Coverage Control for Multiple Event Types with Heterogeneous Robots

Transcription

1 Coverage Control for Multple Event Types wth Heterogeneous Robots Armn Sadegh Stephen L. Smth Abstract Ths paper focuses on the problem of deployng a set of autonomous robots to effcently montor multple types of events n an envronment. There s a densty functon over the envronment for each event type representng the weghted lkelhood of the event at each locaton. The robots are heterogeneous n that each robot s equpped wth a set of sensors and t s capable of sensng a subset of event types. The obectve s to deploy the robots n the envronment to mnmze a lnear combnaton of the total sensng qualty of the events. We propose a new formulaton for the problem whch s a natural extenson of the homogeneous problem. We propose dstrbuted algorthms that drve the robots to locally optmal postons n both contnuous envronments that are obstacle-free, and n dscrete envronments that may contan obstacles. In both cases we prove convergence to locally optmal postons. We provde extenson to the case where the densty functons are unknown pror to the deployment n contnuous envronments. Fnally, we present benchmarkng results and physcal experments to characterze the soluton qualty. I. INTRODUCTION In ths paper, we focus on the problem of deployng multple heterogeneous robots to cover an envronment wth dfferent event types. Each event type has a dfferent spatal dstrbuton n the envronment and can be sensed only wth a specfc type of sensor. The robots are heterogeneous n that each of the robots s equpped wth a subset of sensors. Thus, each robot s capable of measurng a subset of event types. The qualty of sensng for an event at a locaton s a decreasng functon of the dstance of the sensor from the locaton. The obectve s to deploy the robots to maxmze the total coverage qualty of the events of dfferent types. Ths appears n several applcatons ncludng reconnassance, survellance [1] and montorng [2], as well as n the deployment of vehcles wth dfferent capactes and/or capabltes n urban transportaton systems [3], [4]. In convex envronments, Cortes et al. [5] propose a dstrbuted algorthm for homogeneous robots that utlzes Vorono parttonng and the Lloyd descent algorthm. The proposed control law allows the robots to converge to a local maxmum sensng qualty over the events n the envronment and requres communcaton only between neghborng robots n the Vorono partton. Several studes address dfferent types of heterogenety n the robots. In [6], the authors consder the problem of sensng an event where the sensors have dfferent functons governng ther sensng qualty. The approach defnes a generalzed Vorono partton based on the sensng functons and provdes a Lloyd descent type algorthm for controllng each robot. Ths research s partally supported by the Natural Scences and Engneerng Research Councl of Canada (NSERC). The authors are wth the Department of Electrcal and Computer Engneerng, Unversty of Waterloo, Waterloo ON, N2L 3G1 Canada (a6sadegh@uwaterloo.ca; stephen.smth@uwaterloo.ca) Other types of heterogenetes for moble sensors addressed n the lterature nclude the sensors wth dfferent sensng ranges [7], [8], [9] and dfferent addtve weghts on the sensng qualty of robots [10]. In [7], the authors address the coverage problem for crcular sensors wth dfferent rad. In [9], authors provde a gradent descent algorthm for the dstrbuted control of crcular sensors wth dfferent rad n a non-convex envronment. All the aforementoned studes consder the coverage control for a sngle event type. In [11], the authors ntroduced the coverage control problem for multple event types, each wth a dfferent densty functon. An event of a certan type can be sensed by a robot f the robot s equpped wth the requred sensor. The proposed approach consders a Vorono partton generated by all the robots, and the obectve functon s a convex combnaton of the sensng qualty of each robot for the events nsde ts Vorono cell and the sensng qualty over the whole envronment. In ths paper, we consder the same heterogeneous problem, but we pose a dfferent obectve functon, and thus dstrbuted control law. As such, we compare our approach to [11] n detal, and demonstrate ts advantages n smulaton. In our approach, we defne a Vorono partton of the envronment