MANY applications involve robots that are deployed in a. Multi-Robot Data Gathering Under Buffer Constraints and Intermittent Communication

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1 Mult-Robot Data Gatherng Under Buffer Constrants and Intermttent Communcaton Meng Guo Student Member, IEEE and Mchael M. Zavlanos Member, IEEE arxv:17.9v [cs.ma] 3 Oct 17 Abstract We consder a team of heterogeneous robots whch are deployed wthn a common workspace to gather dfferent types of data. The robots have dfferent roles due to dfferent capabltes: some gather data from the workspace (source robots) and others receve data from source robots and upload them to a data center (relay robots). The data-gatherng tasks are specfed locally to each source robot as hgh-level Lnear Temporal Logc (LTL) formulas, that capture the dfferent types of data that need to be gathered at dfferent regons of nterest. All robots have a lmted buffer to store the data. Thus the data gathered by source robots should be transferred to relay robots before ther buffers overflow, respectng at the same tme lmted communcaton range for all robots. The man contrbuton of ths work s a dstrbuted moton coordnaton and ntermttent communcaton scheme that guarantees the satsfacton of all local tasks, whle obeyng the above constrants. The robot moton and nter-robot communcaton are closely coupled and coordnated durng run tme by schedulng ntermttent meetng events to facltate the local plan executon. We present both numercal smulatons and expermental studes to demonstrate the advantages of the proposed method over exstng approaches that predomnantly requre all-tme network connectvty. Index Terms Networked Robots, Lnear Temporal Logc, Moton and Task Plannng, Intermttent Communcaton. I. INTRODUCTION MANY applcatons nvolve robots that are deployed n a workspace to gather dfferent types of data and upload them to a data center for processng. For nstance, teams of unmanned ground vehcles (UGV) can montor the temperature, humdty, and stand densty n large forests or teams of unmanned aeral vehcles (UAV) can montor the behavor of anmal flocks and growth of the crops n farmlands [1]. Due to heterogeneous sensng and moton capabltes, the robots n these applcatons can gather dfferent types of data n dfferent regons wthn the workspace. Thus the robots can be assgned local data-gatherng tasks that vary across the team [1]. In ths work, we employ Lnear Temporal Logc (LTL) as the formal language to descrbe complex hgh-level tasks beyond the classc pont-to-pont navgaton. A LTL task formula s usually specfed wth respect to an abstracton of the robot moton [], [3]. Then a hgh-level dscrete plan s found usng off-the-shelf model-checkng algorthms [], and s executed through low-level contnuous controllers [5]. Ths framework can be extended to allow for both robot moton and actons n the task specfcaton []. The authors are wth the Department of Mechancal Engneerng and Materals Scence, Duke Unversty, Durham, NC 77 USA. Emals: meng.guo, mchael.zavlanos@duke.edu. Ths work s supported n part by the NSF awards CNS #11 and CNS #13. The above framework has also been appled to mult-robot systems ether n a top-down approach where a global LTL task formula s assgned to the whole team of robots [3], [7] [9], or n a bottom-up manner where an ndvdual LTL task formula s assgned locally to each robot [1], [11]. Here, we favor the latter formalsm as t provdes a more natural framework to model ndependent temporal tasks wthn large teams of robots that have heterogeneous capabltes. Specfcally, we consder two types of robots: source robots that are assgned local tasks to gather dfferent types of data n dfferent regons n the workspace, and relay robots that receve data from source robots and upload them drectly to a data center. All robots have a lmted buffer to store the data. Thus the data gathered by source robots should be transferred to relay robots before the buffers overflow. Moreover, all robots have a lmted communcaton range, so that they can only communcate when they are suffcently close to each other. Communcaton n the feld of moble robotcs has typcally reled on constructs from graph theory, wth lne-of-sght models [1], [13] and proxmty graphs [1] [19] ganng the most popularty. In most of these problems, the property of nterest s connectvty of the communcaton network as ths allows relable delvery of nformaton between any par of robots. Approaches that ensure connectvty for all tme ether mantan all ntal communcaton lnks between the robots provded that the ntal communcaton network s connected [1], [19] [1], or allow for addton and removal of communcaton lnks whle ensurng that the connectvty requrement s not volated [15] [1], [], [3]. Realstc communcaton models have recently been proposed n [] [] that take nto account path loss, shadowng, and multpath fadng. The above approaches enforce all-tme connectvty thus are rather restrctve. Intermttent communcaton frameworks, on the other hand, allow the robots to occasonally dsconnect from the team and accomplsh ther tasks free of communcaton constrants. Intermttent communcaton n mult-agent systems has been studed n consensus problems [7], coverage problems [], and n delay-tolerant networks [9], [3]. The common assumpton n these works s that the communcaton network s connected nfntely often. In our recent work [31] [33], we proposed an ntermttent connectvty control strategy that ensures the whole team s connected nfntely often for coverage and path optmzaton problems. However, local hgh-level temporal tasks are not consdered there nor s a model of nter-robot data transfer. The constrant of lmted buffer sze s of practcal mportance especally for tme-crtcal data-gatherng applcatons and for local temporal tasks that requre an nfnte sequence

2 of data-gatherng actons. The work n [3] consders a sngle robot transferrng data between locatons. The proposed approach mnmzes the tme nterval between two consecutve data-uploadng tme nstants. But t does not explctly model the evoluton of the robot s buffer or the nter-robot communcaton. Smlar buffer constrants are consdered n [35] for mult-robot fronter-based exploraton. However locallyassgned data-gatherng tasks descrbed by LTL formulas are not consdered there, nor are communcaton constrants. Another related area s temporal logc task plannng under resource constrants. The work n [3] consders a global survellance task performed by multple aeral vehcles subject to battery chargng constrants. The mult-vehcle routng problem consdered n [37] proposes a soluton based on Mxed- Integer Lnear Programmng (MILP), whch can potentally be extended to nclude resource constrants. The man contrbuton of ths work les n the development of an onlne dstrbuted framework that jontly controls local data-gatherng tasks and data transfer communcaton events, so that the buffers at every robot never overflow. The proposed framework guarantees the satsfacton of all local tasks specfed as LTL formulas, wthout mposng all-tme connectvty on the communcaton network. The effcency of the proposed framework compared to a centralzed approach and two statc approaches s demonstrated va numercal smulatons and expermental studes. To the best of our knowledge, ths s the frst dstrbuted data-gatherng framework under ntermttent communcaton that s also onlne. Ths work s bult on prelmnary results presented n [3]. Compared to [3], the real-tme control and coordnaton algorthm presented here s more effcent as t allows the relay robots to swap meetng events n order to faster servce the source robots, whle t also accounts for robot falures, dynamc robot membershp, and fxed data centers. Furthermore, more extensve numercal smulatons are presented, as well as expermental results showng the capabltes of our method. The rest of the paper s organzed as follows: Secton II ntroduces some prelmnares on LTL and Büch Automata. Secton III formulates the problem. Secton V dscusses the proposed dynamc approach to jont data-gatherng and ntermttent communcaton control. Numercal smulatons and experment studes are shown n Sectons VII and VIII, respectvely. We conclude n Secton IX. II. PRELIMINARIES ON LTL Atomc propostons are Boolean varables that can be ether true or false. The ngredents of an LTL formula are a set of atomc propostons AP and several boolean and temporal operators, wth the followng syntax []: ϕ ::= p ϕ 1 ϕ ϕ ϕ ϕ 1 U ϕ, where True, p AP and (next), U (untl),. For brevty, we omt the dervatons of other useful operators lke (always), (eventually), (mplcaton). The semantcs of LTL s defned over the nfnte words over AP. Intutvely, σ AP s satsfed on a word w = w(1)w()w(3)... ( AP ) ω f t holds at w(1),.e., f σ w(1). Formula ϕ holds true f ϕ s satsfed on the word suffx that begns n the next poston w(), whereas ϕ 1 U ϕ states that ϕ 1 has to reman true untl ϕ becomes true. Fnally, ϕ and ϕ are true f ϕ holds on w eventually and always, respectvely. We refer the readers to Chapter 5 of [] for the full defnton of LTL syntax and semantcs. The language of words that satsfy an LTL formula ϕ over AP can be fully captured through [] a Nondetermnstc Büch automaton (NBA) A ϕ, defned as A ϕ = (Q, AP, δ, Q, F ), where Q s a set of states; AP s the set of all allowed alphabets; δ Q AP Q s a transton relaton; Q, F Q are the set of ntal and acceptng states, respectvely. The process of constructng A ϕ can be done n tme and space O( ϕ ), where ϕ s the length of ϕ []. There are fast translaton tools [39], [] to obtan A ϕ gven ϕ. A. Robot Model III. PROBLEM FORMULATION Consder a team of N dynamcal robots where each robot N {1,,, N} satsfes the uncycle dynamcs: ẋ = v cos(θ ), ẏ = v sn(θ ), θ = ω, (1) where p (t) = (x (t), y (t)) R, θ (t) ( π, π] are robot s poston and orentaton at tme t >. The control nputs are gven by u (t) = (v (t), ω (t)) as the lnear and angular velocty. Each robot has a reference lnear and angular veloctes denoted by v ref and ω ref, whch are used later to estmate the travelng tme. The workspace s a bounded D area W R, wthn whch there are clusters of obstacles O W. The free space s denoted by F = W\O. Note that all robots are assumed to be pont masses and robot collson s not consdered here. As mentoned n Secton I, the robots are categorzed nto two subgroups, denoted by N l, N f N so that N l N f = N and N l N f =. Every robot N f s equpped wth short-range wreless unts and can only send and receve data from other robots j such that p (t) p j (t) r, where r > s the communcaton range, j N. On the other hand, robots n N l are equpped wth long-range wreless unts and have the extra functon to upload ther stored data to a remote data center. In other words, robots n N f are responsble for gatherng data about the workspace whle robots n N l are n charge of uploadng these data to the data center. In the sequel, we smply refer to robots n N f as source robots and robots n N l as relay robots. Note that there s at least one source and relay robot,.e., t holds that N f, N l 1. Remark 1. The fact that the relay robots can upload ther stored data mmedately to the data center s due to ther long-range communcaton capabltes. Ths assumpton can be relaxed by choosng several fxed data centers wthn the workspace that the relay robots need to vst and upload ther data. More detals are provded n Secton VI-A. B. Data-gatherng Tasks Each source robot N f has a local data-gatherng task assocated wth dfferent regons n the freespace. Denote

3 by Π = {π,1, π,,, π,m } the collecton of these regons, where π,l F, l = 1,,, M and M >. They contan nformaton of nterest. Moreover, there s a set of data-gatherng actons that robot can perform at these regons, denoted by G = {g,, g,1, g,,, g,k }, where g,k means that type-k data s gathered by robot, k = 1,,, K and K 1. By default, g, means dong nothng. The tme needed to perform each acton by robot N f s gven by functon Z : G R +. Wth a slght abuse of notaton, we denote the set of robot s atomc propostons by AP = {π,l g,k, π,l Π, g,k G }, where each proposton π,l g,k stands for robot gathers type-k data at regon π,l. Over these atomc propostons, we can specfy a hgh-level data-gatherng task, denoted by ϕ, followng the LTL semantcs n Secton II. Smply speakng, ϕ specfes the desred sequence of datagatherng actons to be performed at certan regons of nterest wthn the workspace. Note that LTL formulas allow us to specfy data-gatherng tasks of fnte or nfnte executons. For nstance, ϕ = ((π,1 g, ) (π,3 g, )) means that robot should gather type- data at regon 1, then type- data at regon 3, or ϕ = (π, g,7 ) (π,7 g, ) means that robot should nfntely often gather type-7 data at regon and type- data at regon 7. Remark. It s worth mentonng that relay robots j N l do not have local tasks as ther goal s to communcate wth source robots and upload data to the data center. Ths assumpton can be relaxed and s part of our future work. C. Buffer Sze and Communcaton Constrants Each robot N has a lmted buffer to store data. To smplfy the formulaton, we quantfy the data sze nto unts,.e., robot has a buffer to store a maxmum number of B > unts of data, N. Furthermore, denote by b (t) N the number of data unts stored n the buffer of any robot N at tme t. Note that b () =, N. It must hold that b (t) B, t such that the buffer of robot does not overflow. Whenever robot N f performs a data-gatherng acton g,k G at tme t, b (t) changes as follows: b (t + ) = b (t ) + D (g,k ), () where D : G Z + s the number of data unts gathered by performng acton g,k G ; b (t ) and b (t + ) are the number of data unts at robot s buffer before and after the acton g,k s performed at tme t. If b (t + ) > B, then ths acton g,k can not be performed as t wll lead to buffer overflow. We assume that D (g,k ) B, g,k G, meanng that any acton can be performed when the buffer s zero. Moreover, any two robots can send and receve data when they are wthn each other s communcaton range. In partcular, denote by c j : R Z + the data transfer map from robot to robot j at tme t >. When robot transfers c j (t) unts of data to robot j, ther stored data unts change by: b (t + ) = b (t ) c j (t) and b j (t + ) = b j (t ) + c j (t), (3) where b (t + ) and b (t ) (or b j (t + ) and b j (t )) are the stored data unts of robot (or robot j) before and after the data transfer. To allow ths transfer, two condtons must hold: () c j (t) b (t ) so that robot has enough data to transfer; and () b j (t + ) B j so that robot j s buffer does not overflow. At last, as mentoned earler, any relay robot j N l has an extra functon to upload ts stored data to the remote data center. Denote by d j : R Z + the upload functon of robot j at tme t >. When robot j uploads d j (t) unts of data to the data center, ts stored data changes as follows: b j (t + ) = b j (t ) d j (t), () where b j (t + ) and b j (t ) are defned smlarly as before. Clearly, the uploaded data must not be more than the stored data,.e., d j (t) b j (t ) and b j (t + ). D. Problem Statement Consder a team of N robots, consstng of N f source robots and N l relay robots, that all satsfy the dynamcs (1). Each robot N has a lmted communcaton rage r and a maxmum buffer sze B. The robots onboard buffers change accordng to ()-(). Furthermore, each source robot N f s assgned a data-gatherng task captured by an LTL formula ϕ over AP. The problem we address n ths paper s () the desgn of moton controllers u and acton events D that satsfy the local tasks ϕ, N f ; as well as () the desgn of sequences of communcaton events c j and d j that ensure data delvery to the data center wthout buffer overflow, N f and j N l. Moreover, we seek a soluton that s dstrbuted and onlne, meanng that there s no central coordnator that collects all nformaton and determnes the robots moton and actons. Note that, even though data storage s nowadays very cheap and for many practcal purposes can be consdered unlmted, settng a buffer lmt has the advantage that t forces the robots to relay the gathered data to the data center more frequently, before ths lmt s reached. Thus, buffer constrants can be used to model urgency for communcaton and they are mportant n case of tme crtcal tasks. Such tasks can range from mult-robot survellance where the urgency to collect nformaton to a data center s related to qucker response tmes to possble stuatons, to cooperatve transportaton where buffer lmts can be used to model the loads that the robots can carry and transport to each other. Note that mposng tme constrants (compared to buffer constrants that ndrectly model urgency to delver data) would change completely the problem formulaton addressed n ths paper and s part of our future work. Remark 3. Note that dfferent from top-down approaches [7], [9], here the data-gatherng tasks are assgned locally to each source robot, not to the whole team. Each source robot does not need to know the number of the other source robots or ther local tasks. IV. CENTRALIZED OPTIMAL SOLUTION In ths secton, we present a centralzed soluton to the consdered problem, whch s also the optmal soluton.

