An emergence-oriented approach to circular formation

TL;DR

A control law is proposed that enables the spontaneous formation of circular formations through only local measurements and is extended to distance-dependent neighborhoods.

Abstract

In this paper, we study the emergence of circular formation for agents in cyclic pursuit. Each agent is a unicycle traveling at a fixed common forward speed. We first establish a necessary and sufficient condition for the existence of circular formation in cyclic pursuit. Building on this theoretical foundation, we propose a control law that enables the spontaneous formation of circular formations through only local measurements. Notably, key geometric features -- the radius and agent spacing -- are not imposed externally but emerge naturally from the initial conditions of the group. This occurs because the closed-loop system possesses infinitely many non-isolated equilibria, each corresponding to a particular circular formation, and none are asymptotically stable. Consequently, analyzing individual equilibria is no longer informative, and attention is instead directed to the full invariant set (the set of all equilibria). Globally, it is disconnected. Locally, however, each equilibrium together with its neighboring equilibria forms a connected invariant set. This motivates a local stability analysis formulated at the level of invariant sets that are maximally connected. An accompanying stability criterion is then derived and applied to analyze small agent groups ($n \leq 3$), providing insights into the convergence mechanism. Finally, the proposed control law is extended to distance-dependent neighborhoods. Under this setting, the group converges into several clusters, most exhibiting a complete-graph topology. A preliminary stability analysis is then conducted for the case of a complete graph with $n=3$.

Figures (8)

• Figure 1: Illustration of $\tilde{z}_j\times\tilde{\mathtt{r}}_j=0$.
• Figure 2: Illustration of $2\alpha_i+\beta_i= \pi$, corresponding to counterclockwise rotation.
• Figure 3: Two arrangements of counterclockwise formations with three vehicles. The second formation is obtained by reversing the rotation direction in the first one and viewing from inside the paper outward.
• Figure 4: $n=5$ unicycles controlled by \ref{['eq-omega-1']} with $v=1$ and $k=\pm5$ converge to a circular formation. Under identical initial conditions, $k<0$ results in counterclockwise rotation (left), whereas $k>0$ results in clockwise rotation (right). Both formations are regular.
• Figure 5: Three unicycles controlled by \ref{['eq-omega-1']} with $v=1$ and $k=-2$ converge to a regular counterclockwise formation.

Theorems & Definitions (31)

• Definition 2.1
• Proposition 2.2
• Proposition 2.3
• Remark 2.4
• Corollary 2.5
• Remark 2.6
• Definition 2.7
• Proposition 2.8
• Proposition 2.9
• Definition 2.10