A new method for distributing power usage across a sensor network

Available online at www.sciencedirect.com

Ad Hoc Networks 6 (2008) 1258–1280

www.elsevier.com/locate/adhoc

A new method for distributing power usage acrossa sensor network

Patrick Vincent a,*, Murali Tummala b, John McEachen b

a Computer Science Department, United States Naval Academy, 572M Holloway Road Stop 9F, Annapolis,

MD 21402-5002, United Statesb Department of Electrical and Computer Engineering, Naval Postgraduate School, United States

Available online 26 December 2007

Abstract

We present a method for more uniformly distributing the energy burden across a wireless ground-based sensor networkcommunicating with an overhead unmanned aerial vehicle (UAV). A subset of sensor nodes, termed a transmit cluster,receives and aggregates data gathered by the entire network, and forms a distributed antenna array, concentrating the radi-ated transmission into a narrow beam aimed towards the UAV. Because these duties are power-intensive, the role of trans-mit cluster must be shifted to different nodes as time progresses. We present an algorithm to reassign the transmit cluster,specifying the time that should elapse between reassignments and the number of hops that should be placed between suc-cessive transmit clusters in order to achieve three competing goals: first, we wish to better and more broadly spread theenergy load across the sensor network while, second, minimizing the energy expended in moving the transmit cluster,all the while, third, reducing to the extent practicable the time to bring the UAV and the sensor network’s beam into align-ment. Additionally, we present a method for reconfiguring the communication burden between the ground-based sensornetwork and the UAV. We describe and analyze two alternative strategies to bring the UAV and the sensor network’sbeam into alignment, while minimizing the energy expended by the sensor network. The performance of the two strategiesis compared in terms of probability of beam-UAV alignment as a function of time, and the expected time to alignment. Weexamine the performance tradeoff between the choice of strategy and parameters of the sensor network that affect powerconservation.Published by Elsevier B.V.

Keywords: Sensor networks; Energy management; Unmanned vehicles

1. Introduction

A wireless sensor network is an interconnectedset of sensor nodes that monitor and collect informa-tion about the environment and transmit the col-

1570-8705/$ - see front matter Published by Elsevier B.V.

doi:10.1016/j.adhoc.2007.11.014

* Corresponding author. Tel.: +1 410 293 6810; fax: +1 410 2932686.

E-mail address: [email protected] (P. Vincent).

lected data to another location for processing andinterpretation. Each individual sensor node in thenetwork consists of one or more sensors, a radiotransceiver, a microprocessor and a small batteryhoused in a common hardware unit, as shown inFig. 1. The individual sensors within a sensor nodeare designed to detect one or more aspects of thephysical environment, such as motion, temperature,sound, the presence of nearby metal objects, etc.

mailto:[email protected]

Fig. 1. Sensor node composition.

P. Vincent et al. / Ad Hoc Networks 6 (2008) 1258–1280 1259

Once a detection occurs, a sensor node transmits thesensed data using its on-board transceiver, and thedata arrives at the ultimate end-user after traversingone or more wireless links.

Sensor networks have attractive military applica-tions since they can be deployed in dangerous,remote or inhospitable environments. For example,a large number of sensor nodes can be droppedfrom an aircraft, densely covering a large area ofinterest. It is generally assumed that there is noexisting infrastructure to support communication,except for the sensor nodes themselves. Ideally, thesensor nodes would collaborate among themselvesto dynamically form a wireless ad hoc network,thereby transmitting the collected information backto the end-users located far from the scene of dan-ger. Thus, each sensor node, in addition to perform-

Fig. 2. Use of

ing its own sensing and communication duties,should also be able to serve as a router to supportmultihop communications for other sensor nodesthat cannot communicate directly.

In military scenarios, it is advantageous to trans-mit the sensor network’s collected data to the ulti-mate end-users via an unmanned aerial vehicle(UAV) that acts as an airborne relay. As shown inFig. 2, a communication signal might proceed fromthe sensor network to the UAV (via a line-of-sightlink), then from the UAV to the intended receiver(via a second line-of-sight link). An airborne relayis advantageous for three reasons. First, since a sen-sor node’s transmitting antenna is only inches offthe ground, the distance to the radio horizon isseverely limited. Use of an airborne relay removesthis impediment. Second, if the sensor network isdeployed within an urban environment, buildingsand other structures effectively block the communi-cations line-of-sight in every direction except ‘‘sky-ward”, meaning that the only way for the sensornetwork to communicate is via an aircraft overhead.The third reason that an airborne relay is advanta-geous relates to propagation loss: a signal’s powerloss is much greater for antennas near the ground[1], and thus it is much more power-efficient to movecommunications ‘‘off the ground” by handing offthe data stream to a UAV.

An examination of the obstacles preventing easydeployment of military sensor networks starts withthe sensor nodes themselves. For military applica-tions, sensor nodes should be small, so as to be

a UAV.

1260 P. Vincent et al. / Ad Hoc Networks 6 (2008) 1258–1280

covert. Fig. 3 shows a commercially available sensornode next to a US 25¢ coin. Furthermore, mostmilitary applications presume that sensor nodes willnot be serviced in the field. Thus, when a sensornode’s small battery fails, the whole node is ren-dered inoperable. It is important, therefore, thatsensor nodes be inexpensive so that employmentof a large-scale network is cost-effective.

Therein lies a significant challenge: a small, inex-pensive sensor node necessitates use of a small, inex-pensive battery that can not be recharged. And useof such a battery, in turn, places a strong premiumon power management in order to extend the life-time of the network. Having decided to use aUAV as a relay, communication from a sensor net-work still presents a formidable challenge since thestrong premium placed on power managementplaces severe restrictions on a node’s transmitpower, and consequently, on the maximum rangeat which a node’s transmissions can be successfullyreceived.

Any individual sensor node has an omnidirec-tional antenna that radiates power uniformly in alldirections. Most of the transmitted power is wasted;only the small amount that happens to propagate inthe direction of the UAV is useful. We have pro-posed [2] having multiple sensor nodes coordinatetheir transmissions, each sending the same signalexcept for phase and amplitude offsets. The propa-gating electromagnetic waves will interfere and –through careful selection of the amplitude andphase offsets – the total radiated power from themultiple nodes can be focused in preferred direc-tions. The ability to focus power in certain direc-tions is referred to as antenna gain, G, while thesolid angle through which the radiated power isfocused is referred to as the beam, and the partici-

Fig. 3. Crossbow MICA2DOT sens

pating transmitting antennas are collectively termedan antenna array.

The military sensor network thus should operateas follows. The sensor nodes are deployed over dan-gerous or remote terrain – perhaps dropped from anaircraft – and land on the ground (Fig. 4a). Uponawakening, the nodes partition themselves intonon-overlapping clusters. Each cluster has a fixedsingle node designated as the clusterhead (CH)which oversees and manages the nodes in its cluster.Fig. 4b, for example, shows the sensor network par-titioned into four clusters, with the clusterheadsshown as darkened nodes. (An actual networkmight have hundreds or thousands of nodes andmany clusters.) A specified cluster, termed the trans-

mit cluster, is selected from among the many clustersto act as the transmission point for the sensor net-work. That is, any sensed data originating anywherein the network is routed to the transmit cluster’s CH(hereafter termed the transmit CH), where it is con-solidated with any data already collected, and pre-pared for transmission to the UAV. The transmitCH organizes a subset of its cluster nodes into a dis-tributed antenna array, coordinating their transmis-sions to direct the beam towards the UAV, asshown in Fig. 4c. Once the beam is aligned withthe UAV (Fig. 4d) and the communication link isestablished, the sensor network’s data are transmit-ted to the UAV. Because these duties are power-intensive, the role of transmit cluster must be shiftedto different clusters as time progresses in order tobalance the energy load as evenly as possible acrossthe network to extend the lifetime of the network(Fig. 4e).

The key condition for this approach to work isthat the sensor network’s narrow transmission beammust be directed such that the UAV falls spatially

or node next to US 25¢ Coin.

Fig. 4. Sensor network operation.


within it, as shown in Fig. 4d. If the direction ofapproach of the UAV (i.e., its elevation and azi-muth angles) is known a priori, then a number oftechniques are available to determine the magni-tudes and phase offsets necessary to synthesize thedesired radiation pattern to point the beam towardsthe UAV [3]. Military applications, however, mustpresume that the sensor network does not know a

priori where the UAV is, nor does the UAV knowthe direction in which the sensor network has aimedits transmission beam. The sensor network, lackingGPS capability, is, in fact, not even aware of anabsolute geographic coordinate system.

