Achieving High Accuracy and Efficiency in Localizing Mobile Sensor Nodes

IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 9, NO. 6, JUNE 2010 897

Accurate and Energy-Efficient Range-Free
Localization for Mobile Sensor Networks
Shigeng Zhang, Jiannong Cao, Senior Member, IEEE,
Lijun Chen, and Daoxu Chen, Member, IEEE

Abstract—Existing localization algorithms for mobile sensor networks are usually based on the Sequential Monte Carlo (SMC)
method. They either suffer from low sampling efficiency or require high beacon density to achieve high localization accuracy. Although
papers can be found for solving the above problems separately, there is no solution which addresses both issues. In this paper, we
propose an energy efficient algorithm, called WMCL, which can achieve both high sampling efficiency and high localization accuracy in
various scenarios. In existing algorithms, a technique called bounding-box is used to improve the sampling efficiency by reducing the
scope from which the candidate samples are selected. WMCL can further reduce the size of a sensor node’s bounding-box by a factor
of up to 87 percent and, consequently, improve the sampling efficiency by a factor of up to 95 percent. The improvement in sampling
efficiency dramatically reduces the computational cost. Our algorithm uses the estimated position information of sensor nodes to
improve localization accuracy. Compared with algorithms adopting similar methods, WMCL can achieve similar localization accuracy
with less communication cost and computational cost. Our work has additional advantages. First, most existing SMC-based
localization algorithms cannot be used in static sensor networks but WMCL can work well, even without the need of experimentally
tuning parameters as required in existing algorithms like MSL*. Second, existing algorithms have low localization accuracy when nodes
move very fast. We propose a new algorithm in which WMCL is iteratively executed with different assumptions on nodes’ speed. The
new algorithm dramatically improves localization accuracy when nodes move very fast. We have evaluated the performance of our
algorithm both theoretically and through extensive simulations. We have also validated the performance results of our algorithm by
implementing it in real deployed static sensor networks. To the best of our knowledge, we are the first to implement SMC-based
localization algorithms for wireless sensor networks in real environment.

Index Terms—Mobile sensor networks, localization, sequential monte carlo methods.

Ç

1 INTRODUCTION

W IRELESS sensor networks (WSNs) have been used in
many fields, including environmental and habitat
monitoring, precision agriculture, animal tracking, and
WSNs is to use existing localization techniques, e.g.,
attaching a Global Positioning System (GPS) receiver on
every sensor node. However, as the scale of sensor networks
disaster rescue. In many applications, it is essential for becomes larger and larger, these methods become infeasible
nodes to know their positions. For example, data should be because of their high cost or inconvenience. Many localiza-
labeled with the positions where they are collected to help tion algorithms have been proposed in the past several years,
the scientists perform corresponding analysis. Position [4], [5], [8], [10], [11], [12], [16], [18], [19], [20], [21], [22], [23],
information of nodes are also necessary in many network [28], [30]. Depending on whether absolute range measure-
protocols, e.g., clustering and routing which depend on the ments (point-to-point distances, angles, etc.) are used or not,
geographical information of nodes. For example, geogra- they can be roughly classified into two categories [11]: range-
phical routing protocols such as Greedy Perimeter Stateless based and range-free. Range-based algorithms usually need
Routing (GPSR) [14] need to know nodes’ position some special hardware to obtain accurate absolute range
information in order to select the next-hop relaying node. measurements and can achieve higher localization accuracy
The procedure through which the nodes obtain their than range-free algorithms. Range-free algorithms, on the
positions is called localization. In localization, the nodes in a other hand, do not need special hardware and are low costly
sensor network can be categorized into two types: beacon and more attractive in recent years.
nodes which are aware of their positions and sensor nodes In some recently emerging applications such as animal
which need to determine their positions using a localization monitoring and tracking [13], [29], sensor nodes may move
algorithm. A straightforward method for localization in after deployment. These nodes form mobile sensor networks in
contrast to traditional static sensor networks in which sensor
nodes remain stationary after deployment. The motion of
. S. Zhang, L. Chen, and D. Chen are with the State Key Laboratory for sensor nodes makes most existing localization algorithms
Novel Software Technology, Nanjing University, 22 Hankou Road,
Nanjing 210093, China. designed for static sensor networks inapplicable to mobile
E-mail: zsg@dislab.nju.edu.cn, {chenlj, cdx}@nju.edu.cn. sensor networks. The simple idea of executing these
. J. Cao is with the Department of Computing, Hong Kong Polytechnic algorithms periodically in a mobile sensor network is
University, Hung Hom, Kowloon, Hong Kong. infeasible, because this will incur high communication cost
E-mail: csjcao@comp.polyu.edu.hk.
and/or high computational cost [23]. There are some
Manuscript received 25 Dec. 2008; revised 9 June 2009; accepted 10 Sept. localization algorithms specially designed for mobile sensor
2009; published online 23 Feb. 2010.
For information on obtaining reprints of this article, please send e-mail to:
networks, [2], [7], [12], [20], [23], [24], [27], [32]. All of them are
tmc@computer.org, and reference IEEECS Log Number TMC-2008-12-0513. based on the Sequential Monte Carlo (SMC) method. This is
Digital Object Identifier no. 10.1109/TMC.2010.39. because the posterior distribution of a sensor node’s position
1536-1233/10/$26.00 ß 2010 IEEE Published by the IEEE CS, CASS, ComSoc, IES, & SPS
Authorized licensed use limited to: Francis Xavier Engineering College. Downloaded on August 11,2010 at 18:28:57 UTC from IEEE Xplore. Restrictions apply.

