An Event-Triggered Virtual Force Algorithm for Multi-Agent Coverage Control with Obstacles

Transcription

1 2018 Annual American Conrol Conference (ACC) June 27 29, Wisconsin Cener, Milwaukee, USA An Even-Triggered Virual Force Algorihm for Muli-Agen Coverage Conrol wih Obsacles Mohanad Ajina Cameron Nowzari Absrac In his work, we propose an even-riggered algorihm based on a virual force deploymen approach o address he muli-agen coverage conrol problem in he presence of obsacles. Unlike mos works ha consider his problem, we are mainly ineresed in reducing he amoun of communicaion and moion required by he agens o reach a configuraion ha increases he coverage hroughou an environmen of ineres. In paricular, mos works ha consider his problem assume agens are in consan communicaion wih each oher. Insead, he even-riggered algorihm we propose allows agens o decide for hemselves when communicaion is necessary while sill achieving he primary goal of covering he environmen and ensuring collisions are avoided. Several simulaions illusrae he resul of our algorihm wih and wihou he presence of an obsacle and compares i agains a similar algorihm ha does no consider even-riggered communicaion. I. INTRODUCTION The opic of wireless sensor neworks (WSN) is receiving close aenion in many research areas oday wih applicaions like sensors monioring for emperaure, raffic, healh saus, and many ohers [1], [2], [3], [4]. The number of sensors in he WSN varies depending on he applicaion. The WSN can be buil wih only a few sensors such as home securiy WSN [5] or several housand such as environmenal monioring WSN [6]. As he size of he neworks and number of possible applicaions increase, researchers have been working o improve performance of WSN and o reduce heir coss. To address hese challenges, effors have been made o maximize sensor area coverage while reducing he amoun of communicaion beween sensors. In his paper, we refer o sensors as agens, and o WSN as a muli-agen sysem. Our work sudies he rade-offs of he muli-agen sysem performing an area coverage conrol ask when he communicaion beween agens is conrolled by an evenriggered law. To achieve his goal, we designed an evenriggered law ha allows agens o decide auonomously when updaed informaion abou neighbors locaions is needed o complee he ask. Our moivaion comes from he need o reduce he performance cos by reducing unnecessary communicaion. Lieraure review: There are wo common sraegies ha have been uilized o address he area coverage conrol problem. These sraegies are compuaional geomery based and force based [7]. An example of he compuaional geomery based sraegy is a Voronoi diagram or a Voronoi pariion. This approach divides an environmen ino coverage regions The auhors are wih he Deparmen of Elecrical and Compuer Engineering, George Mason Universiy, Fairfax, VA, 22030, USA, majina, cnowzari}@gmu.edu based on he disance beween agens; more deails can be found in [8]. The auhors in [9] successfully applied he Voronoi diagram sraegy o achieve opimal deploymen for mobile sensing neworks. However, hey assumed ha each agen has access o all he agens locaions a all imes. In [10], [11], he assumpion was removed, and he Voronoi diagram was implemened based on local informaion provided by he agen s neighbors. In [12], he Voronoi diagram wih he heoreical echniques was used o calculae he bes and wors coverage cases. The above presened Voronoi diagram work assumed free environmen. The auhors in [13] proposed exension o he Voronoi diagram sraegy ha addressed he issue of area coverage wih herogeneous agens, and hey generalized heir approach for non-convex environmen. In [14], he auhors developed a new approach for area coverage in he presence of known obsacles of non-convex polygonal environmens. In [15], obsacles are unknown, and he Voronoi diagram sraegy is applied o maximize area coverage. The second common sraegy is force based, ha allows agens o move based on aacked aracive and/or repulsive forces. An early work of he force based sraegy is proposed in [16] ha ensures collision avoidance. In [17], he auhors used a repulsive force algorihm o deploy agens in a building. In [18], a arge involved virual force algorihm is proposed o locae more agens closer o an area of ineres and o keep agens far away from obsacles. In [19], a virual force algorihm (VFA) is proposed o increase he coverage area afer iniially placing agens randomly in he environmen in he presence of obsacles. However, hey assumed ha each agen has access o all agen s locaions a all imes. The auhors in [20] eliminae his assumpion, and hey proposed an improved VFA (IVFA) and exponenial VFA (VFA). Boh algorihms performed beer area coverage ask han VFA even when agens