There are a number of different techniques thatan antenna array can employ to estimate the direc-tion of arrival of the UAV, some of which aresummarized in [4]. All of these methods share a fea-ture in common: They are very complex. In one ofthe simplest methods, for example, a UAV flies overthe sensor network transmitting a known referencesignal. The antenna array steers a very narrow beamcontinually in space, analyzes the spatial spectrumas a function of position, and then selects the direc-tion that yields the highest power as the direction ofthe UAV, assuming that this power arises fromthe UAV’s transmission. For this method to beeffective, the transmit CH will have to continuallycalculate different sets of amplitudes and phasesfor the participating sensor nodes comprising theantenna array (so as to steer the beam), transmitthese values to the participating nodes, and thesesensor nodes, in turn, will have to continually sendtheir transceiver receptions to the transmit cluster-head for analysis of the spatial spectrum. Whenenergy constraints require that the role of transmitcluster be shifted to a different cluster, all relevantdata collected up to this point must be passedthrough the network to the new transmit CH, orthe entire process of finding the UAV must beginanew. As all these operations involve a large num-ber of radio transmissions between sensor nodes,and a large number of computations performedby the transmit CH(s), they expend scarce batteryresources, thereby limiting the lifetime of theinvolved nodes.

Thus, we see that the problem of effecting com-munication between a wireless sensor network anda UAV is extraordinarily complex. We divide ourproblem into two sub-problems as follows. First,we determine how the transmit cluster functionalityshould be moved about the sensor network in amanner that broadly spreads the energy load acrossthe sensor network, while minimizing the energyexpended in moving the transmit cluster. That is,we examine how the actions of Fig. 4e should becarried out, momentarily deferring the details ofhow the transmit cluster’s beam is aligned with theUAV. This first task is examined in Section 2.

Once this problem has been examined, we thentake up the matter we deferred: we determine howthe transmit cluster’s beam should be aligned withthe UAV in a manner that minimizes energy expen-diture. That is, we examine how the actions ofFig. 4d should be carried out. This second problemis examined in Section 3.


2. Transmit cluster movement to extend network

lifetime

Before communication can occur, the sensornetwork’s transmit CH must form a distributedantenna array and align the transmit beam withthe UAV. We will examine this task in Section 3.We assume for the present discussion that this taskhas been carried out, and our aim is in this section isto examine how the transmit cluster functionalityshould be shifted to other portions of the networkas time progresses in order to more evenly distributethe energy load across the network.

2.1. The need to move the transmit cluster

The transmit cluster is tasked with aggregatingall sensor data originating anywhere throughoutthe network. Consider Fig. 5 which displays a por-tion of a larger sensor network, where the transmitcluster consists of the transmit CH and nodes 1–4.The transmit CH constitutes the data sink for thenetwork; all sensor data gathered by any node inthe network is routed to the transmit CH via oneof its adjacent nodes. Thus, a very heavy burden isplaced on nodes 1–4, since all the data from thenetwork must be transmitted by one of them tothe transmit CH. These transmissions expend scarcebattery resources. As a result, extending the net-work’s lifetime also requires that the transmit clus-ter be shifted among the clusters of the network astime progresses to balance the energy load. Sincethe transmit cluster is shifted about only to extendthe lifetime of the sensor network, it is important

Fig. 5. A deployed s

that the reassignment itself be mindful of thisobjective.

In this section, we present and analyze an algo-rithm to reassign the transmit cluster, specifyingthe time that should elapse between reassignmentsand the number of hops that should be placedbetween successive transmit clusters in order toachieve three competing goals: First, we wish to bet-ter and more broadly spread the energy load acrossthe sensor network while, second, minimizing theenergy expended in moving the transmit cluster,all the while, third, reducing to the extent practica-ble the time to bring the UAV and the sensor net-work’s beam into alignment.

Three related questions must be considered.First, how many hops away from the previoustransmit CH should the new one be? Second, howmuch time should elapse between transmit CH reas-signments? Finally, how should this reassignment becarried out and, specifically, what protocol shouldbe used?

Let h0 denote the number of hops separating anewly assigned transmit CH from the previoustransmit CH. Our first question is thus rephrased:what is a good value for h0? Referring to Fig. 5,we see that h0 = 2 would not be a wise selection;if, for example, node 5 was designated as the newtransmit CH (assuming it is indeed a CH in a sepa-rate cluster), then node 2 would continue to beheavily taxed since it would remain adjacent to thenetwork sink. Thus, h0 P 3. But we also cannotmake h0 as large as practicable. There is a tradeoff:on the one hand, the objective of balancing theenergy load across the network militates towards

ensor network.


the selection of a large h0; on the other hand, as weshall see, the energy expended in shifting the trans-mit CH increases as h0 increases, which militatestowards the selection of a smaller h0.

Let Thold denote the time between shifts in thelocation of the transmit cluster. That is, once a clus-ter (with its associated CH) assumes the role as thesensor network’s transmit cluster, it will retain theseduties for a time Thold before the transmit clusterrole is passed on to a different cluster. If Thold isarbitrarily large, the nodes in the transmit clusterwill deplete their batteries. Thus, in the interests ofextending the network’s lifetime, it is necessary toplace an upper limit on the value of Thold. But, aswith h0, there is a tradeoff in the selection of Thold:a small value of Thold ensures the nodes in a trans-mit CH are not excessively depleted and works todistribute the energy load more evenly across thenetwork, but a too-small value of Thold keeps thetransmit cluster’s upwards-pointing beam alwaysshifting, thereby lengthening the time to beam-UAV alignment. Thus, a transmit cluster shouldretain its duties for both a maximum time, as wellas for a minimum time.

2.2. Algorithm to transfer the transmit CH

To formalize these ideas, consider the followingsimple algorithm for shifting the role of transmit clus-ter from one cluster to another. The protocol assumesthat each node in the sensor network has a unique ID-number, and that the network has already been par-titioned into non-overlapping clusters, with eachcluster managed by a single fixed clusterhead.

Algorithm to reassign transmit cluster

Step 1. Upon assuming the duties of transmit CH,the node starts a timer.

Step 2. When the timer reaches Thold, the transmitCH generates a transmit cluster nominate(TCN) message, which invites anothernode to have its cluster assume the dutiesof the transmit cluster. The TCN messagecontains a sequence number which,together with the ID-number of the trans-mit CH, uniquely identifies the message.(If the node should subsequently have totransmit a new TCN, it will increment thesequence number.) The TCN message alsocontains a hop-counter, h, initiated to aninteger start value, h0.

Step 3. The TCN message propagates by flooding;that is, the transmit CH sends the TCNmessage to all of its neighbors, which inturn forward the data to all of their neigh-bors, and so on, subject to the provisionthat a node will only forward a givenTCN message the first time it is received.(In a wired network, a node would not for-ward the TCN message back to the neigh-bor from whom it received the data; inwireless networks, the broadcast nature ofthe radio channel does not permit selectivetransmission to a subset of neighbors.)When a node receives a TCN message, itdecrements the hop-counter by one andappends its address to the message (forthe purpose of enabling reverse tracing ofthe route followed by the message) priorto transmitting it onward. TCN messagesare flooded only until h = 1.

Step 4. When a node receives a given TCN messagewith a hop-count h = 1 (for the first time), ittransmits a transmit cluster response(TCR) message, nominating its cluster asthe new transmit cluster (and its cluster-head as the new transmit CH). The TCRmessage propagates back to the currenttransmit CH along the reversely tracedroute contained in the received message.

Step 5. Upon receipt of the first arriving TCRmessage, the current transmit CH trans-mits a transmit cluster assignment (TCA)message back to the node that initiated thisTCR message. Any additional TCR mes-sages that might be received (from othernodes that also received a copy of theTCN message with h = 1) are ignored. Inaddition to serving to transfer the transmitcluster duties to the new cluster, the TCAmessage contains all collected sensor datathat has been aggregated up to that point.That is, the TCA message contains the his-torical sensor network data that the retir-ing transmit CH would have transmittedto the UAV, had the UAV been withinits beam. If the destination node for theTCA message is not itself a clusterhead,then the destination forwards the TCAmessage to its clusterhead.

Step 6. Upon receiving the TCA message, the newtransmit CH transmits a message announc-ing itself as the new transmit CH. This


message, a transmit cluster declaration(TCD) message, propagates by floodingto all nodes in the sensor network.

Step 7. Return to Step 1.

