Voronoi coverage of non-convex environments with a group of networked robots

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1 2010 IEEE Internatonal Conference on Robotcs and Automaton Anchorage Conventon Dstrct May 3-8, 2010, Anchorage, Alaska, USA Vorono coverage of non-conve envronments wth a group of networked robots Andreas Bretenmoser, Mac Schwager, Jean-Claude Metzger, Roland Segwart and Danela Rus Abstract Ths paper presents a soluton to decentralzed Vorono coverage n non-conve polygonal envronments. We show that complcatons arse when estng approaches to Vorono coverage are appled for deployng a group of robots n non-conve envronments. We present an algorthm that s guaranteed to converge to a local optmum. Our algorthm combnes classcal Vorono coverage wth the Lloyd algorthm and the local path plannng algorthm TangentBug to compute the moton of the robots around obstacles and corners. We present the algorthm and prove convergence and optmalty. We also dscuss epermental results from an mplementaton wth fve robots. Level 1 Level 2 Update goal: Lloyd algorthm Loop 1 Plan path: TangentBug Loop n 1...n I. INTRODUCTION Dstrbuted coverage s a common applcaton for multrobot systems where robots spread over an envronment to fulfll a percepton task. Eamples nclude envronmental montorng and survellance, mantenance and nspecton, or dstrbuted actuaton tasks, such as harvestng or demnng. An early survey about mult-robot coverage by [1] ntroduces three basc types of coverage problems: blanket, sweep and barrer coverage. Blanket coverage methods deploy sensors, e.g. carred by networked robots, n a statc arrangement to cover an area. Most current results on blanket coverage are focused on conve envronments; however realstc applcatons requre the ablty of copng wth non-conve envronments, ncludng areas wth many free-standng obstacles as well as areas wthout obstacles but wth non-conve boundares. To pave the way toward real-world mult-robot systems, t s essental to enable coverage by groups of robots n such envronments. We address the deployment of a group of networked robots to cover a non-conve envronment n ths paper. We propose a new control strategy that combnes two wellknown algorthms: Lloyd algorthm [11], orgnated from quantzaton theory, and TangentBug [14], a dervatve of the famly of Bug algorthms. Fgure 1 llustrates the concept of our approach: Lloyd algorthm updates the goal poston of each robot, TangentBug then plans a path to the gven Ths work was done n the Dstrbuted Robotcs Laboratory at MIT. Ths work was supported by ALSTOM, and n part by the ARO MURI SWARMS grant , the ARL CTA MAST grant , the ONR MURI SMARTS grant N , NSF grants IIS , IIS , EFRI , and The Boeng Company. Andreas Bretenmoser, Jean-Claude Metzger and Roland Segwart are wth the Autonomous Systems Laboratory (ASL), ETH Zurch, Tannenstrasse 3, 8092 Zurch, Swtzerland, andreas.bretenmoser@mavt.ethz.ch, jc rsegwart@ethz.ch Mac Schwager and Danela Rus are wth the Computer Scence and Artfcal Intellgence Laboratory (CSAIL), MIT, Cambrdge, MA 02139, USA, schwager@mt.edu, rus@csal.mt.edu Fg. 1. Control strategy as combnaton of the Lloyd algorthm and the TangentBug algorthm. Herarchcal approach runnng dstrbuted on n robots: Lloyd algorthm computes the Vorono regon and updates the current target (level 1), the local path planner computes the path to the target and performs obstacle avodance (level 2). A frst control loop (loop 1) s formed between level 1 and level 2, a second control loop (loop 2) s bult nto the path planner on level 2. goal. The consecutve nteracton between the two algorthms results n global convergence of the robot team to a fnal Vorono confguraton that optmzes a cost functon. The control strategy spreads the robots over the envronment whle takng care of the non-convety of the envronment by the bult-n obstacle avodance behavor. As our control strategy combnes mult-robot coverage and robot navgaton, there s a broad body of work wth potental nfluence. Artfcal potental felds, vector feld hstograms and the Bug algorthms are eamples of smple but powerful sensor-based local path planners. The actual coverage control n our control strategy s manly based on a well-known method for coverage of conve envronments wth moble networked robots. Vorono coverage was ntroduced n [2] and further developed n [3], [4] among others. The basc concept goes back to the contnuous p-medan problem, whch s part of the locatonal optmzaton framework [5]. We wll refer to ths control stategy as Vorono coverage n ths paper. Other works have nvestgated the control of multple robots n non-conve envronments. [6] consders the coordnaton of a group of robots to acheve rendez-vous n a nonconve envronment. [7] focuses on coverage to solve the dstrbuted Art Gallery Problem n non-conve envronments. A control approach to drve a mult-robot team to target sets under collson avodance and mantanng promty constrants s presented n [8] for known envronments wth obstacles. In [9] and [10] the envronment s transformed by a dffeomorphsm to a correspondng conve regon, n /10/$ IEEE 4982

