An Adaptive Coverage Control Algorithm for Deployment ofNonholonomic Mobile Sensors

Jose Marcio Luna, Rafael Fierro, Chaouki Abdallah and John Wood

Abstract— We show the Lyapunov stability and convergenceof an adaptive and decentralized coverage control for a teamof mobile sensors. This new approach assumes nonholonomicsensors rather than the usual holonomic sensors found inthe literature. The kinematics of the unicycle model and anonlinear control law in polar coordinates are used in orderto prove the stability of the controller applied over a team ofmobile sensors. The convergence and feasibility of the coveragecontrol algorithm are verified through simulations in Matlab.Furthermore, some experiments are carried out using a teamof four Pioneer 3-AT robots sensing a piecewise constant lightdistribution function.

I. INTRODUCTION

Literature related to oil spills [1] and forest fires [2], [3]shows that some systems present motion dynamics that makethem more complicated to overcome without putting humansin danger. Currently, the use of manned aerial vehicles forfirefighting requires skillful enough pilots to avoid crashingin the attempt to put out a fire. Moreover, firefighters mustget a qualitative estimation of the fire dynamics almost bydirect observation. In another scenario, Cortez et al., exposedin [4] that building radiation maps involving nuclear materialis still done using people for taking measurements close tothe radiated area.

Coverage controllers become promising with the latestdevelopments on wireless communications, material science,new sensors and the constant improvement of computationalpower. The possibilities to send small unmanned aerial,terrestrial or underwater vehicles which coordinate actionsto sense and map an area of interest are increasing as the re-search in decentralized algorithms and hardware progresses.

Sensor networks are groups of simple sensors that can beused to monitor, track or survey an area of interest. Everysensor collects data and shares the information with theothers in the group. The sensor networks we are interestedin are dynamic [1], [5], [6]. Mostofi and Sen propose in [7]a compressive sensing approach to build a map of spatialvariations of certain parameter of interest in its environment,applying the Fourier slice theorem. Hou and Slotine proposein [8], a scalable dynamic region following formation controlfor a swarm of robots, which is able to adjust a formationwith a desired geometric shape by choosing the right functionto describe it.

J. M. Luna, R. Fierro and C. T. Abdallah are withDepartment of Electrical & Computer Engineering, TheUniversity of New Mexico, Albuquerque, NM 87131-0001, USA{jmarcio,rfierro,chaouki}@ece.unm.edu

J. Wood is with Department of Mechanical Engineering, The Universityof New Mexico, Albuquerque, NM 87131-0001, USA jw@unm.edu

Based on [9], the coverage control problem is associatedto a cost function which determines a problem-dependentmetric of the coverage performance. We can implement acontroller to determine the optimal placement of the sensorsin an environment. Lee et al., present in [10] an approach ofcoverage control with environmental sensing where a teamof robots should position themselves over an area such thatthey concentrate themselves in the area with the greatestamount of a chemical using an adaptive triangular mesh.In [11] Schwager et al., propose an optimization criterion todistribute a team of hovering robots with downward facingcameras to obtain the best view of an environment. Theauthors propose a metric based on the minimum informationper pixel in order to elaborate the cost function.

Because of the complexity found in the analysis of net-works of dynamic systems, it is common to consider simpledynamical models such as the single integrator [5], or thedouble and higher order integrators [12]. Recently, Kwokand Martinez in [13] have used a hybrid system approachto attack the decentralized control problem described in [5]using nonholonomic sensors. The authors model the problemas a hybrid automaton with a set of states implementing somemotion behaviors in a team of unicycle agents with fixed andvariable forward velocity. The agents are assumed to have aprevious knowledge of the sampling space.

A. Contributions

This work is motivated by the one presented by Schwageret al., in [14] where the authors describe the development ofan adaptive coverage control for mobile sensor networks andprovide the stability analysis of the controller. The authorsassume that the mobile sensors do not have nonholonomicconstraints and that the estimated density function is static.However, several real world vehicles such as aircrafts atcruising attitude, sea vessels and skid-steered mobile robotshave nonholonomic constraints and several phenomena can-not be considered static. The performance of the results givenfor holonomic mobile sensors can be severely affected oreven invalidated [13], when they are adapted to nonholo-nomic mobile sensors.

