Coverage_control_for_mobile_sensing_networks-gpl.pdf

IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 20, NO. 2, APRIL 2004 243

Coverage Control for Mobile Sensing NetworksJorge Cortés, Member, IEEE, Sonia Martínez, Member, IEEE, Timur Karatas, and

Francesco Bullo, Senior Member, IEEE

Abstract—This paper presents control and coordination algo-rithms for groups of vehicles. The focus is on autonomous vehiclenetworks performing distributed sensing tasks, where each vehicleplays the role of a mobile tunable sensor. The paper proposesgradient descent algorithms for a class of utility functions whichencode optimal coverage and sensing policies. The resultingclosed-loop behavior is adaptive, distributed, asynchronous, andverifiably correct.

Index Terms—Centroidal Voronoi partitions, coverage control,distributed and asynchronous algorithms, sensor networks.

I. INTRODUCTION

A. Mobile Sensing Networks

THE deployment of large groups of autonomous vehiclesis rapidly becoming possible because of technological ad-

vances in networking and in miniaturization of electromechan-ical systems. In the near future, large numbers of robots will co-ordinate their actions through ad-hoc communication networks,and will perform challenging tasks, including search and re-covery operations, manipulation in hazardous environments, ex-ploration, surveillance, and environmental monitoring for pol-lution detection and estimation. The potential advantages of em-ploying teams of agents are numerous. For instance, certaintasks are difficult, if not impossible, when performed by a singlevehicle agent. Further, a group of vehicles inherently providesrobustness to failures of single agents or communication links.

Working prototypes of active sensing networks have alreadybeen developed; see [1]–[4]. In [3], launchable miniature mo-bile robots communicate through a wireless network. The ve-hicles are equipped with sensors for vibrations, acoustic, mag-netic, and infrared (IR) signals as well as an active video module(i.e., the camera or micro-radar is controlled via a pan-tilt unit).A second system is suggested in [4] under the name of Au-tonomous Oceanographic Sampling Network. In this case, un-derwater vehicles are envisioned measuring temperature, cur-

Manuscript received November 4, 2002; revised June 26, 2003. Thispaper was recommended for publication by Associate Editor L. Parker andEditor A. De Luca upon evaluation of the reviewers’ comments. This workwas supported in part by the Army Research Office (ARO) under GrantDAAD 190110716, and in part by the Defense Advanced Research ProjectsAgency/Air Force Office of Scientific Research (DARPA/AFOSR) underMURI Award F49620-02-1-0325. This paper was presented in part at the IEEEConference on Robotics and Automation, Arlington, VA, May 2002, and inpart at the Mediterranean Conference on Control and Automation, Lisbon,Portugal, July 2002.

J. Cortés, T. Karatas, and F. Bullo are with the Coordinated Science Lab-oratory, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA(e-mail: [email protected]; [email protected]; [email protected]).

S. Martínez is with the Escola Universitària Politècnica de Vilanova i laGeltrú, Universidad Politécnica de Cataluña, Vilanova i la Geltrú 08800, Spain(e-mail: [email protected]).

Digital Object Identifier 10.1109/TRA.2004.824698

rents, and other distributed oceanographic signals. The vehiclescommunicate via an acoustic local area network and coordi-nate their motion in response to local sensing information andto evolving global data. This mobile sensing network is meantto provide the ability to sample the environment adaptively inspace and time. By identifying evolving temperature and cur-rent gradients with higher accuracy and resolution than currentstatic sensors, this technology could lead to the development andvalidation of improved oceanographic models.

B. Optimal Sensor Allocation and Coverage Problems

A fundamental prototype problem in this paper is that of char-acterizing and optimizing notions of quality-of-service (QoS)provided by an adaptive sensor network in a dynamic environ-ment. To this goal, we introduce a notion of sensor coverage thatformalizes an optimal sensor placement problem. This spatialresource-allocation problem is the subject of a discipline calledlocational optimization [5]–[9].

Locational optimization problems pervade a broad spectrumof scientific disciplines. Biologists rely on locational optimiza-tion tools to study how animals share territory and to charac-terize the behavior of animal groups obeying the following in-teraction rule: each animal establishes a region of dominanceand moves toward its center. Locational optimization problemsare spatial resource-allocation problems (e.g., where to placemailboxes in a city or cache servers on the internet) and play acentral role in quantization and information theory (e.g., how todesign a minimum-distortion fixed-rate vector quantizer). Othertechnologies affected by locational optimization include meshand grid optimization methods, clustering analysis, data com-pression, and statistical pattern recognition.

Because locational optimization problems are so widelystudied, it is not surprising that methods are indeed available totackle coverage problems; see [5], and [8]–[10]. However, mostcurrently available algorithms are not applicable to mobilesensing networks because they inherently assume a centralizedcomputation for a limited-size problem in a known static envi-ronment. This is not the case in multivehicle networks which,instead, rely on a distributed communication and computationarchitecture. Although an ad-hoc wireless network provides theability to share some information, no global omniscient leadermight be present to coordinate the group. The inherent spatiallydistributed nature and limited communication capabilities ofa mobile network invalidate classic approaches to algorithmdesign.

C. Distributed Asynchronous Algorithms for Coverage Control

In this paper, we design coordination algorithms imple-mentable by a multivehicle network with limited sensing and

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244 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 20, NO. 2, APRIL 2004

communication capabilities. Our approach is related to theclassic Lloyd algorithm from quantization theory; see [11] for areprint of the original report and [12] for a historical overview.We present Lloyd descent algorithms that take into carefulconsideration all constraints on the mobile sensing network. Inparticular, we design coverage algorithms that are adaptive, dis-tributed, asynchronous, and verifiably asymptotically correct.

