Ad-hoc wireless network coverage with networked robotsthat cannot localize

Citation Correll, N. et al. “Ad-hoc wireless network coverage withnetworked robots that cannot localize.” Robotics and Automation,2009. ICRA '09. IEEE International Conference on. 2009. 3878-3885. ©2009 Institute of Electrical and Electronics Engineers.

As Published http://dx.doi.org/10.1109/ROBOT.2009.5152742Publisher Institute of Electrical and Electronics Engineers

Version Final published versionAccessed Mon Feb 27 15:47:15 EST 2012Citable Link http://hdl.handle.net/1721.1/53729Terms of Use Article is made available in accordance with the publisher's policy

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Detailed Terms

Ad-hoc Wireless Network Coverage with Networked Robots that

Cannot Localize

Nikolaus Correll, Jonathan Bachrach, Daniel Vickery and Daniela Rus

Abstract— We study a fully distributed, reactive algorithmfor deployment and maintenance of a mobile communicationbackbone that provides an area around a network gateway withwireless network access for higher-level agents. Possible applica-tions of such a network are distributed sensor networks as wellas communication support for disaster or military operations.The algorithm has minimalist requirements on the individualrobotic node and does not require any localization. Thismakes the proposed solution suitable for deployment of largenumbers of comparably cheap mobile communication nodesand as a backup solution for more capable systems in GPS-denied environments. Robots keep exploring the configurationspace by random walk and stop only if their current locationsatisfies user-specified constraints on connectivity (number ofneighbors). Resulting deployments are robust and convergenceis analyzed using both kinematic simulation with a simplifiedcollision and communication model as well as a probabilisticmacroscopic model. The approach is validated on a team of9 iRobot Create robots carrying wireless access points in anindoor environment.

I. INTRODUCTION

We wish to deploy inexpensive robots with minimalistcapabilities over an area and use the group as a computation,communication, and sensing backbone. The swarm willdisperse autonomously to create a mobile communicationnetwork with maximum coverage. In this paper, we studythis problem when the swarm of robots is minimalist intheir resources and information — that is, the robots op-erate by using wireless connectivity only to estimate thenetwork topology, have no means of localization, and avoidobstacles using bumper sensors. Applications include firstresponse operations in areas where the existing computationand communication infrastructure has been destroyed, andmonitoring remote areas with no infrastructure.

The robots’ information and processing capabilities willaffect the quality of the communication network that theycan establish. The critical resources are (1) knowledge aboutthe position of other robots (relative range-and bearing),(2) knowledge about the placement of the robot within theenvironment (global localization), and (3) knowledge aboutthe environment in terms of maps. We see a hierarchy ofcapabilities, which ranges from no map, no localization, and

N. Correll, D. Vickery, and D. Rus are with the ComputerScience and Artificial Intelligence Laboratory, MassachusettsInstitute of Technology, Cambridge, MA 02478, USA{nikolaus|rus}@csail.mit.edu,drv@mit.edu

J. Bachrach is with Makani Power Systems Inc. Alameda, CA 94501,USA jrb@pobox.com

This work was supported in part by the Swiss NSF under contract numberPBEL2118737, MURI SWARMS project W911NF-05-1-0219, NSF IIS-0426838 and EFRI 0735953 Intel and Boeing. We are grateful for thissupport.

Fig. 1. A team of iRobot Create robots equipped with IEEE 802.11bwireless nodes running MIT Proto is deployed to create an ad-hoc networkthat covers an area with network access.

local communication to full information on the environment,global localization, and complete knowledge of the positionof each member in the swarm.

Our long-term goal is to precisely quantify the perfor-mance benefits as we move across this hierarchy towardincreasingly more capable robots. In this paper, we focuson studying the lowest level of the hierarchy, that is a groupof robots that can move in space, avoids obstacles, and canestimate the topology of the network within the range ofits wireless communication device. While such a minimalistapproach offers the advantage to allow for deployment oflarge numbers of small and cheap units, building up a systemfrom the bottom-up will also help us to construct more robustalgorithms that deal better with sensor and actuator noisepresent in more capable systems. More generally, designinga minimalist algorithm helps us to understand better theinformation structure of a robot task, that is the informationand ressources which are necessarily needed by a robotsystem to accomplish a task.