have limied communicaion ranges, bu on paper, no resuls considered he presence of obsacles. Recenly, in [21], he obsacle avoidance virual force algorihm (OAVFA) proposed o maximize he area coverage and o minimize he average moving disance. The OAFVA performed beer over IVFA and EVFA wih and wihou he presence of an obsacle. However, all he above work assume coninuous or periodic communicaion and/or coninuous or periodic sensing among agens. This brings us o he oher area of relevance o his work which are disribued even- and self-riggered coordinaion sraegies. Boh ypes of riggered laws have been proposed o reduce he amoun of communicaion beween agens while mainaining some desired sysem level properies. The /$ AACC 1009

2 auhors in [22] propose a self-riggered algorihm o save communicaion power for he opimal deploymen problem wihou obsacles. In [23], an even-riggered algorihm was able o reduce he amoun of communicaion beween agens when performing he muli-agen rendezvous ask. In [24], he proposed even-riggered algorihm saved a significan communicaion power for he average consensus problem. In [25] and [26], he even-riggered conrol lowered he number of communicaion beween agens for a leaderfollowing consensus problem, and in [27], for a muli-agen sysems consensus problem. Thus, we are ineresed in combining he disribued riggering sraegies wih a force based coverage conrol sraegy in order o improve he performance of he sysem in erms of reduced communicaion. More specifically, our work builds on he obsacle avoidance virual force algorihm (OAVFA) proposed in [21], where he agens are always aware of heir neighbors posiions. Insead, we are ineresed in combining his moion conrol algorihm wih an evenriggered communicaion sraegy o reduce communicaion while achieving he same level of coverage as he OAVFA ha requires perfec informaion a all imes. Saemen of conribuions: The main conribuion of our work is he design of he even-riggered virual force algorihm ha reduces communicaion while agens sill complee he primary coverage conrol ask. We firs design a moion conrol law ha allows agens o deermine heir conrol inpus based on a virual force deploymen approach. Second, we design a decision conrol law ha allows agens o deermine when communicaion wih neighbors is needed o complee he ask. The even-riggered virual force algorihm combines boh laws o allow agens o deploy in he environmen wih less communicaion performed. Our algorihm does no require periodic communicaion as in [21] while sill achieving he same level of coverage. Various simulaions illusrae he performance of he even-riggered virual force algorihm wih and wihou he presence of obsacles. II. PROBLEM STATEMENT Consider a nework of n agens moving in a recangle environmen S R 2 wih some saic obsacles O S. More specifically, we consider N o disinc obsacles o 1,..., o No such ha m 1,...,No}o m = O. We denoe S S o be he of se environmen s boundaries, and denoe he posiion of agen i 1,..., n} a discree ime Z 0 o be p i. The collecion of all agen posiions a ime is hen given by P (p 1,..., p n ) S n. We consider a simple kinemaic model wih bounded velociy V max, p i +1 = p i + u i, where u i V max is he conrol inpu of agen i a ime and is he acual ime beween wo discree ime seps. We assume ha he agens are iniially unaware of he posiions and number of obsacles in he environmen S. Insead, he agens are able o sense obsacles up o a disance R S away. We also assume he agens are only able o communicae wih neighbors ha are agens R C disance away, R C = 2R S. The goal of he agens is now o reach a configuraion o cover as much of he unoccupied environmen S \ O as possible, while keeping he oal moving disance and communicaion among agens as low as possible. More specifically, we consider a binary disk sensing model (BSM) due o is simpliciy and effeciveness in modeling covered area [21], [28], [29]. Given he posiion of an agen p i, we say ha an arbirary poin in he environmen c S is covered by agen i a ime if i is wihin he sensing range, i.e., p i c R S. Formally, we define he indicaor funcion BSM(p i 1, if p i c R S, c) = 0, oherwise. which reurns 1 if he poin c is covered by he agen a p i, and 0 oherwise. Then, given he vecor of all agen posiions P a some ime, we define he coverage raio as he raio beween areas of all poins in he unoccupied domain S \ O ha are covered and he enire area, C Raio S\O max i 1,...,n} BSM(p i, c)dc. Area(S \ O) We calculae he moving disance of agen i beween is curren and las locaions, and he average moving disance a ime sep, D Ave is averaging of all agens moving disances. The D Ave is defined formally: n D Ave i=1 = (pi p i 1) n More specifically, our goal now is o maximize C Raio while keeping D Ave