International Journal of Robotics Research

Paper number: 383444

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Unifying Geometric, Probabilistic, andPotential Field Approaches toMulti-robot Deployment

The International Journal ofRobotics Research00(000):1–12©The Author(s) 2010Reprints and permission:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/0278364910383444ijr.sagepub.com

Mac Schwager1, Daniela Rus2, Jean-Jacques Slotine3

AbstractThis paper unifies and extends several different existing strategies for deploying groups of robots in an environment. Acost function is proposed that can be specialized to represent widely different multi-robot deployment tasks. It is shownthat geometric and probabilistic deployment strategies that were previously seen as distinct are in fact related throughthis cost function, and differ only in the value of a single parameter. These strategies are also related to potential field-based controllers through the same cost function, though the relationship is not as simple. Distributed controllers are thenobtained from the gradient of the cost function and are proved to converge to a local minimum of the cost function. Threespecial cases are derived as examples: a Voronoi-based coverage control task, a probabilistic minimum variance task,and a task using artificial potential fields. The performance of the three different controllers are compared in simulation.A result is also proved linking multi-robot deployment to non-convex optimization problems, and multi-robot consensus(i.e. all robots moving to the same point) to convex optimization problems, which implies that multi-robot deployment isinherently more difficult than multi-robot consensus.

Keywords

1. Introduction

One of the fundamental problems of multi-robot controlis how to deploy a group of robots over an environmentto carry out sensing, surveillance, data collection, or dis-tributed servicing tasks. We use the term deployment toencompass any task in which the robots move out over anenvironment to reach some final fixed configuration, forexample, coverage, herding, or formation control. A num-ber of control strategies have been proposed to accomplishdifferent multi-robot deployment tasks in a distributed andefficient way. In this paper we introduce a unifying prin-ciple that ties together many of these strategies. We showthat many of the existing methods can be described asspecial instances of gradient descent on a cost function.We propose a cost function that specializes to give severalcommon algorithms for multi-robot deployment, includingVoronoi-based controllers for sensor coverage, as in Cortéset al. (2004), controllers based on probabilistic models, asin Li and Cassandras (2005), and artificial potential field-based controllers for herding, flocking, and consensus1, asin Howard et al. (2002), Jadbabaie et al. (2003), Gazi andPassino (2004), and Tanner et al. (2007).

Controllers for multi-robot deployment are useful formany applications involving distributed sensing and dis-tributed actuation. For example controllers can be usedto deploy underwater robots evenly over a coral reef tomonitor coral health or to deploy wheeled robots with cam-eras to spread out over a room for surveillance. Groups of

robots can also be deployed to carry out actuation tasks,for example, oil clean-up robots can be deployed over anoil spill, or de-mining robots can be positioned to service amine field. We describe a framework that is relevant to bothsensing and actuation tasks. We argue that Voronoi methodsare best suited to distributed actuation tasks, while a contin-uous approximation to the Voronoi decomposition is moreappropriate for distributed sensing tasks. Furthermore, evenin distributed actuation tasks, using a continuous approxi-mation to the Voronoi cell improves the robustness of thecontroller to numerical integration errors.

The controllers we describe are provably convergent,robust to individual robot failures, and can adapt to envi-ronments that change slowly with respect to the speed of therobots. They require that robots know the geometry of theenvironment and they know their own position in it using,for example, GPS or an indoor localization system. We also

1 GRASP Laboratory, University of Pennsylvania, 3330 Walnut St,Philadelphia, PA 19106, USA2 Computer Science and Artificial Intelligence Laboratory, MIT, 32 VassarStreet, Cambridge, MA 02139, USA3 Nonlinear Systems Laboratory, MIT, 77 Massachusetts Avenue, Cam-bridge, MA 02139, USA

*Corresponding author:GRASP LaboratoryUniversity of Pennsylvania3330 Walnut St, Philadelphia, PA 19106, USAEmail:schwager@seas.upenn.edu

2 The International Journal of Robotics Research 00(000)

discuss how to accommodate the constraints of a commu-nication network topology, but do not analyze this aspectof the problem in detail. In our setting, all the robots haveidentical dynamics and capabilities.

1.1. Related Work

Cortés et al. (2004) introduced a controller for multi-robotsensor coverage that works by continually driving the robotstoward the centroids of their Voronoi cells. This inher-ently geometric strategy has seen many recent extensionsto robots with a limited sensing radius (Cortés et al. 2005),to heterogeneous groups of robots and non-convex environ-ments (Pimenta et al. 2008; Breitenmoser et al. 2010), andto incorporate learning of unknown environments (Schwa-ger et al. 2009b). A recent text that presents much of thiswork in a cohesive fashion is Bullo et al. (2008) and anexcellent overview is given in Martínez et al. (2007). Cov-erage controllers also have been successfully implementedon robotic systems in Schwager et al. (2008, 2009a). In thiswork we adopt notational conventions from the Voronoi-based coverage control literature. Other common methodsfor multi-robot deployment take a probabilistic perspec-tive. For example Li and Cassandras (2005) proposes analgorithm for positioning robots to maximize the probabil-ity of detecting an event that occurs in the environment.Distributed dynamic vehicle routing scenarios are consid-ered in Arsie and Frazzoli (2007) and Smith et al. (2010),in which events occur according to a random process andare serviced by the robot closest to them. Another com-mon method is for robots to drive away from or towardsone another by following the negative gradient of artificialpotential fields. These have been used for sensor cover-age (Howard et al. 2002), flocking and herding (Gazi andPassino 2004; Tanner et al. 2007), and consensus (or ren-dezvous) (Jadbabaie et al. 2003). Despite the rather dif-ferent models and objectives in these works, there are twocommon points which motivate us to find a unifying prin-ciple: (1) they all rely upon an optimization, and (2) theyall use controllers that solve this optimization through theevolution of a dynamical system.