Our main goal in this work is to provide the necessarymathematical background to use nonholonomic mobile sen-sors along with the adaptive coverage control presented in[14], and guarantee the stability of the system. It is worthmentioning that extensions to nonholonomic systems arein general non-straighforward. Furthermore, we will applyour coverage control over dynamic density functions whoseparameters are modeled as piecewise constant functions.

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The paper is organized as follows: Section II shows themathematical background related to Voronoi partitions andlocational optimization and the adaptive coverage controlfor holonomic sensor networks which inspired this work.In Section III we present our main theoretical result whichshows the stability of the adaptive coverage control fornonholonomic sensor networks. Section IV shows simulationresults obtained using Matlab. Section V illustrates theexperimental results obtained by using a team of four Pioneer3-AT robots sensing a dynamic light distribution. Lastly,Section VI summarizes the main conclusions and limitationsof our approach as well as future work to overcome thoselimitations.

II. MATHEMATICAL BACKGROUND

In this section we introduce the theoretical backgroundinvolved in the development of our adaptive and decentral-ized controller for nonholonomic robots. We are interested inemploying dynamic sensor networks to achieve an estimationof a density function which represents a concentration ofa measurable phenomenon. The sensory function can beconsidered static (e.g., a stable methane concentration in agarbage dump) or dynamic (e.g., an oil spill [1] or a forestfire [2]).

In some works as the one presented in [15], we findapplications that involve Voronoi partitions. They are atypical feature of several biological systems [16] and recentlyhave received special attention by mathematicians for theirapplication in disciplines such as cellular biology, imagecompression, statistics and robotics among others. Beforeany further discussion, let us start with some necessarydefinitions.

A. Voronoi Diagrams

We based the following definition on the one in [16]Definition 1: Given an open set Q ⊆ RN , the set {Vi}k

i=1

is called a Voronoi tesselation or diagram of Q if Vi∩Vj = ∅for i 6= j and

⋃ki=1 Vi = Q. Given a set of points {pi}k

i=1

belonging to Q, the Voronoi region Vi corresponding to thepoint pi is defined by

Vi = {x ∈ Q | ‖x− pi‖ < ‖x− pj‖for i, j = 1, . . . , k, j 6= i} .

Where ‖ ·‖ denote the Euclidean norm on RN . The points{pi}k

i=1 are called generator points, and Vi is the Voronoiregion associated to the generator point pi.

Although Voronoi diagrams can be defined using severaldistance functions such as the geodesic distance describedin [17]. For this particular problem we are using Euclideandistance, and Q is considered a convex polytope in an N -dimensional Euclidean space.

B. Locational Optimization

Based on [18], let Q ⊂ RN be a convex polytope includingits interior. Assume a mapping φ(q) : Q 7→ R+ with q ∈ Qcalled a distribution density function (or sensory function)

which represents a measurement of the probability of aspecific event on Q. The locational optimization functionis then defined as

H(P ) =n∑

i=1

∫

Vi

f(‖q − pi‖)φ(q)dq, (1)

where P is the set of all the n generator points{p1, . . . , pn} ∈ Q and Vi is the Voronoi partition of thei-th robot.

Now, based on [5] we can adapt some physical conceptsnamely, the mass MVi

, the first moment LVi, the polar

moment of inertia JV,p and the centroid CVi of a Voronoiregion Vi. Their definitions are given by the followingequations,

MVi=

∫

Vi

φ(q)dq,

LVi=

∫

Vi

qφ(q)dq,

JV,p =∫

Vi

‖q − pi‖2 φ(q)dq,

CVi =1

MVi

∫

Vi

qφ(q)dq. (2)

From [5], if we define f(‖q−pi‖) = ‖q−pi‖2 and replaceit in (1), after applying a partial derivative with respect to pi

we have that∂HV (P )

∂pi=

∫

Vi

∂

∂pif(‖q − pi‖)φ(q)dq

= 2MVi(pi − CVi). (3)

Therefore, all the Voronoi tessellations in Q where thegenerator points are at the same time the centroids oftheir Voronoi partitions minimize the locational optimizationfunction. These tessellations are usually called centroidalVoronoi tessellations [16].

