DTIC ADA528943: Quality Tradeoffs in Object Tracking with Duty-Cycled Sensor Networks

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by  Defense Technical Information Center

Quality Tradeoffs in Object Tracking with
Duty-Cycled Sensor Networks
Sadaf Zahedi and Mani B. Srivastava Chatschik Bisdikian Lance M. Kaplan
Electrical Engineering Department IBM T. J. Watson Research Center U.S. Army Research Labratory
University of California Los angeles Hawthorne, NY, USA Adelphi, MD, USA
Los Angeles, CA, USA [email protected] [email protected]
{szahedi, mbs}@ee.ucla.edu
Abstract—Extending the lifetime of wireless sensor networks
requiresenergy-conservingoperationssuchasduty-cycling.How-
ever,suchoperationsmayimpacttheeffectivenessofhigh-fidelity
real-time sensing tasks, such as object tracking, which require
high accuracy and short responsetimes. In thispaper, wequan-
tifytheinfluenceofdifferentduty-cycleschemesonthebearings-
onlyobjecttrackingefficiency.Specifically,weusetheMaximum
Likelihood localization technique to analyze the accuracy limits
of object location estimates under different response latencies
consideringvariablenetworkdensityandduty-cycleparameters.
Moreover,westudythetradeoffsbetweenaccuracyandresponse
latencyundervariousscenariosandmotionpatternsoftheobject.
We have also investigated the effects of different duty-cycled
schedules on the tracking accuracy using acoustic sensor data
collected at Aberdeen Proving Ground, Maryland, by the U.S.
Army Research Laboratory (ARL). 1
I. INTRODUCTION
Real-time, accurate object tracking is key to many applica- Fig.1. Trackingscenario
tionssuchasvehicletrackingandsecuritysurveillance.Recent
developmentsinWirelessSensorNetwork(WSN) capabilities
make them suitable for many applications with real-time hard, (2) besides the real-time responsivenessof the network,
constraints and tight deadlines on the network responses. other issues such as energy efficiency and response integrity
Deadlines on the network responses to the task of interest arealso neededto beconsidered.Trackingaccuracy,response
may arise for various reasons such as the necessity for fast latency, and energy efficiency are among the most important
actionstothe presenceof a particularobjectin somestrategic design metrics for tracking applications [1].
locationsinthefield.Mostofthetime,thesenetworksinvolve
Our goal is to study the accuracy vs. response latency
a large number of sensors distributed in a vast geographical
performance trends for an object tracking application with
area with limited accessibility once deployed. Therefore, it
energy constraints. The key question of interest is: “what
is crucial to analyze and understand the expected real-time
is the impact of different parameters of an energy saving
performance of WSNs before the actual deployment.
design on the performance quality and responsiveness of
The real-time guarantee for WSN performance makes its
the tracking task?”. As an energy efficient scenario, in this
design very challenging due to various reasons such as,
work, we consider a duty-cycled network with random wake-
(1) physical events usually have uncertain and unpredictable
up schedules for different sensor nodes, and focus on the
spatio-temporal properties which make the modeling process
delays that system may face from the duty-cycled scheme.
An end-to-end delay in the network system also contains
1Research was sponsored by US Army Research laboratory and
communication delay from network traffic and packet losses
the UK Ministry of Defense and was accomplished under Agreement
NumberW911NF-06-3-0001. The views and conclusions contained in this whicharenotofourfocusinthiswork.Weusebothtimeliness
documentarethoseoftheauthorsandshouldnotbeinterpretedasrepresenting and tracking quality metrics to quantify the performance of
the official policies, either expressed or implied, of the US Army Research
the tracking process for each set of duty-cycle parameters.
Laboratory, the U.S. Government, the UK Ministry of Defense, or the UK
Government. TheUSandUKGovernments areauthorized toreproduce and Such a multi-attribute quality consideration is in-line with
distribute reprints for Government purposes notwithstanding any copyright Quality of Information (QoI) principles for sensor network
notation hereon.
established in [2]. The QoI metrics are highly affected by
TheauthorswouldalsoliketothankDr.RajuDamarlaforavailing usthe
ARLdataset. many factors including (1) object characteristics (the number
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Quality Tradeoffs in Object Tracking with Duty-Cycled Sensor Networks
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Proceedings the 31st IEEE Real-Time Systems Symposium , November-December 2010.
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14. ABSTRACT
Extending the lifetime of wireless sensor networks requires energy-conserving operations such as
duty-cycling. However such operations may impact the effectiveness of high-fidelity real-time sensing tasks,
such as object tracking, which require high accuracy and short response times. In this paper, we quantify
the influence of different duty-cycle schemes on the bearingsonly object tracking efficiency. Specifically, we
use the Maximum Likelihood localization technique to analyze the accuracy limits of object location
estimates under different response latencies considering variable network density and duty-cycle
parameters. Moreover, we study the tradeoffs between accuracy and response latency under various
scenarios and motion patterns of the object. We have also investigated the effects of different duty-cycled
schedules on the tracking accuracy using acoustic sensor data collected at Aberdeen Proving Ground,
Maryland, by the U.S. Army Research Laboratory (ARL).