for each event type, each generated by the robots wth the sutable sensors. Wth these parttons, we are able to ensure that each event s sensed by the closest robot wth the requred sensor. Moreover, our formulaton captures the case where sensors for dfferent events have dfferent sensng functons. Several studes consdered the coverage control n an envronment where the event densty s unknown pror to deployment [12], [13]. In [12], the authors assume a bass functon approxmaton for the event densty wth adaptve weghts for each robot and propose a decentralzed algorthm that updates the weghts estmatng the true densty. The authors n [13] provde a smple stochastc gradent descent algorthm wthout estmatng the real event densty and prove convergence to a locally optmal soluton. Bult on the results n [13], we extend our analyss to envronments wth multple events types each wth an unknown densty. In a dscrete envronment represented by a graph, the obectve of the coverage problem s to sense events occurrng on the vertces of the graph. A closely related problem s the p-medan problem [14], [15] where a set of servce unts are located on the vertces of a graph servcng demands arrvng on the vertces wth the obectve of mnmzng the total servce tme. In [16], a dstrbuted algorthm s proposed for parttonng and coverage of a non-convex envronment. Yun and Rus [17] presented a dstrbuted vertex swap algorthm for the robots that converges to locally optmal solutons wth two-hop communcaton. To the best of our knowledge, the exstng lterature on the dstrbuted coverage control on

2 graphs s lmted to homogeneous robots sensng a sngle event type. In contrary, we consder multple event types wth heterogeneous robots both n the sensng capablty and qualty. Contrbutons: The contrbutons of ths paper are threefold. Frst, we propose a new formulaton for the problem of coverage control for heterogeneous moble robots, whch s a natural extenson of the homogeneous formulaton. Second, we provde a dstrbuted algorthm for maxmzng the sensng qualty wth the new formulaton. Thrd and fnally, we extend the results to dscrete envronments wth dfferent event types. II. PROBLEM FORMULATION In ths secton, we formulate the coverage problem for contnuous and dscrete envronments. We consder a set of m robots n an envronment wth k dfferent event types. Each event type s measured wth a dfferent sensor n S, and a robot s equpped wth a subset of these sensors,.e., S S. The obectve s to poston the robots n the envronment maxmzng the total sensng qualty of k event types. A. Contnuous Envronments Consder a convex envronment D R 2. Let [k] denote the set {1,..., k}. The densty functon of an event type s φ : D R +, [k], whch represents the measure of nformaton or the probablty of an event type occurrng over D. Let p D be the poston of robot n the envronment and P = {p 1,..., p m }. The sensng qualty of a sensor, denoted by f, s a decreasng functon of the dstance of the robot to the measured pont. Thus, robot s assgned to measure the events of type S at the ponts closest to p,.e., for each event type S robot measures the events n the Vorono cell = {q D q p q p r, S S r, r [m]}. For smplcty we let V denote the set of all the Vorono cells. Now we defne the heterogeneous deployment problem n contnuous envronments as follows: Problem II.1 (Coverage n Contnuous Envronments). Gven a set of m robots wth dynamcs p = u [m] where u s the control nput, fnd a set of locatons P = {p 1,..., p m } that maxmzes the total sensng qualty functon,.e., m H(P, V ) = f ( q p )φ (q)dq. (1) =1 S The sensng qualty measure H(P, V ) s the total sensng qualty of k event types over the envronment by the robots located at P. Contrary to the formulaton n [11], by defnng k Vorono parttons for the envronment, we ensure that an event of type at locaton q s sensed by the closest robot wth the requred sensor. Moreover, the defnton of H(P, V ) captures dfferent sensng qualty functon f for each event type. Observe that wth k = 1, the qualty measure H(P, V ) becomes the sensng qualty for homogeneous robots as proposed n [5]. Fgure 1 llustrates an nstance of the coverage problem wth two event types. In [11], the authors consdered the same heterogeneous coverage problem, but wth the obectve H het (P) = q p 2 φ S dq [m] V + (1 ) q p 2 φ S dq [m] where (0, 1], V