898 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 9, NO. 6, JUNE 2010

after move can be naturally formalized using a nonlinear We have evaluated the performance of the proposed
discrete time model and the SMC method provides simple algorithms both theoretically and through extensive simula-
simulation-based approaches to estimating the distribution. tions. We have validated the convergence of the proposed
Previous SMC-based localization algorithms either suffer algorithms in real environments by implementing it in a real
from low sampling efficiency or require high beacon density deployed sensor network. We also have implemented some
to achieve high localization accuracy. In such localization other SMC-based localization algorithms and compared their
algorithms, a sensor node’s position distribution is repre- performance with our proposed algorithms. The results
sented with a set of weighted samples. In order to obtain show that, even in static sensor networks, our proposed
enough valid samples to accurately characterize the distribu- algorithms can effectively improve the localization accuracy.
tion of its position, a sensor node needs to repeat the sampling The rest of this paper is organized as follows: In Section 2,
step (generate candidate samples) and the filtering step we introduce the background of SMC-based localization
(evaluate candidate samples and filter out invalid samples) algorithms and review related work. In Section 3, we describe
many times. However, generating candidate samples is the WMCL algorithm in detail. WMCL-A, WMCL-B, and the
usually a very costly operation. For example, in Section 6 Iterative WMCL (IWMCL) are also presented in this section.
we will show that, on the Micaz [1] platform the cost of We theoretically analyze the localization accuracy of our
generating a candidate sample is much higher than evaluat- proposed algorithms in Section 4. Then we evaluate the
ing it. Because sensor nodes usually have limited computa- performance of the proposed algorithms under different
tional ability, it is necessary to improve the sampling parameter values by extensive simulations and carefully
efficiency in order to reduce the computational cost. Another analyze the results in Section 5. We also compare the
problem of most existing SMC-based localization algorithms performance of our proposed algorithms with other SMC-
is that they only rely on increasing beacon density to improve based algorithms and present the results in this section. In
localization accuracy. However, beacon nodes are usually order to validate the performance results of SMC-based
more expensive than sensor nodes. Because there are much localization algorithms, we implement the proposed algo-
more sensor nodes than beacon nodes in a sensor network, it rithms and some other algorithms in real deployed static
will be very beneficial if sensor nodes can be used to improve sensor networks and report the results in Section 6. Finally,
the localization accuracy. we conclude this paper in Section 7.
In this paper, we propose an energy efficient algorithm
which addresses both aforementioned issues. The algorithm
is based on the sequential Monte Carlo Localization (MCL) 2 BACKGROUND AND RELATED WORK
algorithm proposed in [12] and is named Weighted MCL 2.1 Background
(WMCL). WMCL achieves high sampling efficiency and We consider a wireless sensor network in which both the
achieves high localization accuracy even when the beacon
beacon nodes and the sensor nodes can move. We assume the
density is low. However, WMCL incurs much more commu-
motion is markovian, i.e., a sensor node’s future position is
nication cost than the original MCL algorithm. In order to
only determined by its current position and is independent
reduce the communication cost, we propose two approx-
imate algorithms, WMCL-A and WMCL-B, which can with its past positions. Assume the time is divided into
achieve nearly the same localization accuracy as WMCL but discrete time units. A sensor node’s position flt ; t 2 N can be
Ng
incur much less communication cost. In fact, compared with modeled as a Markov process of initial distribution pðl0 Þ and
MCL, WMCL-A, and WMCL-B incur only slightly additional transition equation pðlt jltÀ1 Þ. A sensor node obtains some
communication cost (tens of bytes) but achieve much higher observations about its neighbors denoted by ot in time unit t.
localization accuracy with much less computational cost. Given the process flt ; t 2 N the observations fot ; t 2 N Ã g
Ng, N
WMCL uses the following techniques to improve the are assumed to be conditionally independent and of marginal
sampling efficiency and localization accuracy. In existing distribution pðot jlt Þ. Then, a sensor node’s position distribu-
algorithms, a technique called bounding-box is used to tion is described as
improve the sampling efficiency by reducing the scope of
the selecting of candidate samples. By using two-Hop pðl0 Þ
beacon neighbors’ negative effects and sensor neighbors’ pðlt jltÀ1 Þ for t ! 1
estimated position information, WMCL further reduces the
pðot jlt Þ for t ! 1;
average size of the sensor nodes’ bounding-boxes by a
factor of up to 87 percent and, consequently, improves the and we want to compute pðlt jo1;...;t Þ.
sampling efficiency by a factor of up to 95 percent. By using The SMC method provides a set of simulation-based
estimated position information of sensor nodes, WMCL approaches to computing this posterior distribution. In this
greatly improves the localization accuracy. Compared with paper, we use an approach called particle filter [9] because it
MSL* [20] which uses similar method, WMCL achieves is simple and easy to implement. In this approach, the
similar localization accuracy but is much more efficient in distribution is represented with a set of N weighted
terms of both communication cost and computational cost. samples [20]:
Further more, it can be used in static sensor networks
ÈÀ ðiÞ ðiÞ ÁÉ
without the need of experimentally tuning parameters as in pðlt jo1;...;t Þ % lt ; wt ;
i¼1;...;N
algorithms such as MSL*. Taking advantage of this
ðiÞ ðiÞ
property, we propose a new algorithm in which WMCL is where lt is a sample of this distribution and wt is its
P ðiÞ
iteratively executed with different assumptions on the normalized weight ( N wt ¼ 1).
i¼1
nodes’ speed. The new algorithm can dramatically improve Algorithm 1 shows a generic framework of SMC-based
localization accuracy when nodes move very fast. localization algorithms. There are three steps in the