4 R 1 R R 3 Source R R 5 Relay m,l π,l meanng that the next transton can be taken only f all robots have completed ther current transton. Not only does ths ntroduce heavy communcaton overhead for synchronzaton but also ths all-tme synchronzaton may be nfeasble due to lmted communcaton range consdered here. More numercal analyses can be found n Secton VII. V. DYNAMIC DATA-GATHERING AND INTERMITTENT COMMUNICATION CONTROL Fgure 1: Illustraton of the proposed soluton n Secton V. Each source robot (n magenta, green, orange) syntheszes ts own dscrete plan, whch ncludes the regons of nterest (n grey-flled crcles), the wayponts n between (n black-flled crcles) and actons to perform dfferent regons. They coordnate wth relay robots (n blue and red) to meet, transfer and upload the gathered data (ndcated by blue and red arrows), before ther buffers overflow. Note that each source robot can coordnate wth multple relay robots, and vce versa. The centralzed soluton conssts of three major steps: () the constructon of a composed transton system for the whole team, whch encapsulates all robots moton and actons (ncludng data gatherng, data upload and data exchange). Partcularly, the composed FTS s defned as T a (S a, a, S a,, T a, AP, L a ), (5) where S a = (Π 1 B 1 ) (Π B ) (Π N B N ) s the set of composed states and B = {, 1,, B }. Namely, state s S a ndcates the poston and buffer sze of each robot. The transton relaton a S a S a s defned by (s, s ) a where s = (π 1, b 1 ) (π N, b N ) and s = (π 1, b 1) (π N, b N ) f robot s allowed to transton from π to π, and f the change n buffer sze from b to b satsfes both the communcaton-range constrants and the buffer dynamcs defned n ()-(), N. The ntal state S a, S a s gven by the ntal poston and buffer sze of the robots. The transton cost T a : a R + measures the tme that each transton takes. AP = N f AP s the set of propostons. Lastly, the labelng functon L a : S a AP reflects the data-gatherng actons that have been performed by the source robots at the regons of nterest. () The conjuncton of all source robots local tasks s defned as ϕ a = N ϕ, f and the correspondng NBA s derved as A ϕa, as descrbed n Secton II. () Standard model checkng algorthms [] can be used to search for a lasso-shaped path of T a that satsfes ϕ a. These nvolve constructng the product automaton P a between T a and the NBA A ϕa. To optmze the total plan cost both n the plan prefx and plan suffx, defned as the accumulated travel tme, the synthess algorthm from our earler work [1] can be used. Note that the above soluton has two serous drawbacks: frst, t s computatonally ntractable for systems wth large numbers of robots and complex tasks, due to the combnatoral sze of composed system and the double-exponental complexty of the model-checkng process []. Second, the derved plan needs to be executed n a fully-synchronzed way, The proposed soluton, as shown n Fgure 1, conssts of three man parts: () the workspace abstracton and the synthess of local dscrete plans; () the coordnaton of meetng events between source and relay robots, ncludng the ntal coordnaton and the real-tme coordnaton; and () the executon of local dscrete plans and the data transfer protocol. A. Local Dscrete Plan Synthess Intally at tme t =, each source robot N f syntheszes ts local dscrete plan to satsfy ts local task ϕ. Ths plan s gven as an nfnte sequence of regons to vst and the datagatherng actons to perform at each regon. 1) Road Map Constructon: Frst, an abstracton of the freespace F s constructed as a roadmap on whch all robots n N can move. Defnton 1. The roadmap over the freespace F s a weghted and undrected graph M = (M, H, W ), where M s the set of wayponts m R, m M, H M M ndcates whether two wayponts are connected, and W : H R + s the Eucldean dstance between two wayponts. To construct the roadmap M, n ths work, we rely on the trangulaton algorthm for polygons wth holes, see Chapter n [1] and the package polytr n []. We omt the algorthmc detals due to lmted space and refer the nterested readers to [3] and [3], [] for dfferent algorthms. An example s shown n Fgure. Ths roadmap allows the robots to move among the wayponts wthout crossng the obstacles. Usng the roadmap M, we can construct a fnte transton system (FTS) to abstract the moton of each source robot N f among ts regons of nterest wthn the freespace. Denote ths moton model by T = (Π,, Π,, T ), where Π s the set of regons of nterest, Π Π denotes the transton relaton, Π, Π s the regon robot starts from ntally, T : R + approxmates the tme each transton takes. Partcularly, consder two regons of nterest of robot denoted by π,s, π,f Π. Denote by m,s, m,f M the closest wayponts to the center ponts of π,s and π,f, respectvely. Then, (π,s, π,f ) f there exsts a path n M startng from m,s to m,f wthout crossng any other waypont m,l M that belongs to any other regon π,l Π wth l s, f. Denote the shortest of those paths by Γ,sf = m,s m,s+1 m,f, whch can be obtaned from a graph search over M between m,s and m,f. Furthermore, for each transton (π,s, π,f ), the tme for robot to traverse the

5 y(m) 1 1 y(m) 1 r7 r r r r r9 r 1 Fgure : Left: example of the constructed roadmap for the workspace model n Secton VII. Blue areas are boundares and obstacles. The wayponts and edges are shown by red ponts and lnes; Rght: example of the dscrete plans for three source robots a, a 1, a (n blue, purple, yellow), wth regons of nterest marked by ther labels. assocated path Γ,sf s computed by T (π,s, π,f ) = ( f 1 m,k, m,k+1 ) /v ref k=s + ( f Θ(m,s+1 m,s, m,s+ m,s+1 ) ) /ω ref k=s where v ref, ω ref are the reference lnear and angular veloctes as defned n Secton III, and the functon Θ : R R ( π, π] computes the angle between two D vectors. Note that T ( ) s only an estmate of the tme t takes for robot to travel along each edge. Gven the moton abstracton T and the data-gatherng actons n G, the complete robot model can be constructed as shown below; more detals can be found n []. Defnton. The complete robot model s the FTS R = (Π,R,,R, AP, L, Π,,R, T,R ), where Π,R = Π G s the full state;,r Π,R Π,R s the transton relaton such that ( π,s, g,l, π,f, g,k ),R f () π,s, π,f and g,k = g,, or () π,s = π,f and g,l, g,k G ; AP are the atomc propostons from Secton III-B; the labelng functon s defned as L ( π,s, g,l ) = {π,s, g,l }, π,s, g,l Π,R ; Π,,R = Π,, g, s the ntal state; and T,R ( π,s, g,l, π,f, g,k ) = T (π,s, π,f ) + Z (g,k ), ( π,s, g,l, π,f, g,k ),R s the tme measure. ) Local Plan Synthess: The local plan of robot, denoted by τ,r, s an nfnte path of R whose trace satsfes ts local task ϕ. We rely on the automaton-based model checkng algorthm [], [1] to synthesze τ,r, whose descrpton we omt here due to lmted space. Partcularly, the local plan τ,r has the prefx-suffx structure below and the mnmum total cost as the summaton of prefx and suffx costs: τ,r = π,r π,r 1 π k 1,R (πk,r πk+1,r r1 r, r5 r3 r () πk,r )ω, (7) where the state π,r k Π,R, k =, 1,, K and K > s the total length, π,r π1,r πk 1,R s the prefx executed only once and π k,r πk+1,r πk,r s the suffx to be repeated nfntely often. τ,r provdes an nfnte sequence of moton and data-gatherng actons to be performed by robot. Software mplementaton detals can be found n [1], []. Example 1. Consder the roadmap shown n Fgure wthn a clustered workspace. Three robots are deployed wth dfferent local tasks. For nstance, robot a needs to vst r 1, r and r 3 n sequence and perform the acton g 1 at each regon. The resultng dscrete plan τ,r s shown n Fgure. Note that each source robot N f syntheszes τ,r locally wthout coordnaton wth other robots. Thus, robot mght not execute τ,r successfully by tself wthout the help of relay robots to transfer data, due to the nfnte sequence of datagatherng actons n τ,r and ts lmted buffer sze. B. Coordnaton of Intermttent Meetng-Events To execute the plan of each source robot N f, we need to ensure that ts stored data s transferred to at least one relay robot j N l before ts buffer overflows. The man dffculty les n the lmted communcaton range for both source and relay robots, meanng that both data transfer and coordnaton are only possble when two robots are wthn each other s communcaton range. As dscussed n Secton I, nstead of mposng all-tme connectvty as n most related work [1], [1], [1] [3], we propose here a dstrbuted onlne coordnaton scheme where the communcaton network s allowed to become dsconnected. The key dea s to desgn a method that allows source and relay robots each tme they meet (.e., connect to each other) to negotate when and where they should meet the next tme, whle mnmzng the watng tme at the new meetng locaton. Afterwards, they move ndependently wthout communcaton, untl they meet agan at the agreed locaton and tme, and the same procedure repeats. In the sequel, we present a dstrbuted coordnaton scheme for both the source robots and relay robots to schedule meetng events, whch s based on onlne request and reply message exchanges, for four dfferent scenaros: () the ntal coordnaton phase; () the real-tme coordnaton for the next meetng event; () the spontaneous meetng event; and (v) the swappng of meetng events. 1) Intal Coordnaton: Intally at t =, each source robot needs to coordnate ts frst meetng event wth at least one relay robot. Denote by N (t) N the set of robots that robot N can communcate wth at tme t,.e., N (t) = {j N p (t) p j (t) r}. Then, denote by N l(t) = N () N l the set of relay robots that a source robot N f s connected to at tme t =. We mpose the followng assumpton on the ntal confguraton: Assumpton 1. At tme t =, each source robot N f s connected to at least one relay robot j N l : N l (). Meetng requests by source robots: To begn wth, every source robot N f needs to estmate where and when t needs to meet wth a relay robot j N l, gven ts dscrete plan τ,r. We consder the followng problem. Problem 1. For each source robot N f, fnd the frst waypont n the τ,r and the assocated tme that robot needs to meet wth a relay robot and transfer data, before robot s buffer overflows.