A. Contribution of this paper

We develop a minimalist distributed algorithm that pro-vides a scalable and robust solution to maximizing the cov-erage area of a wireless mesh network solely by relying onthe number of other robots within communication range. Wecompare its performance to a hypothetical optimal solution.Convergence of the algorithm as a function of differentparameter choices is analyzed using kinematic simulationand a probabilistic model. The results presented in thispaper establish a baseline for the coverage performancethat can be achieved with a robotic platform in its most

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basic configuration, and will help us to analyze benefits andcosts of more sophisticated algorithms requiring additionalcapabilities. Finally, we present the development of a low-cost, mobile networking robotic platform running MIT Proto,an open-source programming language and virtual machineaimed at swarm robotic applications, which is used forvalidating the proposed algorithm in an indoor mesh networkdeployment scenario.

B. Related work

Algorithms for deploying large teams of agents can bewell classified using the requirements they pose on the indi-vidual robotic nodes. For instance, some algorithms requirenoise-less, global localization and reliable, global informa-tion exchange among the nodes. Assuming perfect globallocalization and reliable information exchange allows forleveraging tools from graph and control theory for controlleranalysis and design (see also the special issue on networkedcontrol systems [1]). Particularly noteworthy with respectto the case study considered in this paper are distributedcontrol approaches which exploit global metrics of graphconnectivity for deriving local control laws [2].

Another class of algorithms are fully decentralized andreduce the requirements on the individual robotic node toprovide the relative positions of its immediate neighbors andlocal information exchange [3]–[6].

Assumptions on the individual robotic platform are furtherrelaxed by approaches that solely rely on the signal strengthinformation between nodes [7], [8]. Only few contributionshave adressed the network coverage problem using minimal-ist robotic nodes. For instance, [9] proposes an algorithmand analysis for coalescence of a team of robots to a staticgateway based on random walk. As robots never resumemotion once they stopped close to the gateway, this workis a special case of the algorithm and analysis presented inthis paper.

II. EXPERIMENTAL SETUP

We study the proposed algorithm at four levels of abstrac-tion: a hardware implemention (Figure 1), a dynamical modelimplemented in the Proto simulator (Figure 2), a kinematicmodel implemented in Matlab, and an analytical probabilisticmodel. Experiments are conducted in the GPS-denied groundfloor of the MIT Stata Center.

A. Hardware

For computation and networking we use an Atherosradio-on-a-chip MIPS platform running OpenWRT 1 Linux(180MHz, 32MB RAM, 8MB Flash). The transmissionstrength of the radio interface has been artificially limitedto 5 dBm. For locomotion we are using a iRobot Createdifferential wheel robot (diameter around 30cm) that pro-vides coarse odometry, bumpers and cliff sensors. The MIPSCPU communicates with the Create via a serial interfacein intervals of 50ms. The current retail cost of the overall

1http://www.openwrt.org

system, including NiMH battery and GPS is below $200 andhas potential for further miniaturization.

B. Proto

MIT Proto2 [10] is an open-source language and toolkitthat makes it easy to write complex programs for spatialcomputers. A spatial computer is a collection of devices dis-tributed to fill space, where the difficulty of communicatingbetween devices is strongly dependent on their geometricdistance. Examples include sensor networks, robotic swarms,cells during morphogenesis, FPGAs, ad-hoc wireless sys-tems, biofilms, and distributed control systems. Proto is a lan-guage we have developed for programming spatial computersusing a continuous space abstraction. Rather than to describethe behavior of individual devices, the programmer viewsthe space filled by the devices as an amorphous medium—aregion of continuous space with a computing device at everypoint—and describes the behavior of regions of space. Theseprograms are automatically transformed into local actionsthat are executed approximately by the actual network ofdevices. When the program obeys the abstraction, these localactions reliably produce an approximation of the desiredaggregate behavior. MIT Proto is our implementation ofProto, comprising a compiler, cross-platform virtual machineincluding a simulator, code libraries, and tutorial material. Ascreenshot of the Proto simulator is shown in Figure 2.