Some existing approaches do not fit under the frame-work we propose in this paper. A significant body of workhas looked at multi-agent deployment as a motion planningproblem. A survey of this work can be found in Choset(2001), and some significant contributions can be found in,for example, Butler et al. (2000) and Latimer et al. (2002)and the citations therein. Other authors have proposed infor-mation theoretic algorithms for sensor networks which con-sider placing static sensors sequentially rather than drivingrobots with sensors using a distributed controller. Workssuch as Hernandez et al. (2004) and Krause and Guestrin(2007) position sensor nodes to maximize information forthe sake of estimating a Gaussian random process in theenvironment.

1.2. Contributions

In the present work we focus on multi-agent deploymentas an optimization problem. This is advantageous because

it is amenable to geometric, probabilistic, and potentialfield interpretations, all of which have been seen in aseparate light in the past. Our optimization approach tiestogether many existing methods that were previously seenas unrelated. Specifically, our contributions are:

1. We propose a cost function, putting particular emphasison the role of a mixing function, a previously unrec-ognized component that captures critical assumptionsabout the deployment task.

2. We introduce a family of mixing functions with a freeparameter, α, and show that different values of theparameter correspond to different assumptions aboutthe deployment task, specifically showing that a min-imum variance solution (i.e. a probabilistic strategy)is obtained with a parameter value of α = −1, andVoronoi coverage (a geometric strategy) is recovered inthe limit α → −∞. A broad family of potential field-based herding and consensus controllers are recoveredwhen α = 1, and by specializing two other componentsof the cost function.

3. We prove a new result linking the convexity of a costfunction to the multi-agent phenomenon of consensus.We show that deployment tasks are fundamentally dif-ferent from consensus, and that they require the opti-mization of a non-convex cost function. This suggeststhat gradient descent controller designs, which are per-vasive in the literature, can only be proved to convergeto a local minimum of the cost function.

The paper is organized as follows. In Section 2 we intro-duce the cost function, describing the purpose of each ofits parts including the mixing function. We then producea class of provably stable distributed coverage controllersby taking the gradient of the cost function. In Section 3we derive three special cases of the controller: a Voronoicontroller, a minimum variance controller, and a potentialfield controller. Section 4 presents our results on the relationbetween the convexity of a cost function, and multi-robotconsensus. Simulation results are given in Section 5 andconclusions are in Section 6.

2. Generalized Deployment

In this section we introduce a general multi-robot cost func-tion. We will use this cost function to define a new classof multi-robot controllers by introducing a mixing func-tion, which describes how information from different robotsshould be combined. We use the cost function to derive astable gradient descent controller.

2.1. Cost Function

Let there be n robots2, and let robot i have a state pi ∈ P ⊂R

dp , where P is the state space of a robot, and dp is thedimension of the space. The vector of all robot states isdenoted P = [pT

1 , . . . , pTn ]T ∈ Pn, and we will call P the

configuration of the robots. We want our robots to deployover a bounded region Q ⊂ R

dq , which may or may not be

Schwager et al. 3

the same as the state space P of the robots. For example, therobots may be constrained to move in the space over whichthey are deployed, so that P = Q as in Cortés et al. (2004),or the robots may hover over a planar region that they coverwith cameras, so P ⊂ R

3 and Q ⊂ R2, as in Schwager et al.

(2009a).For each robot, a cost of sensing, or servicing, a point

q ∈ Q is given by a function f ( pi, q). For simplicity ofanalysis we assume that f ( pi, q) takes on only non-negativevalues, and that, with respect to pi, it isdifferentiable3. Thesensor measurements of the n robots are combined in afunction g( f ( p1, q) , . . . , f ( pn, q) ), which we will call themixing function. The mixing function embodies assump-tions about the task, that is, by changing the mixing functionwe can derive Voronoi-based coverage control, probabilisticcoverage control, and a variety of other kinds of distributedcontrollers.

Combining these elements, we propose to use a costfunction of the form

H( P) =∫

Qg( f ( p1, q) , . . . , f ( pn, q) ) φ( q) dq, (1)

where φ : Rdq �→ R>0 is a weighting of importance

over the region Q (we use the notation R>0 to mean theset of positive real numbers and R

d>0 the set of vectors

whose components are all positive, and likewise for R≥0

and Rd≥0). Intuitively, the cost of the group of robots sensing

at a single arbitrary point q is represented by the integrandg( f ( p1, q) , . . . , f ( pn, q) ). Integrating over all points in Q,weighted by their importance φ( q) gives the total cost of aconfiguration of the robots. We want to find controllers thatstabilize the robots around configurations P∗ that minimizeH. We will see in Section 4 that for coverage, and manyother multi-agent problems, H is necessarily non-convex,therefore gradient-based controllers will yield locally opti-mal robot configurations. The cost function (1) will beshown to subsume several different kinds of existing multi-robot deployment cost functions. Drawing out the relationsbetween these different deployment algorithms will suggestnew insights into when one algorithm should be preferredover another.

Although this cost function is general enough to encom-pass a broad range of deployment tasks, there are someconstraints inherent in the way we have formulated H. Weimplicitly assume that the robots have identical capabili-ties, since f ( pi, q) is the same function for all robots. Also,since f ( pi, q) does not depend on the positions of the otherrobots, it cannot capture the effect of one robot obscuringor interfering with another.