Adaptive: Our coverage algorithms provide the networkwith the ability to address changing environments, sensingtask, and network topology (due to agents’ departures, ar-rivals, or failures).Distributed: Our coverage algorithms are distributed inthe sense that the behavior of each vehicle depends onlyon the location of its neighbors. Also, our algorithms donot require a fixed-topology communication graph, i.e.,the neighborhood relationships do change as the networkevolves. The advantages of distributed algorithms are scal-ability and robustness.Asynchronous: Our coverage algorithms are amenableto asynchronous implementation. This means that thealgorithms can be implemented in a network composed ofagents evolving at different speeds, with different compu-tation and communication capabilities. Furthermore, ouralgorithms do not require a global synchronization, andconvergence properties are preserved, even if informationabout neighboring vehicles propagates with some delay.An advantage of asynchronism is a minimized communi-cation overhead.Verifiable Asymptotically Correct: Our algorithms guar-antee monotonic descent of the cost function encoding thesensing task. Asymptotically, the evolution of the mobilesensing network is guaranteed to converge to so-called cen-troidal Voronoi configurations (i.e., configurations wherethe location of each generator coincides with the centroidof the corresponding Voronoi cell) that are critical pointsof the optimal sensor-coverage problem.

Let us describe the contributions of this paper in some detail.Section II reviews certain locational optimization problemsand their solutions as centroidal Voronoi partitions. Section IIIprovides a continuous-time version of the classic Lloyd algo-rithm from vector quantization and applies it to the setting ofmultivehicle networks. In discrete time, we propose a familyof Lloyd algorithms. We carefully characterize convergenceproperties for both continuous and discrete-time versions(Appendix I collects some relevant facts on descent flows).We discuss a worst-case optimization problem, we investigatea simple uniform planar setting, and we present simulationresults.

Section IV presents two asynchronous distributed imple-mentations of Lloyd algorithm for ad-hoc networks withcommunication and sensing capabilities. Our treatment care-fully accounts for the constraints imposed by the distributednature of the vehicle network. We present two asynchronousimplementations, one based on classic results on distributedgradient flows, the other based on the structure of the coverageproblem. (Appendix II briefly reviews some known results onasynchronous gradient algorithms.)

Section V-A considers vehicle models with more realisticdynamics. We present two formal results on passive vehicledynamics and on vehicles equipped with individual local con-trollers. We present numerical simulations of passive vehiclemodels and of unicycle mobile vehicles. Next, Section V-B de-scribes density functions that lead the multivehicle network topredetermined geometric patterns. We present our conclusionsand directions for future research in Section VI.

D. Review of Distributed Algorithms for Cooperative Control

Recent years have witnessed a large research effort focusedon motion planning and coordination problems for multivehiclesystems. Issues include geometric patterns [13]–[16], formationcontrol [17], [18], gradient climbing [19], and conflict avoid-ance [20]. It is only recently, however, that truly distributed co-ordination laws for dynamic networks are being proposed; e.g.,see [21]–[23].

Heuristic approaches to the design of interaction rules andemerging behaviors have been throughly investigated within theliterature on behavior-based robotics; see [17], and [24]–[28].An example of coverage control is discussed in [29]. Alongthis line of research, algorithms have been designed for sophis-ticated cooperative tasks. However, no formal results are cur-rently available on how to design reactive control laws, ensuretheir correctness, and guarantee their optimality with respect toan aggregate objective.

The study of distributed algorithms is concerned withproviding mathematical models, devising precise specificationsfor their behavior, and formally proving their correctness andcomplexity. Via an automata-theoretic approach, the references[30] and [31] treat distributed consensus, resource allocation,communication, and data consistency problems. From a nu-merical optimization viewpoint, the works in [32] and [33]discuss distributed asynchronous algorithms as networkingalgorithms, rate and flow control, and gradient descent flows.Typically, both these sets of references consider networks withfixed topology, and do not address algorithms over ad-hoc dy-namically changing networks. Another common assumption isthat any time an agent communicates its location, it broadcastsit to every other agent in the network. In our setting, this wouldrequire a nondistributed communication setup.

Finally, we note that the terminology “coverage” is also usedin [34] and [35] and references therein to refer to a differentproblem called the coverage path-planning problem, where asingle robot equipped with a limited footprint sensor needs tovisit all points in its environment.

II. FROM LOCATION OPTIMIZATION TO

CENTROIDAL VORONOI PARTITIONS

A. Locational Optimization

In this section, we describe a collection of known factsabout a meaningful optimization problem. References includethe theory and applications of centroidal Voronoi partitions,see [10], and the discipline of facility location, see [6]. In thepaper, we interchangeably refer to the elements of the networkas sensors, agents, vehicles, or robots. We let be the set


CORTÉS et al.: COVERAGE CONTROL FOR MOBILE SENSING NETWORKS 245

Fig. 1. Contour plot on a polygonal environment of the Gaussian densityfunction � = exp(�x � y ).

of nonnegative real numbers, be the set of positive naturalnumbers, and .

Let be a convex polytope in , including its interior,and let denote the Euclidean distance function. We calla map a distribution density function if it repre-sents a measure of information or probability that some eventtakes place over . In equivalent words, we can consider tobe the bounded support of the function . Letbe the location of sensors, each moving in the space . Be-cause of noise and loss of resolution, the sensing performance atpoint taken from th sensor at the position degrades with thedistance between and ; we describe this degrada-tion with a nondecreasing differentiable function .Accordingly, provides a quantitative assessment ofhow poor the sensing performance is (see Fig. 1).

