930 IEEE TRANSACTIONS ON ROBOTICS, VOL. 29, NO. 4, AUGUST...

930 IEEE TRANSACTIONS ON ROBOTICS, VOL. 29, NO. 4, AUGUST 2013

Constrained Interaction and Coordination inProximity-Limited Multiagent Systems

Ryan K. Williams, Student Member, IEEE, and Gaurav S. Sukhatme, Fellow, IEEE

Abstract—In this paper, we consider the problem of control-ling the interactions of a group of mobile agents, subject to a setof topological constraints. Assuming proximity-limited interagentcommunication, we leverage mobility, unlike prior work, to enableadjacent agents to interact discriminatively, i.e., to actively retainor reject communication links on the basis of constraint satisfac-tion. Specifically, we propose a distributed scheme that consists ofhybrid controllers with discrete switching for link discrimination,coupled with attractive and repulsive potentials fields for mobilitycontrol, where constraint violation predicates form the basis fordiscernment. We analyze the application of constrained interac-tion to two canonical coordination objectives, i.e., aggregation anddispersion, with maximum and minimum node degree constraints,respectively. For each task, we propose predicates and control po-tentials, and examine the dynamical properties of the resultinghybrid systems. Simulation results demonstrate the correctness ofour proposed methods and the ability of our framework to generatetopology-aware coordinated behavior.

Index Terms—Connectivity control, distributed robot systems,dynamic networks, topology control.

I. INTRODUCTION

THE ubiquity of advanced communication technologies,coupled with the continued scaling of processing capa-

bilities, has generated intense interest in the analysis and con-trol of networked systems, and, particularly, in systems of mo-bile agents. Specifically, networks of cooperating intelligentagents have become a natural evolution of centralization, yield-ing gains in efficiency, scalability, and robustness when com-pared with classical solutions. Applications of mobile agentnetworks are multidisciplinary and highly varied; examples in-clude formation control [1]–[3], adaptive sampling [4]–[6], andtarget tracking [7]–[9].

Central to the success of distributed systems is the modelingof coordinated objectives and the derivation of control schemesthat rely only on limited local information. In this paper, weconsider a realistic model of interaction consisting of agents

Manuscript received June 29, 2012; revised January 7, 2013; accepted April4, 2013. Date of publication May 2, 2013; date of current version August 2,2013. This paper was recommended for publication by Associate Editor V. Islerand Editor G. Oriolo upon evaluation of the reviewers’ comments. This workwas supported in part by the Office of Naval Research Multidisciplinary Univer-sity Research Initiative Program under Award N00014-08-1-0693, the NationalScience Foundation CPS Program (CNS-1035866), and a fellowship to R. K.Williams from the USC Viterbi School of Engineering.

The authors are with the Department of Electrical Engineering and the De-partment of Computer Science, University of Southern California, Los Angeles,CA 90089 USA (e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TRO.2013.2257578

with proximity-limited communication and local sensing capa-bilities. When coupled with agent mobility, such interactionsinduce topological variations that can be modeled by the dy-namic network (or dynamic graph) framework. In particular,we are concerned with controlling the topology of such a dy-namic network of agents, while respecting a set of constraintsthat are relevant to system objectives or some performance met-rics. This goal is motivated by recent results indicating the pro-found effect that the network topology has on the performanceand robustness of networked algorithms, for example, in theanalysis of formation stability [10], consensus seeking [11],and swarming behaviors [12]. Consensus algorithms are par-ticularly interesting as they underly many distributed processesincluding filtering [13]–[15], optimization [16]–[18], and esti-mation [19]–[21]. In [11], [22], and [23], Olfati-Saber showedthat network connectivity, which is expressed as the spectrum ofthe graph Laplacian matrix, fundamentally impacts the conver-gence rate, time-delay stability, and the robustness of consensus,yielding a distinct tradeoff in the network topology design. Itis then natural to consider the controlling topology in order toyield performance gains in networked processes like consensusor to otherwise shape network information flow [10].

The connectivity (topology) control problem has been ad-dressed primarily in two manners, by maintaining the connec-tivity of initially connected networks, or through the regulationof global and local connectivity measures. In [24], a robust localmeasure of connectivity is proposed along with connectivity-preserving controls to solve broadcast optimization and sensorcoverage problems. Alternately, in [25], a hybrid automata ap-proach is taken, incorporating market-based auctions and a lo-cal connectivity measure to define distributed controls for con-nectivity preserving link deletion. In [26], centralized potentialfields over the determinant of a reduced Laplacian are employedto guarantee connectivity. Distributed controls and algebraicconnectivity estimation are considered in [27] and [28], whilein [29], a distributed power iteration is formulated together witha distributed gradient controller for connectivity maximization.

As opposed to previous work that is centralized in nature[26], [30], solutions that limit connectivity control to link reten-tion [31]–[33] or unconstrained maximization [27], [29], andapproaches that regulate network links and not agent configu-ration [25], we propose a distributed1 control framework thatmodulates topology through agent mobility and manages dis-criminatively the addition and deletion of network links in or-der to shape agent interaction under constraints.2 Further, direct

1Allowing for efficient scaling in network size and communication cost.2Previous work on constrained connectivity is relatively sparse (see, e.g., [34]

and [35]).

1552-3098/$31.00 © 2013 IEEE

WILLIAMS AND SUKHATME: CONSTRAINED INTERACTION AND COORDINATION IN PROXIMITY-LIMITED MULTIAGENT SYSTEMS 931

control over network links coupled with constraints that reflectnetwork connectedness allows decisions on agent interactionto control network connectivity.3 Specifically, we construct lo-cal controllers with discrete switching for link discrimination,whereby adjacent agents are classified as either candidates forconstraint-aware link removal or addition, or as constraint vio-lators requiring either the retention or denial of communicationlinks. Attractive and repulsive potentials fields establish the localcontrols that are necessary for link discrimination, and constraintviolation predicates form the basis for discernment. We exam-ine constrained interaction by considering two common coor-dination objectives, i.e., aggregation and dispersion, under twointeraction constraints: maximum and minimum node degree.For each task, we propose predicates and continuity-preservingpotential fields, analyze the dynamical properties of the result-ing hybrid system, and present simulation results demonstratingthe correctness of our control formulation.

The primary motivations for our fully mobility-based formu-lation are the implications of proximity-limited systems, par-ticularly with regard to a recent surge in swarm robotics (see,e.g., [38]), in terms of modes of interaction. Such realistic sys-tems rely intimately on spatial interaction and communicationthat is effectively broadcast-based, rendering the direct switch-ing of links between adjacent agents difficult (the approach takenby [25]). Further motivation can be derived from the implica-tions of spatial control over constraints on the swarm behaviorsof the collective. As the constraints are rendered spatially forboth link addition and deletion, there exists a direct correspon-dence between discrete topological constraints and the agentconfiguration in space. Thus, it appears promising the possi-ble applications of our spatial controls on the behavior of thecollective beyond the analyzed aggregation and dispersion be-haviors. In summary, the primary contributions of this paper areas follows: an analysis of the structure of memory in constrainedinteraction (i.e., discernment sets and associated logical switch-ing; see Section III-A), an analysis of aggregative and disper-sive swarming behaviors under discrete topological constraints(including swarm bounds; see Sections IV and V), and the iden-tification of preemptiveness and worst-case topologies as vitalin predicate construction for constrained spatial interaction (seeSection III).

The outline of this paper is as follows. In Section II, we pro-vide preliminary materials including agent and network models,a spatial model of interaction, and problem statements. A gen-eral framework for constrained agent interaction, including aswitching model for discriminative connectivity and constraint-aware mobility control, is presented in Section III. Sections IVand V present an analysis of constrained aggregation and dis-persion behaviors, respectively. Simulation results are providedin Section VI, and concluding remarks, as well as directions forfuture work, are stated in Section VII.

3In this paper, connectivity control is accomplished in the study of constrainedaggregation (see Section IV). In our upcoming work [36], [37], connectivitycontrol in the proposed constrained interaction framework is explored, includingan extension to nonlocal constraints.

II. MODELING AND PROBLEM FORMULATION

Consider a system of n mobile agents operating in Rm

each capable of computation, interagent communication, andlocalized sensing. Denote by xi(t) ∈ Rm the position of the ithagent at time t ∈ R+ , and consider the single integrator agentdynamics

xi(t) = ui(t) (1)

where ui(t) ∈ Rm is the control input for the ith agent. Wecan alternatively view the system in terms of the compositenetwork dynamics given by

x(t) = u(t) (2)

where x(t) = [x1(t), . . . , xn (t)]T ∈ Rmn and u(t) =[u1(t), . . . , un (t)]T ∈ Rmn are formed by stacking vectorcomponents.4 It is assumed that the agents have some primarycoordination objective (in this paper, we consider aggregationand dispersion) together with interaction constraints that formthe basis of the local control inputs.

