3560 ieee transactions on signal processing, vol. 60, …ymostofi/papers/tsp12.pdf · 3560 ieee...

3560 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 7, JULY 2012

Path Planning for Networked Robotic SurveillanceAlireza Ghaffarkhah, Student Member, IEEE, and Yasamin Mostofi, Member, IEEE

Abstract—In this paper, we consider a robotic surveillanceproblem where a fixed remote station deploys a team of mobilerobots to spatially explore a given workspace, detect an unknownnumber of static targets, and inform the remote station of theirfindings. We are interested in designing trajectories (local motiondecisions) for the robots that minimize the probability of targetdetection error at the remote station, while satisfying the require-ments on the connectivity of the robots to the remote station.We show how such a design is possible by co-optimization ofsensing (information gathering) and communication (informationexchange) when motion planning. We start by considering the casewhere the robots need to constantly update the remote station onthe locations of the targets as they learn about the environment.For this case, we propose a communication-constrained motionplanning approach for the robots. We next consider the case wherethe remote station only needs to be informed of the locations ofthe targets at the end of a given operation time. By buildingon our communication-constrained results, we propose a hybridmotion planning approach for this case. We consider realisticcommunication channels that experience path loss, shadowing andmultipath fading in the paper. Then, our proposed communica-tion-aware motion planning approaches evaluate the probabilityof connectivity at unvisited locations and integrate it with thesensing objectives of the robots. We mathematically characterizethe asymptotic behavior of our motion planning approaches anddiscuss the underlying tradeoffs. We finally devise strategies toincrease their robustness to multipath fading and other channelestimation uncertainties.

Index Terms—Communication-aware motion planning, mobilesensor networks, networked robotic surveillance.

I. INTRODUCTION

O VER the past few years, considerable progress has beenmade in the area of robotic/mobile sensor networks. Dif-

ferent aspects of robotic networks have been addressed in recentyears, such as control and planning of the motion [1]–[8], coop-erative data fusion and signal processing [9]–[13], and powermanagement [14]–[16].In this paper, we consider a networked robotic surveillance

operation where a number of mobile robots are deployed tosurvey an environment, for the possible presence of an unknownnumber of static (stationary) targets, and inform a remote sta-tion of their findings. We discretize the environment into sev-

Manuscript received May 27, 2011; revised October 06, 2011; acceptedMarch 16, 2012. Date of publication April 16, 2012; date of current versionJune 12, 2012. The associate editor coordinating the review of this manuscriptand approving it for publication was Dr. Kainam Thomas Wong. This workwas supported in part by ARO CTA MAST project # W911NF-08-2-0004 andNSF award # 0812338. A small portion of this paper was presented at the 2011American Control Conference (ACC’11), San Francisco, CA, July 2011.The authors are with the Cooperative Network Lab, Department of Elec-

trical and Computer Engineering, University of NewMexico, Albuquerque, NM87131 USA (e-mail: [email protected]; [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TSP.2012.2194706

Fig. 1. A schematic of the robotic surveillance operation considered in thispaper.

eral nonoverlapping cells. The cells are assumed small enough,such that there exists at most one target in each cell. The robotsdetect the targets along their trajectories, using their collectedsensory data. Each robot is equipped with an omni-directionalsensor with a limited sensing range. The sensing quality withinthe sensing range also degrades as the distance to the sensor in-creases. To inform the remote station, the robots send fixed-sizebinary vectors, referred to as target maps, to the remote station.In a target map, a one (zero), at any element, indicates that therobot has detected a target (or not) in the corresponding cellof the discretized version of the environment. Additionally, thecommunication links between the robots and the remote stationare realistic narrowband communication channels that experi-ence path loss, shadowing and multipath fading. The remotestation then fuses the target maps received from the robots, byrunning its target detection algorithm, and builds a more reli-able map of targets over the entire environment. Fig. 1 showsa schematic of the robotic surveillance operation considered inthis paper.In this paper we start with analyzing the impact of the trajec-

tories of the robots and the resulting sensing and communica-tion qualities on the probability of target detection error at theremote station. We then proceed with solving the main problemconsidered in this paper, which is stated as follows:Problem Statement: How can each robot plan its trajectory

such that it explores the environment and gathers as much infor-mation as possible regarding target locations, while maintainingthe required connectivity to the remote station?Following the terminology of [17], [18], we refer to such

a motion design as communication-aware motion planning inthis paper. This is a challenging task that requires 1) evaluatingthe probability of connectivity to the remote station at unvis-ited locations and 2) co-optimization of sensing (information

1053-587X/$31.00 © 2012 IEEE

GHAFFARKHAH AND MOSTOFI: PATH PLANNING FOR NETWORKED ROBOTIC SURVEILLANCE 3561

gathering) and communication (information exchange) throughproper trajectory design. In this paper, we develop the mathe-matical framework of communication-aware motion planningfor networked robotic surveillance. To the best of our knowl-edge, this is the first time that such a framework is designed forrobotic surveillance in realistic fading environments. Next, wediscuss the related work and explain the main contributions ofthis paper in more details.

A. Related Work

There has recently been considerable interest in networkedrobotic surveillance, exploration, coverage, field estimationand environmental monitoring. In this part, we provide a briefsummary of the state of the art in this area, as related to thispaper.In [2], the authors consider a surveillance scenario, using

a team of unmanned aerial vehicles (UAVs) and unmannedground vehicles (UGVs), for detecting and localizing an un-known number of features within a given search area. Theythen design an information-theoretic framework for coordi-nation of the UAVs and UGVs, which maximizes the mutualinformation gain for target localization. However, only sensingobjectives are considered for coordination. In the roboticexploration/coverage context, two related works are [3] and[4], where the authors propose gradient-based controllers tonavigate the robots along trajectories that provide the bestsensing coverage performance [3] or guarantee explorationof the entire environment asymptotically [4]. Spatial fieldestimation is also studied in several works, such as [5], [6]. In[5], a distributed kriged Kalman filter is used to estimate thespatio-temporal variations of a field. The author then proposesa gradient-based motion controller to find the maxima of thefield. A similar field estimation approach, based on kriging,is also considered in [6], where the authors solve a dynamicprogram to find the optimal trajectories. In the environmentalmonitoring context, the authors in [7] address the problemof adaptive exploration for an autonomous ocean monitoringsystem. Feedback control laws are then derived to coordinatethe robots along the trajectories that optimize a predefinedexploration performance metric. Designing speed controllersfor the persistent monitoring and surveillance of a time-varyingenvironment is also studied in [8].In the current literature on robotic surveillance and explo-

ration, the authors effectively consider the sensing objectives,i.e., goals that are aimed at maximizing the exploration andcoverage performance of the robots when planning the motion.However, proper communication objectives, i.e., goals that areaimed at maximizing the probability of connectivity to the re-mote station, are not taken into account [2]–[8]. Most of the cur-rent research on the motion planning of robotic networks typi-cally assumes unrealistic communication links. For instance, itis common to assume either perfect links [2] or links that areperfect within a certain radius of a robot [19], [20], a significantoversimplification of communication channels (see Fig. 2 fora real wireless channel). Recently, a number of papers startedto highlight the importance of considering realistic communi-cation links when planning the motion, in scenarios different

Fig. 2. Underlying dynamics of the received signal power in dB (the summa-tion of the TX and channel powers in dB) across a route in the ECE building ofthe University of New Mexico. shows the distance to the remote station.

from the one considered in this paper. For instance, in our pre-vious work in [17], [18], [21], we considered remote trackingof a moving target by communicating over realistic communi-cation links. Another example is the work of [22], where theauthors co-design the communication and motion policies suchas transmissions rate, motion speed, and stop times, in order tomaximize the amount of information a robots sends to a remotestation as it travels along a predefined trajectory and under en-ergy and time constraints. In this paper, we propose a commu-nication-aware motion planning framework for robotic surveil-lance. Our framework enables robust networked operation inrealistic communication settings. We next summarize the maincontributions of the paper.