Remark 2.1: As an example, consider mobile robotsequipped with microphones attempting to detect, identify, andlocalize a sound source. How should we plan the robots’ motion

in order to maximize the detection probability? Assuming thesource emits a known signal, the optimal detection algorithmis a matched filter (i.e., convolve the known waveform withthe received signal and threshold). The source is detected de-pending on the signal-to-noise-ratio (SNR), which is inverselyproportional to the distance between the microphone and thesource. Various electromagnetic and sound sensors have SNRsinversely proportional to distance.

Within the context of this paper, a partition of is a collec-tion of polytopes with disjoint interiors,whose union is . We say that two partitions and areequal if and only differ by a set of -measure zero, forall .

We consider the task of minimizing the locational optimiza-tion function

(1)

where we assume that the th sensor is responsible for measure-ments over its “dominance region” . Note that the function

is to be minimized with respect to both 1) the sensors lo-cation , and 2) the assignment of the dominance regions .The optimization is, therefore, to be performed with respect to

the position of the sensors and the partition of the space. Thisproblem is referred to as a facility location problem and, in par-ticular, as a continuous -median problem in [6].

Remark 2.2: Note that if we interchange the positions of anytwo agents, along with their associated regions of dominance,the value of the locational optimization function is not af-fected. Equivalently, if denotes the discrete group of per-mutations of elements, then

for all . Toeliminate this discrete redundancy, one could take natural actionof on , and consider as the configuration spacefor the position of the vehicles.

B. Voronoi Partitions

One can easily see that, at a fixed-sensors location, theoptimal partition of is the Voronoi partition

generated by the points

We refer to [9] for a comprehensive treatment on Voronoidiagrams, and briefly present some relevant concepts. The setof regions is called the Voronoi diagram for thegenerators . When the two Voronoi regionsand are adjacent (i.e., they share an edge), is called a(Voronoi) neighbor of (and vice versa). The set of indexesof the Voronoi neighbors of is denoted by . Clearly,

if and only if . We also define theface as . Voronoi diagrams can be defined withrespect to various distance functions, e.g., the 1-, 2-, , and

-norm over , see [36]. Some useful facts about theEuclidean setting are the following: if is a convex polytopein an -dimensional Euclidean space, the boundary of eachis the union of -dimensional convex polytopes.

In what follows, we shall write

Note that using the definition of the Voronoi partition, we havefor all .

Therefore

(2)

that is, the locational optimization function can be interpretedas an expected value composed with a “min” operation. This isthe usual way in which the problem is presented in the facilitylocation and operations research literature [6]. Remarkably, onecan show [10] that

(3)

i.e., the partial derivative of with respect to the th sensoronly depends on its own position and the position of its Voronoineighbors. Therefore, the computation of the derivative ofwith respect to the sensors’ location is decentralized in the sense



of Voronoi. Moreover, one can deduce some smoothness prop-erties of : since the Voronoi partition depends at least con-tinuously on , for all , the functionis at least continuously differentiable on

for some .

C. Centroidal Voronoi Partitions

Let us recall some basic quantities associated with a regionand a mass density function . The (generalized) mass,

centroid (or center of mass), and polar moment of inertia aredefined as

Additionally, by the parallel axis theorem, one can write

(4)

where is defined as the polar moment of inertia ofthe region about its centroid .

Let us consider again the locational optimization problem (1),and suppose now we are strictly interested in the setting

(5)

that is, we assume . The parallel axistheorem leads to simplifications for both the function andits partial derivative

Here, the mass density function is . It is convenient todefine

Therefore, the (not necessarily unique) local minimum pointsfor the location optimization function are centroids of theirVoronoi cells, i.e., each location satisfies two properties si-multaneously: it is the generator for the Voronoi cell , and itis its centroid

Accordingly, the critical partitions and points for are calledcentroidal Voronoi partitions. We will refer to a sensors’ config-uration as a centroidal Voronoi configuration if it gives rise to acentroidal Voronoi partition. Of course, centroidal Voronoi con-figurations depend on the specific distribution density function

, and an arbitrary pair admits, in general, multiple cen-troidal Voronoi configurations. This discussion provides a proofalternative to the one given in [10] for the necessity of centroidalVoronoi partitions as solutions to the continuous -median lo-cation problem.

III. CONTINUOUS AND DISCRETE-TIME LLOYD DESCENT FOR

COVERAGE CONTROL

In this section, we describe algorithms to compute the loca-tion of sensors that minimize the cost , both in continuous andin discrete time. In Section III-A, we propose a continuous-timeversion of the classic Lloyd algorithm. Here, both the positionsand partitions evolve in continuous time, whereas the Lloyd al-gorithm for vector quantization is designed in discrete time. InSection III-B, we develop a family of variations of Lloyd algo-rithm in discrete time. In both settings, we prove that the pro-posed algorithms are gradient descent flows.

A. A Continuous-Time Lloyd Algorithm

Assume the sensors location obeys a first-order dynamicalbehavior described by

Consider a cost function to be minimized and impose thatthe location follows a gradient descent. In equivalent con-trol theoretical terms, consider a Lyapunov function, andstabilize the multivehicle system to one of its local minima viadissipative control. Formally, we set

(6)

where is a positive gain, and where we assume that thepartition is continuously updated.

Proposition 3.1 (Continuous-Time Lloyd Descent): For theclosed-loop system induced by (6), the sensors location con-verges asymptotically to the set of critical points of , i.e., theset of centroidal Voronoi configurations on . Assuming this setis finite, the sensors location converges to a centroidal Voronoiconfiguration.

Proof: Under the control law (6), we have

By LaSalle’s principle, the sensors location converges to thelargest invariant set contained in , which is precisely theset of centroidal Voronoi configurations. Since this set is clearlyinvariant for (6), we get the stated result. If consists ofa finite collection of points, then converges to one of them(see Corollary 1.2).