ZHANG ET AL.: ACCURATE AND ENERGY-EFFICIENT RANGE-FREE LOCALIZATION FOR MOBILE SENSOR NETWORKS 899

algorithm: initialization, importance sampling, and filtering. In shrink its polygon. When sensor nodes are mobile, a sensor
the initialization step, N samples are randomly drawn from node dilates its polygon before broadcasting it. In the
the deployment area. In time unit t, in the importance algorithm proposed in [31], a sensor node records the
sampling step, candidate samples are drown based on the history beacon information it has heard and uses a mobility
samples in the previous time unit and their weights are pattern to predict its position.
computed using the observations collected in time unit t. In There are localization algorithms specially designed for
the filtering step, samples with weight 0 are filtered out. The mobile sensor networks [2], [7], [12], [20], [23], [24], [27], [32].
importance sampling step and the filtering step may repeat MCL [12], the first of such algorithms, assumes that all the
several times in order to obtain enough number of candidate nodes are mobile and know their maximum speed. The time
samples with weight greater than 0. Then N samples are is divided into discrete time units and in each time unit all the
selected from these candidate samples and their weights are sensor nodes update their positions. While achieving high
normalized. At last, the weighted average of the N samples localization accuracy, MCL left spaces for improvement in
is used as the estimation of a sensor node’s position in the two aspects: Its sampling efficiency is low and it relies on high
current time unit. beacon density to achieve high localization accuracy. The
Monte Carlo localization Boxed (MCB) algorithm [2] im-
Algorithm 1. A framework of SMC-based localization proved MCL’s sampling efficiency by using bounding-box
algorithms (called anchor box in their paper) to restrict the scope from
1: Step 1: Initialization which the candidate samples are drawn. Considering that
2: t 0 when many beacon nodes are observed a small number of
3: for i 1; N do samples will be enough to characterize the distribution of a
ðiÞ
4: Sample l0 $ pðl0 Þ sensor node’s position, the authors in [27] proposed the
5: end for Sample Adaptive Monte Carlo Localization (SAMCL) algo-
ð1Þ ðNÞ rithm which can adaptively determine the number of
6: L0 fðl0 ; 1=NÞ; . . . ; ðl0 ; 1=NÞg
7: t 1 samples hence can reduce the computational cost. WMCL
8: Lt ¼ fg further reduces the size of the bounding-box constructed in
MCB by using two-Hop beacon neighbors’ negative effects
9: while jLt j < N do
and sensor nodes’ estimated position information. The
10: Step 2: Importance Sampling
average size of the bounding-boxes is reduced by a factor of
11: Ct ¼ fg up to 87 percent. This makes WMCL achieve higher sampling
12: for i 1; N do efficiency than MCB even when the filtering conditions in
ðiÞ ðiÞ
13: Sample lt $ pðlt jltÀ1 Þ WMCL are much stricter than that in MCB. The techniques
ðiÞ ðiÞ ðiÞ
14: Evaluate the weight of lt as wt ¼ pðot jlt Þ
~ used in [27] can be used as a complementarity to WMCL to
dynamically determine the number of needed valid samples.
ðiÞ ðiÞ
15: Ct ¼ Ct [ fðlt ; wt Þg
~ In MCL a sensor node only uses its beacon neighbors
16: end for within two hops as its observation. Some works improve
17: Step 3: Filtering MCL’s localization accuracy by using more information as
ðiÞ ðiÞ ðiÞ ðiÞ ðiÞ observation. The Mobile and Static sensor network Localiza-
18: Ct0 ¼ fðlt ; wt Þjðlt ; wt Þ 2 Ct and wt > 0g
~ ~ ~
19: Lt ¼ Lt [ Ct0 tion (MSL*) algorithm [20] uses neighboring sensor nodes’
estimated position information to improve the localization
20: end while
accuracy of MCL. In MSL*, a sensor node uses all its neighbors
21: Lt ¼ choose(Lt , N) ==choose N valid samples
wi
~ (both the beacon nodes and sensor nodes) within two hops to
22: Normalize the weights of samples in Lt : wi ¼ PN t i
t compute its position. In order to keep a low computational
~
w
23: t t þ 1, goto line 8 i¼1 t
cost, MSL* adopts a resampling strategy different from that of
MCL. In MSL*, a sample in last time unit could be reserved in
2.2 Related Work the current time unit with a probability proportional to its
Works can be found on applying localization algorithms weight; new samples are drawn only when there are not
designed for static sensor networks to mobile sensor enough samples inherited from last time unit. This makes
networks. In [25], the authors investigated that, when such MSL* more suitable for sensor networks in which nodes move
an algorithm was applied to a mobile sensor network, how slowly and perform poorly when nodes move very fast, as
often that algorithm should be executed to achieve required the simulation results in [20] show. In order to reduce the
localization accuracy with minimum energy cost. In [3], the communication cost incurred in MSL*, the authors of [20] also
authors proposed an algorithm to localize mobile nodes proposed the MSL algorithm. MSL incurs much less com-
with only two beacon nodes whose transmission range munication cost but achieves lower localization accuracy
could cover the whole network. The distance between a than MSL*. Both MSL* and MSL suffer from low sampling
sensor node and a beacon node is derived from the power efficiency and use some parameters whose values need to be
of received signals and averaged on a given window to experimentally determined. In WMCL no such parameters
counteract interference and fading. Then triangulation is are used. Compared with MSL*, WMCL achieves much
used to compute the sensor node’s position. In [6], the higher sampling efficiency and similar localization accuracy
authors proposed an algorithm which could be used in both while incurring only about 2/3 communication cost. We
static and mobile sensor networks. Each sensor node also propose two approximate algorithms, WMCL-A and
constructs a polygon that contains all its possible positions WMCL-B. They incur slightly higher communication cost
and broadcasts the description of this polygon to its than MSL but achieve much higher localization accuracy
neighbors. A sensor node uses its received information to with much less computational cost.



