Simultaneous Intermittent Communication Control and PathOptimization in Networks of Mobile Robots

Yiannis Kantaros and Michael M. Zavlanos

Abstract— In this paper, we propose an intermittent com-munication framework for mobile robot networks. Specifically,we consider robots that move along the edges of a connectedmobility graph and communicate only when they meet at thenodes of that graph giving rise to a dynamic communicationnetwork. Our proposed distributed controllers ensure inter-mittent connectivity of the network and path optimization,simultaneously. We show that the intermittent connectivityrequirement can be encapsulated by a global Linear TemporalLogic (LTL) formula. Then we approximately decompose itinto local LTL expressions which are then assigned to therobots. To avoid conflicting robot behaviors that can occur dueto this approximate decomposition, we develop a distributedconflict resolution scheme that generates non-conflicting dis-crete motion plans for every robot, based on the assigned localLTL expressions, whose composition satisfies the global LTLformula. By appropriately introducing delays in the executionof the generated motion plans we also show that the proposedcontrollers can be executed asynchronously.

I. INTRODUCTION

Communication among robots has been typically modeledusing proximity graphs and the communication problem isoften treated as preservation of graph connectivity [1]–[4].Common in the above works is that point-to-point or end-to-end network connectivity is required to be preserved for alltime. However, this requirement is often very conservative,since limited resources may hinder robots from accomplish-ing their assigned goals. Motivated by this fact, in thispaper we propose a distributed intermittent communicationprotocol for mobile networks. In particular, we consider thatrobots move along the edges of a connected mobility graphand communicate only when they meet at the nodes of thisgraph giving rise to a dynamic communication network.We design distributed controllers that ensure intermittentcommunication of the network while minimizing at the sametime the distance traveled between meeting points. We showthat intermittent communication can be captured by a globalLinear Temporal Logic (LTL) formula that forces robotsto meet infinitely often at the rendezvous points. Givensuch an LTL expression, existing model checking techniques[5], [6] can be employed in order to implement correct byconstruction controllers for all robots.

LTL-based control synthesis and task specification formobile robots build upon either a bottom-up approach whenindependent LTL expressions are assigned to robots [7]–[9]

This work is supported in part by the NSF awards CNS #1261828 andCNS #1302284.

Yiannis Kantaros and Michael M. Zavlanos are with the Department ofMechanical Engineering and Materials Science, Duke University, Durham,NC 27708, USA. {yiannis.kantaros,michael.zavlanos}@duke.edu

or top-down approaches when a global LTL describing acollaborative task is assigned to a team of robots [10], [11],as in our work. Bottom-up approaches generate a discretehigh-level motion plan for all robots based on a synchronousproduct automaton among all agents and, therefore, theyare resource demanding and scale poorly with the numberof robots. To mitigate these issues, we propose a noveltechnique that approximately decomposes the global LTLformula into local ones and assigns them to robots. Sincethe approximate decomposition of the global LTL formulacan result in conflicting robot behaviors we develop a dis-tributed conflict resolution scheme that generates discretemotion plans for every robot based on the assigned localLTL expressions. By appropriately introducing delays inthe execution of the generated motion plans we show theproposed controllers can also be executed in an asynchronousfashion. In contrast, most relevant literature assumes thatrobot control is performed in a synchronous way [10],[11]. Asynchronous robot mobility is considered in [12] byintroducing ‘traveling states’ in the discretized abstraction ofthe environment decreasing in this way the scalability of theproposed algorithm.

The most relevant works to the one proposed here arepresented in [13]–[16]. A centralized intermittent commu-nication control scheme is presented in [13] that ensurescommunication among robots infinitely often but it does notscale well with the number of robots. In [14] a distributedintermittent communication control scheme is proposed thatrequires synchronization among robots, unlike the approachdeveloped here. Moreover, [14] considers a priori determinedcommunication points while in this paper, rendezvous pointsare selected optimally, to minimize the distance traveledby the robots. [15] proposes a distributed synchronizationscheme that allows robots that move along the edges of abipartite mobility graph to meet periodically at the verticesof this graph. Instead, here we make no assumptions on thegraph structure on which robots reside or on the communica-tion pattern to be achieved. On the other hand, [16] proposesa receding horizon framework for periodic connectivity thatensures recovery of end-to-end connectivity within a giventime horizon. As the number of robots or the size of the timehorizon grows, this approach can become computationallyexpensive. To the contrary, our proposed method scales wellto large numbers of robots and can handle situations wherethe whole network can not be connected at once, by ensuringconnectivity over time, infinitely often.