B. Statement of Contribution

The main contributions of this paper are summarized asfollows:• The first key contribution of this paper is introducing acommunication-aware motion planning framework forrobotic surveillance. The proposed framework consistsof two decentralized switching approaches to satisfy therequirements on the connectivity of the robots to theremote station. Our communication-constrained approachplans the motion of each robot such that it explores theworkspace while maximizing its probability of connec-tivity to the remote station during the entire operation.This approach is appropriate for the case where the re-mote station needs to be constantly informed of the mostupdated map of the targets, which puts a constraint on themotion of the robots to constantly maintain their connec-tivity. Constant connectivity, however, is not required ifthe mission is such that the remote station only needs tobe informed of the map of the targets at the end of a givenoperation time. In this case, the robots can explore theenvironment with less connectivity constraints, providedthat they get connected to and inform the remote stationat the end of the given operation time. Our hybrid motionplanning approach is then appropriate for this case. Thisapproach builds on our communication-constrained one


and allows the robots to explore the area more extensivelythan the communication-constrained approach, whilemaximizing their probability of connectivity at the end ofthe operation. Both approaches make use of a probabilisticchannel assessment framework to predict the path lossand shadowing components of the channel at unvisitedlocations, based on a small number of channel samplesthat are collected online or a priori [17], [23], [24]. Aconsiderable part of this paper is then dedicated to de-signing these two switching approaches, mathematicallycharacterizing their asymptotic behaviors, and discussingtheir underlying tradeoffs.

• Another important contribution of this paper is proposingstrategies to increase the robustness of the proposedcommunication-constrained and hybrid approaches tomultipath fading. Such strategies are specially desiredsince multipath fading cannot be predicted efficientlyusing our channel assessment framework and, therefore,is a source of uncertainty.1

It should be noted that in this paper, we are not concernedwith the interference among the robots. We assume that thenumber of robots is small enough, with respect to the availableresources, such that each mobile robot can have a pre-assignedslot for communication to the remote station. Furthermore, wedo not consider coordination among the robots, when motionplanning. Instead, each robot optimizes its motion locally, tomaintain proper connectivity to the remote station, while ex-ploring the area efficiently. Similarly, we are not concerned withobstacle avoidance. However, our framework can be extendedto include obstacle avoidance, as we have considered such casesin our previous work [17], [26].The rest of the paper is organized as follows. In Section II,

we describe our sensing and communication models and brieflysummarize the probabilistic multi-scale modeling of a channel.In Section III, we mathematically characterize target detectionquality of both mobile robots and the remote station. This formsthe base of our analysis for Section IV, where we introduceour communication-constrained and hybrid motion planningapproaches. The performance of these approaches is analyzedmathematically in Section V. We present our simulation resultsin Section VI, followed by conclusions in Section VII.

II. SYSTEM MODEL

Consider a closed and convex workspace , whichcontains an unknown number of fixed targets that need to bedetected. Let denote a partition (or tessellation) ofinto nonoverlapping subsets, such that and

for . We refer to each as a cell inthis paper. We assume small enough cells such that there existsat most one target in each cell. Furthermore, we assume thatthe events, corresponding to the presence of a target in differentcells, are independent.The remote station deploys robots to detect the targets in. Each robot , for , uses a local detection al-

1Multipath fading typically gets uncorrelated over small distances [25]. Thus,it cannot be predicted at unvisited locations based on a few spatial samples ofthe channel.

gorithm to fuse its gathered sensory data at any time and up-date its binary target map. The target map of the th robot attime refers to a binary vector , inwhich (or ) indicates that the th robothas detected a target (or not) in the th cell, based on its obser-vations up to time . The robots then send their most updatedmaps to the remote station, over realistic wireless communica-tion links, that experience path loss, shadowing and multipathfading [25]. The remote station fuses the most updated targetmaps received from the robots and builds its more reliable targetmap . The goal is for the remote sta-tion to obtain an accurate assessment of all the cells that containthe targets, using the information received from the robots asthey move along their trajectories.2

The trajectories of the robots affect both their sensing/explo-ration and communication link qualities, impacting the overallperformance at the remote station. While some motion trajecto-ries could result in the best sensing quality, they may not satisfythe constraints on connectivity of the robots to the remote sta-tion. Similarly, the trajectories that solely optimize connectivitymay result in poor sensing/exploration quality and a resultinghigh probability of target detection error. Thus, the desired tra-jectories are the ones that combine sensing and communicationobjectives to satisfy the constraints on connectivity of the robotsto the remote station, while improving the sensing quality of therobots, as we mathematically characterize in this paper. Basedon the requirement on the connectivity of the robots to the re-mote station, we propose two switching motion planning ap-proaches, called communication-constrained and hybrid, to im-prove the performance of the remote station, in the presence ofrealistic fading channels. These two approaches are explainedin details in Section IV.

A. Sensing and Dynamical Models of the Robots

Assuming small enough cells, each cell can be effectivelyrepresented by a single position . We assume omni-di-rectional onboard sensors such that a target in the th cell canbe sensed by the th robot if , wheredenotes the position of the th robot at time and isits sensing radius. The time-varying set

, referred to as the footprint of theth robot, then contains the indices of all the cells that are in thesensing range of the th robot at time .Let hypothesis refer to the case that there exists

a target (there is no target) in a cell. Also, let representthe observation of the th robot of the th cell at time , when

. Under hypotheses and , the distribution ofis given by two probability density functions

and , respectively. Note that andare time-varying functions that depend on the

positions of the th robot and the th cell. A well-knownexample is the Gaussian observation model:under , and under , where

2Note that in case the distribution of the targets is known to be very sparse,instead of sending its whole target map, each robot can only send the differencebetween its current map and the one that was last received by the remote station.The proposed framework of this paper can be used in this case too, with minormodifications.


is a zero-mean white Gaussian noise representing the effectof sensing error.3 The variance of is then given by

, where is a nondecreasingfunction of its argument [3], [4].4

As for the dynamics of the robots, we consider holonomicrobots [27] with the following first order dynamics:5

, where is themotion control input of the th robotat time . We assume , where denotesthe maximum step size of the th robot. We furthermore assumethat the robots are small enough, compared to the dimension ofthe workspace, such that each robot can be considered a pointin .

B. The Communication Model and ProbabilisticCharacterization of Wireless Links

Each robot updates the remote station on its current targetmap, at a number of points along its trajectory. In case of real-istic communication settings, the remote station receives a cor-rupted version of the transmitted target map. Let de-note the instantaneous received signal-to-noise ratio (SNR) inthe transmission from the th robot to the remote station at time.We have , where represents thetransmission (TX) power of the th robot at time , isthe channel power in the transmission from the th robot to theremote station at time , is the channel bandwidth and isthe power spectral density (PSD) of the receiver thermal noise[25]. In this paper, we consider a packet-dropping receiver atthe remote station, where a received packet from the th robot iskept if , for a fixed threshold , and isdropped otherwise. Then, at any time , we refer to the th robotas connected (disconnected) if

. Note that in practice, the receiver drops the packetsbased on the quality of decoding. However, the experimentalresults of [28] suggest that this is equivalent to having a re-ceived SNR threshold (for the case that additive receiver noiseis considered), which is the model we use in this paper. To facil-itate the mathematical derivations, we furthermore assume that

is large-enough such that the packets that are kept atthe remote station can be assumed to be error-free.Assumption 1: The packet-dropping threshold is

large enough, or equivalently the performance of the decodingalgorithm at the remote station is good enough, such that thepackets that are kept at the remote station can be considerederror-free.As shown in the communication literature [25], can be

probabilistically modeled as a multiscale nonstationary randomprocess, with three major dynamics: path loss, shadowing(shadow fading) and multipath fading. Let denote thechannel power in the transmission from a robot at position

to the remote station, such that .We then have the following characterization for (in

3Note that under , instead of the observation of the robot can be anynon-zero real value.4Our proposed framework is general in terms of the sensing model and does

not presume the validity of the Gaussian model. This model is, however, usedextensively in our simulation results.5By a holonomic robot, we refer to a robot whose degrees of freedom are all

controllable, i.e., it can be controlled to move in any direction by changing itscontrol input [27].

dB), using a 2D nonstationary random field model thatcharacterizes all the three dynamics of the channel [25]:

,where , is the Euclidean dis-tance from to the remote station, and arepath loss parameters and and are independentrandom variables, representing the effects of shadowing andmultipath fading in dB, respectively. In this model, is aconstant that depends on the antenna characteristics and isthe path loss exponent [25]. The multipath fading term, in thispaper, denotes the impact of multipath fading after normaliza-tion in the linear domain and subtraction of its average in thedB domain. The distributions of and , as wellas their spatial correlations, are typically given by empiricalchannel models. For instance, a Gaussian distribution, withan exponential correlation, is a good fit for the distribution of

in dB domain. Nakagami, Rician, Rayleigh and log-normal distributions are also proven to match the distributionof in the linear domain. For more details on wirelesschannel modeling, see [17] and [23]–[25].6 A real wirelesschannel, measured across a sample route in our basement, isalso shown in Fig. 2, where the three dynamics of the channelare marked.

