Distributed Coverage Control with Sensory Feedback for Networked Robots

Transcription

1 Dstrbuted Coverage Control wth Sensory Feedback for Networked Robots Mac Schwager, James McLurkn, and Danela Rus Massachusetts Insttute of Technology Computer Scence and Artfcal Intellgence Lab Cambrdge, MA USA Emal: and Abstract Ths paper presents a control strategy that allows a group of moble robots to poston themselves to optmze the measurement of sensory nformaton n the envronment. The robots use sensed nformaton to estmate a functon ndcatng the relatve mportance of dfferent areas n the envronment. Ther estmate s then used to drve the network to a desrable placement confguraton usng a computatonally smple decentralzed control law. We formulate the problem, provde a practcal control soluton, and present the results of numercal smulatons. We then dscuss experments carred out on a swarm of moble robots. (a) Numercal Smulaton (b) Robot Swarm Experment I. INTRODUCTION We consder the problem of controllng networked groups of moble robots n a decentralzed fashon. Such moble sensor networks promse the ablty to collect nformaton over dstrbuted, large-scale domans wth mnmum nfrastructure mantenance. Ths technology wll enable scentfc studes on geologcal and ecologcal scales prevously beyond practcal reach, and provde tools for a host of securty and survellance applcatons. In ths paper, we present a decentralzed control law for producng ncreased robot densty n areas of hgh mportance and decreased densty n areas of low mportance. Specfcally, we consder a group of robots that s dspatched over a bounded regon of nterest. The group s task s to sample a sensory functon over the regon. The sensory functon s a scalar functon unknown to the robots that ndcates the relatve mportance of dfferent areas n the regon. Our soluton composes an approxmaton of ths functon from sensory measurements. A decentralze control law then uses ths approxmaton, as well as neghbor postons, to drve the robots to a confguraton such that the samplng of the sensory functon s near-optmal. Ths enables the network to record observatons about the envronment wth varyng resoluton, so that areas wth hgh values of the sensory functon receve hgher-densty data observatons than areas wth low values. A. Relatonshp to Pror Work Ths problem belongs to the general category of coverage problems. Varous strateges have been ntroduced for controllng coverage wth networked moble robots, and our work bulds on several mportant results n ths category. In [9], moble sensng agents are controlled usng potental functons for nter-agent nteractons. Stablty results are derved, but Fg. 1. The control strategy was mplemented n numercal smulaton and on a swarm of moble robots. the optmalty of the network confguraton s not addressed. Smlarly, n [2] an algorthm s proposed that allows for agents to concentrate n areas of hgh event densty whle mantanng area coverage constrants. The algorthm s proved to mantan sensor coverage for a lmted case wthout addressng optmalty. Most relevant to ths paper s a body of results reported n [1], [5], and [4]. In ths work decentralzed control laws are derved for postonng moble sensor networks optmally wth respect to a known event probablty densty. Ths approach s advantageous because t guarantees that the network (locally) mnmzes a cost functon relevant to the coverage problem. However, the control strategy requres that each agent have a complete foreknowledge of the event probablty densty, thus t s not reactve to the sensed envronment. Our control strategy s an extenson to the one reported n [5]. We re-nterpret the problem n a non-probablstc framework, and derve a local control law whch requres that each agent can measure the value and gradent of a sensory functon. In contrast to [5], the robots do not requre foreknowledge of ths functon. Instead, the robots approxmate the functon from sensor measurements whle mantanng or seekng a near-optmal sensng confguraton. Also n contrast to [5], our functon approxmaton yelds a control law wth a computatonally effcent closed form. Ths elmnates the need for numercal ntegraton of an arbtrary functon over a polygonal doman at every tme step. The control law results n near-optmal, as opposed to optmal, performance, though the dfference s shown to be neglgble n practce. We demonstrate the effectveness and computatonal smplcty of