2016 IEEE 55th Conference on Decision and Control (CDC)ARIA Resort & CasinoDecember 12-14, 2016, Las Vegas, USA

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II. PROBLEM FORMULATION

Consider a team of N robots that move in a workspaceW ⊂ Rn according to xi(t) = ui(t), where xi(t) ∈ Rn

is the position of robot i, i ∈ {1, 2, . . . , N}, at time t andui(t) ∈ Rn is a control input. Also, consider L locationsin W denoted by `j , j ∈ {1, 2 . . . , L} located at positionsµj ∈ W and paths γij : [0, 1] → Rn that connect twolocations `i and `j such that γij(0) = µi and γij(1) = µj .The union of locations µj and paths γij gives rise to anundirected graph G = {V, E}, where the set of nodes V ={1, 2, . . . , L} is indexed by the set of locations `j and theset of edges E ⊆ V × V is determined by the paths γijsuch that an edge (i, j) ∈ E exists if and only if a path γijexists. Hereafter, we call G a mobility graph and we assumethat the robots move along its edges to possibly accomplisha high-level task; see Figure 1(a). For example, robots maytravel along their assigned paths to monitor different parts ofa region and then coordinate to transmit their measurementsto a user in a multi-hop fashion. In what follows, we assumethat the mobility graph G is connected.

A. Intermittent Communication

We assume that the robotic team is divided into Msubgroups. The indices i of robots that belong to the m-th subgroup are collected in a set denoted by Tm, m ∈{1, 2, . . . ,M} while every robot can belong to more than onesubgroups. Robots in a subgroup Tm can communicate onlywhen all of them are present simultaneously at a commonlocation `j . The locations `j where communication can takeplace for the robotic team Tm are collected in a set Cm. Thisway, a dynamic robot communication graph Gc = {Vc, Ec}is constructed where the set of nodes Vc is indexed byrobots, i.e., Vc = {1, 2, . . . , N} and Ec ⊆ Vc × Vc isthe set of communication links that emerge when, e.g., allrobots in a team Tm meet at a common rendezvous point`j ∈ Cm simultaneously. Given the robot teams Tm, a graphGT = {VT , ET } is constructed whose set of nodes VT ={1, 2, . . . ,M} is indexed by the teams Tm and set of edgesET consists of links between nodes m and n if Tm∩Tn 6= ∅;see Figure 1(b). We assume that the graph GT is connected inorder to ensure dissemination of information in the network.Furthermore, we can define the set of neighbors of nodem ∈ VT by NTm = {n ∈ VT |(n,m) ∈ ET }. Also, forrobot i we define the set that collects the indices of allother robots that belong to common teams with robot ias Ni = {j|j ∈ Tm,∀m ∈ Si} \ {i}, and the set thatcollects the indices of teams that robot i belongs to asSi = {m|i ∈ Tm, m ∈ {1, 2, . . . ,M}}.

At a rendezvous point `j ∈ Cm communication takes placewhen all robots in Tm are present there, simultaneously. Thenthe communication graph Gc is defined to be connected overtime if all robots in Tm meet infinitely often at a region`j ∈ Cm, for all m ∈ VT . Such a requirement can be capturedby the following global LTL expression:

φ = ∧∀m∈{1,2,...,M}(�♦

(∨`j∈Cm(∧∀i∈Tmπ

`ji )))

, (1)