15. SUBJECT TERMS
16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF
ABSTRACT OF PAGES RESPONSIBLE PERSON
a. REPORT b. ABSTRACT c. THIS PAGE Same as 10
unclassified unclassified unclassified Report (SAR)
Standard Form 298 (Rev. 8-98)
Prescribed by ANSI Std Z39-18
of concurrent objects, their velocities, and moving patterns), The work in [6] studies the quality and energy tradeoffs
(2) sensor characteristics (activation schedule, measurement for mobile object tracking by a wireless sensor network. It
quality, measurement type, radio range, sampling rate and discussesdifferentactivationstrategiesincludingsynchronized
transmission rate), and (3) network characteristics (topology duty-cycled activation. Moreover, it is assumed that sensors
and communication protocol). are binary detectors (with classical disk model) and, at any
In this work, we focus on an object tracking scenario given time, the centroid of all detecting sensors is used as
using a network of acoustic sensor (e.g. microphone) arrays an estimate for the object location. It is also shown that the
which are distributed in the observation region, see Fig. 1. duty-cycled activation offers a flexible and dynamic tradeoff
At any time, the Line of Bearing (LoB) of the object vs. between the energyexpenditureand tracking error when used
the sensor array center may be estimated by processing the in conjunction with the selective activation based on the
measurements of all the sensors in one array using Minimum trajectory prediction. Differently from our focus in this work,
Variance Distortionless Response (MVDR) [3], Multi Signal authors in [6] do not analyze the effect of energy preserving
Classification (MUSIC) [4], or any other similar algorithm. schemes on the response latency and achievable tracking
Having the LoB measurements of at least two sensor arrays accuracy of the system.
not collinear with the object, the fusion node can process the The work in [1], [7], [8] presents the real-time design
information to localize and, ultimately, track the object(s) of and analysis of VigilNet, a large-scale sensor network system
interest.Bearings-onlyobjecttrackinghasrecentlyraisedalot whichdetectsandclassifiesobjectsinatimelyandenergyeffi-
ofinterestincriticaltrackingapplications[5]duetoseveralad- cient manner. It provides a design framework and derivations
vantagessuchas:(1)foranobjectwithasmoothtrajectory,the to guarantee the requested end-to-end deadline. The power
LoB does not change abruptly, which enables outlier removal management system composes of both sentry and non-sentry
from the measurements, (2) by using the MVDR or MUSIC nodes where non-sentry nodes are sleep unless sentry nodes
method, circular acoustic sensor arrays with M microphones activate them. The sensing model and tracking algorithm are
can provide information on bearing measurements of up to similar to those assumed in [6]. The contributionof our work
M−1objectsinthefield(MUSICrequiresanaprioriestimate compared to [1], [7], [8] is in considering the problem of
ofthenumberofobjects).Thiscapabilityisadesirablefeature bearings-only tracking and also analyzing the accuracy of
in multi-object tracking scenarios. tracking vs. response latency to the tracking queries. We also
Analyzingthelatencyandaccuracymetricsforthetracking analyze the effect of duty-cycle parameters on the quality
scenario of our interest is very challenging due to many metrics.
reasons includingthe randomactivation schedule of the duty- In [9], using Bayesian estimation techniques, the effect of
cycled sensors, the unknown time of the object presence, packet loss and network transport delays on the quality of
and the uncertain object trajectory. This work is concerned object localization is studied. Sensors in [9] are always on
in addressing the above issues. In particular, our main con- and, hence, contrary to our work, the effect of delay due
tributions are: (1) performance analysis of QoI attributes and to duty-cycling is not taken into consideration. As another
theirtradeoffsforbearings-onlyobjecttrackinginduty-cycled example of recent efforts on delay analysis for real-time
WSNs using Maximum Likelihood techniques; (2) extension detection, [10] quantifies the tradeoff between the detection
ofthesnapshot-baselocalizationtotheinterval-basedoneand delay and the false alarm rate. Besides classical disk model,
derivation of their performance bounds; and (3) computation a probabilistic detection model for low SNRs is used in [10]
of the probability distribution of the delay that an object in order to analyze the minimum required network density
trackingtaskmayexperiencebeforetheavailabilityofenough satisfying quality performance requirements.
number of applicable measurements. Next, we go over the system assumptions and problem
The remainder of the paper is organized as follows. Sec- specifications.
tion II presents the related work. Section III overviews the
III. PROBLEMOVERVIEW
problem setting and our assumptions. Section IV describes
theQoIanalysisofbearings-onlyobjecttrackingbasedonthe Throughout this work, we particularly study the tradeoffs
Maximum Likelihood estimation techniques, and Section V between the response latency and the tracking accuracy of
discusses the simulation results. Finally Section VI concludes the system at times that a fusion node does not necessarily
the paper. hold a high quality prior information on the object state (its
recentlocationandvelocity).Thisanalysisiscriticalformany
different circumstances which we mention two of them next.
II. RELATEDWORK
Thefirst case occurswhenanobjectinitially appearsinthe
Research on high quality tracking systems with close to field, and the applicationhas deadlinerequirementsto receive
real-time responsiveness has been highly investigated by the the object location information. In this case, it is essential to
sensor network community in recent years. Various effective analyze that how fast and with what accuracy the tracking
techniques and approaches have been proposed and verified. system can track the object.
We here briefly review only a handful of them that are most The second case happens in query-based tracking appli-
related to this study. cations where the tracking system may have two different
operating phases. The first phase corresponds to the time
that no query initiates from the user for that period of time.
In this phase, sensor nodes may perform a low-cost local
detection and report the available results to the fusion node
with a moderate rate. Consequently, the fusion node exploits
the global knowledge on sensor node locations and estimates
the object location. The second phase starts when at time t
Q
auserinitiatesaquerylike:“reporttheobjectpositionattime
t∗ by the deadline D”. Receiving this query, sensor nodes
switch to a higher sampling rate, perform a more advanced
Fig.2. Relativegeometryoftheobjectbearinglinewithrespecttoasensor
local processing and send their local results to the fusion
node.