2 the algorthm n numercal smulatons and n experments on swarms of robots (see Fgure 1). II. PROBLEM FORMULATION In ths secton, we buld a functon representng the sensng cost assocated wth a network confguraton. A network s sad to have optmal coverage f t mnmzes ths cost functon over the regon of nterest. Followng standard results n the feld [5], we show that all confguratons of a certan type, namely centrodal Vorono confguratons, correspond to local mnma of the cost functon. The sensor network conssts of a group of dentcal robots, each wth some degree of moblty and the capacty for measurng a sensory functon from the envronment. The sensory functon ndcates the relatve mportance of dfferent areas n the envronment. It may be a quantty that s sensed drectly, such as temperature, or t may demand more elaborate processng of sensory data, such as would be requred to detect the concentraton of a chemcal, or the ntensty of sound of a partcular frequency. 1 In addton, we assume that a robot can measure the postons of ts Vorono neghbors relatve to tself, and that t can detect the boundares of the regon of nterest. We revew the formalsm ntroduced n [5] to rgorously model the scenaro descrbed above. Let there be n robots n a known, convex polytope Q R N. An arbtrary pont n Q s denoted q, the poston of the th robot s denoted p, and the set of all robot postons s denoted P = {p 1,..., p n }. Let W = {W 1,..., W n } be a partton of Q such that one robot at poston p les wthn each regon W. Defne the sensory functon, φ(q) : Q R +, as a scalar functon wth contnuous frst dervatves over Q. The functon φ(q) s not known by the robots n the network. Let the unrelablty of the sensor measurement be defned by a functon f(x) whch s strctly ncreasng. Specfcally, f( q p ) descrbes how unrelable s the measurement of the nformaton at q by a sensor at p (henceforth,. s used to denote the l 2 -norm). Ths form of f(x) s physcally appealng snce t s reasonable that sensng wll become more unrelable farther from the sensor. We can formulate the cost ncurred by one robot sensng over one regon W as h (p, W ) = f( q p )φ(q)dq. W Notce that unrelable sensng s expensve and hgh values of φ(q) are also expensve. Summng over all robots, a functon representng the overall sensng cost of a gven network confguraton can be wrtten n H(P, W) = f( q p )φ(q)dq. (1) W =1 An optmal network confguraton corresponds to a partcular par (P, W) whch mnmzes (1). 1 In contrast, Cortés et al [5] use a probablty densty functon descrbng the lkelhood of an event occurrng n a partcular area. To solve ths mnmzaton problem, we must ntroduce the noton of a Vorono partton. The Vorono regon, V, of a gven robot s the regon of ponts that are closer to that robot than to any other, that s V = {q Q q p q p j, j }. The dvson of an area nto such regons s called a Vorono partton, denoted V(P ), and s a functon of the robot postons. We wll use the shorthand H V (P ) = H(P, V(P )). Because the functon f(x) s strctly ncreasng, the Vorono partton, V, mnmzes the cost functon, H, for any fxed confguraton, P, of robots. Ths s clear snce, for an arbtrary pont q Q, q V gves the smallest value of f( q p ) over, and therefore the smallest contrbuton to H. Thus we have mn H = mn H V. P,W P To fnd local mnma of H V, we examne solutons to the expresson H V = [ H T V ] T = 0. p It s clear that each partal dervatve must be zero for a local mnmum. Applyng a mult-varable generalzaton of Lebnz Rule, 2 we can move the dfferentaton nsde the ntegral sgn n (1) to get H V p = f( q p ) φ(q)dq + V p f( q p )φ(q) V j n j dq + V j p j N V f( q p )φ(q) V p n dq, (2) where V denotes the boundary of the regon V, n (q) denotes the outward facng normal of V, and N s the set of ndces of the neghbors of p, excludng tself. Note that all the ntegrals n (2) are N 1 vectors snce V p s an N N matrx and n s an N 1 vector. We assert that the last two terms of (2), n fact, sum to zero. A proof can be outlned as follows. Snce p only affects V j at the shared boundary of V j and V, we have that V j = V. p p j N Also, an nward normal, n, for V s equal to an outward normal, n j, for any of ts neghbors V j, at the boundary whch they share. Ths leads to f( q p )φ(q) V j n j dq = V j p j N V f( q p )φ(q) V p n dq, 2 Ths procedure s known n flud mechancs as the Reynolds Transport Theorem.

