everyone even though they lack information about others' intentions. For comparison, we use a Centralized, full information policy as the benchmark. One contribution of this paper is proving feasibility for the Centralized, PCCA, and CCS policies. Monte Carlo simulations show that decentralized, host-only control policies and CCS lack liveness while the PCCA policy performs as well as the Centralized. Next, we explain the observed results by considering two agents negotiating the passing order through an intersection. We show that the structure and stability of the resulting equilibria correlates with the observed propensity to gridlock -- the policies with unstable equilibria avoid gridlocks while those with stable ones do not.

Motivation & Objective

• To address the trade-off between safety and liveness in multi-agent systems using CBF-based controllers.
• To analyze why decentralized CBF policies exhibit gridlock while centralized and co-optimization policies do not.
• To prove feasibility of centralized, PCCA, and CCS control policies in a distance-CBF framework.
• To explain the observed liveness differences through equilibrium stability analysis in a two-agent joint state space.
• To demonstrate that unstable equilibria correlate with reduced gridlock and faster convergence, even with incomplete information.

Proposed method

• Formulates a centralized CBF-based quadratic program (QP) for multi-agent collision avoidance, proving its feasibility for distance-CBF constraints.
• Introduces the PCCA algorithm as a co-optimization policy that predicts and corrects for other agents’ actions using a disturbance model.
• Develops the Complete Control Set (CCS) policy as a co-optimization alternative that computes all agents’ controls without using other agents’ actions.
• Analyzes equilibrium structures in the joint state space of two agents negotiating a merge point or intersection.
• Uses a 1D kinematic model with state variables (position, velocity) and applies CBF constraints to enforce safety via QP solutions.
• Employs Monte Carlo simulations with five agents to compare liveness, collision rates, and convergence times across policies.

Experimental results

Research questions

• RQ1 Why do decentralized CBF policies like DF and DR exhibit gridlock in multi-agent scenarios despite being safe?
• RQ2 How does the stability of equilibria in the joint state space influence liveness and convergence in CBF-based multi-agent control?
• RQ3 Can co-optimization policies such as PCCA and CCS achieve liveness comparable to centralized control while maintaining decentralized implementation?
• RQ4 What is the role of equilibrium structure—specifically stable vs. unstable manifolds—in determining the propensity to gridlock?
• RQ5 Is it possible to design a decentralized CBF policy that induces unstable equilibria to avoid gridlock, and if so, how?

Key findings

• The centralized CBF policy is always feasible and achieves zero gridlock, with only a measure-zero set of initial conditions leading to equilibrium.
• PCCA achieves liveness performance indistinguishable from the centralized policy, with only 0.002% of simulations resulting in gridlock.
• Decentralized policies (DF/DR) and CCS exhibit significant gridlock—15.4% and near-100%, respectively—due to stable equilibria in the joint state space.
• The stable manifold for the centralized policy is a 1D line in 2D state space, making gridlock unlikely; for PCCA, it is a 2D manifold in 4D space, further reducing probability.
• CCS, despite co-optimization, behaves like decentralized policies due to stable equilibria, resulting in high gridlock and poor liveness.
• The paper demonstrates that unstable equilibria—especially exponentially unstable ones—enable rapid escape from near-gridlock states, explaining PCCA’s superior liveness.