6 To solve Problem 1, robot N f needs to search through the future sequence of states n τ,r and determne the frst state where the data stored n ts buffer wll exceed ts buffer sze B f t has not met any relay robot to transfer ts data n the meanwhle. Denote by π ke,r τ,r ths state and by π kt,r τ,r the current state of robot, where k e > k t. Specfcally, the ndex k e of π ke,r τ,r s the ndex such that ke k=k t D (g,lk ) B, k e+1 k=k t D (g,lk )>B, () where π,r k = π,s k, g,lk, k t k k e and D (g,lk ) s the number of data unts gathered by acton g,lk from (). Thus, the buffer s less or equal to ts full capacty up to π ke,r, but t wll overflow at π ke+1,r after performng acton g,l ke+1. Then, robot calculates the route and the assocated tme to transton from π ke,r to πke+1,r. Wthout loss of generalty, let π ke,r Π = π,s and π ke+1,r Π = π,f. The shortest path from π,s to π,f s gven by Γ,sf = m,s m,s+1 m,f from Secton V-A1 and the assocated tme of reachng each waypont m,s Γ,sf s denoted by t,k T,sf, where T,sf = t,s t,s+1 t,f and s k f. The tme sequence T,sf s calculated usng the reference lnear and angular veloctes by (). As a result, the request message from a source robot N f to a relay robot j N l () at tme t =, denoted by Req j (), s gven by Req j () = (Γ,sf, T,sf ), j N l (), (9) where Γ,sf and T,sf are defned above. Smply speakng, robot s requestng that robot j should come to meet at any of the wayponts wthn Γ,sf at the assocated tme n T,sf. Reples by relay robots: Upon recevng the requests from all source neghbors N f j (), where N f j () N j() N f, each relay robot j N l should decde the locaton and tme to meet each source robot N f j () and reply accordngly. Denote by Rep j () the reply message from robot j to robot at tme t =, whch has the followng structure: Rep j () = (m j, t j ), N f j () (1) where m j M s the waypont where robots, j wll meet and t j > t s the tme of the meetng event. Partcularly, gven the requests Req j () = (Γ,sf, T,sf ) by (9), N f j (), we ntend to fnd a path Γ j() = m j,1 m j, m j,sj, where m j,sj M, s j = 1,,, S j and an assocated tme sequence T j () = t j,1 t j, t j,sj such that the followng two condtons hold. Condton one: Γ j should ntersect wth Γ,sf exactly once,.e., there exsts exactly one waypont m j Γ,sf that m j Γ j, N f j. Wthout loss of generalty, let m j = m,kj where s k j f and m j = m j,sj where 1 s j S j ; Condton two: Γ j should mnmze the sum of the dfferences n the predcated meetng tme between robot j and each N f j (),.e., N t f,kj t j,sj, where t,kj T,sf and t j,sj T j are j the correspondng tme nstances of reachng m j n Γ,sf () and Γ j (), respectvely. Formally, we state the problem below. Problem. Gven Req j () = (Γ,sf, T,sf ), N f j (), compute Γ j () such that both condtons above hold. Problem s closely related to the well-known travelng salesman problem (TSP) [5] but wth three dstnctons: the set of wayponts to be vsted s to be determned by the soluton; there s no need to return to the startng waypont; and the cost s defned as the total watng tme over each waypont nstead of the total travel dstance. The above problem s NPhard [1] as t contans the TSP as a specal case. A smlar formulaton appears n the computer wrng problem as dscussed n [5]. To fnd the exact soluton to Problem, we can transform t nto a generalzed TSP. In partcular, let N f j,+ = N f j {j} {ν}, where N f j s the set of source neghbors that robot j s connected to and ν s an artfcal node. Recall that the requests Req j () = (Γ,sf, T,sf ) satsfy Γ,sf = m,s m,s+1 m,f and T,sf = t,s t,s+1 t,f. For ease of notaton, let I sf {s, s + 1,, f }. As mentoned n condton one, Γ j ntersects wth Γ,sf exactly once, N f j. Let ths happen at the (k ) th element of Γ,sf, where k I sf, N f j. Let us defne frst the set of wayponts Υ = {m j,, m ν,, m,k, N f, k I sf }, whch ncludes all wayponts wthn each source robot s path segment Γ,sf and m j, = m ν, (x j (), y j ()). Note that ν s an artfcal node at the end of Γ j. Also, to smplfy the notaton, we set I sf j {k j, } and I sf ν {k ν, }, where k j, = k ν, denote the frst and the only element assocated wth nodes j and ν, respectvely. Furthermore, we defne a cost functon c : Υ Υ R between any two nodes n Υ such that: () for all, h N f, t holds that c kk h t,k + T j (m,k, m h,kh ) t h,kh, (11) where k I sf and k h I sf h, where t,k, t h,kh are the assocated tme nstants of m,k, m h,kh obtaned from T,sf and T h,sh f h and the functon T j ( ) s the tme t takes robot j to travel from m,k to m h,kh, whch can be computed smlarly to (); () for all N f {j}, c kk ν =, k I sf ; and () for all N f, c k νk = +, k I sf, and c kνk j =. Furthermore, let β kk h B be a Boolean varable so that β kk h = 1 f Γ j contans a segment from m,k to m h,kh, and s otherwse, k I sf Gven the above notatons, we can formulate the followng, k I sf and, h N f j,+. nteger lnear program (ILP) on the varables {β kk h }: mn {βk k h } c kk h β kk h (1) s.t. k, k h I sf,h ;,h N f j,+ k h I sf h ; h N f j,+ k, k h I sf,h ; h N f j,+ β kh k = h N f j,+ ; k h I sf h β kk h, k I sf, N f j,+ ; (1a) β kk h = 1, N f j,+ ; (1b) α k α kh + (N f j + 1) β k k h N f j, (1c) k, k h I sf,h ;, h N f j {ν}; where the notaton k, k h I sf,h s equvalent to k I sf and k h I sf h, smlar arguments hold for k, k j I sf,j ; and α k Z s used to avod the exstence of multple cycles,

7 k I sf and N f j {ν}. The frst two constrants (1a)- (1b) ensure that exactly one element of Γ,sf s ntersected by Γ j, N f j. The last constrant (1c) and the defnton of varables {α k } ensure that all the wayponts m k and m kh that satsfy β kk h should belong to one bg cycle where m ν, s the last waypont and s connected to m j,. Smply speakng, assume that an addtonal cycle of wayponts (excludng m j, ) wth length N c > appears n Γ j. Summng up the nequaltes wthn (1c) for all wayponts contaned n that cycle would yeld N c (N f j +1) N c N f j, leadng to a contradcton. More detals can be found n [5]. The ILP problem by (1) has ( ) ˆN Boolean varables and ˆN nteger varables, where ˆN = N Γ f,sf s the total j,+ number of wayponts and ( ) ˆN s the bnomal coeffcent. Thus the complexty of (1) s closely related to the number of source robots that each relay robot ntally connects to and ther request messages. Note that (1) always has a soluton as each relay robot j N l can reach any waypont n M (thus any waypont n Γ,sf ). Lastly, gven the solutons Γ j and T j, the reples Rep j () can be derved as: m j = m,k and t j = t,k, N f j (). Note that durng the transton (m j,s, m j,s+1 ) Γ j, robot j can ntersect wth Γ,sf more than once but no data exchange wll take place wth robot. Remark. If the watng tme of some source robots are penalzed more than other source robots, we can readly ncorporate ths aspect by mposng statc prortes wthn the source robots n Problem, by addng dfferent weghts n front of the watng tme c kk h β kk h. Confrmaton by source robots: Upon recevng the reples Rep j () from all relay robots j N l (), each source robot N f evaluates these reples and sends confrmatons back. In partcular, denote by Conf j () the confrmaton message from the source robot to robot j N l () at tme so that Conf j () = f robot confrms the meetng locaton and tme wth robot j, whle Conf j () = f robot refuses the reply and thus s not commtted to the meetng event wth robot j. Gven the reples Rep j () = (m j, t j ), j N l(), robot chooses the relay robot j N l () that yelds the mnmum watng tme for tself at the frst meetng event,.e., j = argmn j N l () t j t,kj, (13) where s k j < f satsfes that m,kj = m j. Then, Conf j () =, for j above, whle Conf j () =, j N l() and j j. Thus robot marks m,k j as the meetng locaton wth robot j at tme t,kj. On the other hand, after recevng the confrmaton messages Conf j () from source robots N f j (), each relay robot j N l removes the meetng event wth each source robot from ts path Γ j () that was computed by (1) f the confrmaton message from the source robot satsfes Conf j () =, N f j (). In other words, each relay robot j N l s only commtted to meet the source robots that have confrmed the meetng event. ) Coordnaton for Next Meetng Event: After the ntal coordnaton, robots and j wll meet at the waypont m j at tme t = t j, N f. For the ease of notaton, we replace j by j n ths secton. Then, the data at robot s buffer wll be transferred to robot j s buffer and wll be uploaded