s the Vorono partton generated by the postons of all robots and φ S = S φ. To llustrate the dfferences wth the proposed obectve n Equaton (1), Fgure 2 shows a coverage problem on a lne wth three event types. Each robot senses the event type of the same color. The colored polygons represent the densty functon of each event type. As shown n Fgure 2a, the obectve n [11] has locally optmal confguratons n whch one event type s covered sub-optmally. Ths s n contrast to the confguraton shown n Fgure 2b, where a partton s generated for each event type. B. Dscrete Envronments In the dscrete case, the envronment s represented by an undrected graph G = (V, E, c). The graph could represent a roadmap of an envronment wth obstacles. Let V be the set of vertces, and let E be the edge set, whch represents the paths between vertces. The cost functon c : E R + assgns a cost c(e) for traversng each edge e E. The events occur on the vertces of the graph, and the functon φ u gves the mass of event type at vertex u V. Let d(u, v) be the length of the shortest path from u to v on graph G. The goal s to deploy robots to a set of vertces of the graph. We do not consder deployments n whch robots are located on the edges of the graph, and ths s motvated by the result on the propertes of locatng robots on vertces gven n Lemma IV.1. An event of type s assgned to the closest robot among the robots that are equpped wth the sensor type. If there exsts an even wth two equally close robots, [m], we assgn the event to robot f > and robot otherwse. The subset W (P) s the set of vertces assgned to robot to sense events of type S on those vertces. Observe that W s the dscrete analogue of. For smplcty, we let W denote the set of all the subsets W. Now we defne the heterogeneous deployment problem n dscrete envronments as follows: Problem II.2 (Coverage n Dscrete Envronments). Gven a set of m robots fnd a set of locatons on the graph P = {p 1,..., p m } such that maxmzes the total sensng qualty functon,.e., m H(P, W ) = f (d(p, v))φ v. (2) D =1 S v W Observe that the obectve functons s the adaptaton of the obectve functon (1) to dscrete envronments. Also note that wth k = 1, the qualty measure H(P, W ) becomes the sensng qualty for homogeneous robots as proposed n [17].

3 (a) Event densty of event type 1,.e., φ 1. The sensor capable of sensng ths event s shown wth a crcle on the robots. (b) Event densty of event type 2. The sensor capable of sensng ths event s shown wth a trangle on the robots. Fg. 1: The Vorono parttons of an envronment for each event type. A robot consdered n a Vorono partton f t s equpped wth the requred sensor for the event type. (a) Locally optmal confguraton for [11] wth < 1 shown on top and = 1 shown below. (b) Locally optmal confguraton for Equaton (1) Fg. 2: Instance of coverage of three event types on a lne. Trangles (resp. rectangle) show the densty functon for event type 1(resp. 2 and 3). Robots are capable of sensng the event types wth the same color. III. GRADIENT ASCENT ALGORITHM FOR CONTINUOUS ENVIRONMENTS In ths secton, we provde a dstrbuted control law for robots wth a proof of convergence to a local maxmum of H(P, V ) n Problem II.1. A well-known approach to maxmze the sensng qualty for each robot to ascend the gradent of H. Let δ be the boundary of the Vorono cell, and let δvk be the common boundary of the Vorono cells and k. Let n k (q) be the unt normal to δ k at q n the outward drecton of. Fnally, we defne the neghbors of a robot n Vorono partton as N = {l [n] δv δ l }. Then we establsh the results on the dervatve of H wth respect to the poston of robots n Lemma III.1. The results n Lemma III.1 s an extenson to the homogeneous case n [5] and the proof follows closely. Lemma III.1. For dfferentable sensng functons f, the dervatve of H wth respect to the poston of robot s H p = m =1 S f ( q p ) φ (q)dq. p Proof. By the Lebntz theorem [18] we have, H m ( f ( q p ) = φ (q)dq p p =1 S + l N + l N Observe that n l = n l follows mmedately. f ( q p )n l (q) (δ p k ) (q)φ (q)dq f ( q p )n l (q) (δ k ) p (q)φ (q)dq ). for all l N. Then the result An nterestng observaton from ths s that unlke the homogeneous verson of the coverage problem, two robots can share a locaton n D f they do not have sensors n common,.e., the dervatve s defned for any P D