2.2. Mixing Function

The mixing function gα : Rn≥0 �→ R describes how infor-

mation from different robots should be combined to givean aggregate cost of the robots sensing at a point q. This isshown graphically in Figure 1 where the overlap of the two

Fig. 1. The mixing function is illustrated in this figure. Themixing function determines how information from the sensorsof multiple robots is to be combined, shown graphically as theintersection of the two circles in the figure.

sensors is shown for illustrative purposes as the intersectionof two circles. We propose a mixing function of the form

gα( f1, . . . , fn) =( n∑

i=1

f αi

)1/α

, (2)

with a free parameter α ∈ R. The arguments fi ≥ 0 are realvalued, and in our context they are given by evaluating thesensor function f ( pi, q), hence the notation fi. To be precise,when α < 0 the expression in (2) is undefined if fj = 0for some j, therefore in this case we define gα by its limit,gα( f1, . . . , 0, . . . , fn) = limfj→0

(∑ni=1 f α

i

)1/α = 0.This mixing function has several important properties.

Firstly, notice that for α ≥ 1 it is the p-norm of the vec-tor [f1 · · · fn]T. Specifically, it is convex for α ≥ 1 and asα → ∞, gα( ·) → maxi( ·), which is the �∞ norm. How-ever, we are interested in the regime where α < 1. In thiscase gα( ·) is not a norm because it violates the triangleinequality. In this regime it is also non-convex, leading toa non-convex cost function, which is a necessary attributeof deployment problems, as we will prove in Section 4. Onecan readily verify4 that as α → −∞, g( ·) → mini( ·). Froman intuitive point of view, with α < 1, gα( ·) is smaller thanany of its arguments alone. That is, the cost of sensing ata point q with robots at pi and pj is smaller than the costof sensing with either one of the robots individually. Fur-thermore, the decrease in gα from the addition of a secondrobot is greater than that from the addition of a third robot,and so on. There is a successively smaller benefit to addingmore robots. This property is often called supermodularity,and has been exploited in a rather different way in Krauseand Guestrin (2007). Surface plots of gα( f1, f2) for α = −1,1, and 2 are shown in Figures 2(a), 2(b), and 2(c), respec-tively, and the decrease in gα( ·) as the number of argumentsgrows is shown in Figure 2(d). In this paper we considerthe number of robots to be fixed, but it is useful to illus-trate the supermodularity property of the mixing functionby considering the successive addition of new robots.

4 The International Journal of Robotics Research 00(000)

Fig. 2. The proposed mixing function with α = −1, 1, and 2 is shown in 2(a), 2(b), and 2(c), respectively. The function is convex forα ≥ 1 and non-convex otherwise. The non-linear decrease in the function as more sensors are added, a property known assupermodularity, is shown in Figure 2(d).

Including this mixing function in the cost function from(1) gives

Hα =∫

Q

( n∑i=1

f ( pi, q)α)1/α

φ( q) dq. (3)

To model scenarios with a finite sensor footprint, we canalso let f ( pi, q) be infinite in some areas, in which caseto keep the cost function bounded and differentiable itbecomes necessary to include a prior w in the mixing func-tion, yielding the variation gα( f1, . . . , fn) = ( ∑n

i=1 f αi +

wα)1/α

. An application of this case was explored in Schwa-ger et al. (2009a) to design a controller for positioningmultiple flying robots with downward facing cameras.

2.3. Gradient Control

In order to derive a gradient descent controller, we take thederivative of the cost function Hα with respect to the stateof robot i to get

∂Hα

∂pi=

∫Q

(f ( pi, q)

gα

)α−1∂f ( pi, q)

∂piφ( q) dq.

To provide some intuition about the meaning of this func-tion, notice that in the case that f ( pi, q) is strictly increasing,the function inside the integral ( f ( pi, q) /gα)α−1 gives anapproximation to the indicator function5 of the Voronoi cellof agent i, the approximation improving as α → −∞. Thisis shown graphically in Figure 3. It can be readily verifiedthat this function is continuous, and that at f ( pi, q) = 0 ittakes the value 1, and at f ( pj, q) = 0 and j = i it takes thevalue 0.

For simplicity, we choose the function f ( pi, q) to be

f ( pi, q) = 1

2‖q − pi‖2, so that

∂f ( pi, q)

∂pi= −( q − pi) .

Other choices of f ( pi, q) were investigated in Cortés et al.(2005) and could be used here as well, including functionswith discrete jumps that model a finite senor footprint. Thisfunction represents the cost of a single robot i sensing atthe position q. Therefore the quadratic form is appropriatefor light-based sensors, such as cameras or laser scanners.Light intensity drops off as the inverse square of the dis-tance from the source, so it is reasonable for the cost to beproportional to the square of the distance. For tasks in which

Schwager et al. 5

Fig. 3. Contour plots of ( f ( pi, q) /gα)α−1 are shown for a configuration of 10 agent positions. The Voronoi tessellation is also shownfor comparison. As the parameter α approaches −∞, ( f ( pi, q) /gα)α−1 becomes closer to the indicator function of the Voronoi cell Vi.

robots have to drive to a point q for servicing, and we wantthe cost to be proportional to the distance traveled, it wouldbe more appropriate to use f ( pi, q) = ‖q−pi‖, for example.

We propose to use a gradient-based controller

pi = −k∂Hα

∂pi= k

∫Q

(f ( pi, q)

gα

)α−1

( q − pi) φ( q) dq,

(4)

where k > 0 is a positive control gain. We assume that therobots have integrator dynamics, pi = ui, so we can con-trol their velocity directly. We have found experimentally,in Schwager et al. (2008) for ground vehicles and Schwa-ger et al. (2009a) for quadrotor air vehicles, that this is a fairassumption as long as a fast inner control loop is in place totrack the desired pi.

We can equivalently express the n coupled equations in(4) as a single equation using the configuration vector P as

P = −kdHα

dP.

Our multi-robot system is therefore a gradient system,meaning the right-hand side of the governing differentialequation is proportional to the negative gradient of thescalar-valued cost function Hα . Gradient systems have par-ticularly simple and powerful convergence and stabilityproperties, the most important of which will be given here.

Theorem 1 (Global Convergence). Let � = {P∗ |dHα/dP |P∗= 0} be the set of all critical points of Hα .All trajectories of the system P = −kdHα/dP convergeasymptotically to �.

