Decentralized Estimation of Topology Changesin Wireless Robotic Networks

Nicola Bezzo, Francesco Sorrentino, and Rafael Fierro

Abstract— In this paper we present a novel decentralizedtechnique to detect changes in the topology of a mobile roboticnetwork. The network topology evolves due to the relativemotion of the robots and the presence of obstacles in theenvironment. We consider that each robot is equipped witha chaotic oscillator whose state is propagated to the otherrobots through wireless communication. Under certain condi-tions, the states of the oscillators may synchronize, i.e., theyconverge on the same dynamical time evolution. The key ideaof our approach is that by receiving and conveniently usingan aggregate signal from the surrounding neighbors, eachnode may become able to detect changes in the local networktopology. We introduce an adaptive strategy that each robotindependently implements to: (i) estimate the net coupling fromall the oscillators in its neighborhood, and (ii) synchronize thestate of its oscillator with the others. We show that by usingthis strategy, synchronization can be attained and changes ofthe network topology can be detected. Numerical simulationsvalidate the proposed methodology.

I. INTRODUCTION

In the last decade the coordination of networked multi-agent systems has been intensively investigated. Both therobotic and communication research communities have beenworking on how to properly integrate wireless communi-cation in motion planning algorithms, considering the ran-dom properties of the rf channels and the mobility of theautonomous agents. A typical goal of the research in thisarea is to achieve a certain mission goal while maintainingconnectivity of the robotic network. Because of the randomnature of the communication channel and the complexityof the environment, the network topology may change intime and unwanted disconnections may occur. Moreover,maintaining connectivity becomes even more challengingwhen a decentralized approach is considered, in which eachagent has only local information about its connections andthe surrounding environment.

Fig. 1 shows a pictorial representation of the situationenvisioned in this paper, in which a heterogeneous roboticnetwork maintains line of sight communication betweenaerial and ground robots, by using light pulses (e.g., laserstransmitters and receivers). We further assume that the robotsmove in an unknown environment to accomplish a coordi-nated mission. The agents have no a priori knowledge of the

N. Bezzo is with the PRECISE Center, Department of Computer andInformation Science, University of Pennsylvania, Philadelphia, PA 19104,USA. E-mail: nicbezzo@seas.upenn.edu

F. Sorrentino is with the Department of Mechanical Engineering, Uni-versity of New Mexico, Albuquerque, NM 87131-0001, USA. E-mail:fsorrent@unm.edu

R. Fierro is with the MARHES Lab, Department of Electrical and Com-puter Engineering, University of New Mexico, Albuquerque, NM 87131-0001, USA. E-mail: rfierro@ece.unm.edu

Fig. 1. A virtual environment with an heterogeneous network. The linesbetween the robots represent line-of-sight communication paths.

topology of the network and need to estimate it throughoutthe mission. They also operate in a context characterized bylimited availability of information.

In this work we assume that each robot is equipped witha chaotic oscillator whose state is propagated to the othersby wireless communication. We propose a decentralizedadaptive strategy that each agent implements in order toreach and maintain synchronization while estimating changesin the local connectivity of the network. We show that byusing this strategy, the oscillators may synchronize on achaotic time evolution; moreover, each node may becomeable to detect changes in the local network topology, suchas deletion and aggregation of connections. We note that thechaotic signals appear as noise to any external party andthus provide improved security of the communication whenconflicting agents operate in the same environment, such asin pursuit and evasion games.

A. Related WorkMultiple mobile robotic systems and wireless communi-

cations have been individually and extensively studied forseveral years. Recently, roboticists have recognized the needto consider realistic communication models when designingmulti-robot systems [1], [2], [3], [4]. For instance, the authorsof [1] formally analyze the properties of the communicationchannel and use them to optimally navigate autonomousagents to improve the communication performance in termof signal-to-noise ratio and bit error rate. In [2] the authorspropose a modified traveling salesperson problem to navigatean underwater vehicle in a sensor field, using a realisticmodel that considers acoustic communication fading effects.