and he amoun of communicaion required by he agens as small as possible. In paricular, our work builds on he work of [21], where an algorihm is proposed o solve his problem bu requires consan communicaion among he agens. Our main goal is o relax his assumpion o improve efficiency of he nework while sill achieving good overall performance in erms of he merics defined above. III. EVENT-TRIGGERED ALGORITHM DESIGN In his secion, we design he even-riggered virual force algorihm ha allows agens o decide for hemselves when communicaion wih neighbors is necessary o complee he global ask. The has wo componens: a moion conrol law ha deermines he conrol inpu of each agen, and a decision conrol law ha decides when communicaion is required. A. Moion conrol law We firs design he moion conrol law ha allows agens o deermine how o move in he environmen based on a virual force approach. Each agen is exposed o hree ype of forces: 1. a neighbor force, F ij, ha could be an 1010

3 aracive or a repulsive force, 2. an obsacle force, F i,o m ha is a repulsive force, 3. a boundary force, F i,b, ha is a repulsive force. These forces are heavily effeced by he disance beween an agen and neighbors, obsacles, and boundaries, respecively. We adop he following force equaions from [21], where he auhors only consider deploymen over a field of uniform densiy. Modifying he algorihm o deploy over non-uniform fields will be reserved for fuure work. The neighbor force F ij is defined as F ij 0, if d ij > R C, K A (d ij d h ), if R C d ij ij )( pj pi d ij 0, if d ij K B (d h ), if d ij ij dij )( pi pj d ij = d h ij, < d h ij, > d h ij,, where d h i,j = 3R C /2 = 3R S, and K A and K B are consans. The d i,j is a disance beween an agen and a neighbor. Agen j is a neighbor of agen i if and only if p i p j R C. If agen i has a neighbor wihin is communicaion range, he agen communicae wih he neighbor o collec is curren locaion. When he neighbor s locaion is received, he agen i calculaes he disance o he neighbor d i,j. In addiion, he obsacle force F i,om is defined as F i,om 0, if d i,om d h (K r1 (d h o d i,om ), α i,om + π), if d i,om o, < d h o, where d h o = 3R S /2, m 1,..., N o }, and K r1 is a consan. The d i,om is he shores disance beween an agen and an obsacle ha is wihin agen i sensing range. If agen i senses an obsacle, o m, he agen calculaes he shores disance o he obsacle d i,om. Furhermore, The boundary force F i,b is defined as F i,b 0, if d i,b d h (K r2 (d h b di,b ), α i,b + π), if d i,b b, < d h b, where d h b = 3R S /2 and K r2 is a consan. The d i,b is he perpendicular disance beween an agen and a boundary ha is wihin he agen s sensing range. Le b S o be a poin on he environmen s boundary, and he closed segmen [p i, b] S o be a line such ha [p i, b] S. If agen i senses an environmen s boundary, he agen calculaes he perpendicular disance o he boundary. In a recangle environmen, he F i,b is he oal sum of he four boundaries forces. Formally: F i,b = F i,bx+ + F i,bx + F i,by+ + F i,by Therefore, he Ne-Force ha aacks an agen a ime, F i, is he sum of all forces. The F i defined as: F i j N i F ij + N 0 m=1 F i,om + F i,b, (1) where N i P is he se of agen i s neighbors a ime, and N 0 is he number of obsacles in he environmen. The Ne-Force of an agen could be 0 if no neighbor is wihin he agen s communicaion range and no obsacle and boundary are wihin is sensing range. Also, F i could be weak if he forces are conradicory or neighbors, obsacles and/or boundaries are far from an agen bu wihin is communicaion and sensing ranges. On he oher hand, F i could be srong if an agen has very close neighbors, obsacles and/or boundaries. More specifically, F i is he desired displacemen of agen i a posiion p i. Thus, i simply moves wih velociy v i owards his poin by seing u i where is velociy v i is given by F i F i vi, (2) v i = minv max, α F i }, (3) where α (0, 1). The moion conrol law in shor, a every insan of ime, each agen calculaes is F i and moves in he direcion as fas as possible if F i > V max. Oherwise, i moves in F i direcion a slower speed. The simple moion conrol law is described formally in Algorihm 1 Algorihm 1: moion conrol law Agen i 1,..., n} performs a all imes Z 0 : 1: receives posiions p j from neighbors j wihin a disance R C 2: senses boundary S and deecs obsacles o m wihin a disance R S 3: compues F i according o (1) 4: ses v i according o (3) 5: compues u i according o (2) 6: compues p i +1 = pi + ui 7: moves o p i +1 by vi B. Decision conrol law We are now ineresed in improving he moion conrol law which requires all agens o be in communicaion each ime a conrol signal is compued by relaxing he need for consan communicaion. Le p i even be an inermediae goal ha agen i can reach in muliple imeseps and define i as p i even p i + F i. We aim o allow agens o ravel owards p i even wihou consanly communicaing wih neighbors by designing he decision conrol law ha combines wo even-rigger condiions. A rivial even-rigger condiion, Condiion1, o be ha an agen moves o p i even in muliple wihou communicaing wih ohers. When he agen reaches p i even, he agen communicaes wih neighbors o updae is F i, p i even and u i. However, his condiion is problemaic in some scenarios such as here could an obsacle blocking he way o p i even in