3 gvng the desred result. Usng ths fact, (2) can smply be wrtten H V f( q p ) = φ(q)dq. p p V We can evaluate the partal dervatve of f(x) usng the chan rule, and move p outsde of the ntegral to gve H V q df(x) = p V q p dx φ(q)dq + q p 1 df(x) p V q p dx φ(q)dq (3) q p Next we defne two propertes analogous to mass-moments of rgd bodes. The mass of V s defned as 1 df(x) M V = V q p dx φ(q)dq, (4) q p and the centrod of V s defned as C V = 1 q df(x) M V q p dx φ(q)dq. (5) q p V Note that f(x) strctly ncreasng and φ(q) strctly postve mply both M V > 0 V { } and C V V \ V (C V s n the nteror of V ). Thus M V and C V have propertes ntrnsc to physcal masses and centrods. Substtutng these quanttes nto (3) gves H V p = M V (C V p ). (6) Equaton (6) mples that local mnma of H V, and therefore H(P, W), correspond to the confguratons, P, such that p = C V, that s, each agent s located at the centrod of ts Vorono regon. We wll denote the set of all such centrodal Vorono confguratons as P C. Thus, the optmal coverage task s to drve the group of robots to a centrodal Vorono confguraton, P P C. III. ESTIMATED ERROR FEEDBACK CONTROL We wll desgn a control law to take advantage of the surprsngly smple result n (6). The control law wll drve the network to an estmated centrodal Vorono confguraton usng sensory data avalable to the robots to form an on-lne approxmaton of the centrods of ther Vorono regons. Assume that the dynamcs of each robot can be modeled by the frst-order equaton p = u, (7) where u s the control nput. Ths mght mean smply that a low-level controller s n place to enforce frst-order dynamcs. Next, prescrbe the lnear proportonal control law u = k(ĉv p ), (8) where ĈV s an estmate of C V based on the nformaton avalable to robot. To nvestgate the stablty of such a control law we propose to use H V (P ) as a Lyapunov-lke functon. Takng the tme dervatve of H V (P ) along the trajectores of (7) gves Ḣ V = and usng (6) and (8) we get Ḣ V = k T H V p, p n M V e T ê, (9) =1 where e = (C V p ), and ê = (ĈV p ). The actual error, e, s unknown. Notce, however, that f an estmate, ê, can be found such that the nner product of the two errors s postve, convergence of ê to zero wll be guaranteed. Geometrcally, ths means that the angle between ê and e must reman less than π/2 rad for all tme. We use ths nsght to desgn a centrod estmate, Ĉ V, usng only sensed nformaton local to robot. Henceforth we wll deal wth the specfc case n whch f(x) = 1/2x 2. Ths causes the factor 1 q p df(x) dx q p to become unty, makng the proceedng expressons more transparent. The method presented, however, s vald for any strctly ncreasng f(x) wth smooth frst dervatves. A. Control Usng Lnear Approxmatons Consder a stuaton n whch the values of the sensory functon, φ(p ), and ts gradent, φ p, are avalable contnuously to robot at ts current poston. In ths case, the avalable nformaton motvates a lnear estmate of C V over the known regon V. We defne the lnear approxmaton to C V calculated from an agent at poston p as where Ĉ V = 1 q ˆM V V ˆφ (q)dq, (10) + ˆM V = ˆφ (q)dq, (11) = V {q ˆφ (q) 0}, and (12) ˆφ (q) = φ(p ) + φ T p (q p ). (13) The above formulaton follows naturally from the defnton of C V and M V n (5) and (4). The ntegrals are taken over to avod calculatng values of ˆMV 0 and values of ĈV outsde of the regon V. We can prove the stablty of the proposed controller n the case of lnear φ(q). In ths case the functon φ(q) s fully parameterzed by ts value and gradent as measured at any pont, therefore ˆφ (q) = φ(q). Ths specal case becomes mathematcally equvalent to that treated n [5]. Then ĈV = C V and from (9) we have that n Ḣ V = k M V e T e. =1

