Achieving Range-free Localization Beyond Connectivity
Ziguo Zhong
Department of Computer Science
University of Minnesota
zhong@cs.umn.edu
Tian He
Department of Computer Science
University of Minnesota
tianhe@cs.umn.edu
Abstract
Wireless sensor networks have been proposed for many
location-dependent applications. In such applications, the
requirement of low system cost prohibitsmany range-based
methods for sensor node localization; on the other hand,
range-free localization depending only on connectivity may
underutilize the proximity information embedded in neighborhood
sensing. In response to the above limitations, this
paper presents a range-free approach to capturing a relative
distancebetween1-hopneighboringnodesfromtheirneighborhood
orderings that serve as unique high-dimensional
location signatures for nodes in the network. With little
overhead, the proposed design can be conveniently applied
as a transparent supporting layer for many state-of-the-art
connectivity-based localization solutions to achieve better
positioningaccuracy. Weimplementedourdesignwiththree
well-knownlocalizationalgorithmsandtesteditintwotypes
ofoutdoortest-bedexperiments: an850-foot-longlinearnetwork
with 54 MICAz motes, and a regular2D networkcovering
an area of 10000 square feet with 49 motes. Results
show that our design helps eliminate estimation ambiguity
with sub-hop resolution, and reduces localization errors by
as much as 35%. In addition, extensive simulations reveal
an interestingfeature of robustnessfor our design underunevenly
distributed radio propagation path loss, and confirm
itseffectivenessforlarge-scalenetworks.
Categories andSubject Descriptors
C.2.4 [Computer-Communication Networks]: DistributedSystems
General Terms
Algorithms,Design,Performance,Experimentation
Keywords
WirelessSensorNetworks,Range-freeLocalization
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SenSys’09, November 4–6,2009, Berkeley,CA,USA.
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1 Introduction
Wireless sensor networks (WSN) have been considered
as a promising tool for many location-dependent applications
[1, 2], e.g., battlefield surveillance [3], environment
datacollection[4],eventorhumanlocalization[5,6]. Inaddition,someoftheroutingprotocols[7,8]andnetworkmanagement
mechanisms proposed for such networks are built
ontheassumptionthatgeographicparametersofeachsensor
node are available. Althoughsensor node localization plays
an important role in all those systems, it is itself a challenging
problem due to extremely limited resources available at
eachlow-costandtinysensornode.
Many ideas have been proposed for node localization in
WSN. Based on whether accurate ranging is required, there
are basically two types of methods: (i) range-based localization
and (ii) range-free localization. Range-based localization
could achievegoodaccuracybut costly for requiring
eitherper-noderanginghardware[10,12,14,16,22]orcareful
system calibrationand environmentprofiling[9, 11, 40],
andthusisnotappropriateforlarge-scaleoutdoorsensornetworks.
Range-freeapproacheslocalize nodesbased on simple
sensing, such as wireless connectivity [26, 27, 29, 32,
33],anchorproximity[25,28,30],orlocalizationeventsdetection
[36, 37]. Amongthese, connectivity-basedsolutions
featurealowoverallsystemcost,however,bysacrificinglocalizationaccuracy.
Our work is motivated by the finding that localization by
means of mere connectivity may underutilize the proximity
information available from neighborhood sensing. Although
radio signal strength (RSS) is not considered a good
choice for physical distance estimation in many scenarios
because of unknown radio path loss factors, multi-path effects,
hardware discrepancies, antenna orientation, and so
forth [40, 41, 42], it does provide some useful distancerelated
information beyond indicating connectivity among
neighboringnodes. Our experimentalstudy confirmsthat in
outdoor open-air scenarios, the radio signal strength weakensapproximatelymonotonicallywiththephysicaldistance,
especially from the viewpoint of a single node, where RSS
might provide heuristic information about which neighboringnodeiscloserandwhichis
further.
Based on our empirical study, this paper introduces
a novel range-free approach to extracting relative distanceinformationfromneighborhoodorderingswhichserve
as unique high-dimensional location signatures for sensor
281nodes in the network. Instead of offering yet another new
localization method, the design described in this paper can
beconvenientlyappliedasatransparentsupportinglayerfor
many state-of-the-art connectivity-based localization algorithms,providingalow-costbuteffectivesolutionforsystem
accuracy.
We augmented three range-free localization algorithms,
i.e., MDS-MAP [26], DV-Hop [27], RPA [33], with our design,
and evaluated the effectiveness of the proposed design
in two types of outdoor test-bed systems: an 850-footlong
linear network with 54 MICAz motes, and a regular 2-
dimensional network covering an area of 10000 square feet
with49motes. Systemevaluationshowednoticeableperformancegainsincludingeliminatingestimationambiguityand
reducinglocalizationerrorsbyasmuchas35%. Inaddition,
extensive simulation demonstrated the effectiveness of our
design for large-scale networks and revealed an interesting
feature of robustnessto the unknownand spatially unevenly
distributedradiopathloss.
The rest of the paper is organized as follows: Section 2
surveys related work. Section 3 explains the motivation behindthepaperwithempiricaldata.
Themaindesignisintroduced
in Section 4. Section 5 briefs three range-free protocolsonwhichwe
evaluatedourwork. Section6reportsoutdoor
test-bed experiments. Section 7 discusses results from
simulation. Finally,Section8concludesthe paper.
2 Related Work
Based on whether ranging is conducted at the resourceconstrained
sensor nodes, most of the previous work about
node localization can be categorized into one of following
two classes: (i) range-based [9, 10, 11, 14, 12, 15, 16, 17,
18, 19, 20, 21, 22, 24], and (ii) range-free localization [25,
26, 27,28, 30,29, 31,32, 34,35, 36,37, 38].
Range-based solutions try to estimate absolute distances
or angles among randomly deployed sensor nodes and then
apply triangulation or multilateration for location calculation.
Many range-based methods use techniques such as
Time of Arrival (ToA) [13], Time Difference of Arrival
(TDoA) (e.g., Cricket [10], AHLos [11], TPS [12]) and
Angle of Arrival (AOA) (e.g., APS [22], SpinLoc [18]) to
measure distance or angles among nodes and anchors (also
called beacons or reference nodes with pre-known location
information). Those methods can be accurate but costly by
adding per-node additional hardware [10, 12, 14], requiring
intensive tuning [17] or not suitable for large-scale systems
due to their limited effective range [10]. Although some researchhastriedtoutilizeRSS(ReceiveSignalStrength)with
noise filtering for distance estimation or for wireless fingerprint
matching (e.g., Radar [9], wMDS [43], SpotOn [45],
Indoor GPS [46], Sequence [48], Ranking [49]), empirical
studies [39, 40, 41] have concluded that unless careful calibrationandenvironmentprofilingcanbeaccomplished,RSS
isnotagoodchoiceforaccurateranging.
Range-free methods have applied many smart ideas for
pursuinga low-costdesign. Earlyrange-freesolutionsmade
use of the proximity information to anchor nodes. Typical
examples are Centroid [25], APIT [28], Concave [35] and
Self [30], in which the high cost of anchors is supposed to
beamortizedwithalargenumberoflow-costordinarysenor
nodes. To achieve a good accuracy, however, a high anchor
density is required, which is impractical for large-scale systems.
Concurrently, wireless connectivity-based protocols
such as DV-Hop [27], MDS-MAP [26], RPA [33], Amorphous
[32] and so on, proposed using local neighborhood
sensing to build hop-based virtual distances for large-scale
sensor network localization. In those systems, only a small
numberof anchorsare necessary for constructingthe global
coordinates,whichsignificantlyreducesthesystemcost. Recent
work helps solve problems of “holes” [31, 34] and
“complex shapes” [29], contributing to connectivity-based
solutions in practical irregular node deploymentwith obstacles.
However, we found that localization by means of connectivity
alone does not make full use of information availablefromlocalneighborhoodsensing.
Realizing the limitations of previous works, this paper
presents the idea of regulated signature distance (RSD), a
metric of the proximity among 1-hop neighboring nodes.
Acting as a transparent supporting layer, our design can
effectively improve the system accuracy of state-of-the-art
connectivity-basedlocalizationwithlittle extracost.
