IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 4, APRIL 2007 1765

Fluids in Electrostatic Fields: An Analogyfor Multirobot Control

Luciano C. A. Pimenta, Miguel L. Mendes, Renato C. Mesquita, and Guilherme A. S. Pereira

Departamento de Engenharia Elétrica, Universidade Federal de Minas Gerais, Belo Horizonte 31270-901, Brazil

This paper addresses the problem of controlling a large group of robots in a 2-D pattern generation task. Different from previousmethodologies, our approach can be used in generic static environments, where obstacles may appear. This approach is based on theanalogy with the simulation of fluids in electrostatic fields. By means of a weak coupling between the smoothed particle hydrodynamicsand the finite element method we derive a scalable solution where decentralized controllers are provided.

Index Terms—Electric fields, hydrodynamics, mobile robots.

I. INTRODUCTION

ARELEVANT problem in robotics is the formation of geo-metric patterns with multiple robots [1]. Several tasks such

as surveillance and cordoning off hazardous areas require a scal-able solution for this problem. In our context, a scalable solu-tion means a methodology which is efficient when large groups(about hundreds) of robots are considered. To obtain such so-lutions we must derive decentralized approaches, in the sensethat all decisions taken by each robot are based only on local in-formation. Besides, it is desirable that this approach considersall the robots as anonymous entities, i.e., a robot interacts withothers without the need of knowing their identities.

In [1], a scalable solution for the problem of 2-D pattern gen-eration is proposed. This approach uses implicit functions tomodel the geometric patterns. The idea is to build functions byinterpolating from points placed at the pattern in such a way thepattern characterizes a region of minimum of these functions.Therefore, by applying control laws based on the descend gra-dient of those functions the patterns are formed by the robots.Actually, artificial repulsive forces among the robots are alsoadded to the control laws to avoid inter-robot collisions. A lim-itation of this strategy is the fact that the environment where therobots navigate must be free of obstacles.

In this paper, we propose a scalable solution which is ableto handle static previously modeled obstacles. Similarly to [2],that uses the kinetic theory of gases, we control large groups ofrobots by modeling them as fluids. However, differently fromprevious methodologies, we model a group as an incompress-ible fluid immersed in a region where an electrostatic field isdefined. In our approach, the smoothed particle hydrodynamics(SPH) method is applied to model a “robotic fluid,” more specif-ically, to model the interactions among the robots of the group.This method has been extensively used to solve problems in thefield of fluid dynamics, mainly because of its ability to easilyhandle moving boundaries and large deformations. The finiteelement method (FEM) is also used in this work to compute the

Digital Object Identifier 10.1109/TMAG.2007.892514

electrostatic fields that determine external forces to the fluid. Infact, these fields are responsible for driving the robots towardsthe geometric pattern and also for treating the environmentalobstacles in a proper manner. Since those obstacles may havegeneric geometries the use of FEM allows for efficiency in thefield computation. By means of a weak coupling between FEMand SPH we derive decentralized controllers for the robots. Inthe next section, we present the formal problem definition andthe assumptions and simplifications we adopted.

II. PROBLEM DEFINITION

Given robots and their initial spatial distribution, the mapof a static environment defining a compact domainwhere the robots are placed, and a curve , where

. We must control the robots, without colliding with theobstacles and each other, to approximate the pattern describedby .

To simplify the problem we make some assumptions. First,for obvious reasons, we assume the number of robots is suffi-ciently large to properly approximate the desired pattern, and atthe same time it is sufficiently small to guarantee enough spacealong the curve for all the robots. Second, we consider the robotsare point robots with fully actuation in their acceleration

(1)

where is the robot velocity and is a control input. We mustclarify that this second simplification is not a limitation of ourapproach. A real robot can be treated as a point if we build itsconfiguration space [3]. In this space, the obstacles are grownby the size of the robot and by controlling only one fixed pointplaced at the robot we can guarantee a collision free navigation.Also, it is possible to design controllers for real robots that allowthe use of (1) under certain acceleration limits.

Finally, we assume the robots are able to estimate positionsand velocities of themselves and of other robots of the grouplocated at distances smaller than . This is not a strong as-sumption, since most of real robots have sensors and commu-nication equipments which can be used to obtain the requiredinformation.