4 As was noted prevously, M V > 0, therefore ḢV 0 p, and ḢV = 0 ff p = C V. Addtonally, P C s the largest nvarant set snce, from (7) and (8), ṗ = 0 P P C. Then by LaSalle s theorem lm t p = C V whch mples P P C. In the case of nonlnear φ(q), we observe that the control law wth a lnear estmaton causes the robots to converge to the estmated centrod, Ĉ V, of ther Vorono regon. We call such confguratons near-optmal. It s dffcult to bound the error between the estmated centrod and the actual centrod n the general N-dmensonal case above. It s however possble to do so for 1-dmensonal systems, where robots can move along an arbtrary curvlnear segment n three-space (e.g. a robot movng along a track.) B. Effcent Computaton of Integrals We wll use the convenent form of the centrod estmaton above to derve an analytcal soluton to the centrod ntegral. The analytcal expresson wll make the control law feasble for robot platforms wth lmted computatonal resources. Ths elmnates the need to descretze the Vorono regon and approxmate the ˆM V and ĈV ntegrals n a computatonallyexpensve numercal procedure. Snce the estmated ˆφ s polynomal n q, we can use the results from [3] to fnd the ntegrals for ˆMV and ĈV as polynomals n the vertces of the Vorono regon V. Frst, from (10), (11), and (13) we can dvde the ntegral expresson for ĈV and ˆM V nto monomal components to get where Ĉ V = 1 ( φ(p ) φ ˆM T ) p p qdq + V 1 q ˆM V V T qdq φ p, (14) + ˆM V = ( φ(p ) φ T ) p p p φ T p dq + qdq. (15) These expressons can be smplfed further by ntroducng the constants c 1 = ( ) [ ] φ(p ) φ T T p p, and c1 c 2 = φ p, and by defnng a general ntegral of a monomal over a polygon as I αβ = x α y β dxdy, (16) where q = [ x Ĉ V = c 1 ˆM V where y ] T. Then we can wrte (14) and (15) as ] [ ] + 1 I 20 I 11 [ c2 ˆM V c 3 [ I 10 I 01 ˆM V = I 00 [ c 1 + I 10 I 11 I 01 I 02 ] [ c 2 c 3 ], (17) ]. (18) The soluton of the ntegral n (16) s gven n analytcal form by equaton (4) from [3]. The cases specfcally requred for the computaton of (17) are smplfed and enumerated below: I 00 I 01 I 10 I 11 I 02 I 20 = 1 2 = 1 6 m (y +1 y )(x +1 + x ), =1 m [(x x +1 )(y y +1 y + y 2 ) + =1 3(x +1 y+1 2 x y 2 )], = 1 m (y +1 y )(x x +1 x + x 2 ), 6 = 1 24 =1 m [(y +1 y )(2x +1 x (y +1 + y ) + =1 x 2 +1(3y +1 + y ) + x 2 (y y ))], = 1 m [(x x +1 )(y y y + =1 y +1 y 2 + y 3 ) + 4(x +1 y+1 3 x y 3 )], = 1 m (y +1 y )(x x x + =1 x +1 x 2 + x 3 ), where m s the number of vertces, [ x y ] T, of, and where the ndex m + 1 s nterpreted as 1. The vertces must be ordered counter-clockwse around the permeter of. The expresson n (17) wth the assocated expressons for the ntegral terms can be computed drectly to gve the actual value of ĈV and ˆM V. C. Practcal Algorthms A practcal method for mplementng the proposed control law on a network of robots s detaled n Algorthm 1. We assume that the robots have access to a procedure for obtanng ther own Vorono regon. Several such algorthms exst, for example those gven n [5], [6]. In our hardware mplementaton, we use the Delaunay computaton from [7] and buld Vorono regons usng knowledge of the Delaunay neghbors. Ths algorthm s decentralzed, fully dstrbuted, and requres mnmal communcaton between neghborng robots. It can be used on teams of large robots, on teams of small robots such as [8], or on moble sensor network nodes wth lmted computaton and storage capabltes such as the moble Mca Motes descrbed by [10]. A. Implementaton IV. NUMERICAL SIMULATIONS Smulatons were carred out n a Matlab envronment. The dynamcs n (7) wth the control law n (8) for a group of robots were modeled as a system of coupled dfferental equatons. Vorono computaton was carred out n a centralzed fashon usng standard Matlab Vorono functons. The centrod was calculated usng the analytcal soluton to the centrod