3 Empirical Dataas Motivation
Thispaperismotivatedbyourexperimentaldatashowing
thatin theoutdoorenvironments,
• Network-wide monotonic relationship between radio
signalstrengthandphysicaldistancedoesnothold,but
• Per-node monotonic RSS-Distance relationship holds
well, i.e., any single node’s RSS sensing results for its
neighboring nodes can be used as an indicator for the
relative“near-far”relationshipamongneighbors.
In the following, we first explain results from a preliminary
test, and then provide data obtained from large-scale
outdoorexperimentsforverification.
3.1 Preliminary Experiments
Figure 1 shows RSS sensing results from MICAz nodes
inseveraloutdoorexperimentsconductedintwotypesofenvironment:
grasslandandparkinglot. Inthetest, we placed
9 sender nodes at different distances from a receiver node.
Each sender node broadcast 100 packets with 0dBm sending
power, and the receivernode recorded the RSS upon receivingthepacket.
Inthegrass-landscenario,weperformed
the test twice with two differentreceivernodesplacedat the
samelocationandwithoutmovingorswitchingsendernodes
(Grass Land Test1 and Grass Land Test 2, respectively). In
RSS (in dBm)
−55
−60
−65
−70
−75
−80
−85
Grass Land Test 1
Grass Land Test 2
Parking Lot Test 1
Parking Lot Test 2
−90
0 10 20 30 40 50 60 70 80
Distance (in feet)
Figure1. ExperimentalResults: RSS vs. Distance
282RSS (in dBm)
−60
−80
RSS vs. Physical Distance in Linear Network
−90dBm
↓
−100
0 16 32 48 64 80 96 112 128 144
Distance Between Node Pairs (in feet)
(a) SystemLevel View: RSSvs. Physical Distance
RSS (in dBm)
−60
−80
RSS vs. Physical Distance in Regular 2D Network
−90dBm
↓
−100
0 16 32 48 64 80 96 112
Distance Between Node Pairs (in feet)
Similarity
1
0.5
RSS Ordering vs. Distance Ordering in Linear Network
Similarity
1
0.5
RSS Ordering vs. Distance Ordering in Regular 2D Network
0
0 5 10 15 20 25 30 35 40 45 50 55
node ID
the parking lot scenario, identical sets of nodes were tested
during day-time (Parking Lot Test 1) and at night (Parking
LotTest2). Testswereconductedmultipletimes,andresults
did notshow significant changesin the overallshapesof the
curvesshowninFigure1.
Figure 1 tells that at the system level, using absolute values
of RSS for distance estimation is not reasonable since
identical RSS values may correspond to different distances.
However,for each individualcurve(i.e., from the viewpoint
of a single node), RSS values mostly decreased monotonically
with increasing distance, conveyinginformation about
relative“near-far”relationshipsamong1-hopneighbors.
3.2 Large-scale Experiments
We then conductedlarge-scale outdoorexperimentswith
two types of networksto verify the phenomenafound in the
preliminary test. The first experiment was a linear network
containing54MICAznodeswitha16-footintermediatedistance
between adjacent nodes covering a 850-foot length
along a road. In the second experiment, we constructed a
regular 2D network with a 7 ×7 grid-shaped layout including49nodesoccupyinganopen-airparkinglotareaof10000
squarefeet. Thesetupof thelarge-scaleexperimentswill be
furtherdetailedin Section6.
Figure2reportstheempiricaldataobtainedfromthetwo
test-beds. Figure 2(a) plots the sensed RSS value for each
pair of nodes against the distance between them, which verifiesthatmonotonicRSS-distancerelationshipdoesnothold
for the whole network. In both the linear network and the
regular 2D network, on one hand, RSS may vary dramatically
for identical distance. For example, as shown in the
right sub-figureof Figure 2(a), RSS rangesfrom -60dBmto
-90dBm for a 16-foot distance in the 2D network. On the
other hand, a single RSS value may correspond to a wide
range of distances. For instance, as shown in the left subfigure,
-90dBm could range from 32 feet to 112 feet in the
linear network; even worse, -90dBm RSS covers almost all
ofthedistancespectrum,i.e.,from16feetto112feet,inthe
2Dnetworkshowingin therightsub-figureofFigure2(a).
However,examiningthedatafromtheviewpointofasingle
node tells a different story. For any node, say ui, we
can obtain an ordered node list, say A, by listing ui’s 1-hop
neighbors according to their RSS values sensed at ui in decreasingorder;andanothernodelist,
sayB, byorderingui’s
(b) EachNode’s Point of View: Feature of RSSMonotonic Attenuation
Figure2. Empirical DatefromLargeScaleExperiments
0
0 5 10 15 20 25 30 35 40 45 50
node ID
1-hop neighbors with increasing physical distance. Ideally,
if the sensed RSS decreases monotonically with increasing
distance,AandBshouldbeidentical. Wedefinethesimilarity
betweentwo lists AandBasthepercentageofaccordant
node pairs between them. For example, let A = (u1,u2,u3)
and B = (u1,u3,u2), then {u1, u2} is an accordantnode pair
betweenAandBsincenodeu1 isorderedaheadofu2 inboth
A and B; while {u2, u3} is not since their ordering gets reversedfromAtoB.
WecanseethatifAandBareconsistent
withtheirsimilarityclose to1,themonotonicfeatureholds.
Figure 2(b) shows the similarity results for all nodes in
twotest-beds. We canseefromtheleftsub-figurethatin the
linear network, most of the nodes have a similarity close to
1 (the minimum, mean and maximum similarities are 0.86,
0.96 and 1, respectively). It means that in the linear network,fromsinglenode’spointofview,RSSvaluesfor1-hop
neighbors are approximately monotonic with the distance.
This finding still holdsfor the 2D regularnetwork as shown
in the right of Figure 2(b), where the minimum, mean and
maximumsimilaritiesare0.81,0.88and0.96,respectively.
Above experiments confirm that the monotonic RSSdistance
relationship does not hold at the system level, but
approximatelyholdsfromtheviewpointofasinglenode.
3.3 AnalysisandDiscussion
In addition to the physical distance between two nodes,
there are many factors that affect RSS sensing results. Table
1 lists some major aspects. We marked an aspect with
a “ √ ” if pre-deploymentengineeringeffortscouldpossibly
be applied to reduce its impact, or a “×” if it would be hard
orcostlytoaddress.
At the sender side, besides the sending power, the carrier
frequency, modulation, baud rate and etc. determine
the band-width, center frequency and spectrum shape [47],
which all affect the RSS at the receiver side. Most of those
parameters can be configured with small offset errors and
maintained relativelystable duringthe runtime. Antenna is-
Table 1. MajorFactorsAffectingRSS Sensing
Typesof Factors
P
√
RFTransmit Parameters: Sending Power,Carrier Frequency, Modulation ...
√
AntennaIssues: TX/RXGain,Radiation Pattern, Orientation, Height ...
√
Random Noise: Interference,Mobile Effects,ElectronicPulse ...
Propagation Path Loss: Terrain,Vegetation,Obstacle,MagneticField ... ×
Node-levelSensing Discrepancy: LNA,ADCRef. Voltage,Ground Noise ... ×
283sues such as isotropic gain, orientation and etc. can also be
carefully engineered in the design phase. For transient random
noise, traditional filtering methods are able to help reduceits
impact. All ofthe aboveare consideredaddressable
withoutsignificantin-fieldcalibration.
On the contrary, unpredictable environmental factors are
much harder to handle. For example, radio path loss is unknown
and costly to profile in most cases since it is temporally
dynamic and spatially unevenly distributed. Another
difficult issue is the sensing hardware discrepancy among
different nodes. For example, a tiny bias of the ADC reference
voltage or small variance of LNA (low noise amplifier)
gain caused by different ground noise levels, may lead
to different RSS values at two nodes, even when their received
signal strengthsare equivalent. Runtime sensing discrepancy
among nodes is caused by many dynamic reasons
andper-nodein-fieldcalibrationcouldbe verycostly.
The above two “×” factors are hard to address in a deployedsystem,however,fromtheviewpointofasinglenode,
theirimpactscouldbelesssevere. Firstly,a1-hopneighborhoodareais
muchsmallerthanthe wholeregioncoveredby
the network,so onenode’slocal sensingalleviatesthe problem
of spatially unevenly distributed path loss. In addition,
RSS from a single node’s sensing avoids the issue of nodelevel
receiverside hardwarediscrepancy. Therefore,as confirmed
by our empirical data, the monotonic RSS-distance
relationshipholdsmuchbetterin thecase ofonenode.