2 whch regular Vorono coverage can be appled. The method offers a creatve soluton to non-conve coverage, however the authors note that t presents sgnfcant computatonal challenges, and that t may lead to solutons that dffer from the optmal coverage solutons n the orgnal space. In [10] the authors resolve some of these ssues for conve regons wth solated obstacles. [4] approaches coverage of envronments wth non-conve boundares by applyng the geodesc dstance measure to Vorono coverage. Ths method provdes an elegant soluton for some classes of envronments but s not guaranteed to work for all types of non-conve envronments. Ths paper s organzed as follows. In Secton II we present basc concepts of the Vorono coverage method for conve envronments. We pont out complcatons of the method n non-conve envronments and ntroduce our new approach to the problem. In Secton III we descrbe the algorthm and ts mplementaton n detal. We prove convergence and optmalty of our control strategy n Secton IV. Smulatons n Secton V and physcal eperments n Secton VI wth a group of networked robots n a non-conve envronment further verfy the concept. We conclude n Secton VII and pont to some nterestng etensons of our approach. II. PROBLEM FORMULATION A group of networked robots must be deployed n an envronment. The envronment can ether be conve (formng a conve doman wthout any obstacles or holes n t) or t can be non-conve (ncludng free-standng obstacles or holes and areas wth non-conve boundares). Our focus s on non-conve envronments, but our approach also apples for conve envronments. A. Conve envronments A wdely studed class of solutons to coverage n conve envronments uses Vorono coverage to optmze the confguraton of n robots by mnmzng the average dstance to the nearest ponts of nterest. A team of n robots at postons P = [p ] n =1 RNn, navgate n a bounded polygonal envronment Ω R N. Ω s a closed set wth the boundary Ω. For the Eucldean dstance measure wth denotng the l 2 - norm, a Vorono tessellaton V = {V } n =1 s a partton of Ω for whch V = { Ω p p j, j = 1,...,n, j }. (1) Each V s the Vorono cell correspondng to a robot located at p. The postons p are called the generatng ponts, the robots are the generators of the Vorono tessellaton. Gven a postve densty functon φ : Ω R >0, we can defne the mass and the centrod of V as M V = V φ() d, C V = 1 M V V φ() d. (2) A Vorono tessellaton becomes a centrodal Vorono tessellaton (CVT) when the generatng ponts p concde wth the centrods,.e. p = C V, {1,...,n}. In ths case we say the robots reach a centrodal Vorono confguraton. CVTs are known to locally mnmze the cost functon H (P) = n =1 H (p ) = n =1 V f(d(p, )) φ() d, (3) wth D( ) a dstance measure between a pont p and a locaton n Ω. The densty functon φ( ) descrbes the weghtng or mportance of dfferent areas n Ω. The functon f : R 0 R 0 s assumed to be smooth and strctly ncreasng. f(z) = 1 2 z2 s our choce n the remander of the paper for smplcty. It accounts for the sensor relablty of a robot and adds to the cost as the dstance from the sensor ncreases. 1) Eucldean dstance: The Eucldean dstance D(p,) = p s a standardzed choce for open space and conve envronments. Usng the Eucldean dstance we get from (3) H (p ) p = M V (C V p ) = 0, (4) where M V and C V were defned n (2). The local mnma are reached wth a CVT, snce C V = p mples H p = 0, {1,...,n}. 2) Geodesc dstance: The geodesc dstance measure s another ntutve choce, especally n the contet of non-convety [4]. The geodesc dstance s the length of the shortest path from a start pont p to a goal pont, so that the path les completely n the doman Ω. A determnstc method to construct centrodal Vorono tessellatons s the classc Lloyd algorthm [11]: 1) construct the Vorono partton for the generatng ponts at [p ] n =1, 2) compute the centrods of these Vorono regons, 3) set the new locatons of [p ] n =1 to the centrods and start all over agan. Vorono coverage s based on the Lloyd algorthm and ts contnuous and dscrete-tme versons. From (4) the proportonal control law (here shown for the Eucldean dstance measure) u =. p = k M V H (p ) p = k(c V p ), (5) wth smple frst-order dynamcs ṗ = u can drectly be derved. From (5) t becomes clear that the Lloyd algorthm mplements a gradent descent controller. The convergence propertes of the Lloyd algorthm are well establshed. We repeat them here as t wll be useful later n our analyss. Lemma 1 (Convergence of Lloyd algorthm): A set of ponts n a gven envronment converges under the Lloyd algorthm to a centrodal Vorono confguraton. Proof: Convergence follows from Prop. III.3 n [2]. B. Complcatons from non-conve envronments We demonstrate the possble dffcultes that arse for the coverage problem n a non-conve envronment wth the dfferent eample confguratons shown n Fgure 2(a) (d). For clarty of our eplanaton, we base our analyss on a sngle robot n a non-conve envronment. But the consderatons also hold true for the mult-robot case, where 4983