III. MULTIROBOT SURVEILLANCE IN THE PRESENCE OFFADING CHANNELS

In order to find the optimum communication-aware motiondecisions and characterize the underlying sensing and commu-nication tradeoffs, we need to first derive expressions for thedetection performance of a robot and the remote station, at anytime instant, as a function of channel and sensing qualities. Inthis section, we design target detection algorithms, for both therobots and the remote station, and analyze their performance.In Section III-A, a sequential detection algorithm is proposedfor the mobile robots. A target detection algorithm for the re-mote station is then introduced in Section III-B, where the exactprobability of error at the remote station, as well as its Chernoffbound, are characterized.

A. Optimal Sequential Detection at the Robots

Let denote the prior probability that a targetexists in the th cell, in the absence of any observation. Also, let

be the set of all the time instants,up to and including time , when the th cell has been observedby the th robot. The th robot uses a maximum a posteriori(MAP) test to decide whether there exists a target in the th cell,based on its set of observations . Wehave , if , and ,otherwise, where and are the a pos-teriori probabilities of having a target or not in the th cell [29].In the Bayesian paradigm,

and , whereand . Assuming i.i.d. observations, we cansee that the MAP test is equivalent to the following likeli-hood ratio test (LRT) [29]: ,

6We will discuss our assumed distributions for these variables in SectionIV-A.


where is the likelihood ratio usingobservation . This LRT can also be implemented in a se-quential fashion as follows: , where

denotes the updated posteriorof the th robot, regarding the presence of a target in the thcell, using its observations up to time instant . Note thatwe set for . Also, in case ,we set , which results in . Finally,if the th cell has not been observed by the th robot for allthe time instants , i.e., , the decision regardingthe presence of a target is made solely based on the value ofthe initial prior, i.e., if andotherwise. More details on sequential likelihood ratio testingcan be found in [29].The performance of the local detectors of the robots is char-

acterized by their detection, false-alarm and the correspondingerror probabilities at each time instant. Let , , and

denote the detection, false-alarm and error probabilitiesof the th robot, for detection of a target in the th cell at time. We have ,

and. These probabilities can be

characterized using the sensing model of the robot. Note thatsince and depend on the position ofthe th robot at time , the probabilities , , and asa direct result , will be functions of the whole trajectoryof the th robot from the beginning up to time . To showthis, consider the Gaussian observation model of Section II-A,when (no initial prior on the positions of the tar-gets). It is easy to confirm that in this case

, whereis the Q-function (the tail probability of the Gaussian distribu-tion) [1], [29]. This example shows how the target detectionperformance of each robot can be a function of its entire tra-jectory. Finally, for the special case of , we have

, as expected.

B. Optimal Detection at the Remote Station

Define the binary variable as follows:

otherwise. Also, let denote the last time

instant that the th robot was connected to the remote station,up to time : .Note that since , we define to indicatethe case where the th robot has not been yet connectedto the remote station up to time , i.e., for

. At any time instant, the remote station fusesthe last received decisions of all the robots (if available),regarding the presence of a target in a cell. Let us define

as theset of the robots that have been connected to the remotestation at least once up to time and visited the thcell at least once before their last connection to the remotestation. Assuming independent received observations fromthe robots, we then have the following LRT at the remote

station:7 , wheredenotes the LR corresponding to the last received obser-vation from the th robot and can be easily shown to be:

, using the assumption that

the packets that are kept at the remote station are error-free.Note that in case , we set . Then

if and otherwise.Detection, false-alarm and error probabilities at the remote

station, as functions of target detection performance of therobots and their channel to the remote station, are also impor-tant to characterize. Let

,and denote thedetection, false-alarm and error probabilities at the remotestation. Next we show how to characterize these quantities.1) Mathematical Characterization of the Performance at the

Remote Station: Let us define, with the size of . This

set contains all the possible binary decisions of the robots of theth cell, up to the last time they were connected to the remotestation (if , then the th robot will not participate inthe decision making process at the remote station regarding theth cell). By calculating the probability of occurrence of eachmember of , under the both hypotheses and , andconsidering the cases where , wehave

(1)

where , is the Heaviside step func-

tion. The detection error probability, , is then calcu-lated as . Also,similar to optimal detection at the robots, if ,we have . Note that forthe special case where (no initial prior) and

7We implicitly assume that the remote station is aware of the sensing modeland current positions of the robots and can calculate their corresponding detec-tion and false-alarm probabilities.


,8 the detection error probability takesa more simplified form. By settingand in (1), we conclude that

and, therefore,

. The derived expressions for

the performance at the remote station are useful for analysis.However, their computational complexity may deem these in-feasible for real-time motion planning applications. Therefore,we next characterize the Chernoff bound [29] on the probabilityof error at the remote station, which can be calculated moreefficiently and is more suitable for motion planning scenariosof Section IV. We will then devise motion planning approachesthat are based on minimizing this upper bound in the rest of thepaper.2) Chernoff Bound on the Probability of Error at the Remote

Station: With the assumption of independent received observa-tions from the robots, the probability of detection error at theremote station is upper bounded by its Chernoff bound, as ex-plained in details in [29]: , where

(2)

Finding the optimum in (2) is not easy in general. However,for the special case of and ,the optimum exponent is , as proved by the followinglemma:9

Lemma 1: Assume for , where. Then,

.

Proof: Define. Then, the optimum is the one that minimizes for

. After some straightforward calculations, We have

(3)

where . It can be seen that for ,

for and for ,which completes the proof.Using Lemma 1, the Chernoff bound, for the case of(no initial prior) and , is given

by . Finally, in

8This is the case for several realistic sensing models, such as the Gaussianobservation model introduced in Section II-A.9In case is used, the Chernoff bound is called the Bhattacharyya

bound [29].

case , the Chernoff bound on the probability of errorwill be ,

which is the same as what we found previously for the proba-bility of error in this case.

IV. MOTION PLANNING AND ROBUSTNESS STRATEGIES FORMINIMIZING THE DETECTION ERROR PROBABILITY AT THE

REMOTE STATION

Based on the explanations of the previous section, a commu-nication-aware surveillance problem in a robotic network canbe stated as follows: Given a limited operation time, , lim-ited average transmission powers for the robots, , for

, and their dynamical models, find the positions of therobots, , as well as their instantaneous TX powers, ,for , over the entire time interval ,such that one of the following holds:(i) is minimized, while maximizingthe probability of connectivity of all the robots during theentire operation, i.e., , for and overthe entire time interval .

(ii) is minimized, while maximizing theprobability of connectivity of the robots at the end of theoperation, i.e., , for .

As explained before, problem (i) is applicable to the casewhere the remote station requires a constant update on the targetpositions. Then, the robots are required to maximize their proba-bility of connectivity to the remote station during the entire op-eration and improve their exploration performance within thisconnectivity constraint. In problem (ii), on the other hand, max-imizing the probability of connectivity during the entire oper-ation is not a goal and the probability of error at the end ofthe operation is the only performance measure. In this case,the robots can freely explore the environment, provided thattheir probability of connectivity, at the end of the operation, ismaximized.Solving problems (i) and (ii) is considerably challenging,

without any approximation. Furthermore, the distribution ofthe estimated channel at unvisited locations, which is used tocalculate , is time-varying. This makes solving theproblem even more challenging. Therefore, more efficient butsub-optimal approaches are desired.In this section, we propose a communication-aware motion

planning framework for sub-optimally solving problems (i) and(ii).10 The proposed framework consists of two switching ap-proaches: communication-constrained approach for problem (i)and hybrid approach for problem (ii). This framework can ac-count for the time-varying distribution of channel variations,using a probabilistic assessment of wireless channels. We alsointroduce robustness strategies, such as TX power adaptation,

10Our proposed approaches are sub-optimal for two reasons. First, instead ofdirectly minimizing the probability of detection error, we minimize its Chernoffbound. Second, we devise greedy motion strategies that minimize the detectionerror probability of the next step, as opposed to planning the whole trajectory.


motion jittering and robustness margins, to increase the robust-ness of the proposed framework to multipath fading and channelassessment errors.11

Next, we start by briefly summarizing our previously pro-posed channel assessment framework [17], [23], [24]. We thenuse this framework and devise our communication-constrainedand hybrid motion planning approaches.