The coordination of such a system lies in agent interactionand in the exchange of information, by means of passive sens-ing and explicit connectedness. The literature examines a broadrange of dynamically interacting systems, from centralized sce-narios to all-to-all communication assumptions. Here, we adopta realistic model of interaction that assumes that the agentsoperate without contact to a centralized controller and thatsensing and communication occur only within limited prox-imities. Specifically, denoting the distance between agents i and

j by ‖xij‖�= ‖xi − xj‖, the threshold ρ2 acts as the interaction

radius, beyond which agent interaction becomes impossible.When ‖xij‖ ≤ ρ2 , we assume that the agents can sense displace-ment ‖xij‖ and communicate over an established communica-tion link, denoted (i, j). The creation of communication linksoccurs within the connection radius ‖xij‖ ≤ ρ1 < ρ2 , gener-ating a region of width ρ2 − ρ1 within which adjacent agentsdiscern interactions (see Section III for more details). Finally,we require no collisions between agents5; thus, we define theradius ‖xij‖ ≤ ρ0 < ρ1 , within which action is taken to avoidinteragent collisions. Such a spatial partitioning reflects the re-ality of practical interacting systems: limited communication,finite range sensing, and discerned interaction. Fig. 1 depictsthe proposed interaction model, ignoring for now the detailsrelated to discernment.

Agent mobility coupled with the described model of interac-tion induces a time-varying (or switching) element to the net-work topology, i.e., as the system evolves, agents may dynam-ically enter or exit the interaction radii of others and establishcommunication. As is common, we associate a dynamic graphobject with the underlying topology of the network6:

G(t) = (V, E(t)) (3)

4To simplify notation, in the sequel, we drop explicit notation for time andstate dependence when it is contextually apparent.

5For point agents, this implies ‖xij ‖ > 0, ∀ i �= j .6This is also referred to as a proximity graph or disk graph [39], [40].


Fig. 1. Agent interaction model. Agents can sense one another and communi-cate within the interaction radius ρ2 . In the discernment region, ρ1 < ‖xij ‖ ≤ρ2 , an agent i determines, relative to local constraints, whether adjacent agentsare candidates for link addition (k) or removal or should be attracted (j) orrepelled (l) due to constraint violation. In the connection region with radiusρ1 < ρ2 , agents establish communication, with collision avoidance being acti-vated within a radius ρ0 < ρ1 .

with vertices (nodes) V = {1, . . . , n} indexed by the set ofagents and time-varying edge set E = {(i, j) | i, j ∈ V} definedby

(i, j) ∈ E ⇔ (‖xij‖ ≤ ρ2) ∧ σij (t) (4)

with switching signals [25], [33], [41]

σij (t+) ={

0, (i, j) /∈ E ∧ ‖xij‖ > ρ11, otherwise

(5)

where the notations t− and t+ refer to the state transitions of theswitching signal, and the symbol∧ represents the Boolean ANDoperation.7 Switching signals (5) define our assumed mecha-nism of delayed link addition at the connection boundary andinduce a lag in topology changes. The further implication isthat agents detect potential collaborators within some region ofsufficient communication, yet remain in a disconnected state.As discussed in [33], such a construction generates a hysteresiseffect in link addition and loss, with a switching threshold8 mod-ulated by ρ2 − ρ1 . The hysteresis in topology switching is vitalto properly formalizing controls for link discrimination, as willbe shown in Section III. Specifically, the application of infinitepotential fields in controlling mobility requires a lag in topologychange to guarantee that the controls attain infinity only in un-desired states (link loss or link addition, relative to constraints).Further, such a dwell time allows a proximity-limited agent toengage in detect-react behavior, a foundation of the proposedconstraint-aware interaction framework.

The graph edges (4) take on the switched behavior of themobile network topology, reflecting our assumed interactionmodel. We assume bidirectional communication and that there

7In the sequel, we will also use the symbols∨ and¬, representing the BooleanOR and NOT operations, respectively.

8Specifically, a dwell time between topology switches is introduced which isfundamental to certain systematic stability proofs [25].

exists no self-loops,9 implying that set E will contain only undi-rected edges and that the induced graph G will itself be undi-rected. Agents that are connected by a link are called neighbors,and the set of neighbors of an agent i is given by Ni = {j ∈V | (i, j) ∈ E}, which for an undirected graph induces the sym-metry j ∈ Ni ⇔ i ∈ Nj . We can further define the notion of aninteraction set for agent i, Ii = {j ∈ V | ‖xij‖ ≤ ρ2}, i.e., theset of agents within the interaction radius of i. The interactionset is also symmetric with j ∈ Ii ⇔ i ∈ Ij . Finally, the con-nectedness property of the underlying graph G is of particularimportance as it reflects a notion of global information flow overcommunicating agents.

Definition 2.1 (Graph Connectivity): The graph G(t) is con-nected at time t if for every pair of vertices i, j ∈ V , there existsa path from i to j, i.e., a sequence of distinct, adjacent vertices.We denote by Cn the set of connected graphs of n nodes.

We now come to our primary goal in this paper, i.e., the con-strained interaction of proximity-limited agents. Agent mobilitycoupled with communication, sensing, and processing capabil-ities motivates our desire to shape agent interaction, and thusinformation flow, through topological constraints. Although in-tuitive, it has been made clear in the literature that the topologyprofoundly impacts the diffusion of information in networks, thestability of coordinated control processes, etc. [10]–[12]. Quitesimply, the issue of interaction lies at the core of most dynamicnetwork processes. In this paper, we examine constrained inter-action by considering two common coordination objectives, i.e.,aggregation and dispersion, under two interaction constraints,i.e., maximum and minimum node degree, respectively. The spe-cific problems that are considered in this paper are summarizedas follows.

Problem 1 (Max Degree Aggregation): Assuming an initialtopology G(0) ∈ Cn , design agent controls ui such that inter-agent displacements are decreasing, i.e., aggregation occurs.Further, we require G(t) ∈ Cn for all t ≥ 0, while satisfyingconstraints |Ni | ≤ βi ∀ i ∈ V .

Problem 2 (Min Degree Dispersion): Assuming an initialtopology G(0) ∈ Cn , design agent controls ui such that intera-gent displacements are increasing, i.e., dispersion occurs. Fur-ther, G(t) for all t ≥ 0 must satisfy constraints |Ni | ≥ ωi ∀ i ∈V , while establishing no new communication links.

In the sequel, we will propose a general formulation to con-strain the interactions of proximity-limited agents and the ex-amination of methods and related proofs to solve the proposedconstrained coordination problems.

III. CONSTRAINING AGENT INTERACTION

In a system comprising information sharing agents, interac-tion is fundamentally a matter of communication. The implica-tion is that in order to constrain interaction, one must control theunderlying communication topology, i.e., discriminately createand delete links, and act to deny or retain new or establishedlinks, respectively. Our proposition is the notion of discrimi-native connectivity among agents in the system, leveraging a

9That is, (i, i) /∈ E ∀ i ∈ V .


spatial model of interaction (see Section II) and agent mobil-ity to generate an environment in which agents actively regu-late their neighbors based on interagent constraints. Generally,the constraints may not be explicitly spatial in nature (e.g., aconstraint on interagent mutual information); however, we con-sider the satisfaction of constraints to be a spatial task; that is,unlike works that simply activate or deactivate links (see, e.g.,[25]), we force link discrimination through spatial organization.

To achieve discriminative connectivity, we begin by describ-ing further the spatial model of Section II. Specifically, considerthe region of width ρ2 − ρ1 between the interaction and connec-tion radii, which we deem the discernment region. Within thiszone, each agent discerns, relative to local constraints, betweencandidates for link addition (nonneighbors) or deletion (neigh-bors), and agents which must be either attracted (link retention)or repelled (link denial). The discernment region is an intuitiveconstruction, as in a realistic interacting system, communicationdoes not occur at the instant of discovery, it is typical to deter-mine or negotiate a connection. Our model is simply an analogof this notion applied to a proximity-limited mobile system,i.e., interactions should always be tempered. Fig. 1 illustratesour proposed model and the varying scenarios that may occurduring discernment.10

A. Switching Model for Discriminative Connectivity

We are now prepared to present a formal construction for thesatisfaction of interaction constraints through agent mobility.First consider the following notion of a predicate [25].

