Coverage Control for Multi-Robot Teams With Heterogeneous SensingCapabilities

Marıa Santos1, Yancy Diaz-Mercado2 and Magnus Egerstedt1

Abstract— This paper investigates how mobile agents withqualitatively different sensing capabilities should be organizedin order to effectively cover an area. In particular, by encodingthe different capabilities as different density functions in thelocational cost, the result is a heterogeneous coverage controlproblem, where the different density functions serve as away of both abstracting and encapsulating different sensingcapabilities. However, different density functions imply thatmass is not conserved as the agents move and, as a result, thenormal cancellations that occur across boundaries betweenregions of dominance in the homogeneous case no longer takeplace when computing the gradient of the locational cost. Asa result, new terms are needed if the robots are to executea descent flow in order to minimize the locational cost, andwe show how these additional terms can be formulated asboundary-disagreement terms that are added to the standardLloyd’s algorithm. The results are implemented on a realrobotic platform for a number of different use cases.

Index Terms— Multi-robot systems, networked robots, dis-tributed sensor networks.

I. INTRODUCTION

Coverage control concerns itself with the problem ofdistributing a collection of mobile sensor nodes across adomain in such a way that relevant environmental featuresand events are detected by at least one sensor node (withsufficiently high probability), e.g., [1], [2]. Different waysof encoding this have been proposed, including the construc-tion of networks with particularly effective topologies, e.g.,triangulations [3], [4], deployment according to spatial pointprocesses with desired probability characteristics [5], and thepartition of the domain into useful regions of dominance,where each node is in charge of covering its own region [6].

In particular, if a team of N planar robots with positionspi ∈ D ⊂ R2, i = 1, . . . , N , are to cover a convex domainD, one natural choice is to let Robot i be in charge of thepoints in D that are closest to pi, i.e., to let Robot i’s regionof dominance be given by the Voronoi cell

Vi(p) = q ∈ D | ‖q − pi‖ ≤ ‖q − pj‖ , ∀j ∈ N,

where p is the combined positions of all the robots[pT1 , . . . , p

TN ]T and N is the index set 1, . . . , N.

Given a Voronoi partition of the domain into regions ofdominance, one can now ask how well the team is actually

This research was supported by the Office of Naval Research under Grant#N00014-15-1-2115.

1M. Santos and M. Egerstedt are with the School of Electrical andComputer Engineering, Georgia Institute of Technology, Atlanta, Georgia,USA maria.santos,magnus@gatech.edu

2Y. Diaz-Mercado is with The Johns Hopkins University Applied PhysicsLaboratory, Laurel, Maryland, USA yancy.diaz@gatech.edu

covering the area. This question is typically asked relativeto an underlying density function φ : D → [0,∞), whichcaptures the relative importance of points in the domain,with φ(q) > φ(q) meaning that q is more important, hasa higher probability of being a place where an event willoccur, or contains more relevant features than point q, asdiscussed in [6]. If we furthermore assume that the qualityof the measurements that Robot i makes is higher for pointsthat are closer to Robot i, the quality of the coverage obtainedin region Vi(p) can be encoded through the cost∫

Vi(p)

‖q − pi‖2φ(q) dq,

with a better coverage corresponding to a lower cost. Sum-ming over all agents thus yields the so-called locational cost

Hhom(p) =∑i∈N

∫Vi(p)

‖q − pi‖2 φ(q) dq, (1)

as described in [2] as a way of capturing the coverageperformance, and where the subscript hom refers to the factthat all the robots have the same sensing capabilities, i.e.,the team is homogeneous.

The question pursued in this paper is how to introduceheterogeneity into this formulation in a way that reflects thecapabilities of the different robots in a natural manner whenthe sensing modalities are qualitatively different. A numberof approaches to the heterogeneous coverage problem havebeen proposed, focusing on sensor ranges [7], [8], robot foot-prints [9], and motion performance [10], as the differentiatingfeatures among the robots, encoded through weights in thepower diagram [11]. Heterogeneity has also been consideredin anisotropic sensor networks, where the domain partitionsaccomodate the specific geometry of the sensor footprint[12], [13]. However, in these cases the sensors still measurethe same types of features and, as a result, the densityfunction φ(q) is common to all the agents.