(a) Mobility Graph G (b) Graph GT

Fig. 1. A graphical illustration of the problem formulation. Figure 1(a)depicts the mobility graph G for a network of N = 5 robots (black dots)divided into M = 5 teams. The robot teams are selected to be: T1 = {1, 2},T2 = {2, 3}, T3 = {3, 4}, T4 = {2, 4, 5}, and T5 = {1, 5}. Red dotsrepresent communication points and the black dashed lines between twosets Cm and Cn imply that there is at least one path γij that connectscommunication points `i ∈ Cm and `j ∈ Cn. Figure 1(b) depicts theassociated graph GT .

where π`ji is an atomic proposition that is true when robot i

is sufficiently close to µj . In (1), ∧ and ∨ are the conjunc-tion and disjunction operator, respectively, while � and ♦stand for the temporal operators ‘always’ and ‘eventually’,respectively. For more details on LTL, we refer the readerto [5], [6]. In words, the LTL expression in (1) requires allrobots in a team Tm to meet infinitely often at at least onecommunication point `j ∈ Cm, for all m ∈ VT .

B. Discretized Abstraction of the Workspace

We can model the environment in which robot i residesby a weighted transition system (wTS) denoted by wTSi thatis defined as follows

Definition 2.1 (weighted Transition System):A weighted Transition System wTSi is a tuple(Qi, q

0i ,Ai,→i, wi,AP, Li

)where: (a) Qi =

{q`ji }∀`j∈Cm,∀m∈Si is the set of states, where a stateq`ji indicates that robot i is at `j ; (b) q0i ∈ Qi is the initial

state of robot i; (c) Ai is a set of actions. The availableactions at state q

`ji are “wait” and “go to state q`ki ” for

every k such that (j, k) ∈ E ;1(d) →i⊆ Qi ×Ai ×Qi is thetransition relation; (e) wi : Qi×Qi → R+ is a cost functionthat assigns weights/cost to each possible transition in wTS.These costs are associated with the distance between twostates q`ji and q`ki ; (f) AP is the set of atomic propositions;and (h) Li : Qi → 2AP is an observation/output relationgiving the set of atomic propositions that are satisfied in astate.

An infinite path τi of wTSi is an infinite sequence ofstates, τi = τi(1)τi(2)τij(3) . . . such that τi(1) = q0i ,τi(n) ∈ Qi, and (τi(n), ai, τi(n + 1)) ∈→i, for someai ∈ Ai, ∀n.2 The cost of an infinite path τi of a wTSi

is Ji(τi) =∑∞

n=1 wi(τi(n), τi(n + 1)). Similarly, the cost

1Throughout the rest of the paper, for the sake of simplicity we assumethat the mobility graph G is constructed so that there is a path from anylocation `j to any `k , where q

`ji ∈ Qi and q

`ki ∈ Qi, that does not

pass though a location `e ∈ Cm for some m /∈ Si, for all robots i. Thisassumption can be relaxed if we incorporate all communication points `jin the set of states Qi.

2A finite path of wTSi can be defined accordingly. The only differencewith the infinite path is that a finite path is defined by an finite sequenceof states of wTSi.

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incurred by the composition of N infinite paths τi denotedby τ = ⊗∀i∈{1,2,...,N}τi is given by

J(τ) =∑∞

n=0

∑N

i=1wi(τi(n), τi(n+ 1)), (2)

The problem we address in this paper can be stated as:Problem 1: Given any initial configuration of the robots

in the mobility graph G determine motion plans τi for allrobots i that satisfy global LTL expression given in (1),i.e., communication graph Gc is connected over time andminimize the total distance traveled by robots captured bythe objective function (2).

III. INTERMITTENT COMMUNICATION CONTROL

To solve Problem 1, known centralized model check-ing techniques can be employed, that typically rely on adiscretized abstraction of the environment captured by awTS and the construction of a synchronized product systemamong all robots of the network. As a result, such approachesare resource demanding and scale poorly with the size ofthe network. Therefore, a distributed solution is preferredwhereby discrete high-level motion plans for every robot canbe computed locally across the network. For this purpose,notice first that although the global LTL formula (1) is notdecomposable with respect to robots, it can be decomposedin local LTL formulas φTm associated with a robot teamTm, which are coupled with each other by the conjunctionoperator ∧. Specifically, we can write φ = ∧m∈VT φTm ,where φTm is defined as

φTm = �♦(∨`j∈Cm(∧∀i∈Tmπ

`ji )), (3)

and forces all robots i ∈ Tm to meet infinitely often at atleast one rendezvous point `j ∈ Cm.