Definition 3.1 (Predicate): Let X be a finite set of variables.A predicate over X , which is denoted P (X ) : X → {0, 1}, isa Boolean-valued function defined as the finite conjunction ofstrict or nonstrict inequalities over X .

That is, P (X ) is a logical statement over the elements of

X . For example, a relevant predicate for this work is P (X )�=

|Ni | ≤ βi over the set X ∈ Z+ , which returns 1 if agent i hasa degree bounded above by βi and 0 otherwise. The predicateforms the basis of discernment for each agent, where for brevity,we omit the predicate domain notation in subsequent usage.11

Assume that there exists some global constraint on agentinteraction that is defined as the finite conjunction of local con-straint predicates assigned to each agent. Specifically, we havea global constraint predicate

P�=

n∧i=1

pi∧k=1

Pki (6)

that indicates the constraint satisfaction status of the system,where each agent possesses pi local constraint predicates, whichare denoted by Pk

i and indexed by k. Thus, the global constraintis assumed to have a separability quality across the system orthere exists an agreement mechanism in the local constraint

10The scenarios depicted in Fig. 1 would occur, for example, if the agentswere under maximum and minimum degree constraints (as will be examined inthis paper).

11Further, we use varying typefaces and subscripts/superscripts to distinguishglobal and local predicates and predicate purpose. We also use the notation P (t)to indicate a predicate evaluated at some time t.

formulation, e.g., consensus or an auction. The local predicatesPk

i indicate the satisfaction of the kth local constraint, andthe conjunction

∧pi

k=1 Pki represents the constraint satisfaction

status of the ith agent. It is assumed that the predicates are wellposed and that they are topological in nature, i.e., they can bemanipulated directly or indirectly through network link additionor removal.

In order to appropriately regulate topology, each agent is as-signed predicates for link addition and deletion, i.e., Pa

ij and Pdij ,

which indicate possible constraint violations by the addition ordeletion of link (i, j). Necessitated by the hysteresis in topol-ogy switching, these discernment predicates enable the agents toproperly determine constraint-aware topology changes and actthrough mobility. We then associate switching signals {ϕi}n

i=1 ,{γi}n

i=1 , {ξi}ni=1 , and {ζi}n

i=1 with addition candidacy, dele-tion candidacy, attraction, and repulsion, where ϕi ∈ {0, 1}n ,γi ∈ {0, 1}n , ξi ∈ {0, 1}n , and ζi ∈ {0, 1}n are associated withagent i. We denote by ϕj

i , γji , ξj

i , and ζji the jth element of the re-

spective signals, where we assume that ϕii = 0, γi

i = 0, ξii = 0,

and ζii = 0. The signals indicate for the ith agent the member-

ship of nearby agents j in link addition and deletion candidatesets

Cai = {j ∈ V |ϕj

i = 1}, Cdi = {j ∈ V | γj

i = 1} (7)

and attraction and repulsion sets

Dai = {j ∈ V | ξj

i = 1}, Dri = {j ∈ V | ζj

i = 1} (8)

dependent on constraint violation. Discriminative connectivityis then achieved by defining constraint-dependent dynamics forthe sets (7) and (8) through their respective signals in order toregulate the system topology by attraction and repulsion. Theproposed candidate signal updates (per element) are then

ϕji (t

+) =

I︷︸︸︷((‖xij‖ ≤ ρ2) ∧ (j /∈ Ni))

∧(ϕj

i (t−) ∨

(¬ζj

i (t−) ∧ ¬Paij ∧ ¬Pa

ji

))︸︷︷︸

II

(9)

for link addition, and

γji (t+) =

III︷︸︸︷((‖xij‖ > ρ1) ∧ (j ∈ Ni))

∧(γj

i (t−) ∨(¬ξj

i (t−) ∧ ¬Pd

ij ∧ ¬Pdji

))︸︷︷︸

IV

(10)

for link deletion. The attraction and repulsion signal updatestake forms complimentary to (9) and (10) given by

ξji (t

+) = ((‖xij‖ > ρ1) ∧ (j ∈ Ni))

∧(ξji (t

−) ∨(¬γj

i (t−) ∧(Pd

ij ∨ Pdji

)))(11)

and

ζji (t+) = ((‖xij‖ ≤ ρ2) ∧ (j /∈ Ni))

∧(ζji (t−) ∨

(¬ϕj

i (t−) ∧

(Pa

ij ∨ Paji

)))(12)


Fig. 2. Model for discriminative connectivity, including signals and associated switching conditions for controlling (a) link addition and (b) link deletion.

respectively. Notice that the updates (9)–(12) take a straightfor-ward form. Considering for illustration the candidate signals (9)and (10), Terms I and III represent discernment management,i.e., nearby agents are determined for candidacy only if they liewithin the discernment region and have the appropriate neighborstatus. Terms II and IV then operate to assign agents within thediscernment region to the appropriate candidate set on the basisof constraint violation. The dynamics of (9)–(12) are illustratedin Fig. 2. The salient properties of the candidate, attraction, andrepulsion sets are summarized in the following.

Proposition 3.1 (Discernment set properties): The sets (7) and(8) under dynamics (9)–(12) possess the following properties:

1) Symmetry: The inclusion of Paji and Pd

ji in the set dy-namics of discerning agent i induces a symmetry in thediscernment sets of the interacting agents:

j ∈ Cai ⇔ i ∈ Ca

j , j ∈ Cdi ⇔ i ∈ Cd

j

j ∈ Dai ⇔ i ∈ Da

j , j ∈ Dri ⇔ i ∈ Dr

j (13)

That is, there exists a notion of agreement or cooperationin interactions.

2) Exclusivity: For a discerning agent i, a nearby agent j withρ1 < ‖xij‖ ≤ ρ2 belongs to one and only one discernmentset of i:

j ∈ Cai ⇒ j /∈

(Cd

i ∪ Dai ∪ Dr

i

)j ∈ Cd

i ⇒ j /∈ (Cai ∪ Da

i ∪ Dri )

j ∈ Dai ⇒ j /∈

(Ca

i ∪ Cdi ∪ Dr

i

)j ∈ Dr

i ⇒ j /∈(Ca

i ∪ Cdi ∪ Da

i

). (14)

3) Persistence: For a discerning agent i, a nearby agent jwith ρ1 < ‖xij‖ ≤ ρ2 that belongs to a discernment setof i remains a member of the set for all time until either‖xij‖ ≤ ρ1 or ‖xij‖ > ρ2 , i.e., j leaves the discernmentregion. Together with the exclusivity property, it is impliedthat during this period of persistence agent, j does notbecome a member of any other discernment set of i, i.e.,switching between discernment sets does not occur.

We point out several important characteristics of the pro-posed constraint satisfaction formulation. In the construction ofdiscernment set dynamics (9)–(12), we have forced the discrim-ination of links to occur on the boundaries of the discernmentregion, before the actual addition or deletion of a link. Specif-ically, switching in the elements of sets (7), (8) occurs onlywhen displacement ‖xij‖ transits the ρ1 or ρ2 radii. As willbe shown in Section III-B, the constraint violation controls ap-proach infinity at the boundaries of the discernment region, andthus, our preemptive policy ensures a hysteresis effect similarto the switching signal (5) for link addition. Further, withoutpreemptively determining violators and applying appropriatemobility controls, a situation could arise where multiple agentsreach the connection or interaction boundary simultaneously,generate a constraint violation, and experience infinite controlforce. Finally, in the vein of preemption, we require the fol-lowing general assumption on the construction of discernmentpredicates.

Assumption 1 (Preemption): As the invocation of Paij and Pd

ij

occurs on the boundaries of the discernment region, while possi-ble link addition occurs at some future time relative to ρ2 − ρ1 ,we require the following assumption on predicate formulationto hold for all agents i:

{(∃ t2 | j ∈ Dr

i (t2)) ⇔(∃ t1 |Pa

ij (t1) ∨ Paji(t1)

)}∧ {(j ∈ Dr

i (t)) ⇒ ∃ εr > 0 | ‖xij (t)‖ ≥ ρ1 + εr}∧

{(∃ t2 | j ∈ Da

i (t2)) ⇔(∃ t1 |Pd

ij (t1) ∨ Pdji(t1)

)}∧ {(j ∈ Da

i (t)) ⇒ ∃ εa > 0 | ‖xij (t)‖ ≤ ρ2 − εa}

⇒pi∧

k=1

Pki (15)

where t1 ≤ t2 , and separate instances of t1 and t2 are treatedindependently. In words, the discernment predicates must beconstructed to be preemptive in nature, i.e., in tandem withappropriate link discrimination, they predict the potential for


constraint violation upon link addition or deletion before it oc-curs, ensuring topological constraint satisfaction.12

B. Constraint-Aware Mobility Control

The discernment of interacting agents into either candidatesor constraint violators allows us now to define attractive andrepulsive controls between agents comprising a violating inter-action (one or both agents violate a predicate). First, consideran attractive potential field ψa

ij : R+ → R+ for the purpose ofretaining established neighbors that present a constraint viola-tion risk if lost. We consider here the case of an unboundedpotential and require ψa

ij to be constructed with the followingproperties [32].