Unfortunately,this heuristic correlation is not utilized by
previous localization methods based on mere connectivity,
where only a binary “1” or “0” is evaluated for either connectedornot,resultingin
adegradedsystemaccuracy.
4 Design: a RelativeDistance
Thissection presentsthe main design of a range-freerelativedistanceamong1-hopneighboringnodes.
4.1 Neighborhood Ordering as a Signature
Given the RSS sensing results for neighboring nodes, a
nodecanobtainaneighborhoodorderingwithtwo steps:
• Sorting its 1-hop neighbors according to their signal
strengthbydecreasingorder,and
• Addingitselfasthefirstelementinthesortednodelist.
A simple example is shown in Figure 3. In this figure,
graph G on the left illustrates the connectivity of the network.
On the right, each node generates a node list starting
with itself and containing all its 1-hop neighbors which are
orderedbydecreasingsignalstrength,i.e.,byincreasingdistanceinanideal
case.
Connectivity Graph G
6
2
1
3
Example �eighborhood Ordering
Node 1 S1 : 1 6 2 4 5 3
Node 2 S2 : 2 1 6 3
Node 3 S3 : 3 2 1
Node 4 S4 : 4 5 1 6
Node 5 S5 : 5 4 6 1
4 Node 6 S6 : 6 1 5 2 4
5
Figure3. NeighborhoodOrdering
Foranynodeui,weconsideritsneighborhoodorderingSi
as a high-dimensionalsignature of the node in the network.
Si has a vector format and contains all 1-hop neighbors of
nodeui withthreeimportantfeatures:
• Si isuniqueforeachnodeui.
• Si isposition-dependent. Si embedslocation-relatedinformationonbothconnectivityandproximity.
• Si isobtainedwithoutranging;instead,itisarange-free
sensingobservationbeyondconnectivity.
For the sake of clarity, in the following design part, we
first use ideal neighborhood orderings for conveying ideas.
Namely,Si isconsistentwiththeorderingaccordingtophysical
distance. Latersectionswill verifythe effectivenessand
robustnessofourdesigninpracticalnoisyscenariosthrough
bothtest-bedandsimulationexperimentation.
4.2 SD: Signature Distance
The high-dimensional signatures of sensor nodes can be
obtained easily via local signal strength sensing. In this
section, we explain the concept and rationale of signature
distance (SD) which quantifies the difference between two
high-dimensionalsignatures. SDisthefirststeptowardarelative
distance that effectively reflects the physical distance
relationshipsamongneighboringnodesin thenetwork.
4.2.1 Formation,DefinitionandCalculationofSD
Say that a pair of nodes um and un get flipped between
two signatures Si and Sj, if the ordering of um and un in Si
gets reversed in Sj. For example, as shown in Figure 4, the
ordered node pair {1,6} in S2 = (2,1,6,3) gets reversed to
{6,1}inS5 = (5,4,6,1).
S2 : 2 1 6 3 S2 S5
S5 : 5 4 6 1 1 6 6 1
Figure4. 1Explicit Node-PairFlip
Thereare threetypesofpotentialnode-pairflipsbetween
two signatures Si and Sj : (i) explicit flip, (ii) implicit flip,
and (iii) possible flip. If node um and un appear in both Si
and Sj, then we can easily tell whether this node pair gets
flipped or not, as the example shows for node pair {1,6} in
Figure 4. This type of flip is called explicit flip. Implicit
flips and possible flips are somewhat tricky, as explained in
thefollowingwithexamples.
As shown in the left part of Figure 5, S2 and S5 have different
sets of node elements. For example, S2 = (2,1,6,3)
containsnode2while S5 = (5,4,6,1)doesnot. In thiscase,
many node pairs in S2 do not have corresponding counterparts
in S5. For instance, {2,1},{2,6}in S2 have no related
node pairs in S5 since node 2 is absent in S5. We solve this
problem by attaching “wildcards” to S2 and S5, as depicted
S’2
S’5
S2
S5
S’2 S’5 S’2 S’5
Figure5. 10Implicit Node-PairFlips
284by gray squares □ in Figure 5. Formally, for Si and Sj, a
numberof |Si ∪Sj −Si| wildcardsare attachedto Si to make
S ′ i . In S′ i , each wildcard can stand for any node u ∈ Sj but
/∈Si, namely ∀u ∈ (Si ∪Sj −Si). For example, in Figure 5,
eachgraysquarein S ′ 2 canstandforeither5or4,andagray
square in S ′ 5
can be substituted with either 2 or 3. Since
“wildcardnodes”attachedtoSi arenaturallyregardedasfurtherawaythanneighborsofui
inSi,S ′ i maintainsthefeatures
as a location-dependentsignature without violating proximity
relationships embedded in the original Si. Figure 5 lists
all implicit node pair flips from S ′ 2 to S′ 5
. We call them implicit
flips since they are not as obviousas the explicit flips.
For example, node pair {2,1} in S ′ 2 can only have a counterpart
node pair {1,□} in S ′ 5
, where □ stands for 2 in this
case. {2,1} and {1,□} is a flip from S ′ 2 to S′ 5
because an
orderreversionoccursnomatterwhich □in S ′ 5 standsfor2.
S’2 : 2 1 6 3
S’5 : 5 4 6 1
S’2 S’5
2 3 3 2
4
5
5 4
Figure6. 2 Possible Node-PairFlips
Figure 6 gives examples of the possible node-pair flip.
Formally, if a node pair {um,un} appears in Si but neither
um nor un is in Sj, we consider it possible that {um,un} gets
reversed in Sj. For example, as shown in Figure 6, {2,3}
from S ′ 2 can only have a counterpart {□,□} in S′ 5
. With no
additionalinformation,this nodepair givesapossible nodepairflipwith
50%probability.
Based on the above explanations, we now define the signaturedistancebetweenSi
andSj asfollows:
Definition1: thesignaturedistanceSD(Si,Sj)isequaltothe
summationofthenumberofexplicitflipsFe(Si,Sj), implicit
flips Fi(Si,Sj), and possible flips Fp(Si,Sj) times 0.5 (50%
probabilityofflip forpossiblenodepairs),namely,
SD(Si,Sj) =Fe(Si,Sj)+Fi(Si,Sj)+Fp(Si,Sj) ×0.5 (1)
TakingS2andS5inFigure4asanexample,Fe(S2,S5) =1
as shown in Figure 4, Fi(S2,S5) = 10 as listed in Figure 5,
and Fp(S2,S5) = 2 as depicted in Figure 6. According to
definition1,we haveSD(S2,S5) =1+10+2×0.5=12.
In fact, each node-pairflip from S(i) to S(j) corresponds
to one and only one node-pair flip from S(j) to S(i), so the
definitionofSD guaranteesSD(Si,Sj) ≡SD(Sj,Si).
Algorithm1illustratesamethodforcomputingthesignature
distance. First of all, Si and Sj get sorted at line 1 and
2 with complexity O(Klog(K)), where K = |Si ∪Sj| is the
total numberofneighborsoftwonodesui anduj. Thefunction
wildCard()at Line3attaches (K − |Si|)wildcardsto Si
andfillsthesewildcardswithelements (Si ∪Sj −Si)that are
ordered the same as they are in Sj. Line 4 performs similarly
to Sj to obtain S ′ j . Line 3 and 4 have a cost of O(K)
with sorted input ˜Si and ˜Sj. Line 5 computes the total number
of explicit and implicit node-pairflips using a variant of
heap-sortalgorithm[51]withcomplexityO(Klog(K)). Line
6calculatesthenumberofpossibleflips. Line7givestheresult
ofSD(Si,Sj)basedonEquation1. Thetime complexity
ofAlgorithm1isO(Klog(K)). NormallyK ≪n,wherenis
thetotal numberofnodesin thenetwork.