Given the decomposition of φ into local LTL formulasφTm , every robot i ∈ Tm needs to develop motion plans τiso that the composition of plans τi, ∀i ∈ Tm denoted byτTm = ⊗i∈Tmτi satisfies the local LTL expression φTm , forall m ∈ Si. In this way, we can ensure that the compositionof τi, ∀i ∈ {1, 2 . . . , N}, satisfies the global LTL expression(1), since all local LTL expressions φTm are satisfied.

Motion plans τTm |= φTm , ∀m ∈ VT , can be constructedusing existing tools from model checking theory [5], [6].However, notice that constructing plans τTm and τTn , ∀n ∈NTm independently cannot ensure that the robots’ behaviorin the workspace will satisfy the global LTL formula (1).The reason is that the local LTL formulas φTm in (3) are notindependent from the local LTL expressions φTn for whichit holds n ∈ NTm , since they are coupled by robots’ statein their respective transition systems. For instance, assumethat n ∈ NTm and that robot i belongs to teams Tm and Tn.Then robot i is responsible for communicating with the otherrobots that belong to teams Tm and Tn at a communicationpoint in Cm and Cn, respectively. This equivalently impliesthat the LTL expressions φTm and φTn are coupled due torobot i through the atomic propositions π`j

i , ∀`j ∈ Cm andπ`ki , ∀`k ∈ Cn. Consequently, generating plans τTm |= φTm

that ignore the LTL expressions φTn , ∀n ∈ NTm may result

in conflicting robot behaviors, since the projection of motionplans τTm and τTn onto wTSi may result in two differentmotion plans τi for a robot i ∈ Tm ∩ Tn, n ∈ NTm . Thismeans that cases where a robot i needs to be in more thanone states in wTSi simultaneously may occur.

To circumvent these issues, we propose a distributedalgorithm in Section III-B that implements free-of-conflictdiscrete motion plans τi, for all robots i, so that the globalLTL expression φ is satisfied. These motion plans will beconstructed based on the prefix parts of motion plans τTm |=φTm , constructed in Section III-A, for all m ∈ Si.

A. Optimal Automata-based Model Checking

Given an LTL formula φTm and the wTSi of all robotsi ∈ Tm a motion plan τTm |= φTm can be implemented usingexisting automata-based model checking methods [5], [6].First the weighted synchronous Product Transition System(wPTS) wPTSTm = ⊗ik∈TmwTSik , k = 1, 2, . . . , |Tm| isconstructed [5], whose state space is QTm = ×ik∈TmQik

and essentially captures all the possible combinations ofrobots’ states in their respective wTSi, ∀i ∈ Tm. Next theLTL formula φTm is translated into a Nondeterministic BuchiAutomaton (NBA) over 2AP denoted by BTm [17].

Once the wPTS wPTSTm and the NBA BTm are con-structed, a motion plan τTm |= φTm can be found by check-ing the non-emptiness of the language of the Product BuchiAutomaton (PBA) PTm = wPTSTm ⊗BTm [5]. To check thenon-emptiness of the language of PTm and, consequently,to find a motion plan that both satisfies φTm and at thesame time minimizes J(τTm), we can employ existing modelchecking methods that are based on graph search algorithms;see, e.g., [18]. Such motion plans can be written in a prefix-suffix structure τTm = τ pre

Tm [τ sufTm ]ω , where the prefix part τ pre

Tmis executed only once and the suffix part τ suf

Tm is repeatedinfinitely.

