A Rendezvous Strategy With $\mathbb{R}^{2}$ Reachability for Kinematic Agents

Abstract

In this article, we present a consensus strategy based on cyclic pursuit that ensures rendezvous at any desired point in 2-D space ( $\mathbb{R}^{2}$ reachability) starting with any nonsingular initial configuration of the agents. Choice of a negative gain expands the reachable set beyond the convex hull of the initial configuration of agents. However, the entire $\mathbb{R}^{2}$ is not reachable in general. We first find the necessary and sufficient condition for the initial configuration of the agents, which ensures $\mathbb{R}^{2}$ reachability. We show that any desired point can be reached with finite controller gains under basic cyclic pursuit if the initial configuration of agents forms a nonconvex polygon. Next, we employ a control strategy using deviated cyclic pursuit to reach the special configuration mentioned above starting from an arbitrary initial configuration. With a switching between these two strategies we can achieve $\mathbb{R}^{2}$ reachability with finite gains. We also present an implementation algorithm to realize this strategy including computation of the finite gains.