Algorithm1 SignatureDistance
Input: Si andSj
Output: SD(Si, Sj)
1: ˜ Si=sort(Si); %O(Klog(K))
2: ˜Sj = sort(Sj); % O(Klog(K))
3: S ′ i =wildCard(Si, ˜Si, ˜Sj); %O(K)
4: S ′ j = wildCard(Sj, ˜Sj, ˜Si); % O(K)
5: Fe+i =HeapSort(S ′ i ,S′ j ); % O(Klog(K))
6: Fp = (K−|Si|)(K−|Si|−1)
2
+ (K−|Sj|)(K−|Sj|−1)
2
; % O(1)
7: SD(Si,Sj) =Fe+i +Fp ×0.5; % O(1)
4.2.2 Insightsinto theSignatureDistance
In a signatureSi, each orderednodepair containsaproximity
relationship. For example, as shown in Figure 7(a),
S2 = (2,1,6,3), the ordered node pair {1,3} in S2 means
that from node 2’s point of view, node 1 is closer than node
3. In other words, if we divide the plane with B(1,3) (depicted
as a dashed line), which is the perpendicularbisector
of the line segment L(1,3) connecting node 1 and 3, the ordering
of {1,3} in S2 indicates that node 2 is located on the
leftside ofB(1,3).
Based on the aboveexample,we can see that a node-pair
flip between two signatures correspondsto passing a bisector
line. For example, as shown in Figure 7(a), S2 contains
nodepair {1,3}whichgetsreversedto {3,1}inS3,meaning
that node 2 and node 3 are on different sides of B(1,3). So
goingfrom node 2 to node 3 along the straight line segment
L(2,3) needs to pass B(1,3), as shown in the figure. Figure
7(b) illustrates an opposite case, in which signature S2
and S3 have an accordant node pair {2,1}, indicating that
node 2 and 3 are located at the same side of B(1,2) and
L(2,3) does not intersect B(1,2). Figure 7(c) shows an ex-
,nodepair {6,3}hasacoun-
ampleoftheimplicitflip. InS ′ 2
terpart {3,□}inS ′ 3
,where □isawildcardstandingfornode
6 in this case. This implicit flip corresponds to an intersectionofL(2,3)andB(3,6),asshowninthefigure.
6
S2 : 2 1 6 3
S3 : 3 2 1
2
L(2, 3) 3
L(2, 3) 3
L(2, 3)
1
5
L(1, 3)
4
B(1, 3)
S2 : {1,3} S3 : {3,1}
Passing B(1, 3)
6
S2 : 2 1 6 3
S3 : 3 2 1
2
1
5
B(1, 2)
(a) Explicit Flip (b) Non-Flip (c) Implicit Flip
Figure7. The Insight ofa Node-PairFlip
4
S2 : {2,1} S3 : {2,1}
Without Passing B(2, 1)
6
S’2 : 2 1 6 3
S’3 : 3 2 1 6
2
1
5
4
3
B(3, 6)
S2 : {6,3} S3 : {3,6}
Passing B(3, 6)
Ingeneral,we havethefollowingobservation:
Observation 1: a node-pair flip {um,un} ⇒ {un,um} from
Si to Sj indicates that the line segment L(ui,uj) passes the
perpendicularbisectorlineB(um,un).
285One remark about the above observation is that there is
only one intersection between L(ui,uj) and B(um,un) when
node-pair flip happens. This is because two straight lines
(segments)haveat mostonecrossingpoint.
On the other hand, based on the definition of signature
distance, SD(Si,Sj) evaluates the difference between two
signatures Si and Sj by counting the total number of nodepairflips.
Thus,we canconcludefromObservation1that
Observation2: SD(Si,Sj)isequivalenttothenumberofbisector
lines we need to pass if goingfromneighboringnode
ui to uj alongthe linesegmentL(ui,uj).
Anotherkeyobservationconcerningthephysicaldistance
betweentwo neighboringnodesisthat
Observation3: underroughlyuniformbisectorlinedensity,
forneighboringnodesui anduj,thenumberofbisectorlines
passed by line segment L(ui,uj) is approximately proportional
to the physical distance between ui and uj, i.e., the
lengthofL(ui,uj), denotedasPD(ui,uj).
The insight offeredby Observation 3 is that longer physical
distances provide a higher probability of passing more
bisector lines. Here bisector line density is defined as the
number of lines exist between two positions with unit distance.
Figure 8 shows an example for this observation. Figure8(a)drawsthelayoutofperpendicularbisectorlines(denoted
as dashed lines) for all node pairs, and line segments
connectingnode1withits1-hopneighbors(denotedassolid
lines). Figure 8(b) lists line segments, corresponding number
of bisector lines they passing, and signature distances
between their terminal nodes, respectively. We can see that
the numberof bisector lines passed by a line segment is approximately
proportional to the length of the line segment,
i.e.,thephysicaldistancebetweentwo nodes.
6
2
1
5
4
3
Line
Segments
Bisectors
Passing
(a)
(b)
Figure8. PhysicalDistancevs. BisectorLinesPassing
1
1
6
2
3
4
1 4 6
1 5 8
1 3 8
Combining the above three observations, we have a
heuristic rule as follows: for two neighboring nodes ui and
uj, their signature distance is approximatelyproportionalto
thephysicaldistancebetweenthem,namely
SD
3
4.5
6.5
8.5
8.5
SD(Si,Sj) ∝PD(ui,uj) (2)
An important remark is that Equation 2 is not valid for
non-neighboring node pairs most of the time. This is because
SD(ui,uj) only counts the number of passed bisector
linesgeneratedbynodepairsfromsetSi ∪Sj. Figure9shows
an example for explaining this remark. In this figure, ui, uj
and uk are located far from each other, and their neighboring
areas (denoted with dashed circles) do not overlap. As
a result, SD(Si,Sj) and SD(Si,Sk) both are determined only
Neighborhood of ui
ui
Neighborhood of uj
uj
Network Area
Neighborhood of uk
Figure9. Far-awayNodePairs
by possible flips that depend on the number of nodes in Si,
Sj and Sk. Under similar 1-hop radio range, SD(Si,Sj) and
SD(Si,Sk) could have similar values, although the physical
distances of these two node pairs may be dramatically different.
In a word, the heuristic relationship of Equation 2 is
meaningfulonlyfor1-hopneighboringnodes.
BasedonEquation2,signaturedistancecanbeutilizedas
a relative distance for localization purposes because it approximately
reflects the “near-far” relationships among 1-
hop neighbors. However, in some cases, SD can be biased
duetospatiallynon-uniformbisectorlinedensitythroughout
thenetworkarea,aviolationoftheconditioninObservation
3. In the next section, we propose a more robust relative
distance,i.e.,RegulatedSD,to addressthisproblem.
4.3 RSD:Regulated Signature Distance
This section introduces the regulated signature distance,
orRSD forshort,asarefinedversionofSD.
4.3.1 TheRationalebehindSD Refinement
Spatially non-uniform bisector line density could affect
the effectiveness of SD as a relative distance. This problem
comes from two aspects: (i) local node placement; and
(ii) network wide neighborhood size, both of which are explainedin
thefollowingwith examples.
As shown in Figure 10(a), L(2,3) passes 4 bisector lines
andL(6,1)intersects3. However,L(2,3)ismorethantwice
as long as L(1,6). This inconsistency is caused by spatially
unbalanced bisector line density within the local area. For
example, the area close to node 1 has a higher bisector line
density, while boundary areas close to node 2 and 3 have a
low density. This micro-level observation indicates that SD
needstoberefinedconsideringthelocalbisectorlinedensity.
L(6, 1)
6
2
1
5
L(2, 3)
4
3
(a) Local Bisector Density (b) Neighborhood Size
Figure10. MotivationforSD Regulation
uk
ui
W
v
uj
c
B(v, c)
At the macro level, for the same physical distance, the
value of signature distance could be different, depending
on the neighborhood size. For example, as shown in Figure
10(b), two nodes ui and uj has a constant physical distance
W. When they have a large 1-hop radio range illustrated
by two big circles, both of them have node v and
286c as neighbors included in their signatures. In this case,
SD(ui,uj) counts the passing of B(v,c) which is the bisectorfornodepair
{v,c}. However,if ui anduj haveasmaller
1-hopradiorange(e.g.,duetostrongambientnoise)denoted
by gray-filled circles. Node v, u are absent from signatures
of ui and uj, so SD(ui,uj) doesnot countB(v,c), resultinga
smaller value of SD(ui,uj) comparing to the previous case.