B. Conflict Resolution Coordination

As discussed in the beginning of Section III, constructingmotion plans τTm for all m ∈ Si, for all robots i can resultin conflicting robot behaviors. To overcome this issue, wepropose a distributed algorithm that resolves any conflicts inthe robot behavior introduced by the motion plans τTm andconstructs free-of-conflicts motion plans τi for all robots iusing the prefix parts τ pre

Tm constructed in Section III-A. Thegeneral form of these motion plans is

τi =τi(1)τi(2) · · · = [τi(n)]∞n=1

=

[X . . .XΠ|wTSip

kTm1

X . . .XΠ|wTSipkTmj

X . . .XΠ|wTSipkTm|Si|

X . . .X

]∞k=1

=[pki

]∞k=1

, (4)

such that τi(1) = q0i . In (4), pkTmjis a finite path of wPTSTm ,

where mj ∈ Si and j ∈ {1, . . . , |Si|}. Also, X stands for afinite path in which robot i waits at its current state in wTSi.In (4), the concatenation of the paths X and pkTmj

, ∀mj ∈ Sigives rise to the finite path pki . Hereafter, the e-th finite path

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in pki is denoted by pk,ei where 1 ≤ e ≤ `, where ` stands forthe number of finite paths that appear in pki . The parameter` is a priori selected to be ` = max {dTm}

Mm=1 + 1 for all

robots, where dTm denotes the degree of vertex m in thegraph GT . This particular choice for the parameter ` ensuresthe construction of free-of-conflict motion plans, as it willshown in Proposition 3.1. Moreover, we denote by pki (n)the n-th state in the finite path pki , e.g., p1i (1) = p1,1i (1) =τi(1) = q0i , by construction of τi. The same notation extendsto the infinite path τi.

In what follows we first describe the construction of thefinite paths pkTmj

and then we show how these finite pathsare ordered in pki giving rise to a free-of-conflict motionplan τi. First, for the finite path pkTmj

it holds that pkTmj=

τ pre,kTmj

, where τ pre,kTmj

is the prefix part of the motion planτkTmj

|= φTmjconstructed as per Section III-A. The index

k is introduced in τ pre,kTmj

to point out that the prefix partis recomputed as the index k, introduced in (4) changes.The reason it needs to be recomputed is because the initialstate of wPTSTmj

changes as k changes. Particularly, theinitial state of wPTSTmj

is the state q0,kTmj= pkTmj

(1).The state Π|wTSi

pkTmj(1) is selected so that all sub-paths

Π|wTSipkTmj

in pki chain up consistently. Hence, we selectthe state Π|wTSi

pkTmj(1) to be the final state of the previous

sub-path that appeared in pki . Consequently, the state pki (1)coincides with the the final state of the finite path pk−1i . Ifk = 1, then pki (1) refers to the initial position of robot i inthe workspace.

The finite paths pki are constructed sequentially across thenodes `j ∈ V , as follows. Let S = {`1, . . . , `j , . . . } be anordered sequence of the nodes in the mobility graph G, sothat consecutive nodes (communication points) `j , `e in S areassociated with teams Tn and Tm, respectively, that belongto neighboring nodes in the graph GT , i.e., `e ∈ Cm, `j ∈ Cnand m 6= n and m ∈ NTn . We assume that S is known byall robots and that every robot i is initially located at the firstcommunication point `e ∈ Cm, m ∈ Si that appears in S.Assume that paths have been constructed for all nodes in Sthat precede `e ∈ Cm and that currently all robots i ∈ Tm arelocated at node `e and coordinate to construct the paths pki .Since the mobility graph G is connected consecutive nodesin S are connected by a path in G, this means that there isat least one robot j ∈ Tn ∩ Tm, n ∈ NTm , which previouslyconstructed its path pkj by placing at its nTmj -th entry of

the finite path Π|wTSjpkTm , i.e., p

k,nTmj

j = Π|wTSjpkTm . Then

robot i constructs the path pki based on three rules. Accordingto the first rule, the path Π|wTSi

pkTm will be placed at thenTmi -th entry, which is selected to be equal to nTmj , whichis common for all robots j ∈ Tm [line 1, Alg. 1]. Thisensures that robot i and all other robots j ∈ Tm will meetat a communication point that belongs to Cm, as it will beshown in Proposition 3.3. The next step is to place the pathsΠ|wTSi