Remark 3.2: If is finite, and , then asufficient condition that guarantees exponential convergence isthat the Hessian of be positive definite at . Establishingthis property is a known open problem, see [10]. Note that thisgradient descent is not guaranteed to find the global minimum.For example, in the vector quantization and signal processingliterature [12], it is known that for bimodal distribution densityfunctions, the solution to the gradient flow reaches local minimawhere the number of generators allocated to the two region ofmaxima are not optimally partitioned.



B. A Family of Discrete-Time Lloyd Algorithms

Let us consider the following variations of Lloyd algorithm.Let be a continuous mapping verifying thefollowing two properties:

1) for all ,, where denotes the th component of ;

2) if is not centroidal, then there exists a such that.

Property 1) guarantees that, if moved according to , theagents of the network do not increase their distance to its cor-responding centroid. Property 2) ensures that at least one robotmoves at each iteration and strictly approaches the centroid ofits Voronoi region. Because of this property, the fixed points of

are the set of centroidal Voronoi configurations.Proposition 3.3 (Discrete-Time Lloyd Descent): Let

be a continuous mapping satisfying properties1) and 2). Let denote the initial sensors’ location.Then, the sequence converges to the setof centroidal Voronoi configurations. If this set is finite, thenthe sequence converges to a centroidalVoronoi configuration.

Proof: Consider as an objective func-tion for the algorithm . Using the parallel axis theorem,

, andtherefore

(7)

as long as for all ,with strict inequality if for any , .In particular, , with strict inequality if

, where denotes the set of centroids of the partition. Moreover, since the Voronoi partition is the optimal one for

fixed , we also have

(8)

with strict inequality if .Now, because of property 1) of , inequality (7) yields

and the inequality is strict if is not centroidal by property 2)of . In addition

because of (8). Hence, , and the inequalityis strict if is not centroidal. We then conclude that is adescent function for the algorithm . The result now followsfrom the global convergence Theorem 1.3 and Proposition 1.4

in Appendix I.Remark 3.4: Lloyd algorithm in quantization theory [11],

[12] is usually presented as follows. Given the location ofagents, : 1) construct the Voronoi partition corre-sponding to ; 2) compute the mass centroidsof the Voronoi regions found in step 1). Set the new locationof the agents to these centroids, and return to step 1). Lloyd

algorithm can also be seen as a fixed-point iteration. Considerthe mappings for

Let be defined by .Clearly, is continuous (indeed, ), and corresponds toLloyd algorithm. Now, ,for all . Moreover, if is not centroidal, thenthe inequality is strict for all . Therefore, verifiesproperties 1) and 2).

C. Remarks

1) Note that different sensor performance functions in (1)correspond to different optimization problems. Provided oneuses the Euclidean distance in the definition of [cf. (1)],the standard Voronoi partition computed with respect to theEuclidean metric remains the optimal partition. For arbitrary

, it is not possible anymore to decompose into the sumof terms similar to and . Nevertheless, it is stillpossible to implement the gradient flow via the expression forthe partial derivative (3).

Proposition 3.5: Assume the sensors location obeys a first-order dynamical behavior, . Then, for the closed-loopsystem induced by the gradient law (3), , thesensors location converges asymptotically tothe set of critical points of . Assuming this set is finite, thesensors location converges to a critical point.

2) More generally, various distance notions can be used todefine locational optimization functions. Different performancefunction gives rise to corresponding notions of “center of aregion” (any notion of geometric center, mean, or average is aninteresting candidate). These can then be adopted in designingcoverage algorithms. We refer to [36] for a discussion onVoronoi partitions based on non-Euclidean distance functions,and to [5] and [8] for a discussion on the correspondinglocational optimization problems.

3) Next, let us discuss an interesting variation of the originalproblem. In [6], minimizing the expected minimum distancefunction in (2) is referred to as the continuous -median

problem. It is instructive to consider the worst-case minimumdistance function, corresponding to the scenario where no in-formation is available on the distribution density function. Inother words, the network seeks to minimize the largest possibledistance from any point in to any of the sensor locations, i.e.,to minimize the function

This optimization is referred to as the -center problem in [6]and [7]. One can design a strategy for the -center problemanalog to the Lloyd algorithm for the -median problem. Eachvehicle moves, in continuous or discrete time, toward the centerof the minimum-radius sphere enclosing the polytope. Werefer to [37] for a convergence analysis of the continuous-timealgorithms.

In what follows, we shall restrict our attention to the -medianproblem and to centroidal Voronoi partitions.



Fig. 2. Notation conventions for a convex polygon.

D. Computations Over Polygons With Uniform Density

In this section, we investigate closed-form expression for thecontrol laws introduced above. Assume the Voronoi regionis a convex polygon (i.e., a polytope in ) with vertexeslabeled such as in Fig. 2. It isconvenient to define . Furthermore, weassume that the density function is . By evaluatingthe corresponding integrals, one can obtain the followingclosed-form expressions:

(9)

To present a simple formula for the polar moment of inertia, letand , for .

Then, the polar moment of inertia of a polygon about its centroidbecomes

The proof of these formulas is based on decomposing thepolygon into the union of disjoint triangles. We refer to [38] foranalog expressions over .

Note also that the Voronoi polygon’s vertexes can be ex-pressed as a function of the neighboring vehicles. The vertexesof the th Voronoi polygon that lie in the interior of are thecircumcenters of the triangles formed by and any two neigh-bors adjacent to . The circumcenter of the triangle determinedby , , and is

(10)

where is the area of the triangle, and .Equation (9) for a polygon’s centroid and (10) for the Voronoi

cell’s vertexes lead to a closed-form algebraic expression forthe control law in (6) as a function of the neighboring vehicles’location.