1) ψaij is a function of the distance between adjacent agents

i and j, i.e., ψaij

�= ψa

ij (‖xij‖), ensuring that interactionsrequire the knowledge of only local information.

2) ψaij → ∞ as ‖xij‖ → ρ2 , guaranteeing neighboring

agents that present a constraint violation risk are retained.3) The potential is continuously differentiable over ‖xij‖ ∈

(ρ1 , ρ2 ].4) For ‖xij‖ ≤ ρ1 , we have ψa

ij = 0 and ∂ψaij /∂xi = 0.

5) For ‖xij‖ > ρ1 , we have ∂ψaij /∂‖xij‖2 > 0, and for

‖xij‖ ≤ ρ1 , we have ∂ψaij /∂‖xij‖2 = 0. That is, the po-

tential is attractive within the discernment region and in-effectual otherwise.

The dependence of ψaij on ‖xij‖ induces the symmetry prop-

erties ψaij = ψa

ji and ∂ψaij /∂‖xij‖2 = ∂ψa

ji/∂‖xji‖2 . An ap-propriate attractive potential that we adopt for this paper takesthe following form:

ψaij =

⎧⎨⎩

0, ‖xij‖ ∈ (0, ρ1)1

ρ22 − ‖xij‖2 + Ψa , ‖xij‖ ∈ [ρ1 , ρ2)

(16)

where Ψa(‖xij‖) = a‖xij‖2 + b‖xij‖ + c is chosen suchthat (16) is smooth over the ρ1 transition, i.e., Ψa(ρ1) =∂Ψa/∂‖xij‖(ρ1) = 0.

Now, consider a repulsive potential field ψrij : R+ → R+ for

the purpose of repelling nonneighbors that present a constraintviolation risk upon link creation. We consider again the case ofan unbounded potential and require ψr

ij possess the followingproperties.

1) ψrij is a function of the distance between adjacent agents

i and j, i.e., ψrij

�= ψr

ij (‖xij‖).2) ψr

ij → ∞ as ‖xij‖ → ρ1 , guaranteeing that communica-tion is not established with nonneighboring agents thatpresent a constraint violation risk.

3) The potential is continuously differentiable over ‖xij‖ ∈[ρ1 , ρ2).

4) For ‖xij‖ > ρ2 , we have ψrij = 0 and ∂ψr

ij /∂xi = 0.5) For ‖xij‖ ≤ ρ2 , we have ∂ψr

ij /∂‖xij‖2 < 0, and for‖xij‖ > ρ2 , we have ∂ψr

ij /∂‖xij‖2 = 0. That is, the po-

12An implication of the predicate formulation is that it will require the inclu-sion of candidate set state, as will be shown. We make no claims concerning asystematic way of constructing predicates; this is reserved for future work.

tential is repulsive within the discernment region and in-effectual otherwise.

Again, the dependence of ψrij on ‖xij‖ induces the symme-

try properties ψrij = ψr

ji and ∂ψrij /∂‖xij‖2 = ∂ψr

ji/∂‖xji‖2 .Similar to the attractive potential (16), the repulsive potentialtakes the form

ψrij =

⎧⎨⎩

1‖xij‖2 − ρ2

1+ Ψr , ‖xij‖ ∈ (ρ1 , ρ2)

0, ‖xij‖ ∈ [ρ2 ,∞)(17)

where Ψr is again a second-order polynomial in ‖xij‖ chosento guarantee that ψr

ij is smooth over the ρ2 transition.Finally, our proposed constraint-aware mobility controls are

given by

ui = −∇xi

⎛⎝∑

j∈Ii

ψoij +

∑j∈Da

i

ψaij +

∑j∈Dr

i

ψrij

⎞⎠

= −∇xiψ(‖xij‖) (18)

where we consider for now a generalized coordination objectivedriven by potential field ψo

ij : R+ → R+ over agents in theinteraction set. We assume that the coordination potential hasthe following properties.

1) ψoij is a function of the distance between adjacent

agents i and j, ψoij

�= ψo

ij (‖xij‖), i.e., it constitutes adistributed coordination protocol. This also guaranteesthe symmetry properties ψo

ij = ψoji and ∂ψo

ij /∂‖xij‖2 =∂ψo

ji/∂‖xji‖2 .2) The potential is continuously differentiable over ‖xij‖ ∈

[0, ρ2), specifically with respect to switches over the in-teraction set (or any further partitioning of the interactionset, as we will see).

3) The potential is collision avoiding, i.e., ψoij → ∞ as

‖xij‖ → 0.The overall system consisting of proximity-limited agent dy-

namics (1), along with interaction constraints and objectivesdriven by potential field controls (18), can be treated as a hy-brid system13 in which discrete transitions occur at times whennew network edges are added or removed, or when switchesin the discernment sets occur. The analysis of such a systemcan be treated using the common Lyapunov function and an ex-tension of LaSalle’s invariance principle from hybrid systemtheory [42]–[45]. First, the common Lyapunov function can bedefined as follows.

Definition 3.2 (Common Lyapunov function): If we denotethe switching system generally by the differential equationx = fs(x), where s indexes the system transitions, a commonLyapunov function is a positive-definite, smooth function V suchthat ∇V (x)fs(x) < 0 for all s [45].

Additionally, we make the following assumption concerningthe switching behavior of the system, as induced by the con-straint and predicate construction, and agent dynamics:

13In this paper, we have avoided the specific nomenclature and representationsof formal hybrid automata. We reserve such a treatment for future work.


Assumption 2 (Zeno behavior): An execution of a hybrid sys-tem is called finite if there exists a finite sequence of switchingtimes ending with a compact interval and infinite if the sequenceof switches is either an infinite sequence, or if the time betweenswitches is infinite. An execution is called Zeno if it takes an infi-nite number of discrete transitions in a finite amount of time [42].In this paper, we assume that the composition of agent dynamicswith the system constraints and predicates generates switchingwithout Zeno behavior.

We begin our analysis by defining the set of feasible initialconditions

F �= {x ∈ Rmn | ‖xij‖ ∈ [0, ρ1), ∀ (i, j) ∈ E}

∩ {x ∈ Rmn | ‖xij‖ ∈ [ρ2 ,∞], ∀ (i, j) /∈ E}∩ {x ∈ Rmn |P = 1} (19)

where we specifically denote the constraint satisfaction set by

Fp�= {x ∈ Rmn |P = 1} �= ∅. Note that it is implied by (19)

and dynamics (9)–(12) that in F , we have Cai = Cd

i = Dai =

Dri = ∅, ∀ i ∈ V . The first result of this paper now follows.Theorem 3.1: Consider the system (1) under the previously

described interaction model, driven by local controls (18), andstarting from feasible initial conditions (19). Then, the systemconverges to an equilibrium configuration, where ui = 0, ∀ i ∈V .

Proof: First, consider the Lyapunov function V : Rmn →R+ , which is given by

V =12

n∑i=1

⎛⎝∑

j∈Ii

ψoij +

∑j∈Da

i

ψaij +

∑j∈Dr

i

ψrij

⎞⎠ (20)

and observe that at times where the system switches, V andcontrols u are continuously differentiable by construction, withvalues that remain constant.14 Specifically, as ψo

ij is smooth overswitches in Ii , ψa

ij is smooth over switches in Dai , and ψr

ij issmooth over switches in Dr

i ; V is smooth in x over all systemtransitions, satisfying the requirements of Definition 3.2. Thus,V serves as a common Lyapunov function for the hybrid system,and analysis can proceed in the standard manner. Now, for anyc > 0, define the set ΩV = {x ∈ Rmn |V ≤ c}, noticing thatΩV is compact due to the continuity of V and as a consequenceof potential construction.