pkTg for all g ∈ Si\{m}, at the nTgi -th entry of pki . The

index nTgi will be determined by one of the two following

rules. Specifically, according to the second rule if there existrobots j ∈ Ni∩Tg that have already constructed the paths pkj ,then the index nTgi is selected to be equal to nTgj , which iscommon for all j ∈ Tg [line 4, Alg. 1]. Otherwise, accordingto the third rule the path Π|wTSi

pkTg can be placed at anyfree entry of pki indexed by n

Tgi , provided that the nTgi -th

entry of all paths pkj of robots j ∈ Ni that have alreadybeen constructed does not contain states Π|wTSj

pTh(k) withh ∈ NTg [line 6, Alg. 1]. To highlight the role of this ruleassume that h ∈ NTg . Then this means that there exists atleast one robot r ∈ Th∩Tg . Then notice that without the thirdrule [line 5], at a subsequent iteration of this procedure, robotr ∈ Th ∩ Tg would have to place the paths Π|wTSr

pkTg andΠ|wTSrp

kTh at a common entry of pkr , i.e., nTgr = nThr , due to

the two previous rules and, therefore, a conflicting behaviorfor robot r would occur. In all the remaining entries of pki ,Xs are placed [line 8, Alg. 1].3 This procedure is repeateduntil all robots i ∈ Tm have constructed their respective pathspki . Once this happens, all robots i ∈ Tm depart from node`e ∈ Cm and travel to the next communication point `c ∈ Cv ,that appears in S that satisfies v ∈ Si [line 9, Alg. 1]. Atthat point, all robots associated with the next communicationpoint in S are present at that node, and can coordinate tocompute their respective paths, as before. The procedure isrepeated sequentially over the nodes in S until all robotshave computed their paths.

When all robots have constructed their finite paths, theyexchange a set of indices denoted by Xi that collects theindices nXi at which p

k,nXi

i = X . If there exist pathspk,nX

ii = X , for some nXi ∈

⋂∀i Xi, they are discarded,

since in these paths all robots i wait at their current states.Also, notice that in general, the finite paths pk,ei for somee ∈ {1, . . . , `} may have different lengths across the robotsi. Consequently, this implies that two robots i, j that belongto a team Tm may start executing the finite paths Π|wTSi

pkTmand Π|wTSjp

kTm at different time instants, assuming that

the robots pick synchronously their next states in theirtransition systems. Avoiding such a case is crucial to ensureintermittent communication within team Tm, as it will beshown in Proposition 3.3. Therefore, given any index e wecan introduce states at the end of the finite paths pk,ei wherethe robots wait in their current states so that all finite pathspk,ei , for all robots i, have the same length. Communicationbetween the robots in the last two stages of the algorithmcan happen in the order defined by S, as before.

3If `j = `1 ∈ Cq , for some q ∈ VT , then initially, a randomly selectedrobot j ∈ Tq creates arbitrarily its path pkj by placing the paths Π|wTSj p

kTq

at the nTq

j -th entry of pkj , for all q ∈ Sj . Then the procedure previouslydescribed follows. Moreover, depending on the structure of the graph GTit is possible that a communication point `j ∈ Cm appears more than oncein S. In this case, robots i ∈ Tm construct the finite paths pki only the firsttime that `j appears in S.

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Algorithm 1 Construction of motion plans τi = [pki ]∞k=1 atnode `e ∈ CmRequire: Already constructed finite paths pkj of robots j ∈Ni;

Require: All robots in Tm are located at node `e ∈ Cm;1: p

k,nTmi

i := Π|wTSipkTm , nTmi = nTmj , ∀j ∈ Tm;

2: for g ∈ Si \ {m} do3: if there exist constructed paths pkj , j ∈ Ni ∩ Tg then

4: pk,n

Tgi

i := Π|wTSipkTg , nTgi = n

Tgj , ∀j ∈ Tg;

5: else6: p

k,nTgi

i := Π|wTSipkTg provided either pk,n

Tgi

j = X ,

or pk,nTgi

j = Π|wTSjpkTh with h /∈ NTg , ∀j ∈ Ni;