4 To avoid collisions, we inroduce an addiional mechanism o rigger an even. Le X i R 2 be he union of neighbors, obsacles, and boundaries wihin agen i s sensing range, and define R T as he riggering radius. Then, given agen i s curren posiion p i a ime, we le a riggering sensing model (TSM) o be TSM(p i, X i ) = 1 if x X i s.. p i x R T 0 oherwise. which reurns 1 if agen i deecs an objec x X i wihin his riggering range R T of p i, and 0 oherwise. This riggering sensing model (TSM) does no guaranee no collisions unless R T is bounded. In our analysis, he wors case scenario is ha when wo agens are raveling in he opposie direcion by V max. The disance ha will guaranee no collision beween he agens mus be more han 2V max. Therefore, we lower bounded by R T > 2V max. For agens o move wihou a collision, Condiion2, hey are required o communicae wih neighbors o updae heir F i, p i even and u i if hey sense an objec wihin heir riggering sensing ranges. Noe, he smaller he R T, he less communicaion performed. Thus, we le R T as small as possible. The decision conrol law combines boh condiions and is described formally in Algorihm 2. Algorihm 2: decision conrol law Agen i 1,..., n} performs a every riggered even: 1: receives posiions p j from neighbors j wihin a disance R C 2: senses boundary S and deecs obsacles o m wihin a disance R S 3: compues F i according o (1) 4: compues p i even pi + F i 5: se T SM = 0 6: while ( p i pi even = 0 & T SM 1) do 7: compues p i +1 8: move o p i +1 9: se p i pi +1 10: sense all objecs X i wihin riggering sensing range 11: compue TSM according o (4) 12: end while C. The even-riggered virual force algorihm Here, we synhesize he even-riggered sraegy ha helps agens o deermine a each when updaed informaion is needed o complee he ask. Our designed algorihm is a combinaion of moion conrol law of Secion III- A and decision conrol law of Secion III-B wih a procedure o acquire communicaion when he condiions are me. [Informal descripion]: Agen i communicaes wih neighbors o collec heir locaions, and senses he surrounding o locae obsacles and boundaries ha are wihin is sensing range. When, he aracive and/or repulsive forces are calculaed, he agen compues is F i. When F i is compued, he agen calculaes p i even, ses is v i = min(v max, α F i / ), and compues is u i. Then, he agen calculaes p i +1 and moves oward i. When p i +1 is reached, he agen senses for an objec wihin is riggering sensing range and updaes (4) T SM. If he T SM reurned 0, he agen calculaes a new p i +1, moves o p i +1, senses for an objec a p i +1 unil i reaches he p i even. If p i even is reached or he T SM reurned 1, he agen communicaes wih neighbors o updae is F i, p i even and u i. The even-riggered virual force algorihm is described formally in Algorihm 3. Algorihm 3 : even-riggered virual force algorihm Agen i 1,..., n} performs a every riggered even: 1: receives posiions p j from neighbors j wihin a disance R C 2: senses boundary S and deecs obsacles o m wihin a disance R S 3: compues F i according o (1) 4: compues p i even pi + F i 5: ses v i according o (3) 6: compues u i according o (2) 7: ses T SM = 0 8: while ( p i pi even = 0 & T SM 1) do 9: if u i pi pi even hen 10: compues p i +1 = pi + ui 11: moves o p i +1 by vi 12: updaes p i pi +1 13: sense all objecs X i wihin riggering sensing range 14: compue TSM according o (4) 15: else 16: ses p i +1 = pi even 17: moves o p i +1 by vi 18: updaes p i pi +1 19: end if 20: end while IV. SIMULATION In his secion, we provide simulaions of he. All simulaions are done in MATLAB wih n = 40 agens moving in 100m by 100m square environmen and 300 ime seps. All simulaion are averaged over 100 runs, and only few simulaions are shown due oo limied paper number. The parameers of he simulaions are given in TABLE 1. TABLE I SIMULATION PARAMETER grid size =100mX100m V max = 0.5m/s K A = K B = 0.2 K r1 = 0.8 K r2 = 0.8 R S = 10m R C = 20m R T = 1.1m α = 0.5 We adop a communicaion power model from [30]. Specifically, he oal power P i used by agen i o communicae, in dbmw power unis is defined as: P i = 10 log 10 n j 1,...,n},i j β Pi j+β2 pi p j where β 1 and β 2 are posiive real parameers ha depend on he characerisics of he wireless medium, and P i j is he power received by agen j of he signal ransmied by agen i. In our simulaions, all hese values are se o 1. We sar by comparing he performance of wih OAVFA in [21]. The even-riggered virual 1012