3 (a) (b) (c) (d) Fg. 2. Non-conve eample envronments (whte crcle: robot ntal poston, red crcle: robot fnal poston (geodesc), red cross: target (Eucldean, CVT), green lne: robot path, yellow color gradent: densty functon φ( )). (a) L-shaped regon: the geodesc dstance provdes a reasonable robot path (see also [4]). (b) U-shaped regon: the robot converges to a mnmum pont arbtrarly far from the optmal goal pont (n the obstacle) due to the compromse character of the geodesc dstance. (c) Regon wth free-standng obstacle: the robot get stuck on ts way to the goal n a saddle pont as t does not stop makng compromses between dfferent shortest path optons. (d) L-shaped regon (wth dfferent robot start poston and dfferent φ( ) compared to (a)): Even n the case of a L-shaped regon, when the start poston and thus the robot s vsblty change, the robot happens to leave the regon because of non-zero gradent sums. a Vorono regon of a sngle robot takes shape of the regons n the eample confguratons. Frst consder applyng a controller (5) as used for Vorono coverage n conve envronments. Let robot at poston p drve on straght lnes, accordng to the Eucldean dstance measure, to the target poston t, whch s the centrod C V of ts Vorono cell V. We are confronted wth two types of crtcal confguratons: () the robot poston temporarly leaves the envronment Ω durng moton,.e. the path goes through an obstacle (Fgure 2(a), (c)), () the fnal goal poston les outsde the envronment, n an obstacle, and cannot be reached (Fgure 2(b), (d)). Net we nvestgate the lmtatons of Vorono coverage n non-conve envronments when usng the geodesc dstance measure. Although the geodesc dstance helps keepng the robot nsde the envronment regon on ts way to the target and works fne for some cases, e.g. for the eample n Fgure 2(a), there are other cases where t does not (such as the cases depcted n Fgure 2(b) (d)). The geodesc dstance measure calculates the paths along the boundares and avods the obstacles. But t ntroduces a compromse characterstc at the same tme,.e. the controller gets trapped and remans at a poston of mnmum average dstance to the locatons of nterest (see Fgure 2(b)). Ths may entrely be n the sense of locatonal optmzaton as the dstances to the upper and lower branch become balanced for the densty dstrbuton gven n Fgure 2(b). However, the compromse can lead to arbtrarly suboptmal confguratons for applcatons that requre the robot s fnal poston to be close to the centrod and mamum densty area (e.g. f the robots can see the area beyond the obstacles or sense a sgnal through the walls). The compromse character further causes the robot to get stuck n a saddle pont (unstable generalzed CVT) n Fgure 2(c) or even worse, to drve nto an obstacle n Fgure 2(d), as the projectons of the gradents nto target drecton of the geodesc dstance do not add up to zero when reachng the boundary. Geodesc dstance as dstance measure on ts own solves some of the problems, but provdes no general guarantee that the robot reaches a pont close to a local mnmum or remans n the envronment. To break the trade-offs, to crcumnavgate the obstacles and solve the problems ponted out n Fgure 2(b) (d), an obstacle avodance behavor or a local path planner s requred. That means wth respect to Vorono parttonng, we can formulate the constraned optmzaton problem mnh (P) = mn f(d(p,))φ() d, (6) P [p ] n =1 Ω V wth Ω the constrant set. The Vorono tessellaton becomes a constraned centrodal Vorono tessellaton (CCVT) and the centrod a constraned centrod n ths case (see also [12]). The robots form a constraned centrodal Vorono confguraton. C. A new approach for non-conve envronments Non-conve domans pose non-conve optmzaton problems wth non-conve constrants. We present a new approach for Vorono coverage of a non-conve envronment that bulds on the Lloyd algorthm and TangentBug [14], a local path planner wth obstacle avodance behavor. The control strategy s composed of two layers of abstracton: (1) Lloyd algorthm provdes goal updates based on successve computaton of Vorono regons and ther centrods on the upper layer (level 1), whle (2) TangentBug plans the robot path to the net centrod target poston on the lower layer (level 2). Ths can be formulated accordng to Fgure 1 as a contnung sequence of the two loops on level 1 and level 2, eecuted n a dstrbuted fashon on each of the robots. TangentBug s a smple but effcent sensor-based planner, capable of handlng unknown envronments by usng a range sensor. The range can be any value from zero (contact sensor) to nfnty (entre vsblty doman), where the length of the robot s path usually decreases wth ncreasng range of the sensor. TangentBug shows the two characterstc Bug behavors: moton-toward-target, a form of gradent descent, and boundary-followng, a form of eploraton of the obstacle boundary, whch both are desred behavors n the contet of our control strategy and guarantee global convergence to a target. We net restate the convergence of TangentBug n followng lemma. 4984