Proof: The theorem follows as a corollary to LaSalle’sInvariance Principle (LaSalle 1960; Slotine and Li 1991).Let Hα be the Lyapunov function candidate. Then Hα =−k‖dHα/dP‖2 ≤ 0, and since Hα is radially unbounded,the trajectories of the system are bounded, therefore byLaSalle’s Invariance Principle all trajectories converge tothe largest invariant set contained in �. By the definitionof the dynamics, � itself is an invariant set, therefore alltrajectories converge to �.

Remark 1. This result does not necessarily imply that thetrajectories converge to a single point in �. However, thisis true if � is a set of isolated points. Furthermore, if thesystem ever reaches a point P∗ ∈ �, it will stay at thatpoint for all time, whether or not it is an isolated criticalpoint, since P = 0 ∀t ≥ 0 at such a point.

The following useful result pertains to the local stabilityof critical points of Hα .

Theorem 2 (Isolated Minima are Locally Stable). Let P∗ bea critical point of Hα . Then P∗ is a locally asymptoticallystable equilibrium of the gradient system P = −kdHα/dPif and only if P∗ is an isolated minimum of Hα .

6 The International Journal of Robotics Research 00(000)

Proof: Please see Hirsch and Smale (1974) Chapter 9,Section 4, corollary to Theorem 1.

Remark 2. Theorem 1 is concerned with all critical pointsof Hα—maxima, minima, and saddle points. However, it isintuitively clear that the system ought to prefer minima. Thisintuition is made precise in Theorem 2. There are initialconditions for which the system will converge to a saddlepoint or a maximum, but these critical points are not locallystable. That is, a perturbation will cause the system to leavethe critical point. Minima, on the other hand, are locallystable. They are robust to perturbations.

Remark 3 (Network Requirements). The computation ofthe controller requires that robot i knows the states of allthe robots in the network. For this to be feasible there musteither be a global supervisor or a fully connected networkcommunication topology. It would be more useful if the con-troller depended only upon the states of robots with which itcommunicates. We suggest two methods to accomplish this,but we do not analyze them in detail in this paper. First,robot i can approximate its control law simply by comput-ing (4) using only the states of the robots with which it is incommunication. We expect this to give a good approxima-tion because the function ( f ( pj, q) /gα)α−1 depends weaklyupon the states of agents that are not Voronoi neighbors,especially for small values of α, as evident from Figure 3.However, a rigorous stability analysis of this approximationscheme is difficult. A second option is for a robot i to usean estimated configuration vector, P, in its calculation ofthe control law. The estimated configuration can be updatedonline using a standard distributed consensus algorithm (aso called “consensus estimator”). We expect that such ascheme may be amenable to a rigorous stability proof asits architecture is similar to adaptive control architectures.The investigation of these matters is left for future work.

Remark 4 (Unknown Environments). The controllerrequires prior information about the environment such asthe importance weighting φ( q), the sensor function f ( pi, q),and the geometry of the environment Q. One may naturallywonder what can be done if any of this information is lack-ing. In the case of an unknown weighting φ( q), a stablecontroller can be formulated to approximate φ( q) on-linewhile carrying out the deployment task, as described inSchwager et al. (2009b). We are currently looking to extendthis method to unknown sensor functions f ( pi, q) and tounknown environment geometries Q. It would also be par-ticularly useful to extend to the case of environments withunknown obstacles.

3. Deriving Special Cases

In this section we show how the cost function (1) canbe specialized to give three common kinds of deploymentcontrollers: a Voronoi controller, which is geometric innature; a minimum variance controller, which has a prob-abilistic interpretation; and a potential field controller. Weconjecture that other deployment objectives beyond these

three can be achieved with different choices of the mixingfunction parameter α.

3.1. Voronoi Coverage, α → −∞The Voronoi-based coverage controller described in Cortéset al. (2004) is based on a gradient descent of the costfunction

HV =n∑

i=1

∫Vi

1

2‖q − pi‖2φ( q) dq,

where Vi = {q ∈ Q | ‖q − pi‖ ≤ ‖q − pj‖, ∀j = i} isthe Voronoi cell of robot i and the use of the subscript V isto distinguish it from H and Hα . The Voronoi partition canequivalently be written using the min function as

HV =∫

Qmin

i

(1

2‖q − pi‖2

)φ( q) dq,

because a point q is in the Voronoi cell Vi if and only if‖q − pj‖ is minimized for j = i. As noted in Section2.2, limα→−∞ gα( f1, . . . , fn) = mini fi. Therefore HV is aspecial instance of (3) with the mixing function g−∞ =limα→−∞ gα and f ( pi, q) = 1/2‖q − pi‖2.

The choice of the min function for a mixing function nowwarrants some reflection. Consider a distributed actuationscenario in which we want to position robots so as to servicean event that occurs randomly at some point in the environ-ment q. Suppose any robot is equally capable of renderingthe service, robots have to physically travel to the event torender the service, and our objective is to service an event asquickly as possible. Naturally, an event should be servicedby the robot that is closest to it, as it will reach the event thequickest. In this case, the min function is the appropriatechoice for a mixing function. By using the min function weare saying that the cost incurred by all the robots due to theevent at q is the same as that incurred by the robot that isclosest to q.

On the other hand, consider a sensing task in which anevent of interest occurs randomly at a point q and is sensedat a distance by sensors located on the robots. In this casethe use of the min function is more difficult to justify. Usingthe min function in this instance would imply that eventhough both pi and pj have some sensory information aboutthe event, the cost function only counts the informationfrom the one that is closest to q. This seems to be a poorchoice of cost function for sensing, since in such cases wewould want to capture the intuition that two sensors are bet-ter than one. The mixing function (2) captures this intuition.Furthermore, even in distributed actuation tasks, using acontinuous approximation to the Voronoi cell improves therobustness of the controller. The discrete, geometric natureof the Voronoi computation combined with the continuouscontroller can lead to chattering, and small sensing errorscan result in large changes in the control input. Fortunately,the Voronoi tessellation can be approximated arbitrarilywell by choosing a small value of α, thereby preserving theVoronoi controller behavior while improving robustness.

