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A Leader-Follower Approach for The Attitude Synchronization of Multiple Rigid Body Systems on $SO(3)$

Yiliang Li, Jun-e Feng, Abdelhamid Tayebi

TL;DR

The paper addresses leader-follower attitude synchronization for multiple rigid bodies on $SO(3)$ with a constant reference attitude $R_0$ available only to a single leader. It develops a distributed control law that combines leader-relative and neighbor-consensus terms with angular-velocity damping on an undirected tree topology, and proves almost global asymptotic stability of the desired synchronized state. The stability proof employs a Lyapunov function and Chetaev's theorem to show undesired equilibria are unstable with zero-measure stable manifolds, ensuring convergence to the desired equilibrium where all $R_i\to R_0$ and $\omega_i\to 0$. Simulations on a 7-agent network validate the theoretical results, demonstrating complete attitude synchronization to $R_0$ and damping of angular velocities across the network.

Abstract

This paper deals with the leader-follower attitude synchronization problem for a group of heterogeneous rigid body systems on $SO(3)$ under an undirected, connected, and acyclic graph communication topology. The proposed distributed control strategy, endowed with almost global asymptotic stability guarantees, allows the synchronization of the rigid body systems to a constant desired orientation known only to a single rigid body. Some simulation results are also provided to validate the theoretical developments and illustrate the performance of the proposed control strategy.

A Leader-Follower Approach for The Attitude Synchronization of Multiple Rigid Body Systems on $SO(3)$

TL;DR

The paper addresses leader-follower attitude synchronization for multiple rigid bodies on with a constant reference attitude available only to a single leader. It develops a distributed control law that combines leader-relative and neighbor-consensus terms with angular-velocity damping on an undirected tree topology, and proves almost global asymptotic stability of the desired synchronized state. The stability proof employs a Lyapunov function and Chetaev's theorem to show undesired equilibria are unstable with zero-measure stable manifolds, ensuring convergence to the desired equilibrium where all and . Simulations on a 7-agent network validate the theoretical results, demonstrating complete attitude synchronization to and damping of angular velocities across the network.

Abstract

This paper deals with the leader-follower attitude synchronization problem for a group of heterogeneous rigid body systems on under an undirected, connected, and acyclic graph communication topology. The proposed distributed control strategy, endowed with almost global asymptotic stability guarantees, allows the synchronization of the rigid body systems to a constant desired orientation known only to a single rigid body. Some simulation results are also provided to validate the theoretical developments and illustrate the performance of the proposed control strategy.
Paper Structure (8 sections, 2 theorems, 38 equations, 5 figures)

This paper contains 8 sections, 2 theorems, 38 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Graph Theory
  5. Problem formulation
  6. Main result
  7. Simulation
  8. Conclusion

Key Result

Lemma 1

Consider the matrix $L\in\mathbb{R}^{3N\times 3(N-1)}$ associated to the interconnection graph $\mathcal{G}$, with the block $L_{ik}$ defined in L_matrix. Then $L_2x=\mathbf{0}_{3(N-1)}$ implies $x=\mathbf{0}_{3(N-1)}$, where $L_2$ is obtained from $L$ by removing the first three rows, i.e.,

Figures (5)

  • Figure 1: An interconnection graph $\mathcal{G}$ of 7 rigid body systems
  • Figure 2: The time evolution of the relative attitude associated with each edge
  • Figure 3: The time evolution of the angular velocity error associated with each edge
  • Figure 4: The time evolution of the relative attitude between each rigid body system and the leader
  • Figure 5: The time evolution of the angular velocity associated with each rigid body system

Theorems & Definitions (2)

  • Lemma 1
  • Theorem 1