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Research Papers

# Swarm Motion as Particles of a Continuum With Communication Delays

[+] Author and Article Information
Hossein Rastgoftar

Mechanical Engineering and Mechanics,
Drexel University,
3141 Chestnut Street 115 B,
e-mail:  hossein.rastgoftar@drexel.edu

Distinguished Professor
Mechanical Engineering and Mechanics,
Drexel University,
3141 Chestnut Street 115 B,

1Corresponding author.

2Deceased.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received September 8, 2014; final manuscript received May 31, 2015; published online August 14, 2015. Assoc. Editor: Srinivasa M. Salapaka.

J. Dyn. Sys., Meas., Control 137(11), 111008 (Aug 14, 2015) (13 pages) Paper No: DS-14-1364; doi: 10.1115/1.4030757 History: Received September 08, 2014

## Abstract

In this paper, we give an upper bound for the communication delay in a multi-agent system (MAS) that evolves under a recently developed continuum paradigm for formation control. The MAS is treated as particles of a continuum that transforms under special homeomorphic mapping, called a homogeneous map. Evolution of an MAS in $ℝn$ is achieved under a special communication topology proposed by Rastgoftar and Jayasuriya (2014, “Evolution of Multi Agent Systems as Continua,” ASME J. Dyn. Syst. Meas. Control, 136(4), p. 041014) and (2014, “An Alignment Strategy for Evolution of Multi Agent Systems,” ASME J. Dyn. Syst. Meas. Control, 137(2), p. 021009), employing a homogeneous map specified by the trajectories of $n+1$ leader agents at the vertices of a polytope in $ℝn$, called the leading polytope. The followers that are positioned in the convex hull of the leading polytope learn the prescribed homogeneous mapping through local communication with neighboring agents using a set of communication weights prescribed by the initial positions of the agents. However, due to inevitable time-delay in getting positions and velocities of the adjacent agents through local communication, the position of each follower may not converge to the desired state given by the homogeneous map leaving the possibility that MAS evolution may get destabilized. Therefore, ascertaining the stability under time-delay is important. Stability analysis of an MAS consisting of a large number of agents, leading to higher-order dynamics, using conventional methods such as cluster treatment of characteristic roots (CTCR) or Lyapunov–Krasovskii are difficult. Instead we estimate the maximum allowable communication delay for the followers using one of the eigenvalues of the communication matrix that places MAS evolution at the margin of instability. The proposed method is advantageous because the transcendental delay terms are directly used and the characteristic equation of MAS evolution is not approximated by a finite-order polynomial. Finally, the developed framework is used to validate the effect of time-delays in our previous work.

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## Figures

Fig. 1

Initial distribution of an MAS evolving in the X–Y motion plane

Fig. 2

(a) X components of the leaders’ positions, (b) Y components of the leaders’ positions, (c) entries of Q, and (d) entries of D

Fig. 3

Parameters Pi,1, Pi,2, and Pi,3

Fig. 4

(a) X components of r8,HT(t) and r8t and (b) Y components of r8,HT(t) and r8(t)

Fig. 5

(a) X components of r8,HT(t); X component of r8 for different communication delays h=0.1,0.25,0.50,1.00; (b) Y components of r8,HT(t); Y component of r8 for different communication delays h=0.1,0.25,0.50,1.00

Fig. 6

X-coordinate of position of follower agent eight for two different values for time-delay h8=0.19s and h8=0.23s, when β=0

Fig. 7

Y-coordinate of position of follower agent eight for two different values for time-delay h8=0.19s and h8=0.23s, when β=0

Fig. 8

X-coordinate of position of follower agent eight for two different values for time-delay h8=0.21s and h8=0.25s when β=1

Fig. 9

Y-coordinate of position of follower agent eight for two different values for time-delay h8=0.21s and h8=0.25s, when β=1

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