Schwager et al. 7

3.2. Minimum Variance Deployment, α = −1

We show in this section that setting the mixing functionparameter to α = −1 causes the robots to minimize theexpected variance of their measurement of the location of atarget of interest. As a side effect, we will formulate an opti-mal Bayesian estimator for the location of the target giventhe measurements of the agents.

Suppose our agents are equipped with sensors that give anoisy measurement of the position of a target in the environ-ment. Let the target position be given by a random variableq that takes on values in Q, and agent i gives a measurementyi = q + w, where w ∼ N( 0, I2

√f ( pi, q)) is a bi-variate

normally distributed random variable, and where I2 is the2 × 2 identity matrix. The variance of the measurement,f ( pi, q), is a function of the position of the sensor and thetarget. Intuitively one would expect a sensor to localize atarget with more precision the closer the target is to the sen-sor. Then the measurement likelihood of agent i is P( yi |q : pi) = 1/( 2π f ( pi, q) ) exp{−‖yi − q‖2/( 2f ( pi, q) ) }, andthe notation P( · : pi) is to emphasize that the distributionis a function of the agent position. Assume the measure-ments of different agents conditioned on the target positionare independent. Also, let φ( q) be the prior distribution ofthe target’s position. Then Bayes rule gives the posteriordistribution,

P( q | y1, . . . , yn) =∏n

i=1 P( yi | q : pi) φ( q)∫Q

∏ni=1 P( yi | q : pi) φ( q) dq

. (5)

One can use the posterior to obtain a Bayesian estimate ofthe position of the event q given the measurements. Forexample, one may choose to estimate q using the mean, themedian, or the maximum of the posterior in (5).

Our interest here, however, is not in estimating q. Insteadwe are interested in positioning the robots so that whateverestimate of q is obtained is the best possible one. To thisend, we seek to position the robots to minimize the vari-ance of their combined sensor measurements. The productof measurement likelihoods in the numerator of (5) can besimplified to a single likelihood function, which takes theform of an un-normalized Gaussian

n∏i=1

P( yi | q : pi) = A exp

{−‖y − q‖2

2g−1( ·)}

,

whose variance is equivalent to our mixing functiong−1( ·) = (∑n

i=1 f ( pi, q)−1)−1

. The values of A and y arenot important in this context, though we state them forcompleteness:

y = g−1( ·)n∑

i=1

f ( pi, q)−1 yi, and

A = 1

( 2π )n∏n

i=1 f ( pi, q)

exp

{1

2‖y‖2g−1( ·) −1

2

n∑i=1

‖yi‖2f ( pi, q)

}.

If we want to position the robots so as to obtain themost decisive information from their sensors, we should

move them to minimize this variance. Notice, however, thatg−1( f ( p1, q) , . . . , f ( pn, q) ) is a random variable since itis a function of q. Taking the expectation over q of thelikelihood variance gives our original cost function,

H−1 = Eq[g−1( f ( p1, q) , . . . , f ( pn, q) ) ]

=∫

Qg−1( f ( p1, q) , . . . , f ( pn, q) ) φ( q) dq. (6)

Thus we can interpret the coverage control optimizationas finding the agent positions that minimize the expectedvariance of the likelihood function for an optimal Bayesestimator of the position of the target.

A more theoretically appealing criterion would be toposition the agents to minimize the variance of the poste-rior distribution in (5). This gives the considerably morecomplicated cost function

Var[q | y1, . . . , yn]

=∫

Q

∏ni=1 P( yi | q : pi) φ( q) qqT dq∫

Q

∏ni=1 P( yi | q : pi) φ( q) dq

− qqT, (7)

where

q = E[q | y1, . . . , yn] =∫

Q

∏ni=1 P( yi | q : pi) φ( q) q dq∫

Q

∏ni=1 P( yi | q : pi) φ( q) dq

.

The complication of this cost function and the fact that gra-dients can not be easily computed makes it a less practicaloption.

3.3. Potential Field Herding,φ( q) = ∑n

i=1 δ( ‖q − pi‖)

The third type of deployment controller we consider is sig-nificantly different from the previous two in that it does notinvolve an integral over the environment. Instead it relieson the idea that robots should push away from one anotherto spread out over an environment, but should not movetoo far from one another or else they will become discon-nected. Surprisingly, however, we will show that this ratherdifferent deployment philosophy can be reconciled with ourgeneralized cost function H in (1).

Let the importance function, φ( q), be given as a sum ofdelta-Dirac functions centered at each of the robot positions

φ( q) =n∑

i=1

δ( ‖q − pi‖) .

Substituting this for φ( q) in (1), the integral in H can thenbe evaluated analytically to give

Hpot =n∑

i=1

g( f ( p1, pi) , . . . , f ( pn, pi) ) ,

and setting g( f ( p1, pi) , . . . , f ( pn, pi) ) = ∑nj=1,j =i f ( pj, pi)

gives a cost function for potential field-based herding.

Hpot =n∑

i=1

n∑j=1,j =1

f ( pj, pi) , (8)

8 The International Journal of Robotics Research 00(000)

where f ( pj, pi) can be interpreted as an inter-agent potentialfunction. One choice for f ( pj, pi) is

f ( pj, pi) = 1

6‖pj − pi‖−2 − ‖pj − pi‖−1 (9)

which, taking the gradient of (8), yields the controller

pi = kn∑

j=1,j =i

(‖pj − pi‖−2 − 1

3‖pj − pi‖−3

)pj − pi

‖pj − pi‖ .