For this paper, we implemented a virtual machine onthe MIPS platform that is able to execute Proto bytecode.Common to all platforms is the implementation of primitivesfor sending and receiving raw packets over the (platformspecific) radio link. On a robotic platform, Proto requiresadditional opcodes for driving the robot given a vector aswell as reading its sensor values that are then available asProto expressions. The Proto simulator is a virtual machinerunning on Linux and implements the most commonly usedsensors and actuators used across the supported platforms.

C. Networking

The Atheros wireless card is configured to provide two vir-tual devices that both operate on the same physical channel.One device is used by Proto in monitor mode to exchangeconnectivity information, including the number of hops to thegateway, with its neighbors. A second device is configuredin ad-hoc mode and associates with neighboring nodes. Thisdevice is used by an implementation of the OLSR algorithm(OLSRd3) to provide TCP/IP routing for the resulting meshnetwork.

III. PROBLEM STATEMENT AND PERFORMANCE

METRICS

We are interested in a network that both maximizesthe area covered by the wireless signal as well as beingsufficiently dense in order to be robust to unreliable linksand failure of individual nodes, while being connected to astatic gateway node through a finite number of hops. We are

2http://groups.csail.mit.edu/stpg/proto.html3http://www.olsr.org

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Fig. 2. 10 networked mobile robots in the Proto 3D simulator. The robotproviding the network gateway is highlighted in the center (blue disc), linesbetween robots show radio connectivity, and circles around the robots depictcommunication range.

thus interested in the sparsest possible deployment, whichstill satisfies a certain minimal number of neighboring nodesfor maintaining robustness.

We can summarize these requirements by the followingconstraint optimization problem

maxP

C(P ) (1)

s.t. ! ! "Ni" ! ", #i $ [1; n0]

ig < %, &g $ G

where C(P ) is the effective area being covered given thepositions of the robots P : R2n0 of all n0 robots. C(P ) isgiven by the union of area covered by all robots

C(P ) =!

i![1;n0]

Ai(pi) (2)

with Ai the coverage area of robot i at its position p i $ Pand Ni is the set of robots in communcation range of nodei, with "Ni" its cardinality. The set G comprises a set ofgateway nodes to which a route must exist and ig denotesthe minimal distance from robot i to robot g in hops.

A. Optimal Performance

The theoretical optimal coverage is bounded by A in0. Inthis case, however, robots are not connected, as robot-to-robot distance must not exceed R (assuming a disc-shapedcommunication model). In case of a topology with at leastone and not more than two links per robot, which allowsmaximal spread, each robot’s coverage area A i is reduced

by 13#R2'

"3

4 R2 (see also the illustration in Figure 3, left),and the effective coverage area is given by

!

i![1;n0]

Ai ( Ain0 ' 2n0

"

1

3#R2 '

)3

4R2

#

(3)

A near-optimal distribution which coverage comes closeto (3) is illustrated in Figure 3, middle. When the topology

!120o

"3

2 R2

Fig. 3. Relation of effective area coverage and individual coverage for2-link topologies (middle and left) and a 3-link topology (right).

requires at least 3 links per node, overlap between nodesincreases and effective coverage is even smaller, which isillustrated in Figure 3, right.

IV. AN ALGORITHM FOR DEPLOYMENT AND

OPTIMIZATION OF A WIRELESS AD-HOC NETWORK

In this section we develop the algortihm that provides afully distributed solution to (1). Let Ni be the set of robotswithin communication range of robot i. We assume that eachrobot can estimate the number of other robots within signalrange "Ni", avoid collisions with other robots and obstacles,and that the environment is bounded. We will first describethe algorithm (Section IV-A) and then propose a kinematicmodel (Section IV-C). Convergence of the algorithm is thenproven using a probabilistic model (Section IV-D). Whereasthe kinematic model bases on physical first principles and canbe simulated in MATLAB, the probabilistic model is derivedby associating probabilities with possible state transitions ofthe individual robot and allows us to capture the populationdynamics of the whole swarm by a set of difference equa-tions.