to the data center, see Secton V-C. When ths happens, the two robots wll need to coordnate n order to determne ther next meetng event followng the procedure descrbed below. Frst, robot needs to determne agan the segment of ts future plan when t should meet wth a relay robot, before ts buffer overflows. The same equaton as n () can be appled gven that the robot s current buffer sze s zero and π kt,r s the current state. Denote the new request message by Req j (t) = (Γ,sf, T,sf ), where Γ,sf = m,s m,f and T,sf = t,s t,f are defned analogously as before. Then, after recevng the request, robot j needs to reply wth ts preferred next locaton and tme to meet wth robot, denoted by m + j and t + j, respectvely. Let Γ j(t) = m j,kj m j,fj be the remanng path obtaned by (1) at tme t, and the assocated sequence of tme nstants s T j (t) = t j,kj t j,fj. Thus, the last commtted meetng locaton and tme are gven by m j,fj and t j,fj. Then m + j can be chosen among Γ,s f such that movng from m j,fj to m + j yelds the mnmum watng tme for robot. Thus, t holds that m + j = m,s + and t + j = t j,s + j, where the ndex s + j satsfes that s + j = argmn s s j f t j,fj t,sj + T j (m j,fj, m,sj ), (1) where T j (m j,fj, m,sj ) s the tme to navgate from waypont m j,fj to m,sj. s + j can be found by teratng through all wayponts n Γ,sf to fnd the mnmum watng tme. Therefore, the reply message from robot j to s gven by Rep j = (m + j, t+ j ). After recevng the reply message, robot wll send back the confrmaton as Conf j = and mark m + j as the next meetng locaton wth robot j. On the other hand, after the confrmaton, robot j wll concatenate ts path Γ j wth the shortest path from m j,fj to m + j wthn M and mark m + j as the next meetng locaton wth robot. 3) Spontaneous Meetng Events: When there are more than one relay robots n the team, t s possble that robot N f meets wth another relay robot j N l on ts way to meet the confrmed relay robot j. We call ths stuaton a spontaneous meetng event. In ths case, robot transfers the stored data n ts buffer to robot j, and coordnates wth j for the next meetng event n a smlar way as descrbed n Secton V-B, but now robot takes nto account the fact that t wll meet wth j at m + j as prevously confrmed. Thus, the next path segment of Γ where robot needs to meet wth a relay robot should be calculated as n () by settng π kt,r = (m+ j, g ),.e., robot s buffer s zero after meetng robot j at m + j. After the coordnaton wth robot j, robot contnues to meet robot j. In ths way, a source robot can meet and transfer data through all relay robots t has met, nstead of beng restrcted to the relay robot t was connected to ntally. Each tme t coordnates wth a new relay robot, t takes nto account the fact that t wll meet wth all the relay robots t has commtted to and partcularly ts buffer wll be empty after the last meetng event. It s crucal that the source robot stll meets ts ntally confrmed relay robot j (even wth an empty buffer), after a

8 y(m) Orgnal Γ j1 Orgnal Γ j y(m) Updated Γ j1 Updated Γ j Fgure 3: Vsualzaton for Example of robots j 1, j s paths before and after the swappng algorthm presented n Secton V-B. Intal poston of robots j 1 and j are ndcated by flled stars. spontaneous meetng wth another relay robot j N l. Due to the lmted communcaton range, robot can not nform robot j to cancel the confrmed next meetng. If robot smply skps that meetng, robot j wll wat for robot at the confrmed regon ndefntely, whch leads to a deadlock. Remark 5. Note that source robots are not allowed to transmt data to each other even when they are wthn the communcaton range. Ths assumpton can be relaxed and s part of our ongong work. ) Relay Robots Swap Meetng Events: Untl now, we have dscussed the communcaton between source and relay robots. In ths part, we dscuss how relay robots can communcate wth each other and swap ther commtted meetng events wth source robots. Partcularly, assume that two relay robots j 1, j N l meet at tme t >. The remanng path and the assocated tme stamps of robot j 1 are gven by Γ j1 (t ) = m j1,k j1 m j1,f j1 and T j1 (t ) = t j1,k j1 t j1,f j1, respectvely. Smlarly, Γ j (t ) = m j,k j m j,f j and T j (t ) = t j,k j t j,f j for robot j. Our goal s to rearrange the entres n Γ j1 and Γ j such that the total watng tme for source robots s further reduced. Clearly, the optmal way to rearrange Γ j1 and Γ j1 that yelds the mnmum watng tme s to formulate a nteger lnear problem smlar to (1). It can be thought of as a travelng salesman problem wth two salesmen. Here we propose a greedy algorthm that takes advantage of the ordered structure of Γ j1 and Γ j1. Frst, we construct a new sequence of -tuples Υ = (m 1, t 1 )(m, t ) (m L, t L ), where L = Γ j1 + Γ j. It holds that m l = Γ j1 [l 1 ] and t l = T j1 [l 1 ] wth the ndex l 1 that satsfes k j1 l 1 f j1, or m l = Γ j [l ] and t l = T j [l ] wth the ndex l that satsfes k j l f j, l = 1,, L. More mportantly, Υ s ordered by t 1 t t L,.e., an ncreasng tme order accordng to whch each waypont should be vsted. Second, let Υ 1 and Υ be two subsequences of Υ that we want to construct. They are ntalzed by Υ 1 = (m j1 (t ), t ) and Υ = (m j (t ), t ), where m j1 (t ) and m j (t ) are the wayponts robots j 1 and j are located, respectvely. Then we terate over each entry of (m l, t l ) Υ and evaluate the watng tme usng (1) f the paths of robots j 1 or j contan ths entry as ther last meetng event. If robot j 1 yelds a smaller watng tme, we add (m l, t l ) to the end of Υ 1 ; otherwse, f robot j yelds a smaller watng tme, we add (m l, t l ) to the end of Υ. At last, Υ 1 s decomposed nto the new Γ j1 and T j1 for robot j 1, whle Υ s decomposed nto the new Γ j and T j for robot j. Snce the above algorthm s greedy, we can compare the total watng tme under the new paths Γ j1 and Γ j, whch s then compared to the orgnal total watng tme. If the total watng tme s reduced, the updated Γ j1 and Γ j wll be used; otherwse, the paths reman unchanged. In ths way, some of the meetng events are swapped between relay robots j 1 and j and the total watng tme s reduced. Example. Consder two relay robots j 1 and j wth tmed paths (Γ j1, T j1 ) and (Γ j, T j ) as shown n Fgure 3. The reference veloctes are gven n Secton VII. The paths are updated by the above algorthm to swap ther meetng events. The total watng tme s reduced from 3.3s to 1.1s. C. Real-tme Executon Real-tme executon of the system conssts of two essental components: () the local plan executon of source robots and () the meetng events between source and relay robots. 1) Plan Executon: After the system starts, each source robot N f executes ts dscrete plan τ,r = π,r π1,r πk 1,R (πk,r πk+1,r πk,r )ω, where π,r k = π,sk, g,lk Π,R, k =, 1,, K, whch was derved n Secton V-A. Startng from the ntal poston π,s, robot frst navgates to regon π,s1 through the correspondng path Γ,ss 1. The control nputs follow the turn-and-forward swtchng control: (C.1): v = and ω = ω ref ; and (C.): v = v ref and ω =. The controller (C.1) s actvated to turn robot towards the next waypont n Γ,ss 1 and then, (C.) drves t forward wth the reference speed. Once robot reaches π,s1, t performs the data-gatherng acton g,l1 there. After the acton s completed, robot navgates to regon π,s through Γ,s1s and performs acton g,l there. Ths procedure repeats tself untl robot reaches the (k e ) th state π ke,r accordng to (). Durng ths perod of tme, the amount of data unts stored n robot s buffer s ncreased ncrementally by D (g,lk ) usng (), k =, 1,, k e. Then on ts way from state π ke,r to πke+1,r, robot meets wth robot j at waypont m j. It s ensured by the formulaton of () that the buffer s never overflowed and all data-gatherng actons can be performed before reachng π ke+1,r. After the meetng, robot contnues executng the rest of ts plan untl the next meetng event wth j or another relay robot. Smlarly, any relay robot j N l starts by executng the path Γ j derved from (1) at tme, whch s then modfed by addng new segments each tme robot j coordnates wth a source robot about the next meetng event. Remark. Note that f a dfferent controller s used, such as PID-based lne followng, then () needs to be updated to reflect the estmaton of travelng tme between wayponts. Furthermore, more complex robot dynamcs can also be ncorporated as long as the robot s travelng tme between two wayponts can be well estmated.