A. Probabilistic Assessment of the Spatial Variations of aWireless Channel

The probabilistic channel assessment framework of [17],[23], [24] allows each robot to probabilistically assess thechannel at unvisited locations, based on a small number ofchannel measurements in the same environment. It also pro-vides a mathematical characterization of the channel assess-ment uncertainty at each robot (how much each robot can trustits channel assessment). Note that the multipath fading com-ponent of the channel is a source of uncertainty in this frame-work, as it is generally unpredictable, based on sparse spatialsamples of the channel. As such, in Section IV-D, we proposestrategies to increase the robustness to multipath fading andimprove the overall performance at the remote station. Fig. 3summarizes our proposed strategy for dealing with realisticcommunication channels, which consists of 1) probabilisticassessment of the shadowing and path loss components and2) strategies to increase robustness to multipath fading andother modeling errors. We will discuss the robustness strate-gies in Section IV-D.Let , for , de-

note the (time-varying) set of the positions correspondingto the small number of channel power measurements avail-able to the th robot at time instant . These measure-ments could be gathered by the th robot along its trajec-tory, could be available through an offline survey of thechannel, or could be a combination of the two.12 The stackedvector of the received channel power measurements (indB), available to the th robot, can then be expressed by

, where ,denotes the -dimensional vector of all ones,

,

, for , is the Euclidean distance from

to the remote station, is the vector of

path loss parameters,

and . Based on the

11Note that in this paper, we are not concerned with energy conservation.Instead, our goal is to schedule a given limited average TX power for commu-nication. In case energy conservation is a goal, the current framework can beextended to consider both communication and motion powers and optimize thetrajectories to minimize the total energy consumption.12We assume symmetric uplink and downlink channels, i.e., the channel from

a robot to the remote station is taken identical to the one from the remote sta-tion to the robot. This is the case, for instance, if both transmissions occur in thesame frequency band and are separated using Time Division Duplexing (TDD).If uplink and downlink use different frequency bands, then we assume that afew uplink channel measurements are sent back to the robot, using a feedbackchannel, as is common in the communication literature [25]. These uplink mea-surements then form the basis of uplink channel assessment.

Fig. 3. Illustration of the proposed probabilistic channel assessment frameworkand robustness techniques.

commonly used lognormal distribution for shadow fadingand its reported exponential spatial correlation [17], [23],[24], we take to be a zero-mean Gaussian randomvector with the covariance matrix , where

, for , ,

with , and denoting the variance of the shadow fadingcomponent in dB and its decorrelation distance, respectively.As for multipath fading, distributions such as Rayleigh, Rician,Nakagami, and lognormal are shown to match its probabilitydensity function (pdf) (in non-dB domain), depending on theenvironment [17], [23], [24]. In this section, we assume log-normal multipath fading and a resulting Gaussian distributionfor . We also take the elements of to be uncorrelated.As a result, we take to be a zero-mean Gaussian randomvector with the covariance , where is the power ofmultipath fading component (in dB) and is the -di-mensional identity matrix. Note that our channel assessmentframework does not attempt to predict the multipath fadingcomponent and only estimates its variance. Thus, multipathfading is a source of uncertainty for channel assessment. Fora detailed discussion on validation of this model using realchannel measurement, the readers are referred to [25], [30].As we proved in [17], [23], [24], based on the measurements

available to the th robot at time and conditioned on thechannel parameters, the assessment of the channel at an unvis-ited position is given by a Gaussian distributionwith mean and vari-ance . Wethen haveand , where

,

and .

Note that to calculate and , the th robotalso needs to estimate the underlying channel parameters. We,however, skip the details of the estimation of the underlyingparameters and refer the readers to our previous work in[17], [23], [24], where we explain the estimation of theunderlying channel parameters and analyze the performanceof our channel assessment framework using real and simulatedchannel measurements.


B. Communication-Constrained Motion Planning

Consider planning the motion of the robots in order toconstantly update the remote station on the positions of thetargets, while improving the exploration performance of eachindividual robot. Assume , for , and

, for and . Byusing the optimal , we get the following for the logof Chernoff bound of (2) at time :

.Let denote the Kullback-Leibler (KL) distance[29] between two discrete distributions Bern(0.5) and

, where represents the Bernoulli dis-tribution with the success probability of . We have

. Using the definition ofand , we have

, wherewe set for (the case where the throbot has not yet been connected to the remote station up totime ). The average of , conditioned on thechannel values up to time , is then given by the following:

(4)

where . Using the probabilisticchannel assessment framework of the previous section, the throbot can assess the probability of being connected to the remotestation at time as follows:

(5)

where denotes the (possibly time-varying)channel power threshold of the th robot at time (indB), which depends on the TX power adaptation strategythat we will introduce in Section IV-D-1. We then proposethe following (decentralized) motion planning optimizationframework for the th robot at time , based on minimizing itscontribution to the average detection error probability of thenext time step at the remote station:

(6)

where is a constant (is not a func-

tion of ) and ,, for a positive

threshold . Note that larger will result in a moreconservative (in terms of connectivity maintenance) but lesssensing-effective strategy. In other words, by increasing ,the set will become smaller. Since the motion planner in

(6) is forced to choose the next optimal position in , themotion of the robots will be limited to a smaller area, whichdegrades the exploration performance. However, largerresults in more robustness to multipath fading due to forcingthe robots to move in the areas with better estimated channelquality. Also, note that the sensing part of (6) is always nonneg-ative as , for every .Equation (6) is a key equation that shows: 1) the separation of

communication and sensing objectives for the purpose of nav-igation; 2) that solely from a sensing perspective, each robotshould minimize its surveillance uncertainty at the next step bymaximizing its KL distance to Bern(0.5); 3) that solely froma communication perspective, each robot should maximize theprobability of being connected to the remote station at the nextstep; and 4) that the optimal trajectory is the one that providesthe right balance between these objectives. In other words, theoptimal communication-constrained navigation strategy is theone that minimizes the sensing uncertainty while maximizing theprobability of being connected to the remote station at the nextstep, based on what the robot can assess about the connectedregions.One drawback of the localized motion planning strategy of

(6) is its feasibility issues. Based on the available knowledge onchannel link qualities, there may be situations where(for example the case where the robot starts in a disconnectedarea far from the remote station). In such cases, the th robotcan get stuck in a region with a poor link quality. Furthermore,even if the problem does not become infeasible, if istoo small, the proposed approach will be less robust to channelassessment errors. In order to avoid such undesired conditions,next we propose a switching strategy to navigate each robotto unvisited locations with good link qualities, in case the linkquality is poor in the local region around the current position ofthe robot.1) A Switching Strategy for Avoiding Undesired Local Ex-

trema: Let , for , denote the index of the cell thatcontains , i.e., the unique such that . Also, let us define

and , where is asdefined before and is another positive threshold. The set

denotes the set of points in the workspace where the prob-ability of connectivity to the remote station, based on the cur-rent assessment of the channel at the th robot, is larger than

. The set also contains the unexplored (or poorlyexplored) points, where the probability of detection at the throbot is less than . The idea behind the switching ap-proach is to add another mode of operation, referred to as thesensing-aware connection seeking mode, to navigate the throbot to the closest point in whenever

, for a small positive . In other words, it navigates theth robot towards the closest unexplored (or poorly explored)region, where the probability of connection (using the currentassessment of the channel at the robot) is good enough. Then,as soon as we have , we switch back to themotion planning strategy of (6), which we refer to as commu-nication-aware exploration mode. The control input of the th


Fig. 4. Illustration of the proposed communication-constrained (left) and hy-brid (right) motion planning approaches.

robot in the sensing-aware connection seeking mode can thenbe found as follows:

(7)

Using the control strategy of (7), whenever, the th robot is navigated directly towards ,

which is the closest point of to the current positionof the robot. Fig. 4 (left) summarizes the proposed switchingstrategy, which prevents the robot from getting stuck in unde-sired local extrema. Furthermore, it increases the robustnessto channel assessment errors (mainly caused by multipath),by introducing extra design parameters such as . InSection IV-D, we discuss more strategies for improving theperformance and increasing the robustness of the switchingapproach.