2013 American Control Conference (ACC)Washington, DC, USA, June 17-19, 2013

978-1-4799-0176-0/$31.00 ©2013 AACC 5919

In [5] we consider heterogeneous robotic networks withrealistic sensing and communication constraints. A powercontrol algorithm is proposed to improve the signal-to-interference plus noise ratio between the agents of thesystem. In [6] the authors optimize routing probabilities toensure desired communication rates while using a distributedhybrid approach. In [7] the authors analyze connectivity inconsensus networks with dynamical links, while [4] presentsa set of algorithms to repair connectivity within a networkof mobile routers.

Similar to the work presented in this paper, reference[8] investigates the problem of estimating the connectiontopology in a complex dynamical network by means of asteady-state control identification method. In [9] the authorspropose an algorithm inspired by the Kirchhoff’s laws usedin electrical circuits, to detect losses (also called “cuts”) inthe connectivity of large sensor networks. The same problemis also investigated in [10], in which a subset of sensorscalled “sentinels”, communicate on a regular basis to a basestation. The authors show that the base station is able toestimate the network topology based only on detections ofcommunication failures with the sentinel nodes.

On the other hand, a large literature has studied syn-chronization of large networks of interconnected systems.In [11] networked Lagrangian systems are synchronizedusing a binary consensus technique and a signum protocolthat estimate the relative heading differences between themobile agents. References [12] and [13] present an adaptivetechnique to synchronize a sensor network. This strategy istested in an experimental network of coupled opto-electronicsystems in [14].

In what follows we consider an extension of the strategyin [12] and [13] and we show that a similar approach canbe used to detect changes in networks of coupled mobilerobotic systems. As we consider mobile-agent systems thatmove in a decentralized fashion in a cluttered environment,the coupling may be unavailable or imperfect. Thus in thispaper we propose an adaptive technique that can track andestimate the evolution of the network topology.

The remainder of this paper is organized as follows.In Section II we give some preliminary graph theoreticaldefinition that will be use throughout the paper. In SectionIII we describe the models for the robots, the communi-cation connectivity strategy, and the sensing constraint. InSections IV and V we present the adaptive strategy based onsynchronization of chaotic oscillators followed by extensivesimulation results in Section VI. Finally conclusions andfuture work are drawn in Section VII.

II. PRELIMINARIES

In this section we present the graph theoretical tools [15]that will be used hereafter.

Without loss of generality, let pi ∈ p denote the positionof robot i, i = 1 . . .N . Then the network of N robots givesrise to a dynamic graph G(p).

Definition 2.1: (Dynamic Graph): We call G(p) =(V, E(p)) a dynamic graph consisting of• a set of vertices V = {v1, · · · , vN } indexed by the set

of robots, and

• a set of edges E(p) = {(i, j)|dij(p) < δ}, where dij =‖pi − pj‖2 is the Euclidean distance between robots iand j, and δ > 0.

Definition 2.2: (Graph Connectivity): A non-empty graphG is called connected if any two of its vertices are linked bya path in G.

Definition 2.3: (Adjacency Matrix): Given a non-emptygraph G with vertices V = {v1, · · · , vN } and edges in theset E , we define the adjacency matrix A = {Aij} such that,Aij = 1 if (vi, vj) ∈ E , and Aij = 0 otherwise.

Definition 2.4: (Weighted Adjacency Matrix): A weightedadjacency matrix is an adjacency matrix whose off diagonalentries are weighted according to a specific metric. In thispaper, each Aij ≥ 0, reflects the signal strength between areceiver node i and a transmitter node j 6= i.

In this work we consider that the weighted adjacencymatrix changes in time based on the motion of the robotson a plane. Hence we will write A(t) = {Aij(t)}.

III. SYSTEM MODELS

In this section we present the dynamical model for themotion of the mobile robots, along with the communicationand sensing capabilities needed for the mobile agents tomove in an obstacle populated environment.

