Cooperative event-based rigid formation control

TL;DR

This work presents two event-based strategies for stabilizing rigid formations: a centralized scheme with a global trigger and a distributed scheme where each agent triggers locally. By leveraging the distance error $e$, rigidity matrix $R(p)$, and a Lyapunov candidate $V=\tfrac14\|e\|^2$, the authors prove local exponential convergence and ensure Zeno-exclusion, with a simple per-agent trigger in the distributed case and a modified trigger to guarantee Zeno-free operation for all agents. A key insight is the SE($N$) invariance and the ability to implement controllers in local coordinate frames, enabling GPS-denied deployment. Simulations on a double-tetrahedron in $\mathbb{R}^3$ demonstrate rapid convergence and reduced communication, underscoring the practical relevance for multi-robot coordination and networked systems.

Abstract

This paper discusses cooperative stabilization control of rigid formations via an event-based approach. We first design a centralized event-based formation control system, in which a central event controller determines the next triggering time and broadcasts the event signal to all the agents for control input update. We then build on this approach to propose a distributed event control strategy, in which each agent can use its local event trigger and local information to update the control input at its own event time. For both cases, the triggering condition, event function and triggering behavior are discussed in detail, and the exponential convergence of the event-based formation system is guaranteed.

Figures (7)

• Figure 1: Examples on rigid and nonrigid formations. (a) non-rigid formation (a deformed formation with dashed lines is shown); (b) minimally rigid formation; (c) rigid but non-minimally rigid formation.
• Figure 2: Simulation on stabilization control of a double tetrahedron formation in 3-D space with centralized event controller. The initial and final positions are denoted by circles and squares, respectively. The initial formation is denoted by dashed lines, and the final formation is denoted by red solid lines. The black star denotes the formation centroid, which is stationary.
• Figure 3: Exponential convergence of the distance errors with centralized event controller.
• Figure 4: Performance of the centralized event-based controller. Top: evolution of $\|\delta\|$ and $\|\delta\|_{\max} = \gamma \|R(t)^T e(t)\|$. Bottom: event triggering instants.
• Figure 5: Simulation on stabilization control of a double tetrahedron formation in 3-D space with distributed event controller. The initial and final positions are denoted by circles and squares, respectively. The initial formation is denoted by dashed lines, and the final formation is denoted by red solid lines. The black star denotes the formation centroid, which is not stationary.

Theorems & Definitions (33)

• Definition 1
• Definition 2
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• proof
• Lemma 5
• Theorem 1
• proof