9 ) Meetng Events Executon: Assume that Γ,sf = m,s m,s+1 m,f s the path that robot follows to navgate from m,s to m,f, and assume also that ts confrmed meetng waypont wth robot j s m,s. Startng from m,s, robot moves towards m,s. Then two cases are possble: () f robot j s already watng at m,s, then robot contnues movng towards m,s untl robot j s wthn ts communcaton range. When ths happens, robot transfers all the data stored n ts buffer to robot j. As a result, the stored data unts n the buffers of robots and j are updated accordng to b (t + ) = and b j (t + ) = b j (t )+b (t ). When the data transfer s completed, robot j uploads all the data n ts buffer to the data staton mmedately. Thus ts stored data s updated accordng to b j (t + ) =. If the stored data at robot s more than robot j s buffer sze B j, these data are dvded nto smaller batches, whch are then transferred to robot j sequentally; () f robot j has not arrved at m j yet, then robot wats untl robot j enters ts communcaton range and then follows the same procedure as n (). Note that due to the watng procedure descrbed above, an exact synchronzaton on the meetng tmes s not requred between the source and relay robots. Namely, f ether robot or j arrves at a meetng locaton later than the agreed meetng tme t j (e.g., due to uncertanty n robot velocty), the other robot that arrves early wll wat untl the data exchange happens. Therefore, the proposed method can handle uncertanty n the travelng tmes defned n (). Furthermore, delays on current meetng events do not propagate to the future meetng events snce all subsequent meetng events defned n (1) are always coordnated usng the current meetng tmes. In other words, delays are always reset to zero whenever two robots meet. A numercal robustness analyss of the proposed approach can be found n Secton VII. Proposton 1. Under Assumpton 1 statng that each source robot s connected to at least one relay robot ntally, the above framework ensures that each source robot N f can satsfy ts local task ϕ and also that ts buffer wll not overflow. Proof. Frst, the correctness of the local plan for each source robot s guaranteed by the model-checkng algorthm, see [], [1]. Moreover, snce all local tasks are ndependent, these local plans can be executed ndependently. Thus we need to show that the plan can be executed successfully by each source robot,.e., the data-gatherng actons can be performed and the data buffer never overflows. Intally, each source robot s confrmed to meet wth one relay robot by (1). When the two robots meet, the stored data can be transferred and uploaded, before the source robot s buffer overflows due to the formulaton of (). Then executon of the meetng events above ensures that every source robot always wats to meet a relay robot and transfer the stored data before performng the next gatherng acton that leads to buffer overflow. Smlarly, the spontaneous meetng events descrbed n Secton V-B3 ensure that all data-gatherng actons up to the next meetng tme can be performed and the data buffer never overflows. The same procedure repeats tself and holds for all source robots. VI. DATA CENTER CONSTRAINTS, ROBOT FAILURES, AND DYNAMIC ROBOT MEMBERSHIP In ths secton, we dscuss how the proposed framework can be extended to account for a fxed data center, robot falure, and dynamc membershp. The later two characterstcs enhance the robustness of the proposed approach. A. Fxed Locaton of Data Center As mentoned n Remark 1, assume that a relay robot j, nstead of uploadng ts stored data mmedately after meetng a source robot, needs to vst a fxed data center H j M wthn the workspace to upload the data, j N l. Then the proposed scheme can be modfed as follows. Consder the meetng between robot j and the source robot N f. Frst, durng the executon of the meetng event as dscussed n Secton V-C, robot j s moton plan needs to be modfed to nclude vstng the data center. In partcular, f the amount of data robot needs to transfer s less than robot j s buffer sze, robot j can receve all the data at once and then travel to the data center va the shortest path to upload the data. On the other hand, f the amount of data robot needs to transfer s more than robot j s buffer sze, robot j can receve the data n batches that equal to ts buffer sze, and then travel to the data center multple tmes. Consequently, for fxed data center locatons, t may be benefcal to par up source and relay robots of smlar buffer szes to reduce the number of tmes that relay robots need to travel to the data center. In ths case, Algorthm 1 can be modfed by redefnng c kk h as follows: c kk h t,k +T j (m,k, m h,kh )+N j T j (m,sj, H j ) t h,kh, where N j B j /B s the number of tmes robot j needs to travel to the data center H j, B j /B s the rato between robot j and robot s buffer sze, the functon returns the prevous largest nteger; and T j (m,sj, H j ) s the tme t takes for robot j to navgate from waypont m,sj to H j. As a result, the coordnaton obtaned by the soluton of problem (1) now consders the extra tme that s needed for robot j to travel to H j to empty robot s buffer gven robot j s buffer lmt. Second, regardng the coordnaton of the next meetng event as dscussed n Secton V-B, robot j s choce of the next meetng locaton from (1) can be modfed as follows: s + j = argmn s s j f t j,fj t,sj + T j (m j,fj, m,sj ) + N j T j (m,sj, H j ), (15) where N j and T j (H j, m,sj ) are defned above. Now (15) takes nto account the extra tme that s needed for robot j to travel to H j n order to empty robot s buffer. Last but not least, f there are multple data centers that robot j can choose from, we can easly modfy (15) to fnd the optmal one. B. Robot Falures Let us assume frst that a source robot N f fals. If robot can stll communcate wth all relay robots t has commtted to meet, then robot can ntate a cancel message to each of them to cancel the commtted meetng events. In ths way, these relay robots can skp the meetng wth robot

10 and contnue meetng the next source robot (nstead of watng ndefntely for robot ). However, f robot fals when t s not n the communcaton range of one or more relay robots, then to avod deadlock we can ntroduce a maxmum watng tme T max >, so that f a robot wats at a confrmed meetng locaton for a perod of tme longer than T max, then t assumes that ths meetng s canceled and contnues executng ts dscrete plan untl the next meetng event. Assume now that a relay robot j N l fals. If robot j can stll communcate wth the source robots t s commtted to meet, t can cancel the meetng events drectly as before. However, n ths case, the source robot N f j can not smply skp ths meetng event and contnue ts plan executon as ts buffer wll overflow. Instead, robot needs to navgate to ts next meetng locaton drectly, upload ts stored data wth relay robot j N l and more mportantly keep the next meetng event wth robot j unchanged. In other words, robot needs to meet wth robot j consecutvely twce. Last but not least, f robot j s the only relay robot that robot s commtted to, robot may have to wat untl t meets another relay robot to upload ts data. Ths can only happen spontaneously as robot has no knowledge of the locaton of other relay robots due to lmted communcaton range. Ths stuaton can be solved by allowng source robots to relay data to each other or exchange nformaton about ther meetng events, whch s part of our ongong work, see also Remark 5. At last, f several source or relay robots fal, the procedure descrbed above wll be performed for each fault robot. Moreover, f a robot recovers after falure, t wll be treated as a robot that newly jons the system, as dscussed below. C. Dynamc Membershp By dynamc membershp, we mean that () exstng robots wthn the team can leave the team wthout resultng n a deadlock; and () new robots can jon the team seamlessly wthout the need to restart the system. The frst case can be acheved n a smlar way as descrbed n Secton VI-B to handle robot falures. Partcularly, before a source robot leaves the team, t needs to meet wth each relay robot that t s commtted to meet, but wthout coordnatng the next meetng event. In the same way, before a relay robot leaves the team, t stll needs to meet wth each source robot that t s commtted to meet, wthout coordnatng the next meetng event. Secondly, due to the dstrbuted and onlne nature of the proposed scheme, new source or relay robots can be easly added to the system durng run tme. If the new relay robot j that just joned the team s connected to an exstng source robot N f, robot wll treat ths meetng as a spontaneous meetng event as descrbed n Secton V-B3. The same procedure apples when an exstng relay robot meets a new source robot that just joned the team durng run tme. However, a new source robot must be connected to at least one relay robot when t jons the team. VII. CASE STUDY Ths secton presents smulaton results for a team of 1 data-gatherng robots. All algorthms are mplemented n Python.7. Gurob [] and polytr [] are external packages and P_MAS_TG [] s developed by the authors. All smulatons are carred out on a laptop (3.GHz Duo CPU and GB of RAM). A. System Descrpton All 1 robots satsfy the uncycle dynamcs (1). There are 9 source robots (denoted by a, a,, a ) and 3 relay robots (denoted by l 1, l, l 3 ). The workspace has sze 1m 1m and contans three polygonal obstacles, as shown n Fgure. The trangular partton s derved from []. All robots communcaton ranges are set to 1m. The reference lnear and angular veloctes are chosen randomly between [.5,.]m/s and [.1,.3]rad/s. The buffer sze of all source robots s chosen randomly between [3, 5] data unts, whle all relay robots have a buffer sze of 5 data unts. To smplfy the task descrpton, we dvde the source robots nto three categores: () the frst category (a, a 1, a ) gathers type-1 data n regon r 1, type- data n regon r and type-3 data n regon r 3 (n any order), nfntely often. Ths specfcaton can be expressed by the LTL formula ϕ c1 = (r g ) (r 1 g 1 ) (r 3 g 3 ); () the second category (a 3, a, a 5 ) gathers type- and type- 5 data n regons r, then type- data n regon r (n ths order) and also type-5 data n regon r 5, nfntely often,.e., ϕ c = (((r g ) (r g 5 )) (r g )) (r 5 g 5 ); () the thrd category (a, a 7, a ) gathers type- data n regons r 7, r 9 and type-7 data n regon r, nfntely often,.e., ϕ c3 = (r g 7 ) (r 7 g ) (r 9 g ). The actons g, g 3, g, g gather unts of data, whle actons g 1, g 5, g 7 gather 1 unt. Moreover, any data-gatherng acton takes 1s whle the data transfer or upload actons take s. Intally, robots a, a 3, a, l 1 start from (.5m,.m), robots a 1, a, a 7, l start from (5.m, 5.m), and robots a, a 5, a, l 3 from (.m,.3m). Thus every source robot s connected to at least one relay robot, as requred by Assumpton 1. B. Smulaton Results Frst, the roadmap of each robot s constructed usng a trangular partton of the workspace, as descrbed n Secton V-A1. For robots a, a 1, a, the FTS R has 1 nodes and 11 edges, the NBA A ϕ has nodes and 13 edges, and the product P has nodes and 7 edges. For robots a 3, a, a 5, the FTS R has 1 nodes and 7 edges, the NBA A ϕ has 7 nodes and 3 edges, and the product P has nodes and 3 edges. For robots a, a 7, a, the FTS R has 1 nodes and 7 edges, the NBA A ϕ has nodes and 13 edges, and the product P has nodes and 31 edges. Then each source robot syntheszes ts dscrete plan usng the algorthm n [1] and the package []. It took approxmately.3s,.5s and.1s for the above three groups to synthesze ther dscrete plans. For nstance, a has prefx cost 57. and suffx cost.1, whle a 3 has prefx cost. and suffx cost 5.9. It took.3s by Gurob [] to fnd the optmal soluton of (1), whch determnes the ntal paths of all relay robots. The dscrete plans are executed accordng to Secton V-C1, whle the data are transferred and uploaded