In this paper, we explicitly try to maintain some of thestructural advantages afforded by the formulation of thecoverage problem through a locational cost, while capturingqualitatively different sensing capabilities distributed acrossthe robots. To this end, let S be a set of sensory modalities,with each robot being equipped with a subset of thesesensors, denoted by s(i) ⊂ S. Moreover, for each sensorj ∈ S, there is a corresponding density of events or featuresin D that this particular sensor can detect. For example, acamera can detect color variations associated with wiltingcrops on a farm field, while chemical gas sensor arrays canbe used to measure soil conditions [14], [15]. As a result,

we no longer have a single density function, but rather aclass of functions φj : D → [0,∞), j ∈ S, with the densityassociated with point q, as it pertains to Robot i, being givenby

φs(i)(q) =⊕j∈s(i)

φj(q), (2)

where ⊕ is an appropriately chosen composition operator.The choice of composition operator reflects how the densitiesfrom the different sensors on the robot should be combined inorder to compute the overall density function. For example,one simple way to combine the density functions is a directsummation, ⊕

j∈s(i)

φj(q) =∑

j∈s(i)

φj(q),

where the relative importance of a point is reflected bythe sum of its importance among different sensors. Anotherpossible composition is to pick the maximum density valueamong the sensors on Robot i,⊕

j∈s(i)

φj(q) = maxj∈s(i)

φj(q).

This choice would ensure that the density associated witha point corresponds to the highest relative importance mea-sured by its sensors.

This paper investigates what the implications are whenintroducing qualitatively different sensing capabilities for thepurpose of coverage control. The outline of the paper is asfollows: In Section II, we recall how the standard, homo-geneous locational cost formulation lends itself to a veryelegant descent algorithm for coverage control, known asLloyd’s algorithm, and formally introduce the heterogeneouslocational cost. The gradient to this new cost is derivedin Section III together with a gradient-based, distributedcontroller that minimizes the cost. Section IV presents aseries of experiments on a real robotic platform that allowsus to make observations about the optimality of the proposedcontrollers. Lastly, Section V provides conclusions.

II. LOCATIONAL COSTS

Recalling the locational cost for homogeneous coveragein (1), one relevant question is how the robots should movein order to minimize this cost. An approach to this could beto let the individual robots move against the gradient of thecost, i.e., to let

pi = −γi(p)∂Hhom(p)

∂pi

T

, i ∈ N ,

for some positive, possibly state-dependent, gain γi(p), withthe result that

dHhom(p)

dt= −∂Hhom(p)

∂pΓ(p)

∂Hhom(p)

∂p

T

= −

∥∥∥∥∥∂Hhom(p)

∂p

T∥∥∥∥∥

2

Γ(p)

,

where Γ(p) = diag(γ1(p), . . . , γN (p)) is a positive definitediagonal matrix with the individual gains on the diagonal.

This descent formulation has two highly desirable prop-erties, as discussed in [2]. On the one hand, it directlyturns Hhom into a Lyapunov function, amenable to theapplication of LaSalle’s invariance principle as a way ofshowing convergence to a stationary point. On the otherhand, the distributed nature of the team is encoded througha Delaunay adjacency relationship [2] - Robots i and j onlyhave to exchange information if they share a boundary inthe Voronoi tessellation (as long as Γ(p) does not introduceadditional dependencies).

Now, in order to compute the gradient to Hhom(p),Leibniz integral rule must be applied, which contains termsinvolving the derivative of the integrands as well as thedomains over which the integrals are defined. However, eventhough a small change in pi results in a correspondingchange to the boundary of the Voronoi cell Vi(p), the netcontribution from this change to the locational cost is offsetby the corresponding changes to the locational cost fromthe boundaries of the adjacent Voronoi cells, given that thedensity function, φ(q), is common to all the agents and thetotal mass is preserved across D. As a result, the domainterms in Leibniz rule cancel among neighbors and only theintegrand terms must be considered when computing thegradient [1], [16], given by

∂Hhom(p)

∂pi= 2

∫Vi(p)

(pi − q)Tφ(q) dq.