A. Deployment Algorithm

Maximizing *Ai is equivalent to minimizing +Ai, i.e. theintersection between the coverage areas and can be obtainedby having the robots randomly walk and mutually avoideach other for a sufficient amount of time. In a boundedenvironment, such behavior eventually leads to a uniformprobability distribution for the robots in the environment.In order for satisfying the constraints of (1), we let robotsonly move if the constraints are not satisfied, i.e. the numberof neighbors of a robot is not within the window given by[!; "], or a node is not connected to a gateway. By this,the random walk acts as an unbiased, distributed searchon the configuration space. The algorithm is illustrated byan example in Figure 4 and summarized in pseudo-code(Algorithm 1). Robots move with speed vi.

B. Proto implementation

The Proto implementation of this algorithm that will laterbe executed on the robots is as follows

(mov (if (> distance-to (gateway)) threshold)(brownian)(if (< (num-nbr) alpha)

(brownian)(if (> (num-nbr) beta)(brownian)(tup 0 0)

))

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Algorithm 1: DEPLOYMENT ALGORITHM (PSEUDO

CODE)

Data: Number of robots !Ni! within range R of robot i.Result: A locally optimal deployment.while true do1

if not connected to gateway or !Ni! > ! or !Ni! < "2

thenheading = random([0; 2#[)3

speed = vi4

else5

speed=06

move(speed,heading)7

)

Proto is a purely functional language written using s-expressions very similar to Scheme. A Proto expression pro-duces a field that maps every point in space to a value. In ouralgorithm, the hop-count to the gateway (distance-to)and the number of neighbors (num-nbr) at the currentlocation of the robot are mapped into a 2-tupel that definesdirection and speed of the robot to move. If the number ofneighbors is within the range given by alpha and beta,the expression evaluates to a Null-vector ((tup 0 0)).Otherwise—or when the distance to the gateway exceeds acertain threshold (usually the maximum number of robots)—the expression evaluates to a random vector (brownian) onthe unit circle.

C. Kinematic Model

The kinematics of an individual node are given by

pi(k + 1) = pi(k) + vi

$

cos$i

sin $i

%

(4)

where pi(k) is the position of robot i at time interval k, vi

is a scalar speed, and $i is a random heading ($i $ [0; 2#[).The algorithm is defined in discrete time and one timeinterval corresponds to the update speed T of the neighborcount estimate. In order not to penetrate the boundary of theenvironment B , R2 that can be detected using the on-boardcollision sensor, we set

(pi(k + 1) ' pi(k)) |B = 0 (5)

and boundaries are avoided by choosing a new randomheading $i.

The speed vi of each robot is given by

vi =

&

0 if ! ! "Ni" ! " - ig < %, g $ G

vi otherwise(6)

which reflects the constraints from (1).

D. Analysis of Convergence

We will now show convergence of Algorithm 1 by provingthe convergence of a probabilistic model of the roboticsystem.

1

3

2

4

56 7

9

10

80

11

Fig. 4. Possible configuration during execution of the algorithm. Whitenodes are moving (an arrow indicating their random heading), whereas blackand gray nodes are static. The gray node is the static gateway node, linesdepict bi-directional one-hop connections. In this example, robots move ifthey have less than ! = 2 or more than " = 4 connections or are notconnected to the gateway node by a finite number of hops. Nodes 1–4 arenot connected to 0, whereas nodes 5 and 7 do not satisfy the connectednessconstraints given by ! and ", respectively.

Theorem 1: A collective of robots with the behavior de-scribed in Algorithm 1 will converge into a single cluster,when " > !, !, " $ Z+ and " < n0 ' 1.