E. Numerical Simulations

To illustrate the performance of the continuous-timeLloyd algorithm, we include some simulation results. Thealgorithm is implemented in as a single cen-tralized program. For the setting, the code computes thebounded Voronoi diagram using the package

, and computes mass, centroid,and polar moment of inertia of polygons via the numericalintegration routine . Careful attention was paid tonumerical accuracy issues in the computation of the Voronoidiagram and in the integration. We illustrate the performanceof the closed-loop system in Fig. 3.

IV. ASYNCHRONOUS DISTRIBUTED IMPLEMENTATIONS

In this section, we show how the Lloyd gradient algorithmcan be implemented in an asynchronous distributed fashion. InSection IV-A, we describe our model for a network of roboticagents, and we introduce a precise notion of distributed evolu-tion. Next, we provide two distributed algorithms for the localcomputation and maintenance of the Voronoi cells. Finally, inSection IV-C, we propose two distributed asynchronous im-plementations of Lloyd algorithm. The first one is based onthe gradient optimization algorithms, as described in [32], andthe second one relies on the special structure of the coverageproblem.

A. Modeling an Asynchronous Distributed Network of Mobile

Robotic Agents

We start by modeling a robotic agent that performs sensing,communication, computation, and control actions. We are in-terested in the behavior of the asynchronous network resultingfrom the interaction of finitely many robotic agents. A frame-work to formalize the following concepts is the theory of dis-tributed algorithms; see [30].

Let us here introduce the notion of robotic agent with com-

putation, communication, and control capabilities as the th ele-ment of a network. The th agent has a processor with the abilityof allocating continuous and discrete states and performing op-erations on them. Each vehicle has access to its unique identifier. The th agent occupies a location and it is ca-

pable of moving in space, at any time for any period oftime , according to a first-order dynamics of the form

(11)

The processor has access to the agent’s location and deter-mines the control pair . The processor of the th agenthas access to a local clock , and a scheduling sequence,i.e., an increasing sequence of times suchthat and . Theprocessor of the th agent is capable of transmitting informationto any other agent within a closed disk of radius . Weassume the communication radius to be a quantity control-lable by the th processor and the corresponding communica-tion bandwidth to be limited. We represent the information flowbetween the agents by means of “send” (within specified radius

) and “receive” commands with a finite number of arguments.



Fig. 3. Lloyd continuous-time algorithm for 32 agents on a convex polygonal environment, with the Gaussian density function of Fig. 1. The control gain in (6)is k = 1 for all the vehicles. The left (respectively, right) figure illustrates the initial (respectively, final) locations and Voronoi partition. The central figureillustrates the gradient descent flow.

We shall alternatively consider networks of robotic agents

with computation, sensing, and control capabilities. In this case,the processor of the th agent has the same computation andcontrol capabilities as before. Furthermore, we assume the pro-cessor can detect any other agent within a closed disk of radius

. We assume the sensing radius to be a quantitycontrollable by the processor.

Remark 4.1: We assume all communication between agentsand all sensing of agents’ locations to be always accurate andinstantaneous.

Consider the closed-loop system formed by the evolution ofthe agents of a network, according to (11). The network evo-lution is said to be Voronoi-distributed if eachcan be written as a function of the form ,with , . It is well known that thereare, at most, neighborhood relationships in a planarVoronoi diagram [9, Sec. 2.3]. As a consequence, the numberof Voronoi neighbors of each site is, on average, less than orequal to six, i.e., . (Recall that sites are Voronoi neigh-bors if they share an edge, not just a vertex.) Accordingly, weargue that Voronoi-distributed algorithms lead to scalable net-works. Finally, note that the set of indexes for aspecific generator of a Voronoi-distributed dynamical systemis not the same for all possible configurations of the network.In other words, the identity of the Voronoi neighbors changesalong the evolution, i.e., the topology of the closed-loop systemis dynamic.

B. Voronoi Cell Computation and Maintenance

A key requirement of the Lloyd algorithms presented inSection III is that each agent must be able to compute its ownVoronoi cell. To do so, each agent needs to know the relativelocation (distance and bearing) of each Voronoi neighbor. Theability of locating neighbors plays a central role in numerousalgorithms for localization, media access, routing, and powercontrol in ad-hoc wireless communication networks; e.g., see[39]–[41] and references therein. Therefore, any motion-con-trol scheme might be able to obtain this information from theunderlying communication layer. In what follows, we set outto provide a distributed asynchronous algorithm for the localcomputation and maintenance of Voronoi cells. The algorithmis related to the synchronous scheme in [41], and is based onbasic properties of Voronoi diagrams.

TABLE IADJUST-SENSING-RADIUS ALGORITHM

Fig. 4. An execution (from left to right) of the adjust-communication-radiusalgorithm. The sensing disk B(p ;R ) is in light gray, and the Voronoi cellestimate W (p ;R ) is the darker gray region.

We present the algorithm for a robotic agent with sensingcapabilities (as well as computation and control). The processorof the th agent allocates the information it has on the positionof the other agents in the state variable . The objective is todetermine the smallest distance for agent which providesenough information to compute the Voronoi cell . Define thefollowing convex set:

(12)

where and the halfplanes are

Provided is twice as large as the maximum distance betweenand the vertexes of , one can show that all Voronoi

neighbors of are within distance from and the equalityholds. The minimum adequate sensing radius

is, therefore, . This ar-gument guarantees the correctness of the adjust-communica-



TABLE IIMONITORING ALGORITHM

tion-radius algorithm in Table I. The execution of this algorithmis illustrated in Fig. 4.