node with a higher rate. Then, fusion node uses recent local
informationandthepriorknowledgeofthenetworktoprovide
ahigh(er)qualityestimateoftheobjectlocation.Inthecaseof Gaussian noise,
not receiving any new measurement between t and t +D,
Q Q θ˜(t)=θ (P(t))+n (t), n (t)∼N(0,σ2). (2)
new location is estimated based on the prior information at i i i i i
the fusion node. It is assumed that sensor nodes are sufficiently separated so
Next, before focusing on the quality analysis, we introduce that the noise terms are independent. The value of bearing
the parameters and assumptions used through the rest of the measurementvarianceσ2 dependson the signal to noise ratio
i
paper. As illustrated in Fig. 1, it is considered that N sensor (SNR) of the received acoustic signal at node i and is a
arrays, which duty-cycle both their radios and samplers, are function of the relative distance d between this node and the
i
distributed in an observation region. They all have the same object. In our numerical analysis, we use σ2 = (Kdα)2 for
i i
scheduling period T, and duty-cycle β = τon (0 < β ≤ 1), the bearing measurement variance [11], where K is selected
T
where τ is the duration of “on” (awake) state. All the such that bearing error at 1 km from the object reaches 5o.
on
sensors in one array have synchronized wake-up times and From [11] α can be chosen to be 0, 1 or 2, and we set α=0
are considered as one sensor node which provides LoB mea- in our simulations in Section V-A.
surements with rate λ when it is on. Wake-up times for Wealsoassumethatonlyoneobjectappearsatanylocation
s
different nodes can be either selected randomly or through a inthe observationregionanditsspeedis lowenoughto allow
systematic process. As it is shown in [1] and [8], at relatively sufficient time for awake sensor nodes to sense the object.
large β (e.g., β > 0.05), the difference between the random Moreover, the object does not have any knowledge of the
and optimal scheduling can be practically ignored. Since the sensor node positions and, therefore, it can not intentionally
random scheduling of the wake-up times does not need any choose a trajectory that reduces the detection probability. We
extra controlmessage and it is not affected by time drifts, we furthermoreuseadiskmodelforsensing,inwhichaLoBofan
choose a random scheduling for this work. In other words, objectcanbemeasuredbyasensornodei,withmeasurement
it is assumed that different sensor nodes select their wake- noise variance σ2, if it is located inside a circle of radius R
i
up schedules independent of each other and with a uniform aroundthesensornodecenter.Using a localizationalgorithm,
distribution over the cycle of T. the fusion node estimates the object location based on the
Fig. 2 illustrates the geometry of the object location vs. recent bearing measurements and available information on
a sensor node at one snapshot. The object location and the object moving patterns. For the rest of the paper, we
velocity at time t ≥ 0 are P(t) = [P (t),P (t)]T and assumethatthefusionnodeusesaMaximumLikelihood(ML)
x y
V(t) = [V (t),V (t)]T, respectively. The ith sensor node localization algorithm. The ML localization method can be
x y
location is S(i) = [S (i),S (i)]T, and the object state at usedeithertoinitiallyestablishatrackortolocalizetheobject
x y
time t is denoted by Z(t)=[P(t),V(t)]T. The true bearing at any snapshot during the tracking task. Next, we describe
angle θ (P(t)) reflects the Direction of Arrival (DoA) of the the ML estimation techniques for both snapshot and interval-
i
object at the time that object emitted the acoustic signal. It is based bearings-only object tracking and analyze their quality
assumed that the object speed is small enough such that the performance.
time retardation factor due to changes in the object location
IV. QUALITYANALYSIS FORBEARINGS-ONLY OBJECT
during the propagation delay is negligible. At any time t, the
TRACKING
true bearingvalue fromthe ith sensor nodeis computedfrom
In this section, we study the accuracy and latency qual-
ity metrics for the ML estimation using measurements of
P (t)−S (i)
θ (P(t))=arctan y y . (1) both a single snapshot and an interval. Since localization
i
P (t)−S (i)
(cid:18) x x (cid:19) performance of an object is highly depend on the availability
of the LoB measurements, we also investigate the statistics
The bearingmeasurementattimet fromthe ith sensor node, of available measurements at any snapshot and during each
θ˜(t), i = 1,...,N, is subject to the additive zero-mean interval. Moreover, we analyze the distribution of a latency
i
that the localization task may face before having enough Similartoderivationsin[12],theFisherInformationMatrix
measurements for a well-posed tracking task. (FIM) J for estimation problem (3) is
Na 1
A. Single snapshot-based tracking J = ∇θ (∇θ )T, (4)
σ2 Ik Ik
As the first case, we consider a localization based on the Xk=1 Ik
T
estimated bearing measurements of one snapshot at time t. where ∇= ∂ ∂ . Further simplification results in
This case is useful for making fast responses to the tracking ∂Px(t)∂Py(t)
h i
querieswithshortdeadlines.Whenaqueryrequeststheobject Na 1
location at a later time t∗ (t∗ > tQ), the snapshot t can be J = σ2 d2 Ak , (5)
selected either veryclose to t∗ which results in highaccuracy k=1 Ik Ik !
X
localization, or closer to tQ which results in a faster response in which dIk is the relative distance between the object and
to the query. For the case that tQ > t∗, the snapshot t can sensor node Ik and
abveaislealbelceteindfocrlmosaetitoontoQn.thIenoabnjyecctamseo,vifnogrptat6=tertn∗s,,tuhseinogbjtehcet Ak = −sins(iθn2()θcIoks)(θ ) −sinc(oθsI2k()θcos)(θIk) . (6)
location Pˆ(t∗) at t∗ can be estimated from the computed (cid:18) Ik Ik Ik (cid:19)
IfC isthelocalizationerrorcovarianceofEq.(3),CRLB can
object location at time t, Pˆ(t).
be computed by taking inverse of J, and therefore we have
In order to have a well-posed localization problem, it is
C >J−1.