The time derivative of V in ΩV is

V =12

n∑i=1

⎛⎝∑

j∈Ii

ψoij +

∑j∈Da

i

ψaij +

∑j∈Dr

i

ψrij

⎞⎠ (21)

wheren∑

i=1

∑j∈Ii

ψoij = 2

n∑i=1

∑j∈Ii

xTi ∇xi

ψoij

n∑i=1

∑j∈Da

i

ψaij = 2

n∑i=1

∑j∈Da

i

xTi ∇xi

ψaij

14This analysis is inspired by the application of a smooth, switching Lyapunovfunction to a hybrid system in a similar context by [32].

n∑i=1

∑j∈Dr

i

ψrij = 2

n∑i=1

∑j∈Dr

i

xTi ∇xi

ψrij (22)

due to symmetry of potentials ψoij , ψa

ij , and ψrij , and sets Ii , Da

i ,and Dr

i . Thus

V =n∑

i=1

xTi ∇xi

⎛⎝∑

j∈Ii

ψoij +

∑j∈Da

i

ψaij +

∑j∈Dr

i

ψrij

⎞⎠

= −n∑

i=1

‖∇xiψ(‖xij‖)‖2 ≤ 0 (23)

which implies the positive invariance of the level sets ΩV ofV . The extension of LaSalle’s invariance principle for hybridsystems15 together with (23) and initial conditions (19) guaran-tees that the system converges to the largest invariant subset of{x ∈ Rmn |∇xi

ψ(‖xij‖) = 0}. As ui = −∇xiψ(‖xij‖), we

must have at equilibrium ui = 0, ∀ i ∈ V . �The collision avoidance and discernment set reachability

properties are established in the following lemma.Lemma 3.1: Assume that the system (1) evolves accord-

ing to control laws (18) and starts from initial conditions

(19). Then, the sets Fc�= {x ∈ Rmn | ‖xij‖ > 0, ∀ i ∈ V, j ∈

Ii}, Fa�= {x ∈ Rmn | ‖xij‖ ≤ ρ2 − εa , ∀ i ∈ V, j ∈ Da

i }, and

Fr�= {x ∈ Rmn | ‖xij‖ ≥ ρ1 + εr , ∀ i ∈ V, j ∈ Dr

i } are in-variant for the trajectories of the closed-loop hybrid systembetween instances of switching transitions. Further, invarianceis preserved at the instance of a system transition, i.e., the switchof a discernment set or neighbor set.

Proof: It follows from (23) that the time derivative of Vremains nonpositive for every x(0) ∈ F and for all t ≥ 0.Since V (x(0)) < ∞, we must have V (x(t)) < ∞, ∀ t ≥ 0. As‖xij‖ → 0+ for j ∈ Ii , ‖xij‖ → ρ−2 for j ∈ Da

i , and ‖xij‖ →ρ+

1 for j ∈ Dri , we have V → ∞. Thus, we can conclude that

there exists εa > 0 and εr > 0 such that x(t) ∈ Fc ∩ Fa ∩ Fr

for all t with 0 < t1 < t < t2 , where t1 and t2 are switchinginstances. Now, upon a switch in Ni , Da

i , or Dri , we have by

construction of the interaction region ‖xij‖ ≥ ρ1 , ‖xij‖ = ρ1 ,and ‖xij‖ = ρ2 , respectively. Thus, by inspection, the invari-ance of Fc , Fa , and Fr holds upon transitions in Ni , Da

i , andDr

i . �We now come to our primary result concerning the constraint

satisfaction properties of the system.Theorem 3.2: Consider the multiagent system (1) controlled

by (18) and starting from initial conditions (19). Further, as-sume that predicates Pa

ij and Pdij are constructed according to

Assumption 1. Then, the constraint satisfaction set Fp is invari-ant for the trajectories of the closed-loop hybrid system.

Proof: By dynamics (11), initial condition Dai (0) = ∅, ∀ i ∈

V , and the exclusivity and persistence properties of Dai , we con-

clude that j ∈ Dai (t2) at time t2 for some agent i if and only

if there exists a time t1 ≤ t2 such that Pdij (t1) ∨ Pd

ji(t1) = 1with ‖xij (t1)‖ = ρ1 and j ∈ Ni(t1), i.e., a link deletion pred-icate for established link (i, j) was violated on the transit of

15We can invoke this extension as the system exhibits no Zeno behavior [42].


a neighboring agent j over the connection boundary ρ1 . Sim-ilarly, by dynamics (12), initial condition Dr

i (0) = ∅, ∀ i ∈ V ,and the exclusivity and persistence properties of Dr

i , we con-clude that j ∈ Dr

i (t2) at time t2 for some agent i if and only ifthere exists a time t1 ≤ t2 such that Pa

ij (t1) ∨ Paji(t1) = 1 with

‖xij (t1)‖ = ρ2 and j /∈ Ni(t1), i.e., a link addition predicate forpotential link (i, j) was violated on the transit of a nonneigh-boring agent j over the interaction boundary ρ2 . Now, invokingLemma 3.1, we conclude that proper link discrimination occurs,i.e., there exists no member j ∈ Da

i that becomes disconnectedfrom an agent i, and likewise, there exists no member j ∈ Dr

i

that becomes a neighbor of agent i. Finally, having satisfiedthe requirements of Assumption 1 concerning predicate formu-lation, we can conclude that

∧pi

k=1 Pki (t) = 1, ∀ i ∈ V, t ≥ 0,

and the result follows. �It is clear that the proper execution of the system rests primar-

ily on the construction of the discernment predicates, specifi-cally with respect to the requirements imposed by Assumption1. We dedicate the remainder of this paper to the investigationof predicates for realistic multiagent problems, related systemanalysis, and simulation.

IV. AGGREGATION UNDER MAX DEGREE CONSTRAINTS

Our first coordination objective calls for the distributed sys-tem of agents to aggregate,16 i.e., from some initial position,the group of agents must enter asymptotically into a boundedregion of Rm . Further, in the spirit of this study, we requirethat the agents respect upper bounds on the degree of localconnectivity, i.e., |Ni | ≤ βi, ∀ i ∈ V , while maintaining over-all network connectedness.17 We suggest that by constrainingdynamically the connectivity of each agent, we yield realisticaggregation behaviors in terms of limiting local communicationburden and bolstering the performance of information diffusionin the network.18

Consider first a potential field ψaggij : R+ → R+ to achieve

the aggregation objective:

ψaggij = K1‖xij‖2 (24)

where K1 > 0 is a gain parameter. Our desire is to apply ψaggij

over only j ∈ Ni ; however, note that as ψaggij (ρ1) �= 0, a discon-

tinuity occurs upon the network switch on link addition at ρ1 .We, thus, require an additional potential field ψcan

ij : R+ → R+that stitches together the aggregation potential for link additioncandidates in order to preserve continuity over the topologyswitch at ρ1 . That is, we define

ψcanij = a‖xij‖2 + b‖xij‖ + c (25)

with constants a, b, c chosen such that the following propertiesapply.

16This behavior is also referred to as swarming and the agent collective as aswarm. Without some collision avoidance requirement, aggregation is typicallytreated as the common rendezvous problem.

17Our contributions here go beyond the investigation in [32], which considersunconstrained aggregating behaviors.

18In [23], it is shown that large node degrees induce time-delay sensitivity innetworked consensus processes.

Fig. 3. Composite of interagent potentials for constrained aggregation, in-cluding aggregation, collision avoidance, and link retention (blue, j ∈ Ni ), linkdenial (green, j ∈ Dr

i ), and continuity-preserving attraction (red, j ∈ Cai ). The

interaction model parameters ρ2 , ρ1 , and ρ0 are illustrated by dashed lines.

1) To ensure that the transition over ρ1 is sufficientlysmooth, we require ψcan

ij (ρ1) = ψaggij (ρ1) and ∂ψcan

ij /∂‖xij‖(ρ1) = ∂ψagg

ij /∂‖xij‖(ρ1).2) ∂ψcan

ij /∂‖xij‖(ρ2) = 0, ensuring continuity overswitches in the ρ2 transition.

For the purposes of collision avoidance, consider the potentialfield ψcol

ij : R+ → R+ , which is defined as [32]

ψcolij =

⎧⎨⎩

K2 log(

1‖xij‖2

), ‖xij‖ ∈ (0, z)

Ψcol(‖xij‖), ‖xij‖ ∈ [z, ρ0)(26)

where 0<z<ρ0 and Ψcol =a‖xij‖3 + b‖xij‖2 + c‖xij‖ + d,with constants a, b, c, d chosen such that

1) to ensure continuity over the potential transi-tion at z, we require Ψcol(z) = K2 log

(1/z2

)and

∂Ψcol/∂‖xij‖(z) = −2K2/z;2) Ψcol(ρ0) = 0 and ∂Ψcol/∂‖xij‖(ρ0) = 0, ensuring con-

tinuity of the potential over switches in the ρ0 transition.As we will see in the sequel, this seemingly peculiar definition

of ψcolij allows us to bound the potential derivative and derive

bounds on the swarm size of the system.The final control laws are now given by

ui = −∇xi

⎛⎝∑

j∈Ni

ψaggij +

∑j∈Ca

i

ψcanij +

∑j∈Π i

ψcolij

+∑j∈Da

i

ψaij +

∑j∈Dr

i

ψrij

⎞⎠ (27)

where Πi = {j ∈ V | ‖xij‖ ≤ ρ0} is the collision avoidance setfor agent i. A composite of the potentials comprising controllaws (27) is depicted in Fig. 3. It is important to note that thecontrols (27) are continuously differentiable over switches inthe network topology and transitions in the discernment sets, asrequired for analysis with a common Lyapunov function.


