Formation Shape Control using the Gromov-Wasserstein Metric

TL;DR

This work addresses formation shape control for multi-agent systems by formulating a minimum-energy discrete-time OCP that uses the $\mathsf{GW}$ distance as a shape-sensitive terminal cost. To overcome the NP-hardness of $\mathsf{GW}$, the authors develop a tight $\mathrm{SDP}$ relaxation that provides global optimality certificates under a rank condition and yields a computable surrogate objective $\mathrm{GWSDP}$. Numerical experiments with 10 agents demonstrate that the relaxed solution achieves a GW cost close to the true optimum (ratio near 1) while maintaining low control effort, and show the method's flexibility by employing alternative GW costs based on graph structures. The approach offers a principled, optimality-guaranteed pathway to shape-based formation control and points to sparsification, entropy-regularized variants, and unbalanced GW as avenues for practical scalability and robustness.

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.

Figures (3)

• Figure 1: Approximation ratio
• Figure 2: In the top row, the left-hand and right-hand subfigures depict the target shape and the final shape, respectively. The bottom row depicts the randomly chosen initial points (squares) and their trajectories up to the final formation (circles), with $\varepsilon = 0.5$.
• Figure 3: The left-hand subfigure depicts the target graph structure considered in \ref{['eq:graph_distance']}, the right-hand subfigures shows the trajectory achieved, transitioning from squares to circles, with $\varepsilon = 0.5$. Note that in the final shape, the agents are successfully separated into 2 groups, respecting the graph structure. This is ensured by the term $\mathsf{GW}_c(x_d)$.

Theorems & Definitions (7)

• remark 1
• proposition 1
• proof
• remark 2: Approximation ratio and strong duality
• theorem 1
• proof
• remark 3