A similar adjust-communication-radius algorithm can be de-signed for a robotic agent with communication capabilities. Thespecifications go as in the previous algorithm, except for the factthat steps 2 and 6 are substituted by

send (request to reply within radius

receive (response from all agents within radius

Further, we have to require each agent to perform the followingevent-driven task: if the th agent receives at any time a “re-quest to reply” message from the th agent located at position

, it executes

send (response within radius

Next, we present the monitoring algorithm (cf. Table II),whose objective is to maintain the information about theVoronoi cell of the th agent, and detect certain events. Weconsider robotic agents with sensing capabilities. We callan agent active if it is moving, and we assume the th agentcan determine whether any agent within radius is activeor not. It turns out that (only) the following two events areof interest: 1) a Voronoi neighbor of the th agent becomesactive; and 2) an active agent becomes a Voronoi neighbor ofthe th agent. In both cases, we record the event by settinga Boolean variable event to true (as we shall later show, thiswill trigger an appropriate control action). The map weight

in Table II is defined by

if and is activeif and is not activeif

We denote by weight the th component of weight. The algo-rithm is designed to run for times .

The correctness of the monitoring algorithm is guaranteed bythe following argument: if an event of type 1) occurs at time

, i.e., an agent (say, the th) that is a Voronoi neighborof the th agent becomes active, then weight ,

TABLE IIICOVERAGE BEHAVIOR ALGORITHM I

and therefore, event is set to . Similarly, if an event of type2) occurs at time , i.e., a new active agent(say, the th) becomes a Voronoi neighbor of the th agent, thenweight , and event is set to .

C. Asynchronous Distributed Implementations of Coverage

Control

Let us now present two versions of Lloyd algorithm for thesolution of the optimization problem 1) that can be implementedby an asynchronous distributed network of robotic agents. Forsimplicity, we assume that at time 0, all clocks are synchronized(although they later can run at different speeds), and that eachagent knows at 0 the exact location of every other agent. Thecoverage behavior algorithm I (cf. Table III) is designed forrobotic agents with communication capabilities, and requiresthe adjust-communication-radius algorithm (while it does notrequire the monitoring algorithm).

As a consequence of the results in [32, Th. 3.1 and Cor. 3.1](see Appendix II, Theorem 2.2 below for a brief exposition), wehave the following result.

Proposition 4.2: Let denote the initial sensorslocation. Let be the sequence in increasing order of allthe scheduling sequences of the agents of the network. Assume

. Then, there exists a sufficiently smallsuch that if , the coverage behavior algo-

rithm I converges to the set of critical points of , that is, theset of centroidal Voronoi configurations.

Next, we focus on distributed asynchronous implementationsof Lloyd algorithm that take advantage of the special structureof the coverage problem. The coverage behavior algorithm II(cf. Table IV) is designed for robotic agents with sensing capa-bilities; it requires the monitoring and the adjust-sensing-radiusalgorithms. Two advantages of this algorithm over the previousone are that there is no need for each agent to exactly go towardthe centroid of its Voronoi cell nor to take a small step at eachstage.



TABLE IVCOVERAGE BEHAVIOR ALGORITHM II

Remark 4.3: The control law in step of the coverage be-havior algorithm II can be defined via a saturation function. Forinstance,

ifif

Then set .With respect to the correctness of the coverage behavior algo-

rithm II, one can consider the time instants when the computa-tion of the centroid of the Voronoi region of any agent is made,together with the time instants when any agent decides to stop,and regard the execution of this algorithm as a discrete-timemapping. Resorting to the discussion in Section III-B on theconvergence of the discrete Lloyd algorithms, one can prove thatthe coverage behavior algorithm II verifies properties 1) and 2).As a consequence of Proposition 3.3, we then have the followingresult.

Proposition 4.4: Let denote the initial sensors lo-cation. The coverage behavior algorithm II converges to the setof critical points of , that is, the set of centroidal Voronoiconfigurations.

V. EXTENSIONS AND APPLICATIONS

In this section, we investigate various extensions and appli-cations of the algorithms proposed in Sections II–IV. We ex-tend the treatment to vehicles with passive dynamics and wealso consider discrete-time implementations of the algorithmsfor vehicles endowed with a local motion planner. Finally, wedescribe interesting ways of designing density functions to solveproblems apparently unrelated to coverage.

A. Variations on Vehicle Dynamics

Here, we consider vehicle systems described by more generallinear and nonlinear dynamical models.

Coordination of vehicles with passive dynamics. We start byconsidering the extension of the control design to nonlinear

Fig. 5. Coverage control for 32 vehicles with second-order dynamics. Theenvironment and Gaussian density function are as in Fig. 3. The control gainsare k = 6 and k = 1.

control systems whose dynamics are passive [42]. Relevantexamples include networks of vehicles and robots withgeneral Lagrangian dynamics, as well as spatially invariantpassive linear systems. Specifically, assume that for each

, the th vehicle state includes the spatial vari-able , and that the vehicle’s dynamics are passive with input

, output , and storage function . Furthermore,assume that the input preserving the zero-dynamics manifold

is .For such systems, we devise a proportional derivative (PD)

control via

(13)

where and are scalar positive gains. The closed-loopsystem induced by this control law can be analyzed with theLyapunov function

yielding the following result.Proposition 5.1: For passive systems, the control law (13)

achieves asymptotic convergence of the sensors location to theset of centroidal Voronoi configurations. If this set is finite,then the sensors location converges to a centroidal Voronoiconfiguration.

Proof: Consider the evolution of the function

By LaSalle’s principle, the sensors location converges to thelargest invariant set contained in . Given the assump-tion on the zero dynamics, we conclude that for

, i.e., the largest invariant set corresponds tothe set of centroidal Voronoi configurations. If this set is finite,LaSalle’s principle also guarantees convergence to a specificcentroidal Voronoi configuration.

In Fig. 5, we illustrate the performance of the control law (13)for vehicles with second-order dynamics and storagefunction .