In fact, under random node deployment with uniform density,
the signature distance for node ui and uj should be regarded
as a relative value in terms of the dimension of their
neighborhoodareathataffectsthenumberofavailablebisectorlinescountedforpassinginthemap.
Based on the above analysis, we propose the Regulated
SignatureDistance,orRSD forshort,definedasfollows:
√
K
RSD(ui,uj) =SD(Si,Sj) · (3)
K(K −1)/2
√
K
, where
K(K−1)/2
Equation 3 refines SD(Si,Sj) with a factor
K = |Si ∪Sj|isthetotalnumberofnodesintheneighborhood
of node ui and uj combined. In this equation, K(K −1)/2
calculatesthenumberoflocalbisectorlines,usedtonormalize
SD(Si,Sj) with the local bisector density; √ K estimates
the diameter of this neighborhood, which puts the factor of
neighborhood size into consideration. We formally derive
andexplainthisequationinthe nextsection.
4.3.2 RegulationFactorDerivation
For neighboringnodesui and uj, let |Si ∪Sj| =K. There
are a total of nB = K(K −1)/2 bisector lines generated by
node pairs in Si ∪Sj. According to the Pie-Cutting Theorem[50],nB
bisectorlinesdividethelocalareaintonR small
regions,where
nR =O((n 2 B +nB +2)/2) =O(n 2 B) (4)
Let S be the size of the local area occupiedby the neighborhoods
of ui and uj. The expected size and diameter of each
small region,denotedasE[sR] andE[dR]respectively,are
E[sR] = S
nR
= S
O(n 2 B ),
E[dR] = α · √ E[sR] = α√ S
O(nB)
where αisaconstantfactordeterminedbytheshapemodelingofthesmall
region.
Residual
ui
Passing �B(ui,uj) Bisectors
...
uj
Residual
(�B(ui,uj) − 1) Small Regions
L(ui,uj)
Bisector Line
Figure11. BisectorLines andSmall Regions
Suppose that line segment L(ui,uj) intersects NB(ui,uj)
bisector lines, then L(ui,uj) passes (NB(ui,uj) −1) small
regionsasshowninFigure11. Addingresidualsatbothends
(eachcountedasahalfregion),wegetanapproximation
(5)
PD(ui,uj) ≈NB(ui,uj) ·E[dR] (6)
meaning that the distance between ui and uj approximately
equalsthenumberofsmallregionstimesexpecteddiameter.
SD(ui,uj) estimates the number of bisector lines that
L(ui,uj)passes, i.e.,SD(ui,uj) ≈NB(ui,uj). So wehave
PD(ui,uj) ≈SD(Si,Sj) ·E[dR] =SD(Si,Sj) · α√ S
O(nB)
Foruniformrandomnodedeployment,theexpectednumber
of nodes in an area is proportional to the area size,
namely E[K] = φ ·S where φ is the node density. Since
nB =K(K −1)/2,wecanrewriteEquation7 as
ϕ
PD(ui,uj) ≈SD(Si,Sj)·
√ K
K(K −1)/2 , where ϕ = α √ (8)
φ
ϕ is a constant scaling factor that can be eliminated without
violating near-far relationship among different neighboring
nodes. Thus,weobtaintheproposedrelativedistanceRSD:
√
K
RSD(ui,uj) =SD(Si,Sj) · (9)
K(K −1)/2
4.3.3 RSDvs. SD
Figure 12 compares SD and RSD serving as a relative
distance. This figure is obtained from the simulation of a
network composed of 150 randomly deployed nodes covering
a 500 ×500 square feet area with 100-foot radio range.
For each node pair in the network, both SD and RSD are
computed,andplottedagainsttheirphysicaldistanceinFigure
12(a) and Figure 12(b) respectively. Figure 12 conveys
two major points: (i) within 1-hop radio range, RSD offers
a much better linear correlation with physical distance than
SD;(ii)bothfiguresconfirmsourearlierremarkaboutEquation2that
aftera1-hopradiorange,signaturedistanceis no
longercorrelatedwithphysicaldistance. Fromthisexample,
we can see that RSD is a better choicethan SD to serve as a
metricofrelativedistance.
SD
800
600
400
200
← 1−Hop Radio Range
0
0 100 200 300 400 500 600 700
Physical Distance (in feet)
(a) SDvs. PhysicalDistance (b) RSDvs. Physical Distance
Figure12. Correlationwith PhysicalDistance
RSD
6
5
4
3
2
1
(7)
← 1−Hop Radio Range
0
0 100 200 300 400 500 600 700
Physical Distance (in feet)
5 Designas aSupporting Layer
The design of RSD can be implemented as a supporting
layer that is transparent to the localization algorithms. As
showninFigure13,wesimplyusethesmallestaccumulated
RSDalongapathbetweentwonodesinsteadoftheshortestpath
hop count as the estimated relative distance. Specially,
theaccumulatedRSD betweentwo nodesisdefinedas
• For 1-hop neighboring nodes ui and uj, accumulated
RSD equalsRSD(ui,uj)computedwithEquation9;
• Fornon-neighboringnodesuianduj,accumulatedRSD
is calculated as the summation of the RSD values of
neighboringnodesalongapathbetweenui anduj.
287Hop-Based
Distance between
two nodes
=
Least number of
hops
Localization Algorithm
RSD
Distance between
two nodes
=
Smallest
accumulated RSD
Relative Distance Estimation Layer
PHY: Neighborhood Sensing
Figure13. RSD Design Embedding
In the following, we briefly describe three connectivitybasedlocalizationschemesstudiedinourevaluation.
5.1 Connectivity-Based Schemes
The key idea in connectivity-based localization is to use
the number of communication hops between two nodes to
evaluate the physical distance between them. The following
threeschemesaretypicalexamples:
• MDS-MAP[26]byY. Shang,W. Ruml,et al.
• DV-Hop[27] byD.NiculescuandB. Nath.
• RPA [33] byC. Savarese,J. M.Rabaey,et al.
Here we provide brief descriptions of these algorithms and
detailscanbefoundin[26, 27,33].
MDS-MAP [26] first forms a distance matrix A in which
thevalueattheithrowandthe jthcolumnistheshortesthopbased
distance betweennodeui and uj. The algorithmcomputes
a “relative map” of the network with the MDS (multidimensionalscaling)
[52] techniquegivendistance matrixA
as input. After that, an “absolute map” can be obtained by
scaling and rotating the “relative map” according to the absolutecoordinatesbuiltwithat
least threeanchornodes.
DV-Hop[27]usesamechanismthatissimilartoclassical
distance vector routing. After beacon flooding from more
than three anchors in the network. The algorithm estimates
theexpectedphysicaldistancefor1-hopwith
HopSize = ∑i̸=jDistance(vi,vj)
∑i̸=jHops(vi,vj)
(10)
where vi,vj are anchors. Distance(vi,vj) and Hops(vi,vj)
are the physical distance and the least number of hops betweenvi
andvj,respectively. Foreachsensornodeui,itestimates
its distance to anchor vi with HopSize ×Hops(ui,vi).
Finally, each node’s location is computed with least-square
multilaterationonavailableanchors.
RPA [33], which is proposed independently from DV-
Hop, uses a similar mechanism of hop-based distance estimation
called Hop-TERRAIN for its first step. Besides, it
introduces an iterative refinement step for position adjustmentbasedonlocal
sensingresults. Basically,at iterationk,
thepositionofnodeui isrecomputedbasedontheestimated
positionsofitsneighborsobtainedfromiterationk−1. More
sophisticated approaches such as confidence based filtering
arealso proposedinRPA.
5.2 DesignEmbedding
With local RSS sensing results, RSD can be calculated
at each node or in a localization server. For range-freeconnectivity
based localization algorithms such as MDS-MAP,
DV-Hop, and RPS, applying RSD is convenient and incurs
little overhead. Without modifying the major design of the
originalalgorithm,theRSDvaluecanbeusedinsteadof“1”
(indicating connection) for neighboringnodes. Specifically,
inMDS-MAP,forexample,everythingremainsthesameexceptthatthedistancematrixnowholdsthesmallestaccumulatedRSDvaluesalongapathinsteadoftheshortesthopdistance.
For DV-Hop, similarly, the relative distance turns to
thesmallestaccumulatedRSDinsteadofshortest-pathhops.
