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Angle-Constrained Formation Control under Directed Non-Triangulated Sensing Graphs (Extended Version)

Kun Li, Zhixi Shen, Gangshan Jing, Yongduan Song

TL;DR

The paper tackles angle-constrained formation control under a directed, non-triangulated sensing graph using a Leader-First Follower architecture. It develops a distributed controller based on angle constraints that achieves global exponential convergence to the target shape using only local relative-position measurements, and it provides maneuvering laws that preserve shape while translating, rotating, and scaling the formation. A key insight is that the convergence rate depends only on the follower angles in the target framework, and the method is validated through simulations and Robotarium experiments. The work extends angle-based formation control beyond undirected triangulated graphs, reduces sensing requirements, and points to future 3D extensions and collision-avoidance considerations.

Abstract

Angle-constrained formation control has attracted much attention from control community due to the advantage that inter-edge angles are invariant under uniform translations, rotations, and scalings of the whole formation. However, almost all the existing angle-constrained formation control methods are limited to undirected triangulated sensing graphs. In this paper, we propose an angle-constrained formation control approach under a Leader-First Follower sensing architecture, where the sensing graph is directed and non-triangulated. Both shape stabilization and maneuver control are achieved under arbitrary initial configurations of the formation. During the formation process, the control input of each agent is based on relative positions from its neighbors measured in the local reference frame and wireless communications among agents are not required. We show that the proposed distributed formation controller ensures global exponential stability of the desired formation for an nagent system. Furthermore, it is interesting to see that the convergence rate of the whole formation is solely determined by partial specific angles within the target formation. The effectiveness of the proposed control algorithms is illustrated by carrying out experiments both in simulation environments and on real robotic platforms.

Angle-Constrained Formation Control under Directed Non-Triangulated Sensing Graphs (Extended Version)

TL;DR

The paper tackles angle-constrained formation control under a directed, non-triangulated sensing graph using a Leader-First Follower architecture. It develops a distributed controller based on angle constraints that achieves global exponential convergence to the target shape using only local relative-position measurements, and it provides maneuvering laws that preserve shape while translating, rotating, and scaling the formation. A key insight is that the convergence rate depends only on the follower angles in the target framework, and the method is validated through simulations and Robotarium experiments. The work extends angle-based formation control beyond undirected triangulated graphs, reduces sensing requirements, and points to future 3D extensions and collision-avoidance considerations.

Abstract

Angle-constrained formation control has attracted much attention from control community due to the advantage that inter-edge angles are invariant under uniform translations, rotations, and scalings of the whole formation. However, almost all the existing angle-constrained formation control methods are limited to undirected triangulated sensing graphs. In this paper, we propose an angle-constrained formation control approach under a Leader-First Follower sensing architecture, where the sensing graph is directed and non-triangulated. Both shape stabilization and maneuver control are achieved under arbitrary initial configurations of the formation. During the formation process, the control input of each agent is based on relative positions from its neighbors measured in the local reference frame and wireless communications among agents are not required. We show that the proposed distributed formation controller ensures global exponential stability of the desired formation for an nagent system. Furthermore, it is interesting to see that the convergence rate of the whole formation is solely determined by partial specific angles within the target formation. The effectiveness of the proposed control algorithms is illustrated by carrying out experiments both in simulation environments and on real robotic platforms.
Paper Structure (14 sections, 6 theorems, 27 equations, 3 figures)

This paper contains 14 sections, 6 theorems, 27 equations, 3 figures.

Table of Contents

  1. INTRODUCTION
  2. Problem Formulation
  3. Graph-Related Notions
  4. Agent Dynamics and Sensing Capability
  5. Problem Statement
  6. Angle-Constrained LFF Formation Control
  7. Angle Constraints in Target Framework
  8. Formation Stabilization
  9. Formation Maneuver Control
  10. EXPERIMENTS
  11. Formation Shape Stabilization
  12. Formation Maneuver Control
  13. Acknowledgment
  14. CONCLUSION

Key Result

Lemma 1

Given a target formation $(\mathcal{G}_f,p^*)$ with $n$ vertices in $\mathbb{R}^2$ satisfying Assumption af, the shape of $(\mathcal{G}_f,p^*)$ can be uniquely determined by angle constraints (clm) in $(\mathcal{G}_f,p^*)$, i.e., $q\in\mathscr{E}(p^*)$ for $q=[q_1^\top,...,q_n^\top]^\top\in\mathbb{R

Figures (3)

  • Figure 1: The formation graph. (a) the sensing graph $\mathcal{G}_s$; (b) the target formation ($\mathcal{G}_f,p^*$).
  • Figure 2: The formation shape control law (\ref{['u']}) was implemented on a team of six differential drive robots. (a) the target formation ($\mathcal{G}_f,p^*$); (b) the sensing graph $\mathcal{G}_s$; (c) initial positions of six robots on the Robotarium; (d) the six robots eventually form the target formation shape; (e) the angle errors $\|e_1\|=\sum_{\alpha_{ijk} \in \mathcal{A}_{\mathcal{G}_f}}\|\alpha_{ijk}-\alpha^*_{ijk}\|$ converge to zeros asymptotically.
  • Figure 3: The formation shape control law (\ref{['man']}) was implemented on a team of six differential drive robots. (a) initial positions of six differential-drive robots on the Robotarium; (b) the formation at time $t=50s$, $v^{*}_{r}(t)=[0,0.02]^\top$,$\delta^*_{12}(t)=[0.4,0.4]^\top$; (c) the formation at time $t=90s$, $v^{*}_{r}(t)=[0.05,0]^\top$, $\delta^*_{12}(t)=\mathcal{R}(\frac{\pi}{2})[0.4,0.4]^\top$; (d) the formation at time $t=120s$, $v^{*}_{r}(t)=[0.04,0]^\top$, $\delta^*_{12}(t)=0.7*\mathcal{R}(\frac{\pi}{2})[0.4,0.4]^\top$; (e) evolution of angle errors $||e_2||=\sum_{\alpha_{ijk} \in \mathcal{A}_{\mathcal{G}_f}}\|\alpha_{ijk}-\alpha^*_{ijk}\|$.

Theorems & Definitions (12)

  • Remark 1
  • Remark 2
  • Lemma 1: Uniqueness of the target formation
  • Remark 3
  • Lemma 2
  • Theorem 1
  • Remark 4
  • Remark 5
  • Lemma 3: Collision-free between agent $i$ and its neighbors
  • Lemma 4
  • ...and 2 more