2.3. Algorithm motivation

Before analyzing the algorithm, we discuss itsmotivation and features. First, note that floodingis used in steps three and six of the algorithm (forTCN and TCD messages). Flooding results in alarge number of unnecessary transmissions, a phe-nomena termed the implosion problem [5] and illus-trated in Fig. 6: node A, the current transmit CH,transmits a TCN message that is received by nodesB, C and D, each of whom transmit the message tonode E. Node E acts on the first reception of theTCN message, but disregards the two subsequent(duplicate) receptions. Since power is expended foreach transmission, and many transmissions underflooding are unnecessary, this routing algorithmwould seem to be a poor choice for a power-con-strained network.

Notwithstanding this inefficiency, flooding has anumber of virtues in its favor and is, in fact,employed as the routing algorithm in many sensornetworks. Flooding is a simple algorithm that isfault-tolerant and robust in the face of changes inthe network topology (since it does not depend onthe network topology at all). Much more impor-tantly, the use of efficient point-to-point routingalgorithms would require that sensor nodesexchange control messages with each other in orderto learn something about the underlying network

Fig. 6. The implosion problem.

topology. In some applications, these control mes-sages will be small compared to the actual data tobe transmitted later, and so the investment sunkinto transmitting and processing these controlmessages – which represents an inefficiency sincethese messages do not convey actual informationthat end-users are interested in – is more than madeup for by the efficient transmission of large blocks ofdata later. But TCN and TCD messages may bequite small, consisting of tens of bytes. Certainly itwould not be energy efficient to transmit a largeamount of control information just so that thesevery short blocks of data can be efficiently trans-ferred later. Methods that might be employed toconserve power while employing flooding have beenproposed in [5,6].

Note that TCR messages (step 4) and TCA mes-sages (step 5), on the other hand, use point-to-point

routing. Each flooded TCN message records andupdates the route traveled as the message progressesthrough the network. Thus, when a node receives aTCN message with a hop-counter equal to one, thearriving TCN message contains the route back tothe transmit CH, which can then be employed asthe point-to-point route used for the TCR message.This same route can then be used by the transmitCH to route the TCA message to the applicablenode. Efficient routing of TCA messages is impor-tant since these messages are presumably very large,containing all the network’s historical sensor dataaggregated up to that time.

The algorithm is designed such that the initialvalue of the hop-counter, h0, determines how far(in hops) the newly assigned transmit CH will befrom the prior transmit CH. Recall that the nodeseeking to relinquish its duties as transmit CH –say, node u – will transmit a transmit cluster nomi-nate (TCN) message with h0 initiated to an integervalue that is subsequently decremented on eachhop as the TCN message floods through the net-work. A node v that receives the TCN message withh = 1 will transmit a transmit cluster response(TCR) message, nominating its cluster head as thenew transmit CH. If v is itself a clusterhead, andits TCR message is the first to arrive at u, then thenew clusterhead (v) will be h0 hops from the priorclusterhead. If v is not a clusterhead, and is j hopsaway from its clusterhead, then v will be h0 ± j hopsfrom u (i.e., it may be the case that node v’s CH iscloser to node u than is node v itself). Many cluster-ing algorithms in the literature (see e.g., [7–9])ensure all nodes are within one hop of their cluster-


heads, and our algorithm implemented on such aclustered network would ensure h0 ± 1 hopsbetween successive transmit CHs.

2.4. Analysis of the algorithm

We are interested in minimizing the amount ofenergy expended in moving the transmit CH. Howmuch energy is expended by the algorithm in trans-ferring transmit cluster duties from one cluster toanother? We answer this question by first noting thatcommunication consumes much more power thandoes data processing. A concrete example describedin [10] illustrates this point: for a specific ground-to-ground transmission scenario, a sensor node trans-mitting 1000 bits of data to a node 100 m awayexpended 3 J of energy, the same amount of energyrequired for a modest processor to execute 300million CPU instructions. Thus, in calculating theenergy cost in transferring the duties of the transmitcluster, we only consider the power expended incommunicating, neglecting power expended onprocessing that takes place within any node.

The algorithm has, in essence, two types of mes-sages: small control messages (transmit cluster nom-inate (TCN) messages in steps 2 and 3, transmitcluster response (TCR) messages in step 4 and trans-mit cluster declaration (TCD) messages in step 6)and long data messages (transmit cluster assignment(TCA) messages in step 5). The length of the shortcontrol messages is denoted m1 bits, while the lengthof the long data message is denoted m2 bits(m2� m1). We assume all transmissions occur at arate of R bits/s.

We model the sensor network as an undirectedgraph G = (V, E), where V denotes the set of vertices(sensor nodes) and E denotes the set of edges, wherean edge exists between two vertices if the correspond-ing sensor nodes are within direct communicationrange of each other. The total number of sensor nodesis denoted as N; that is jVj = N. Noting that thedegree of a node is equal to the number of edges inci-dent to it, we denote the maximum vertex degree inthe graph as D, and the average degree as D. Sensornodes communicate with each other at a transmitpower P, and the maximum transmission range fromone sensor node to another at this power P is denotedd. Since no node will transmit a flooded message morethan once, a message flooded throughout the entiresensor network will require at most N transmissions.

We calculate the energy expended in moving thetransmit CH by first counting the number of occur-

rences of each type of message (TCN, TCR, TCA,and TCD) generated during the reassignmentalgorithm.

TCN messages: In steps 2–3 of the algorithm,TCN messages are not flooded throughout theentire network; the presence of the counter thatdecrements on each hop ensures that only nodeswithin h0 � 1 hops of the transmit CH will transmita TCN message. Thus, for steps 2–3, the number ofmessage transmissions will be equal to the totalnumber of nodes within h0 � 1 hops of the transmitCH. Let the number of TCN messages generated bythe algorithm be denoted as NTCN. We calculate anupper-bound on NTCN by simple induction. First, ifh0 = 1, there is only a single transmission (from thetransmit CH), and NTCN = 1. Next consider thecase h0 = 2. Now, the initial transmission from thetransmit CH is received by at most D nodes, eachof which decrements the hop-counter to one andtransmits the message onward to each of its neigh-bors. Thus, in this case, NTCN 6 1 + D. Next, con-sider h0 = 3. As before, we have the initialtransmission, followed by at most D transmissionsfrom the transmit CH’s adjacent nodes. Now eachof these D new transmissions is, in turn, receivedfor the first time by at most D-1 nodes. Thus,NTCN 6 1 + D + D(D � 1). Similarly, if h0 = 4, wefind NTCN 6 1 + D + D(D � 1) + D(D � 1)2, andcontinuing in like manner, we find that for anyvalue of h0, NTCN 6 1þ D

Ph0�2j¼0 ðD� 1Þj. Employing

the formula for the geometric sum, and noting thatthe total number of transmissions cannot exceed thenumber of nodes, N, we find:

NTCN 6 min 1þ DD� 2

� �ð½D� 1�h0�1 � 1Þ;N

� �

h0 P 1: ð1Þ

TCR messages: Only nodes that receive a TCN mes-sage with a hop-count equal to one will transmit atransmit cluster response (TCR) message. Notingthe development above, the maximum number ofnodes that transmit a TCN message with a hop-count of one is DðD� 1Þh0�2. Now, each of thesemessages is received by at most D � 1 nodes forthe first time, and these are the nodes that transmitTCR messages. Each of these TCR messages prop-agates via a direct h0-hop path back to the transmitCH. Thus, the number of transmissions required bythe algorithm, NTCR, is upper-bounded by

NTCR 6 min h0DðD� 1Þh0�1;Nh0

� �: ð2Þ

Fig. 7. Energy expended to shift the transmit CH.


TCA messages: Since the TCA message generatedby the transmit CH propagates directly to the firstnode whose TCR message was received, and thatnode, if not a clusterhead, forwards the TCA mes-sage to its clusterhead located at most one hopaway, the number of TCA transmissions, NTCA, isupper-bounded as

NTCA 6 h0 þ 1: ð3ÞTCD messages: The transmit cluster declarationmessage propagates by flooding. Thus, since nonode will transmit a flooded message more thanonce, the number of TCD messages, NTCD, isupper-bounded as

NTCD 6 N : ð4ÞIn light of the foregoing, we provide an upper-

bound on the energy expended in transferring atransmit CH.