Expectedphysicaldistancefor1unit ofRSD isgivenby
RSDunitSize = ∑i̸=jDistance(vi,vj)
∑i̸=jRSD(vi,vj)
(11)
ForRPA, besidesembeddingRSDintheHop-TERRAIN
step,therefinementstepalsobenefitsfromapplyingRSDfor
localiterativepositionadjustment.
5.3 Complexity ofRSD Embedding
InvolvingRSD in localization introduceslittle additional
cost. Algorithm 1 costs O(Klog(K)) for SD calculation,
where K is the maximum length of a signature sequence in
the network, and normally K ≪n where n is the total number
of nodes. Even in a centralized localization system, additional
overhead of RSD calculation is O(nK 2 log(K)) (n
nodesandeachhasatmostK −1neighbors),giventhecomplexityoftheothercomputationalcomponents.
Forinstance,
MDS has a complexity of O(n 3 ) for the step of matrix decompositionalone[44].
Aboutcommunication,theonlyadditionaloverheadisforsignatureexchangeamongneighboringnodes.
Signaturesareveryshortandcanbepiggybacked
economically on messages during the network initialization
phase. Importantly, signature exchange only occurs within
1-hopneighborhoodwithout multi-hopflooding. Therefore,
embedding RSD in a connectivity-based localization does
notaffectthe scalabilityofthesystem.
6 Test-bed Experimentation
In this section, we report two types of outdoor experimentswith54and49MICAzmotesrespectively.
6.1 Experiment I:LinearNetwork
We start our test-bed evaluation from a linear network
widelyappliedintransportation-relatedroadnetworks.
6.1.1 ExperimentSetup
As shown in Figure 14, 54 MICAz motes were deployed
on the grass covered curb along a road (with surrounding
obstacles including trees, metal fence and parked vehicles).
Each node was left 8 inches above the ground and
the distance between two immediate nodes was about 16
feet. Everynodebroadcasts100packetswithcarrier-sensing
and back-off time intervals set from 200ms (millisecond)
to 1690ms individually for collision avoidance. The radio
sending power was 0dBm at channel 26 (Fc = 2480MHz)
to avoid possible WiFi interference from the surroundings.
Each packet contained the sender’s node ID and a sequence
number of the packet. When a node received a packet, it
logged the sensed signal strength, sender’s ID, and the sequencenumberofthepacketintoitsflash
memory.
Table 2 lists some of the collected information about the
linear network. The whole length of the network was about
848 feet with 15 hops between two terminal nodes (node#1
288�ode #54
�ode #1
In-field View
Figure14. Test-bed ExperimentsI:Linear Network
Hops vs. Distance (within 1−Hop)
2
3
RSD vs. Distance (within 1−Hop)
15
Hops vs. Distance (Multi−Hop)
30
RSD vs. Distance (Multi−Hop)
Hop Distance (in hops)
1.5
1
0.5
ρ = 0
RSD
2.5
2
1.5
1
0.5
ρ = 0.89
Hop Distance (in hops)
10
5
ρ = 0.98
RSD
25
20
15
10
5
ρ = 0.99
0
0 20 40 60 80 100 120
Physical Distance (in feet)
(a)
0
0 20 40 60 80 100 120
Physical Distance (in feet)
(b)
0
0 200 400 600 800 1000
Physical Distance (in feet)
Figure15. Distance CorrelationComparison: RSD vs. Hop(LinearNetwork)
and node#54). The maximum 1-hop radio communication
range in this experimentwas about 144 feet and the average
1-hopneighborhoodsize was6.7(nodes).
Table 2. StatisticsoftheLinear Network
Scale Max. Hops Max. 1-Hop RFRange Neighborhood Size
≈848(feet) 15 ≈144(feet) 6.7(nodes)
6.1.2 DistanceCorrelations
Foreitherthe traditionalhop-baseddistance orthe newly
proposed RSD, their effectiveness as a relative distance in
localization is determined by the correlation between their
valuesandthe physicaldistance.
Figure 15 illustrates the correlations between hop-based
distance, RSD, and the physical distance. Figure 15(a)
and 15(b) plot all the node pairs within 1-hop communication
range in the linear network. In both figures, the X-axis
isthephysicaldistancebetweentwonodes,andtheY-axisis
the hop-based distance and RSD, respectively. Figure 15(a)
reveals that all 1-hop node pairs have an identical hop distance
of 1, thusthe correlationcoefficient is ρ =0 within 1-
hop range. On the contrary, as shown in Figure 15(b), most
of the node pairs with different distances can be differentiated
from each other according to their RSD value. The
empirical data shown in Figure 15(b) have a correlation coefficient
ρ = 0.89. Comparing Figure 15(a) and 15(b), we
canconcludethatRSD ownsasub-hopresolutionthatisnot
availablefromthetraditionalhop-baseddistances.
Figure 15(c) plots all the node pairs (both 1-hop and
multi-hop node pairs) according to their hop distance and
physical distance. Similarly, Figure 15(d) plots RSD (accumulatedRSD
formulti-hopnodes)againstphysicaldistance
for all node pairs. Comparing these two figures, although
the correlation coefficients are close (ρ = 0.98 for hop distance
and ρ = 0.99 for RSD), RSD provides better resolution.
InFigure15(c),aphysicaldistancecanonlybemapped
(c)
0
0 200 400 600 800 1000
Physical Distance (in feet)
toanintegerhopdistanceinadiscretemanner,whileinFigure15(d),themappingiscontinuous.
Localizationresultsin nextsection showthat RSD’s subhop
resolution can nicely solve the ambiguity problem, i.e.,
mappingcloselylocatednodesto identicalpositions.
6.1.3 LocalizationPerformance
We use the terminology “MDS-Hop”, “DV-Hop” and
“RPA-Hop” for the original hop-based approaches and
“MDS-RSD”, “DV-RSD” and “RPA-RSD” for correspondingmethodsembeddedwithRSD.
Let us first look at DV-Hop and DV-RSD. Figure 16(a)
showsthelocalizationresultsfrombothalgorithmswithtwo
terminal nodes of the linear network as anchors. In the figure,blackline
segmentsare plottedstarting fromnodes’deployedpositions(depictedas
blue dots) andpointingto correspondingestimatedlocations,
forclear observation. In the
left subfigure for DV-Hop, many nodes are mapped to identical
estimated locations, i.e., the ambiguity problem; while
DV-RSD does not encounterthis issue as shown in the right
subfigure. ThisresultconfirmsthatRSDoffersauniquesubhopresolutionwhilehop-baseddistancedoesnot.
Figure16(b)illustratesthe resultsofRPA-Hop andRPA-
RSD, both with two iterative refinement steps. Both RPA-
Hop and RPA-RSD achieved unique position estimation for
each node. For RPA-Hop, this is credited to the refinement
stepthatsmoothstheestimatedpositionofeachnodeoverits
neighbors, naturally solving the problem of clustered mapping.
From Figure 16(b), we can observe that RPA-RSD
achievesbetterlocalizationaccuracythanRPA-Hop.
Figure 16(c) shows the localization results from MDS-
Hop and MDS-RSD. We found that 2-dimensional MDS
is not stable for linear or close-to-linear networks because
of singularity issues in matrix decomposition. Here, 1-
dimensional MDS is applied first, and then two terminal
nodes are used as anchor nodes for map scaling and rota-
(d)
289DV−Hop Localization Results
DV−RSD Localization Results
Y Axis(in feet)
Y Axis(in feet)
Y Axis(in feet)
150
100
Deployed Positions
↓
50
0
↑
Estimated Positions
−50
0 100 200 300 400 500 600 700 800 900
X Axis (in feet)
150
100
50
0
RPA−Hop Localization Results
−50
0 100 200 300 400 500 600 700 800 900
X Axis (in feet)
150
100
50
0
MDS−Hop Localization Results
−50
0 100 200 300 400 500 600 700 800 900
X Axis (in feet)
Y Axis(in feet)
150
100
(a) DV-Hopvs. DV-RSD
Y Axis(in feet)
Deployed Positions
↓
50
0
↑
Estimated Positions
−50
0 100 200 300 400 500 600 700 800 900
X Axis (in feet)
150
100
50
(b) RPA-Hopvs. RPA-RSD
Y Axis(in feet)
0
RPA−RSD Localization Results
−50
0 100 200 300 400 500 600 700 800 900
X Axis (in feet)
150
100
(c) MDS-Hopvs. MDS-RSD
50
0
MDS−RSD Localization Results
−50
0 100 200 300 400 500 600 700 800 900
X Axis (in feet)
Figure16. Localizationin Linear Networks: Hop-BasedDistancevs. RSD
tion to obtain the 2-dimensionalabsolute map. Figure 16(c)
tells that MDS algorithmitself is notable to solvethe ambiguity
problem; while MDS-RSD shows better performance
withoutclusteredmapping.