7: end if8: Put Xs in the remaining entries;9: Transmit path pki to a robot in Tm that has not

constructed its motion plan. If there are not suchrobots, all robots i ∈ Tm depart from node `e ∈ Cm;

10: end for

C. Correctness of the Proposed Algorithm

In this section, we show that the composition of the dis-crete motion plans τi generated by Algorithm 1 satisfies theglobal LTL expression (1), i.e., that the network is connectedover time. To prove this result, we need first to show thatAlgorithm 1 can develop non-conflicting motion plans τi,for which we have the following two results. Proposition3.1 can be proved by following the steps of the proof ofProposition 3.2 in [14] for the graph GT . Proposition 3.2holds by construction of the finite paths pki and its proof,which is omitted due to space limitations, is along the linesof the proof of Proposition 3.3 in [14].

Proposition 3.1: Algorithm 1 can always construct finitepaths pki that consist of ` finite paths pk,ei where ` ≤max {dTm}

Mm=1 + 1.

Proposition 3.1 shows also that the finite paths pki andconsequently, the motion plans τi depend on the node degreeof graph GT , and not on the size of the network.

Proposition 3.2: Algorithm 1 generates admissible dis-crete motion plans τi, i.e., motion plans that are free ofconflicts and satisfy the transition rule →i of wTSi.

The proofs of the next two propositions are omitted dueto space limitations.4

Proposition 3.3: The composition of motion plans τi gen-erated by Algorithm 1 satisfies the global LTL expression (1),i.e., connectivity of the robot network is ensured over time,infinitely often.

Proposition 3.4: Algorithm 1 generates discrete motionplans τi for all robots i in a prefix-suffix structure, i.e.,τi = τ pre

i

[τ sufi

]ω= [p1i . . . p

kp−1i ][p

kp

i . . . pksi ]ω .

4The proofs can be found at http://arxiv.org/abs/1609.07038.

IV. ASYNCHRONOUS INTERMITTENT COMMUNICATION

In section III, we showed that if all robots i pick syn-chronously their next states in wTSi according to the motionplans τi, then the LTL expression (1) is satisfied. In thissection, we show that the generated motion plans can beexecuted asynchronously, as well, by appropriately introduc-ing delays in the continuous-time execution of τi. We omita formal proof of this result due to space limitations, andinstead validate the proposed asynchronous scheme throughnumerical simulations in Section V.

Due to the asynchronous execution of the controllers, themotion plans τi can be written as in (4) replacing the indicesn and k with ni and ki, respectively, which allows us tomodel the situation where the robots pick asynchronouslytheir next states in wTSi. In the asynchronous execution ofthe infinite paths τi robot i moves from state τi(ni − 1) ∈Qi to τi(ni) ∈ Qi according to a continuous-time motioncontroller ui(t) ∈ Rn that belongs to the tangent space ofγij at xi(t). Without loss of generality, assume that i ∈ Tmand τi(ni) = q`ei , for some `e ∈ Cm. When robot i arrives atstate τi(ni) it checks if τj(ni) = Π|wTSj

pTm(fi) = q`ej forall j ∈ Tm. If this is the case, then robot i waits at node `euntil all other robots j ∈ Tm are present there. When thishappens, or if there is at least one robot j ∈ Tm such thatτj(ni) 6= Π|wTSjpTm(fi) = q`ej , then robot i moves towardsthe next state τi(ni + 1).

V. SIMULATION STUDIES

In this section, a simulation study is provided that illus-trates our approach for a network of N = 5 robots thatmove along the edges of the mobility graph with L = 20communication points as shown in Figure 2. The network isdivided in M = 5 teams as in the example shown in Figure1. Also, the mobility graph is constructed so that there is apath γij from any node `i to any other node `j . Therefore,the finite paths pkTm constructed as per Section III-A havethe form pkTm = q0,kTmqTm , where q0,kTm is the initial state ofwPTSTm constructed as defined in Section III-B and qTm =