A. Predicate Formulation for Max Degree Constraints

Together with controls (27), we require predicates Paij and Pd

ij

to dictate the switching behavior of discernment sets Dai and Dr

i

in order to satisfy the maximum degree constraints |Ni | ≤ βi

of Problem 1. In particular, we must choose predicates thatmeet the specifications of Assumption 1 to guarantee constraintsatisfaction under the dynamic network topology. It is implied bythe predicate formulation requirements that the agents possessthe capability to either evaluate or predict the impact of a linkaddition or deletion before it occurs and be able to appropriatelytemper that knowledge against local constraints. In the case ofbounding node degree, the interaction constraint is explicit tothe existence of the link itself, and thus, we can formulate ina straightforward manner constraints that are compliant withAssumption 1. Specifically, we consider a worst-case networktopology as it relates to the max degree constraints, yielding linkaddition predicate

Paij

�= |Ni | + |Ca

i | ≥ βi. (28)

In terms of a maximal degree bound, predicate (28) evaluatesa worst-case network topology by considering all candidatesfor connection, i.e., j ∈ Ca

i , as virtual neighbors of discerningagent i. This way, the preemptive property is guaranteed asdiscernments always evaluate over a worst-case degree boundednetwork topology.

Problem 1 also dictates that connectivity be preserved overthe trajectories of the system. For this purpose, we define thelink deletion predicate

Pdij = 1 (29)

such that all links in the network are retained and initial con-nectivity implies dynamic connectedness. Although the main-tenance of all established links for the purposes of guarantee-ing connectivity is by nature conservative, we point out thatin an aggregation objective, the addition of links is of primaryconcern.

B. Stability Analysis of Constrained Aggregation

We are now prepared to analyze the behavior of dynamics(1) under controls (27) and predicates (28), (29), with initial

conditions in F ∩ FC , where FC�= {x ∈ Rmn | G ∈ Cn} is the

set of agent configurations that are connected. The equilibrium,collision avoidance, and discernment set reachability propertiesof the system can be determined by applying the results ofSection III as follows.

Theorem 4.1: Assume that the agents (1) are subject to con-trols (27) starting from initial conditions x(0) ∈ F ∩ FC . Then,the system converges to an equilibrium configuration, whereui = 0, ∀ i ∈ V , and the sets Fc and Fd are invariant for thetrajectories of the closed-loop hybrid system.

Proof: By construction, the objective potential field

∑j∈Ii

ψoij

�=

∑j∈Ni

ψaggij +

∑j∈Ca

i

ψcanij +

∑j∈Π i

ψcolij (30)

meets the standards of a generalized coordination objective,i.e., (30) is continuously differentiable over ‖xij‖ ∈ [0, ρ2) withrespect to switching in Ni , Ca

i , and Πi . It is then clear thataggregation controls (27) are equivalent in nature to (18), andwe can invoke Theorem 3.1 and Lemma 3.1, yielding our desiredresult.19 �

Now, we establish the constraint satisfaction properties of thesystem in the following.

Theorem 4.2: Consider the system (1) under control laws(27) and discernment predicates (28) and (29), with x(0) ∈F ∩ FC . Further, assume that the local constraint predicates

are P1i

�= |Ni | ≤ βi, ∀ i ∈ V . Then, the constraint satisfaction

set Fp is invariant for the trajectories of the closed-loop hybridsystem, and the network remains connected for all t ≥ 0.

Proof: Following Theorem 3.2 with link addition predi-cate (28), we can conclude that nonneighboring agents i andj become mutual candidates for link addition only when|Ni | + |Ca

i | < βi and |Nj | + |Caj | < βj hold. In conjunction

with proper discrimination of link additions (guaranteed byTheorems 4.1 and 3.2) and the fact that |Ni(0)| + |Ca

i (0)| ≤ βi ,it follows that |Ni(t)| ≤ βi − |Ca

i (t)|, ∀ i ∈ V, t ≥ 0. Further,considering link deletion predicate (29), it is clear that no j ∈ Ni

becomes a candidate for link deletion, and all established linksare maintained. Noting that link retention has no effect on P1

i ,we can finally conclude that

∧ni=1 P1

i (t) = 1 and x(t) ∈ FC

(due to link maintenance over an initially connected network)for all t ≥ 0, which is our desired result. �

From the preceding analysis, we can conclude that the sys-tem is convergent to a static state, and that the desired col-lision avoidance and constraint satisfaction requirements areabided. However, it remains to determine whether the systembehavior constitutes aggregation, i.e., at equilibrium have theagents entered into a bounded region of the workspace. We be-gin our analysis of the coordination behavior by defining thecenter of the agent collective, known as the swarm center, given

by x�= 1

n

∑ni=1 xi . Notice that as ui =

∑j f(xi − xj ), and by

construction, antisymmetry f(xi − xj ) = −f(xj − xi) holds,it follows that ˙x = 0, i.e., the swarm center is invariant over thetrajectories of the system [32], [33]. This property holds dueto the symmetry of all sets that undergo switching (neighborand discernment sets), guaranteeing that despite switching, thecontribution of agent controls is balanced and the swarm centerremains fixed.20 Therefore, without loss of generality, for thefollowing analysis, we choose x as the origin of our coordinatesystem and define the swarm energy function

Vs =12

n∑i=1

‖xi‖2 . (31)

Note that Vs is positive semidefinite and Vs → 0 as xi →x, ∀ i ∈ V , i.e., upon agent rendezvous. The time derivative of

19Notice that the aggregating application satisfies Assumption 2 as all linksare retained, i.e., the switching is finite.

20Compare, for example, with the convergence of average consensus in adynamic graph, as is reviewed in [33].


Vs is then

Vs =12

n∑i=1

∂‖xi‖2

∂‖xi‖

(∂‖xi‖∂xi

)T

xi =n∑

i=1

xTi ui. (32)

Applying the chain rule to controls (27), plugging into (32),and exploiting the potential and discernment set symmetry thenyields

Vs = −n∑

i=1

⎛⎝∑

j∈Ni

δagg‖xij‖2 +∑j∈Ca

i

δcan‖xij‖2

−∑j∈Π i

|δcol|‖xij‖2 +∑j∈Da

i

δa‖xij‖2

−∑j∈Dr

i

|δr |‖xij‖2

⎞⎠ (33)

where for clarity, we let δ��= ∂ψ�

ij /∂‖xij‖2 , and we usethe fact that δcol ≤ 0, ∀ j ∈ Πi and δr ≤ 0, ∀ j ∈ Dr

i . Now,consider the derivative bounds δagg = K1 , δcan ≥ 0, δa ≥0, |δcol| ≤ K2/‖xij‖2 , and |δr | ≤ κ, and agent displacementbound ‖xij‖ ≤ ρ2 , ∀ j ∈ Dr

i , which when applied to (33) givesthe following bound on the time derivative of the swarm energy:

Vs ≤ −K1

n∑i=1

∑j∈Ni

‖xij‖2 +n∑

i=1

(K2 |Πi | + κρ2

2 |Dri |

). (34)

Now, we can derive an upper bound on the swarm size bydetermining the conditions under which the swarm energy isdecreasing, Vs < 0. That is

n∑i=1

∑j∈Ni

‖xij‖2 >1

K1

n∑i=1

(K2 |Πi | + κρ2

2 |Dri |

). (35)

By Theorem 4.2, we know that for all time G is connected, andthus, we can assume

x�= max

i,j∈V‖xij‖2 ≤

n∑i=1

∑j∈Ni

‖xij‖2 (36)

where x is the squared swarm size. We then have

x >1

K1

n∑i=1

(K2 |Πi | + κρ2

2 |Dri |

)←→ Vs < 0. (37)

It is important to notice that the left-hand side of (37) switcheswith the dynamics of the network topology and discernment sets.To obtain a bound over all possible network and discernmenttopologies, we consider the worst case of (37), yielding the finalswarm bound

x ≤ 1K1

n∑i=1

maxG

(K2 |Πi | + κρ2

2 |Dri |

). (38)

As the terms of (38) are finite, we can conclude that the agentsenter into a bounded region of the workspace, i.e., the desiredaggregation behavior is achieved.