required to have LoB measurements from at least two sensor
Similar to [14], and with a small notation adaptation, we
nodes non-collinear with the object. Following a query at
set the expected localization error metric
time t , due to random wake up schedules, the system may
Q
experience a delay before the object is simultaneously within ρ(U)=[C]1,1+[C]2,2, (7)
thesensingrangeoftwoawakesensornodes.Thedistribution
which is a function of the awake sensor nodes set U =
of such delay is studied in Section IV-A1. Moreover, at any {S ,...,S }. Here [C] and [C] are the first and
snapshot t, we may have different number of awake sensor secIo1nd diagINonaal elements 1o,f1 matrix C2,2and are the error
nodes in the relative distance of R from the object. In this
variancesinestimationofthexandycoordinatesoftheobject.
section, we first analyze the localization accuracy of a fixed
Results of Theorem 1 in [14] show that the expected position
number of measurements at the snapshot and then investigate
error for localization based on N measurements is bounded
a
the effect of randomness from wake up schedules.
from below by
LetN bethetotalnumberofsensornodesintherangeRof
theobjectlocationP(t). Atanytimet dueto therandomness Na 1 −1
ρ(U)≥4 =ρ (U), (8)
oawf awkaekese-unpsorscnhoeddueslesp,rowveidimngayNhavLeoNBam(e0as≤uremNeant≤s. FNo)r (cid:16)kX=1σI2kd2Ik(cid:17) L
a
confirming that the lower bound of the error decreases as the
a feasible localization, we require to have N ≥ 2 awake
a numberofN availablemeasurementsincreases.Ontheother
sensor nodes. The set of awake sensor nodes is denoted by a
hand,themeasurementsfromfarthersensornodestotheobject
U = {S ,...,S } where indices I ,...,I are sensor
I1 INa 1 Na ormeasurementswithlargernoisevariance,havemoreimpact
node IDs. Similar to [12] and with some notation adaptation
on increasing the lower bound.
and considering the possible non-equal measurement noise
Next, we use the result in (8) to quantify the lower bound
variances, from the Gaussian assumption of the measurement
onthe overallexpectedposition errorof(3) where,dueto the
noise, ML localization at snapshot t can be formulated as
randomness of wake-up schedules, the value of N may vary
a
Pˆ(t)=argmin Na 1 |θ˜ −θ (P(t))|2 . (3) eavllesnufbosretasfiwxiethdaβt.lFeoasrtthtweoerarmoroanngalNysisse,nwsoernneoeddetso. Wcoenshiadveer
P σ2 Ik Ik
(t)kX=1 Ik ! M = 2N −N −1 of subsets Ui with 1 ≤ i ≤ M. With the
duty-cycle β, the probability of having each sensor node in
In (3), Pˆ(t) = {Pˆ (t),Pˆ (t)} is the object location estimate
x y “on”state is β. Therefore,the probabilityof havingthe active
antotdiemIekt,,aθ˜nIdk(θtI)k(isPt(hte))noisisiytsbceoarrriensgpomnedainsugrheympeontthfersoizmedsebnesaorr- set Ui is P(Ui,β)=βγ(1−β)N−γ, (9)
ing computed from (1). In addition, σ2 is the measurement
Ik where γ = |U |. Considering all the possible combinations
noise variance of sensor node I . Due to the nonlinearity of i
k
θ (P(t)), Eq. (3) is a nonlinear Least-Squares estimator. for the awake sensor nodes and their probabilities, we can
Ik conclude the following Proposition.
The ML estimator in (3) is asymptotically unbiased [13],
Proposition 4.1: If U refers to the set of all sensor nodes
and its error covariance is bounded below by the Cramer-
in the object range, for a network with duty-cycleβ, the total
RaoLowerBound(CRLB).Inaddition,thecovarianceofML
expected object localization error ρ (U,β) satisfies
estimatorconvergestoCRLBwhenN goestoinfinity,andfor T
a
measurements with high SNR the error variance approaches ρ (U,β)≥ P(U ,β)ρ (U )=ρ (U,β), (10)
T i L i L
the CRLB with smaller values of N [13].
a UXi∈U
cases that “on” states of two sensor nodes have overlaps. For
the sake of not adding an extra parameter, we assume that
the object presents in an area coveredby N nodes, each with
a asynchronous duty-cycle β. The probability of having no
intersection of equal to or greater than ǫ is
0
P (N,β,ǫ )=
Ø 0
1 β ≤ ǫ0,
T (11)
1−N(β− ǫ0) N−1 ǫ0 <β < 1 + ǫ0,
 T T N T
 0 β ≥ 1 + ǫ0.
(cid:0) (cid:1) N T
Forβ≤ ǫ0,sincethemaximumoverlapsizeisβT,twosensor
T
nodes can not have overlaps of size equal or greater than ǫ .
0
For ǫ0 <β < 1+ǫ0,theprobabilityiscomputedbasedonthe
T N T
factthatforthecaseofnointersectionequaltoorgreaterthan
ǫ the distance between every two consecutive starting points
0
Fig.3. Overlapbetween“on”statesoftwodifferentsensornodesforthree of the awake states must be greater than βT −ǫ0. Moreover,
differentcases.Case1isanexampleofoverlapsforǫ0/T <β≤ 21+ǫ0/T. if we fix the first starting point as the start of the cycle, the
Case 2 and 3 are examples of two different possible situations for β >
12 +ǫ0/T, where overlap is one segment for case 2 and two segments for distance between the last starting point in the cycle and the
case3. end of the cycle must be greater than βT −ǫ0. Now that first
starting point is selected, we are not allowed to put any other
starting point in two intervals of size βT −ǫ at the first and
0
where P(U ,β) and ρ (U ) are derived from (9) and (8), endofthecycle.Therefore,alltheotherN−1startingpoints
i L i
respectively. In practice, due to the hardware response and have to be located in an interval of size T −2(βT −ǫ ). The
0
sampling delay, a sensor node may experience an initial minimum required distance between the start points of any
activation delay (start-up time) ǫ at the beginningof its “on” two (among N −1) consecutive awake periods is βT −ǫ .