Theorem. The amount of energy expended in execut-

ing the algorithm to transfer the transmit CH from

one node to another is upper-bounded by:

Etransfer 6 P min 1þ DD� 2

� �ð½D� 1�h0�1� 1Þ;N

� ��

� m1

R

� �þ P ½minðh0DðD� 1Þh0�1

;Nh0Þ�

� m1

R

� �þ PNm1

Rþ Pðh0þ 1Þm2

R: ð5Þ

Proof. An upper-bound on the total number ofTCN, TCR and TCD messages generated by thealgorithm is given by (1), (2) and (4), respectively.Each of these messages, being control messages,takes time m1/R to transmit. By (3), at most h0 + 1TCA messages are transmitted during the courseof the algorithm, each taking time m2/R. Each trans-mission occurs at power P, and the energy expendedby a transmission equals the power multiplied bythe duration of the transmission. h

Note that the upper-bound (5) depends criticallyon the value of h0, the number of hops to transferthe CH, as well as the maximum node degree D.The computed upper-bound is OðDh0Þ. As an exam-ple, Fig. 7 displays the upper-bound on the totalenergy expended in implementing the algorithm toshift the transmit CH as a function of h0, assumingthe sensor network consists of 10,000 CrossbowMICAz motes, communicating with each other ata minimal RF power of �24 dBm (4 lW) [11]. Atlow values of h0, the last two terms in (5) dominate,

i.e., (5) is �(PNm1 + Ph0m2 + Pm2)/R. As h0

increases, the first two terms of (5), growing expo-nentially, come to dominate.

With the exception of the maximum node degreeD, all of the parameters in (5) are presumed known a

priori. For example, the transmission rate (R) andtransmit power (P) are determined by the trans-ceiver design, the number of deployed nodes (N) isknown, and the average packet sizes (m1 and m2)as well as the hop-counter start value (h0) are pre-programmed. Since the maximum node degree isnot known and can not be readily determined, andan energy calculation based on the maximum nodedegree may not be very illuminating (e.g., a networkmay have a single node of maximum degree, sayD = 10, with the next highest-degree node being ofdegree 5), we replace D in (5) with D, the averagenode degree. Of course this modification means that(5) will no longer be an upper-bound, but willinstead serve as a more useful estimate of the energyexpended in transferring the transmit CH from onelocation to another.

We estimate D using the geometry of the prob-lem. The transmission power of a node, P, will cor-respond to a maximum reception range d. That is, anode transmitting at power P will be within commu-nication range of all nodes within a distance d away.We assume that the sensor network is known to beconfined to a specific region – termed the sensor

region – of size A. Consider N nodes distributed uni-formly over this region. Focusing on a single nodewithin the interior of the region, all nodes within acircle of radius d will be within communicationrange of the node. The average number of nodeswithin this region is N(pd2/A). Since this quantity


includes the node under consideration, an estimateof D is

D ¼ Npd2

A

� �� 1: ð6Þ

We note that this estimate of the average nodedegree is still an over-estimate, since nodes nearthe boundaries of the region will have fewer neigh-bors, since a portion of their reception zone willlie outside the sensor region. As a simple numericalexample, consider the sensor network of 10,000MICAz motes from Fig. 7, uniformly distributedover a 10 km � 10 km region. Using a transmissionrange of d = 100 m, we find from (6) that D ¼2:1416. Fig. 8 displays an estimate of the totalenergy expended in implementing the algorithm toshift the transmit CH as a function of the start valueof the hop-counter, h0, assuming the same sensornetwork parameters from Fig. 7, but using D insteadof D. We see in Fig. 7 that the energy expended intransferring the transmit CH first exceeds 2 mJwhen h0 � 6. Comparing this to Fig. 8, we note thatthe energy expended in transferring the transmit CHfirst exceeds 2 mJ when h0 � 35.

Based on (5) (and its behavior as depicted inFigs. 7 and 8), the value of h0 chosen for use inthe algorithm to reassign the transmit cluster shouldnot exceed the point where the exponentially grow-ing terms (the first two terms in (5)) exceed the sumof the constant term (the third term in (5)) and thelinearly increasing term (the last term in (5)). Thus,in order to more evenly distribute the energy loadacross the network while minimizing the energyexpended in moving the transmit cluster, we should

Fig. 8. Energy expended to shift the transmit CH.

select h0 to be the maximum value of h such that thefollowing inequality is satisfied:

ðD� 1Þh�1 DD� 2

þ hD

� � m2

m1

h

6 N þ m2

m1

þ DD� 2

� 1: ð7Þ

For the scenario depicted in Fig. 7, (7) yields h0 = 6,while for the scenario depicted in Fig. 8, (7) yieldsh0 = 36.

In the next section we describe in detail anenergy-conserving technique to align the transmitCH’s upward-pointing beam, and present a seriesof examples illustrating the utility of the clustershifting algorithm presented in this paper.

3. An energy efficient approach for information

transfer from sensor networks

3.1. Solution approach for establishing a

communication link with a UAV

Recall that the transmit CH organizes a subset ofits cluster nodes into a distributed antenna array,calculating the magnitude and phase offsets to beapplied to the otherwise identical transmissions ofthe participating nodes in order to form a radiationpattern that concentrates the transmitted power intoa narrow beam. The key condition for this approachto work is that the sensor network’s narrow trans-mission beam must be directed such that the UAVfalls spatially within it. As mentioned in Section 2,military applications must presume that the sensornetwork does not know a priori where the UAVis, nor does the UAV know the direction in whichthe sensor network has aimed its transmissionbeam.

So, how can the sensor network ‘‘find” the UAV?In Section 1 we noted that while an antenna arraysuch as that formed by the transit CH can, in the-ory, employ a variety of techniques to estimate thedirection of arrival of the UAV, these methodsentail a large number of transmissions betweensensor nodes. Each of these transmissions expendsscarce battery resources and thereby limits the life-time of the involved nodes. Still, regardless of thedifficulties, it remains a fact that before communica-tion can occur, the sensor network’s transmit CHmust align the transmit beam with the UAV.

Our approach to the problem begins byrecognizing that the sensor nodes are very


power-constrained, having limited transmissioncapacity. The UAV, on the other hand, is highlycapable. Not as limited in size and weight, it carriesa much larger battery, and, unlike sensor nodes, itwill fly back to its base and have its batteryrecharged once its mission is complete. Neverthe-less, despite the strength of the UAV, and the limi-tations of the sensor nodes, algorithms thatdetermine the proper steering of the beam wouldplace all of the communication burden on the sensornetwork. The very capable UAV does nothing butfly around, waiting to be ‘‘found”. Our approachis to reverse these efforts: the sensor network willinvoke a simple algorithm that requires minimalcommunication and only a single amplitude/phaseoffset calculation. Specifically, the transmit CH willdirect a fixed beam, pointing straight up, at 0� eleva-tion. The burden will then be placed on the morecapable UAV to fly over the sensor network andfind this beam.

The basic approach is illustrated in Fig. 9. Thetransmit CH has organized a subset of its nodes intoa distributed antenna. The transmit CH calculatesthe proper amplitude and phase offsets needed forthe resultant beam of the desired gain to point

Fig. 9. Solution approach.

straight up, and these values are passed to each par-ticipating node. No transmission to the UAV isattempted at this time since there is no reason tobelieve that the UAV will be overhead within thebeam; the amplitude and phase values, calculatedonly once, are retained by the array elements in casea transmission should be required later. In fact, thedistributed antenna ‘‘stands down”, and the trans-mit CH now functions as an omnidirectionalantenna again, waiting for the UAV to find it. Inthe meantime, sensor nodes continue to pass sensordata to the transmit CH, for eventual transmissionto the UAV.

The UAV, meanwhile, travels over the sensorregion as shown in Fig. 9a, continuously transmit-ting a known reference signal, pointing its transmis-sion beam straight down. If the transmit CH shouldfall within the UAV’s beam, the clusterhead willsend the data stream (containing the consolidateddata gathered by the sensor network) to the ele-ments in the antenna array, and these elements will,in turn, transmit the signal using the precomputedamplitudes and phase offsets, as shown in Fig. 9b.If the distributed antenna array’s gain is close tothe UAV’s antenna gain (and employs a similar con-ically shaped beam), we can be assured of thescheme’s success since whenever the transmit CHfalls within the UAV’s transmission beam, theUAV will fall within the sensor network’s transmis-sion beam.

In short, instead of having the sensor network’sbeam search for the UAV, we simply propose havingthe UAV search for the sensor network’s beam.Thus, the power-constrained sensor nodes are notburdened with finding the highly capable UAV;instead, the burden of aligning the transmissionbeam is shifted to the UAV. Since the role of transmitcluster must be shifted to different locations as timeprogresses, it can be expected that the transmit CHwill move during the course of the UAV’s flight. Thismovement of the transmit CH, discussed in the priorsection, will be revisited in the present context afterdefining the system model and two rival search plans.