Remark 4.1 (Bounded repulsion): The bound κ on δr can bederived using the method proposed by Ji and Egerstedt [33].

Briefly, we consider the Lyapunov function as in (20) and de-termine the maximum energy that can be generated over thepossible network and discernment topologies. We then allocatethis maximal energy to ψr

ij (dmin), where dmin is the minimumdistance from ρ1 that can be achieved. That is, we solve

maxG

(V ) = ψrij (dmin) (39)

for dmin , yielding bound κ = δr (dmin). Of course, one couldalso apply a bounded potential field for repulsion; however,constraint satisfaction would no longer be guaranteed.

Remark 4.2 (Explicit bound): To obtain an explicit formof bound (38), the problem maxG

(K2 |Πi | + κρ2

2 |Dri |

)can

be solved as follows. Notice that by Theorem 4.2, we have|Ni(t)| ≤ βi, ∀ i ∈ V, t > 0, requiring |Πi | ≤ βi and |Dr

i (t)| ≤n − 1 − βi, ∀ i ∈ V, t > 0. We then directly have the solution

maxG

(K2 |Πi | + κρ2

2 |Dri |

)= K2βi + κρ2

2(n − 1 − βi) (40)

which yields an explicit bound when applied to (38).

V. DISPERSION UNDER MIN DEGREE CONSTRAINTS

We now consider solving Problem 2 by defining local controlsthat ensure the agents disperse, while respecting lower boundson the degree of local connectedness, |Ni | ≥ ωi, ∀ i ∈ V . Anal-ogous to the aggregation behavior, realistic dispersive dynamicsare generated by forcing a minimal level of agent connectivity,as information exchange processes are guaranteed a minimumlevel of performance.21 More generally, we can also envisiona system in which the agents act to maximize coverage areathrough dispersion (e.g., threat monitoring, environmental sam-pling, etc.), while maintaining certain critical links in the net-work as defined by some mission objective or metric.

The dispersion objective is achieved by defining potentialfield ψdis

ij : R+ → R+ , which is given by

ψdisij =

K3

‖xij‖2 + a‖xij‖2 + b‖xij‖ + c (41)

where K3 > 0 is a gain parameter, and we choose constantsa, b, c such that ψdis

ij (ρ2) = 0 and ∂ψdisij /∂‖xij‖(ρ2) = 0, en-

suring sufficient smoothness of the potential on link loss at ρ2 .Notice that in this case, collision avoidance is implicit to thepotential (41). Our control laws for constrained dispersion arethen

ui = −∇xi

⎛⎝∑

j∈Ni

ψdisij +

∑j∈Da

i

ψaij +

∑j∈Dr

i

ψrij

⎞⎠ . (42)

Fig. 4 depicts a composite of the potentials composing controllaws (42).

Predicates for constraint satisfaction are derived in a man-ner similar to aggregation, by considering worst-case networktopologies relative to the minimum degree constraint. Specifi-cally, the link deletion predicate takes the form

Pdij

�= |Ni | − |Cd

i | ≤ ωi. (43)

21In [11], it is shown that increased connectedness is correlated with improvedconsensus in networked systems.


Fig. 4. Composite of interagent potentials for constrained dispersion, includ-ing dispersion (blue, j ∈ Ni ), link retention (red, j ∈ Da

i ), and link denial(green, j ∈ Dr

i ). The interaction model parameters ρ2 , ρ1 , and ρ0 are illus-trated by dashed lines.

By considering all candidates for deletion, i.e., j ∈ Cdi , as virtual

nonneighbors, predicate (43) discerns link loss over a worst-case network topology with respect to the minimum degreeconstraint. Problem 2 also requires that no links are added duringthe system execution. For that purpose, we define link additionpredicate

Paij = 1 (44)

guaranteeing that all nonneighbor agents that enter the discern-ment region are repelled.

A. Stability Analysis of Constrained Dispersion

We now determine the behavior of dynamics (1) under con-trols (42) and predicates (43) and (44). The equilibrium, colli-sion avoidance, and discernment set reachability properties ofthe system can be determined by applying the results of Sec-tion III as follows.

Theorem 5.1: Assume that the agents (1) are subject to con-trols (42) starting from initial conditions x(0) ∈ F ∩ FC . Then,the system converges to an equilibrium configuration, whereui = 0, ∀ i ∈ V , and the sets Fc and Fd are invariant for thetrajectories of the closed-loop hybrid system.

Proof: By construction, the objective potential field∑j∈Ii

ψoij

�=

∑j∈Ni

ψdisij meets the standards of a generalized

coordination objective, i.e., ψdisij is continuously differentiable

over ‖xij‖ ∈ [0, ρ2) with respect to switching in Ni . It is thenclear that dispersion controls (42) are equivalent in nature to(18), and invoking Theorem 3.1 and Lemma 3.1, the resultfollows.22 �

Now, we establish the constraint satisfaction properties of thesystem in the following.

Theorem 5.2: Consider the system (1) under control laws(42) and discernment predicates (43) and (44), with x(0) ∈F ∩ FC . Further, assume that the goal constraint predicates are

22Notice that the dispersion application satisfies Assumption 2 as no links areadded, and thus, switching must be finite.

P1i

�= |Ni | ≥ ωi, ∀ i ∈ V . Then, the constraint satisfaction set

Fp is invariant for the trajectories of the closed-loop hybridsystem.23

Proof: Following Theorem 3.2 with link deletion predicate(43), we can conclude that neighboring agents i and j becomemutual candidates for link deletion only when |Ni | − |Cd

i | > ωi

and |Nj | − |Cdj | > ωj hold. In conjunction with proper dis-

crimination of link deletions (guaranteed by Theorems 5.1and 3.2) and the fact that |Ni(0)| − |Cd

i (0)| ≥ ωi , it followsthat |Ni(t)| ≥ ωi + |Cd

i (t)|, ∀ i ∈ V, t ≥ 0. Further, consider-ing link addition predicate (44), it is clear that no j /∈ Ni be-comes a candidate for link addition, and thus, no new links areestablished. Noting that link denial has no effect on P1

i , we canfinally conclude that

∧ni=1 P1

i (t) = 1 for all t ≥ 0, which is ourdesired result. �

With knowledge that the system is convergent and respectsour desired collision avoidance and constraint satisfaction re-quirements, we now determine whether the induced coordi-nation behavior is indeed dispersive. Consider once again theswarm energy Vs , in this context having time derivative

Vs =n∑

i=1

⎛⎝ ∑

j∈Dai

(|δdis | − δa) ‖xij‖2 +∑j∈Δ i

|δdis |‖xij‖2

+∑j∈Θ i

|δdis |‖xij‖2 +∑j∈Dr

i

|δr |‖xij‖2

⎞⎠ (45)

where we further partition the interaction region by defining sets

Δi = {j ∈ V | j ∈ Ni , ‖xij‖ ≤ ρ1}Θi = {j ∈ V | j ∈ Ni ∩ Da

i , ‖xij‖ > ρ1} (46)

with Dai denoting the complement of Da

i . Now con-sider derivative bounds, |δdis | ≥ 0, δa ≤ τ, ∀ j ∈ Da

i , |δdis | ≥|δdis(ρ1)|, ∀ j ∈ Δi , |δdis | ≥ 0, ∀ j ∈ Θi , and |δr | ≥ 0, ∀ j ∈Dr

i , together with agent displacement bound ‖xij‖ ≤ ρ2 , ∀ j ∈Da

i , yielding

Vs ≥ −τρ22

n∑i=1

|Dai | + |δdis(ρ1)|

n∑i=1

∑j∈Δ i

‖xij‖2 (47)

Thus, for dispersive action, i.e., Vs > 0, we require

n∑i=1

∑j∈Δ i

‖xij‖2 >τρ2

2

|δdis(ρ1)|

n∑i=1

|Dai |. (48)

As the network is assumed to be initially connected with|Da

i |(0) = 0, ∀ i ∈ V and 0 < ‖xij (0)‖ ≤ ρ1 , ∀ i, j ∈ E , wehave Vs > 0 if

n∑i=1

∑j∈Δ i

‖xij‖2 > 0 (49)

23Notice that unlike Theorem 4.2, there is no guarantee of network con-nectedness. Investigations into initial conditions and interaction constraints thatguarantee connectedness in a dispersive objective is a line of our future research.