5 Algorthm 1 The Coverage Control Algorthm Requre: Each robot can compute ts local Vorono regon Requre: Each robot s sensor can measure φ(p ) and φ p Requre: Each robot can sense the locaton of the boundary of the space of nterest, Q loop Measure coordnates of neghborng robots Compute local Vorono regon, V Measure φ(p ) and φ p Truncate V to get Evaluate analytcal expresson n (17) to compute centrod approxmaton, Ĉ V Apply control nput u = k(ĉv p ) end loop (a) Intal Confguraton (b) Robot Trajectores ntegral gven n (17). A custom, fxed-tme-step numercal solver was used to ntegrate the equatons of moton of the group of robots. The sensory functon, φ(q), was bult from Gaussans. Specfcally, two cases were nvestgated. For the frst case, the functon was chosen as φ(q) = γ ( ) σ e (q µ 1 ) 2 2σ 2 + e (q µ 2 )2 2σ 2. (19) 2π A functon wth multple maxma was used to llustrate the robustness of the control scheme to complcated sensng envronments. For the second case, a sngle Gaussan was used of the form φ(q) = γ σ (q µ 1 ) 2 2π e 2σ 2. (20) Ths φ(q) was chosen to gather statstcal data about convergence propertes over a number of runs. The parameters used for the Gaussan functons were γ = 1, σ = 1/ 2, µ 1 = (.2,.2), and µ 2 = (.8,.8). The control gan was k = 5, the regon Q was set to be a square of 1 meter on each sde, and q was a Cartesan pont [ x y ] T Q. B. Results Fgure 2 shows the results of a typcal smulaton run wth the φ(q) gven by (19). The ntal confguraton of the network s shown n Fgure 2(a), the trajectores of the robots (dashed lnes) n Fgure 2(b), and the fnal confguraton n Fgure 2(c). The centers of the Gaussan functons, µ 1 and µ 2, are marked wth x s. In Fgure 2(d), the tme evoluton of the x and y coordnates of one robot are shown. The performance of the control scheme s clearly demonstrated n the smulaton. To assess convergence propertes of the algorthm, statstcal results were compled over a number of smulaton runs wth random ntal confguratons usng the φ(q) gven by (20). Three batch runs were executed: one batch of 1000 smulaton runs wth 10 robots, one of 100 smulaton runs wth 100 robots, and one of 10 smulaton runs wth 1000 robots. Each smulaton run was sad to converge f the mean normed error of the group of robots was found to be less than m. The mean and standard devatons of the convergence tmes are shown n Table I. (c) Fnal Confguraton (d) Poston of One Robot Fg. 2. The ntal confguraton of the network s shown n 2(a), the trajectores of the agents (dashed lnes) n 2(b), and the fnal confguraton n 2(c). The Gaussan centers of φ(q) are marked by x s. In 2(d), the tme evoluton of the x and y coordnates of one agent are shown. Mean (s) Standard Devaton (s) 10 Robots 1000 Trals Robots 100 Trals Robots 10 Trals TABLE I MEAN AND STANDARD DEVIATION OF CONVERGENCE TIMES The mean normed errors are shown n Fgure 3 for all smulaton runs for each of the three batches. The plots are gven on sem-log axes to emphasze that after some tme, the convergence rate of the closed-loop system appears to be nearly exponental. It s nterestng to note that, although covergence rates decrease as the network sze ncreases, the tme to reach a partcular small error value does not necessarly ncrease, as shown n Table I. One can nterpret ths effect as the result of two opposng factors. Wth few robots, global movement of the network can occur rapdly because the moton of one robot quckly propagates to nfluence all of the others. Ths propagaton effect becomes more sluggsh as the network sze ncreases. However, wth a large number of robots, there s a hgh lkelhood that the ntal poston of any one robot s close to ts fnal poston. Thus each robot has less dstance to cover. The push-and-pull of these two factors creates the effect that s evdenced n the data. Notce also that the varance appears to decrease wth ncreasng network sze. Ths s a statstcal sde-effect. Because the plots show mean normed error, a large network wll be more lkely to have a mean