Proof: For convergence of the whole team into asingle cluster, we need to show that the number of robotsconnecting to the cluster is larger than the number of robotsleaving the cluster for all time intervals k. For this we willfirst develop a probabilistic model from the microscopicperspective of an individual agent (see for instance [11],[12]), which is inspired by the Master equation in physics[11], and models the probability for a robot to be in a certainstate as a function of the other robots’s states. We will thenderive a set of macroscopic rate equations, which describethe average number of robots in a cluster. Convergence isfinally shown by considering certain special cases, whichhelp us highlight desired properties of the system.

1) Modeling Assumptions: Our analysis bases on thefollowing assumptions. The environment is bounded, i.e. arandomly moving node will eventually visit every point inthe environment with non-zero probability [13], and robotsare randomly distributed in the environment according toa uniform distribution, i.e. the likelihood for encounteringa specific robot is independent from its location in theenvironment. Although these assumptions seem to be re-strictive, they correspond to the worst-case scenario of theproposed algorithm and predictions from models based onthese assumptions can thus be understood as lower boundon the performance. Also, if we prove that robots convergefrom a random distribution, they will also converge froma centralized deployment. Using the assumption of spatialuniform distribution, we assume a constant probability p toencounter any other node in the environment during one timeinterval. The probability p is a function of the environmentsize, the sensor range, and the speed with which an individualnode is moving through the environment and can either bemeasured in the real system or calibrated as proposed in [12].

2) Microscopic Model: In the algorithm described inSection IV-A, a node can be in one of the following states:

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N0(k)

N!(k) Ni(k) N"(k)

pNc(k)

pii!1'

j=0(Nc(k) ! j)

p"N

c (k)"

pN0 (k)

pN0(k

! 1)N"(k

! 1)!

"E"(k

)"

pN0(k) pN0(k)

pN0(k ! 1)N"(k ! 1) ""E"(k)

"pN0(k ! 1)N"(k ! 1) i"E"(k)

"

Fig. 5. Markov chain describing the degree distribution dynamics basedon possible states of an individual node.

being mobile or being part of the cluster connected to theinternet via any allowed number of neighbors (given by theparameters ! and " for the lower and upper bound, respec-tively). Thus, the total number of states of an individualrobot is bounded. The probabilistic finite state machine issummarized in Figure 5, and its state transition probabilitiesare developed as follows.

Given the probability p to encounter another robot (withpn0 < 1), the probability to encounter one and only oneout of Nc objects in the cluster is given by pNc, and theprobability for a mobile node to encounter exactly i othernodes in the same time-step is given by

p(i|0) = pii#1(

j=0

(Nc(k) ' j), i $ [!; "] (7)

When a mobile node encounters i other nodes that arealready part of the cluster, it will gain i neighbors. A mobilenode joining the cluster, however, changes the status of allother robots it connects with. From the perspective of a robotwith i connections, the probability that it is hit by a searchingrobot is pN0(k) with N0(k) the number of mobile robots attime interval k. Hence, the probability for a robot with iconnections to get an additional connection during a timeinterval is given by

p(i + 1|i) = pN0(k), i $ [!; "[ (8)

In the case of a robot already having " connections, thisrobot will leave the cluster with probability

p(0|") = pN0(k) (9)

When this happens, all nodes previously connected tothe leaving node will notice a lost connection in the nexttime interval. The probability for this event to happen to anindividual node corresponds to the number of robots thatleft the cluster during the last time interval (pN0(k'1)). Asconnections are removed from all nodes in the cluster withequal probability, the probability for an individual node toloose a connection needs to be normalized by the ratio ofown connections and total connections in the cluster ( i

$E$(k) ).