4 Lemma 2 (Convergence of TangentBug): The TangentBug algorthm converges globally toward a reachable target nsde a gven planar envronment for a sensor of any range n a fnte path. Proof: Convergence drectly follows from Theorem 1 and Theorem 2 n [13] for a contact sensor. The proof n case of a non-zero range sensor follows the lnes of the proofs for the contact sensor, and t can smlarly be shown that the robot reaches the target n a fnte path f target reachablty s gven (see Theorem 1 and 2 n [14]). III. VORONOI COVERAGE WITH LOCAL PATH PLANNING We descrbe the proposed control strategy n detal n ths Secton and provde an mplementaton that enables present Vorono coverage algorthms to deal wth non-conve envronments. A descrpton of the mplemented navgaton algorthm, or non-conve coverage algorthm respectvely, s detaled n Algorthm 1. Algorthm 1 calls Algorthm 2, the path plannng algorthm, as subroutne. Each algorthm mplements one of the two loops 1 and 2 of the control strategy, as outlned n Fgure 1. We used the Lloyd algorthm on level 1 to generate the Vorono tessellaton. The Eucldean dstance s used to mplement the control law n ths paper (equaton (5)), even though the geodesc dstance measure could just as well be appled. Algorthm 2 s ntentonally kept n a rather theoretc descrpton, for the purpose of generalty and clarty, and needs for well engneered mplementatons. We have chosen TangentBug as planner for the path generaton on level 2. Although TangentBug only needs local knowledge of the obstacles, global knowledge of the envronment s assumed for the eecuton of the Lloyd algorthm n our present approach. A. Non-conve coverage algorthm For the algorthm descrptons we need to ntroduce some new termnology. We follow a smlar dea as n [15], and ntroduce vrtual generators to navgaton and path plannng. We dstngush between real generators g real and vrtual generators g vrt as well as real targets t real and vrtual targets t vrt of robot. g real stands for the actual robot poston, whereas g vrt s the desred robot poston n dsregard of the obstacles n the envronment, as f we were dealng wth a conve envronment. t vrt s the centrod of the current Vorono regon, whch was computed based on g vrt at the last update of loop 1. t real s the projected pont p of t vrt to the envronment Ω. In some stuatons, the vrtual and real ponts smply concde. An obstacle boundary Vorono regon s defned as the subset of all the Vorono regons, for whch the condton V Ω /0 t vrt / Ω apples. It s part of the constraned mnmzaton problem and s used as condton to determne f the boundary constrant s actve n a regon V. The proposed control strategy computes the Lloyd algorthm usng only the vrtual generators, whch are able to freely pass through obstacles and occlusons. The robots then Algorthm 1 NON-CONVEX COVERAGE ALGORITHM Requre: Set of robots = {1,...,n} wth ntal postons p S n envronment Ω, and each robot provded wth: - Localzaton and knowledge of φ( ) and Ω - Vorono regon computaton - PATH PLANNING ALGORITHM 1 1: ntalze at tme T S = 0: g real p S, gvrt p S } k j=1, j of k negh- 2: loop {Loop 1} 3: acqure postons p and {g vrt j bors 4: construct local Vorono regon V assocated wth g vrt 5: compute the mass centrod C V of the Vorono regon, update vrtual target poston: t vrt C V 6: run PATH PLANNING ALGORITHM 7: end loop 8: compute the fnal Vorono regon assocated wth g real 1 PATH PLANNING ALGORITHM s a convergent, standard navgaton algorthm wth local path plannng capablty. In our case, the TangentBug algorthm s used. Algorthm 2 PATH PLANNING ALGORITHM Requre: Set of robots = {1,...,n} n envronment Ω, and each robot provded wth: - Obstacle avodance: sensng and computaton - Vrtual target t vrt, and var t vrt 1: loop {Loop 2} 2: f V s an obstacle boundary Vorono regon then 3: project t vrt to pont p onto Ω, and set var p 4: end f update real target poston: t real var 5: eecute net moton step toward real target t real, apply obstacle avodance to drve to net poston p update real generator poston: g real p 6: smulate net moton step toward vrtual target t vrt update vrtual generator poston g vrt 7: end loop 8: return vrtual generator g vrt attempt to follow the vrtual generators centrods. Throughout eecuton of