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1766 IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 4, APRIL 2007

III. SMOOTHED PARTICLE HYDRODYNAMICS

SPH was introduced in [4] and [5] to solve problems in as-trophysics. It is a mesh-free, lagrangian, and particle numer-ical method. SPH equations are derived from the continuumgoverning equations by interpolating from a set of disorderedparticles. In fluid dynamics problems, each particle representsa small volume of the fluid and the interpolation is performedby using differentiable interpolation kernels, , which approx-imate a delta function. The continuum equations are convertedto a set of ordinary differential equations, where each one con-trols the evolution of an attribute of a specific particle. The SPHequations of conservation of mass, momentum, and energy, fora particle are [6]

(2)

where is density, is velocity, is energy, is pressure,is the position vector of particle is the gradient taken withrespect to the coordinates of particle is the sum of externalforces normalized by the mass , and .

If we use a kernel with compact support the smoothing length,, defines where the kernel is nonzero. This fact improves the ef-

ficiency, since it is no longer necessary to compute the sums overall particles but only over the “neighbors” of . In this paper,we use cubic splines smoothing functions, as recommended by[7], for two dimensions

(3)

where and is the euclidean norm. One shouldnotice that this function is zero for points located at distancessuperior than from the center. This is the compact supportcharacteristic, which allows for efficiency as we said before.

The term in (2) is an artificial viscosity added to handleshocks and there are several variants for it. The most used oneis [7]

(4)

where

(5)

In (4), is the average between the densities of particlesand and are viscosity constants, is the average soundspeed, and . The term is only used to avoidsingularities and it should be small enough to avoid severesmoothing of the viscosity term [7].

The sound speed for a particle is given by [8]

(6)

where

(7)

is the maximum fluid depth, is the gravitational constant,and is a constant related to specific heats and it is usually equalto 7.

Besides differential equations, we also use a suitable stateequation for modeling an incompressible fluid like water [9]

(8)

where is the initial density. In the simulations presented inthis paper, we used 1000 kg/m , which is the same as water.

IV. ELECTROSTATIC ANALOGY

Our approach is composed by two steps. In the first step, wesolve an analogous electrostatic problem. This is similar to [10]where this analogous problem could be used to compute col-lision free trajectories for a robot. The trajectories were deter-mined by the analogous electric field since this field was usedas the robot velocity. The second step consists of controllingthe robots by using control laws based on the analogous elec-tric field weakly coupled to SPH equations. In this sense, weconsider that our approach is analogous to simulating fluids inelectrostatic fields. In this section, we describe the first step.

Our electrostatic analogous problem considers a homoge-neous isotropic medium in the absence of density of charge.Therefore, we model this problem with Laplace’s equation andwe also use constant Dirichlet boundary conditions

(9)

where is the scalar electric potential, ,and (see Fig. 1) is a safety factor added to guarantee robust-ness, since there is no perfect localization system for the robots.By using the boundary conditions presented in (9) we guaranteeconvergence to the desired pattern. This is due to the fact that theboundary conditions assigned to the pattern ( and ) definea region of global minimum for the electric potential, while theboundary conditions of the external contour and the obsta-cles define the maxima. When we have a closed pattern wealso add a point of maximum, , inside the pattern to guarantee

PIMENTA et al.: FLUIDS IN ELECTROSTATIC FIELDS: AN ANALOGY FOR MULTIROBOT CONTROL 1767

Fig. 1. Analogous problem boundaries.

convergence from the interior of the pattern. Therefore, a robotthat follows the electric field lines defined by reaches thetarget (the region delimited by and ) in a finite time withoutany collision with obstacles. To solve (9) numerically we pro-pose to use FEM as in [10]. This is because the obstacles mayhave generic geometries.

V. CONTROL LAW

The second stage of our methodology consists of applying de-centralized controllers that drive the robots to the region wherethe pattern is located and distribute them inside it. These con-trollers are derived by considering each robot as a SPH par-ticle subjected to an external force proportional to the analogouselectric field. The control law for robot is

(10)

where

(11)

(12)

where , and are positive tuning constants. In (11), theSPH conservation of momentum equation is considered and allthe terms shown are the same as in Section III. The first term in(12) corresponds to the normalized electric field that guaranteesconvergence to the pattern. On the other hand, the second termrepresents a damping which stabilizes the system by filteringhigh velocity derivatives.