11 y(m) 1 r7 a 7 r a r t=. s r1 a l 3 a r3 1 sendreceve r r9 a a send g l 1 recever r r r5 r 1 Fgure : A snapshot of the smulaton at s. source and relay robots are red and green squares, whle the stored data unts are ndcated by black crcles. The data-gatherng actons, data transfer and upload actons are shown by flled green text boxes, e.g., g, send, receve, upload. All robots and regons of nterest are labeled by ther names. Buffer Sze a a a a a l 1 a 1 a 3 a 5 a 7 l l 1 T me(s) Fgure 5: Stored data at each robot s buffer durng the smulaton. The buffer szes of robots a, a 1,, a and l, l 1, l are set to [, 5, 3,, 5, 5,, 5, 3, 5, 5, 5], whch are respected for all tme. durng the meetng events as descrbed n Secton V-C. The coordnaton for the next meetng event and spontaneous meetngs follow Sectons V-B and V-B3. We smulate the system for 1s. A snapshot of the smulaton at s s shown n Fgure, where we show the number of data unts stored at each robot s buffer and the acton taken by each robot. The evoluton of the stored data unts at each robot s buffer s shown n Fgure 5. The maxmum number of connected robots remans below 5 durng most of the smulaton, as shown n Fgure. Thus the communcaton network among the robots s almost never connected. Furthermore, we also montor the tmes that relay robots l, l 1, l swap meetng events as descrbed n Secton V-B. Fgure 7 shows the reducton n total watng tme after two relay robots swappng ther meetng events. In total, 137 unts of data are uploaded, as shown n Table II and Fgure 1. The complete smulaton vdeos can be found n [7]. Furthermore, n order to demonstrate the robustness of the proposed approach to uncertantes n the robots travelng tmes, we have smulated the case where the travelng velocty of all robots s subject to addtve random nose (wth zero mean and varance equal to % of the velocty value.). As shown n Fgure 1, the delays n the meetng events caused by a 3 a 5 a l Maxmal Components t(s) Fgure : The evoluton of the sze of maxmal components of the communcaton graph (of sze 1),.e., the maxmal number of connected robots, gven the unform communcaton range 1m. Watng Tme(s) l l 1 l T me(s) Fgure 7: Hstory of relay robots l, l 1, l swappng meetng events durng the smulaton. The hgh and low ponts of the error bar ndcate the total watng tme before and after the swappng, respectvely. uncertan travelng tmes are not propagated across the network and the total amount of gathered data wthn 1s n ths case s 9 (close to 137 n the nomnal case). Last but not least, as dscussed n Secton VI and shown n Fgure, the proposed scheme can be easly extended to take nto account other scenaros, e.g., fxed data center, robot falures and new members. Frst, we choose a fxed data center located at coordnate (.3, 7.) that all relay robots need to vst to upload ts stored data. Instead of uploadng the data drectly, a local plannng module s used by each relay robot to navgate to ths fxed data center as proposed n Secton VI-A. Second, we ntroduce faults to source robots a, a, a 1 and relay robot l at tme 5s, when they all stop movng and reman statc. Moreover, three new source robots a 3, a 7, a 11 and one relay robot l 3 are added to the system (thus 1 robots n total), wth the source robots havng the same task descrpton as three groups descrbed earler. The evoluton of the stored data at each robot s buffer s shown n Fgure 9. It shows that the buffer of these faulty robots remans unchanged after the faults occur, whle the rest of the team (along wth the new members) follow the reconfguraton scheme from Sectons VI-B and VI-C whle respectng the buffer constrant. It can be seen from the smulaton results that for the clustered workspace consdered here, the meetng events wth a faulty robot are canceled once the maxmum watng tme s reached and furthermore the new robots can easly jon the network va the spontaneous meetng events. Smulaton vdeos under these extended scenaros can be found n [7].

12 y(m) 1 t=5. s r1 r a1 l a receve 3 send lr3 l 1 r7 a g7 r a a 7 r r r9 r 1 r r5 r a 5 a 3 a y(m) 1 t=9. s r1 a r 3 r3 a a 1 l r r7 r r l upload 1 r a 9 ar9 a 11 g r l 1 a5 r5 g5 a r g 7 Fgure : Left: a snapshot of the smulaton where a fxed upload center s gven for each relay robot (marked by the blue star); Rght: a snapshot of the smulaton where exstng robots fal (n black squares) and new robots jon the team (marked by yellow squares). Buffer Sze a, a 1, a, a 5, a, a 9, l, l 1 a, a, a 1, l a 3, a 7, a 11, l 3 1 T me(s) Fgure 9: Stored data at each robot s buffer for the scenaro where three source robots a, a, a 1, l fal at tme t = 5s (n black lnes). At the same tme robots a 3, a 7, a 11, l 3 jon the team (n red lnes). The other robots are shown n blue lnes. The buffer szes are set to [, 5, 3, 5,, 5, 5, 5,, 5, 3, 5, 5, 5, 5, 5], whch are all respected. C. Comparsons to Other Approaches In ths part, we compare the data-gatherng performance of the proposed scheme to the centralzed approach and two statc approaches ntroduced below. Smulaton vdeos for all three approaches can be found n [7]. 1) Centralzed Approach: As mentoned n Secton IV, the centralzed soluton provdes the optmal soluton n terms of total dstance traveled. For ths case study, the product moton model has approxmately states and transtons. The product Büch automaton has approxmately states and transtons. Thus to construct the product automaton for the whole system s computatonally nfeasble. Moreover, we provde a numercal analyss to compare the optmalty and computatonal complexty of the proposed approach to the centralzed method, for problems of smaller sze that can be handled usng the centralzed method. The results are shown n Table I. It can be seen that () for small systems (wth 3 5 robots) the centralzed method provdes an optmal soluton that has a slghtly smaller total cost of the plan suffx than the proposed approach. However, as mentoned n Secton IV, ths centralzed plan can only be executed n a synchronzed way and s not robust to robot falures; () for larger systems (wth more than 3 robots), the centralzed method fals to provde a soluton wthn reasonable tme (where P has more than 1 a Method (N l, N f ) P C suf Tme[s] Proposed Centralzed (1, 1) (,.7e3) 37..1s (1, ) (1, 9.e3) s (1, 3) (3.e3, 1.3e).7.1s (5, 15) (1.e,.7e) s (7, 1) (.7e, 9.e) s (1, 1) (.5e3,.e) s (1, ) (3.1e5,.e7) h (1, 3) > (5.e, 1.3e9) * > h Table I: A comparson of optmalty and computatonal complexty between the proposed method and the centralzed approach. The notaton aeb a 1 b for a, b >. For the proposed method, P s the summaton of all local product P between R and A ϕ, C suf s the maxmum length of the plan suffx among all robots, and the synthess tme s manly the tme needed to solve the MILP problem for ntal coordnaton. For the centralzed case, P s the product of T a and A ϕa from Secton IV, C suf s the mnmum length of ts plan suffx, and the synthess tme s manly the tme needed for the model-checkng process. bllon states), whle n contrast our approach can scale much better, even to system wth 7 relay robots and 1 source robots. ) Statc Approaches: Alternatvely, a straghtforward soluton to the data-gatherng problem consdered n ths paper s to requre that all relay robots reman statc at ther ntal postons for all tme. As a result, as long as each source robot s nformed about the locaton of at least one relay robot, every source robot can smply navgate to the closest relay robot once t has gathered enough data that needs to be transferred and uploaded. Ths statc approach s always feasble for the problem consdered here, but can be very neffcent f the workspace s large and many relay robots are located close to each other. The optmal placement of relay robots can only be determned n a centralzed way as descrbed n Secton IV. We mplement the above approach and smulate the system for 1s under the same settngs presented n Secton VII-A. As a result, 5 unts of data are uploaded n total, as shown n Table II and Fgure 1, compared wth 137 unts va the proposed dynamc approach. The dfference s that n our approach every relay robot can actvely navgate to meet multple source robots that need to transfer data whle mnmzng the total watng tme. Fnally, another smple soluton s to force all source and relay robots to move as a group that s wthn communcaton range for all tme. In ths case the source robots can follow a predefned statc order to execute ther local plans. Snce all relay robots are wthn the communcaton range, the data gathered by any source robot can be transferred to any relay robot and uploaded drectly. Ths statc approach mposes all-tme connectvty of the communcaton network. It can also be very neffcent snce the source robots can not execute ther local plans smultaneously and ndependently, whle relay robots are not fully utlzed regardng ther data-uploadng ablty. Ths predefned statc order can be also optmzed n a centralzed way, as descrbed n Secton IV, by addng the constrants that all robots are wthn each other s communcaton range.