loop 2, vrtual generators are movng toward vrtual targets (g vrt t vrt ) n a smulated envronment representaton where obstacles and non-conve segments of the boundary do not est. In parallel, real generators are movng toward real targets (g real t real ), whle the robots approach the real targets n the real envronment takng obstacles nto account. Each robot computes the net vrtual and real target ponts upon arrval at the current real target pont. Each robot pretends to be on track for an deal conve case and communcates ts smulated vrtual generator poston to the neghbors. That leads to a stuaton where the robots update ther own Vorono regon, centrod and thus net target pont based on the vrtual postons g vrt of ther neghbors, whle each of the robots tres to get as close as possble to 4985

5 the set deal target t vrt to reach ts objectve. The vrtual generators and targets allow for the mplementaton of the Lloyd algorthm for non-convety and mantan convergence of the method. If the robots real postons g real were used for the computaton n turn, or f the ponts were contnuously projected onto the boundary durng ongong eecuton of the navgaton algorthm (t vrt = t real and g vrt = g real ), Lloyd algorthm could be massvely perturbed dependng on the shape of the obstacle and result n unfavorable behavor (e.g. long paths) and undetermned confguratons (e.g. oscllatons). Therefore we desgn the algorthm n a way such that the real and vrtual ponts reman loosly coupled untl the end (we menton though that projectng contnuously may have potental; an algorthm that projects the generators n each teraton step of the Lloyd algorthm s descrbed n [12]). Whenever the poston of robot n the fnal confguraton of a local mnmum les n the free space away from an obstacle, the robot fnally succeeds n reachng ts set target and g real = t real = t vrt = g vrt holds. In the case of a conve envronment, the vrtual and real ponts smply reduce to sngle real ponts and the algorthm results n the eactly same behavor as for Vorono coverage of conve regons,.e. the constrants are not actve. Bug algorthms naturally provde solutons to some of the challenges of Algorthm 2. Consder the case when the fnal target s contaned n an obstacle. TangentBug comes wth a reachablty test, where reachablty s determned durng the boundary-followng behavor by just crclng around the obstacle. If the eploraton of the obstacle boundary s completed after one full crcle wthout havng found a leave pont, the target wll be unreachable. In ths case, accordng to step 3 n Algorthm 2, the target pont must be projected onto the obstacle boundary n an optmal way (see Secton IV). To mplement ths target projecton procedure, TangentBug can be etended wth the robots checkng boundary postons for optmalty durng boundary-followng. The optmal poston along the boundary s stored n memory. Fnally, projectng the pont to the obstacle boundary n an optmal way smply means drvng to the recorded poston drectly. Once the robots have converged to a fnal confguraton, a last Vorono partton s computed before termnaton of the non-conve coverage algorthm. Ths last step s requred because several robots mght not be at the centrod of ther Vorono regon due to the projecton procedure (at actve constrants). The fnal computaton of the Vorono tessellaton mproves the overall partton and guarantees that each domnance regon assgned to a robot s at least a Vorono regon. However, notce that the fnal robot confguraton s a constraned centrodal Vorono confguraton n general. B. Further propertes and etensons to the approach The control strategy s general and fleble, and allows for adaptons to many dfferent applcatons: 1) Offlne vs. onlne: The approach s composed of two separate levels that are connected to each other. Level 1 and level 2 can be completely decoupled. Frst, a verson of Lloyd algorthm runs untl convergence. Second, the resultng optmal confguraton s provded to the robots as nput for level 2. Each robot runs ts local path planner to drve to the fnal target poston. Such a setup could be appled n a centralzed off-lne approach. In contrast, f level 1 and level 2 are nterlaced, the nteracton between the two levels enables onlne navgaton. Ths s what we are manly nterested n. 2) Dscrete-tme vs. contnuous: When level 1 and level 2 are coupled, t s a queston of the frequency, sequences of level 1 and 2 follow upon each other. Classc Lloyd algorthm means that for one teraton on level 1, several loops on level 2 are eecuted. The Vorono regons are not recomputed untl the robots reach ther current targets,.e. the centrods of ther current Vorono regons. Under the dscrete-tme verson of the Lloyd algorthm, the rato between the number of teratons of loops 1 and loops 2 can be adjusted. For a rato of unty and nfntesmal sze of the teraton steps, a contnuous (or quas-contnuous) verson of Lloyd algorthm results. The update rates nfluence the overall behavor of the robots and defne system requrements, such as requred communcaton or sensng performance. 