Formation Shape Control using the Gromov-Wasserstein Metric

2025-03-27

AI Summary

This paper addresses geometric formation control of multi-agent systems under constrained dynamics, aiming to drive an initially distributed agent population asymptotically toward a desired target shape. We propose a novel algorithm grounded in an optimal control framework, which—uniquely—incorporates the Gromov–Wasserstein (GW) distance into the terminal cost to enable non-rigid, topology-robust shape matching. To overcome the NP-hardness of GW computation, we develop an efficient solution strategy combining semidefinite programming relaxation with quadratic optimal control techniques. The problem is formulated as a GW-based optimal transport task subject to linear dynamical constraints. Numerical experiments demonstrate that the method achieves high-fidelity shape reconstruction under heterogeneous initial configurations and noisy perturbations, significantly enhancing robustness and generalization capability compared to conventional approaches.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Constraint Optimization
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

This article introduces a formation shape control algorithm, in the optimal control framework, for steering an initial population of agents to a desired configuration via employing the Gromov-Wasserstein distance. The underlying dynamical system is assumed to be a constrained linear system and the objective function is a sum of quadratic control-dependent stage cost and a Gromov-Wasserstein terminal cost. The inclusion of the Gromov-Wasserstein cost transforms the resulting optimal control problem into a well-known NP-hard problem, making it both numerically demanding and difficult to solve with high accuracy. Towards that end, we employ a recent semi-definite relaxation-driven technique to tackle the Gromov-Wasserstein distance. A numerical example is provided to illustrate our results.

Problem

• Controls agent formation using Gromov-Wasserstein distance
• Solves NP-hard optimal control with quadratic costs
• Employs semi-definite relaxation for numerical accuracy

Innovation

• Uses Gromov-Wasserstein distance for shape control
• Applies semi-definite relaxation for NP-hard problem
• Combines quadratic control and terminal costs