Distributed Multi-agent Coordination over Cellular Sheaves

TL;DR

The paper develops a unified framework for distributed coordination of heterogeneous multi-agent systems by casting coordination problems as nonlinear homological programs on cellular sheaves, with edge potentials encoding coordination tasks. A distributed ADMM-based solver is derived, leveraging the nonlinear sheaf Laplacian to perform local computations and neighbor communications while enforcing a global coordination constraint L^{∇U}_{F} x = 0. The approach is instantiated in a multi-domain vehicle case study (UAV, USV, UUV) and demonstrated through numerical MPC experiments for consensus, stationary formation, flocking, and moving formation, illustrating scalable, decentralized coordination with convexity guarantees under appropriate conditions. The work contributes a mathematically principled, flexible framework that unifies consensus, formation, and flocking under a single sheaf-theoretic optimization paradigm, with practical implications for heterogeneous, networked systems. Potential impacts include enabling robust, scalable coordination in cyber-physical and robotic systems with diverse agents and communication modalities.

Abstract

Techniques for coordination of multi-agent systems are vast and varied, often utilizing purpose-built solvers or controllers with tight coupling to the types of systems involved or the coordination goal. In this paper, we introduce a general unified framework for heterogeneous multi-agent coordination using the language of cellular sheaves and nonlinear sheaf Laplacians, which are generalizations of graphs and graph Laplacians. Specifically, we introduce the concept of a nonlinear homological program encompassing a choice of cellular sheaf on an undirected graph, nonlinear edge potential functions, and constrained convex node objectives, which constitutes a standard form for a wide class of coordination problems. We use the alternating direction method of multipliers to derive a distributed optimization algorithm for solving these nonlinear homological programs. To demonstrate the applicability of this framework, we show how heterogeneous coordination goals including combinations of consensus, formation, and flocking can be formulated as nonlinear homological programs and provide numerical simulations showing the efficacy of our distributed solution algorithm.

Figures (3)

• Figure 1: The cellular sheaf encoding the evolution of a discrete LTI system from an initial condition $x(1)=c$ over $T$ time-steps. Note that $!\colon \mathbb{R}^0\to\mathbb{R}^n$ is the unique map from $\mathbb{R}^0$, and $\pi_1$ is the first projection. The graph which the sheaf is defined on is given on the bottom with the stalks and restriction maps lying above. The blue section uses an edge potential function to fix the initial condition to $x(1)=c$. All other edge potential functions are the standard consensus potentials so we omit them from the diagram for clarity. The black section then encodes the evolution of the dynamics from $c$ for $T$ time-steps.
• Figure 2: Multi-domain operation: UAVs, USVs, & UUVs.
• Figure 3: (a) Agents perform a hybrid goal of consensus in $x$ and tracking in $y$. (b) Agents perform the goal of reaching a triangular formation centered at the origin. (c) Agents perform the goal of flocking. (d) Agents perform a moving formation goal. In (c) and (d), the leader agent (1) tracks a constant rightward velocity vector.

Theorems & Definitions (15)

• Definition 1: Cellular Sheaf
• Example 1: Constant Sheaf
• Definition 2: Global Sections
• Definition 3: Nonlinear Sheaf Laplacian
• Definition 4: Nonlinear Homological Program
• Example 3: Distributed Optimization
• Theorem 1
• Example 4: Single-Agent Optimal Control
• Remark 1
• Theorem 2