There are some other variants of MCL, for example, the
dual and Mixture MCL [24], Multihop-based Monte Carlo
Localization (MMCL) [32], and Range-based MCL [7]. The
dual and Mixture MCL improves the localization accuracy
of MCL by exchanging the probability functions used in the
sampling step and in the filtering step. It incurs higher
computational cost than MCL. MMCL and Range-based
MCL use multihop sensor-beacon distances to improve the
localization accuracy and to reduce the number of needed
beacons. Compared with them, WMCL doesn’t use multi-
hop sensor-beacon distances so incurs much less commu- Fig. 1. How MCL [12] works: 1) The left figure shows the sample set LtÀ1
in time unit t À 1. 2) Sampling based on LtÀ1 and filtering using
nication cost. observations in time unit t.

3 THE PROPOSED WMCL ALGORITHM model to study the performance of the proposed
algorithm under irregular communication models.
In this section, we describe our proposed WMCL algorithm
The Q-UDG model will be introduced in Section 5.4.
in detail. We first introduce the network model, then
3. Random motion of nodes. We assume that all the
describe the three main parts of WMCL: bounding-box
nodes can move after deployment and they have
construction, samples’ weights computing, and maximum
some limited knowledge about their motion. We
possible localization error computing.
assume that every node knows its maximum speed
The WMCL algorithm follows the generic framework
defined by vmax .
presented in Algorithm 1 and makes the following mod-
ifications. Before starting sampling (line 8), each sensor For the sake of simplicity in description, we give some
node constructs a bounding-box from which the candidate definitions here.
samples are drawn (Section 3.2). For each candidate sample, Definition 1 (Node/Beacon Density/Degree). The node
its weight is computed with our proposed weight comput- density n and the beacon density s are defined as
ing methods (Section 3.3). Finally, before broadcasting the
estimated position information to other sensor nodes, each n m
n ¼ ; s ¼ ; ð1Þ
sensor node computes its maximum possible localization S S
error with our proposed maximum possible localization respectively. The node degree nd and the beacon degree sd
error computing method (Section 3.4).
represent the average number of nodes and of beacon nodes in a
The iterative WMCL (IWMCL), which can dramatically
node’s communication range and are defined as
improve the localization accuracy when nodes move very
fast, is described at the last of this section. nd ¼ n Ã r2 ; sd ¼ s Ã r2 ; ð2Þ
3.1 Network Model respectively.
Our algorithm can be applied to diverse network models.
Definition 2 (SðsÞ; T ðsÞ; USðsÞ). For a sensor node s, we use
However, in order to compare the performance of our
SðsÞ to denote the set of its one-Hop beacon neighbors, T ðsÞ to
algorithm with the existing ones, we adopt a network
denote the set of its two-Hop beacon neighbors, and USðsÞ to
model consistent with them [2], [12], [20], [23], [24], [27]. We
have the following assumptions: denote the set of its one-Hop neighboring sensor nodes. When
it is clear which sensor node is referred to according to the
1. Uniform node deployment. We assume there are context, we use S; T ; US as their abbreviations.
n nodes of which m are beacon nodes uniformly
deployed in a planar rectangle area with size S. 3.2 Building the Bounding-Box
Beacon nodes are assumed to be aware of their Fig. 1 illustrates how MCL works. There are two areas
exact position information all the time. The node involved in MCL: the candidate samples area and the valid
deployment strategy determines the initial distribu- samples area. The candidate samples area is used to draw
tion of sensor nodes’ positions, say, pðl0 Þ. new candidate samples and the valid samples area is used
2. Unit Disk Graph (UDG) connectivity model. In the to filter out invalid samples. When the candidate samples
UDG model, two nodes p and q can directly area is large and the valid samples area is small, candidate
communicate with each other if and only if they samples drawn in the sampling step have high probability
are within the communication range defined by a to be filtered out in the filtering step. In MCL, the possible
radius r. If two nodes p and q are within r, we say q locations of a sensor node after move lie in a disk with
is a one-Hop neighbor of p. If p and q are within 2r radius vmax . So the size of the candidate samples area will
but not r, we say q is a two-Hop neighbor of p. The increase when vmax increases. On the other hand, when sd
connectivity model determines the margin distribu- increases, the size of the valid samples areas will decrease.
tion of observations, say, pðot jlt Þ.
Denote by Vt the total number of candidate samples drawn
The UDG model represents the ideal case. We use
in the sampling step in time unit t and define the sampling
this model to derive a lower bound on the localiza-
tion accuracy of the proposed algorithm in Section 4. efficiency in t as
However, the performance of the proposed algo- jLt j
rithm will be affected by the irregularity of commu- et ¼ ;
Vt
nications. In our simulation, we use the Q-UDG



Fig. 2. Reduce the size of the bounding-box: The shadowed area should
be eliminated.
Fig. 3. The explanation of approximate method 1. The probability of
observing s at li equals to the probability of two nodes locating at Rt;s
then in MCL the sampling efficiency will decrease when t
and li can communicate with each other, say, the probability of Rt;s
t
vmax or sd increases, which will cause high computational resides in area II. This probability can be approximated as the probability
cost accordingly. In order to improve the sampling that RtÀ1;s resides in area I.
efficiency, Baggio and Langendoen [2] use a technique
called bounding-box (called anchor-box in their paper) to 3.3 Weighting the Samples
reduce the size of the candidate samples area. Assuming 3.3.1 Three Approximate Methods
that a sensor node has n one-Hop beacon neighbors, a In WMCL, ot ¼ S [ T [ US. So the weight of a candidate
bounding-box ðxmin ; xmax ; ymin ; ymax Þ can be built as follows: sample is computed as
À Á Y À Á Y À Á
xmin ¼ maxn fxi À rg; xmax ¼ minn fxi þ rg
i¼1 i¼1 wi ¼ p ot jli ¼
~t p sjli p sjli : ð6Þ
ð3Þ t t t
ymin ¼ maxn fyi À rg; ymax ¼ minn fyi þ rg;
i¼1 i¼1 s2S;T s2US