Fig. 5. Aggregation simulations of an n = 10 agent system with interaction model ρ2 = 10, ρ1 = 7, and ρ0 = 2, under maximum degree constraints |Ni | ≤ 6[(a)–(c)] and |Ni | ≤ 4, |Ni | ≤ 5 [(d)–(f)]. (a) Random initial configuration; (b) intermediate configuration |Ni | ≤ 6; (c) final converged configuration |Ni | ≤ 6;(d) initial ring network configuration; (e) final converged configuration |Ni | ≤ 4; and (f) final converged configuration |Ni | ≤ 5.

which always holds. We, therefore, have the following intu-itive result. For any valid initial configuration x ∈ F ∩ FC , theagents are guaranteed to disperse until a point when some agentj becomes a member of a set Da

i , an occurrence that resultsfrom j being a minimum degree violation risk.

Remark 5.1 (Bounded attraction): The bound τ on δa canbe derived using the steps described in Remark 4.1, i.e., byapplying the method proposed by Ji and Egerstedt [33]. In thiscase, however, we would determine a maximal distance dmaxachievable between an agent i and a neighbor j under attractivepotential ψa

ij .

VI. SIMULATION RESULTS

In this section, we present simulation results of our proposedcontrol scheme for the constrained interaction of proximity-limited mobile agents and show that the desired interactiv-ity constraints, collision avoidance, and connectivity proper-ties hold. In particular, we consider the coordination scenariosof aggregation and dispersion analyzed in Sections IV and V,subject to maximum and minimum degree constraints, respec-tively. For all simulations, we consider n = 10 agents operating

in a bounded workspace over R2 , with interaction model pa-rameters ρ2 = 10, ρ1 = 7, and ρ0 = 2. First, we consider theagents driven by constrained aggregation controls (27) and sub-ject to discernment predicates (28) and (29). Two initial agentconfigurations are considered, a random placement of agentsin the workspace [see Fig. 5(a)], and a ring network of radius10 [see Fig. 5(d)], each satisfying initial condition requirementx ∈ F ∩ FC . To the randomly placed network, we apply max-imal degree bound βi = 6, ∀ i ∈ V , yielding intermediate con-figuration [see Fig. 5(b)] and final configuration [see Fig. 5(c)].For the ring network, we assume the cases of βi = 4, ∀ i ∈ Vand βi = 5, ∀ i ∈ V , yielding final configurations in Fig. 5(e)and (f), respectively. As illustrated by the maximum node de-gree and swarm size over the simulations [see Fig. 7(b)], theswitching discernment controls satisfy the desired interactionconstraints, avoid collisions, and maintain the connectednessof the resulting network, while driving the agents to perform acoordinated aggregation objective.

Now, we consider the agents driven by constrained disper-sion controls (42) and subject to discernment predicates (43)and (44). In this case, a fully connected ring network of radius


Fig. 6. Dispersion simulations of an n = 10 agent system with interaction model ρ2 = 10, ρ1 = 7, and ρ0 = 2, under minimum degree constraints |Ni | ≥ 5[(b)] and |Ni | ≥ 3 [(c)]. (a) Initial fully connected ring network configuration; (b) final converged configuration |Ni | ≥ 5; and (c) final converged configuration|Ni | ≥ 3.

2.5 is considered for the initial agent configuration [see Fig.6(a)], again satisfying initial condition x ∈ F ∩ FC . We as-sume the cases of minimal degree bounds ωi = 5, ∀ i ∈ V andωi = 3, ∀ i ∈ V , yielding final configurations in Fig. 6(b) and(c), respectively. From the minimum node degree and swarmsize over the simulations [see Fig. 7(b)], we see that the controlssatisfy our desired interaction constraints, while avoiding col-lisions, resulting in a topology-aware, coordinated dispersionobjective.

VII. CONCLUSION AND FUTURE WORK

In this paper, we have considered the problem of controllingthe interactions of a group of mobile agents, such that a setof topological constraints were satisfied. Assuming proximity-limited communication between agents, we leveraged mobilityto enable adjacent agents to achieve discriminative connectiv-ity, i.e., to actively retain established links or reject link creationon the basis of constraint satisfaction. Specifically, we haveproposed a distributed scheme consisting of controllers withdiscrete switching for link discrimination, whereby adjacentagents were classified as either candidates for constraint-awarelink removal or addition, or as constraint violators requiringeither link retention through attraction or link denial throughrepulsion. Attractive and repulsive potentials fields establishedthe local controls necessary for link discrimination, and con-straint violation predicates formed the basis for discernment.We analyzed the application of constrained interaction to twocanonical coordination objectives in distributed multiagent sys-tems, i.e., aggregation and dispersion behaviors, with maxi-mum and minimum node degree constraints, respectively. Foreach task, we proposed predicates and continuity-preservingpotential fields and proved the dynamical and constraint sat-isfaction properties of the resulting hybrid system. Simulationresults demonstrated the correctness of our proposed methodsand the ability of our framework to generate topology-awareself-organization.

Fig. 7. Node degree bounds and swarm size for aggregation and dispersionsimulations depicted in Figs. 5 and 6. (a) Maximum node degree (top) andswarm size (bottom) for |Ni | ≤ 6 (blue [see Fig. 5(c)]), |Ni | ≤ 4 (red [see Fig.5(e)]), and |Ni | ≤ 5 (green [see Fig. 5(f)]); and (b) minimum node degree (top)and swarm size (bottom) for |Ni | ≥ 3 (blue [see Fig. 6(c)]) and |Ni | ≥ 5 (red[see Fig. 6(b)]).


Directions for future work include investigating the satis-faction of nonlocal topological constraints (e.g., k-hop con-nectivity; see [37]) through local discernment and interagentcommunication, determining existential guarantees for predi-cate formulations that solve certain classes of constraint prob-lems, and exploring constraints that are not explicitly topo-logical in nature (e.g., mutual information, agent similarity,compatibility measures, etc.). Further, it is conjectured thatby examining compositions of local and nonlocal interactionconstraints in the proposed framework, there exists a foun-dation for rich coordinated behaviors that extend dynamicalcomplexity beyond aggregation and dispersion. Finally, itshould be noted that the preemptive formulation of the dis-cernment predicates in the interaction region appears promis-ing in approaching directed interaction between agents, due tosensor errors, communication failure, etc. As the preemptivecharacteristic creates a spatial buffer in interaction (i.e., a dwellbetween decisioning and topology change), there may exist er-ror/failure bounds from which constraint satisfaction can berecovered.

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Ryan K. Williams (S’11) received the B.S. degreein computer engineering from Virginia PolytechnicInstitute and State University, Blacksburg, VA, USA,in 2005. As a Viterbi fellowship recipient, he is cur-rently working toward the Ph.D. degree in electricalengineering with the University of Southern Califor-nia, Los Angeles, CA, USA.

His current research interests include control, co-operation, and intelligence in distributed multiagentsystems, consensus methods for modeling distributedcooperative phenomena, and distributed algorithms

for optimization, estimation, inference, and learning. He also has a patent pend-ing for his work on high-speed autonomous underwater vehicles.

Gaurav S. Sukhatme (F’11) received the B.Tech.degree in computer science and engineering from theIndian Institute of Technology Bombay, Mumbai, In-dia, and the M.S. and Ph.D. degrees in computer sci-ence from University of Southern California (USC),Los Angeles, CA, USA.

He is currently a Professor of Computer Science(joint appointment in Electrical Engineering) withUSC. He is the Co-Director of the USC RoboticsResearch Laboratory and the Director of the USCRobotic Embedded Systems Laboratory, which he

founded in 2000. His research interests include robot networks with applica-tions to environmental monitoring. He has published extensively in these andrelated areas.

Dr. Sukhatme has served as the Primary Investigator (PI) on numerous Na-tional Science Foundation (NSF), Defense Advanced Research Projects Agency,and National Aeronautics and Space Administration grants. He is a Co-PI onthe Center for Embedded Networked Sensing: an NSF Science and TechnologyCenter. He received the NSF CAREER Award and the Okawa Foundation Re-search Award. He is one of the founders of the Robotics: Science and SystemsConference. He was the Program Chair of the 2008 IEEE International Confer-ence on Robotics and Automation and the Program Chair of the 2011 IEEE/RSJInternational Conference on Intelligent Robots and Systems. He is the Editor-in-Chief of Autonomous Robots and has served as an Associate Editor of the IEEETRANSACTIONS ON ROBOTICS AND AUTOMATION, the IEEE TRANSACTIONS ON

MOBILE COMPUTING, and on the Editorial Board of IEEE Pervasive ComputingMagazine.

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