m \Q m where Q m s the set of ponts n whch robots wth common sensor type overlap,.e., Q m = {{p 1,..., p m } D m, [m] p = p,, S S }. To derve a dstrbuted control law, we frst rewrte the dervatve of H n the followng form. H = 2 (p q) df (x) p S d(x 2 ) q p φ (q)dq. (3) Let the mass and centrod of a Vorono cell respectvely be df M V = V d(x 2 ) p q φ(q)dq, C V = 1 df M V d(x 2 ) p q φ(q)dq. V q Then we have H p = 2 S M V the followng smple control law 1 u = c M S M V V S (p C V ). Now consder (p C V ) [n], (4) where c s a postve gan such that p + u remans n the convex set S. Robot s the generator of a cell n each of the Vorono parttons of the events n S, and the control law u moves the robot to the average of the

4 centrods of ts Vorono cells. Observe that computng ths control law only requres the communcaton between robot and ts neghbors n each Vorono partton. We assume that the communcaton range s suffcent for the robots to communcate wth ther neghborng robots. The followng shows the result on the convergence of the proposed dstrbuted control law n Equaton (4). Proposton III.2 (Contnuous-tme Lloyd Ascent). Under control law (4), the robots converge to the unon of Q m and the set of crtcal ponts of H. Proof. Under control law (4), f c of robot at some tme nstance becomes zero then robot s on the boundary of S and concdng wth another robot. Therefore, the robots have converged to Q m. Otherwse, the robots startng from a collson free confguraton n D m \Q m reman nsde D m \ Q m at any tme nstance, then the set D m \ Q m s a postvely nvarant set under control law (4). The dervatve of H wth respect to tme s m d dt H(P(t), V ) = H m 2c p = ( H ) 2. p p =1 =1 S M V Therefore, observe that the drecton of control law (4) concdes wth the gradent of H. The rest of the proof follows from the proof of Proposton 3.1 n [19], whch uses LaSalle s Invarance Prncple to show at each step of the algorthm H monotoncally ncreases and converges to the largest nvarant set contaned n C m ={P D m ( H(P) p ) 2 = 0 [n]} Although control law (4) converges to a set consstng of Q m, smulaton results n Secton V show that by resolvng collsons wth local a controller, the system remans n D m \ Q m and converges to the set of crtcal ponts of H. Remark III.3 (Unknown Densty Functons). Consder the scenaro where the robots do not have access to the densty functons of the event types and they only observe events of dfferent types arrvng over tme. A control law s proposed n [13] for the homogeneous case whch drves the closest robot to the observed event and converge to a locally optmal postons. In the heterogeneous case, each observed event Z = [z d, z c ] s a random vector consstng of a dscrete component z d [k], denotng the observed event type, and a contnuous component z c D representng the locaton of the event. The events of dfferent types arrve wth equal frequences and accordng to ther spatal densty functons. We assume that the relatve mportance of the event types, denoted by Φ = D φ (q)dq s known. By extendng the results n [13], the control law become p,t+1 = (5) df {p,t γ t Φ dx 2 z c t p,t (zt c p,t ) f zt d S, zt c V zd t otherwse. p,t Observe that the expected value of the drecton df dx 2 x= z c t p,t (zt c p,t ) concdes wth the drecton of the dervatve of H wth respect to p n Equaton (3). The control law (5) drves the closest robot wth the requred equpment towards the observed event. The convergence of the control law to locally optmal postons arrve from the proof of Theorem 3 n [13]. IV. GRADIENT ASCENT ALGORITHM FOR DISCRETE ENVIRONMENTS In ths secton, we dscuss the coverage problem for heterogeneous robots n a dscrete envronment wth multple event types (Problem II.2 n Secton II). Gven a graph, the goal s to poston the robots on the vertces of the graph such that the total sensng functon H s maxmzed. We defne two confguratons of the robots on the graph. The frst confguraton Q V s a subset of sze m of the vertces of the graph whch represents postonng the robots on the vertces. The second confguraton D E [0, 1] represents the placement of the robots on the edges. For a placement of robot on edge (u, v),.e., p = ((u, v), γ) (see Fgure 3), parameter γ s the fracton of the path from p to v along the edge (u, v). The dstance of robot from vertex w V s d (p, w) = mn{γc(u, v) + d(u, w), (1 