6 (a) 10 Robots, 1000 Trals (b) 100 Robots, 100 Trals (a) 10 Robots, 1000 Trals (b) 100 Robots, 100 Trals (c) 1000 Robots, 10 Trals Fg. 3. The trajectores of the mean normed error of the robots are shown for 10, 100, and 1000 robots over 1000, 100, and 10 trals respectvely. The plots are on log-log axes to show the near exponental error decay. (c) 1000 Robots, 10 Trals Fg. 4. The steady state densty of robots as a functon of dstance from the Gaussan center are shown for the three batch runs. close to the ensemble mean than a small network. The fnal dstrbuton of the robots wth respect to dstance from the Gaussan center s shown for all three batches n Fgure (4). The denstes appear more jagged for smaller networks because there are fewer centrodal Vorono confguratons for smaller groups of robots. The denstes are not precsely Gaussan, nor are they meant to be. But there are hgh concentratons of robots where the φ(q) s large, ndcatng an area of sensory nterest, and low concentratons where φ(q) s small, where there s lttle sensory nterest. V. HARDWARE EXPERIMENTS We mplemented and tested the algorthm on a group of moble SwarmBots. A lght source was used to create a sensory functon φ(q) of lght ntensty. The robots used readngs from lght sensors and neghbor localzaton to carry out Algorthm 1. A. The Swarm Hardware Each SwarmBot (Fgure 5) s autonomous and s equpped wth bump sensors, lght sensors, and an nfra-red nterrobot communcaton and localzaton system called ISIS. The lght sensors detect the sensory nput. The ISIS nter-robot localzaton system provdes local neghbor postons used to compute the Vorono cells. We lmted the range of the localzaton system to one meter to ncrease the dameter of the network and mnmze the number of nter-robot packet collsons. Ths algorthm uses no communcatons other than the localzaton messages, allowng us to test t wth many robots n a small physcal space. Fg. 5. The Robot SwarmBot s desgned for dstrbuted algorthm development. Each SwarmBot has an nfra-red localzaton and communcaton system called ISIS whch enables nearby robots to communcate and determne the bearng, orentaton, and range of ther neghbors. An omn-drectonal bump skrt provdes low-level obstacle avodance. A 40 MHz 32-bt ARM Thumb mcroprocessor provdes enough processng power for our algorthms. B. Implementaton of the Control Algorthm The mplementaton of Algorthm 1 on the SwarmBot system requred several modfcatons. The robots had a lmted communcaton range of radus one meter, therefore the Delaunay neghbors computed by any robot were only those Delaunay neghbors wthn the one meter dsk of the robot. These local Delaunay graphs mght not be the same as ther global counterparts. We assumed that edges of greater than length one meter are uncommon n the trangulated graphs we consder and the effects of removng them have lttle mpact

7 Sngle Neghbor r r r r On The Edge Fg. 6. The method used for closng the unbounded Vorono regon of a robot wth 1 neghbor s shown on the left. The method for an unbounded Vorono regon of a robot wth any two consecutve neghbors separated by more than π rad s shown on the rght. on the fnal result. 3 In addton, Algorthm 1 requres that each robot can detect and localze the boundares of the regon Q. Ths s necessary to prevent the Vorono regon of any robot from beng unbounded. We dd not mplement boundary sensng on the SwarmBot. Instead, we developed three heurstc rules for truncatng nfnte Vorono regons. Unbounded Vorono regons occur n three dstnct cases for robots on the boundary of the network. These are enumerated below along wth the assocated truncaton technque. 1) No Vorono Neghbors: The centrod s computed to be the robot s current poston, thus the robot does not move untl t acqures a neghbor. 