A. Mobile Robot DynamicsThe dynamics of the ith robotic agent can be approximated

by using the following model

pi = vi (1)vi = ui, i = [1, . . . ,N ],

where pi = [p(x)i p

(y)i ]T ∈ R2 is the position vector of

the ith agent relative to the base station, vi ∈ R2 andui ∈ R2 denote the velocity and acceleration (control input),respectively, for each agent i ∈ N . The workspace in whichthe agents move, W , is populated with No fixed polygonalobstacles {O1, . . . , ONo}, whose geometries and positionsare assumed unknown to the agents.

B. SensingIn describing the mobile agents, we include some sensing

capabilities inspired by real world devices to allow severalfunctionalities. In order to avoid the obstacles we model aray field of view, similarly to a laser range finder footprint,and we create a repulsive potential force that appears onlywhen an obstacle is detected at a distance smaller than ρ0.Hence:

FO,i = ηi

(1

ρ(pi)− 1

ρ0

)1

ρ(pi)2∇ρ(pi), (2)

where ρ(pi) is the distance between the agent and anydetected obstacle in the workspace, ηi is a positive constant,and ∇ρ(pi) is the gradient of the minimum distance betweenthe robot and the closest detected obstacle. Note that for easeof discussion, in this paper we consider simple scenarioswith convex polygons. Therefore, we do not examine casesin which agents can get stuck in local minima. However localminima can be avoided by using e.g., either random-walk orpotential based techniques [16].

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C. Communication ModelOur work is motivated by the use of wireless radio

and optical communication. The signal propagated from atransmitter (Tx) to a receiver (Rx) can be decomposed intothree separated and well-known characteristics: path loss,shadowing, and multipath [17].

Path loss among all the above three modes is the onlynon random behavior because it depends on the distanceseparation between the transmitter and the receiver.

In general a simplified path loss model can be built tocapture the essence of signal propagation. The followingequation

Pr = Ptk

[d0d

]τ+ ω, (3)

represents a generalized approximation of a real channelwhere Pt is the transmitted power and Pr (d) is the receivedpower which is function of the distance d between Txand Rx. The constant k depends on the antenna/emittercharacteristics and channel attenuation, d0 is a referencedistance, τ is the path loss exponent and ω is a factor thattakes into consideration the random effects due to shadowingand multipath. Clearly, the received signal strength falls off ininverse proportion to the τ th power of the distance d betweenthe transmitter and the receiver.

Finally, our weighted adjacency matrix will be evolved byusing the information on the received signal. Specifically,Aij(t) = Pij(t) with Pij(t) the power received by the ith

agent from the jth transmitter at time t.

D. Virtual Leader PotentialIn this work we assume that the multi-robotic network

is guided by a virtual leader agent, often referred to inthe literature as a pin [18] that acts as a mobile attractivepotential. Let ζ : Rn → R be the leader potential function.With a proper design of ζ(p), the mobile agents can beattracted or repulsed to certain areas of the environment.

We define a time varying attractive potential centered inthe virtual leader position. For the sake of simplicity, weuse an exponential weighted function to represent the leaderattractive function, whose gradient is:

∇piζ(pi) =Aζ`ζ

(pi − cζ(t)) e−‖pi−cζ(t)‖

2

`ζ . (4)

where Aζ ∈ R+, `ζ ∈ R+, cζ(t) ∈ R2 and Bζ ≥ Aζ/2. cζ(t)is the center of the time varying user attractive function andthe term `ζ controls the shape of the exponential function.

E. ControllerBy assembling together all the pieces described in the

previous sections, we obtain the overall control law for theagents

pi = FO,i −∇piζ(pl) i = [1, . . . ,N ]. (5)

We assume that at the initial time, the network is fullyconnected and thus the only zero-entries for the weightedadjacency matrix are on the main diagonal. If the agentsmove too far apart from each other or the line-of-sight path-way between them is crossed by an obstacle, the connection

is assumed lost and thus the weighted adjacency matrix hasoff-diagonal zero elements, too.

In the following sections we present a decentralized tech-nique to track changes in the network topology for multi-agent systems, based on adaptive synchronization of chaoticsystems.