It is possible to express this gradient in a more compactform by defining the mass and center of mass associated withRobot i’s Voronoi cell as

mi(p) =

∫Vi(p)

φ(q) dq, ci(p) =

∫Vi(p)

qφ(q) dq

mi(p),

which yields the gradient

∂Hhom(p)

∂pi= 2mi(p) (pi − ci(p))T .

Moreover, by letting the gain be

γi(p) =κ

2mi(p),

the scaled descent algorithm becomes the well-knownLloyd’s algorithm [17],

pi = −κ(pi − ci(p)), (3)

where κ > 0 is a proportional control gain. In fact, usingLaSalle’s invariance principle, Lloyd’s algorithm has beenshown to asymptotically achieve a centroidal Voronoi tes-sellation (CVT), i.e., a configuration where, asymptotically,pi = ci(p), which in turn is a necessary condition for optimalcoverage, as shown in [1].

As discussed in Section I, the objective behind this work isto introduce heterogeneity in the sensing capabilities throughheterogeneous density functions constructed as in (2),

φs(i)(q) =⊕j∈s(i)

φj(q),

where s(i) ∈ S is the set of sensing modalities associatedwith Robot i, S is the set of all such sensing modalities, andφj is the density associated with (and detectable by) sensorj ∈ S . A direct usage of this formulation in the locationalcost gives

HC(p) =∑i∈N

∫Vi(p)

‖q − pi‖2φs(i)(q) dq. (4)

Note that under this formulation, the original Voronoi par-tition is employed, giving each individual robot the soleresponsibility for its region of dominance. The reason for thisis twofold, namely (a) a desire to recover as much as possiblefrom the homogeneous coverage control case in terms of thestructure of the derivations, and (b) the fact that coordinationemerges explicitly from the regions of dominance – hencethe subscript C.

However, in the heterogeneous case, it is no longer truethat whichever area Robot i does not cover outside of Vi(p)is automatically covered by the adjacent robots. Since therobots may be equipped with different sensor suites, it maybe necessary to let coverage responsibilities encroach onother robots’ cells, i.e., we no longer have a strict partition ofthe domain into regions of dominance. In the extreme case, ifRobot i is the only robot equipped with a particular sensor,and that sensor is needed to cover the whole domain (aswell as possible), it is necessary to define an additional costover the whole domain. As such, in order to let the agentsembrace their domain objectives, denoted by the subscriptO, a different locational cost is needed,

HO(p) =∑i∈N

∫D‖q − pi‖2φs(i)(q) dq, (5)

where each integral measures how well Robot i is coveringthe entire domain D with respect to its particular sensorconfiguration.

Armed with these two different locational costs, we letthe heterogeneous locational cost be given by a convexcombination of the costs in (4) and (5),

Hhet(p) = σHC(p) + (1− σ)HO(p)

= σ∑i∈N

∫Vi(p)

‖q − pi‖2φs(i)(q) dq

+ (1− σ)∑i∈N

∫D‖q − pi‖2φs(i)(q) dq, (6)

where σ ∈ (0, 1] acts as a regularizer between the twocompeting objectives. We do not let σ = 0 since, withthis choice, no coordination among agents is present. Theeffect of selecting different values of σ is further discussedin subsequent sections.

These changes in the locational cost, as compared to thehomogeneous case, have significant implications for howthe gradient should be computed. In subsequent sections,we untangle these implications and present a controller thatachieves convergence to the critical points of the heteroge-neous locational cost in (6), which constitutes a necessarycondition for optimal, heterogeneous coverage.

III. HETEROGENEOUS GRADIENT DESCENT

If we were to obtain the gradient to the heterogeneouslocational cost in (6), a descent direction could be computedfor the robots that achieves a local minimizer. To this end,we compute the gradient to Hhet by considering the twolocational costs HC and HO separately, starting with theformer of the two.