C. Hybrid Motion Planning

The Q-function in the communication part of the objectivefunction in (6) enforces the th robot to move to the positionswith a higher chance of experiencing a better channel quality. Asthe quality of channel learning gets better, becomessmaller. In the limit of perfect channel learning, the Q-functionacts as a hard limiter, enforcing the robot to only explore theconnected regions. This property of the communication-con-strained approach can be considerably useful for the applica-tions that require constant communication of the most updatedbinary maps to the remote station. However, for a limited oper-ation time and a time-invariant field, the robots may only beconcerned with the probability of detection error of the remotestation at the end of operation. In this part, we are interestedin such missions. Minimizing the probability of detection errorof the remote station at the end of operation time will resultin a different motion planning approach, with a different bal-ance between exploration and communication, which we referto as hybrid motion planning. The idea behind the hybrid mo-tion planning approach is to have each robot explore the envi-ronment extensively and communicate its binary target map atthe end of the mission (note that the remote station only uses the

last communication of each robot). This proposed approach hasfour modes of operation:• Mode 1 (sensing-aware exploration mode): In this mode,each robot optimizes its motion only based on its sensingand exploration objectives, without taking communicationconstraints into account. The purpose of this mode is toexplore the environment as much as possible, in the givenoperation time. Each robot , therefore, chooses its nextmotion decision such that its local detection probability,

, is maximized. The controller input of the throbot is then calculated using the following optimizationproblem:

(8)

where the th robot assesses its detection probability basedon its sensing model. In this mode, communication ob-jectives are not considered in local motion planning. Al-though, it is possible that the robot transmits its currenttarget map if it randomly moves to a connected spot. Thiscan increase the robustness, in case the operation was ter-minated abruptly and earlier than planned. However, fromenergy consumption perspective, it may be better if therobot does not communicate in this stage and leaves allits power for optimizing the connectivity at the end ofoperation.

• Mode 2 (local extrema avoidance mode): Using the local-ized exploration strategy of (8), each robot explores the en-vironment as much as possible. However, there might existundesirable situations where it gets stuck in an already ex-plored area, while there are still unvisited areas that can beexplored in the given operation time. Similar to the com-munication-constrained case, to avoid such undesirable sit-uations, we add a mode to navigate a robot to poorly ex-plored locations, whenever the estimated improvement ofits sensing quality is small. Using the definition ofin Section IV-B-1, the robot switches to the local extremaavoidance mode, to move towards the closest point in ,

whenever

, for a small positive and. In other words, we navigate the robot to-

wards the closest poorly explored point in the workspaceto improve its sensing performance. We then switch backto the sensing-aware exploration mode as soon as we have

. The control input of the th robot, in thismode, is also found by following the same approach of thesensing-aware connection seeking mode of the previouscase:

(9)


• Mode 3 (connection seeking mode): Once the environmentis explored extensively and the limited operation time isapproaching, each robot needs to move to positions withhigh chance of connectivity, where it can send its most up-dated target map to the remote station. Based on its mostrecent predicted channel, each robot has an estimate ofhow many steps it takes for it to move to a connected po-sition. Based on this knowledge and by considering theremaining number of operation steps, each robot can de-cide when to switch from the sensing-aware explorationmode or the local extrema avoidance mode to the con-nection seeking mode. Consider the set

, defined inSection IV-B-1. The closest point in is then found asfollows: . Using thefirst-order holonomic dynamics of the robots, theminimumnumber of steps required to get to , at time , is

. The th robot switches to theconnection seeking mode at time if 1) it is not connectedto the remote station at time and 2) ,where is a positive and small offset, which is added intothe operation as a robustness margin. Once a decision toswitch to the connection seeking mode is made, the con-trol input of each robot is found as follows:

(10)

Similar to the communication-constrained approach, as weincrease in , the motion planner becomes morerobust to the variations of multipath fading and other as-sessment uncertainties, by acting more conservatively.

• Mode 4 (communication-aware exploration mode): Oncea robot moves to a region where ,it utilizes the proposed communication-aware explorationmode of Section IV-B-1, to maintain its connectivity to theremote station till the end of operation. In this mode, therobot still senses the connected area, as described in theprevious section. However, its main goal is connectivitymaintenance. Note that in case the th robot is still dis-connected after switching to this mode (possibly due toits poor assessment of the channel in the presence of largemultipath fading), it can take advantage of a jittery move-ment around its current location, to increase its chance ofconnectivity. In the next section, we explain such strate-gies to further increase the robustness of both communica-tion-constrained and hybrid approaches.

Fig. 4 (right) demonstrates an overview of the hybrid approachwhere transition between the modes is illustrated.

D. Further Robustness to Multipath Fading and OtherChannel Assessment/Modeling Errors

In Section IV-A, we explained how our probabilistic channelassessment framework can be used by each robot to 1) learn

the shadowing and path loss components and 2) characterizethe channel assessment uncertainty. This enabled our switchingmotion control approaches in Sections IV-B and -C. As ex-plained in Fig. 3, to increase the robustness of our approachesto unpredictable multipath fading and other channel assess-ment/modeling errors, efficient strategies are required. InSections IV-B and IV-C, we already showed how to increasethe robustness, using a number of design parameters, suchas . This approach is referred to as robustness marginsin Fig. 3. In this section, we introduce two more robustnessstrategies: adaptive TX power and packet-dropping thresholdand jittering. These strategies can be combined with our com-munication-constrained or hybrid motion planners to furtherimprove the performance.1) Adaptive Transmit Power and Packet-Dropping Threshold

for Increasing the Robustness to Multipath Fading: Thethreshold in the formulation of in (5),as well as in the definitions of and , depends on theTX power adaptation strategy of the th robot. Consider thecase where the th robot is given the total average TX power of

. It can use the fixed TX power of duringits operation, independent of the channel quality. In this case,we have ,where is added to increase the robustness tomultipath fading. The idea is that by increasing the packet-drop-ping threshold, the robot is then forced to operate in regionswith higher channel power, resulting in more robustness tomultipath fading.Alternatively, each robot can adapt its TX power to channel

quality. Power adaptation strategies have been heavily exploredin the communication literature [14]–[16]. In this paper, the goalof TX power adaptation is to increase the robustness of the pro-posed motion planning algorithm by saving power at positionswith high channel qualities, with the goal of satisfying the con-nectivity requirements at positions with low channel qualities.Similar to [16], we consider the following simple TX poweradaptation strategy for each robot:

otherwise

(11)

where is the maximum TX power of theth robot and is the average of the remainingpower budget of the th robot at time instant :

. Thechannel power threshold will then be:

. Note that in thispaper, we are not concerned with minimizing the total energyconsumption. In other words, our goal is to schedule a limitedaverage TX power for communication. However, the currentframework can be extended to consider both communicationand motion powers and optimize the trajectories of the robotssuch that the total energy consumption is minimized, whilethe given networked task is accomplished. It is also worthmentioning that, in the hybrid approach, the robots may decide


not to send anything, while operating in modes 1 and 2, tosave power for the end of the operation when they need tobe connected. This strategy is particularly useful for longoperations with limited power budgets. However, if power isnot a constraint, the robots can always send their most updatedtarget maps to the remote station if they move to a connectedspot by chance. This can further increase the robustness, incase the operation terminates abruptly and earlier than planned.2) Jittery Movements for Increasing the Probability of Con-

nectivity in the Presence of Large Multipath Fading: Considerthe communication-constrained approach. This strategy guideseach robot to an area that has a high probability of connectivity.However, a specific position may or may not have the connec-tivity requirement, as our channel learning framework cannotpredict the fine variations of multipath fading and is also proneto errors. Since the robot is guided to an area that has a highprobability of connectivity (high average channel power dic-tated by the lognormal shadowing and path loss components),then small jittering can help the robot find a better locationin terms of connectivity. In case a jittery movement is added,after the proposed communication-constrained or hybridmotionplanner is executed at each step, the robot picks randompoints in a small circular region around its current location. Notethat is a design parameter that depends of the type of robotand the ease of navigation. It then moves to each point one byone, measures the channel there, and then simply chooses thepoint with the best channel quality for sending. The radius ofthe circular region for jittering is chosen small enough such thatthe path loss and shadowing parts of channel remain stationaryin the region, based on the assessment of the channel availableto each robot. Note that in case the adaptive TX power strategyof (11) is used, if none of the points satisfy the connec-tivity requirement, then nothing will be sent at these points.

V. PERFORMANCE ANALYSIS OF THE PROPOSED MOTIONPLANNING STRATEGIES

With some assumptions, it is possible to analyze the final per-formance of the proposed communication-constrained and hy-brid approaches. The following theorems summarize our keyresults.Theorem 1: Assume that: 1) the robots use the fixed TX

power strategy of , for ; 2) basedon the assessment of the channel at all the robots,

(refer to Section IV-B for the definition of ); and3) the sensing radius of each robot and the size of each cell aresmall, compared to the size of the workspace, such that some ofthe cells in cannot be sensed by any robot ,while it moves inside . Then, the average of the proba-bility of error at the remote station (averaged over the space andthe distribution of the channel, conditioned on the channel mea-surements) is lower bounded by a positive value at any time,i.e., there exists such that , forall .