Errors (in feet)
30
25
20
15
10
5
0
Max Error
Mean Error
MDS−Hop MDS−RSD DV−Hop DV−RSD RPA−Hop RPA−RSD
Methods
Figure17. Comparison: RSD vs. HopDistance
In our evaluation, the localization error is defined as the
distance from the true position of a node to its estimated location.
Figure17showsthemaximumandmeanlocalization
errors from the results in Figure 16 for all six methods. We
canclearlyobservethatalltheRSD-embeddedmethods(“*-
RSD”) have smaller errors than their original “*-Hop” versions.
Specifically,themaximumandmeanerrorsof“MDS-
Hop”,“DV-Hop”and“RPA-Hop”getreducedby27%,22%,
35%,23%,29%and24%,respectively.
6.2 Experiment II:Regular2D Network
The second test-bed evaluation targets a 2-dimensional
grid-shaped network with consideration of investigation on
the impactofirregularnetworktopologytolocalizationperformance.
A grid can be easily transformed to irregular
shapesbysimplyremoving“pixels”inthe grid.
6.2.1 ExperimentSetup
Figure18showstheexperimentscenarioofalargeopenairparkinglotatnight.
Weplaced49MICAzsensornodesin
a7×7gridshape,asshownintheleftofFigure19. Thenetwork
coveredan area of about 100 ×100square feet. Rows
Figure18. Test-bedExperimentsII:Regular2DNetwork
Y Axis (in feet)
120
100
80
60
40
20
0
−20
Network Layout
0 50 100
X Axis (in feet)
40
30
20
10
0
7
6
5
4
3
2
1
Y Axis (in row)
Neighborhood Size
1
2 3
4 5
6 7
X Axis (in column)
Figure19. NetworkLayoutand NeighborhoodSize
and columnswere approximately16 feet apart. Similarly to
ExperimentI, each node was left 8 inches above the ground
withantennaspointingto thesky.
Table3listssomekeystatisticsregardingtheexperiment.
This 49-node system is a 3-hop network. 1-hop radio range
varies from about 20 feet to 100 feet among different node
pairs along diverse directions. The right subfigure in Figure
19 shows the neighborhood size of each node. The X-
axis and Y-axis in this figure depict the index of rows and
columns in the network. The height of each bar indicates
the 1-hop neighborhood size of the node at that position.
This figure verifies that nodes in the center of the network
have more 1-hop neighbors while the boundary nodes have
smallerneighborhoodsize.
290Hop Distance (in hops)
Hops vs. Distance (within 1−Hop)
2
1.5
1
0.5
ρ = 0
0
0 20 40 60 80 100
Physical Distance (in feet)
(a)
RSD
5
4
3
2
1
RSD vs. Distance (within 1−Hop)
ρ = 0.78
0
0 20 40 60 80 100
Physical Distance (in feet)
(b)
Hop Distance (in hops)
4
3.5
3
2.5
2
1.5
1
0.5
Hops vs. Distance (Multi−Hop)
ρ = 0.75
0
0 50 100 150 200
Physical Distance (in feet)
Figure20. DistanceCorrelationComparison: RSD vs. Hop(Regular2DNetwork)
(c)
RSD
10
8
6
4
2
RSD vs. Distance (Multi−Hop)
ρ = 0.86
0
0 50 100 150 200
Physical Distance (in feet)
(d)
120
100
120
100
60
50
MDS−Hop: Max Error
MDS−RSD: Max Error
MDS−Hop: Median Error
MDS−RSD: Median Error
Y Axis (in feet)
80
60
40
20
Y Axis (in feet)
80
60
40
20
Error (in feet)
40
30
20
0
0
10
−20
−20 0 20 40 60 80 100 120
X Axis (in feet)
(a) Localization byMDS-Hop
−20
−20 0 20 40 60 80 100 120
X Axis (in feet)
(b) Localizationby MDS-RSD
Figure21. LocalizationResultsofMDS-RSD andMDS-Hop
0
4 5 6 7 8 9 10
Number of Anchors
(c) StatisticalComparison
Table 3. Statisticsofthe 2DNetwork
Statistics 1-Hop 2-Hop 3-Hop
Number of NodePairs 575 519 82
Minimum Distance(in feet) 16 22.63 80
MedianDistance(in feet) 35.78 71.55 97.32
Maximum Distance(in feet) 97.32 124.96 135.76
6.2.2 DistanceCorrelations
Figure 20 shows the correlationsbetween hop-baseddistance,RSD,andphysicaldistanceinthisgrid-shaped2Dnetwork.
Figure 20(a) and 20(b) confirms that RSD providesa
better sub-hop resolution than hop-based distance. Furthermore,
Figure 20(b) also verifies that for MICAz motes using
monopole whip antenna with radiation pattern close to
isotropic [53], neighborhoodordering based on RSS can be
a good heuristic indicator for physical distance. For multihop
distance, Figure 20(c) and Figure 20(d) show that RSD
providesa highercorrelationcoefficientthan hop-baseddistance,
verifying that RSD is superior to hop-based distance
asarelativedistanceforproximityexpression.
6.2.3 LocalizationPerformance
Since the network has only 3 hopsthat is not suitable for
DV-Hop and RPS, to be fair, we only applied MDS-based
algorithms in this evaluation. Performance comparisons for
DV-Hop and RPS in non-linear-shape networks will be investigatedinSection7vialarge-scalesimulation.
Figure 21(a) and 21(b) depict localization results from
MDS-HopandMDS-RSDwith4randomlyselectedanchors
in the system, respectively. Inboth figures,the blue dotsare
the true positions of nodes and a line segment originating
from each dot point to its estimated position. We can see
from these two figuresthat MDS-RSD gives better localizationaccuracythanMDS-Hopin
thiscase.
Inordertoeliminatepossiblebiascausedbyanchorselection,werandomlypickeddifferentnumbersofanchors,from
4to10instepof1,andtriedeachfor1000runs. Figure21(c)
plots the averaged maximum and median errors for both
methods,fromwhichwecanconcludethat(i)MDS-RSDoffersasignificantlybetterperformanceoverMDS-Hopacross
allnumbersofanchors,and(ii)moreanchorgivesslightgain
inlocalizationaccuracy. By embeddingRSD, boththemaximum
and median errors are reduced greatly. For example,
for 4 anchornodes, RSD reducesthe maximumand median
errorsbyabout27%and30%,respectively.
Radio is notorious for irregularity, however, we are able
toachievebetterlocalizationaccuracythanconsideringconnectivity
alone. Test-bed experiments show that RSD provides
sub-hop resolution and correlates more with physical
distancethanthetraditionalhopdistance.
7 SimulationEvaluation
Inthissection,wereportselectedsimulationresultsabout
the performance gain from the RSD design for large-scale
networksunderdifferentsystemsettings.
7.1 The NoiseModel
Inthesimulation,weappliedthewidelyusedlogarithmic
attenuationmodel[43,44, 47,54] forRSS sensing:
Pi,j(t) =P0 −10βlog( PD(ui,uj)
)+Xi(t) (12)
wherePi,j(t)standsforthesensingresultatnodeui fornode
uj attimeinstancet. P0 isthereceivedpoweratashortreferencedistanced0.