2) One Vorono Neghbor: A Vorono regon s constructed from an sosceles trangle of constant sze whose base les on the bsector between the two robots (see the left of Fgure 6). 3) Two Consecutve Vorono Neghbors Separated by an angle π: A lne s added perpendcular to the bsector of the nfnte regon at a constant dstance from the robot (see the rght of Fgure 6). The two ntersecton ponts of ths lne wth the unbounded Vorono regon are taken as vertces of the truncated Vorono regon. Also, the robots lght sensors returned a bearng and dstance. Ths was not enough nformaton to compute φ p drectly. To overcome ths problem, we fxed the magntude of φ p and determned ts drecton from the sensory stmulus. The sgnal strength φ(p ) was measured drectly. Fnally, the SwarmBot had a low-level controller n place whch allowed the robot to move to a partcular pont relatve to ts current poston. We used ths poston control as a substtute for velocty control. In partcular, at each tme nstant, the nput to the poston controller was gven as the estmated centrod value ĈV. Ths turned out not to be a 3 A Vorono based control scheme wth lmted range communcaton was shown n [4] to have smlar convergence propertes as the case wth unlmted communcaton range. dffculty snce the low level control-loop was already well suted for the experments. The computatonal and memory requrements for ths algorthm were small, and modfcaton of the algorthm to run wth nteger calculatons was straghtforward. On the 40MHz ARM Thumb processors used n the SwarmBots, the memory usage for all the steps outlned n Algorthm 1, ncludng the specal cases dscussed above, was 3949 bytes of code and 1284 bytes of RAM. Wth 8 neghbors, one cycle of the algorthm ran at 70ms, fast enough for real-tme poston updates. C. Experments We measured the performance of the algorthm n two sets of experments. For the frst set of experments, a factor was ntroduced to chnage the robots fnal confguraton. In partcular, f we defne φ(p ) as the lght ntensty measured by the robots lght sensors at p, we chose a DC offset, Φ Offset to create a φ functon φ = Φ Offset + φ. Ths had the effect of reducng the nfluence of the measured lght gradent. Thus fnal robot densty vared nversly wth the value of Φ Offset. Three runs were performed for each of sx values of Φ Offset : 800, 400, 200, 100, 50, and 10. In each expermental run, robots were started unformly dspersed n a dark, 8 8 workspace. A lght source was placed at the mddle of one of the permeter walls of the workspace. The robots were gven two mnutes to reach a fnal confguraton. Photographs of the fnal robot confguratons are shown for each Φ Offset value n Fgure 7, along wth plots of the fnal robot denstes as a funton of dstance from the lght source. A second set of experments was carred out to quantfy the actual error of the robots over the course of a sngle run. Twenty robots were started consentrated n one corner of the 8 8 workspace. The lght ntensty was controlled to be as unform as possble over the workspace. Specal statonary robots were evenly spaced along the permeter of the workspace, fve along each wall. These were used to help ensure that the vorono regons of the actve robots remaned bounded throughtout the experment wthout havng to use the heurstcs descrbed above. The 20 actve robots ran algorthm 1 for a total of 3 mnutes. They were stopped at regular tme ntervals throughout the 3 mnutes, and ther postons were measured. The poston measurements were used to compute, off lne, the error between each robot s poston and the centrod of ts Vorono regon. The tme hstory of the mean error measured n ths way s shown n Fgure 9. Note that the error dmnshes over tme as would be expected. VI. CONCLUSION In ths paper, a decentralzed method for controllng coverage n moble sensor networks was presented. The method s related to one proposed n [5], and ntroduces an nnovaton that allows for the network to adapt to unknown sensory envronments. An analytcal expresson for centrod calculatons was derved to make the algorthm computatonally feasble on small robot platforms. The control scheme was demonstrated n numercal smulaton. The control scheme