Proof: For every , we have, where

the expected values are calculated over the distribu-tion of the channel, conditioned on the channel mea-surements. Assuming , for ,

and , for and

, we obtain

, where

and is the set of all the binary vec-tors of size . Note that in case .Consider the communication-constrained approach, where

and . If the sensingradius of each robot and the size of each cell are small, we canfind some cell inside and some ,such that , . This cell can be any cell inside

, which is not visited by any robot during theentire operation or is sensed for a finite number of steps beforeeach robot reaches , for . This implies thatthere exist a positive lower bound onand, by averaging over the conditional channel distribution,on . In other words, there exists

such that . Thesame result can be obtained for the more general case where

, for some , or , forsome and , using (1). This completes theproof.Theorem 2: Assume that: 1) the channel is assessed per-

fectly; 2) the robots are connected to the remote station at thebeginning of the operation; 3) the sensing radius of each robotand the size of each cell are small, compared to the size of theworkspace, such that the maximum area of the cells coveredby a robot , while moving in , is approximately equalto the area of (refer to Section IV-B for the definitionof ), for ; 4) the robots use the same fixedTX power strategy of , for ;5) , for ; and 6) the workspaceis a circular region with the remote station located at thecenter of the circle. Then, in the communication-constrainedapproach, a lower bound for ,averaged over every possible channel, is approximately

given by , where

,

and is the radius of the workspace.Proof: In case the robots start connected and the channel is

assessed perfectly, each robot only explores in the com-munication-constrained approach. If the size of the cells and thesensing radii of the robots are small, we approximately have

. In case ofperfect channel assessment, the area ofis equal to the area of disconnected region. Therefore, by aver-aging over the channel distribution, we get


for , and

.Theorem 3: Assume that the channel is assessed perfectly,

, for , and , forand . In the hybrid approach, if we select

(see Section IV-B for the definition of the ), for anarbitrary small positive , then we have the following for theasymptotic average probability of detection error at the remotestation: , when averagingis done over every possible channel.

Proof: Assume that the robots start with the sensing-awareexploration mode in the hybrid approach. Consider the throbot, for . For this robot, the exploration continues(the spatial average of its detection error probability decreases)until (see Section IV-C for the definition of thevariables). In this case, we have either , which meansthat , for , or . For thelatter, the proposed local extrema avoidance mode of the hybridapproach navigates the robot to a point in . Then, twodifferent cases may happen. If a situation wheredoes not happen, while in the local extrema avoidance mode,the robot will remain in this mode and visit every point in

until . On the other hand, if the robot switchesback to the sensing-aware exploration mode at some point, theexploration continues similar to the previous case, which alsoproves that after some steps. Therefore, we have

, for and greater than a positiveconstant. Then, assuming that the channel is assessed perfectly,we have (see the proof of Theorem 1 for the def-inition of ), for and . This impliesthat , for . By averaging overthe channel distribution, we have ,for , which completes the proof.

VI. SIMULATION RESULTS

In this section, we evaluate the performance of the proposedcommunication-aware surveillance framework using both sim-ulated and real channel measurements. Our results highlightthe underlying tradeoffs in the design space as summarized inFig. 10. For instance, they show that while the hybrid approachoutperforms the communication-constrained one in terms of thefinal probability of networked detection error, the latter has ahigher probability of constant connectivity and is therefore morerobust to the abrupt termination of the operation. Also, we fur-ther compare these two approaches with communication-un-aware strategies (strategies that consider only sensing objec-tives for planning the motion of the robots) and show the ef-fectiveness of the proposed framework in realistic fading envi-ronments.Consider a surveillance scenario where three robots are

tasked to explore a given environment for the possible pres-ence of targets. We use the Gaussian observation model ofSection II-A, where the following form is chosen for the obser-

vation error variance [3], [4]:otherwise

,

for positive constants and and a limited sensing

Fig. 5. Trajectories of three robots for communication-constrained (left) andhybrid (right) cases, with fixed TX powers. The dashed red, solid magenta anddot-dashed green lines correspond to the trajectories of the robot #1, robot #2,and robot #3, respectively. The empty boxes and the filled ones denote the initialand final positions, respectively. The location of the remote station is denotedon the top left corner of the figures. See the pdf file for more visual clarity.

radius for each robot. The communication channel be-tween the robots and the remote station is simulated usingour probabilistic channel simulator, which can simulate pathloss, shadow fading, and multipath fading, with realistic spatialcorrelations. A detailed description of this channel simulatorcan be found in [30] and [31].First we compare the trajectories of communication-con-

strained and hybrid planning approaches, for a fixed TX powercase, where the following parameters are used:for , 2, 3, , , ,

, , , , ,, , for , 2, 3,, for , 2, 3, and ,

with a Rician multipath fading. Note that Rician multipathfading was simulated to make the simulation more realistic.Furthermore, we use the following design parameters whenimplementing our communication-aware and hybrid mo-tion planners: , , ,

, and (see Section IV-Band IV-C for the definition of these parameters). For the pur-pose of channel learning, the robots use 0.01% of the totalchannel samples (64 samples in a 800 800 grid), which areassumed to be randomly collected during an initial learningphase. Then, they collect more samples as they move alongtheir trajectories. The collected samples are used to estimate theunderlying channel parameters and assess the spatial variationsof the channel at unvisited areas. Fig. 5 shows the trajectoriesof the robots for one simulated channel in both cases. Theblack regions in Fig. 5 represent the areas where the robots arenot connected to the remote station (received SNR is belowthe acceptable threshold), given . It can beseen that by using the communication-constrained approach,the robots converge to the regions where they are connected tothe remote station (if they are not connected at the beginning)and stay connected afterwards. In other words, this approachforces the robots to mostly explore the regions with better linkqualities, so that they can constantly update the remote stationand minimize its detection error probability, as we explainedin Section IV-B. The hybrid navigation strategy, on the otherhand, allows the robots to explore the environment more freely.It, however, enables the robots to be connected to the remote


Fig. 6. Impact of the adaptive TX power on connectivity regions. Time-varying connectivity regions (white areas) in communication-constrained (top row) andhybrid (bottom row) cases are shown for one of the robots of Fig. 5, at time steps , , , and (from left to right). The communicationchannel is taken to be the same as the one used in Fig. 5. Empty boxes and filled ones denote the initial and final positions, respectively.

station at the end of operation. Also, as can be seen from thefigure, some regions may be revisited by the robots severaltimes along their trajectories. Note that in both cases, the robotstake advantage of a jittery movement, in areas of radius 0.3 maround their current locations and by testing pointsat each step. In this specific example, jittery movement in thecommunication-constrained case results in 10%, 15%, and 17%increase in the percentage of time that robot #1, robot #2, androbot #3 are connected along their trajectories, respectively.Similarly, in the hybrid approach, by using the jittery movementwe have 24%, 20%, and 41% increase in the percentage of timethat robot #1, robot #2, and robot #3 are connected along theirtrajectories, respectively.Next, consider the case where the adaptive TX power strategy

of (11) is used for both communication-constrained and hy-brid approaches.13 Then, the connectivity regions will becometime-varying and different for different robots (since their TXpowers are different). One would expect that the white regions(connectivity regions), corresponding to the th robot, expandas its instantaneous average TX power, , increases. Fig. 6shows the evolution of the connectivity regions for both strate-gies, for one of the robots of Fig. 5, when the TX power adap-tation technique of (11) is used. The communication channel istaken to be the same as the one used in Fig. 5. In such an adap-tive strategy, the connectivity region of the robot expands withtime with high probability, as can be seen from the figure, fortwo reasons. First, in case of poor channel quality at time , therobot does not send anything, which results in an increase in

and an expansion in its next step connectivity region.Second, in case the robot decides to send its updated target map

13As explained before, we can also use the adaptive TX power strategy of (11)for the hybrid case too. Multiple transmissions in the hybrid case (as opposedto a single transmission at the end of the operation) will increase the robustnessto any possible abrupt termination of the task.

to the remote station using (11), it uses the minimum requiredpower for connectivity, given by , based on its mea-sured channel power at time . Note that we did not simulateany jittery movement in this case.In order to further compare the communication-constrained

and hybrid approaches, Fig. 7 (left) shows the average ofthe final detection error probability at the remote station, i.e.,