Coordination of vehicles with local controllers. Next, con-sider the setting where each vehicle has arbitrary dynamics andis endowed with a local feedback and feedforward controller.The controller is capable of strictly decreasing the distance toany specified position in in a specified period of time .



Fig. 6. Coverage control for 16 vehicles with mobile wheeled dynamics. The environment and Gaussian density function are as in Fig. 3, and k = 3.

Assume the dynamics of the th vehicle are described by, where denotes its state, and

is such that . Assume also that for anyand any , there existssuch that the solution of

verifies .Proposition 5.2: Consider the following coordination algo-

rithm. At time , , each vehicle computesand ; then, for time , the vehicle executes

. For this closed-loop system, the sensors lo-cation converges to the set of centroidal Voronoi configurations.If this set is finite, then the sensors location converges to a cen-troidal Voronoi configuration.

The proof of this result readily follows from Proposition 3.3,since the algorithm verifies properties 1) and 2) of Section III-B.

As an example, we consider a classic model of mobilewheeled dynamics, the unicycle model. Assume the th vehiclehas configuration evolving according to

where are the control inputs for vehicle . Notethat the definition of is unique up to the discreteaction . Given a target point

, we use this symmetry to require the equalityfor all time . Should

the equality be violated at some time , we shall redefineand as from time onward.

Following the approach in [43], consider the control law

where is a positive gain. This feedback law differs fromthe original stabilizing strategy in [43] only in the fact that nofinal angular position is preferred. One can prove that

is guaranteed to monotonically approach the target posi-tion when run over an infinite time horizon. We illustratethe performance of the proposed algorithm in Fig. 6.

B. Geometric Patterns and Formation Control

Here, we suggest the use of decentralized coverage algo-rithms as formation control algorithms, and we present variousdensity functions that lead the multivehicle network to prede-

Fig. 7. Coverage control for 32 vehicles with � . The parameter valuesare k = 500, a = 1:4, b = 0:6, x = y = 0, r = 0:3, and k = 1.

Fig. 8. Coverage control for 32 vehicles to an ellipsoidal disk. The densityfunction parameters are the same as in Fig. 7, and ` = 10, k = 1.

termined geometric patterns. In particular, we present simpledensity functions that lead to segments, ellipses, polygons, oruniform distributions inside convex environments.

Consider a planar environment, let be a large positive gain,and denote . Let , , be real numbers,consider the line , and define the density function

Similarly, let be a reference point in , let , , bepositive scalars, consider the ellipse

, and define the density function

Fig. 7 illustrates the performance of the closed-loop networkcorresponding to this density function. During the simulations,we observed that the convergence to the desired pattern wasrather slow.

Finally, define the smooth ramp function, and the density function

This density function leads the multivehicle network to obtaina uniform distribution inside the ellipsoidal disk

. We illustrate this density function in Fig. 8.It appears straightforward to generalize these types of density

functions to the setting of arbitrary curves or shapes. The pro-posed algorithms are to be contrasted with the classic approach



to formation control, based on rigidly encoding the desired geo-metric pattern. One disadvantage of the proposed approach isthe requirement for a careful numerical computation of Voronoidiagrams and centroids. We refer to [14] and [15] for previouswork on algorithms for geometric patterns, and to [17] and [18]for formation control algorithms.

VI. CONCLUSIONS

We have presented a novel approach to coordination algo-rithms for multivehicle networks. The scheme can be thoughtof as an interaction law between agents, and as such it is imple-mentable in a distributed scalable asynchronous fashion.

This paper leaves numerous important extensions open forfurther research. First, we envision considering the setting ofstructured environments (ranging all the way from simple non-convex polygon to more realistic ground, air, and underwaterenvironments); it might be useful, for example, to design dis-tributed algorithms for the art gallery problem. Second, it isclearly important to consider nonisotropic sensors, such as cam-eras and directional microphones, as well as limited footprintsensors, as studied, for example, in the literature on coveragepath planning. Third, we plan to extend the algorithms to pro-vide collision avoidance guarantees and to vehicle dynamicswhich are not locally controllable. Finally, to investigate theeffect of measurement errors on our proposed algorithms andto quantify their closed-loop robustness, we are implementingthese algorithms on a network of all-terrain vehicles. All theseproblems provide nontrivial challenges that go beyond our cur-rent treatment.

APPENDIX IINVARIANCE AND CONVERGENCE PRINCIPLES

In this section, we collect some relevant facts on descentflows both in the continuous and in the discrete-time settings.We do this following [44] and [45], respectively. We includeProposition 1.4, as we are unable to locate it in the linear andnonlinear programming literature.

A. Continuous-Time Descent Flows

Consider the differential equation , whereis locally Lipschitz and is an open connected set.

A set is said to be (positively) invariant with respect toif implies , for all (resp. ).A descent function for on , , is a continuouslydifferentiable function such that on . We denote by

the set of points in where and by be thelargest invariant set contained in . Finally, the distance from apoint to a set is defined as .

Lemma 1.1 (LaSalle’s Principle): Let be a compactset that it is positively invariant with respect to . Letand be an accumulation point of . Then and

as .The following corollary is [44, Exercise 3.22].Corollary 1.2: If the set is a finite collection of points,

then the limit of exists and equals one of them.

B. Discrete-Time Descent Flows

Let be a subset of . An algorithm is a continuousmapping from to . A set is said to be positively in-variant with respect to if implies . Apoint is said to be a fixed point of if . We de-note the set of fixed points of by . A descent function foron , , is any nonnegative real-valued continuousfunction satisfying for , where the in-equality is strict if . Typically, is the objective functionto be minimized, and reflects this goal by yielding a point thatreduces (or at least does not increase) .