PD(ui,uj)isthephysicaldistancebetween
ui and uj. β is the path loss factor (also called fading factor
in some literatures) and Xi(t) is a randomnoise at timet
for node ui following Xi(t) ∼ N(0,σ2 X ). We set a reference
d0
291Median Errors (in R)
1.2
DV−Hop
1.1
RPA−Hop
MDS−Hop
1
DV−RSD
RPA−RSD
0.9
MDS−RSD
0.8
0.7
0.6
0.5
0.4
0.3
0.2
4 6 8 10 12 14 16
Number of Anchors
Figure22. DifferentNo. ofAnchors
Median Errors (in R)
1
0.9
DV−Hop
0.8
RPA−Hop
MDS−Hop
DV−RSD
0.7
RPA−RSD
MDS−RSD
0.6
0.5
0.4
100 150 200 250 300 350 400
Number of Nodes
Figure23. DifferentNodeDensities
Median Errors (in R)
1.2 DV−Hop
RPA−Hop
1.1
MDS−Hop
1 DV−RSD
RPA−RSD
0.9 MDS−RSD
0.8
0.7
0.6
0.5
0.4
0.3
0.2
200 300 400 500 600 700 800 900 1000
Scale of the Map (Length and Width)
Figure24. DifferentSystem Scales
1-hopradiorangeofR =100feet. Thecorrespondingsignal
strengthwasset asthereceiversensitivitythreshold.
In the simulation, we modeled the area of interest as a
square map without holes where radio can not reach. More
complicatedmapscanbeusedwithworks [29,31,34]. Unlessotherwisementioned,Table4liststhedefaultsimulation
configurations for the following sections. All the statistics
reportedwereaveragedover50runsforhighconfidence.
Table 4. DefaultConfigurationsin Simulation
Parameter
Field Area
NoiseModel
Number of Sensor Nodes
Number of Anchor Nodes
Default Values&Description
500 (in feet)×500 (in feet)
β =4, σX =6for the whole map
200,randomly deployed withuniform distribution
8,randomly deployed
7.2 The Effectiveness ofRSD
This section evaluates the effectiveness of RSD by comparing
localization errorsbetween connectivity-basedmethods
(MDS-Hop, DV-Hop, and RPA-Hop) with corresponding
RSD-embedded versions (MDS-RSD, DV-RSD, and
RPA-RSD) under various system configurations. We normalized
the localization errors respect to a reference radio
range R =100 (feet) for consistency with the evaluationsin
previousliteratures[26].
ImpactofAnchor Density
In this experiment, we increased the number of anchor
nodesinthenetworkfrom4to16instepsof2. Asexpected,
Figure 22 shows that more anchors help improve the localization
accuracy for all methods. More notable is that RSD
embeddingismoreeffectivethanaddingseveralanchors,especially
after 8 anchors when curves become flat. By embedding
RSD, localization error gets reduced constantly by
around 30% for DV-Hop and RPA-Hop, and by about 10%
forMDS-Hop.
ImpactofNode Density
In this experiment, we increased the number of nodes in
the map from 100 to 400 in steps of 50. Figure 23 shows
that: (i) RSD-applied methods always showed better performance
(e.g., about 30% and 10% performancegain from
DV-Hop/RPA-HopandMDS-Hop,respectively);(ii)forDV-
Hop and and RPA-Hop, increasing the node number from
100to200didnotaffectthesystemaccuracytoomuch. This
is because at this stage, higher node density helps estimate
the hop-baseddistance. While after 200 nodes, the hop distancecanhardlybeimprovedbutmorenodesaremappedto
identical estimated positions, bringing up the error statistically;
(iii) for MDS-Hop and MDS-RSD, higher node density
mostly gives smaller localization error. This is because
the system becomes more robust to a single node’s sensing
errorwithmore1-hopneighbors.
ImpactofSystem Scale
In this experiment, we enlarged dimension of the map
from 150 feet (in length and width) to 1050 feet in steps
of 150 feet. The number of nodes in the network was increased
proportionally to maintain the same node density.
The number of anchors was kept constant. Figure 24 shows
theresults. We canseethat(i)RSD-appliedmethodsalways
achieve better localization accuracy than their original versions
based on hop distance; (ii) localization performance
gets worse in larger systems because the number of anchor
nodes were not increased proportionally; and (iii) MDSbased
methods are more robust than DV-based approaches
intermsofsystemscales. ThisisbecauseMDS-basedmethods
utilize both local proximity and the estimated distance
to remote nodes, while DV and RPA depend more on the
estimated distance to remoteanchors, the errorof which accumulateseasily
inlargernetworks.
7.3 The Robustness ofRSD
Inpreviousevaluations,thepathlossfactor βkeepsauniform
value in the map. In real system deployments, radio
path loss factor β is more than unknown but temporally dynamicandspatiallyunevenlydistributedinthemap[47,54].
In this experiment, we generate a β distribution model according
to the state-of-the-art empirical study [47, 54], for
evaluating the robustness of localization algorithms in the
case ofspatiallyunevenlydistributed β. Thesimulationrandomly
deployed 300 nodes with uniform distribution in a
mapofsize1000feet ×1000feet. AsshowninFigure25(a),
β was around 4 but varied gradually across the map, with a
“hill” located close to X = 800,Y = 500 and a “valley” located
around X = 200,Y = 500. Basically, a hill indicates
bigger βandavalleydepictssmaller β.
Figure25(b)showsthetopologyofdeployednodes. Each
node is colored accordingto its true coordinatesin the map,
so similar colors indicate proximity. Figure 25(c) illustrates
the 1-hoplinksin network. Eachline segment(edge)stands
for a 1-hop link. The path loss factor β of each link is assignedbasedonthepatchesofthespatial
modelit traverses.
A link exists only when the signal strength at both terminal
nodes are higher than the default RF sensitivity threshold.
2921000
800
Y Axis (in feet)
600
400
200
0
(a) SpatialModelof β
(b) Node Deployment
0 200 400 600 800 1000
X Axis (in feet)
(c) Connectivity Graph
(d) MDS-Hop (e) DV-Hop (f) RPA-Hop
(g) MDS-RSD (h) DV-RSD (i) RPA-RSD
Figure25. RobustnessofRSD forUnknown SpatialFluctuationofPathLossFactor β
We can see from the graph that the link density is higher in
the left part of the map, especially near X = 200,Y = 500,
where β “valley” exists in Figure 25(a). On the other hand,
links are more sparse in the right part, especially close to
X = 800,Y = 500, where β “hill” is located. This result is
expected since a larger β creates a shorter communication
range,thusnodesclose tothe“hill” arelessconnected.
Figures 25(d), 25(e), and 25(f) show the localization results
given by MDS-Hop, DV-Hop, and RPA-Hop, respectively.
We can see that the overallshape of the networkgets
distorted. Many nodes are closely clustered toward the position
X = 200,Y = 500, while others are sparsely spread
out from the point X =800,Y = 500. This interesting phenomenonis
consistent with the connectivitygraphshown in
Figure 25(c). In fact, lower β value allows nodes to have
a longer 1-hop radio range, so from the viewpoint of hop
distance, it creates an illusion of shorter physical distances
between two nodes; while bigger β values have the opposite
effects. RSD does not fluctuate with the radio range
due to the regulation step in the design. This nice feature
gets confirmed from localization results shown in Figures
25(g), 25(h) and 25(i) where the overall shape of the
network is correct for all RSD-applied methods. Therefore,
RSD provides the important feature that it is robust for unevenpathlossfactorsacrossthearea.
Simulation results tells that (i) RSD offers a nice feature
ofrobustnesstothe spatiallyunevenlydistributedradiopath
loss; (2) embedding RSD helps greatly improve the system
accuracyofconnectivity-basedlocalizationalgorithms.
8 Conclusions
This paper proposes a relative distance for achieving
range-free localization beyond connectivity. Starting from
theconceptofneighborhoodorderingasahigh-dimensional
location-dependent signature for each node in the network,
we presented the design of RSD, which quantifies the difference
between signatures to capture distance relationships
amongneighboringnodeswith sub-hopresolution. With little
overhead, RSD can be conveniently embedded in many
connectivity-basedlocalizationalgorithmsto improvelocalization
accuracy. System evaluations on test-beds demonstratedthatapplyingRSDhelpssolvetheambiguityproblem
293andconsiderablyenhancelocalizationaccuracy. Inaddition,
extensivesimulationrevealsthatRSDoffersanicefeatureof
robustnessfor spatially unevenlydistributed radio path loss,
preventingfromoutputtingandistortednetworktopology.
9 Acknowledgments
We thank our shepherd Prof. Polly Huang and reviewers
for valuable feedback. This research was supportedby NSF
grantsCNS-0626609,CNS-0626614andCRI-0708344.
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