Here, "E"(k) corresponds to the total number of edges in the

system and is given by

"E"(k) =)

i![!;"]

iNi(k) (10)

Hence, the probability for a robot with i connections toloose a connection is given by

p(i ' 1|i) = pN0(k ' 1)N"(k ' 1)i

"E"(k)", i $]!; "]

(11)As robots only leave the cluster when they are exceeding "connections, each time a robot leaves, " other nodes loosea connection. This is reflected by the factors N"(k ' 1)and " in the above equation. For robots that have only! connections (the lower limit of allowed connections),removing a connection will let this robot resume search.Thus,

p(0|!) = pN0(k ' 1)N"(k ' 1)l

"E"(k)" (12)

3) Macroscopic Model: Having established possible nodestates and state transition probabilities (see also Figure 5),we can derive the following set of rate equations:

The number of robots with i connections at time intervalk is given by

Ni(k + 1) = Ni(k) + p(i|0)N0(k) (13)

+ pii#1(

j=0

(Nc(k) ' j)Ni#1(k)

+ pN0(k ' 1)N"(k ' 1)i + 1

"E"(k)"Ni+1(k)

' pN0(k)Ni(k)

' pN0(k ' 1)N"(k ' 1)i

"E"(k)"Ni(k)

and is constructed by multiplying the state transition proba-bilities from and to state i (7), (8) and (11) with the numberof robots in their original state. Equations for N!, N" , andN0 can be constructed in a similar fashion.

4) Special cases: For ! = " = 0, nodes will not beable to connect to the cluster (as " = 0). For ! = 0 and" . 1, nodes will eventually join the cluster, and the rate ofconvergence will be a function of ". For low ", e.g. " = 1,the influx to N1(k) is given by pN1(k)N0(k) and the outfluxby pN0(k)N1(k) + N1(k)pN0(k ' 1) 1

$E$(k) . That is, nodeswill leave the cluster at a higher rate than joining it. Similarly,we can see, that for all ! = " = i, Ni = Nc and thus

pii#1(

j=0

(Nc(k) ' j)N0(k) < pN0(k) + pN0(k ' 1)i

"E"(k),

i $ [0; n0](14)

i.e. the inflow to the cluster is always lower than the outflow.We will now consider the case ! = 0 and " = n0'1. This

corresponds to the scenario described in [9], who considers" = %, i.e. robots never resume motion once connectedto the gateway. Our first observation is that Ni(k) is rather

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Y-a

xis

(m)

X-axis (m)0 50 1000 50 1000 50 100

0 50 1000 50 1000 50 100

0 50 1000 50 1000 50 100

0

50

100

0

50

100

0

50

100

0

50

100

0

50

100

0

50

100

0

50

100

0

50

100

0

50

100

Fig. 6. Randomly selected deployments at time interval k = 5000 fora centralized initial configuration for (! = 1, " = 3) (top row), (! =2, " = 4) (middle row), and (! = 3, " = 5) (bottom row), and robotspeed vi = 1m

T. Gray discs depict wirless range of an individual. Higher

values for ! and " lead to denser clusters, while topological optimizationleads to instable behavior.

growing due to mobile robots connecting to less connectednodes than due to mobile nodes simultaneously connectingto i nodes as

pN0(k) > pii#1(

j=0

(Nc(k) ' j) (15)

with p << 1/n0. This leads to

Ni(k) > Ni+1(k), i $ [!; " ' 1] (16)

Consequently, if "!'"" > 0, the outflow of the system willbe always smaller than the inflow

pN0(k)N"(k) + pN0(k ' 1)N"(k ' 1)N!(k) <

pN0(k))

i![!;"]

Nc!

(Nc ' i + 1)!(17)

as Nc(k) =*

i![!;"] Ni(k) and pN0(k)N"(k) > pN0(k '1)N"(k ' 1) for N"(0) = 0.

V. RESULTS

We performed a series of simulations of the kinematicmodel given by (4)–(6) for both centralized and uniforminitial deployments, as well as bounded and unboundedenvironments. In all our simulations, we use n0 = 30 robots,an arena of 100 · 100m2, and a wireless coverage area of arobot that is given by a circular disc with radius R = 10m.

Randomly selected topologies out of 100 simulations(kinematic model) at time k = 5000 for a centralized initialconfiguration are shown in Figure 6 and provide a qualitativeidea of the resulting topologies. Stable clusters involving alarge part of the population are usually reached within a fewhundred time intervals.