(10)

Controllers similar to this one have been studied in a num-ber of works, for example, Howard et al. (2002), Jadbabaieet al. (2003), Gazi and Passino (2004), Tanner et al. (2007),and Dimarogonas and Kyriakopoulos (2008). There arenumerous variations on this simple theme in the literature.

3.4. Computational Complexity

The gradient controllers described in this work mustinevitably be discretized and implemented in a discrete-timecontrol loop. We show here that the computational com-plexity of one loop of the controller (when computed in adistributed fashion over the robot network) for all valuesof α is O( dqnm), where n is the number of robots, m isthe number of grid squares in the integral computation, anddq is the dimension of the space. For the herding case, thisbecomes O( dqn2) since the integral simplifies to a sum overthe robots.

For the Voronoi controller, we reason as follows. The typ-ical decentralized algorithm for a single robot to computeits Voronoi cell (Cortés et al. 2004) runs in O( dqn) time.The time complexity for computing a discretized integral islinear in the number of grid squares, and at each grid squarerequires a check if the center point is in the Voronoi cell,which is an O( dqn) operation. Therefore the time complex-ity of the integral is in O( dqnm). The Voronoi cell must becomputed first, but the discretized integral dominates giv-ing an overall time complexity of O( dqnm) at each step ofthe control loop.

For other values of α in (4) (including α = −1, whichgives the minimum variance controller described above) thecontroller does not require the computation of a Voronoicell, but it does require the discretized spatial integral overthe environment. We do not have to check if a point is ina polygon, but the integrand we evaluate, namely gα is lin-ear in n. Therefore the integral computation still has timecomplexity O( dqnm), which is the time complexity of thecontroller at each step of the control loop. The controllerwe propose in this paper is therefore significantly simpler inimplementation (since it does not require the Voronoi com-putation), though it has the same computational complexity.It will be shown in Section 5 that the use of the continu-ous function rather than the Voronoi cell provides greaterrobustness to errors introduced by discretized integration.Finally, for the herding case, the integral is simply a sumover the robots, therefore, m = n and we obtain an orderO( dqn2) computation at each times step.

Fig. 4. This schematic shows the geometrical intuition behind theproof of Theorem 4 in a simplified 2D setting. Corollary 1 isproved by noticing that the set of minima is a single point (theconsensus solution) if H is strictly convex.

4. Convexity and Consensus

Since we treat the multi-agent coverage problem as an opti-mization, it is natural to ask what sort of optimization we aredealing with, and what optimization tools can be broughtto bear to solve it. We show in this section that the costfunction in (3) is non-convex, and that non-convexity is arequired feature of a large class of multi-agent problems,however undesirable this may be from an optimization per-spective. Specifically, we demonstrate a link between theconvexity of a cost function and the multi-agent phenomenaknown as consensus. For our purposes, consensus describesa multi-agent configuration in which all agents take on thesame state, p1 = p2 = · · · = pn. Consensus is geometricallyrepresented in the state space Pn as a dp-dimensional hyper-plane that passes through the origin (from the dp( n − 1)independent equality constraints). This is illustrated by thediagonal line in Figure 4 in a simplified 2D setting. We willprove, with some technical assumptions, that a multi-agentproblem with a convex cost function admits at least oneglobally optimal consensus solution.

We begin with some basic definitions and facts from con-vex optimization which can be found in any standard texton the topic, for example, Bertsekas et al. (2003). A set� ⊂ R

n is called convex if, for any two points in �, allpoints along the line segment joining them are also in �.Formally,

αx + ( 1 − α) y ∈ � ∀x, y ∈ � and ∀α ∈ [0, 1].

An important consequence of the convexity of � is that anyconvex combination of points in � is also in �. A convexcombination of m points xi ∈ � is one of the form

x =m∑

i=1

αixi wherem∑

i=1

αi = 1 and αi ≥ 0 ∀i.

A function f : � �→ R is called convex if

f ( αx + ( 1 − α) y) ≤ αf ( x) + ( 1 − α) f ( y)

∀x, y ∈ � and ∀α ∈ [0, 1].

Schwager et al. 9

This is equivalent to saying that the set of all points lying onor above the function f ( x) is a convex set (this set is knownas the epigraph of f ( x)). A function is called strictly convexif the “≤” can be replaced with a “<” in the above relation.Also, we will use the word minimum to mean minimum orinfimum if no minimum exists. We now state a theorem thatfollows from Weierstrass’ Theorem and some well-knownproperties of convex functions.

Theorem 3 (Minima of Convex Functions). For a convexfunction f : � �→ R, where the domain � ⊂ R

n is convex,if any of the following are true:

1. � is bounded.2. There exists a scalar γ such that the level set {x ∈ � |

f ( x) ≤ γ } is non-empty and bounded.3. f is such that lim‖x‖→∞ f ( x) = ∞.

Then the set of global minima of f is non-empty and convex.

We will apply this result to our multi-agent scenario.Consider a continuous multi-agent cost function H : Pn �→R. As before, an agent i has a state pi ∈ P ⊂ R

dp . It will be

more convenient in this section to refer to a configuration ofagents as a tuple ( p1, . . . , pn) ∈ Pn, rather than the columnvector notation used previously. Let us assume that agentsare anonymous with respect to the cost function, by whichwe mean that the positions of any two agents can be inter-changed without affecting the value of the cost function.This is formalized by the following assumption.

Assumption 1 (Anonymity of Agents). The cost functionH is such that

H( . . . , pi, . . . , pj, . . . ) = H( . . . , pj, . . . , pi, . . . )

∀i, j ∈ {1, . . . , n}.Assumption 1 is in keeping with the ethos of complex,

multi-agent systems, where the emphasis is on the globalpatterns that result from the interactions of many identicalagents. Furthermore, let us assume that H and Pn satisfy atleast one of the three properties in Theorem 3. Now we givethe main result of this section.