where ðxi ; yi Þ means the coordinate of the ith beacon When s 2 S or s 2 T , pðsjli Þ
can be easily computed [12],
t
neighbor. Two-Hop beacon neighbors are also used to [20]. If s 2 S, then
reduce the size of the bounding-box in [2] by replacing r À Á Â À Á Ã
p sjli ¼ d li ; s
t t r : ð7Þ
with 2r in (3).
We further reduce the size of the bounding-box as follows: If s 2 T , then
Suppose that a bounding-box ðxmin ; xmax ; ymin ; ymax Þ has been À Á Â À Á Ã
built as above. We use two-Hop beacon neighbors’ negative p sjli ¼ r d li ; s
t t 2r : ð8Þ
effects to reduce the size of the bounding-box. See Fig. 2 for However, when s 2 US, pðsjli Þ is not easy to compute. In
t
the illustration. Assuming q is p’s two-Hop beacon neighbor, this case, what has been observed is not s’s real position, but
then the shadowed region doesn’t contain p; otherwise q will s’s estimated position in the last time unit. The probability
be p’s one-Hop neighbor. So we can eliminate the shadowed can be expressed as
region without any loss of valid samples. À Á È À Á
Now suppose each sensor node knows its maximum p sjli ¼ P r d Rt;s ; li
t t r ^ Rt;s $ pðRt;s jRtÀ1;s Þ
É ð9Þ
localization error in x-axis ERx and maximum localization ^ RtÀ1;s % posEst ;
error in y-axis ERy (how to compute them will be explained
in Section 3.4). Let us use Rt;p and Et;p to denote a sensor where % means RtÀ1;s is estimated as posEst in time unit
node p’s real position and estimated position in time unit t, t À 1. It can be explained as the probability of the following
respectively. Assuming q 2 USðpÞ, we have: event: A sensor node can communicate with s whose position
in the last time unit is RtÀ1;s and current position follows the
jRt;p À Rt;q j r distribution pðRt;s jRtÀ1;s Þ and RtÀ1;s is estimated as posEst.
jRt;q À RtÀ1;q j vmax According to different forms of posEst a sensor node p
ð4Þ receives, we propose three approximate methods to
jXðRtÀ1;q Þ À XðEtÀ1;q Þj ERx;tÀ1 ðqÞ
compute the probability pðsjli Þ:t
jY ðRtÀ1;q Þ À Y ðEtÀ1;q Þj ERy;tÀ1 ðqÞ;
1. (Used in WMCL) If posEst ¼ Ls , then we can use
where XðÁÞ/Y ðÁÞ means the x/y value of Á and ERx;t ðqÞ/ tÀ1
the following formula to compute an approximation
ERy;t ðqÞ means q’s ERx /ERy in time unit t. The bounding-
of pðsjli Þ:
t
box can be further shrunk (take xmin as an example): Â À iÁ Ã
P
À iÁ l2Ls d l; lt r þ vmax
xmin ¼ maxfxmin ; maxq2USðpÞ fXðEtÀ1;q Þ À vmax À r p sjlt % tÀ1
:
ð5Þ jLs j
tÀ1
À ERx;tÀ1 ðqÞgg:
See Fig. 3 for the illustration. The probability of
A sensor node’s estimated position estimation in last observing s at li equals to the probability of two nodes
t
time unit can also be used to shrink the bounding-box. Take locating at Rt;s and li can communicate with each
t
xmin as an example: other, say, the probability of Rt;s resides in area II. This
xmin ¼ maxfxmin ; XðEtÀ1;p Þ À vmax À ERx;tÀ1 ðpÞg: probability can be approximated as the probability
that RtÀ1;s resides in area I.
After the bounding-box is built, the candidate samples Note here we only need to transmit the positions of
are drawn from the bounding-box. samples in Ls without transmitting their weights,
tÀ1