0 0
state. Thus, wheneverapplicable, it is more precise to replace Therefore, the total minimum required distance between all
β in(9)and(10)byβ−ǫ /T.Itisalsonotablethat,thelower consecutive pairs is (N −2)(βT −ǫ ). Having that, we can
0 0
bound in (9) is particularly derived for a case that relative uniformly distribute the start of N −1 awake states on the
distancesbetweentheobjectandsensornodesareknown.This remaining interval T −2(βT −ǫ )−(N−2)(βT −ǫ ). This
0 0
analysisisveryusefulfordesignerstoestimatethelocalization willleadtothesecondequationin(11).Lastly,forβ ≥ 1+ǫ0,
N T
errorboundfor strategic pointsin the observationregion.The there is always at least one overlap of size equal or greater
derivationoferrorlowerboundforanunknownobjectlocation than ǫ .
0
is the subject of our future work. From (11), the probability of presence of at least 2 out of
Next, in Section IV-A1, we study the delay distribution N sensor nodes with “on” states overlap size of equal to or
for the period of time that it takes for at least two sensor greater than ǫ is
0
nodes in the range of the object to be awake at the same
P(N,β,ǫ )=1−P (N,β,ǫ ). (12)
time and take measurements with negligible time differences. 0 Ø 0
Moreover,in Section IV-B we extend the localization method
It is notable that any sensor node pair with overlap of ǫ be-
from snapshot-based to an interval-based, which increases
tweentheir“on”statescanbeconsideredasanewmonitoring
the chance of receiving more samples from network thus
node with a duty-cycle ǫ/T and a cycle T. The distribution
improving the tracking accuracy.
of delay, t , of being monitored by such a node is
d
1) Delay distribution single snapshot tracking response:
ǫ τ
To localize a moving object with no extra information on its F (τ)=P(t ≤τ|overlap size=ǫ)= + , (13)
characteristics,itisrequiredtohave(almost)synchronousLoB td|ǫ d T T
measurementsfromatleasttwonodesintheobjectrange.The where 0≤τ ≤T−ǫ. The first term in (13) refersto the case
sensitivity level of the localization accuracy to the time dif- of t = 0, expressing the concurrency of the object presence
d
ference between the samples from two different sensor nodes (or the query time) with the overlap of the “on” states. The
depends on the object speed. The faster the moving object secondtermin(13)istheprobabilityofpresenceoftheobject
is, the more accurate synchronization between the samples in the field (or receiving the query) at most τ time unit from
is required. Therefore, to study the delay distribution, it is the start of the “on” state.
necessary to investigate the probability of overlaps between Now,weusetheresultsin(12)and(13)toapproximatethe
“on” states of at least two of the sensor nodes in the range of delay distribution in the presence of two sensor nodes. For
the object. ǫ0 < β ≤ G = 1 + ǫ0, the size of the applicable overlaps
T 2 T
Letǫ betheminimumoverlaprequiredbetween“on”states (see an example in Fig. 3, case 1) has a uniform distribution
0
for a successful localization. Fig. 3 shows three examples of on [ǫ ,βT] with the average ǫ¯= (βT +ǫ )/2. Accordingly,
0 0
G) 1 1 Simulation
≤0.8 ASnimaulylatictiaoln β=0.55 G) 0.8 β=0.65 Analytical
β > β=0.75
<0.6 β=0.45 β 0.6
ǫ0τ,T0.4 β=0.35 ≤τ, 0.4 β=0.95 β=0.85
≤td0.2 β=0.25 Pt(d 0.2
(
P β=0.15
00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
τ τ
Fig. 4. Comparison of simulated and analytical results for the tracking Fig. 5. Comparison of simulated and analytical results for the tracking
response latency distribution at the presence of two sensor nodes. The response latency distribution at the presence of two sensor nodes. The
simulated results are from a Monte Carlo simulation with a sample size simulated results are from a Monte Carlo simulation with a sample size
200000forǫ0=0.1,T =1and ǫT0 <β≤G. 200000forǫ0=0.1,T =1andβ>G.
the distribution of delay t is approximated by: W is selected based on the application requirements and the
d
amount of information that we have about the object. Any
ǫ ǫ¯ τ
0
P(td ≤τ, T <β ≤G)≃min T + T,1 (1−PØ(2,β,ǫ0)). information on the object moving pattern such as velocity,
n o (14) acceleration, and also road map of the object may be used in
The second term in (14) confirms the presence of overlaps the modeling of hypothesized bearings of different snapshots
of size equal to or greater than ǫ . The first term has been in the time frame W. With a proper selection of W, we can
0
adopted from (13) and refers to the delay distribution for the assume that the object velocity is constant, during each time
averagecasewithoverlapsizeǫ¯.Theproductofthetwoterms frame, V = [V ,V ]. We further assumed that the velocity
x y
in (14) is due to the independency assumption between the range is such that the time retardation factor is negligible.
detection process and the overlap occurrence. For β > G, For the sake of simplifying the notation, in the remainder
sensor node overlaps are either one or two segments with at of this section, we assume that query time t is the same as
Q
leastonesegmentlargerthanǫ .Fortypeone(seetheexample the time t∗ that the object location requested. In this case, to
0
in Fig. 3, case 2), the overlap size ǫ is uniformly distributed localize an object at time t∗, a fusion node processes all the
1
on [(2β−1)T,βT] and has an average of ǫ¯ =(3β−1)T/2. receivedsensor nodemeasurementstakenduring[t∗,t∗+W].