8 Fg. 7. The upper row of mages show the fnal postons of 40 robots usng the proposed control law. The sensor offset, Φ, s labeled above each pcture. The graph below each mage plots the normalzed densty of the robots as a functon of ther radus from the lght source. A unform densty would appear as a horzontal lne wth a value of one. are possble f more communcaton overhead s allowed. For example, each robot mght estmate the sensory functon from a quadratc ft of ts own and each of ts Vorono neghbors measured values of φ(q). Theoretcal, numercal, and expermental studes of ths and other methods are ongong. (a) Intal Confguraton (b) Fnal Confguraton Fg. 8. The ntal and fnal confguratons for an experment wth 50 robots s shown. Fg. 9. The mean error between the robots postons and the centrods of ther Vorono regons s shown as a functon of tme for an experment wth 20 robots. Because of sensor nose, we used a fxed error threshold of m. Robots wth desred postons wthn ths dstance remaned statonary, lmtng the convergence of the mean error. was also mplemented on a SwarmBot platform usng nteger calculatons n a lean computng envronment. The ablty to mplement the control scheme on such a platform emphaszes ts practcalty and mnmalst nature. Robust performance was demonstrated n a number of expermental trals. Potental extensons to ths control method are numerous. For example, the approach n ths work used mnmal communcaton among robots, only requrng localzaton to produce a Vorono partton. Shared estmates of the sensory functon ACKNOWLEDGMENT Ths project was supported n part by the NSF, the MURI SWARM project, and Boeng. We are grateful for the support of all our sponsors. REFERENCES [1] F. Bullo and J. Cortés. Adaptve and dstrbuted coordnaton algorthms for moble sensng networks. In V. Kumar, N. E. Leonard, and A. S. Morse, edtors, Cooperatve Control. (Proceedngs of the 2003 Block Island Workshop on Cooperatve Control), volume 309 of Lecture Notes n Control and Informaton Scences, pages Sprnger Verlag, New York, [2] Z. Butler and D. Rus. Controllng moble sensors for montorng events wth coverage constrants. In Proceedngs of IEEE Internatonal Conference of Robotcs and Automaton, pages , New Orleans, LA, Aprl [3] C. Cattan and A. Paoluzz. Boundary ntegraton over lnear polyhedra. Computer-Aded Desgn, 22(2): , [4] J. Cortés, S. Martínez, and F. Bullo. Spatally-dstrbuted coverage optmzaton and control wth lmted-range nteractons. ESIAM: Control, Optmsaton and Calculus of Varatons, 11: , [5] J. Cortés, S. Martínez, T. Karatas, and F. Bullo. Coverage control for moble sensng networks. IEEE Transactons on Robotcs and Automaton, 20(2): , Aprl [6] Qun L and Danela Rus. Navgaton protocols n sensor networks. ACM Transactons on Sensor Networks, 1(1):3 35, Aug [7] Xang-Yang L, Grua Calnescu, and Peng-Jun Wang. Dstrbuted constructon of a planar spanner and routng for ad hoc wreless networks. In Proceedngs of the IEEE INFOCOM, pages , New York, NY, June [8] James McLurkn. Stupd robot trcks: A behavor-based dstrbuted algorthm lbrary for programmng swarms of robots. Master s thess, MIT, [9] P. Ogren, E. Forell, and N. E. Leonard. Cooperatve control of moble sensor networks: Adaptve gradent clmbng n a dstrbuted envronment. IEEE Transactons on Automatc Control, 49(8): , August [10] Gabrel T. Sbley, Mohammad H. Rahm, and Gaurav S. Sukhatme. Robomote: A tny moble robot platform for large-scale sensor networks. In Proceedngs of the IEEE Internatonal Conference on Robotcs and Automaton (ICRA), 2002.