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Research Papers

# Continuum Deformation of Multi-Agent Systems Under Directed Communication Topologies

[+] Author and Article Information
Hossein Rastgoftar

Aerospace Engineering Department,
University of Michigan Ann Arbor,
Ann Arbor, MI 48109-2102
e-mail: hosseinr@umich.edu

Ella M. Atkins

Aerospace Engineering Department,
University of Michigan Ann Arbor,
Ann Arbor, MI 48109-2102
e-mail: ematkins@umich.edu

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received July 9, 2015; final manuscript received June 7, 2016; published online September 9, 2016. Assoc. Editor: Dejan Milutinovic.

J. Dyn. Sys., Meas., Control 139(1), 011002 (Sep 09, 2016) (11 pages) Paper No: DS-15-1312; doi: 10.1115/1.4033866 History: Received July 09, 2015; Revised June 07, 2016

## Abstract

A leader follower model has recently been proposed for homogeneous deformation of a multi-agent system (MAS) in $ℝn$. Researchers have shown how a desired homogeneous transformation can be designed by choosing proper trajectories for n + 1 leader agents and can be learned by every follower through local communication. However, existing work requires every follower to communicate with n + 1 adjacent agents, where communication between every two adjacent followers is constrained to be bidirectional. These requirements limit the total allowable number of agents, so an arbitrary number of agents may not be able to acquire a desired homogeneous mapping by local interaction. Additionally, if followers are not allowed to communicate with more than n + 1 neighboring agents, the convergence rate of actual positions to the desired positions (defined by a homogeneous transformation) may not be sufficiently high. The system may then considerably deviate from the desired configuration during transition. The main contribution of this article is to address these two issues, where each follower is considered to be a general linear system. It will be proven that followers can acquire desired positions prescribed by a homogeneous mapping in the presence of disturbance and measurement noise by applying a new finite-time reachability model under either fixed or switching topologies, if: (i) communication among followers is defined by a directed and strongly connected subgraph, (ii) each follower applies a consensus protocol with communication weights that are consistent with the positions of the agents in the initial configuration, and (iii) every follower i is allowed to communicate with $mi≥n+1$ local agents. With this strategy, an MAS with an arbitrary number of agents with linear dynamics can acquire a desired homogeneous mapping in the presence of disturbance and measurement noise, where convergence rate can be enhanced by increasing the number of communication links.

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## Figures

Fig. 1

Leaders' paths: initial formation P and desired final configuration R

Fig. 2

(a) A typical bidirectional communication topology graph G3; (b) a typical graph with nonminimum interagent topology graph G1; and (c) a typical graph with minimum directed interagent topology graph G2

Fig. 3

MAS formations at t = 15 for topologies G1 and G3

Fig. 4

Parameters a10,1,a10,2, and a10,3

Fig. 5

MAS formations at different sample times (a) graph G1 at t = 5 s, (b) graph G1 at t = 10 s, (c) graph G2 at t = 15 s, (d) graph G2 at t = 20 s, (e) graph G3 at t = 25 s, and (f) graph G3 at t = 30 s

Fig. 6

(a) Position of follower ten in the presence of disturbance and measurement noise and (b) position of follower ten in the absence of disturbance and measurement noise

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