Directed Formation Control of n Planar Agents with Distance and Area Constraints

TL;DR

The paper tackles robust, scalable planar formation control for $N$ single-integrator agents by integrating inter-agent distances with signed triangle areas on a directed Leader-First-Follower graph. It develops a gradient-based controller that is distributed and leverages a triangular, triangulated graph structure to couple subsystems, with explicit geometric and gain conditions ensuring asymptotic convergence to the desired formation class $ ext{SCgt}(F^{*})$ from almost all initial states; critical constraints include $ig| (d_{ki}^2-d_{kj}^2)/d_{ji}^2 igr|<2\sqrt{2}$ and a lower bound on $eta_k/\,oldsymbol{a}_k$ derived from quartic polynomial analysis. The key contributions are the explicit, scalable stability guarantees for any $N$ under mild edge-length restrictions, the use of signed area to break formation ambiguities including reflections, and a distributed control law that does not require global coordinates. Overall, the approach advances distance-based formation control by ensuring unique convergence to the correct shape in 2D through area constraints and directed graph structure, with potential for extension to larger multi-agent systems. The results are practically significant for cooperative tasks requiring precise planar formations using only local measurements and uncoordinated frames.

Abstract

In this paper, we take a first step towards generalizing a recently proposed method for dealing with the problem of convergence to incorrect equilibrium points of distance-based formation controllers. Specifically, we introduce a distance and area-based scheme for the formation control of $n$-agent systems in two dimensions using directed graphs and the single-integrator model. We show that under certain conditions on the edge lengths of the triangulated desired formation, the control ensures almost-global convergence to the correct formation.

Figures (7)

• Figure 1: Types of ambiguous frameworks.
• Figure 2: Signed area examples.
• Figure 3: Three-agent system.
• Figure 4: Geometeric interpretation of (\ref{['triangle cond']}).
• Figure 5: a) Potential function $W(q_{3})$ and corresponding counter plot; b) Potential function $V_{3}(q_{3})$ and corresponding counter plot.

Theorems & Definitions (14)

• Definition 1
• Lemma 1
• proof
• Lemma 2
• Corollary 1
• proof
• Remark 1
• Lemma 3
• Lemma 4
• Theorem 1