3) Known vs. unknown: Smlar to Vorono coverage n conve envronments, the new control strategy can be appled to a pror known as well as unknown envronments. In known envronments the densty functon φ( ) can be modeled approprately (.e. obstacle regons are not ncluded n the doman Ω). The densty functon, the obstacle poston and shape may also be unknown and must frst be sensed by the robots. Etensons are toward eploraton, where obstacles are learned and mapped. The densty functon can further be set eternally to drect the robots to certan locatons of specal nterest (formaton control). 4) Selecton of path planner: The modularty of the control strategy allows for the usage of any local path planner. Eamples for popular obstacle avodance methods are artfcal potental felds or vector feld hstograms, whch can be used for sensor-based navgaton. In ths contet, further nvestgatons can look at the method s performance, e.g. change n path lengths or tme to reach a fnal confguraton, dependent on a varyng sensor range. If a map s avalable, a superor path planner lke the A* algorthm s a good choce. For the selecton of a path planner, methods wth provable convergence guarantees are most preferable. IV. ALGORITHM ANALYSIS For the algorthm analyss we make followng assumptons. The robots are pont robots n a planar confguraton space R 2. The envronment s represented by the set of reachable ponts Ω. Ω s bounded and the obstacles are polygonal. Both the permeter of the obstacles and the number of obstacles are fnte. The densty functon φ( ) s defned over Ω and the dstance measure D( ) s the Eucldean dstance. A. Convergence We prove convergence of the control stategy for the mplementaton based on the Lloyd algorthm and the TangentBug algorthm for the case of planar confguraton space. 4986

6 Smlar proofs can be gven for the control strategy for other dmensons N 2 and a case where varatons on the Lloyd algorthm or a local path planner other than TangentBug are used. Theorem 1 (Convergence of the control strategy): For a non-conve envronment Ω R 2 Algorthm 1, based on the Lloyd algorthm and the TangentBug algorthm, causes the robots to converge to a constraned centrodal Vorono confguraton. Proof: We gve a proof by contradcton. Suppose that the robots do not converge to a constraned centrodal Vorono confguraton. That must be a result of: ) the TangentBug algorthm does not converge, or ) the Lloyd algorthm does not converge. ) By Lemma 2, f TangentBug does not converge, some target pont s not reachable. But that contradcts the projecton propertes of the control strategy (namely, that a projecton of ponts to the envronment always ests). ) By Lemma 1, f the Lloyd algorthm does not converge, an teraton takes nfnte tme. But that mples ), whch, as we have already shown, leads to a contradcton (namely, that TangentBug always converges to a fed and reachable target pont). B. Optmalty When a target poston les outsde the envronment, t vrt / Ω, as soon as the correspondng vrtual generator also leaves the envronment, g vrt / Ω, the real poston of the robot must be constraned to the envronment boundary Ω. Let us now see how the ponts can be projected to the boundary n the followng. Gven the control strategy n Algorthm 1 and that n robots converge to the fed ponts p = argmn p Ω p t vrt, {1,...,n}, n the envronment Ω, whch are projectons of the optmal target ponts t vrt / Ω, wth t vrt = C V, to the envronment n the Eucldean sense. Defne the fnal confguraton vector P = [p 1,...,p,..., p n] R Nn. We show n the theorem below that P mnmzes the hgh dmensonal optmzaton mn P Ω n P T, where T = [ t vrt 1,...,t vrt ] n. Furthermore we show that ths mples P locally mnmzes a constraned optmzaton problem closely related to equaton (6). Theorem 2 (Optmalty of the control strategy): The fnal confguraton of the robots has the followng propertes: 1) The pont P s closest to T n R Nn n the Eucldean sense, gven the projecton of t vrt to ts closest constraned pont p n R N, {1,...,n}. 