For quantitatively studying the performance of the pro-posed algorithm, we measure the average area being coveredby the wireless network, i.e. all robots that are connected

k

Are

aC

over

age

0 500 1000 15000

500

1000

1500

2000

2500

3000

Fig. 7. Average coverage area vs. time for centralized deployment of 30robots for (! = 2, " = 4), (! = 1, " = 5), (! = 1, " = 6), (top,middle, and bottom curve, respectively) for vi = 1m

T. 100 runs, error bars

show standard deviation. Broadening the window of allowable number ofneighbors leads to faster convergence but lower coverage.

to the gateway node by a finite number of hops, over 100runs. Figure 7 shows results for increasingly large intervalsof [!; "]. We observe that increasing the range of allowedneighbors leads to slightly faster convergence at cost ofdecreasing coverage. This observation is well reflected inthe probabilistic model as the probability to stop increaseswith the number of possible immobile states.

When exploring other intervals [!; "] we observe anincrease in area coverage for lesser connected clusters, i.e.for lower values of ! and ", as robots need to spread outmore in order to satisfy the constraints.

We then simulated random initial distributions in order totest, whether the algorithm will indeed lead to convergenceindependently of the initial conditions. Qualitative resultsfor random initial positions are depicted in Figure 8 forvarious settings of ! and ". Figure 9 shows average coverageperformance measured over 100 experiments as well as thetheoretical optimal coverage (! = 1, " = 2). We observethat random initial deployments converge slower, but tendto lead to better overall coverage for sparse topologies asrobots tend to connect to the cluster as soon as it becomesvisible and thus better exploit the wireless signal range.This is not the case for dense topologies, which a) yieldsmaller cluster with a lower probability of being encounteredand b) require the robots to come very close to the clusterbefore connecting (due to high values of !). We tested thealgorithm also in unbounded environments for centralizedand random deployments (within the 100m x 100m field asin the other experiments). Figure 10 shows box-plots of theperformance over 100 experiments recorded at time intervalk = 5000. Surprisingly, median coverage is only slightlylower in unbounded environments. These results suggest highrobustness of the proposed algorithm also in open terrain.

We also conducted experiments in an indoor environmentusing real robots. Using a transmission power of 5 dBM,

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Y-a

xis

(m)

X-axis (m)0 50 1000 50 1000 50 100

0 50 1000 50 1000 50 100

0 50 1000 50 1000 50 100

0

50

100

0

50

100

0

50

100

0

50

100

0

50

100

0

50

100

0

50

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0

50

100

0

50

100

Fig. 8. Random samples showing the resulting topology at k = 15000for random initial deployments for (! = 1, " = 3), (! = 2, " = 4),(! = 3, " = 5) (top, middle, and bottom rows, respectively).

k

Cov

erag

eA

rea

0 5000 10000 150000

1000

2000

3000

4000

5000

Fig. 9. Average coverage area vs. time for random deployment of 30robots for (! = 1, " = 3), (! = 2, " = 4), (! = 3, " = 5) (top,middle, and bottom curve pairs, respectively) for random initial distributionsand vi = 1m

T. The more connections are required for robots to rest (high

values of !), the slower the system converges. The straight line correspondsto coverage with a theoretical near-optimal topology.

the actual communication range varied between 10-20m.Figure 11 illustrates the performance of our algorithm andthe Proto virtual machine. Robots were dispersed and shutdown one after the other until we were left with two robots.As soon as the neighbor count reached the upper threshold,the two remaining robots instantaneously stopped movingand resumed motion only after the lower threshold wascrossed. Figure 12 shows sample results from a centralizeddeployment of 9 robots on the ground floor of the MIT Statacenter. In this experiment, convergence was achieved (allrobots stopped) after approx. 35 minutes. The final coveragearea was approximately 500m2. The qualitative behavior ofthe robots was very similar to that observed in simulation.