C. Adaptive Control for Holonomic Sensors

In [14] the authors propose an approach which guaranteesthat the network of mobile agents minimizes the cost functionHV (P ) in (1). They assume that each agent measures thesensory function without requiring a previous knowledge.

In order to deal with the lack of knowledge of the samplingspace they proposed a decentralized adaptive control basedon the following assumptions,

Assumption 1 (Matching Conditions): There exists a pa-rameter vector a ∈ Rm

+ and a vector function K : Q 7→ Rm+

such thatφ(q) = K(q)T a, (4)

where m ∈ N, and (·)T denotes transpose.The parameter vector a is unknown by the agents but K(q)

is available to them.

Assumption 2 (Lower Bound): Given that a(j) is the j-thelement of the vector a and β ∈ R+ then

a(j) ≥ β ∀j = 1, . . . ,m,

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The reason for a lower bound for the parameter vector isto avoid that K(q)T a = φ(q) = 0 leading to a zero in thedenominator of (2).

The sensory function estimated by the i-th agent is givenby φi = K(q)T ai, where ai is the estimation of the parametervector a calculated by the agent i. Furthermore the parametererror vector ai is given by

ai = ai − ai. (5)

In [5] the mobile agents are considered holonomic vehicleswith first-order continuous dynamics, that is

pi = ui, (6)

which is called a single integrator.The control law is defined as

ui = k(CVi − pi), (7)

where CVi is an estimate of the real centroid CVi of the i-thVoronoi region defined by

CVi=

LVi

MVi

=

∫Vi

qφ(q)dq∫

Viφ(q)dq

.

Finally, the adaptation law is given by

˙ai = Γ( ˙aprei − Iproji˙aprei), (8)

with

˙apre = −Fiai − ξ(Λiai − λi)− ζ∑

j∈Ni

(ai − aj), (9)

where ξ, ζ ∈ R+ are scalar gains, Γ ∈ Rm×m is a diagonalpositive definite gain matrix. The variables Fi, Λi, and λi

are given by the following equations,

Fi =[∫

Vi

K(q)(q − CVi)T dq

]pi (10)

Λi =∫ t

0

w(τ)Ki(τ)Ki(τ)T dτ, (11)

λi =∫ t

0

w(τ)Ki(τ)φi(τ)dτ. (12)

Given a set of indexed vertices Ve = {v1 . . . , vn} anda set of edges E = {e1 . . . el}, where ei = {vj , vk} thenNi = {j|{vi, vj} ∈ E} i.e., Ni contains the indexes of thevertices which are neighbors of the vertices associated to theVoronoi partition of the generator point i.

The matrix Iproji(j) is defined as follows

Iproji(j) =

0 for ai(j) > β,

0 for ai(j) = β and ˙aprei ≥ 0,1 otherwise.

(13)

The index j denotes the j-th diagonal element of thematrix Iproji and the j-th element of the vector ai. Thismatrix implements a projection law which prevents theparameter vector ai from taking values less than or equalto the lower bound β.

The function w(t) ∈ L1 is called a weighting function weprovide a detailed discussion in Section II-D.

Lastly, in [14] the authors state and prove the followingconvergence theorem

Theorem 1 (Convergence Theorem): Under Assumption1, for the system of n agents with the dynamics given by(6) and the control law in (7),

limt→∞

‖CVi− pi‖ = 0, ∀i ∈ In,

limt→∞

K(pi(τ))T ai = 0, ∀τ | w(τ) > 0 and ∀i ∈ In,

limt→∞

‖ai − aj‖ = 0, ∀i, j ∈ In,

with In = {1, . . . , n}.