, as a function of the given operation time ,for the three robots and the same system parameters of Fig. 5.14

The averaging is done over the space and 20 different channels.In Fig. 7 (left), for every , the whole trajectory of the robot isgenerated for 20 different channels and the resulting average atthe end of the operation are then plotted. In order to simplify thescenario, TX power is not adaptive in this case and the robots,in both communication-constrained and hybrid approaches,use the constant TX power of at each step.They, however, make use of a jittery movement, at the end ofeach step, to increase their chance of connectivity to the remotestation. It can be seen that the hybrid approach outperformsthe communication-constrained one, in terms of the averageof the final detection error probability at the remote station, asexpected. The figure also shows that as , the averageprobability of error goes to zero in the hybrid case, while itreaches an error floor in the communication-constrained case.However, the communication-constrained approach providessmaller average detection error probability at the remote station,in case the operation terminates earlier than the planned time. To see this, Fig. 7 (right) shows the average of the detectionerror probability at the remote station, i.e., ,as a function of time step , averaged over space and 20different channels. We used as the operation time in

14The spatial average or spatial integration of the quantity under control is acommon choice for the performance measure in robotic surveillance and explo-ration literature [4].


Fig. 7. Probability of detection error at the remote station as a function of (left) the given operation time and (right) time step for communication-constrained and hybrid approaches. TX power is not adaptive in this case.

Fig. 8. Communication and sensing tradeoffs in a robotic surveillance scenario. The figure shows final average probability of detection error at the remote station,averaged over the space and channel distribution, as a function of , for two cases of (left) and (right).

this case. As can be seen, if the operation ends while the hybridapproach is still in the exploration mode, it results in a worseperformance, as it is designed to optimize the operation givena certain time.Fig. 8 compares the performance of three approaches, with

different levels of awareness, in terms of communication andsensing, for the same system parameters of Fig. 5. The resultsare averaged over 20 different channels. Similar to Fig. 7, TXpower is not adaptive. Also, no jittery movement is simulated inthis case, to show the underlying tradeoffs more clearly. For thesake of comparison, the results for a sensing-constrained caseare also shown. A sensing-constrained case only optimizes theexploration objective and communicates with the remote sta-tion anytime its trajectory takes it to the connected regions,as was described in modes 1 and 2 of the hybrid case. Thus,it is unaware of communication objectives. The figure showsthe average of the final detection error probability at the re-mote station, as a function of , for two cases of(left) and (right). Interesting tradeoffs can be ob-

served. Consider the left figure, where is small. In this case,as increases, the chance of connectivity becomes smallover the whole space. As such, the performance of the sensing-constrained approach degrades considerably, for large ,since it does not include communication objectives in its motionoptimization. It can be see that the communication-constrainedand hybrid cases perform almost the same and better than the

sensing-constrained approach. Also, in this case, the averageof the detection error probability in the communication-con-strained and hybrid approaches are large (compared to detectionerror probability of hybrid approach in the right figure). This isbecause the operation time is limited and the requirement onlink qualities is high. The right figure, on the other hand, showsthe performance for the case of . In this case, since thegiven operation time is longer than the left figure, the trajec-tory of the sensing-constrained approach crosses through moreconnected regions by luck, which improves its performance.For the communication-constrained case, the performance de-grades as increases since its exploration is limited toconnected regions, which are shrinking as increases.The hybrid approach, however, results in the best performancein both cases, providing the best tradeoff between sensing andcommunication, as expected.In order to show the performance of our framework with

real channel measurements, Fig. 9 shows a surveillance sce-nario, using one robot and by considering the real channelmeasurements gathered in the basement of our building. Thechannel measurements are collected through a survey of thechannel, using the onboard IEEE 802.11g WLAN card of aPioneer 3-AT robot and for a remote station (an IEEE 802.11gwireless router) located in one of the rooms in the basement.This channel samples are then used to simulate the surveillancescenario. During the surveillance operation, the channel is


Fig. 9. Performance of the proposed communication-aware surveillance framework using real channel measurements in an indoor environment (basement of theElectrical and Computer Engineering building at the University of New Mexico). From left to right, the first and second figures show the trajectory of the robotin the communication-constrained and hybrid approaches, respectively, where the true connectivity map to the remote station is superimposed on the blueprint ofthe basement. The third figure compares the resulting average of the detection error probability at the remote station, as a function of time step . The robot in thehybrid approach travels to the end of the hallway and then comes back towards the connected region. See the pdf file for more visual clarity.

assessed based on 0.1% a priori channel samples (47 samples),as well as the samples the robot collects along its trajectory. TheTX power is kept fixed and a received SNR threshold of 35 dBis chosen for packet dropping. The robot also takes advantageof a jittery movement, in areas of radius 0.15 m around itscurrent location and by testing points at each step.The sensing model is the same as the one used for the previoussimulations, except that we use in this case.The Pioneer robot is modeled by the holonomic dynamicalmodel of this paper, with . A simple obstacleavoidance strategy, similar to the one proposed in [17], is alsoused. From left to right, the first and second figures show thetrajectory of the robot in the communication-constrained andhybrid approaches, respectively, where the true connectivitymap to the remote station is superimposed on the blueprintof the basement. The third figure also compares the resultingaverage of the detection error probability at the remote station,as a function of time step .It can be seen that although multipath fading power is large,

the robot maintains its connectivity along its entire trajectoryin the communication-constrained approach, as expected. Moreprecisely, in this example it maintains its connectivity at 97.3%of the entire operation time. The hybrid approach, however,achieves a smaller average probability of detection error at theend of operation, compared to the communication-constrainedapproach. It also ensures that the robot is connected by the endof the operation.Finally, Fig. 10 summarizes the results and observations of

this paper, in terms of the level of communication and sensingawareness of a motion planning strategy and its impact on theoverall performance. The sensing-constrained approach in Fig.10 is explained previously in Fig. 8. This approach generallyresults in a poor performance, unless the operation time is verylarge (which is not typically the case).

VII. CONCLUSION

In this paper, we considered the scenario where a team of mo-bile robots are deployed by a remote station to explore a givenenvironment, detect an unknown number of static targets and in-form the remote station of their findings.We studied the problemof designing the trajectories of the robots to minimize the prob-

Fig. 10. Comparison of different motion planning approaches, based on thelevel of communication and sensing awareness and its impact on the overallperformance.

ability of target detection error at the remote station, while sat-isfying the requirements on the connectivity of the robots to theremote station. We showed how to design such trajectories byco-optimization of sensing (information gathering) and commu-nication (information exchange). Based on the requirement onthe connectivity of the robots to the remote station, we consid-ered two cases. First, we considered the case where the robotsneed to constantly update the remote station on the locations ofthe targets. For this case, we proposed our communication-con-strained motion planning approach which enables the robots toexplore the workspace while maximizing their probability ofconnectivity to the remote station during the entire operation.We proved that the overall motion optimization objective func-tion, in this case, is a multiplication of a sensing function thatmaximizes the Kullback-Leibler (KL) divergence between themaximum uncertainty state and the current one, by a communi-cation function, that maximizes the probability of connectivityto the remote station.Second, we considered the case where the remote station only

needs to be informed of the locations of the targets at the end of agiven operation time. By building on our communication-con-strained results, we proposed our hybrid motion planning ap-proach for this case. This approach plans the motion of therobots such that they explore the workspace with less connec-tivity constraint on their motion, as compared to the commu-nication-constrained approach, while maximizing their proba-bility of connectivity at the end of the operation. We mathemat-


ically characterized the asymptotic behavior of our proposed ap-proaches under certain conditions. We showed that, in terms ofthe final detection error probability at the remote station, the hy-brid approach outperforms the communication-constrained one,while the latter provides better constant connectivity and, there-fore, is more robust to the abrupt termination of the operation.We finally proposed strategies to further increase the robustnessof both approaches to multipath fading.

REFERENCES

[1] A. Ghaffarkhah and Y. Mostofi, “Communication-aware surveillancein mobile sensor networks,” in Proc. Amer. Contr. Conf. (ACC), SanFrancisco, CA, Jul. 2011, pp. 4032–4038.

[2] B. Grocholsky, J. Keller, V. Kumar, and G. Pappas, “Cooperative airand ground surveillance,” IEEE Robot. Autom. Mag., vol. 13, no. 3, pp.16–25, 2006.