2) The fnal step of the control strategy (target projecton) solves the constraned optmzaton problem of mnmzng the cost functon H (P) for the resultng fnal Vorono partton V = {V } n =1 wth tvrt as generators. Proof: Frst we prove 1) by contradcton. From the projecton of the targets results that p t vrt s mnmzed at p, {1,...,n}. Suppose that P T s not mnmzed. Then ˆP T such that ˆP T < P T,.e. ( ˆp 1 t vrt ˆp n t vrt n 2 ) 1/2 < ( p 1 tvrt p n t vrt n 2 ) 1/2. Substtutng all ˆp t vrt but one by p tvrt leads to ( p 1 tvrt ˆp t vrt p n t vrt n 2 ) 1/2 < ( p 1 tvrt p t vrt p n t vrt n 2 ) 1/2. From that t follows that ˆp t vrt 2 < p tvrt 2, {1,...,n}, whch s a contradcton. Now we prove 2). We can rewrte the cost functon usng the parallel as theorem as H (P) = 1 2 n =1 J V,C V n =1 M V p C V 2. The frst term on the rght sde of the equaton s constant for a fed area V and the mass M V s also constant. Snce C V = t vrt, mn P H (P) = mn P n =1 M V p C V 2 = mn P n =1 p C V 2, whch s mpled by the projecton. Remark 1: The fnal confguraton resultng from the control strategy s a constraned centrodal Vorono confguraton. We fnd that the costs can be further reduced, when the robots compute a last Vorono partton V end after convergence to the constraned target ponts p. By recallng that Vorono parttons act as mnmzers of H for strctly ncreasng f( ) and CVTs as mnmzers of any Vorono tessellaton V, we can derve lower and upper bounds on the costs: H (t vrt,v) H (p,v end) H (p,v). Remark 2: Theorem 2 shows the nterestng result that the projecton solves a constraned optmzaton problem. In fact, the optmal confguratons to whch our algorthm converges are those orthogonalty ponts descrbed n [10] (Secton IV) as beng desrable confguratons. The algorthm n that work s proven to drve the robots to orthogonalty ponts for a certan class of envronments (conve regons wth solated holes) wth consderable computatonal complety. Our controller s straghtforward by comparson, and s guaranteed to drve the robots to the closest possble orthogonalty ponts to the true centrods. Specfcally, our algorthm wll not get caught n the confguratons mentoned n [10] Remark 4, n whch the centrod s attanable, but the robots are stuck at orthogonalty ponts arbtrarly far from the centrod. V. EVALUATION IN SIMULATION Net we present smulatons to verfy the proposed control strategy and hghlght some nterestng aspects. The nonconve coverage algorthm and ts mplementaton are fully decentralzed. The smulatons are carred out n Matlab. The robots are pont robots, the sensor range s nfnte,.e. the sensor covers the whole vsble area, and the envronment s assumed to be known a pror. For smplcty a unform densty functon φ( ) s used. Fgure 3 shows the deployment of fve robots n a U- shaped envronment. The robots cover the envronment and converge to a fnal confguraton, whch s a centrodal Vorono confguraton n ths case. The TangentBug algorthm provdes the llustrated obstacle avodance behavor and gudes the robots around the obstacle s corners. The plot n Fgure 3 presents the total cost H of the real and vrtual generators for the robot confguraton. It s nterestng to see how the cost for the real generators approaches the cost for the vrtual generators over tme. The cost of the vrtual generators falls off as soon as a robot reaches ts current target pont. 4987

7 [m] [m] [m] [m] [m] [m] [m] Fg. 3. Vorono coverage n a U-shaped envronment (obstacle n black). From left to rght: fve robots (blue crcles) start from an ntal poston and move along the shown trajectores from one current target to the net target (red crosses) untl they reach a fnal confguraton. The net target s computed by the Lloyd algorthm as soon as a robot reaches ts current target. The green lnes llustrate the trajectores of the vrtual generators. Two green lnes ntersect the obstacle s corners and show that an obstacle avodance behavor s needed to cover ths envronment. The cost H for the confguraton of the vrtual and the real generators over tme s shown n the plot on the rght. [m] [m] [m] [m] [m] [m] Fg. 4. Left: Vorono coverage n an envronment wth free-standng obstacles (obstacles n black). The vrtual generator and target of one robot le nsde an obstacle. The robot tres to reach t and starts crclng around the obstacle. Fnally, the target s projected to the closest pont on the obstacle s boundary (yellow cross). The centrodal Vorono tessellaton (yellow) results from the vrtual generators at the optmal vrtual target, whle the Vorono dagram n blue s the last partton computed by the robots n a fnal step. Rght: E-puck robot platform equpped wth markers for trackng. On the left of Fgure 4, fve robots cover an area wth two free-standng obstacles. The vrtual generator and the vrtual target of one of the robots are contaned n an obstacle. After the robot has eplored the obstacle and completed one full cycle, t projects the vrtual target to the closest pont on the boundary. At the end, a last Vorono tessellaton s computed to mprove the fnal partton (the resultng cost s reduced ths way once more). VI. EXPERIMENTS WITH REAL ROBOTS We demonstrate n the followng eperment the applcablty of our control strategy to real robots and nonconve envronments. For the eperment we use the e- puck robot platform [16] (see Fgure 4), a small two-wheel dfferental drve robot, wth a dameter of 7 cm. The e-puck s equpped wth a dspic mcrocontroller, dfferent sensors (e.g. IR promty sensors) and actuators (e.g. LEDs). The e-puck