Lemma 1.3 (Global Convergence Theorem): Letbe an algorithm with a compact, positively invariant set

and a descent function . Let and denote, . Let be an accumulation point of the se-

quence . Then ,and as .

Proposition 1.4: If the set is a finite collection of points,then converges and equals one of them.

Proof: Let be an accumulation point of and as-sume the whole sequence does not converge to it. Then, thereexists an such that for all , there is a , suchthat . Let be the minimum of all the distancesbetween the points in . Fix . Since is con-tinuous and is finite, there exists such that ,with , implies (that is, for each ,there exists such , and we take the minimum over ).

Now, since , there exists such that forall , . Also, we know that there is asubsequence of which converges to , let usdenote it by . For , there exists such thatfor all , we have .

Let . Take such that Then

(14)

Now we are going to prove that . If ,then this claim is straightforward, since . If ,suppose that . Since ,then . Therefore, there exists suchthat . Necessarily, . Now, by thetriangle inequality, .Then

which contradicts (14). Therefore, . Thisargument can be iterated to prove that for all , wehave . Let us take now such that

. Since , we have, and therefore

which is a contradiction. Therefore, convergesto .



APPENDIX IIASYNCHRONOUS GRADIENT ALGORITHMS

In this section, we present a brief account of the results in [32]concerning asynchronous gradient optimization algorithms. Wedo not review them in their full generality, but rather formulatethem in a form readily applicable to our setting.

Let be finite-dimensional real vector spaces, andlet . If , , werefer to as the th component of . Let be a set ofprocessors that participate in the computation. The algorithmsconsidered here evolve in discrete time. This restriction doesnot involve any loss of generality, since the events of interest(an update by a processor, a transmission of a message, etc.)may be indexed by a discrete variable. The value stored by theth processor at time (global) is denoted by . This global

clock is only need for analysis purposes. The processors maybe working without having access to it; instead, they may haveaccess to a local clock or to no clock at all.

Consider the specialization setting [32], where eachprocessor updates a particular component of the vectorspecifically assigned to it and relies on the information pro-vided by the other processors for the remaining components.Formally:

1) ;2) processor may update only its own component ;3) processor only sends messages containing elements of

. If processor receives such a message, it uses it toreset the equal to the value received.

Let be the set of all times when processor performs acomputation involving the th component of . If a messagefrom processor , containing an element of , is received byprocessor at time , let denote the time that this messagewas sent. The content of the message is, therefore, .Naturally, it is assumed , and we set . Fi-nally, denotes the set of all times when processor receivesa message from processor .

Let be a -function whose derivative is aLipschitz function. Consider the deterministic gradient algo-

rithm given by

(15a)

(15b)

where , and . The specificconclusions of [32, Th. 3.1 and Cor. 3.1] that we need for thespecialization setting are presented in the following result.

Theorem 2.1: Assume each processor communicates itscomponents to every other processor at least once everytime units, for some constant . Then, there exists a constant

such that if , the deterministic gradientalgorithm (15) verifies

for all .

In the particular case when, for each ,the partial derivative only depends on , with

for certain set , the previous resultcan be restated in the following form.

Theorem 2.2: Assume each processor communicates itscomponents to every other processor in at least onceevery time units, for some constant . Then, there exists aconstant such that if , the deterministicgradient algorithm (15) verifies

for all and all.

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Jorge Cortés (M’03) received the Licenciaturadegree in mathematics from the Universidad deZaragoza, Zaragoza, Spain, in 1997, and the Ph.D.degree in engineering mathematics from the Univer-sidad Carlos III de Madrid, Madrid, Spain, in 2001.

From January to June 2002, he held a Postdoc-toral position at the Systems, Signals and ControlDepartment, University of Twente, Enschede, TheNetherlands. He is currently a Postdoctoral ResearchAssociate at the Coordinated Science Laboratory,University of Illinois at Urbana-Champaign. His

current research interests focus on mathematical control theory and geometricintegration, with a special emphasis on Lagrangian and Hamiltonian systemsand the role of symmetry principles, and motion coordination algorithms forgroups of autonomous vehicles.

Sonia Martínez (M’02) received the Licenciaturadegree in mathematics from the Universidad deZaragoza, Zaragoza, Spain, in 1997, and the Ph.D.degree in engineering mathematics from the Univer-sidad Carlos III de Madrid, Madrid, Spain, in 2002.

From September 2002 to September 2003, sheheld a Postdoctoral position at the Department ofApplied Mathematics IV, Universidad Politécnicade Cataluña, Cataluña, Spain. Since October 2003,she has been a Postdoctoral Fulbright Fellow atthe Coordinated Science Laboratory, University of

Illinois at Urbana-Champaign. Her current research interests include optimalcontrol policies for robotic locomotion, controllability analysis and motionplanning for underactuated systems, and low-complexity representations ofmechanical systems.

Timur Karatas received the B.S. degree in mechan-ical engineering from the Middle East Technical Uni-versity, Istanbul, Turkey, in 1999, and the M.S. degreein general engineering in 2001 from the University ofIllinois at Urbana-Champaign, where he is currentlyworking toward the Ph.D. degree.

His research interests include trajectory generationand path planning for autonomous vehicles, cooper-ative control and distributed algorithms.

Francesco Bullo (S’95–M’98–SM’03) received theLaurea degree magna cum laude in electrical engi-neering from the University of Padova, Padova, Italy,in 1994, and the Ph.D. degree in control and dynam-ical systems from the California Institute of Tech-nology, Pasadena, in 1999.

He is an Assistant Professor with the Departmentof General Engineering and with the CoordinatedScience Laboratory, University of Illinois at Ur-bana-Champaign, Urbana. His research interestsinclude motion planning and coordination for

autonomous vehicles, geometric control of mechanical systems, and distributedand adaptive algorithms.


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