[3] J. Cortés, S. Martínez, T. Karatas, and F. Bullo, “Coverage control formobile sensing networks,” IEEE Trans. Robot. Autom., vol. 20, no. 2,pp. 243–255, 2004.

[4] Y. Wang and I. Hussein, “Awareness coverage control over large-scaledomains with intermittent communications,” IEEE Trans. Autom. Con-trol, vol. 55, no. 8, pp. 1850–1859, Aug. 2010.

[5] J. Cortes, “Distributed Kriged Kalman filter for spatial estimation,”IEEE Trans. Autom. Control, vol. 54, no. 12, pp. 2816–2827, Dec.2009.

[6] J. L. Ny and G. Pappas, “On trajectory optimization for active sensingin Gaussian process models,” inProc. IEEEConf. Decision Contr. Chi-nese Contr. Conf. (CDC-CCC), Dec. 2009, pp. 6286–6292.

[7] N. E. Leonard, D. A. Paley, F. Lekien, R. Sepulchre, D. M. Fratantoni,and R. E. Davis, “Collective motion, sensor networks, and ocean sam-pling,” Proc. IEEE, vol. 95, no. 1, pp. 48–74, Jan. 2007.

[8] S. L. Smith, M. Schwager, and D. Rus, “Persistent robotic tasks: Mon-itoring and sweeping in changing environments,” IEEE Trans. Robot.2011 [Online]. Available: https://ece.uwaterloo.ca/~sl2smith/pa-pers/2010TRO_Persistent(revised).pdf, submitted for publication

[9] R. Olfati-Saber, “Distributed Kalman filtering for sensor networks,” inProc. 46th IEEE Conf. Decision Contr., Dec. 2007, pp. 5492–5498.

[10] B. Chen, R. Jiang, T. Kasetkasem, and P. K. Varshney, “Channelaware decision fusion in wireless sensor networks,” IEEE Trans.Signal Process., vol. 52, no. 12, pp. 3454–3458, Dec. 2004.

[11] B. Liu and B. Chen, “Decentralized detection in wireless sensor net-works with channel fading statistics,” EURASIP J. Wireless Commun.Netw., vol. 2007, no. 1, 2007.

[12] T. M. Duman and M. Salehi, “Decentralized detection over multiple-access channels,” IEEE Trans. Aerosp. Electron. Syst., vol. 34, no. 2,pp. 469–476, Apr. 1998.

[13] Y. Yuan and M. Kam, “Distributed decision fusion with a random-ac-cess channel for sensor network applications,” IEEE Trans. Instrument.Measure., vol. 53, no. 4, pp. 1339–1344, Aug. 2004.

[14] A. J. Goldsmith and S. B. Wicker, “Design challenges for energy-con-strained ad hoc wireless networks,” IEEE Trans. Wireless Commun.,vol. 9, no. 4, pp. 8–27, Aug. 2002.

[15] N. Sarshar, B. A. Rezaei, and V. P. Roychowdhury, “Low latencywireless Ad Hoc networking: Power and bandwidth challenges and asolution,” IEEE/ACM Trans. Netw., vol. 16, no. 2, pp. 335–346, Apr.2008.

[16] T. ElBatt and A. Ephremides, “Joint scheduling and power control forwireless ad-hoc networks,” IEEE Trans. Wireless Commun., vol. 3, no.1, pp. 74–85, Jan. 2004.

[17] A. Ghaffarkhah and Y. Mostofi, “Communication-aware motion plan-ning in mobile networks,” IEEE Trans. Automatic Control, vol. 56, no.10, pp. 2478–2485, Oct. 2011.

[18] Y. Mostofi, “Decentralized communication-aware motion planningin mobile networks: An information-gain approach,” J. Intell. Robot.Syst., vol. 56, no. 2, pp. 233–256, Sep. 2009, Special Issue on Un-manned Autonomous Veh..

[19] D. V. Dimarogonas and K. H. Johansson, “Bounded control of networkconnectivity in multi-agent systems,” IET Contr. Theory Appl., vol. 4,no. 8, pp. 1330–1338, Aug. 2010.

[20] M. Ji andM. Egerstedt, “Distributed coordination control of multiagentsystems while preserving connectedness,” IEEE Trans. Robot., vol. 23,no. 4, pp. 693–703, Aug. 2007.

[21] A. Ghaffarkhah and Y. Mostofi, “Channel learning and communica-tion-awaremotion planning inmobile networks,” inProc. Amer. Contr.Conf. (ACC), Baltimore, MD, Jun. 2010, pp. 5413–5420.

[22] Y. Yan and Y. Mostofi, “Co-optimization of communication and mo-tion planning of a robotic operation in fading environments,” in Proc.Asilomar Conf. Signals, Syst. Comput., Asilomar, CA, Nov. 2011.

[23] Y. Mostofi, M. Malmirchegini, and A. Ghaffarkhah, “Estimation ofcommunication signal strength in robotic networks,” in Proc. IEEEInt. Conf. Robot. Autom. (ICRA), Anchorage, AK, May 2010, pp.1946–1951.

[24] M. Malmirchegini and Y. Mostofi, “On the spatial predictability ofcommunication channels,” IEEE Trans. Wireless Commun., vol. 11, no.3, pp. 964–978, Mar. 2012.

[25] A. Goldsmith, Wireless Communications. Cambridge, U.K.: Cam-bridge Univ. Press, 2005.

[26] A. Ghaffarkhah and Y. Mostofi, “Communication-aware navigationfunctions for robotic networks,” in Proc. Amer. Contr. Conf. (ACC),St. Louis, MO, Jun. 2009, pp. 1316–1322.

[27] R. M. Murray, Z. Li, and S. S. Sastry, A Mathematical Introduction toRobotic Manipulation. Boca Raton, FL: CRC, 1994.

[28] D. Son, B. Krishnamachari, and J. Heidemann, “Experimental study ofconcurrent transmission in wireless sensor networks,” in Proc. 4th Int.Conf. on Embedded Netw. Sens. Syst., 2006, pp. 237–250.

[29] H. V. Poor, An Introduction to Signal Detection and Estimation, 2nded. New York: Springer-Verlag, 1994.

[30] A. Gonzalez-Ruiz, A. Ghaffarkhah, and Y. Mostofi, “A comprehen-sive overview and characterization of wireless channels for networkedrobotic and control systems,” J. Robot., vol. 2011, 2011.

[31] Y. Mostofi, A. Gonzalez-Ruiz, A. Ghaffarkhah, and D. Li, “Charac-terization and modeling of wireless channels for networked roboticand control systems—A comprehensive overview,” in Proc. 2009IEEE/RSJ Int. Conf. Intell. Robots Syst. (IROS), St. Louis, MO, Oct.2009, pp. 4849–4854.

Alireza Ghaffarkhah (S’12) received the B.S. andM.S. degrees in electrical engineering from SharifUniversity of Technology, Tehran, Iran, in 2005 and2007, respectively.Since 2008, he has been working toward the Ph.D.

degree in electrical and computer engineering at theUniversity of NewMexico, Albuquerque. His currentresearch interests include motion planning and con-trol of robotic and mobile sensor networks, controland decision under communication constraints, andhardware/software design for robotic systems.

Yasamin Mostofi (M’12) received the B.S. degreein electrical engineering from Sharif University ofTechnology, Tehran, Iran, in 1997, and the M.S. andPh.D. degrees in the area of wireless communicationsystems from Stanford University, CA, in 1999 and2004, respectively.She is currently an Assistant Professor with the

Department of Electrical and Computer Engineering,University of New Mexico. Prior to that, she wasa Postdoctoral Scholar in control and dynamicalsystems at the California Institute of Technology,

Pasadena, from 2004 to 2006. Her current research lies at the intersectionof the two areas of communications and control/robotics in mobile sensornetworks. Current research projects include communication-aware navigationand decision making in robotic networks, compressive sensing and control,obstacle mapping, robotic routers, and cooperative information processing.Dr. Mostofi received the Presidential Early Career Award for Scientists and

Engineers (PECASE) and the USNational Science Foundation (NSF) CAREERaward. She also received the Bellcore Fellow-Advisor Award from the StanfordCenter for Telecommunications in 1999. She won the 2008–2009 Electrical andComputer Engineering Distinguished Researcher Award from the Universityof New Mexico. She has served on the Control Systems Society conferenceeditorial board since 2008.

3560 ieee transactions on signal processing, vol. 60, …ymostofi/papers/tsp12.pdf · 3560 ieee...

Documents