D. Weighting Functions

There are several options to build the weighting functionw(·) in (11) and (12) as long as it stimulates the parameterconvergence of the adaptation law. Based on [14], if wechoose w(τ) as a square wave, the integral given in (11) doesnot incorporate any other term in the summation after somefixed time determined by the decay time of the square wave.We can soften the elimination of old terms in the integralusing softer decays, e.g., an exponential decay w(τ) = e−τ .If we specifically use the function w(t, τ) = e−δ(t−τ) theintegrals (11) and (12) become first-order systems, introduc-ing a forgetting factor δ which allows the tracking of slowvarying density functions.

III. ADAPTIVE CONTROL FOR NONHOLONOMICSENSORS

The stability analyzes of the controllers in [5] and [14]have been conducted assuming holonomic kinematics, butnow we propose to formally extend the previous results tononholonomic vehicles.

A. Nonlinear Steering Control

In order to incorporate nonholonomic constrains in ourmodel, we propose to use the polar unicycle model kine-matics equations [19] for a differential steering as a suitableapproach. The equations of motion for the i-th agent in theteam of robots are given as follows

ρi

αi

θi

=

−ui cosαi

−ωi + uisin αi

ρi

uisin αi

ρi

, (14)

where

αi = θi − φi, (15)φi = ωi.

where ui and ωi are the linear and angular speeds of the i-throbot respectively.

After an extensive analysis of several controllers fornonholonomic systems available in the literature, the controlalgorithm described in [19] provided a suitable solution toour multivehicle coordination problem.

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iθ

iα

iφ

iρ

X

Y

iω

iu

iVC

iV

jV

kVlV

}{G

Fig. 1. Unicycle model and variables in the goal frame {G}: Noticethe vectors and angles which determine our nonholonomic model in polarcoordinates.

As shown in Fig. 1 the position of the agent inside itsVoronoi cell is represented in polar coordinates where φi isthe heading angle of the vehicle, ρi represents the positionerror between the agent and the centroid point and αi is theangle between the principal axis of the robot and the vectorerror ρi. Now The control law [19] is given by,

ui

ωi

=

(γ cos αi)ρi

kαi + γ cos αi sin αi

αi(αi + hθi)

, (16)

where k, γ and h are positive gains.The control law in (16) allows the agent to reach asymp-

totically the point (0, 0, 0). Therefore if we carry out an axistranslation to set the centroid at the origin of the plane wecan use this control law to drive the robots to their centroidalVoronoi tessellation. For a detailed proof of the stability ofthe steering control see [19].

B. Stability Analysis

The following is our extended convergence theorem for thedistributed and adaptive control for nonholonomic vehicles.

Theorem 2 (Extended Convergence Theorem): IfAssumptions 1 and 2, are satisfied we have that forthe system of n nonholonomic agents with dynamics (14)and control law (16),

limt→∞

K(pi(τ))T ai = 0, ∀τ | w(τ) > 0 and ∀i ∈ In,

limt→∞

ρi, ‖αi‖, ‖θi‖ = 0, ∀i ∈ In,

limt→∞

‖ai − aj‖ = 0, ∀i, j ∈ In,

with In = {1, . . . , n}.

Proof: To carry out the stability analysis we proposethe following Lyapunov function candidate

V = H+n∑

i=1

[12aT

i Γ−1ai +12

(α2

i + hθ2i

)]. (17)

Notice that compared with the Lyapunov function pro-posed in [14], the expression in (17) has the extra term12

(α2

i + hθ2i

)related to the robot orientation (see Fig. 1).

The matrix Γ is the same diagonal positive definite matrixin (8), H is described by (1), and αi, θi and ρi are the statevariables in the dynamics in (14). Lastly, ai is the parameterestimation error given by (5).