Fig. 5. (a) Computing ERx and ERy ; the biggest dot is ðxe ; ye Þ. (b) A way
to reduce ERx and ERy with little risk.
Fig. 4. The explanation of approximate method 2. We can use SIII =SI
as an estimation of the weight computed using approximate method 1.
show that WMCL achieves nearly the same accuracy as
MSL* when vmax is small.
while in MSL* the weights also need to be transmitted.
This can greatly reduce the communication cost. 3.4 Computing ERx and ERy
2. (Used in WMCL-A) The first method needs to know After obtaining N valid samples, a sensor node computes
the whole set of samples of all the sensor neighbors, the weighted average of these samples as its position
which incurs high communication cost. As illu-
estimation. Using the position estimation and the bounding-
strated in Fig. 4, if we know s’s bounding box Bs , we
box, a sensor node can compute its ERx and ERy , as
can approximately compute the probability as
illustrated in Fig. 5.
À Á SB B Assume p’s position estimation is ðxe ; ye Þ and its bound-
p sjli % s ;
t ing-box is ðxmin ; xmax ; ymin ; ymax Þ. It is obvious that ERx is
SBs
maxfxe Àxmin ; xmax Àxe g and ERy is maxfye Àymin ; ymax Àye g.
where B represents the square centering at li with
t We can reduce ERx and ERy with some risks. Assume p’s
side-length r þ vmax .
sample set is fðx1; y1Þ; . . . ; ðxN ; yNÞg. Let x0min ¼ minfx1; . . . ; xNg
3. (Used in WMCL-B) In our previous work [23], we use
and we can further refine xmin to be ðxmin þ x0min Þ=2. Do the
the following formula to compute the value of pðsjli Þ:
t
same to xmax , ymin and ymax we can get reduced ERx and ERy .
À iÁ Â À Á Ã
p sjlt ¼ d EtÀ1;s ; li vmax þ r þ ERtÀ1 ðsÞ ; A more riskily method is to use the smallest rectangle
t
enclosing all of p’s valid samples to compute ERx and ERy .
where ERtÀ1 ðsÞ means the maximum localization This method can improve localization accuracy a lot.
error of s in time unit t À 1 and is computed as However, when using this method the procedure of
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi constructing the bounding-box should be carefully manipu-
ER2 ðsÞ þ ER2 ðsÞ:
x;tÀ1 y;tÀ1 lated. In this case the inequality presented in (4) may not
hold and some inconsistence may happen. For example, it is
We call the algorithms using the three different weight possible that xmin is larger than xmax and consequently the
computing methods WMCL, WMCL-A, and WMCL-B, bounding-box cannot be built.
respectively. After p gets ðxe ; ye Þ and ERx ; ERy , it broadcasts them to
its neighbors. Its neighbors will use these information to
3.3.2 Approximation Analysis compute their position estimation in the next time unit.
In this section, we analyze the effects of different approx-
imate methods. The accurate value of pðsjli Þ defined in (9) is 3.5 Iterative WMCL
t
difficult to compute. A good approximation method is The localization accuracy of WMCL will degrade when vmax
proposed in [20] and is used in the MSL* algorithm. As increases. This is because when vmax increases, the
illustrated in Fig. 3, this probability can be estimated as constraints introduced by sensor nodes’ estimated position
À Á È É information become very weak and cannot be used to
p sjli ¼ P r l 2 Ls ^ l 2 I
t tÀ1 improve the localization accuracy effectively.
X Â À Á Ã
% wi d l; li r þ vmax : In WMCL the candidate samples are directly drawn from
t t
l2Ls
tÀ1
the bounding-box so it is also applicable even when
vmax ¼ 0. Here, we propose IWMCL, the Iterative WMCL,
The weight computing method used in WMCL computes which can alleviate the negative effects of vmax on WMCL’s
an approximate value of this probability. Assume there are localization accuracy. The idea is simple: After all sensor
M valid samples in Ls and M 0 of them reside in area I.
tÀ1 nodes obtain their position estimation in time unit t, they
Then in WMCL pðsjli Þ is approximated as
t broadcast their position estimations to their neighbors.
À Á M0 Upon receiving the new position estimations, the WMCL is
p sjli %
t : executed with the assumption that vmax equals to 0.
M
The IWMCL algorithm is described as follows: There are
In other words, when computing the value of pðsjli Þ in t two phases in IWMCL:
WMCL, we assume that s’s position in last time unit follows
a uniform distribution. So the difference between the real 1. Phase 1: normal run of WMCL. Every sensor node
value of pðsjli Þ and the approximated value computed in
t collects its sensor neighbors’ position estimations in
WMCL depends on the real distribution of s’s position in last time unit and computes its position estimation
last time unit. Our simulation results presented in Section 5 using WMCL.


Z 1 À Á
EðzÞ ¼ z d 1 À eÀðs 8rzþn 4rzÞ
0
r
¼ :
8sd þ 4nd
However, because the sensor nodes’ estimated position
information is inaccurate so we can speculate that it will be
very difficult to achieve this lower bound in a real
implementation of WMCL.
Fig. 6. A lower bound on localization error for connectivity-based Rudafshani and Datta [20] have argued that this lower
localization algorithms [17]. The summation of the area of I and the area
of II is about 4rz. bound is not rigorous when node degree is low (nd 10).
However, as the results in their paper showed, when node
degree is high this lower bound can still give some insights
2. Phase 2: special run of WMCL with the assumption
about what is the best localization accuracy an algorithm
vmax ¼ 0. After every sensor node gets its position
which depends on only local connectivity status can achieve.
estimation in the current time unit, it broadcasts this
information to its neighbors. Then every sensor node
runs WMCL with the assumption that vmax ¼ 0. This 5 PERFORMANCE EVALUATION THROUGH
phase can repeat several times in order to improve SIMULATION
the localization accuracy.
In this section, we evaluate the performance of the proposed
By replacing WMCL with WMCL-A the communication
algorithms through extensive simulations. We modify the
cost can be dramatically reduced.
widely used simulator developed by Hu and Evans [12] to
implement the proposed algorithms. In order to evaluate
4 THEORETICAL ANALYSIS ON THE LOWER BOUND our contributions in improving sampling efficiency, we also
OF THE LOCALIZATION ACCURACY implement the MCL algorithm and the MCB [2] algorithm
and compare their sampling efficiency with our algorithms.
In this section, we will derive a lower bound on the In order to evaluate our contribution in improving
localization accuracy of our algorithm. We use the method localization accuracy, we compare our methods with the
proposed by Nagpal et al. [17]. The main idea of this method methods proposed in MSL [20]. Because MSL* and MSL use
is described as follows: The sensor network can be different resampling strategy from that of MCL [12], for
represented as a graph in which each vertex corresponds consistency in comparison, here we implement variants of
to a sensor node in the network. There is an edge between MSL* and MSL with the same resampling strategy as in
two vertices if and only if their corresponding nodes in the MCL and call them VMSL* and VMSL, respectively.1 The
sensor network can communicate with each other. Because communication cost of VMSL* and VMSL are the same as
each sensor node only uses its neighbors’ position informa- that of MSL* and MSL.
tion to estimate its position, when a sensor node move from We consider the following metrics: 1) localization
one position to another position, the two positions will be accuracy, 2) computational cost, and 3) communication
cost. The main parameters we consider are nodes’ max-
indistinguishable if the set of the sensor’s neighborhood
imum speed vmax , node degree nd , and beacon degree sd , for
doesn’t change. Then the average distance a sensor node can
they are the main factors affecting the algorithms’ perfor-
move without changing its connectivity status gives a lower mance. When considering the impact of communication
bound on the expected localization accuracy. irregularity on the algorithms’ performance we use the Q-
Fig. 6 shows the case in which only one-Hop neighbors UDG model [15]. We tune the value of d in the Q-UDG
are used in the localization. Assuming a UDG connectivity model to simulate different irregularity of communications.
model, the probability that a sensor node can move z We assume that all the nodes are uniformly deployed in a
without affecting its connectivity status equals to the 500 units Â 500 units square region and the communication
probability that at least one other nodes reside in area I range r is set to be 100 units. We use a modified Random
and II. When the nodes are uniformly distributed in the Waypoint mobility model proposed in the MCL algorithm
deployment region, the probability that k nodes reside in a [12] as the motion model. Unless otherwise specified, the
given region R with size a follows a binomial distribution default values of the parameters are: nd ¼ 10, sd ¼ 1,
Bðn; a=SÞ and can be approximated using a Poisson vmax ¼ :2r. The number of effective samples is set to
distribution because n is very large: 50 according to [12], [20]. The value of necessary parameters
needed in VMSL* and VMSL are set to the same value as
ðn aÞk Àn a used in the original MSL* and MSL algorithm [20]. The
P rðk nodes in RÞ ¼ e : ð10Þ localization error is measured as a multiple in r. For different
k!
parameter values, we randomly generate 30 networks and in
The summation of the area of I and the area of II is about 4rz each network we run 1,000 time units and average the
[12]. So we have: metrics between time unit 600 and time unit 1,000. As we
have studied in [33], with this setting the obtained data are
F ðzÞ ¼ 1 À eÀn 4rz : stable in the sense that the difference between the results of
In WMCL, each sensor node uses its beacon neighbors different simulation procedures will be less than 0:01r. All
within two hops and sensor neighbors within one hop to
1. We replaced the weight computing function used in MSL* with our
estimate its position. So for WMCL, the localization proposed methods in the simulation code provided by the authors of [20]
accuracy is bounded below by and found that they achieve similar localization accuracy.