Theorem 4 (Convexity and Consensus). Under Assump-tion 1, if the cost function H( p1, . . . , pn) is convex, Pn isconvex, and one of the conditions in Theorem 3 is satisfied,then H( p1, . . . , pn) has a global minimum such that pi = pj

∀i, j ∈ {1, . . . , n}.

Proof: Our argument rests upon Assumption 1 and the factfrom Theorem 3 that the set of minima of a convex func-tion H is a convex set. Let h∗ be the set of minima, andlet ( . . . , p∗

i , . . . , p∗j , . . . ) be an optimal solution in that set.

By Assumption 1, ( . . . , p∗j , . . . , p∗

i , . . . ) is also an optimalsolution for any i and j. Therefore all permutations of com-ponents in ( p∗

1, . . . , p∗n) are optima. Then by convexity of

h∗, all convex combinations of points in h∗ are in h∗. Inparticular, the point ( p, . . . , p), where p = 1/n

∑ni=1 pi is

an optimal solution (since it is a convex combination ofpermutations of ( p1, . . . , pn)).

We show a geometric schematic of the proof argumentin Figure 4. The proof uses the fact that the convex set ofminima must intersect the consensus hyperplane (the hyper-plane where pi = pj ∀i, j) at at least one point. A simplecorollary follows.

Corollary 1 (Strict Convexity). If the conditions of Theo-rem 4 are met and the cost function H( p1, . . . , pn) is strictlyconvex, then the minimum is unique and is such that pi = pj

∀i, j ∈ {1, . . . , n}.

Proof: A strictly convex function has at most one minimumover a convex domain.

Remark 5 (Consensus versus Non-consensus). Theorem4 suggests that it is futile to search for convex cost functionsfor multi-robot deployment problems other than consensus.It delineates two classes of multi-agent behaviors reminis-cent of complexity classes in the theory of computation.One class, which we will call consensus behaviors, can bedescribed as optimizing a convex cost function. The otherclass, which we will call non-consensus behaviors, is funda-mentally different in that it can only be described with non-convex cost functions. This is important because if we wishto design an optimization to solve a multi-agent problem,and we know that the problem cannot be solved satisfacto-rily by all the agents taking the same state, then we must usea non-convex cost function. Likewise if we observe a multi-agent behavior in nature which cannot be described by allagents reaching the same state (the construction of a termitenest, for example), then an optimization-based explanationof this behavior must be non-convex.

Remark 6 (Coverage is Non-convex). This is directlyapplicable to coverage problems. Indeed, coverage cannotbe achieved with all agents moving to the same place, there-fore coverage problems must involve the optimization of anon-convex cost function. Our parameterized cost functionHα from (3) is non-convex for α < 1, in which regime itcorresponds to a coverage task (e.g. α → −∞ for Voronoiand α = −1 for minimum variance). It becomes convex(assuming f is convex) for α > 1 in which regime it resultsin consensus. Theorem 4 explains why this is the case.

4.1. Implications and Future Directions

Theorem 3 may seem disheartening from an algorithmicpoint of view. Convex optimization has a powerful and wellcharacterized tool set guaranteed to reach global minima,but non-convex optimization requires searching out spe-cial cases and special cost function properties. Often onemust be satisfied with local minima. Distributed coveragecontrollers that use gradient methods (such as those in thispaper) guarantee convergence to local minima, which is allone can expect in a general non-convex setting.

One may wonder what can be done to overcome theinherently local convergence properties of gradient con-trollers. Indeed, the picture is not as bleak as it may seem.It may be that the cost function is non-convex and has mul-tiple local minima which all have an equally low cost (or

10 The International Journal of Robotics Research 00(000)

nearly so). It would seem that the cost function we discussin this paper (1) is of this form. There are several differ-ent locally optimal configurations for the robots, but all arenearly as good as the global optimum. Putting analyticalbounds on the difference between the global minimum andany local minimum has proved difficult, but is an area ofongoing research.

Alternately, one may wonder if any non-convex opti-mization techniques can be implemented on a distributednetwork of robots. There are a number of such techniques,however, the ones that most readily suggest a distributedimplementation are those based on gradient methods. Forexample, simulated annealing and deterministic annealingare optimization techniques in which the state follows thenegative gradient of the cost function plus a perturbationterm. The perturbation term is meant to “bump” the stateout of local minima to explore new areas of the state space.The size of the perturbation decreases with each iteration sothat eventually the optimization terminates. This techniquecan be readily applied to our multi-agent setting. There aresome analytical guarantees of convergence to a global min-imum, however, they generally require strict conditions onthe cost function (Granville et al. 1994). Another possibil-ity is to use branch and bound techniques, in which a lowerand upper bound on the minimum are used to rule out sec-tions of the state space. That is, if the lower bound on theminimum over one section of the state space is higher thanthe upper bound on the minimum for another, there is noneed to search in that section. Once a portion of the statespace has been identified, this method is amenable to themulti-agent setting, since the agents can simply drive them-selves into a configuration that is in the relevant part of thestate space and begin a gradient descent. The difficulty liesin computing which sections of the state space are relevantin a distributed way. It would seem that this would require aglobal supervisor to assign different parts of the state spaceto different agents, and then to compile the results of theupper and lower bounds for the different regions to decidein which regions to search. These are promising areas forfuture research.

5. Simulation Results

The controller for several different values of α and φ( q)were simulated in a Matlab environment. The environmentQ was taken to be a unit square.