Taking the time derivative of (17), we obtain

V =n∑

i=1

[∂H∂pi

pi + aTi Γ−1 ˙ai + (ααi + hθiθi)

]. (18)

Now, replacing (3) in (18) we get

V =n∑

i=1

[MVi(pi − CVi)

T pi + aTi Γ−1 ˙ai

+ (ααi + hθiθi)]. (19)

Furthermore, we can show that

LVi = MViCVi + MVi(CVi − CVi) = MViCVi , (20)

then replacing (20) in (19), we have

V =n∑

i=1

[−Mvi(CVi − pi)T pi

+(MViCVi − MViCVi)pi

+aTi Γ−1 ˙ai + (ααi + hθiθi)

]. (21)

Taking into account that

MViCVi − MViCVi = ai

∫

Vi

K(q)T (q − CVi)dq,

and replacing the adaptation law given by (8)-(12) in (21) thefinal expression for the derivative of the Lyapunov functionbecomes

V = −n∑

i=1

[MVi(CVi − pi)T pi

+ξ

∫ t

0

w(τ)(Ki(τ)T ai)2 dτ

+aTi ζ

∑

j∈Ni

(ai − aj) + aTi Iproj

˙aprei

−(αiαi + hθiθi)]. (22)

The second, third and fourth terms in the summation in(22) have already been proven to be positive semidefinite[14], considering the negative sign before the summation.Now, we are interested in proving that the first and fifthterms are positive semidefinite as well.

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Calculating CVi− pi and based on Fig. 1, we can assert

that

CVi − pi =(

x2 − x1

y2 − y1

)=

(ρi cos θi

ρi sin θi

)

=(

ρi cos(φi + αi)ρi sin(φi + αi)

). (23)

Taking the first term MVi(CVi − pi)T pi of (22) andreplacing CVi

− pi and pi by using the unicycle model in(14) with the control law in (16) we have

MVi(CVi − pi)T pi =

= MVi

(ρi cos(φi + αi)ρi sin(φi + αi)

)T ((γ cos αi)ρi cos φi

(γ cos αi)ρi sin φi

),

= MViρ2

i γ(cos2 φi cos2 αi + sin2 φi cos2)αi,

= MViρ2

i γ cos2 αi.

(24)

Since the mass MViof the i-th Voronoi region and

the control gain γ are non-negative, the first term in thesummation of (22) is non-negative.

MVi(CVi − pi)T pi = MViρ2i γ cos2 α ≥ 0.

Analyzing the fifth term in (22) namely, −(αiαi + hθiθi)we have that replacing the polar kinematics in (14) it givesbased on [19],

−(αi + hθiθi) =

= −[αi

(−wi +

ui sin αi

ρi

)+ hθi

ui sin αi

ρi

],

= −[αi

(−wi +

(hθi + αi)αi

ui sinαi

ρi

)]. (25)

Replacing the control law (16) in (25) we get

−(αiαi + hθiθi) = kα2i ≥ 0,

and the fifth term −(αiαi + hθiθi) in (22) is non-negative.Since V is lower bounded, V is negative semidefinite anduniformly continuous in time we conclude that V → 0 ast →∞ by the Lyapunov-like lemma.

From the Lyapunov function derivative in (22) it is easy tosee that all the limits converge to zero except the third onelimt→∞ ‖θi(t)‖. In [19] the author proved by the Lyapunov-like lemma that αi → 0 as t → ∞ and this implies thatθi → 0 as well. Therefore the controller guarantees theconvergence of the state variables ρi, αi and θi to zero underthe goal frame {G} shown in Fig. 1.

Although the orientation of the robot alignment with theglobal frame is a consequence of the nonlinear steeringcontrol, in this particular case it guarantees that the robotsare not going to spin around their own axis. For practicalpurposes such as energy saving, spinning around is anundesirable behavior. Moreover, in some applications it maybe required to regulate the orientation of the mobile sensorsince a robot may have navigation or monitoring sensors suchas a camera or laser range finder at the front part.

C. Dynamic Density Function

We consider the case of estimating the parameters of atime-varying density function φi(q, t) = K(q)T a(t) wherethe j-th entry aj(t) (j = 1, 2, . . . , m) of a(t) is a piecewiseconstant function aj(t) : Rm

+ 7→ Rm+ and is right continuous.