13 Approach type-1,,3 type-,5 type-,7 Total Proposed Statc One Statc Two Table II: Total amount of dfferent types of data uploaded by the relay robots durng the smulaton of 1s, under the proposed approach and two statc approaches dscussed n Secton VII-C. T otal Data Uploaded 1 5 Proposed Approach wth Uncertan Travel Tme Statc Approach One Statc Approach Two T me(s) Fgure 1: The total amount of data uploaded under the proposed approach and two statc approaches dscussed n Secton VII-C. Smulaton vdeos for all three cases are onlne [7]. We mplement the above approach and smulate the system for 1s under the same settngs. The source robots take turns to execute ther local plans accordng to the order of ther IDs. As shown n Table II and Fgure 1, only unts of data are uploaded n total, compared to 137 unts va our approach. The dfference s that the proposed ntermttent communcaton framework allows all source robots to move and execute ther local plans ndependently. Thus the source and relay robots only meet when they need to transfer data and coordnate ther next meetng event. The above studes show that the proposed dynamc approach has a much less computatonal burden compared to the centralzed approach and mproves greatly the overall data-gatherng effcency compared to the statc approaches. VIII. EXPERIMENTAL STUDY In ths secton, we present the expermental study to valdate the proposed approach. Four dfferental-drven Robots are deployed wthn a.5m.m workspace, as shown n Fgure 11, whose postons and orentaton are tracked va an Opttrack moton capture system. The communcaton among the robots s handled by the Robot operatng system (ROS). A. System Descrpton Three Robots serve as source robots (denoted by a, a 1, a ) whle one serves as the relay robot (denoted by l 1 ). As shown n Fgure 11, there are sx regons of nterest and two obstacles wthn the workspace; and a vsualzaton panel s used to montor the robot data-gatherng actons and communcatons n real tme. For source robots, ther regons of nterest, allowed actons and local tasks are defned as follows: Robot a has two regons of nterest r 1, r and two actons g 1, g assocated wth one type-1 and two type- data unts, respectvely. Its task s to gather type-1 data n regon r 1 and then type- data n regon r (n ths order) nfntely often,.e., ϕ = Fgure 11: A snapshot of the experment setup. Left: regons of nterest are marked by ther IDs on the ground. Trpods n boxed area are obstacles. The relay robot s marked by a yellow tape and the rest are source robots. Rght: the real-tme vsualzaton panel to montor the robot moton and communcaton. Robots a, a 1, a, l 1 are n blue, green, yellow and magenta, respectvely. The stored data unts are ndcated by flled black crcles. The data-gatherng, datatransfer and upload actons are shown by blue text boxes. y(m) r r r Fgure 1: The trajectory of each robot durng the experment, sampled at every 3s. Robots a, a 1, a, l 1 s trajectores are shown n blue, green, yellow and magenta, respectvely ((r 1 g 1 ) (r g )); Robot a 1 has two regons of nterest r, r and two actons g 3, g assocated wth two type- 3 and one type- data unts, respectvely. Its task s to gather type-3 data n regon r and then type- data n regon r (any order) nfntely often,.e., ϕ 1 = (r g 3 ) (r g ); Robot a has two regons of nterest r 7, r and two actons g 5, g assocated wth two type-5 and one type- data unts, respectvely. Its task s to gather type-5 data n regon r 7 and then type- data n regon r (any order) nfntely often,.e., ϕ = (r 7 g 5 ) (r g ). All robots have a lmted buffer sze of data unts and a communcaton range of.m. The ntal poston of robots a, a 1, a, l 1 s gven by (1.1,.), (1.1,.), (.,.7), (1.,.5) n meters, respectvely. Thus the relay robot l 1 s ntally connected to all source robots a, a 1, a, whch satsfes Assumpton 1. The sze of an Robot s around.m n dameter. Gven the cluttered workspace, a local collson avodance scheme s needed for successful pont-to-pont navgaton as an mportant part of the plan executon. In ths work, we rely on the method of recprocal velocty obstacles (RVO) ntroduced n []. However, snce the orgnal algorthm s developed manly for nonholonomc robots, not for the uncycle robots consdered here, we need to ntroduce a transton perod durng whch the r1 r r

14 Buf f er Sze a 5 a1 a l T me(s) 5 Fgure 13: Evoluton of the amount of data stored at each robot s buffer. Note that the buffer sze lmt s set to for all robots. Fgure 15: A snapshot for the experment under the statc approach one, where the relay robot l1 remans statc at all tme. T otal Data U ploaded 3 Proposed Approach Statc Approach One Statc Approach Two T me(s) 5 Fgure 1: The total amount of data uploaded durng the expermental study, under the proposed approach and two statc approaches. robots turn n place towards the desred drecton determned by the RVO method, before movng forward. B. Experment Results Followng the procedure descrbed n Secton V-C, we frst synthesze the offlne plan for each source robot. For robot a, t took.1s for the solver [] to obtan the ntal plan; smlarly for a1, a. For the ntal coordnaton va (1), t took.1s for Gurob [] to fnd the ntal path for robot l1. Once the robots starts movng, the plan executon and coordnaton of meetng events durng run tme follows Secton V-C. Note that swappng meetng events between relay robots s not consdered as there s only one relay robot. The experment was performed for a duraton of 3 mnutes, and the full vdeo can be found onlne at [7]. The sampled trajectory of each robot s plotted n Fgure 1. It can be seen that each robot satsfes ts local task and avods collsons wth the statc obstacles. Moreover, the amount of data stored wthn each robot s buffer s shown n Fgure 13, whch verfes that buffer constrants are always respected. Fnally, durng the experment, 7 data unts were uploaded n total to the data center, as shown n Fgure 1. C. Comparson to Statc Approaches We also compare the performance of our method to the two statc approaches ntroduced n Secton VII-C. The experment vdeos for all three cases can be found n [7]. Frst, as shown n Fgure 15, we conducted an experment usng the statc approach one for a duraton of 3 mnutes. Robot l1 remans stll at ts ntal locaton for all tme, whle robots a, a1, a navgate back to robot l1 once they have gathered enough data that needs to be transferred. As shown n Fgure 1, 17 unts of data are uploaded n total. Second, as shown n Fgure 1, we conducted an experment usng Fgure 1: Snapshot of the experment under the statc approach two, where all robots move as a group, beng connected at all tme. the statc approach two, also for a duraton of 3 mnutes. The robots form a platoon n the order a, a, l1, a1, so that all source robots a, a1, a are always wthn the communcaton range of robot l1. Robots a, a1, a take turns to execute ther local plans by navgatng wth the whole group to ther desred regons to gather data and transfer the data drectly to l1. As shown n Fgure 1, 1 unts of data are uploaded n total, compared to 7 unts usng the proposed dynamc approach. Thus smlar conclusons can be obtaned as n Secton VII-C that the proposed dynamc approach mproves greatly the overall data-gatherng effcency compared to the other two statc approaches. It s worth mentonng that sequence of spontaneous meetng events that happened durng the experment s qute dfferent from the smulated result, due to the nter-robot collson avodance scheme. IX. C ONCLUSION AND F UTURE W ORK In ths work we proposed a dstrbuted onlne framework for multple robots that jontly coordnates local data-gatherng tasks and ntermttent communcaton events so that the collected data at the robots are transferred to a data center whle ensurng that robot buffers do not overflow. Unlke most relevant lterature that reles on all-tme connectvty, the proposed ntermttent communcaton framework allows the robots to operate n dsconnect mode and accomplsh ther tasks free of communcaton constrants, sgnfcantly mprovng on the performance of data acquston and delvery. We valdated our method through numercal smulatons and real experments, and showed that all local data-gatherng tasks are satsfed and the local buffers do not overflow. R EFERENCES [1] M. Dunbabn and L. Marques, Robots for envronmental montorng: Sgnfcant advancements and applcatons, Robotcs & Automaton Magazne, IEEE, vol. 19, no. 1, pp. 39, 1.