IV. SYNCHRONIZATION OF CHAOTIC OSCILLATORS

We consider that each robot i = 1, ...,N is equipped witha nonlinear oscillator, whose dynamics is described by

xi(t) = F (xi(t)) + γΓ[σi(t)ri(t)−H(xi(t))], (6)

where: xi(t) is the m-dimensional state of oscillator i =1, ...,N ; F (x) determines the dynamics of an uncoupled(γ → 0) system (hereafter assumed chaotic), F : Rm → Rm;H(x) is a scalar output function, H : Rm → R. We takeΓ to be a constant m-vector, Γ = [Γ1,Γ2, ...,Γm]T , andthe scalar γ is a constant characterizing the strength of thecoupling. The scalar signal each node i = 1, ...,N receivesfrom the other nodes in the network is

ri(t) =∑j

Aij(t)H(xj(t)). (7)

Note that in what follows, the entries of the adjacencymatrix Aij(t) evolve in time based on both the relativepositions of the mobile robots and the presence of obstaclesalong the line-of-sight pathways of the signals exchangedby the oscillators. We note that if the following condition issatisfied

σi(t) = [∑j

Aij ]−1, (8)

then Eq. (6) admits a synchronized solution

x1(t) = x2(t) = ... = xN (t) = xs(t), (9)

where xs(t) satisfies

xs(t) = F (xs(t)), (10)

which corresponds to the dynamics of an uncoupled system.Stability of the synchronous solution (9) for the case that

the network topology is fixed and known has been studiedin [19]. However, in this paper we are interested in the casethat the network is time varying and that the strength of thesignal received by each oscillator is unknown.

V. ADAPTIVE STRATEGY

In what follows we regard σi(t) as an internal adaptiveparameter at each node i = 1, ...,N . We propose a decentral-ized adaptive strategy to independently evolve σi(t) at eachnode in order to achieve dynamical approximate satisfactionof condition (8). We regard the Aij(t) as unknown at eachnode i, while the only external information available at nodei is the received signal (7). The goal of the adaptive strategyis to adjust σi(t) so as to maintain synchronism in thepresence of slow and a priori unknown time variations ofthe quantities Aij(t). A similar adaptive strategy was firstproposed in [12], [13] and experimentally tested for a staticnetwork of coupled opto-electronic oscillators in [14].

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We assume that each node implements the adaptive strat-egy in a decentralized way. Thus at each system node i,we define the exponentially weighted synchronization errorψi =< (σiri −H(xi))

2 >ν , where

< G(t) >ν=

∫ t

G(t′)e−ν(t′−t)dt′ (11)

and ν−1 is the time-window over which the average isperformed. We then seek to evolve σi(t) so as to minimizeψi, based on the following gradient-descent relation

σi = −κ∂ψi∂σi

, (12)

where κ is a positive scalar. Following [12], we assume thatthe dynamics of the adaptation process (12) is fast. Thustaking κ→∞, we have that the right-hand-side of Eq. (12)rapidly converges to zero, corresponding to minimizing ψi.Hence, by setting ∂ψi/∂σi to zero, we can replace (12) bythe following equations

σi(t) =< H(xi(t))ri(t) >ν

< ri(t)2 >ν=pi(t)

qi(t), (13)

By virtue of d < G(t) >ν/dt = −ν < G(t) >ν +G(t), weobtain the numerator and the denominator on the right handside of Eq. (13) by solving the differential equations

pi(t) = −νpi(t) + ri(t)H(xi(t)), (14a)

qi(t) = −νqi(t) + ri(t)2. (14b)

Since the dynamics of the Aij(t) is imagined to occur ona timescale which is slow compared to the other dynamicsin the network, we can approximate Aij(t) as constant Aij .In any practical situation, fulfillment of this assumption canbe obtained by choosing the dynamics of the individualoscillators to be fast enough, i.e., so as to verify the condition

Tx < ν−1 � TA, (15)

where Tx and TA are the timescales on which the individualoscillators and the network connections evolve, respectively.