3.2. System model and two rival search plans

We address the question: how should a UAV beemployed to scan an area, seeking to place thetransmit CH within its downward pointing beam?We may consider the UAV to be searching for,and trying to detect, the transmit CH. Three criteriaguide our decision: first, the cumulative probability


of detection (as a function of time), PD(t), should bemaximized. Second, the expected time to detect thetransmit CH should be minimized. Finally, the planshould be as simple as possible.

A considerable body of research (see for example[12–14]) exists for determining how search effortshould be allocated over a region to maximize theprobability of detecting a target. Unfortunately,the known solutions offer results that are theoreticalas opposed to practical, since they do not yield flighttrajectories that an aircraft can physically follow.

The theory of search plan design also shares ele-ments in common with the field of robot motionplanning and, in particular, with the subfields of ter-rain acquisition and coverage path planning. Here,the aim is to design algorithms that enable a sen-sor-equipped robot to generate an optimal paththrough an unexplored terrain such that the sensor‘‘passes over” every point in the terrain. Unfortu-nately, the exhaustive search algorithms useful forrobot motion planning offer limited utility for oursearch operations since these algorithms all assumethe target of the search (in our case, the transmitCH) remains stationary during the execution ofthe algorithm. If the target of the search is mobile– a key assumption in our model where the beammust be allowed to be shifted to another transmit-ting cluster to extend the lifetime of the network –these algorithms fail since the target may moveundetected into an already-scanned region, andnever be detected. Examples of such terrain-cover-ing algorithms are presented in [15–18]; for a surveyof recent results, see [19].

To conduct the search, we have a UAV that fliesat a minimum velocity of V. The UAV transmits aknown reference signal, using a conical beampointed downward. Referring to Fig. 9a, we assumethat any sensor node within a radial distance of W/2from the point on the ground directly below theUAV will detect the UAV’s transmission, but willnever detect the UAV’s transmission beyond thisrange. The quantity W, termed the sweep width, isfixed for a given UAV altitude and antenna gain.The UAV is assumed to have the ability to ascertainthe coordinates of its location at any time and, addi-tionally, is aware of the geographic boundaries ofthe search region.

The sensor network is known to be confined to aregion on the ground, termed the sensor region.Specifically, let us assume the sensor network’s loca-tion is confined to a rectangular planar region ofwidth X, length Y and area A = XY. Without loss

of generality, we assign the label X to the shorterside (i.e., X 6 Y) and note that X is almost alwaysmuch larger than the sweep width, W. While weare certain that the transmit cluster (and itsupward-pointing beam) resides within a region ofknown boundaries, we possess no a priori knowl-edge about the transmit CH’s precise locationwithin the region; as such we assume the transmitCH’s location is uniformly distributed over theregion.

We are interested in examining search strategiesthat will ensure that the UAV finds the transmitCH as quickly as possible (i.e., brings the UAVand the sensor network’s beam into alignment asquickly as possible), maximizing the probability ofdetection (as a function of time), using as simple ameans as possible. We will analyze two alternativestrategies that bring the UAV and the sensor net-work’s beam into alignment. Both schemes seek tominimize the energy expended by the sensor net-work in aligning the beam and the UAV. The firstscheme uses a less capable UAV, and, while notabsolutely guaranteeing alignment, will find thebeam more quickly in a variety of scenarios. Thesecond strategy employs a very capable UAV andensures with probability one that the beam isaligned with the UAV, also guaranteeing that it willbe done as quickly as possible, given that alignmentis to occur. The performance of the two strategies iscompared in terms of probability of beam-UAValignment as a function of time, and the expectedtime to alignment, and we examine the performancetradeoffs between the choice of strategy and param-eters of the sensor network that affect powerconservation.

Our first search plan is a simple scheme in whichthe UAV searches randomly. This scheme employsa ‘‘cheap” UAV that has limited processing capabil-ity and has bare-bones navigational equipment,serving only to keep it within the boundaries ofthe sensor region. The UAV in this scheme followsa meandering track, searching randomly withinthe sensor region. This search is hereafter referredto as random search.

Our second scheme uses a significantly morecapable UAV. In this scheme, the UAV flies a pre-defined pattern. Use of predefined patterns canensure that the search is both effective (no interest-ing areas are missed) and efficient (a UAV doesnot unnecessarily duplicate its efforts). Consider,for example, that if the transmit CH is stationary,a UAV can perform parallel sweeps of the sensor


region (where the sweeps are separated by W, thesweep width) and ensure detection in finite time.On the other hand, flying a pattern may proveinflexible in highly dynamic environments, and thepattern – being predictable – might make theUAV more susceptible to shoot-down. We termour second search plan the progressive search forreasons that will become clear shortly.

Our aim is to analyze these two rival UAV searchschemes, quantitatively comparing them using sev-eral performance metrics. The progressive search,if properly designed, will not scan already-clearedareas and will not miss any areas. A random search,though, is cheaper, by virtue of its less capable UAVthat does not have to implement a search plan algo-rithm, with the attendant costs in calculation, navi-gation, etc. In the remainder of this subsection wedescribe in detail the progressive search, and in thenext subsections we compare the two rival schemesconsidering the probability of detection as a func-tion of time and the expected time to find the trans-mit CH.

In the progressive search, the UAV followsequally spaced, straight back-and-forth paralleltracks. The movement of the UAV from one end

Fig. 10. Progress

of the sensor region to the other along the x-axisconstitutes a sweep. Fig. 10 shows two consecutivesweeps (with the second sweep still in progress).Note that the UAV must move all the way to theedge of the sensor region before turning 90�, so asto avoid any gaps in coverage. It is possible thatthe sensor network will move the transmit CH dur-ing the course of the UAV’s search (so as to evenlydistribute the energy load). For example, considerthe case where the transmit CH is initially at Point‘‘A” in Fig. 10, just outside the UAV’s beam as itflies past. Suppose that once the UAV has passed,the transmit CH is moved to Point ‘‘B”. If consecu-tive sweeps did not overlap, the UAV would com-plete the search and never find the transmit CH.To avoid this scenario, a given sweep must havesome overlap with the prior sweep, while makingprogress toward the eventual goal. The overlapbetween consecutive sweeps is designated S (seeFig. 10).

Recall from Section 2 that Thold denotes the timebetween shifts in the location of the transmit CH,and that the new transmit CH will be h0 ± 1 hopsfrom the prior one. If the maximum reception rangebetween two nodes is denoted as d, then, from the

ive search.


UAV’s perspective, the transmit CH moves aroundthe network as time progresses, with a maximumvelocity v:

v ¼ ðh0 þ 1ÞdT hold

: ð8Þ

Although the transmit CH’s maximum velocity isknown from (8), the transmit CH’s actual speed anddirection of movement at any given time areunknown.

To guarantee detection of the transmit CH, wepresume the worst-case scenario shown in Fig. 10:the transmit CH is at the leftmost boundary justbeyond the UAV’s beam (Point ‘‘A”), and movesstraight up at speed v as soon as the UAV haspassed. The time for the UAV to go from the leftedge back to the left edge is (2X + W � S)/V, andduring this time the transmit CH can move a maxi-mum distance of v(2X + W � S)/V into the previ-ously swept region. To assure detection, we requirethat this maximum distance be set equal to the over-lap. Thus S = v(2X + W � S)/V. Solving for S, wedetermine the minimum overlap between sweepsnecessary to assure detection of the transmit CH:

S ¼ ð2X þ W ÞvV þ v

: ð9Þ

Note that when v = 0 (the transmit CH does notmove), S = 0 (no overlap needed), as expected. Ifthe UAV’s downward movement between sweepsis less than W � S, the search will take longer thannecessary, whereas if the downward movementbetween sweeps exceeds W � S, it is possible thatthe transmit CH might not be detected. For thesearch to be completed in finite time, successivesweeps must advance; the overlap between succes-sive sweeps, S, must be less than the sweep width,W. Hence, we term such a search a progressive

search.A progressive search requires that S < W. From

(9), this requires v < (WV/2X). Of course ifY 6W, then only one sweep is required. Thus, thesmallest transmit CH velocity that prevents a pro-gressive search is given by

v ¼WV =2X Y > W

1 Y 6 W

�: ð10Þ

Put another way, alignment is always assured withthe progressive search provided v < WV/2X (ifY > W).