It means that every entry of the function vector a(t) has afinite number of discontinuities and takes on constant valuesbetween two consecutive discontinuities. This is a reasonableapproximation if we consider slow-time varying systems.

Also, we assume that limt→∞ a(t) = ac where ac ∈ Rm+

is a constant value i.e., the density function reaches a steadystate which is reasonable for many real-world phenomenasuch as oil spills [1] and forest fires [2].

From now on, we will call switching time ts, the time wheneach discontinuity happens, where s = 1, · · · , k, and k is thetotal number of switching times before the density functionreaches its final value. This terminology was taken from [20]given the partial similarity with the switching systems.

Moreover let us assume that the adaptation law rate andthe angular and linear speeds of the agents are fast enoughto follow the dynamics of the density function φ(q, t).

From (8) we know that every robot looks for the centroidof its Voronoi cell while taking measurements of the distribu-tion function on its trajectory. During this time, the trackingerror decreases but notice from Theorem 2 that the networkof robots converges to a near optimal coverage configuration.Based on Theorem 2 this behavior does not necessarily implythat the parameter estimation vector a(t) → 0 as t →∞.

Furthermore, since we are dealing with a piecewise con-stant system, the time interval between two switching times∆ts = ts − ts−1 is finite, in contrast with the infinite timenecessary to guarantee full parameter convergence.

IV. SIMULATIONS

In this section we show some simulation results in orderto verify Theorem 2. The simulations were carried outusing Matlab implementing just the kinematic model of theunicycle vehicle and neglecting the dynamics equations.

A. Simulation Design

For this simulation we used a population of 20 unicyclemodels randomly distributed over a sample space Q definedas a unit square. We implement the control law given in (16)with γ = 3 and k = h = 1.

The parameter values we used in the adaptation law givenby (8) and (9) are Γ = I64, ξ = 1000, ζ = 1 and δ = 1. Forthe matrix Iproji defined by (13), we have β = 0.1.

We divided the sampling space Q in a 8 × 8 grid wherethe geometric center of every square cell corresponds tothe mean µi of a bidimensional Gaussian function. Usinga function similar to the one in [14] we have that the i-thentry Ki of the vector function K(q)64×1 is calculated as,

Ki = e−(q−µi)

2

2σ2i , (26)

with σ2i = 0.05.

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1a

2a3a

4a 5a

6a

7a

8a

1p 2p

Fig. 2. Method to create the parametric density function. The two red dotsdetermine the two gaussians which are sampled determining the parametersai(t) for i = 1, 2 . . . 8.

For this simulation we use the team of robots to detect adensity function which behaves as an expanding circle. Thecircle recreates a simplified behavior of a forest fire wherethe higher temperatures are localized at the boundary of thecircle.

The dynamics of the expanding circle are modeled by thefollowing parametric equations,(

x(t)y(t)

)= c1r(t)

(cos φsin φ

)+

(c2

c3

),

with the radius r(t) defined by the differential equation,

r(t) = −c4r(t) + c5,

with the constants r(0) ∈ R+ and ci ∈ R+ for i = 1, 2 . . . 5.In order to assign a height to the expanding circle, we take

m equidistant points at the boundary of the circle. Each pointdetermines the mean of a bidimensional gaussian functionwith variance σ2

k = 0.05 which is sampled by the 8 × 8grid defined above in Q. The heights of each one of the 64samples determine the parameter vector a(t). In Fig. 2 weshow a simplified 1-dimensional version of this calculationwhere the red dots labeled P1 and P2 are equivalent to them dots in the boundary of the circle. The gaussian functionsare indicated in red and assuming we discretized the spacein 8 bins we sample the gaussians so that we get the eightparameters ai(t) with i = 1, 2 . . . 8 as shown in Fig. 2.

Since our approach covers just piecewise constant dynam-ics we assume that the robots are taking measurements of thedensity function at the discrete-time instants 0, 20, 40 and100 s. This means that assuming a slow varying distributionfunction the robots can reach their respective centroids andrest until some problem dependent condition is fulfilled tostart taking measurements again.