γ)c(u, v) + d(v, w)}. For a confguraton D, each event s assgned to the closest robot wth requred equpment. Then total sensng functon for confguraton D and partton W (D) s H (D, W (D)) = m =1 S v W f (d (p, v))φ v. (6) We provde a gradent ascent control law to poston the robots n the graph maxmzng Equaton (2). To motvate the approach, frst, we provde the followng result on the optmal soluton of the dscrete problem. Lemma IV.1. For any placement of the robots n the graph D, there exsts another placement of the robots on the vertces Q such that H (D, W (D)) H(Q, W (Q)). Proof. For any placement of the robots n the graph, f there exsts a robot located on the edge of the graph, we create another placement wthout decreasng the total sensng qualty. Fgure 3 llustrates an nstance n whch a robot s located on the edge (u, v). For each event type S, we partton the vertces n W nto two subset where the frst subset W,u conssts of the vertces n W such that the shortest path from p contans u and smlarly the second subset W,v conssts of the vertces n W such that the shortest path for the robot to reach the vertex passes through v. The total event mass on the frst subset s S w W φ w and smlarly for,u the second subset s S w W φ w. Then we move the,v robot to the vertex wth a larger total event mass. Observe that movng the robot to the vertex wth a larger total event mass can only ncrease the total sensng qualty. Ths result motvates us to consder the cases where the robots are located on the vertces of the graph. The set of admssble controls for robot, U, s lmted to S W (P t)

5 Fg. 3: Robot located on an edge of a graph. to ensure that the robots are requred to communcate wth only ther neghborng robots. The robots, k are called neghbors f there exsts such that the sets W, W k share n edge on graph G. Observe that the control set U s a nonempty set snce t contans the current locaton of the robot. We say the robots are n a locally optmal confguraton f there s no robot can take a control nput n ts control set to mprove the sensng qualty unlaterally. Now we provde the control law on the graph as follows: p,t+1 = arg max u U S w W (Pt) f (d(u, w))φ w. (7) Control law (7) drves the robots to a confguraton on the graph maxmzng the total sensng qualty of the events n the parttons domnated by the robot. Ths resembles the control law n Equaton (4). The followng lemma provdes the results on the convergence of the control law (7). Lemma IV.2. Under the control law (7), the robots converge to a set of locally optmal locatons for the total sensng problem n dscrete envronments. Proof. By the defnton of the control law n Equaton (7) we move robot to the new poston such that t maxmzes the sensng qualty for the vertces n the parttons of robot,.e., W (P t), S. Consderng a fxed partton, by any robot relocatng on the graph, the total sensng qualty mproves,.e., H(P t, W (P t )) H(P t+1, W (P t )). Observe that the sensng qualty s maxmzed when each event s assgned to the closest robot. Then by updatng the partton for the new postons of the robots we have H(P t+1,w (P t ))) H(P t+1, W (P t+1 )). Therefore, the qualty mproves wth each step of the control law and the algorthm converges to a locally optmal soluton of the dscrete total sensng qualty problem. V. EXPERIMENTAL RESULTS In ths secton, we evaluate the performance of the control laws for both contnuous and dscrete envronments. A. Contnuous Envronment The performance of the proposed control law for contnuous envronments s evaluated wth four experments. Each experment conssts of 8 GRITSBots [20] wth dfferent sensng capabltes. The proposed control law s mplemented on the Robotarum [21] usng both smulaton and physcal experments. The densty functon for each event type s gven by a bvarate normal dstrbuton,.e., φ (q) = β 2π Σ exp( 1 2 (q ζ ) T Σ 1 (q ζ ) ), β 1 β = 1 [k] S 1 = S 7 = {3, 4}, S 2 = {2, 4}, S 3 = {1, 2, 3}, S 4 = {1, 3}, S 8 = {4} S 5 = {1, 2, 3, 4}, S 6 = {1, 2} 2 β = 1 [k] S = {1, 2} 4, S = {3, 4} 5 3 β = 1 [k] S = { [k] <= } 4 β = [k] S = [k] Sensng Improvement Percentage Event Type 1 Event Type 2 Event Type 3 Event Type 4 Total Sensng TABLE I: Parameters of the experments Number of Steps S Proposed Algorthm Heterogeneous Lloyd s Algorthm Fg. 4: Performance of the proposed control law for Experment 3 