Fig. 7. How sd , nd , and vmax affects the localization accuracy. (a) sd ; (b) nd ; (c) vmax .

the data points presented here are the average of these position of node s. The value of b is experimentally set to 7
30 independent experiment results. in VMSL. From (6), we can conclude that when closenesss is
large or nd is large, wi ¼ pðot jli Þ will approach to 0. This
~t t
5.1 Localization Accuracy means that in VMSL a sensor node may not obtain enough
Localization accuracy is the most important metric in valid samples to represent its position distribution when nd
evaluating localization algorithms. we study how the increases. In fact, in our simulation the number of valid
localization accuracy of different algorithms varies when samples in VMSL drops to 38 when nd is 26, while in all
sd , nd , and vmax varies. other algorithms the number of valid samples always
Fig. 7a shows how the localization accuracy of different
remain about 50.
algorithms varies when sd increases. When sd increases, in Fig. 7c shows how the localization accuracy varies when
all algorithms the localization accuracy improves. However, vmax increases. From our analysis in Section 4, the localiza-
VMSL* and our proposed algorithms perform much better tion accuracy should not be affected by vmax if enough valid
than other algorithms. This is because VMSL* and our samples can be obtained. However, in Fig. 7c we can find
proposed algorithms use sensor nodes’ estimated position that localization accuracy first improves then degrades.
information to improve localization. We can also see that We explain this as follows: As illustrated in Figs. 8 and 9,
WMCL-A and WMCL-B achieve nearly the same localiza- vmax can impact the “quality” of the generated candidate
tion accuracy as WMCL and VMSL*. samples in the sampling step. In the two figures, the disk
Fig. 7b shows how the localization accuracy varies when denoted by I represents the whole valid samples area of a
nd increases. We can see that the localization accuracy of sensor node in last time unit, and the ellipse denoted by II
VMSL* and our proposed algorithms improve when nd represents the valid samples area covered by the sensor
increases. We can also see that, when nd 15, the accuracy of node’s sample set in last time unit. PtÀ1 and Pt are the real
MCL and MCB is also improved slightly when nd increases. positions of the sensor node in last time unit and in current
The reason is as follows: When nd is small, the network is time unit, respectively. We can see that when vmax is small,
very sparse and maybe some beacon nodes within 2r of a only a small part of the valid samples area in current time
sensor node cannot be found as two-Hop beacon neighbors. unit can be covered by the candidate samples area. Because
When nd 15, almost all the beacon nodes within 2r of a all the valid samples are drawn from this part, the position
sensor node can be found and the increase in nd will not estimation of a sensor node will be inaccurate. When vmax is
affect the localization accuracy of MCL and MCB any more. large, the candidate samples area can cover a larger part of
It is strange to see that for VMSL the localization
the valid samples area. So when vmax increases, the
accuracy first improves then degrades when nd increases.
localization accuracy can be improved. However, after
In fact, in VMSL the value of pðsjli Þ is approximated as [20]
t vmax exceeds a threshold, the candidate samples area can
À iÁ cover the whole valid samples area and the localization
p sjlt ¼ bÀclosenesss ;
accuracy should not be affected by vmax any more.
where closenesss is a metric representing the average
distance between all the valid samples and the estimated

Fig. 8. When vmax is small, the candidate samples area can only contain Fig. 9. When vmax is large, the candidate samples area can contain a
a small part of the valid samples area. large part of the valid samples area.



Fig. 10. When vmax increases, the real beacon degree decreases
Fig. 11. IWMCL can effectively improve localization accuracy when vmax
slightly.
is large (iterate four times).