1) Simulation Results in Matlab: In Fig. 3 we show theaveraged behavior of the parameter estimation error given by

K(pi(τ))T ai(t) ∀τ |w(τ) > 0,

as well as the error distance ρi(t), the angle αi(t) and theconsensus error given by

‖ai(t)− aj(t)‖ ∀i, j ∈ In.

Notice that the switching times of the simulation are indi-cated by the dashed vertical lines in green.

Let us define

¯ai(t) =1n

(n∑

i=1

ai(t)

)∀t > 0, (27)

which is the averaged parameter error vector over all robots.In Fig. 3 (a) we show the parameter estimation errorK(q)T ¯ai(t) averaged over the whole population of robots.In a similar way let us define

ρ(t) =1n

(n∑

i=1

ρi(t)

)∀t > 0, (28)

which is the averaged position error of all the robots in Q,which is plotted in Fig. 3 (b).

Finally, for the consensus error let us define the quantityca as

ca =n∑

i=1

∥∥∥∥∥∥

n∑

j=1

(ai − aj)

∥∥∥∥∥∥

2

, (29)

which shows the summation of the squared norm of thevector

∑nj=1(ai − aj) over the whole population of robots

and is plotted in Fig. 3 (d).In the plots in Fig. 3 (a), (c) and (d) it is easy to note the

asymptotic convergence to zero after every switching timets, but in the case of αi this is difficult to see because theapproximation of the numerical integrals of the centroidsin (2) induces some noise in the trajectory of the robots.Furthermore, notice that the transitions of αi from −π to πlook like spikes in the plot, however, the robots spend themajority of the time oscillating around the angle αi = 0 asthe filtered red signal illustrates in Fig. 3 (b).

In Fig. 4 we present some snapshots of the robot distribu-tion over the piecewise dynamic density function at differenttimes just before the switching time ts happens. Notice thecentroidal Voronoi tessellations shown in Fig. 4 (b) – (d).

V. EXPERIMENTAL VERIFICATION

This section describes the details of the experimentscarried out using our multivehicle testbed.

A. Experimental Design

The experiments were carried out using a population offour P3-AT robots, sensing a white light concentration in arectangular sampling space of 4.7 × 6.6 m. Similar to thesimulations, the sampling space was divided into a 8 × 8grid. The geometric center of each rectangular division cor-responds to the mean of a bidimensional Gaussian functiongiven by (26) with σ2

i = 0.7 m.The adaptation law is a differential equation, which is

difficult to implement in real-time applications, where thearithmetic precision and computational power are limited.

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0 50 100 150-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

Par

amet

er E

stim

atio

n E

rro

r

Time (s)0 50 100 150

-4

-2

0

2

4

Time (s)

An

gle

Err

or

αα αα i

Angle ErrorFiltered Angle ErrorAveraged Angle Error

(rad

)(a) (b)

0 50 100 1500

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Mea

n E

st. P

osi

tio

n E

rro

r (m

)

Time (s)0 50 100 150

0

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nse

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Fig. 3. Plots of the parameter estimation error K(q)T ¯ai(t), the errordistance ρ(t), the angle αi and the consensus error ca for the simulation.

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(c) t = 39 s (d) t = 99 s

Fig. 4. Simulation in Matlab with a population of 20 nonholonomic robotsrepresented by the blue triangles over a dynamic density function indicatedby the multicolor ring-shape contour. The yellow color of the contour meansmaximum density. This results correspond to the simulation changing ai =β at every switching time ts. The Cyan circles represent the estimatedcentroids CVi

of the i-th robot and the green lines surrounding the robotsform their Voronoi cells. Notice the centroidal Voronoi tessellations in (b)–(d).

