Time-Inverted Kuramoto Dynamics for κ-Clustered Circle Coverage

Abstract

In this paper we analyse the equilibrium configurations for the time-inverted Kuramoto Model with homogeneous agents and a fixed ring topology, where time-inverted means that the coupling between the different states is via a negative factor. This model exhibits a dual behaviour with respect to the classic Kuramoto Model with a positive copuling. In the paper, we show the existence of two possibile stable equilibrium configurations: the splay state formation (1-clustered coverage) and the deployment in clusters (κ-clustered coverage). We provide sufficient conditions for the splay state formation and a stability analysis for the networked system. Moreover, we provide some initial results towards the controllability of the final equilibrium configurations. In particular, we lay the foundations to understand the conditions to switch between stable equilibria.

Multiagent Persistent Monitoring via Time-Inverted Kuramoto Dynamics

“…Finally, we provide a set of numerical experiments to validate the effectiveness of the proposed strategy. A preliminary version of the results contained in this paper was presented in [2]. However, compared to [2] we made several advances: we provide Theorem 1, which in the previous version was informally addressed.…”

“…A preliminary version of the results contained in this paper was presented in [2]. However, compared to [2] we made several advances: we provide Theorem 1, which in the previous version was informally addressed. In addition, this paper shows Theorem 2, which was presented in [2] just as a conjecture.…”

“…However, compared to [2] we made several advances: we provide Theorem 1, which in the previous version was informally addressed. In addition, this paper shows Theorem 2, which was presented in [2] just as a conjecture. The core of the novel contribution are the results that link persistent monitoring (Section III) with the properties of the Kuramoto equilibria.…”

A unified Lloyd-based framework for multi-agent collective behaviours

“…The fact that condition (11) never holds allows us to rule out the existence of blind spots. We have therefore to exclude the case in which S i is not adjacent to S i .…”

“…In [7][8][9] the authors proposed coverage solutions based on the visibility set to deal with non-convex spaces. Other approaches are based on potential fields [10], time-inverted Kuramoto dynamics [11], on the maximisation of the joint detection probabilities of random events [12] or address specific application areas [13].…”