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Technical Briefs

# Rigidity-Based Stabilization of Multi-Agent Formations

[+] Author and Article Information
Marcio de Queiroz

Department of Mechanical and Industrial Engineering,
Louisiana State University,
Baton Rouge, LA 70803

The single-integrator model treats each agent as a kinematic point, where the state is position and the control input is velocity. The double-integrator model treats each agent as a point mass, where the states are position/velocity and the control input is acceleration. That is, it is a dynamic model, albeit simplified, of the agent motion.

By generic, we mean the affine span of p is all of $R2$ [22].

Although the argument of the rigidity matrix function is commonly given as q, it is clear from Eq. (2) that R is dependent on $q˜$ only, i.e., $=R(q˜)$.

The basis for this conjecture is the result of simulations where the initialization of F(t) closer to Iso(F*) (respectively, Amb(F*)) resulted in its convergence to Iso(F*) (respectively, Amb(F*)). See Sec. 6 for an example.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received January 25, 2013; final manuscript received August 8, 2013; published online August 30, 2013. Assoc. Editor: Ryozo Nagamune.

J. Dyn. Sys., Meas., Control 136(1), 014502 (Aug 30, 2013) (7 pages) Paper No: DS-13-1040; doi: 10.1115/1.4025242 History: Received January 25, 2013; Revised August 08, 2013

## Abstract

This paper is concerned with the decentralized formation control of multi-agent systems moving in the plane using rigid graph theory. Using a double-integrator agent model (as opposed to the simpler, single-integrator model), we propose a new control law to asymptotically stabilize the interagent distance error dynamics. Our approach uses simple backstepping and Lyapunov arguments. The control, which is explicitly dependent on the rigidity matrix of the undirected graph that models the formation, is derived for a class of potential functions. Specific potential functions are then used as a demonstration inclusive of simulation results.

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## Figures

Fig. 1

An infinitesimally rigid framework F and a flip ambiguous realization where vertex 1' was obtained by flipping edges (1, 2) and (1, 4) over edge (2, 4)

Fig. 2

Desired formation for five agents

Fig. 3

Agent trajectories qi(t), i = 1,…, 5

Fig. 4

Distance errors eij(t),i,j ∈ V*

Fig. 5

Control inputs ui(t)i = 1,…, 5

Fig. 6

Agent trajectories converging to a flip formation

## Errata

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