By assuming satisfaction of (15), we note that Eqs. (6),(13), and (14) admit a synchronized solution, given by Eqs.(9), (10), and

psi = −νpsi + (∑j

Aij)H(xs)2, i = 1, ...,N , (16a)

qsi = −νqsi + (∑j

Aij)2H(xs)2, i = 1, ...,N . (16b)

If the synchronization scheme is stable, we expect thatthe synchronized solution (9), (10), and (16) will be main-tained under slow time evolution of the couplings Aij(t).Stability of this adaptive synchronization strategy has beeninvestigated in [13]. In particular, it was found that if (15)holds, then stability of the synchronized solution depends onthe pairs (ν, ξi), i = 1, ..., (N − 1) where ξi = γ(1 − αi),and {αi}Ni=1 are the eigenvalues of the normalized networkadjacency matrix A′ = {A′ij}, A′ij = (

∑j Aij)

−1Aij ,excluding the one eigenvalue αN = 1. Note that if the matrixA is symmetric (which is always the case in our application

as we defined Aij = Aji to depend on the distance betweennode i and j) the spectrum of the matrix A′ is real [13]. Forexample by choosing the individual systems to be Rossleroscillators [20], m = 3, x(t) = (x(1)(t), x(2)(t), x(3)(t))T

F (x) =

−x(2) − x(3)x(1) + ax(2)

b+ (x(1) − c)x(3)

, (17)

with the parameters a = b = 0.2, and c = 7, and by settingH(x(t)) = x(1)(t), Γ = [1, 0, 0]T , γ = 1, and ν = 0.5, wesee from Fig. 4(b) of Ref. [13] that the condition for stabilityis that 0.7 ≤ ξi ≤ 4.8, i = 1, ..., (N − 1).

VI. SIMULATION RESULTS

In this section we present some simulation results todemonstrate the applicability of the proposed approach de-scribed in the previous sections.

A. Tracking of Variations in the Network Connectivity

In the first case study we consider a network of fiverobots moving in a workspace with an obstacle in its center,as depicted in Fig. 2. Each agent is guided by the virtualnode, toward a location in the environment. The virtualnode is driving from the lower corner of Fig. 2(a) to theupper corner of the workspace (Fig. 2(d)). Because of thepresence of the obstacle, in Figs. 2(b)(c) the network getsdisconnected and is divided into two groups that reconnecttoward the end of the mission (Fig. 2(d)). The obstacle andthe perimeter of the environment in the figure are detectedlocally by each agent as they move attracted by the potentialof the virtual leader. The corresponding dynamics of the

(a) (b)

(c) (d)Fig. 2. Simulation results of five mobile robots moving from one positionto another in a cluttered environment. The lines between agents representline-of-sight communication links.

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oscillators can be observed in Fig. 3. Specifically, Fig. 3(a)shows the time evolution of the five chaotic oscillators. Fig. 4presents the results of Fig. 3(a) in finer detail and will besubject of discussion later on in this section. In Fig. 3(b),the estimated value of σ (dark line) for one of the robots isplotted in comparison with its true value (dashed red line).As can be noticed, after an initial transient, the agent is ableto synchronize with the rest of the group (Fig. 4(a)) andstays on a synchronous orbit until an event occurs. When adisconnection happens, the superimposed signal received bycertain agents decreases creating a jump in the value of σand a consequent loss of synchrony. For example betweentime 200 and 300 in Fig. 3 the network of 5 robots breaksdown into two groups, as depicted in Figs. 2(b)(c).

As the network disconnects, we observe from Fig. 4(b)that each one of the two groups synchronizes on a distincttime evolution (the two times evolutions are clearly distin-guishable in both Fig. 3(a) and Fig. 4(b)). This is due tothe property of the individual oscillators of being chaotic,as slightly different initial conditions or small dynamicalperturbations result in an exponential divergence of the timeevolutions (a property known as sensitive-dependence on theinitial conditions [21]). Subsequently, the two groups recon-nect between time step 600 and 700 (see Fig. 2(d)). Afterthat we observe that the whole mobile network reconnectsand resynchronizes, as can be seen from the close-up inFig. 4(c). By looking at σ1 in Fig. 3(b) we are able to identifyvariations on the net received signal from which informationcan be gathered on the local connectivity of the networkand its time variation. Lost connections can be detected byincreases of this quantity while addition of links throughdecreases of this quantity.