For the first set of simulations, the function φ( q) was setto be the sum of two Gaussian functions, one centered at( 0.2, 0.2) and the other at ( 0.8, 0.8), both with variance 0.2.We expect to see a higher density of robots around areas oflarge φ( q). In our case, the robots group around the Gaus-sian centers. The results of a simulation with 10 robotsusing the Voronoi-based controller, which corresponds toα → −∞, is shown in Figures 5(a) and 5(d). Similarplots are shown for the minimum variance controller, withα = −1, in Figures 5(b) and 5(e), and the controller withα = 1 in Figures 5(c) and 5(f). Comparison of the con-trollers shows that the Voronoi-based controller causes the

Fig. 5. Trajectories and final configurations are shown for 10robots using the gradient control law with three different param-eter values: α = −∞ for the Voronoi controller (5(a), 5(b)),α = −1 for the minimum variance controller (5(b), 5(e)), andα = 1, which leads to consensus (5(c), 5(f)). The weighting func-tion was a sum of two Gaussians, whose centers are marked withred ×s. The Voronoi tessellation is shown for all scenarios forcomparison, even though the right two controllers do not use theVoronoi cells for control.

robots to spread out more, while as α increases, the robotsgroup more closely together. When α ≥ 1, the cost functionbecomes convex, and the robots all move to the same posi-tion, as seen in Figure 5(f), which corroborates our resultsrelating convexity to consensus.

The second simulation scenario is similar to the first, butthe weighting function, φ( q), is set to be uniform, and wesimulate a group of nine robots. In this case, no area in theenvironment is preferred over another, so the robots move toan even 3×3 grid for all values of α < 1. This is illustratedin Figures 6(a) and 6(d) for α = −∞ and Figures 6(b), and6(e) for α = −1.

Notice that even though both controllers converge tonearly a 3×3 grid, the one with α = −1 more closelyapproaches the exact grid. The deviations from the exact

Schwager et al. 11

Fig. 6. Trajectories and final configurations are shown for ninerobots using the gradient control law with three different param-eter values: α = −∞ for the Voronoi controller (6(a), 6(b)),α = −1 for the minimum variance controller (6(b), 6(e)), andα = 1, which leads to consensus (5(c), 6(f)). The weighting func-tion φ( q) in this case was uniform, so the robots move to an even3×3 grid for all α < 1. The Voronoi tessellation is shown for allscenarios for comparison, even though the right two controllers donot use the Voronoi cells for control.

3×3 grid are caused by errors from the numerical integra-tion required by the controller. Comparing these two sim-ulations highlights one of the advantages of using a finitevalue of α, namely the controller is computed with a con-tinuous function rather that a geometric Voronoi cell, andis therefore more robust to inaccuracies caused by numer-ical integration. As before, when α ≥ 1 the cost func-tion becomes convex, and the robots all move to the sameposition, as shown in Figures 6(c) and 6(f).

The third scenario shown in Figures 7(a) and 7(b) usesthe potential field controller from (10). This controller usesa sum of delta-Dirac functions for φ( q), which causes therobots to arrange themselves in the close-packed lattice pat-tern. Even though α = 1 in this scenario, the cost functionis non-convex because the inter-agent potential, f ( pj, pi), isnon-convex as given by (9).

Fig. 7. Trajectories (7(a)) and final configurations (7(b)) areshown for 10 robots using the potential field controller. TheVoronoi tessellation is shown for comparison, even though thecontroller does not use the Voronoi cells for control.

6. Conclusion

In this paper we introduce a unifying optimization frame-work for multi-robot deployment that brings together sev-eral different existing deployment algorithms. We point outthat important properties of the underlying objective areembodied in the way sensor information or actuator capa-bilities are combined from different robots. We proposea parameterized function to accomplish this combination,where different parameter values are shown to lead to dif-ferent kinds of coverage algorithms. Finally, we prove thatfor deployment problems other than consensus, the underly-ing optimization is necessarily non-convex, making globaloptimization an unrealistic objective, especially for gradientdescent controllers.

Our work invites an immediate extension, which is howto approximate the gradient controller over a communica-tion graph. We outlined two methods for doing this. Inthe future the stability and robustness properties of thesemethods should be characterized and other methods shouldinvestigated as well. Also our recognition that deploymentproblems stem from non-convex optimizations suggestssome new research directions. Gradient descent controllers,which are the most common type in the multi-robot deploy-ment literature, can only be expected to find local minima ingeneral. Therefore it is worthwhile to look for cost functionswith special properties that allow for global optimizationdespite being non-convex. Also it would be interesting toinvestigate other non-convex optimization methods that canbe implemented in a multi-agent setting. We expect thatthese open questions will point the way toward new resultsin multi-robot control.

Acknowledgements

This work was done in the Distributed Robotics Labo-ratory at MIT. This work was supported in part by theMURI SMARTS project grant number N00014-09-1-1051,the MURI SWARMS project grant number W911NF-05-1-0219, NSF grant numbers IIS-0513755, IIS-0426838,CNS-0520305, CNS-0707601, EFRI-0735953, the MASTproject, and The Boeing Company.

12 The International Journal of Robotics Research 00(000)

Notes

1. We use the term consensus in this paper to mean that allrobots drive to a common point in the environment. This isalso commonly called rendezvous in the literature.

2. We will use the term robot throughout, though the frame-work is suitable for general mobile sensing agents, includingbiological ones.

3. This requirement can be generalized considerably as in Cortéset al. (2005) to the case where f ( pi, q) is piece-wise contin-uous with a finite number of jump discontinuities. A finitesensor footprint can be modeled with a single jump discon-tinuity.

4. We know limβ→∞[∑

i hβi ]1/β = maxi hi. Write

limα→−∞[∑

i f αi ]1/α as limβ→∞[[

∑i hβ

i ]1/β ]−1 with

hi = 1/fi and β = −α. We have limβ→∞[[∑

i hβi ]1/β ]−1 =

[maxi hi]−1 = [ 1mini fi

]−1 = mini fi.

5. The indicator function for a set S ⊂ Q returns 1 for q ∈ S, and0 otherwise.

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