5 force algorihm achieves he final deploymen as in [21]. Figure 1 illusrae he iniial and final deploymen of he even-riggered virual force algorihm in he presence of an obsacle. Our algorihm achieves maximum coverage area of he environmen. This iniializaion of he agens is based on randomly placing en agens in each corner of he environmen, and noe ha all furher illusraed resuls assume his iniializaion. (a) in he presens of an obsacle is more han 54%. V. CONCLUSIONS In his paper we considered a muli-agen coverage conrol problem in he presence of obsacles. To solve he problem, we have proposed he even-riggered virual force algorihm by combining he moion conrol law and he decision conrol law which allows agens o auonomously decide for hemselves when communicaion is required, in addiion o how o move. Our algorihm allows agens o ravel in he environmen wihou he need o communicae wih neighbors a every insan of ime. An agen only needs o communicae if i reaches is new locaion or if i senses an objec wihin he riggering sensing range. The main conribuion of his work is reducing he amoun of communicaion beween agens while mainaining he desired coverage conrol performance. Our simulaions illusraed ha he even-riggered virual force algorihm had reduced more han 50% of communicaion power compared o he OAVFA in [21] wih and wihou he presence of obsacles, and also achieved maximum area coverage as if agens communicaed periodically. Fuure work will be devoed o include scenarios such as communicaion delays and package drops wih granees on he level of performance and power saving. In addiion, we are ineresed in developing asynchronous implemenaion by idenifying even rigger condiion ha ensure he level of performance. REFERENCES (b) Fig. 1. (a) Iniial deploymen of 40 agens in he environmen wih an obsacle. (b) Final deploymen of (a) iniializaion using even-riggered virual force algorihm Figures 2 (a) and (b) compare he C Raio performance of and OAVFA wihou and wih an obsacle. The resul shows ha boh algorihms achieve maximum area coverage, bu he OAVFA reaches he maximum slighly faser. Figures 2 (c) and (d) compares he D Ave performance wihou and wih an obsacle. The resuls show ha our algorihm has a beer performance. In case of no obsacles, our algorihm has slighly beer D Ave, bu in case of he presence of an obsacle, our performance minimized D Ave more han OAFVA. Figures 2 (e) and (f) illusrae he average communicaion power consumed wihou and wih an obsacle. The figures show he conribuion of our work. The saved unnecessary communicaion beween agens ha resuled in significaion reducion of he communicaion power consumed. The average power saved wihou obsacles is more han 50%, and he average power saved [1] R. Zhang, D. Yuan, and Y. Wang, A healh monioring sysem for wireless sensor neworks, in nd IEEE Conference on Indusrial Elecronics and Applicaions, May 2007, pp [2] I. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci, Wireless sensor neworks: a survey, Compuer Neworks, vol. 38, no. 4, pp , [Online]. Available: hp:// [3] A. Perrig, J. Sankovic, and D. Wagner, Securiy in wireless sensor neworks, Commun. ACM, vol. 47, no. 6, pp , Jun [Online]. Available: hp://doi.acm.org/ / [4] S.-H. Yang, Wireless Sensor Neworks: Principles, Design and Applicaions. London: Springer London, [5] N. Noury, T. Herve, V. Rialle, G. Virone, E. Mercier, G. Morey, A. Moro, and T. Porcheron, Monioring behavior in home using a smar fall sensor and posiion sensors, in 1s Annual Inernaional IEEE-EMBS Special Topic Conference on Microechnologies in Medicine and Biology. Proceedings (Ca. No.00EX451), 2000, pp [6] J. K. Har and K. Marinez, Environmenal sensor neworks: A revoluion in he earh sysem science? Earh-Science Reviews, vol. 78, no. 3, pp , [Online]. Available: hp:// [7] N. A. A. Aziz, K. A. Aziz, and W. Z. W. Ismail, Coverage sraegies for wireless sensor neworks, Inernaional Journal of Elecrical, Compuer, Energeic, Elecronic and Communicaion Engineering, vol. 3, no. 2, pp , [Online]. Available: hp://wase.org/publicaions?p=26 [8] A. Okabe and D. Kendall, Spaial essellaions: conceps and applicaions of Voronoi diagrams, ser. Wiley series in probabiliy and saisics: Applied probabiliy and saisics. Wiley, [9] J. Cores, S. Marinez, T. Karaas, and F. Bullo, Coverage conrol for mobile sensing neworks, IEEE Transacions on Roboics and Auomaion, vol. 20, no. 2, pp , April

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