where ζ s the mean and Σ = [0.1, 0; 0, 0.1] s the covarance matrx. The parameter β s a scalng factor to represent the mportance of each event type. The sensng functon s for all event types f (x) = x 2, [k]. Table I shows the parameters of each experment. In Fgure 4, the sensng qualty of the proposed control law s compared to that of the Heterogeneous Lloyd s algorthm proposed n [11]. The results are the average of 100 nstances of Experment 3 wth dfferent unformly random ntal locatons for the robots. The mean values ζ, [k] of densty functons for each nstance are generated randomly n a 1 1 envronment. The dashed lnes represent the total sensng qualty n Equaton (1) for both algorthms. The sold lnes n Fgure 4 show the sgnfcant mprovement of the sensng qualty for each event type wth respect to the Heterogeneous Lloyd s algorthm. The percentage mprovement s the rato of the dfference between the total sensng qualty of the Heterogeneous Lloyds algorthm to the total sensng qualty of the proposed algorthm. The comparson of the two algorthms on dfferent experments are gven n Table II. The proposed control law outperforms the exstng algorthm n total sensng qualty and n the sensng qualty of each event type. Fgure 5 llustrates the fnal confguraton of the robots n the physcal mplementaton of the proposed algorthm n the Robotarum wth 8 robots for an nstance of Experment 1. In the experment, the proposed algorthm mproved the total sensng qualty by 87.7% n 1 mnute. B. Dscrete Envronment In the dscrete envronment, we evaluate the performance of the proposed control law n a deployment problem motvated by transportaton applcatons. Fgure 6 shows 100 randomly generated pck-up locatons n Manhattan. Requests for rdes arrve at the pck-up locatons. Each request has a capacty requrement n {2, 3, 4, 5}, gvng the sze of the Total Sensng Qualty H

6 Event Event 2 Event 3 Event 4 Total Sensng TABLE II: Average percentage mprovement of the sensng qualty for the proposed algorthm compared to Heterogeneous Lloyd s algorthm over 100 nstances for each experment. The mean devaton from average s denoted by. (a) Event type 1 (b) Event type 2 (c) Event type 3 (d) Event type 4 Fg. 5: Eght robots wth dfferent sensng capabltes n the Robotarum wth the envronment sze of meters. The densty functons of each event type are shown va contours. The colored crcles next to the robots show the generators of each Vorono partton. group seekng a rde. There are 5 vehcles of each capacty {2, 3, 4, 5} requrement. There are then four event types, one for events wth each capacty requrement. Observe that a vehcle wth capacty can serve any request wth a group sze of k. The obectve s to poston the vehcles n the envronment to mnmze the tme to respond to a request. Note, here we are consderng only the ntal deployment problem, whch s to place the vehcles at locatons to best respond to an ntal request. Fgure 7 shows the mprovement n the total servce qualty for dfferent event types and the total servce qualty. The results are the average of 1000 randomly generated mass functons. The lnes show the mean value and the shaded area represent the frst and thrd quartles of each data set. Notce that the event type wth capacty 2 s servced by all the vehcles, and the algorthm shows the best mprovement for ths event type. 30 Percentage Improvement n Servce VI. C ONCLUSION Ths paper consders the problem of coverage control of multple robots wth heterogeneous sensng capabltes n contnuous and dscrete envronments. A new formulaton s ntroduced for measurng the coverage of multple event types wth dfferent event dstrbutons, and a dstrbuted control law s presented for maxmzng the coverage of the robots n the envronment, whch only requres communcaton between neghborng robots. The extensve results show sgnfcant mprovement n the sensng qualty of the events compared to the exstng studes. For future work, we plan to extend the analyss to capture dfferent sensng functons for the same event type on dfferent robots, and more complex event types such as pck-up-and-delvery tasks n rde-sharng applcatons. Fg. 6: Pck-up locatons n Manhattan N.Y. 25 capacty = 2 capacty = 3 capacty = 4 capacty = 5 Total Servce Qualty Number of Robot Relocatons Fg. 7: Percentage mprovement of the servce tme 70

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