The following approximations of (9), (11) and (12) based on[21]

λi(t + 1) = λi(t) +K(pi(t))φi(t), (30)Λi(t + 1) = Λi(t) +K(pi(t))K(pi(t))T , (31)

aipre = ai + ξ(Λiai − λi)− ζ∑

j∈Ni

(ai − aj),(32)

ai = max(aipre , β). (33)

were used in order to carry out the adaptation law calculationin real time. The parameter values we used in the approxima-tion of the adaptation law given by (30) – (33) are ξ = 1000,ζ = 1 and δ = 1. The matrix Iproji

defined by (13) now isreplaced by the max operation in (33) β = 0.1.

The light concentration is dynamic under the assumptionspresented in Section III-C. There is one switching timets to switch between two different light sources at 108 sof the experiment. The gyroscope and the wheel encodersembedded in the robots are used for robot positioning. A setof four Phidgets light precision sensors [22] are set up atthe top of the robots and the network communication withthe robots is carried out using Player 3.0.0 [23] through aLinksys wireless router.

The Voronoi partitions vertices are calculated using thelibrary Voro++ version 0.3 [24] and the centroid integralswere approximated by Riemann summations discretizing theinside of the polygons in an 8×8 grid and adding the volumesof the hexahedra corresponding to every division.

B. Experimental Results

In Fig. 5 (a) and (b) we show the behavior of the errordistance ρ(t) defined in (28), and the consensus error ca

given by (29). Since we do not know the real parameters tomodel the light distribution we cannot calculate ¯ai(t).

Notice the convergence of the signal in Fig. 5 (a) and(b) which are visibly affected by the noise of the realmeasurements and the numeric approximation of the centroidintegrals. In order to make them clear we plot the filteredsignal in red to show convergence. Furthermore, the effectof the adaptive law approximation given by (33) is evidentin Fig. 5 (b), where the constant intervals correspond to themoment when the robots are going to their Voronoi cellcentroid. The discontinuous edges of the signal representthe moment when the robots take a light measurement andcalculate the centroids again.

In Fig. 6 (a) and (b) we illustrate a snapshot of theexperiment and the averaged light distribution estimatedby the robots at t = 107 s. The light distribution doesnot have units in the z axis because the approximation ofthe adaptation law by difference equations does not givethe true distribution function values. However, the peaksof the surface are well located according to the real lightdistribution.

VI. CONCLUSIONS

In this paper we have developed an adaptive controllerfor deployment of nonholonomic sensor networks to carry

CONFIDENTIAL. Limited circulation. For review only.

Preprint submitted to 49th IEEE Conference on Decision and Control.Received April 2, 2010.

0 50 100 150 200 250 3000

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tio

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(a) (b)

Fig. 5. Plots of the error distance ρ(t) and the consensus error ca for theexperiments.

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2-2

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x 1010

Y (m)X (m)

(a) (b)

Fig. 6. The picture in (a) shows a snapshot of an experiment with four P3- AT robots sensing a light distribution at 107 s. The figure in (b) showsthe estimated density function averaged over the four robots.

out a coverage and estimation of a parameterizable densityfunction in a convex sampling space. We provided a stabilitytheorem which states that the robots distribute themselvesin an optimal way over the density function solving thelocational optimization problem. The mobile sensors weremodeled as unicycle vehicles and a nonlinear steering controllaw in polar coordinates was used to drive them and guaran-tee stability. Through simulations in Matlab we verified ourtheoretical results. Some experiments using a team of fourP3-AT robots illustrates the gap between theory and practice.However, in despite of that, the theoretical results still worksince the system curves exhibit the convergence stated byTheorem 2.

VII. ACKNOWLEDGMENTS

The work was supported by DOE URPR (UniversityResearch Program in Robotics) grant DE-FG52-04NA25590and by NSF grants ECCS CAREER #0811347, IIS#0812338, and CNS #0709329.

The authors would like to thank Francisco Rodriguez andAndres Cortez of The MARHES Lab at the University of NewMexico, for their assistance in the experiments reported thiswork.

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CONFIDENTIAL. Limited circulation. For review only.

Preprint submitted to 49th IEEE Conference on Decision and Control.Received April 2, 2010.