Let Ni encode the Delaunay neighborhood of Robot i,i.e., the set of agents whose Voronoi cells share a face withagent i’s Voronoi cell, as was done in [18]. We can nowbreak ∂HC/∂pi down into three terms, namely Robot i’scontribution, the contributions from robots in Ni, and thecontributions from the remaining robots,

∂HC∂pi

(p) =∂

∂pi

(∫Vi(p)

‖q − pi‖2φs(i)(q)dq

)

+∂

∂pi

∑j∈Ni

∫Vj(p)

‖q − pj‖2φs(j)(q)dq

+

∂

∂pi

∑j /∈Ni∪i

∫Vj(p)

‖q − pj‖2φs(j)(q)dq

. (7)

We immediately note that the last term in the aboveexpression does not depend on pi, and as such, will be zero.For the remaining terms, we need to recall Leibniz integralrule:

Lemma 1 (Leibniz Integral Rule [16]). Let Ω(p) be a regionthat depends smoothly on p such that the unit outwardnormal vector n(p) is uniquely defined almost everywhereon the boundary ∂Ω(p). Let

F =

∫Ω(p)

f(q) dq.

Then,

∂F

∂p=

∫∂Ω(p)

f(q)n(q)T∂q

∂pdq,

where∫∂Ω(p)

denotes the line integral over the boundary ofΩ(p).

This expression needs to be connected to the coordinationlocational cost in (4). Assuming that Vi and Vj share aboundary, this boundary will be orthogonal to the lineconnecting the Voronoi cell generators, as is observed in [19].In other words, for any point q on this boundary,(

q − pi + pj2

)T

(pi − pj) = 0.

Differentiating this with respect to pi yields

(pj − pi)T∂q

∂pi= (q − pi)T . (8)

As (pj − pi)/‖pj − pi‖ is the unit outward normal from Vion this shared boundary, by dividing (8) by ‖pj − pi‖ theterm n(q)T ∂q

∂p in the integrand of Lemma 1 is obtained.

Considering coverage control when mass conservation nolonger holds is not new. For example, [8] considers coveragecontrol with visibility constraints and, analogously to whatwas done in [8], we can calculate the gradient to HC byapplying the Leibniz integral rule to (7),

∂HC∂pi

= 2mi (pi − ci)T

+∑j∈Ni

∫∂Vij‖q − pi‖2

(q − pi)T

‖pj − pi‖φs(i)(q) dq

−∑j∈Ni

∫∂Vij‖q − pj‖2

(q − pi)T

‖pj − pi‖φs(j)(q) dq, (9)

where we, for notational convenience, have suppressed theexplicit dependence of p on ∂Vij – the boundary betweenVoronoi cells Vi and Vj – and where

∫∂Vij

refers to the lineintegral evaluated along this boundary. Moreover, mi and ciare the heterogeneous mass and center of mass in Robot i’sVoronoi cell, given by

mi =

∫Viφs(i)(q)dq, ci =

∫Vi qφs(i)(q)dq

mi. (10)

From the definition of the Voronoi tessellation, all pointson a boundary between cells are equidistant from the seedsfor those cells, i.e., for all q ∈ ∂Vij we have that ‖q−pi‖ =‖q − pj‖. Substituting ‖q − pj‖ by ‖q − pi‖ in (9) yields

∂HC∂pi

= 2mi (pi − ci)T

+∑j∈Ni

(∫∂Vij

(q − pi)T‖q − pi‖2

‖pj − pi‖φs(i)(q) dq

−∫∂Vij

(q − pi)T‖q − pi‖2

‖pj − pi‖φs(j)(q) dq

),

where the integral terms simplify to

∑j∈Ni

∫∂Vij

(q − pi)T‖q − pi‖2

‖pj − pi‖(φs(i)(q)− φs(j)(q)

)dq.

(11)From this, we directly see that the gradient of the coordina-tion term differs from the one obtained in the homogeneouscase. Since the densities are no longer the same in adjacentcells, the net increase over Vi(p) caused by a small move-ment in pi is not offset by the changes in adjacent Voronoicells. Note though that if the density functions are identicalfor all robots, φs(i) = φs(j), i, j ∈ N , then the additionalterm cancels out and the homogeneous gradient from SectionII is immediately recovered.