Cov

erag

eA

rea

k=5000

centralized random

1000

2000

3000

4000

5000

Fig. 10. The box-plots show the distribution of coverage at time intervalk = 5000 for experiments in bounded and unbounded environments forvi = 1m

T. Data for unbounded environments is shown on the left of each

pair and data for bounded environments to the right. The three pairs to theleft are from centralized deployments (Figures 7) and the three pairs to theright are from random deployments (Figure 9).

#of

neig

hbor

s

Robot 1

Res

t

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Fig. 11. Relation between number of robots within communication rangeand mobility for two robots. Robots have been systematically removed fromthe experiment, leading to rest of the robots when the number of remainingrobots is within the allowable threshold.

Fig. 12. Deployment of 9 robots on the ground floor of the MIT StataCenter after convergence. Numbers indicate the number of neighbors. Theexperiment lasted around 35 minutes and covers approximately 500m2. Therobot in the center (marked with a black dot) is the static gateway and allrobots were initially deployed around it. The second robot from the rightshowed a radio failure and is not counted by any other robot.

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VI. DISCUSSION

Extensive simulations show robust behavior with wellshaped topologies, in particular for centralized deployments.Results from simulations with random deployments suggestgood performance also for a real-world use case (e.g., wherea human or robotic agent [14] deploys the robots accordingto some heuristic while exploring the environment).

In general we observe that changing the transmissionpower effects more the robustness towards external distur-bances (e.g. people passing by) than the effective range.This makes the space required for conducting meaningfulexperiments rather large.

A. Probabilistic modeling

In addition to exhibiting guaranteed convergence, the dy-namics of the probabilistic model are well in line with thoseof the kinematic simulations. For instance, for ! being large,the model suggests slower convergence as the likelihood ofa mobile node connecting to the cluster with ! connectionsbecomes low. Similarly, for large "" ' !", not only thelikelihood of sucessful connection to the cluster increases butthe cluster will store more nodes, and thus increase speed ofconvergence.

A limitation of the proposed model is that measuringthe degree distribution only indirectly captures the actualperformance metric (area coverage). Also, the proposed statetransition probabilities do not reflect the actual embodiementof the cluster. For instance, robots at the boundary of thecluster will have a much higher probability of connectingto a bypassing mobile robot. As all robots are connected,however, state transitions of the boundary robots will quicklypropagate to the inside of the cluster. In order to capture thiseffect, the dynamics of the system might be better modeledby a spatial model, which describes the number of robots ihops away from the center or captures the spatial probabilitydensity function of the whole system.

VII. CONCLUSION

We studied a wireless coverage algorithm which posesminimalist requirements on the robotic hardware, namelyknowledge of the number of wireless links and bumpersensors for collision avoidance. Parameters of the algorithmare lower and upper bounds on the desired connectivity ofeach individual node and the robot speed during deployment.

We explore properties of the algorithm at four differentmodel abstraction levels. Kinematic simulation with a disc-shape communication model and a probabilistic, macroscopicmodel, which allows us to formally prove convergence.Convergence of the algorithm shows to be extremely robustand independent of the initial deployment of robots, andwhether the environment is bounded or unbounded. Althoughthe proposed algorithm never achieves the theoretical optimalcoverage, all of 100 simulations with topologies with 1to 3 links per node are within 52% and 81% of optimalperformance, and half of the simulations at 64% of theoptimum. Consequently, we observe similar performance

also in a real-world deployment (given sufficient locomotioncapabilities to explore a large enough environment).

In the future, we are interested to study how additionalressources such as odometry, global localization or increasedcommunication will affect performance. Notably, we willexplore odometry and global localization for better-than-random search and communication for locally estimating theglobal topology, which might allow for actively supportingweak connections in the graph or increase the speed ofconvergence by more directed motion. We are also interestedin considering problems in which the gateways themselvesare mobile or where constraints are extended to providetethering

ACKNOWLEDGMENTS

The authors would like to thank A. Derbakova, J. Eriksson,and A. Swaminathan for help with implementing the Protovirtual machine on the Atheros MIPS/Create platform and J.Beal for helpful discussion on the Proto language.

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