Generalized weak rigidity: Theory, and local and global convergence of formations

TL;DR

This paper introduces a generalized weak rigidity theory for formation control, analyzing stability of formations with distance and angle constraints, and demonstrating local and almost global exponential stability in 2D and 3D spaces.

Contribution

It extends weak rigidity theory to include pure inter-agent distances and angles, providing new stability analysis results for formation control.

Findings

Locally exponential stability in 2D and 3D formations.

Almost globally exponential stability for three-agent formations in 2D.

Numerical simulations validate the theoretical results.

Abstract

This paper discusses generalized weak rigidity theory, and aims to apply the theory to formation control problems with a gradient flow law. The generalized weak rigidity theory is utilized in order that desired formations are characterized by a general set of pure inter-agent distances and angles. As the first result of its applications, the paper provides analysis of locally exponential stability for formation systems with pure distance/angle constraints in the $2$- and $3$-dimensional spaces. Then, as the second result, if there are three agents in the $2$-dimensional space, almost globally exponential stability for formation systems is ensured. Through numerical simulations, the validity of analyses is illustrated.