In order to get the gradient expression in a more compactform, we introduce the total mass and center of mass (bothinterpreted in terms of line integrals) on the boundaries

between Voronoi cells using the following notation,

µij =

∫∂Vij

‖q − pi‖2

‖pj − pi‖φs(i)(q) dq,

ρij =

∫∂Vij

q‖q − pi‖2

‖pj − pi‖φs(i)(q) dq

µij.

Plugging these into (11) yields the derivative of (4) withrespect to Robot i’s position

∂HC∂pi

T

= 2mi(pi − ci)

+∑j∈Ni

µij (ρij − pi)− µji (ρji − pi) . (12)

The computation of ∂HO/∂pi is less involved as the areaof integration is the entire domain D, which does not dependon the position of the agents,

∂HO∂pi

T

=∂

∂pi

(∫D‖q − pi‖2φs(i)(q) dq

)= 2Mi(pi − Ci). (13)

Here Mi and Ci denote the mass and center of mass of thedomain according to the density function of agent i, i.e.,

Mi =

∫Dφs(i)(q) dq, Ci =

∫D qφs(i)(q) dq

Mi.

The gradient of the heterogeneous locational cost thusbecomes

∂Hhet

∂pi

T

= σ∂HC∂pi

T

+ (1− σ)∂HO∂pi

T

= 2σmi(pi − ci)

+ σ∑j∈Ni

µij (ρij − pi)− µji (ρji − pi)

+ 2(1− σ)Mi(pi − Ci). (14)

Letting Robot i follow a negative gradient flow establishesthe following heterogeneous gradient descent theorem.

Theorem 1 (Heterogeneous Gradient Descent). Let Robot i,with planar position pi, evolve according to the control lawpi = ui, where

ui = −2κ (σmi(pi − ci) + (1− σ)Mi(pi − Ci))

− σκ∑j∈Ni

(µij (ρij − pi)− µji (ρji − pi)) . (15)

Then, as t→∞, the robots will converge to a critical pointof the heterogeneous location cost in (6) under positive gainκ > 0.

Proof. From (14), we already know the form for the gradient.What remains to be shown is that convergence to a criticalpoint is indeed achieved.

Consider the total derivative of the locational cost

dHhet(p)

dt=∑i∈N

∂Hhet

∂pipi = −κ

∥∥∥∥∥∂Hhet

∂p

T∥∥∥∥∥

2

≤ 0. (16)

For (16) to be identically zero, we need ∂Hhet

/∂p = 0, in

which case the control law becomes pi = 0. By LaSalle’sinvariance principle, the multi-robot system converges to thelargest invariant set contained in the set of all points suchthat dHhet(p)/dt = 0, which are the critical points to theheterogeneous locational cost in (6).

Note that, unlike the homogeneous case, CVTs are nolonger the only critical points to the locational cost. Indeed,as it will be observed in Section IV, in some situations,placing the agents in a CVT may yield higher costs thannon-CVT configurations. Determining whether the achievedcritical point is a local minimizer to the locational cost isdifficult to establish – this remains an open issue even in thehomogeneous case [16].

IV. EXPERIMENTAL RESULTS

The proposed heterogeneous coverage algorithm is imple-mented on the Robotarium [20], a remotely accessible swarmrobotics testbed at the Georgia Institute of Technology,whose arena serves as the region to be covered by therobot team. The team is composed of six GRITSBots [21],which are miniature, differential-drive robots. A webcam-based tracking system provides information about the posi-tion and orientation of the different robots in the team. Thisinformation is fed to the control algorithm, which producesvelocity commands for the robots.