According to (10)–(12) our approach considers the wholegroup of robots as an incompressible fluid subjected to externalforces. The first term in (10) represents the interaction forcesamong the fluid particles (or the robots) and the second one the

Fig. 2. Electric field in an environment with a rectangular obstacle.

external forces. As we said before, the external forces drive therobots to the desired pattern. On the other hand, the interactionforces allow the robots to distribute themselves along the de-sired curve. Furthermore, these interaction forces avoid shocksamong robots and keep them together to move as a group.

In Section II, we said that each robot needed to estimate po-sition and velocity of other robots located at distances smallerthan . Now, we can see this is related to the SPH equation(11), which depends on information of position and velocity ofneighbor particles. Therefore, according to Section III, this as-sumption can be expressed as .

VI. SIMULATIONS

In this section, we illustrate our approach with simulationsdeveloped in C++ and OpenGL. Movies are available in: http://www.cpdee.ufmg.br/~lucpim/cefc06.

Fig. 2 presents the first step of our approach for an environ-ment with a rectangular obstacle. The desired pattern is a circle.The arrows in the figure represent the computed electric fieldafter normalization. As it was expected, the field converges tothe pattern and diverges from the obstacle and from the externalboundaries.

A simulation of 81 robots applying our approach in the en-vironment of Fig. 2 is shown in Fig. 3. The robots are initiallydistributed in a 9 9 matrix where the distance between twocells is . When the group comes close to the ob-stacle it splits into two subgroups due to the presence of a saddleline. Later on, the robots regroup, reach the circle, and distributethemselves along the pattern. The split of the group is related tothe symmetry of the problem. Other simulations considering aninitial distribution close to axis presented a different behavior,where all the robots moved together along the left side of theobstacle.

Our methodology is also able to treat more complex geo-metric patterns. Fig. 4 presents 324 robots, initially distributedin an 18 18 matrix, generating a star shape.

Although the approximated curves we presented here were allclosed curves, it can be shown that our approach is not limitedto these.

1768 IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 4, APRIL 2007

Fig. 3. Eighty-one robots generating a circle pattern in the presence of a rect-angular obstacle. (a) Initial distribution. (b) Split. (c) Final distribution.

Fig. 4. Three-hundred-twenty-four robots generating a star shape. (a) Start.(b) End.

VII. CONCLUSION AND FUTURE WORK

We proposed a scalable methodology for solving theproblem of generating 2-D geometric patterns with largegroups of robots. Differently from previous works our approachis able to treat static obstacles efficiently. Our approach isbased on an analogy with fluids in electrostatic fields. Thederived controllers are decentralized in the sense that only localinformation is necessary: the electric field at the location ofthe robot and position and velocity of neighbor robots. A clearadvantage of our controllers is the fact that there is no needof labeling the robots. Therefore, all the robots run the samesoftware and the success in task execution is not dependent ondetermined special members of the group.

A possible future work is the extension of this methodologyto consider dynamic obstacles. Currently, we are trying to do

so by applying analogies with multiphase flows. We also intendto use our approach to address other robotic problems such aslarge objects manipulation and environment exploration.

Besides its efficiency, we emphasize the adaptive nature of theSPH method, since its formulation does not depend on the spa-tial particles distribution. This distribution changes every timestep which allows for large deformations treatment. Further-more, this method has all the advantages of the lagrangian ap-proaches and also of the mesh free methods. Since all particlescarry material properties and move subjected to internal andexternal forces as if they were alive, this method is ideal formoving boundaries problems and free surface flows [6].

In this paper, we established a weak coupling between SPHand FEM, where FEM is used to compute static external forcesto SPH equations. Due to the number of advantages of usingSPH in fluid dynamics problems, we are also interested in es-tablishing a strong coupling between SPH and FEM to solvereal physical problems. We are working on an iterative methodfor solving problems of fluids and charged particles in electro-static fields. Possible applications are: electrophoresis, aerosolparticle deposition, electrostatic precipitation, etc.

ACKNOWLEDGMENT

This work was supported in part by the Fundação de Amparoà Pesquisa do Estado de Minas Gerais (FAPEMIG), Brazil andin part by the Conselho Nacional de Desenvolvimento Científicoe Tecnológico (CNPq), Brazil.

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Manuscript received April 24, 2006 (e-mail: renato@cpdee.ufmg.br).