1
Fortypetwo (see an exampleinFig. 3, case3),bothoverlaps Consequently,weuseN (W)todenotethenumberofsamples
a
together have a fixed size ǫ = 2βT −T. Considering these receivedduringthistimeframe.TheMLestimatorforthiscase
2
two types of overlapping and their occurrence probabilities is
togetherwith (13), the delaydistributioncanbe approximated
by Zˆ(t∗)=arg min Na(W) 1 |θ˜ −θ (Z(t∗))|2 , (16)
P(t ≤τ,β >G)≃2(1−β)min ǫ¯1+τ,1 + Z(t∗) Xi=k σI2k Ik Ik !
d
(cid:26) T (cid:27) (15) where Zˆ(t∗) = [Pˆ(t∗),Vˆ(t∗)] is an estimate of the location
ǫ +τ
(1−2(1−β))min 2T ,1 . andvelocityattimet∗.IndicesI1,...,INa(W) in(16),referto
(cid:26) (cid:27) thesensornodeIDsofalltheNa(W)receivedmeasurements.
Toverifytheaccuracyofthederivationin(14)and(15)we Moreover, θ˜Ik, 1 ≤ k ≤ Na(W), is the kth received
perform a Monte Carlo analysis with ǫ0 = 0.1 and T = 1. measurementduring the time frameW and θIk(Z(t∗)) refers
The outcome is reported in Figs. 4 and 5. We note that our to the hypothesized LoB measurement. From the constant
approximate is more accurate for the case of ǫ0 < β ≤ G velocity assumption on the frame W, any object location at
T
compared to β > G. It is also notable that, in general, the time t∈[t∗,t∗+W], is given by
analytical result of the distribution of the tracking for two
P(t)=P(t∗)+(t−t∗)V(t∗). (17)
sensor nodes can be used as an upper bound for the delay in
networks with more than two sensor nodes.
From (17) and (1), the hypothesized LoB measurement is
In the nextsection, we analyze the quality of tracking with
time interval measurements. P (t∗)+∆ V −S (I )
θ (Z(t∗))=arctan y Ik y y k . (18)
Ik P (t∗)+∆ V −S (I )
B. Interval-based tracking (cid:18) x Ik x x k (cid:19)
As a result of duty-cycling with random schedules, for Here ∆ = t − t∗ and t is the timestamp of the kth
Ik Ik Ik
some snapshotswe may not haveenoughawake sensor nodes received measurement.
in the object range. Therefore, it is beneficial to extend Next, we present the CRLB for the localization problem
the localization method for single snapshot to fuse all the (16). Similar to the derivation in (4) and (5), the error
measurementsduringa timeframe.In practice,theframesize covariance matrix can be computed from the inverse of the
FIM Inordertostudythelatencyaccuracytradeoffsfordifferent
scenarios, in what follows, we present some test cases.
Na(W) 1 A A ∆
J = Ik Ik Ik , (19)
k=1 σI2kd2Ik (cid:18) AIk∆Ik AIk(∆Ik)2 (cid:19) V. SIMULATION RESULTS
X A. Study cases
where A , σ2 and d are the same as in (6).
Ik Ik Ik In this section, we study the tradeoffs between the local-
Notice that the localization performance improves with
ization accuracy and its response latency through some study
moresamplesN (W)andwiththesmallernoisevarianceσ2 .
a Ik cases.
IncreasingthewindowsizeW,resultsinthelargerN (W)at
a
1) Staticobjectandvariableduty-cycleβ: Similartotheil-
thecostofahigherresponselatencyandpossiblytheviolation
lustratedscenarioinFig.1,N =10sensornodesarerandomly
of the constant velocity assumption in (17). Therefore, it is
deployedwith a uniformdistribution in an observationregion
important to select the proper value for W based on the
of size 10×10. All the sensor nodes have the same sensing
application requirements and constraints. Similar to (8) and
range of 8. Bearing measurements from all sensor nodes are
(10), we can analyze a lower bound for the localization error
assumed to have a zero-mean Gaussian noise with variance
in (16) which is the subject of our future work.
σ2 =0.09unlessspecifiedotherwise.Itisfurtherassumedall
Due to the important role of N (W) in the localization
a
sensor nodes duty-cycle asynchronouslywith the same β and
error and the performance bound, we next study the first two
T.
statistics of N (W) for a fixed duty-cycle. Since different
a
In this case, we have studied the performance of MLE
sensornodesscheduletheirduty-cyclesindependently,wecan
localization in (3) for a static object. We set t∗ = 0 as the
starttheanalysisforthecasewithonlyonesensornodeinthe
time when the object appears in the field. At any snapshot
field. Let N (W) be the number of provided LoBs by one
1,a
t ≥ 0, the fusion announces a new estimate for the object
sensor node. Then, from the uniform distribution of the start
location, utilizing all the received LoBs from t∗ to t. The
time of the awake periodovera cycleT,the averagevalue of
error is defined as the Euclidean distance between the actual
N (W) is
1,a
and estimated object location. Smaller values of the average
E[N (W)]=⌊βWλ ⌋. (20)
1,a s
and variance of the error imply, respectively, more accurate
Next, Var[N (W)] = E[N2 (W)] − E[N (W)]2 is and less uncertain localization.
1,a 1,a 1,a
used to compute the variance value. The complete derivation The result in Fig. 6 is computed from a Monte Carlo
of E[N2 (W)] can be found in Appendix A, and from that simulationwithasamplesize4000ontheactivationschedules
1,a
the Var[N (W)] is, andconfirmthefollowingpoints;(1)Duetothechanceofre-
1,a
ceivinghighernumberofsamples,networkwithahigherduty-
Var[N (W)]=
1,a cycle reaches a higher localization performance in a shorter
λ2sW2(β− 31WT −β2) 0< WT ≤φ time; (2) For different activation schedules, a network with a
 λ2sφ2(TW − 13φT2−W2) φ< WT ≤(1−φ) smallerduty-cycle(e.g.,β =0.2)experiencesmorevariations
 λ2s(T −W)2(β− 13T−TW −β2) (1−φ)< WT ≤1 in the set of awake sensor nodes. Active set variations result
(21) in different localizations and, hence, increase the variance.

