A Note on the Consensus Protocol with Some Applications to Agent Orbit Pattern Generation

Abstract

We propose an extension to the standard feedback control for consensus problems for multi-agent systems in the plane. The proposed extension allows for a richer class of trajectories including periodic and quasi-periodic solutions, as well as agreement to consensus states outside the convex hull of the initial positions of the agents. We investigate in great detail the special case of three agents, which results in non-trivial geometric patterns described by ellipsoidal, epitrochoidal and hypotrochoidal curves. A Generalized Consensus ProtocolConsensus problems have been originally used in distributed computing and management science and, most recently, have found extensive application in multi-agent, mobile network problems [12,16]. In this paper we propose a generalization of the standard consensus algorithm which has been used extensively in the literature [13,3,11]. The proposed extension of the standard consensus protocol leads to the following advantages: first, it can be used to achieve consensus at points that do not necessarily belong to the convex hull of the initial conditions. This may be beneficial in case of obstacle avoidance or as part of deception strategies. Second, as shown in the second part of the paper, it can be utilized to generate intricate geometrical patterns of the agent paths. These paths can be useful for coordinated, distributed surveillance and monitoring applications.

The geometry of map equations for Trochoids

“…Such curves can be represented parametrically using trigonometric functions, and can be easily graphed using a computer. The more recent results can also be found in [1,2,3,11].…”

Untitled

“…In another very interesting research report in [23], authors have proposed to use modified, using visual composition techniques, spirograph patterns for tweet display or other relatively large time-oriented datasets. In the research area of path design for multiple robotic agents in [24], the researchers have proposed use of complex trochoidal path for the coordinated, distributed and obstacle free movements. The development of spirograph based mechanical system has been reported in [25 ].…”

Emergent Hypotrochoidal and Epitrochoidal Formations in a Swarm of Agents

“…Indeed, trochoidal curves are the natural wind-dependent extension of Dubins-type paths and have been studied in the literature as a solution to a boundary value problem to connect two states in the presence of wind [24]. In [25], and in the preliminary work [26], it is shown how elaborate patterns that are closely related to hypocycloid and epicycloid curves can be generated as the paths followed by a team of interacting agents moving on the plane. The agents considered are of the single integrator type and the generation of hypocycloid and epicycloid depends on a suitable gain matrix and the choice of an appropriate connectivity graph between the agents.…”