(q`ji1. . . q

`ji|Tm|

) is a state where all robots of team Tm are lo-cated at a common meeting point `j ∈ Cm. The motion plansτi generated by Algorithm 1 have the following structure:τ1 =

[pk11

]∞k1=1

=[Π|wTS1

τk1

T1 Π|wTS1τk1

T5X]∞k1=1

, τ2 =[pk22

]∞k2=1

=[Π|wTS2

τk2

T1 Π|wTS2τk2

T2 Π|wTS2τk2

T4

]∞k2=1

, τ3 =[pk33

]∞k3=1

=[Π|wTS3

τk3

T3 Π|wTS3τk3

T2X]∞k3=1

, τ4 =[pk44

]∞k4=1

=[Π|wTS4

τk4

T3XΠ|wTS4τk4

T4

]∞k4=1

, τ5 =[pk55

]∞k5=1

=[XΠ|wTS5

τk5

T5 Π|wTS5τk5

T4

]∞k5=1

, which can bewritten in a prefix-suffix structure with ks = 2 and kp = 1,where the indices ks and kp are defined in Proposition 3.4.The motion plans τi defined above are depicted in Figure2. Notice that the distances between any two meeting pointsvary across G and so do the robots’ velocities and, therefore,robots pick asynchronously their next states in wTSi. Conse-quently, this results in waiting times for every robot i ∈ Tm at

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0 2 4 6 8 10 12 14 16

0

2

4

6

8

10

12

14

16

Fig. 2. Intermittent communication of N = 5 robots moving along theedges of an underlying mobility graph. Red dots represent communicationpoints in each set Cm while edges between any two communication pointsexist (not shown). Straight lines depict the robots’ trajectories as determinedby motion plans τi. The suffix part of τi is represented by dashed lines forevery robot i while solid lines depict a part of prefix structure, i.e., a paththat connects robots’ initial states to the respective suffix structure. Theprefix part of robots 4 and 5 coincides with their respective suffix part and,therefore, there are no corresponding solid lines for them.

1 2 3 4 5Teams

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

WaitingTim

e

1

2 22

3

3

4

5

4

1

5

(a) Waiting Time

0 50 100 150 200 250

Iterations

0

1

2

3

4

5

6

7

8

9

10

(b) Consensus

Fig. 3. Figure 3(a) depicts the waiting time of every robot i at a meetingpoint `j ∈ Cm, for all m ∈ Si during a single execution of τ suf

i . Figure3(b) illustrates the consensus of numbers vi(t).

the meeting points `j ∈ Cm, which are non-integer multiplesof each other, for all m ∈ VT , as illustrated in Figure 3(a).Toillustrate that under the proposed motion plans connectivityis ensured over time, we implement a consensus algorithmover the dynamic network Gc. Specifically, we assume thatinitially robots generate a random number vi(t0) and whenall robots i ∈ Tm meet at `j ∈ Cm they perform the followingconsensus update: vi(t) = 1

|Tm|∑

e∈Tm ve(t). Figure 3(b)shows that eventually all robots reach a consensus on thenumbers vi(t). Note also that applying existing LTL-basedplanning would result in PBA constructed for all robots withΠN

i=1|Qi||QB | = 45360|QB | states, where QB correspondsto the state-space of the NBA associated with the global LTLexpression in (1), which is hard to manipulate in practice.This issue becomes more severe as the size of the networkincreases.

VI. CONCLUSION

In this paper we considered the problem of controllingintermittent communication in mobile robot networks. Weassumed that robots move along the edges of a mobilitygraph and they can communicate only when they meet atits nodes, which gave rise to a dynamic communicationnetwork. The network was defined to be connected overtime if communication takes place at the rendezvous pointsinfinitely often which was captured by a global LTL formula.

Then this LTL expression was approximately decomposedinto local LTL expressions which were assigned to robots.To avoid conflicting robot behaviors, we developed a dis-tributed conflict resolution scheme that generated free-of-conflicts discrete motion plans for every robot that ensuredconnectivity over time, infinitely often and minimized thedistance traveled by the robots. We also showed that thegenerated motion plans can be executed asynchronously byintroducing delays in their continuous-time execution.

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