As the descent algorithm ultimately produces desiredvelocities pi, i ∈ N , an implicit assumption behind thisconstruction is that the robot dynamics can be expressedas (or at least can execute) single integrator dynamics.But the differential-drive configuration does not directlysupport single integrator dynamics and, as such, the controlcommands resulting from Theorem 1 must be converted intosuitable, low level inputs for the GRITSBots. To this end,let pi = (xi, yi)

T be the position of Robot i, and θi itsorientation. Then, the differential-drive configuration can bemodeled using unicycle dynamics,

xi = vi cos θi, yi = vi sin θi, θi = ωi,

where vi and ωi are the translational and rotational velocitiesto be commanded to the robot, respectively. Using a modelsimilar to the one in [2], we can approximately convert thesingle integrator dynamics into unicycle dynamics as follows,

vi = kv[cos θi sin θi]pi,

ωi = kω arctan

([− sin(θi) cos(θi)]pi[cos(θi) sin(θi)]pi

),

with kv and kω positive gains.To evaluate the control law in Theorem 1, its performance

is compared to a baseline controller. To this end, we com-pare it to a heterogeneous version of Lloyd’s algorithm,whereby pi = κ(ci(p) − pi), where ci is evaluated usingthe heterogeneous densities as in (10). As the locational costis an instantaneous measure, we moreover add a temporalcomponent by evaluating the total cost of the controllers∫ tf

0

Hhet(p(t))dt

TABLE ISENSOR MODALITIES FOR THE DIFFERENT EXPERIMENTS

Sensor modalities: S Robot sensors

Exp. 1 1 s(i) = 1 ∀i ∈ NExp. 2 1, . . . , 6 s(i) = i ∀i ∈ NExp. 3 1, . . . , 6 s(i) = i ∀i ∈ N

Exp. 4 1, . . . , 4s(1) = s(2) = 1

s(3) = 2, s(4) = 3

s(5) = s(6) = 4

TABLE IIDENSITY PARAMETERS FOR THE DIFFERENT EXPERIMENTS

αi νi (cm)Agent i 1 2 3 4 5 6

Exp. 1 βi = 10

0

Exp. 2 βi = i0

0

Exp. 3 βi = 1−400

−200

0

20

0

−2020

0

40

0

Exp. 4 βi = 1−300

−300

0

20

0

−2030

0

30

0

under identical initial conditions.The experiment consists of four different configurations

both in terms of the sensor suites assigned to the robots, s(i),i ∈ N , and the density functions associated with each sensortype, φj , j ∈ S. The sensory capabilities of each robot aresimulated using the overhead camera, which provides eachrobot with the information that its sensors would measureaccording to the corresponding density functions. Table Ishows the sensor modalities for each experiment. In the firstexperiment, all the robots have the same sensor, thereforebeing in an equivalent configuration to the homogeneouscase. Experiments 2 and 3 reflect situations where each robothas a unique sensor configuration, while in Experiment 4some robots share sensor configurations.

Gaussian radial basis functions have been used in roboticnetworks to model sensors whose noisy signals representphysical quantities, such as magnetic forces, heat, radiosignal, or chemical concentrations [22]. Following alongthese lines, for each sensor j ∈ S, the corresponding densityfunction is modeled as a bivariate normal distribution,

φj(q) =βj

2π√|Σ|

exp

(−1

2(q − νj)T Σ−1(q − νj)

),

where νj is the mean of the density and Σ is the covariancematrix, which is kept constant for all the sensors. βj serves asa scale factor that models the strength of the density function.Table II indicates the density parameters used for each of theexperiments, corresponding to the sensor modalities in TableI. Note that the values for νj are measured with respect tothe center of the Robotarium arena, a 120×70 cm rectangle.

(a) σ = 0.50 (b) σ = 0.70 (c) σ = 0.90

Fig. 1. Effect of the regularizer term, σ, on the final configuration of the robot team for the sensor configuration of Experiment 4. As specified in Table II,Robots 1 and 2 share the same density function, as do Robots 5 and 6. We can observe how, as the value of the regularizer gets smaller, the coordinationbetween agents vanishes, making the robots that share the same objectives crowd together.

A value of σ = 0.9 is used in all four experiments. Thevalue given to the regularizer σ is selected to favor thecoordination component, HC , over the individual, domainobjectives. A comparison of the effect of different regularizervalues on the behavior of the robot team for the sensor con-figuration of Experiment 4 is presented in Fig. 1, where wecan observe that lower values of σ tend to excessively favorthe domain objectives term,HO, cramming the robots aroundtheir individual density functions and therefore reducing thecoordinated nature of the coverage algorithm.