Now, every progressive search consists of M

sweeps, where M, is given by

M ¼ maxY � SW � S

�; 1

� �: ð11Þ

To show this, first assume that Y > W so that thearea cannot be examined in a single sweep. Notingthe x and y axes shown in Fig. 10, we see that duringeach sweep the UAV travels a distance X along thex-axis. With the exception of the final sweep, eachsweep progresses a distance of W � S along the y-axis. During the last sweep, part of the UAV’s beammay fall beyond the sensor region’s boundary. If thelast sweep (which completes the search) covers a dis-tance of Ylast along the y-axis, then, since the totalbreadth in the y-axis direction is Y, we haveM ¼ 1þ Y�Y last

W�S . Now, since S < Ylast 6W, Ylast mustdiffer from W by an amount less than (W � S).Thus, M ¼ 1þ Y�W

W�S

� ¼ Y�S

W�S

� . Finally, we remove

our initial assumption that Y > W by incorporatingthe max function (if Y 6W then M = 1).

The time between when the search begins (i.e.,when the outer periphery of the UAV’s beam firstenters the sensor region) to the time when theUAV itself is first over the sensor region is W/2V.Each sweep except the final sweep takes time(X + W � S)/V. The final sweep takes X/V. Thus,provided the transmit CH velocity does not exceedthe conditions of (10), the total time to completethe progressive search, t, is given as:t ¼ W

2V þ ðM � 1Þ XV þ W�S

V

� �þ X

V . Simplifying

t ¼ MXVþ W � S

V

� �þ 2S � W

2V: ð12Þ

Fig. 11 shows the time to complete the progres-sive search for a sensor network located in a10 km � 10 km region, as a function of the maxi-mum transmit CH speed, v. The UAV moves at120 km/h and has a sweep width of 0.5 km.

3.3. Probability of beam-UAV alignment as a

function of time

The progressive search can assure transmit CHdetection (PD = 1) through the careful selection ofthe amount of overlap between successive sweeps.The tradeoff between the search time t and thetransmit CH velocity is discussed above.

The probability of detection (PD) as a function oftime t for a random search against a randomly mov-ing transmit CH whose initial position is uniformlydistributed within a region of size A km2 was firstderived in [20] and is given by

Fig. 11. Time to complete a progressive search.


P DðtÞ ¼ 1� e�WVt

A ; ð13Þwhere, as before, W is the sweep width and V is theUAV’s velocity. We note from (13) that detection ofthe transmit CH is never assured when employing arandom search, whereas detection is always (eventu-ally) assured with the progressive search (providedthe conditions in (10) are satisfied). Thus, the pro-gressive search is clearly desirable in scenarios whereit is critical to find the transmit CH.

3.4. Expected time to find the beam

The progressive search is designed to guaranteedetection. Is this more expensive scheme so over-designed to account for the worst-case scenario thatit performs poorly under more typical scenarios,where the less costly random search would performjust as well (or better)? To answer this question, wewill compare the expected time to find the transmitCH employing a progressive search to that taken fora random search assuming the transmit CH’s initiallocation is uniformly distributed within the sensorregion. Consider that in the progressive search theUAV starts at one end of the sensor region andgradually and methodically progresses forward,with the far end of the sensor region being examinedfor the first time only at the very end of the search.By contrast, the UAV performing a random searchmight happen to quickly fly to the position of thetransmit CH. Which search will locate a beam morerapidly, on average?

Theorem. Suppose a UAV is engaged in a progres-

sive search for a uniformly distributed transmit CH.

Let TP denote the time that the transmit CH is first

found by the UAV. An upper-bound on the expected

value of TP is given by

E½T P�UB ¼W2Vþ X

2Vþ X

Vþ W � S

V

� �ðM � 1Þ

� 1�M2

W � SY

� �� : ð14Þ

where M, as before, represents the number of sweeps

and is given by (11), and S is the minimum overlap

between sweeps needed to guarantee detection, given

by (9).

Proof. We assume the transmit CH’s initial positionis uniformly distributed over the sensor region, andmoves in random directions and with randomspeeds, up to a maximum speed of v. The UAV per-forms a progressive search of the sensor region,where the overlap between successive sweeps isequal to S. The probability of detection as a func-tion of time, PD(t), will improve as the search pro-gresses, and will be equal to unity when the searchconcludes (at the end of the final sweep). We ask,informally: what is the worst we can do? and, moreformally: what is a lower bound on PD(t)?

Consider the first sweep. At the end of this sweep,a transmit CH within a region of size X(W � S) willhave been detected, and no undetected transmit CH’swill be able to enter this region in the future. We cansay that the first sweep sweeps clean a region of sizeX(W � S) in the sense that no further effort needs tobe explicitly spent in examining this region. Thus,assuming that the transmit CH’s position is uni-formly distributed within the region, the probabilitythat the CH has been detected at the end of the firstsweep is at least equal to X(W � S)/XY = (W � S)/Y. All subsequent sweeps except for the final sweepyield the same result: an additional region of sizeX(W � S) is swept clean, and, at the conclusion ofthe sweep the probability of detection has increasedby at least (W � S)/Y over the prior sweep. The finalsweep differs from the prior sweeps in that some ofthe UAV’s beam may lie outside the sensor region,and the probability of detection will improve by atmost X(W � S) over the penultimate sweep, rising toone at the conclusion of the last sweep.

To further ensure that our estimate of PD(t) isindeed a lower bound, we assume that new areacovered as the UAV repositions between sweeps doesnot contribute to the probability of detection; i.e., thenew area covered as the UAV repositions is taken


into consideration during the analysis of the sub-sequent sweep. Likewise, we assume that the areacovered at the very start of the search, as theperimeter of the UAV’s circular beam first entersthe sensor region at time t = 0, until the UAV itself isover the sensor region at t = W/2V, does notcontribute to the probability of detection. This areais taken into consideration during the analysis of thefirst sweep.

With these considerations in mind, Fig. 12 dis-plays a lower bound for the probability of detectionvs. time for the progressive search. The initial sweeptakes time X/V and sweeps clean a region of sizeX(W � S); at the conclusion of the first sweep, theprobability that the transmit CH has been detectedis (W � S)/Y. The UAV then repositions for thesecond sweep, taking time (W � S)/V, and, asmentioned, we assume that repositioning does notcontribute to the probability of detection. The UAVthen performs the second sweep, again taking timeX/V and sweeping clean an additional area equal toX(W � S); the probability of detection at theconclusion of the second sweep is thus 2(W � S)/Y. The UAV then repositions and sweeps again, in amanner exactly analogous to the second sweep. TheUAV performs a total of M � 1 sweeps prior to thefinal sweep. The probability of detection rises from(M � 1)(W � S)/Y to unity during this final sweep.

Let the random variable TP denote the time atwhich the transmit CH is first detected during the

Fig. 12. Probability of detection vs. time f

progressive search. Then the probability that TP isless than some time t is equal to the probability thatthe transmit CH has been detected by time t, whichis given by Fig. 12; that is, Fig. 12 is also a graph ofP(TP 6 t) vs. t. Since E½T P� ¼

R10 ½1� P ðT P 6 tÞ�dt

[21], we have only to determine the area under thegraph of 1 � PD using the graph of PD (Fig. 12).The graph of 1 � PD is shown in Fig. 13. We dividethe area under the curve into four different regions,as labeled in the figure. Note that region 1 consistsof a single rectangle, region 2 consists of M � 1identically sized triangles, region 3 consists ofM � 1 different-sized rectangles, and region 4 con-sists of a single triangle. We compute the corre-sponding areas as

Region 1 :W2V

Region 2 :M � 1

2

� �XV

� �W � S

Y

� �

Region 3 :XM�1

m¼1

XVþ W � S

V

� �1� m

W � SY

� ��

¼ XVþ W � S

V

� �

� ðM � 1Þ 1�M2

W � SY

� ��

Region 4 :1

2

XV

� �1� ðM � 1ÞðW � SÞ

Y

� �:

or progressive search (lower bound).

Fig. 13. (1 � PD) vs. time for progressive search.


Eq. (14) results from summing these three areas andsimplifying the resulting expression, recognizingthat since our graph of PD(t) is a lower bound,our calculation of E[TP] yields an upper-bound. h

Corollary. If M = 1, i.e., if the UAV is able to com-

plete the search in a single sweep (i.e., W P Y), then

(14) yields E[TP] = (W/2V) + (X/2V). This is as

expected, since if the transmit CH is uniformly dis-

tributed within the search region and only a single

sweep (of duration X/V) is required, then, on average,the CH will be detected halfway through the search.

Theorem. Suppose a UAV is engaged in a progres-

sive search for a uniformly distributed transmit CH.