Fig. 3(c) shows the ratio between the transmitted and thereceived signals for one of the agents, say ρ1 = x1/r1.This graph demonstrates that each agent can track changesin network topology by comparing the received signal withthe transmitted one.B. Connectivity Estimation

As a special case study, we now consider a networkcomposed of three mobile robots moving in an obstaclepopulated workspace. For this particular case, the adjacencymatrix containing the signal strengths between each pairof nodes is a 3 × 3 matrix. Thus each agent receives thesuperimposition of maximum two chaotic signals whichmakes the estimation of the individual links possible. Forthis case, we repeated the same type of experiment as inSection VI and we obtained similar results to those shownin Fig. 3, i.e., each agent was able to synchronize and trackchanges in network topology (not reported here).

The main point of interest for the simulation in Fig. 5is that for this case, knowledge of σ1, σ2, and σ3, allowsus to dynamically reconstruct the whole network adjacencymatrix time evolution [14]. For example, by assuming thateach agent i broadcasts information on σi as well as onxi, the whole network topology can be reconstructed byeach one of the agents. Fig. 6 shows the comparison be-tween the estimated and the real values of A12 = A21,A13 = A31, A23 = A32. The estimated value is given by

Fig. 3. Dynamics of the oscillators for the simulation with five robotsin Fig. 2. a) Plot of the state time evolution of the five oscillators. b)Comparison between the estimated and the true values of σ for one ofthe robots. c) Plot of the ratio between the transmitted and the receivedsignals for the same robot.

Fig. 4. Synchronization of chaotic oscillators during the experiment with fivemobile agents in Fig.2. a) The five chaotic signals start not synchronized andsynchronize around time 40. b) Two synchronized orbits created while thesystem is divided in two groups. c) Resynchronization of the entire grouptoward the end of the simulation.

Aij = (σ−1i + σ−1j − σ−1k )/2, with i, j, k = 1, . . . , 3 andi 6= j 6= k. The proposed strategy performs well as we canboth track changes in the network topology and individuallyestimate the inter-agents signal strengths. This enables us toreconstruct the relative positions between the robots, evenwhen disconnections occur, as for the case shown in Fig. 5.

VII. CONCLUSIONS

In this work we have presented a decentralized frameworkto track variations of the network topology for a dynamicalsystem of coupled mobile robots. We proposed an adaptive

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(a) (b)

Fig. 5. Two snapshots of the simulation of three mobile robots moving alongthree elliptical curves. The triangular obstacle creates discontinuities in theconnectivity between the agents.

Fig. 6. Comparison between estimated and true values for the entries of theadjacency matrix obtained from the simulation with three robots in Fig. 5.

technique based on synchronization of chaotic oscillators. Weshowed that variation of connectivity can be detected basedonly on measurements of the aggregated received signal ateach agent. For the case of a network composed of three mo-bile agents, we demonstrated that we can estimate the inter-agents connection strengths and thus estimate the individualrelative position of the agents. As discussed throughout thepaper, our strategy takes advantage of the property of theindividual oscillators of being chaotic.

Future work will consist in expanding the proposed strat-egy to consider more complex networks with heterogeneouscharacteristics. By using subgroups of different chaotic os-cillators, we can estimate the connections of large networksand discriminate between different categories or internalparameters for some of the agents in the system. More-over we are interested in investigating control algorithms toavoid disconnections while enforcing line-of-sight pathwaysamong the robotic system. Finally, we plan on testing theproposed strategy in an experiment with our test bed ofground and aerial vehicles [22].

ACKNOWLEDGMENTS

This work was supported by NSF grants ECCS # 1027775and IIS # 0812338, and by the Army Research Laboratorygrant # W911NF-08-2-0004.

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