# Controlling and Stabilizing a Rigid Formation Using a Few Agents

## TL;DR

This paper demonstrates that a small subset of agents in a rigid formation can control the entire formation's position and orientation through specific control strategies, ensuring rigidity and desired configuration stability.

## Contribution

It introduces a method where a clique of agents can control the whole formation's pose, extending control capabilities with minimal agents under mild conditions.

## Key findings

- A subset of agents can control formation orientation and position.
- Perturbations induce rigid motions of the formation.
- The control map is locally surjective, enabling flexible formation control.

## Abstract

We show in this paper that a small subset of agents of a formation of n agents in Euclidean space can control the position and orientation of the entire formation. We consider here formations tasked with maintaining inter-agent distances at prescribed values. It is known that when the inter-agent distances specified can be realized as the edges of a rigid graph, there is a finite number of possible configurations of the agents that satisfy the distance constraints, up to rotations and translations of the entire formation. We show here that under mild conditions on the type of control used by the agents, a subset of them forming a clique can work together to control the position and orientation of the formation as a whole. Mathematically, we investigate the effect of certain allowable perturbations of a nominal dynamics of the formation system. In particular, we show that any such perturbation leads to a rigid motion of the entire formation. Furthermore, we show that the map which assigns to a perturbation the infinitesimal generator of the corresponding rigid motion is locally surjective.