Let TP denote the time that the beam is first found

by the UAV. A lower bound on the expected value

of TP is given by

E½T P�LB ¼ E½T P�UB �pW 2

16AW þ X

V

� �

� X þ W � SV

� �ðM � 1Þ S

Y

� �

� X2Vþ W � S

V

� �W ðW � SÞðM � 1Þ

2A:

ð15Þwhere E[Tp]UB is given by (14).

Proof. We sketch the proof. The proof follows thesame procedure as the preceding theorem, except

now we ask, informally: What is the best we can

do? and, more formally: what is an upper-boundon PD(t)? We upper-bound PD(t) by examiningwhat would happen if the transmit CH was station-ary. The initial sweep examines a region of size WX,and the probability that the transmit CH has beendetected at the end of this sweep is W/Y. TheUAV then repositions for the second sweep, takingtime (W � S)/V. The second sweep does not con-tribute as much to the probability of detection asthe first sweep since the second sweep overlaps thefirst by an amount equal to S, and only examinesan additional area equal to X(W � S); the probabil-ity of detection at the conclusion of the secondsweep is W/Y + (W � S)/Y. The final sweep differsfrom the prior sweeps in that some of the UAV’sbeam may lie outside the sensor region, and theprobability of detection will rise to 1 at the conclu-sion of the last sweep. To further ensure that ourestimate of PD(t) is indeed an upper-bound, thenew area covered as the UAV repositions is takeninto consideration at the conclusion of the preceding

sweep. The new area covered by the UAV as it posi-tions for the next sweep is over overestimated by arectangle of size W(W � S)/2 (Fig. 14) and theincremental probability of detection, equal toW(W � S)/2A, is immediately added to the valueof PD(t) at the conclusion of the previous sweep.Likewise, we assume that the area covered at thevery start of the search, as the perimeter of theUAV’s circular beam first enters the sensor region

Fig. 14. New area covered as UAV repositions for next sweep.


at time t = 0, until the UAV itself is over the sensorregion at t = W/2V, immediately contributes aprobability of detection of p(W/2)2/2A at t = 0.With these considerations in mind, Fig. 15 displaysan upper-bound for the probability of detection vs.time for the progressive search. To simplify the axes

Fig. 15. Probability of detection vs. time f

in the figure, the following notation is used: a =(X + W � S)/V, c = W(W � S)/2A, d = (W � S)/Y. As before, Fig. 15 is also a graph of P(TP 6 t)vs. t. Determining the area under the graph of1 � PD, using the graph of PD (Fig. 15), providesE[TP]LB. h

or progressive search (upper bound).

Fig. 16. Antenna radiated power contained within solid angle X.


Lemma. Suppose a UAV is engaged in a random

search for a uniformly distributed transmit CH in a

region of size A. Let TR denote the time at which

the CH is first detected by the UAV. The expectedvalue of TR is given by

E½T R� ¼ A=WV : ð16Þ

Proof. The probability that TR is less than sometime t is equal to the probability that the CH hasbeen detected by time t, which is given by (13).Thus, the probability distribution function for TR

is also given by (13): P DðT R 6 tÞ ¼ 1� e�WVt

A . Theexpected value of TR is calculated as E½T R� ¼R1

0½1� P ðT R 6 tÞ�dt ¼

R10

e�WVt=A ¼ A=WV . h

Theorem. Consider a search for a transmit CH, uni-

formly distributed within a region of size A. The

search is conducted using a UAV. The expected value

of the time to find the CH will be lower (i.e., better)

using a progressive search as long as:

W2Vþ X

2Vþ X

VþW �S

V

� �ðM�1Þ 1�M

2

W �SY

� ��

<A

WV: ð17aÞ

and will be lower using a random search when:

E½T P�UB �pW 2

16AW þ X

V

� �

� X þ W � SV

� �ðM � 1Þ S

Y

� �

� X2Vþ W � S

V

� �W ðW � SÞðM � 1Þ

2A>

AWV

:

ð17bÞ

Proof. Follows immediately from (14)–(16). h

We note that the expected value of the time tofind the transmit CH using a random search maybe between E[TP]UB and E [TP]LB. In this case,expected value of the time to find the transmit CHwill be comparable using either algorithm, andother criteria (e.g., cost of the UAV) might guidethe search plan selection.

3.5. Relationship between antenna gain, UAV height

and sweep width

In this section we derive a relationship betweenthe gain of the sensor array’s distributed antenna(assumed to be approximately equal to the UAV’s

antenna gain), the height of the UAV and theUAV’s sweep width. The notion of concentratingthe radiated power into a conical beam is illustratedin Fig. 16. It can be shown that the relationshipbetween the antenna gain, G, and the solid angle,X, through which the power is concentrated is givenby [22]:

G ¼ 4p=X: ð18Þ

where X is in units of steradians. Generally, if aspecified section of a sphere’s surface has an areabeam area, and this section subtends a solid angleof X steradians, then

X ¼ beam area

z2: ð19Þ

where z is the radius of the sphere (a conical sectionof which is shown in Fig. 16). If the antenna gain islarge (i.e., X is small), the area of the section of thesphere’s surface will be closely approximated by thearea of the base of the cone with the same solid an-gle X and height z:beam area = p(W/2)2, where W isthe width of the beam (Figs. 16 and 9a). Substitut-ing this expression into (19) yields X ¼ pðW =2Þ2

z2 .Substituting this expression into (18) yields a rela-tionship between W and G:

W ¼ 4z=ffiffiffiffiGp

: ð20Þ

3.6. Applications

We now present three scenarios that illustratehow real-world practical considerations are madeanalytically accessible by the foregoing results. Thefollowing assumptions apply to all three scenarios:


� A sensor network consisting of many sensornodes is distributed over a 10 km � 10 km region.� The maximum communication range between

sensor nodes is d = 25 m.� The UAV is a Shadow-200 that will fly over the

sensor region at a speed of V = 120 km/h andhas a maximum endurance of 7 h [23]. Given thatthe UAV base is a certain distance from the sen-sor region, the time that the UAV spends overthe sensor region is limited to 5 h.� To minimize the UAV’s risk of being shot down,

mission planners have decided that the UAVmust be kept at an altitude greater than 4000 feet.The UAV altitude is chosen to be z = 4100 feet(=1250 m).� To communicate with the UAV at the specified

altitude (1250 m), while meeting the required biterror rate for the given modulation scheme, mis-sion planners have calculated that the sensor net-work must form a distributed antenna with again of G = 100. (The details of how this valuewas determined are not of interest.)

Fig. 17. Expected time to find the transmit CH vs. CH velocity.

1. Scenario 1: the UAV must (with certainty)

establish a link with the sensor network

Since we must guarantee link establishment, aprogressive search is required (recall that arandom search cannot assure link establish-ment, see (13)). Since G = 100 andz = 1250 m, we calculate the sweep widthusing (20) as W = 500 m. Using (12), we canreadily calculate the time to complete the pro-gressive search as a function of the transmitCH maximum velocity v. In fact, solving (12)for the given parameters is exactly the scenariothat is depicted in Fig. 11. Since we must limitthe search to 5 h, we readily determine (eitheranalytically using (12) or from Fig. 11) thatthe maximum permissible transmit CH veloc-ity is v = 2 km/h, which corresponds to anoverlap between successive sweeps equal to336 m (using (9)). This represents the maxi-mum distance the transmit CH is permittedto move in (2X + W � S)/V = 10.08 min.Since, by (8), the distance moved is equal to(h + 1)d, we find the maximum number ofhops separating a newly assigned transmitCH from the previously assigned transmitCH is h = 12. Thus, the sensor nodes mustbe programmed such that:

� The time between shifts in the transmitCH,Thold, must be P10.08 min. That is, oncea cluster (with its associated CH) assumes

the role as the sensor network’s transmit clus-ter, it must retain these duties for a period ofat least 10.08 min before the transmit clusterrole is passed on to a different cluster.

� The number of hops separating a newlyassigned transmit CH from the one must be612.

