[Paper Review] Generalized weak rigidity: Theory, and local and global convergence of formations

Seong‐Ho Kwon, Hyo‐Sung Ahn|arXiv (Cornell University)|Sep 7, 2018

Distributed Control Multi-Agent Systems4 citations

TL;DR

This paper introduces generalized weak rigidity (GWR) theory to characterize rigid formations using pure inter-agent distances and angles, enabling local and global convergence analysis via a gradient flow control law. The key contribution is proving locally exponential stability for formations in 2D and 3D, and almost global exponential stability for 3-agent formations in 2D with distance and angle constraints.

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.

Motivation & Objective

• To develop a generalized weak rigidity (GWR) theory that characterizes rigid formations using pure inter-agent distances and angles, avoiding redundant information found in prior weak rigidity frameworks.
• To analyze the local exponential stability of formation control systems governed by a gradient flow law under GWR constraints in 2D and 3D spaces.
• To establish almost globally exponential stability for 3-agent formations in 2D space under pure distance and angle constraints.
• To provide a generic framework for determining rigidity using the rank condition of the weak rigidity matrix, applicable to both distance and angle constraints.
• To demonstrate through simulations the validity and convergence performance of the proposed control law under various formation configurations.

Proposed method

• Proposes generalized weak rigidity (GWR) as an extension of type-2 weak rigidity, using only pure distance and angle constraints without redundant information.
• Defines a generalized weak rigidity matrix and derives its rank condition to determine local rigidity of a formation framework.
• Applies a gradient flow control law to drive inter-agent distances and angles toward desired values, minimizing squared distance and cosine errors.
• Uses Lyapunov stability analysis to prove locally exponential stability for minimally globally infintely weakly rigid (GIWR) formations in 2D and 3D.
• Employs proportional gains on angle error terms to enhance convergence speed in simulations.
• Conducts numerical simulations with 3-, 5-, and 6-agent formations in 2D and 3D to validate theoretical stability results.

Experimental results

Research questions

• RQ1 Can a generalized weak rigidity theory be formulated to characterize rigid formations using only pure inter-agent distances and angles, without redundant information?
• RQ2 Under what conditions is a formation control system with pure distance and angle constraints locally exponentially stable in 2D and 3D spaces?
• RQ3 Is almost global exponential stability achievable for 3-agent formations in 2D when only distance and angle constraints are used?
• RQ4 How does the rank condition of the generalized weak rigidity matrix relate to the rigidity of a formation framework?
• RQ5 What is the role of the gradient flow control law in achieving convergence to desired rigid formations under GWR constraints?

Key findings

• The paper proves that a formation control system governed by the gradient flow law achieves locally exponential stability for minimally GIWR formations in both 2D and 3D spaces.
• For 3-agent formations in 2D space, the system achieves almost globally exponential stability under one distance and two angle constraints, even from collinear initial configurations.
• Numerical simulations confirm exponential convergence of distance and angle errors in 6-agent (2D) and 5-agent (3D) formations with pure distance and angle constraints.
• The simulations show that initial collinear formations can lead to incorrect equilibria, highlighting the importance of initial configuration in achieving desired convergence.
• The generalized weak rigidity theory is shown to be a necessary condition for distance rigidity theory, and GWR/GIWR are generic properties of formation frameworks.
• The use of proportional gains on angle errors significantly improves convergence speed in simulations without compromising stability.