Table III presents a comparison of the total cost observedfor the four sensor configurations, where both the hetero-geneous version of Lloyd’s algorithm and the descent lawin Theorem 1 are executed for a total time of 2 minutes.

TABLE IIICOMPARISON OF THE OBSERVED TOTAL COST

Heterogeneous Lloyd’s Gradient Descent

Exp. 1 0.17 0.17Exp. 2 0.40 0.37Exp. 3 0.51 0.47Exp. 4 0.47 0.40

0 20 40 60 80 100 120

Time (s)

0.004

0.006

0.008

0.01

0.012

0.014

0.016

Hhet(p(t))

Gradient descent

Heterogeneous Lloyd's

Fig. 2. Evolution of the costHhet(p(t)) with respect to time in Experiment4. The difference between the cost for heterogeneous Lloyd’s and theproposed gradient descent results from ignoring the boundary terms in (12)necessary to minimize the heterogeneous cost. Note that the increase in costaround t = 40 is due to the fact that the algorithm assumes single integratordynamics while the actual robots are subject to nonholomic constraints.

Except for the first experiment, which corresponds to thehomogeneous case, the total cost for the proposed algorithmis consistently smaller than the total cost attained by theheterogeneous Lloyd’s algorithm, which confirms that thecontrol law in Theorem 1 is more adequate for teamswith heterogeneous sensing capabilities. The differences inperformance between the two algorithms are also depictedin Fig. 2, where the absence of the boundary terms makesthe heterogeneous version of Lloyd’s algorithm converge toa configuration with a higher final cost, showing that, for aheterogeneous cost, a CVT is not necessarily on its own aminimizer for the cost function.

A group of ten robots is used to illustrate the team behaviorwhen σ = 1 in (15), that is, when the control law is solelydetermined by the gradient of the coordination cost, HC . Inthis case, the movement of a robot only depends on the valuesof its density function within its Voronoi cell and boundaries.Consequently, the team may be deterred from adequatelycover an area associated with a particular density function,φj , if the robots equipped with the associated sensor, j, arelocated in areas with low values of the density φj , and areunable to move to higher density areas due to the positionof their Delaunay neighbors, as shown in Fig. 3b.

In Section II, HO was introduced as an additional loca-tional cost to palliate the lack of coverage of areas outsideeach robot’s region of dominance when the team is equippedwith disjoint sets of sensors. The results from the convexcombination of both locational costs, HO and HC , areshown in Fig. 3a. This situation illustrates how the proposedcontroller, thanks to the introduction of the domain objectiveterm, achieves a better spatial configuration of the agents inthe domain while still allowing each robot to coordinate withthe other members of the team.

V. CONCLUSIONS

In this paper, we presented a new locational cost func-tion that encodes qualitatively different sensing capabilitiesthrough heterogeneous density functions. In order to coverthe areas of interest, we adopted a distributed gradientdescent approach which drives the agents in the team in adirection that decreases the cost. A series of experiments

(a) (b)

Fig. 3. A group of ten GRITSBots executing the control law in Theorem 1 with σ = 0.975, (a), versus a pure coordination algorithm, with σ = 1, (b).An overhead projector is used to visualize relevant information in the robot arena. For Robot i, the filled circle represents the center of mass of its theVoronoi cell, ci, while the centers of mass in the boundary, ρij , j ∈ Ni, are depicted using crosses at the boundaries. For this experiment, each robot hasa unique sensor configuration with only one sensor. The location of the mean of the associated density function, φs(i) = φi, corresponds to the emptycircle labeled with the robot’s numerical identifier. Making σ = 1 implies the sole consideration of the coordination term in the control law, which mayresult in the blockage of some robots in areas with low information density, as in (b). This situation is alleviated by making σ < 1 in the control law andtherefore involving the term HO , which allows the robot team to attain a better spatial configuration in the domain, (a).

were performed on a team of differential drive robots toassess the performance of the proposed gradient descentmethod as compared to a heterogeneous version of Lloyd’salgorithm. The experiments suggest that the additional termsobtained due to the heterogeneous nature of the performancemetric resulted in overall better coverage than a heteroge-neous version of Lloyd’s algorithm for a number of differentdensity configurations.

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