2. Scenario 2: the UAV should establish the link

with the sensor network as quickly as possible

Fig. 17 displays the expected time to find thetransmit CH for both the progressive and ran-dom searches as a function of the CH velocity,for the given parameters (X = 10 km,Y = 10 km, V = 120 km/h and W = 0.5 km).These plots are (14)–(16). If the sensor net-work is programmed as in Scenario 1 above,with the transmit CH maximum velocity setto 2 km/h, Fig. 17 would suggest that, if ourprimary concern is finding the transmit CHas quickly as possible (i.e., we no longerrequire that a link be established with cer-tainty), we would do better by having a‘‘cheap” UAV search the sensor region ran-domly. Prior to deployment, a designer, refer-ring to Fig. 17, might desire a maximumtransmit CH velocity of 1.4 km/h, so that theUAV is certain to find the transmit CH (withprobability one) and in a time better than therandom search. In this case, using the analysisof this section, we readily find that the maxi-mum overlap between successive sweeps islimited to 236.4 m, in a time of 10.132 min.This would entail programming the sensornodes so that a transmit CH will retain its

Fig. 18. Expected time to find the transmit CH vs. CH velocity.


duties for a minimum time of 10.132 min andthen transfer duties to a new CH at most 8hops away. So, the tradeoff in designing thesensor nodes to ensure beam alignment withthe UAV in a time better (on average) thanthat which would occur for the random searchis a decrease in the distance between successivetransmit CH from 12 hops to 8 hops. Thisrestricts the rate at which energy expendituresare distributed throughout the network. (Also,of minor note, the new design requires thetransmit CH retain its duties for a slightlylonger time).

3. Scenario 3: the sensor network can adjust the

gain, G, of its distributed antenna array

Suppose the sensor network can adjust thegain of its distributed antenna (by incorporat-ing more sensor nodes into the array). It maywish to do this in order to improve the BER atthe UAV’s receiving antenna. Increasing thegain has the effect of decreasing the searchwidth W per (20).Fig. 18 displays the expectedtime to find the transmit CH for both the pro-gressive and random searches as a function ofthe transmit CH velocity, for the given param-eters (X = 10 km, Y = 10 km, V = 120 km/h),for three different gains. Note that the curvesshown for the progressive search representthe upper bounds (i.e., (14)). If our networkwas designed with a gain of 100 and a maxi-mum transmit CH velocity of 1.4 km/h, thenFig. 18 shows that the progressive search isexpected to find the transmit CH as fast asthe random search and will surely find it bythe conclusion of the search. If the gain is

increased to 200, and the transmit CH maxi-mum velocity unchanged, Fig. 18 shows thatnow the random search will find the transmitCH faster. If we wish to restore the virtue ofthe progressive search, the transmit CH veloc-ity would need to be reduced to 1 km/h. Thisentails reprogramming the sensor nodes sothat a transmit CH will transfer its duties toa new CH at most five nodes away (whereasthe gain of 100 allows transferring duties toa node eight nodes away).

4. Conclusions

We have presented a method for more uniformlydistributing the energy burden across a wireless sen-sor network communicating with an overheadUAV. A subset of sensor nodes, termed a transmitcluster, receives and aggregates data gathered bythe entire network, and forms a distributed array,concentrating the radiated transmission into a nar-row beam aimed towards the UAV. We presentedan algorithm to reassign the transmit cluster, speci-fying the time that should elapse between reassign-ments and the number of hops that should beplaced between successive transmit clusters in orderto more broadly spread the energy load across thesensor network while minimizing the energyexpended in moving the transmit cluster. The designattempts to reduce to the extent practicable the timeto bring the UAV and the sensor network’s beaminto alignment, while meeting system-level perfor-mance objectives. Additionally, we have analyzedand compared the performance of two strategiesfor reconfiguring the communication and computa-tional burden between a wireless ground-basedsensor network and an unmanned aerial vehicle(UAV). Both strategies bring the UAV and the sen-sor network’s transmission beam into alignment,while minimizing the energy expended by the sensornetwork. The performance of the two strategies iscompared in terms of probability of beam-UAValignment as a function of time, and the expectedtime to alignment. We examined the performancetradeoff between the choice of strategy, and theparameters of the sensor network that affect powerconservation.

We suggest topics for further research that wouldbuild upon the ideas presented in this paper.

In our development, the BER required at theUAV is fixed at some required value. This fixed


receiver BER fixes, in turn, the value of G/h2, whereG is the gain of the sensor network’s distributedantenna, and h is the UAV altitude. So, by requiringa fixed BER, G and h cannot vary independently ofeach other. Further research would develop a morecomplex search scheme that recognizes the tradeoffsbetween BER, sweep width, UAV altitude, proba-bility of UAV shootdown, and the time to completethe search. If the UAV is at a higher altitude, it hasa greater sweep width, faster search time and alower probability of being shot down (all good),but the UAV’s receiver will suffer a greater BER.If a UAV is at a lower altitude, it has less sweepwidth, takes a longer time to complete the search,and stands a greater chance of being shot down(all bad), but the UAV’s receiver has an improvedBER.

Our proposed algorithm moves the transmit clus-ter in order to extend the lifetime of the sensor net-work. The analytical results derived from themathematical model, while intuitively appealingand understandable, might be verified with a simu-lation study. Specifically, we note that the energyto move the transmit cluster can be quite high.The energy benefits inherent in our algorithm mightbe compared to an alternative approach that leavesthe location of the transmit cluster fixed. The life-time of the sensor network under both conditionsmight then be compared under varying networkloads.

In shifting the transmit CH to a new location,the current algorithm to choose the next transmitCH makes a random selection from among thoseCH’s a specified number of hops away. Moresophisticated strategies might be considered: first,a node which previously served as a transmitCH might not ordinarily serve a second timeand, second, the replying cluster whose CH hasthe highest remaining energy might be selectedas the new transmit CH. Additionally, the analysisof the algorithm to shift the transmit cluster mightalso be extended to account for link failures andpacket collisions.

Proper synchronization is required in order toimplement the phase differences required for opti-mal beamforming. Our investigation did notaddress the difficulty in synchronizing the oscillatorsin the various sensor nodes. (Also, we did notaddress an even more basic question: how does asensor node that lands on the ground determinewhich way is up (i.e., skyward)?) Further researchwould also attempt to incorporate representative

element patterns into the analysis of the array pat-tern (we assumed isotropic radiators).

References

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[11] Crossbow MICAz data sheet available at <http://www.xbow.com/Products/Product_pdf_files/Wireless_pdf/MICAz_Datasheet.pdf/> (checked April 15, 2007).1.

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[20] B.O. Koopman, The theory of search II, target detection,Operations Research 4 (5) (1956) 503–531.

[21] H.M. Taylor, S. Karlin, An Introduction to StochasticModeling, Academic Press, California, 1994.

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[23] Department of Defense Office of the Secretary of DefenseMemorandum, Unmanned Aerial Vehicles Roadmap 2005-2030, dated August 4, 2005. Available at <http://www.acq.osd.mil/usd/Roadmap%20Final2.pdf/> (link checked April15, 2007).

Patrick Vincent was awarded the Ph.D.in Electrical Engineering from the NavalPostgraduate School, the ElectricalEngineer degree from UCLA, the MSEEfrom the Naval Postgraduate School andthe BSEE from Polytechnic Institute ofNew York. He is currently an AssistantProfessor of Computer Science at theUnited States Naval Academy. An activeduty naval officer, his research interestsinclude military applications of sensor

networks and ad hoc networks.

Murali Tummala received the M. Techdegree in 1981 and the Ph.D. degree in1984, both in electrical engineering, fromthe Indian Institute of Technology,Bombay. From 1976 to 1979, he was anInstructor at PES College of Engineering(Mysore University), Mandva, India.From 1983 to 1985, he worked as aProject Engineer at the Advanced Centerfor Research in Electronics. IIT, Bom-bay. In 1985 he joined the faculty of the

Naval Postgraduate School, Monterey, CA, where he is nowProfessor of Electrical and Computer Engineering. His research
interests include sensor networks, mobile ad hoc networks,wireless networks, and signal processing.
John C. McEachen, II received theBSEE. from the University of NotreDame, the MSEE from the University ofVirginia, and the M.Phil. and Ph.D.degrees from Yale University. He was aNational Library of Medicine ResearchFellow from 1992 to 1995. In 1995, hewas awarded a National Institutes ofHealth National Research ServiceAward that he performed at Yale Uni-versity. In 1996, he joined the faculty of

the Naval Postgraduate School, Monterey, CA, where he is nowan Associate Professor of Electrical and Computer Engineering.
His teaching interests include computer networks, and commu-nications systems. His research interests include sensor networks,managing routing in computer networks, wireless networkingprotocols, patternless intrusion detection, and steganographiccommunications. He retired as a Commander from the US NavalReserves in 2005 and is a member of the IEEE, Tau Beta Pi andEta Kappa Nu.
http://www.acq.osd.mil/usd/Roadmap%20Final2.pdf

http://www.acq.osd.mil/usd/Roadmap%20Final2.pdf

A new method for distributing power usage across a sensor network

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