where φ = min(β,1 − β). Since the number of available
Average of error Variance of error
samples in a complete cycle T is fixed, the variance has a 2 2.5
β=0.2 β=0.2
periodic pattern. Therefore, for W > T, Var[N1,a(W)] = β=0.4 2 β=0.4
Var[N (w′)] where W ≡w′ (mod T). 1.5 β=0.6 β=0.6
1,a
Next, to compute the average and variance of available β=0.8 1.5 β=0.8
samples from all the sensor nodes in the field, we use the 1 β=1 β=1
1
independencyassumption between the sensor node schedules.
0.5
Foranetworkwithauniformsensornodetopologyofdensity 0.5
ρ, at any time, the average number of sensor nodes in the
relative distance R of the object is πR2ρ. Therefore, the 00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
t,Time t,Time
average and variance of available number of samples from
(a) (b)
all sensor nodes in the range are
Fig.6. (a)Averageoferrorvs.localizationresponselatency,(b)Varianceof
E[N (W)]=(πR2ρ)E[N (W)],and errorvs.localization response latency forafixednoisevariance σ2 =0.09,
a 1,a (22) T = 1, and different duty-cycles. The results are based on a Monte Carlo
Var[N (W)]=(πR2ρ)2Var[N (W)]. simulation with a sample size 4000 and with fixed locations for the object
a 1,a
andsensornodes.
From (21) and (22), Var[N (nT)] = 0 for integer values
a
of n, which results in the minimum variation for the overall 2) Static object and variable noise variance σ2: The goal
localization error of (16). Also the variance value in (22) and of this case is to study the effect of noise variance σ2 on
thereforethelocalizationperformancevariationaremaximized the static object tracking accuracy.For this purpose, a similar
for W =nT/2, where n is an odd integer. scenarioasincase1withafixedduty-cycleβ =0.5andthree
different noise levels σ2 ∈ {0.04,0.09,0.16} is considered. 5
As seen from Fig. 7, to guarantee a minimum localization β=0.2, Known Velocity
accuracy,asystemwithhighernoiselevelexperiencesalarger error4 ββ==00..46 KKnnoowwnn VVeelloocciittyy
response latency. eof3 ββ==01. 8K nKonwonw nV eVloecloitcyity
Average of error Variance of error ag β=0.2, Unknown Velocity
2 σ2=0.04 1.5 σ2=0.04 Aver2 ββ==00..46 UUnnkknnoowwnn VVeelloocciittyy
1.5 σσ22==00..0196 1 σσ22==00..0196 1 ββ==01. 8U nUknnkonwonw nV eVloecloitcyity
0
1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
t,Time
0.5
0.5 Fig. 9. Average ofthe interval-based localization error with W =0.4sec
for a moving object and with a noise variance σ2 = 0.09. The results are
fromaMonteCarlosimulationwithasamplesize10000,fixedsensornode
00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 locations, andrandomwake-upschedules.
t,Time t,Time
(a) (b)
Fig.7. (a)Averageoferrorvs.localizationresponselatency,(b)Varianceof 8
earnrdordvifsf.elroecnatlivzaartiiaonnceres.spTohneserelastuelntscyarfeorbaasfiexdedonduatyM-coynctleeβCa=rlo0.s5im,Tula=tio1n, ror ββ==00..24, KKnnoowwnn VVeeloloccitityy
withasamplesize4000andfixedobjectandsensornodelocations. er6 β=0.6 Known Velocity
of β=0.8 Known Velocity
ce β=1 Known Velocity
3) Static object and variable scheduling cycle T: In this an4 β=0.2, Unknown Velocity
case,westudytheeffectofschedulecyclesT ontheaccuracy ari β=0.4 Unknown Velocity
V β=0.6 Unknown Velocity
of a static object tracking. Again, a similar scenario as in 2
β=0.8 Unknown Velocity
Section V-A1 with a fixed duty-cycle β = 0.5 but different β=1 Unknown Velocity
T values is considered. At any time instance, the parameter 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
T does not affect the availability of each sensor node and the t,Time
probability of any sensor node being awake is β. As a result,
Fig. 10. Variance of of interval-based localization error with W = 0.4
changing T does not make a difference in the snapshot-based secforamovingobject andwithnoisevariance σ2 =0.09.Theresults are
localization. Furthermore, for an interval-based localization, fromaMonteCarlosimulationwithasamplesize10000,fixedsensornode
to achieve the same accuracy level, networks with larger locations, andrandomwake-upschedules.
scheduling cycles experience higher latency. Results in Fig.
8 confirm the above two points at t≃0 and t>0.
case, the variable Z is replaced by P . The Monte Carlo
t t
Average of error Variance of error simulation results under both assumptions are presented in
1.5 T=0.5 1.5 T=0.5 Figs. 9 and 10. Each point with t≥0.4 sec in Fig. 9 reflects
T=1 T=1 the error average of the interval-based localization over the
T=2 T=2
timeframe[t−W,t].Similarly,Fig.10reflectsthelocalization
1 1
error variance. As the results in both figures confirm, higher
duty-cycles β experience less localization error. Moreover,
0.5 0.5 comparing error averages in Fig. 9, we experience less error
when the velocity is known. Furthermore, it is notable that a
network with a smaller β (e.g., β = 0.2) experiences more
00 0.1 0.2 0.3 0.4 0.5 00 0.1 0.2 0.3 0.4 0.5 variations in the localization error at different times. This is
t,Time t,Time
(a) (b) a consequence of smaller N (W) and larger sensitivity of
a
the localization to the randomness of the awake sensor nodes
Fig. 8. (a) Average oferror vs. localization response latency, (b) Variance
of error vs. localization response latency for a fixed duty-cycle β = 0.5, set. Lastly, our localization with an unknown velocity has
σ2=0.09,anddifferentcyclevaluesT.TheseresultsarebasedonaMonte higher variation compared to the one with known velocity as
Carlo simulation with asample size 4000 and fixedobject andsensor node
estimating the velocity, in addition to the location, requires
locations.
larger number of samples.