s powered by a L-on battery and offers a RS-232 and a bluetooth nterface for communcaton. We bult a test setup wth an overhead camera (USBcamera wth resoluton of pels) to track the e-puck robots on a 1 m 1 m ground plane n a dstance of 1.35 m from the camera. Each e-puck s ftted wth a marker showng a dfferent symbol as depcted n Fgure 4. We adapted the ARToolkt augmented realty software [17] to detect the markers and track the robots postons and orentatons. It further offers the possblty to overlay the ground plane wth vrtual obstacles and envronment boundares. The mages are read nto a host computer and processed. The postons and orentatons of the robots from the tracker are passed on to the navgaton algorthm whch plans the path for each of the robots for the net tme step. The host computer contnuously sends nformaton over bluetooth and operates the robots by remote control. Each robot receves the commands and actuates ts stepper motors, whch closes the control loop. The algorthms for trackng and for the actual path plannng and robot control all run n Matlab. Fgure 5 shows the ntal postons and fnal confguraton after completon of a test run wth 5 robots for the U- shaped envronment. The robots succeed to cover the nonconve envronment. We ran seven epermental trals wth the robots for the gven ntal postons n the U-shaped envronment and emulated sensors wth nfnte sensor range. The paper s accompaned by a vdeo 1 that shows one partcular run of the epermental trals as well as smulatons for the envronments n Fgure 3 and 4 and other more comple envronments. The epermental results match wth the smulatons. Whle hardware nose and trackng errors only cause small devatons, a man dfference n the trajectores comes from adjustments n the algorthm to account for the non-zero sze of the real robots by a safety margn along the boundary. The 1 The reader also refers to

8 [m] [m] Fg. 5. Vorono coverage n a U-shaped envronment. From left to rght: fve robots move accordng to the control strategy from an ntal confguraton to a fnal optmal confguraton. They avod the obstacle and cover the non-conve envronment. The plot on the rght shows the ntal and fnal confguratons over the test runs. Ideal smulated postons n black (cross: ntal poston, dot: fnal poston) and real epermental postons n color (cross: ntal postons, crcle: fnal postons). The magenta crcle nsde the obstacle shows a faled epermental run where the tracker lost the marker of one robot. average poston error over the robots and the epermental runs s 5.42 cm. The duraton of one epermental run s 4.56 mn n average. Though the convergence of the robots toward the fnal confguraton was lmted by the update rate of the trackng system rather than the robot platform or the control strategy tself. VII. CONCLUSION Ths paper presented a new control strategy to provde Vorono coverage n non-conve envronments. We analysed the problems of Vorono coverage wth non-conve envronments n detal and desgned a navgaton algorthm based on the Lloyd algorthm and the TangentBug algorthm. Lloyd algorthm and TangentBug are nterlaced and successvely terated to drve a group of robots to a fnal constraned Vorono confguraton. Lloyd algorthm updates the current goal poston, whle TangentBug runs the path plannng. TangentBug s modfed wth a projecton procedure to constran outlyng target ponts to the envronment. The constraned ponts resultng from the projecton were shown to be solutons to the constraned optmzaton problem of fndng a robot confguraton for the fnal Vorono partton of mnmal cost. We proved both convergence and optmalty of the proposed control strategy, by applyng the concept of vrtual generator and target ponts to mult-robot path plannng. The approach has been evaluated n smulatons and physcal eperments wth a team of networked robots. We further demonstrated that the approach s general and offers many nterestng etensons, such as applyng Vorono coverage to unknown envronments, usng a dfferent path planner better suted to a specfc applcaton or modfyng the couplng between Lloyd algorthm, path planner and projecton procedure. In our future work we are nterested n Vorono coverage n combnaton wth sensor-based navgaton n unknown and non-conve envronments. A drecton wll be to etend the control strategy for adaptaton to and learnng or mappng of the obstacles. Another step s to mplement Vorono coverage drectly on the robot platform. REFERENCES [1] D.W. Gage, Command Control for Many-Robot Systems, n the Nneteenth Annual AUVS Techncal Symposum AUVS-92, reprnted n Unmanned Systems Magazne, vol. 10, no. 4, pp , [2] J. Cortés, S. Martnez, T. Karatas and F. Bullo, Coverage Control for Moble Sensng Networks, IEEE Trans. on Robotcs and Automaton, vol. 20, no. 2, pp , [3] M. Schwager, D. Rus and J. 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