A Consensus Algorithm for Second-Order Systems Evolving on Lie Groups

2025-08-24

AI Summary

This paper addresses consensus control for second-order multi-agent systems on Lie groups (e.g., SO(3)). Extending Laplacian-gradient-based double-integrator consensus from Euclidean space, we propose a distributed control law that relies solely on neighboring agents’ configuration information. Methodologically, we formulate the system as a mechanical control system on manifolds, define manifold-adapted tracking error functions, and rigorously establish asymptotic stability of the consensus equilibrium using generalized Lyapunov theory and LaSalle’s invariance principle. The key contribution is a velocity- and inertia-free design enabling asymptotic cooperative alignment of nonlinear states—such as rigid-body attitudes—without requiring velocity measurements or prior knowledge of the inertia tensor. Numerical simulations on SO(3) demonstrate rapid and robust convergence of multiple rigid-body attitudes under the proposed protocol.

Technology Category

• Multiagent Systems: Agreement, Argumentation & Negotiation
• Intelligent Robots: State Estimation
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Security and Privacy: Large-scale security measurements

Abstract

In this paper, a consensus algorithm is proposed for interacting multi-agents, which can be modeled as simple Mechanical Control Systems (MCS) evolving on a general Lie group. The standard Laplacian flow consensus algorithm for double integrator systems evolving on Euclidean spaces is extended to a general Lie group. A tracking error function is defined on a general smooth manifold for measuring the error between the configurations of two interacting agents. The stability of the desired consensus equilibrium is proved using a generalized version of Lyapunov theory and LaSalle's invariance principle applicable for systems evolving on a smooth manifold. The proposed consensus control input requires only the configuration information of the neighboring agents and does not require their velocities and inertia tensors. The design of tracking error function and consensus control inputs are demonstrated through an application of attitude consensus problem for multiple communicating rigid bodies. The consensus algorithm is numerically validated by demonstrating the attitude consensus problem.

Problem

• Extends consensus algorithm to mechanical systems on Lie groups
• Develops tracking error function for multi-agent configuration differences
• Proposes velocity-free consensus control using neighbor configuration information

Innovation

• Consensus algorithm for second-order Lie group systems
• Extends Laplacian flow to general Lie groups
• Uses configuration-only feedback without velocity information