4) Movingobjectandvariableduty-cycleβ: Forthisstudy, 5) Error bound for static object with variable duty-cycle
the scenario parameters are similar to those of previous cases β and noise variance σ2: In this case, we study the re-
and a moving object with constant velocity V =[10,−10] is lationship between the expected localization error and the
considered.Theinterval-basedlocalizationapproachdescribed analytical derived lower bound in (10). A tracking scenario
inSectionIV-BisimplementedwithW =0.4andconsidering with parameters from Section V-A1 is considered. The error
both conditions of known and unknown velocities. We use is defined as the square of the Euclidean distance and the
MLEin(16)forbothconditions,wherefortheknownvelocity average error is computed by a Monte Carlo simulation on
12 Analytic error lowerbound, σ2=0.01 8
nerror 108 SASASininimmmaauuullyylllaaattiiccttteee eedddrr rraaaoovvvrreee llrrrooaaawwgggeeeeerr bbeeeoorrrrrruuooonnrrr ddσσσ,,222 σσ===22000==...00200195..0295 oferror67 W=5 W=10 W=15 W=20
alizatio 46 Average45
oc 2 3
L
00 .2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 22 0 25 30 35 40 45 50 55 60
β t,Time
Fig. 11. Expected localization error ρT(U,β) vs. error lower bound Fig. 13. (a) Average of error vs. time for β = 0.5, T = 20 sec, and
ρL(U,β) for different bearing measurement noise variances and different different window sizes W. The results are from a Monte Carlo simulation
β values. withasamplesize400onsensornodeschedules.
hing) 44..33669989x 106 or30
Nort 44..33669967 err25 β=1 Snapshot−based
Latitude( 4444....3333666699992345 Averageof112050 βββββ=====00100.. ..52I52n 5 5StI en SInrtnevantarepavlrsp−avhsbla−ohal−btos−abte−bsadabes,sa deWes,d deW,= dW5==55
4.3649.10756 4.0758 4.076 4.0762 4.0764 4.0766 4.0768 4.077 4.0772 4.0774 4.0776
Longitude (Easting) x 105 5
5 10 15 20 25 30 35 40 45 50 55 60
Fig.12. Sensorarraypositionsandtrackedvehicletrajectoryintheuniversal time (sec)
transverse mercator(UTM)coordinates.
Fig. 14. Localization accuracy for T = 20 sec, W = 5 sec, and β ∈
{0.25,0.5,1}.TheresultsarefromaMonteCarlosimulationwithasample
size400.
thesensornodewake-upschedules.Fig.11showstheaverage
error from the simulation and the analytical lower bound
from (10) when different values of β and σ2 are considered. on quality metrics for tracking techniques in (3) and (16), we
The results in Fig. 11 confirm the following points: (1) For haveselectedthreedifferentduty-cyclesβ ∈{0.25,0.5,1}and
largerβ values, a largernumberof sensor nodesare awake at created duty-cycled patterns with random wake-up times for
any snapshot resulting in a smaller localization performance all the six sensor nodes.The duty-cycleframe size T is set to
variability, smaller error variance, and closer average error to 20 sec and the tracking performancefor different frame sizes
its lowerbound;(2)For smallermeasurementnoise variances W is studied. Fig. 13 shows the average error for different
σ2, the localization accuracy is higher and the error varies frame size and suggests that, for this tracking problem, the
less by increasing the number of measurements. Therefore, interval-based tracking with W = 5 sec outperforms other
by increasing β, the average error for a smaller σ2 has less studied time frames. Each point in Fig. 13 is the average
change and is closer to the error lower bound. errorcomputedfromaMonteCarlosimulationontheinterval-
based localization with random wake-up schedules. The error
B. Simulation on the real data
is defined as the Euclidean distance between the estimated
In this section we test the localization techniquesdescribed vehiclelocationanditsgroundtruth.Fig.14showstheaverage
in Section IV on the collected data at Aberdeen Proving errorforbothsnapshotandinterval-basedtrackingfordifferent
Ground, Maryland, by the Army Research Lab in June 2002. β values. The results in Fig. 14 confirm that using higher
During the field experiment, six microphone arrays were duty-cyclesandapplyingtheinterval-basedlocalizationhighly
deployed in a 2.5 Km by 2.5 Km field to collect the acoustic improves the accuracy.
signature of some vehicles, see Fig. 12. The diameter of each
sensorarraywas8feetandconsistsof7microphonesarranged VI. CONCLUSION
in a circular configuration. The data was collected at each In this paper, we study the effect of random duty-cycling
sensor in the array at the rate of 1024 samples per second. on the accuracy and response latency of bearings-only object
In this section, we have used the data from a period of tracking.Inordertoinvestigatethetradeoffsbetweenresponse
time (60 sec) in the deployment where a tracked vehicle is latencyanditsaccuracy,asetofclosed-formformulasandap-
traversing a path (shown in Fig. 12) from the north to south proximationsarederived.Extensivesimulationsareconducted
of the field. Each sensor node data is processed to obtain to study the validity and applicability of the analysis both on
LoB using the MVDR method [3]. At the training phase, the synthetic and real data. The provided analytical results in
having the LoB measurements and exploiting the available this paper are of high importance for energy efficient sensor
ground truth on the vehicle trajectory, the noise level of each network designs with multi-attribute Quality of Information
sensor node is estimated. To study the effect of duty-cycling (QoI) design metrics. Designers can apply these results to