On the other hand, when vmax increases, the average much higher than evaluating it. So the sampling efficiency is
number of beacon neighbors a sensor node can hear a very import metric in SMC-based localization algorithms
decreases,2 as shown in Fig. 10. Combing these two factors because higher sampling means less candidate samples
we can explain why vmax affects different algorithms in generation and consequently less computational cost.
different manners. In our proposed algorithms, the bound- Fig. 12a shows how the number of candidate samples
ing-box always contains the whole valid samples area, so it varies when sd increases in different algorithms. As we have
is only affected by the second factor. Other algorithms are pointed out in Section 3.2, the sampling efficiency degrades
affected by both factors so their accuracy first improves fast in MCL and VMSL* when sd increases. On the other
because more possible samples are obtained and then hand, the sampling efficiency in our proposed algorithm
degrades after vmax exceeds a threshold. The reason that and MCB only degrades slightly when sd increases. The
MCB is also affected by the first factor is that in its sampling efficiency in our proposed algorithms is slightly
implementation vmax is used to construct a smaller rectangle higher than that in MCB but the accuracy in our algorithms
from which the candidate samples are drawn. is much higher than that in MCB.
From Fig. 10, we can see that when vmax becomes larger, In Fig. 12a, we can see that the sampling efficiency in
the localization accuracy of our proposed algorithms VMSL* and VMSL is higher than that in MCL. However, in
degrade fast (even worse than MCL when vmax 1:2r). This VMSL* and VMSL a parameter call (set to be .1r in the
is because when vmax is large, the constraints introduced by simulation, the same as in [20]) is added to vmax in
sensor neighbors become very weak so cannot effectively generating candidate samples. This parameter enlarges the
improve the localization accuracy. The IWMCL algorithm candidate samples area so intuitively the sampling effi-
we proposed in Section 3.5 can achieve high localization ciency in VMSL* and VMSL should be lower than that in
accuracy even when vmax is large. In Fig. 11, we can see that MCL. It can be explained as follows: The parameter has
IWMCL and IWMCL-A can effectively migrate the negative two impacts on the sampling efficiency. First it may
effects of vmax on the localization accuracy of WMCL. More degrade the sampling efficiency because it enlarges the
importantly, the computational cost and the communication candidate samples area. Second it may improve the
cost of IWMCL-A is much lower than that of VMSL*. So we sampling efficiency because by adding it the candidate
can use IWMCL-A instead of VMSL* in a network in which samples area can cover more valid samples area, as shown
nodes move very fast. in Figs. 15 and 16. When vmax is small, the second impact
5.2 Computational Cost dominates the first impact, so VMSL* and VMSL achieve
higher sampling efficiency than MCL. When vmax is large,
In SMC-based localization algorithms, the computational
the second impact becomes small and VMSL* and VMSL
cost consists of two parts: the cost in generating candidate
should achieve similar sampling efficiency as MCL, as
samples and the cost in evaluating the candidate samples.
shown in Fig. 12b. To verify our conclusion, we modify
There are two key operations in evaluating the candidate
VMSL* and VMSL with the same transition equation of
samples: computing the distance between two samples and
MCL (without adding ). The results are shown in Fig. 12c.
comparing the distance with a predefined value (e.g., the
We can see that VMSL* and VMSL achieve similar sampling
communication radius r). In VMSL, there are another key
efficiency as MCL. This also explains why in MCL the
operation used in evaluating a candidate samples: comput-
sampling efficiency first improves then degrades when vmax
ing the value of 7x where x is a negative. In this section, we
increases, as shown in Fig. 12b.
first compare the cost in generating candidate samples in
Fig. 12b shows how the number of generated candidate
different algorithms, then compare the cost in evaluating
samples varies in different algorithms when vmax increases.
candidate samples in different algorithms.
We can find that the sampling efficiency in MCL, VMSL*,
5.2.1 Generating Candidate Samples and VMSL degrades dramatically when vmax increases. This
is because when vmax increases, the size of the candidate
According to the cost of different operations listed in
samples area increases dramatically in these algorithms. On
Section 6.1, the cost of generating a candidate sample is
the other hand, in our proposed algorithms and MCB, the
2. The decrease is only observed from the simulation. We cannot explain candidate samples area is bounded by the bounding-box
it theoretically now and consider this as a future work. and so is not effected by vmax .



Fig. 12. How sd , nd , and vmax affects the sampling efficiency. The sampling efficiency is improved by a factor of up to 95 percent (compared with
MCL). (a) sd ; (b) vmax ; and (c) sd without .

Figs. 13 and 14 shows the size of the bounding-box built
in MCB and in our algorithms. We can see that our
algorithms effectively reduce the size of bounding-box. The
improvement factor is about 63 percent when sd varies and
from 4 percent to 87 percent when vmax varies. In Fig. 14 we
can also find that when vmax increases, the improvement
becomes smaller and smaller. This means that the localiza-
tion accuracy of the proposed algorithms degrades to the
localization accuracy in MCB when vmax increases. In this
case, IWMCL and IWMCL-A can be used to improve the
localization accuracy.

5.2.2 Evaluating Candidate Samples
Fig. 13. The average size of bounding-boxes when sd varies.
There are two types of operations in evaluating a candidate
sample: computing the distance3 between two points and
comparing the distance with a predefined value. As listed
in Section 6.1, the cost of the first type of operation is nearly
100 times higher than the cost of the second type of
operation. So in this section, we only compare the number
of distance computing in evaluating candidate samples in
different algorithms. Some algorithms need some other
special type of operations. For example, in the bounding-
box building phase of our proposed algorithms, it needs to
compute the square root of a number; in the VMSL
algorithm, it needs to compute a power of a number. We
also compare such computational costs.
Fig. 17 shows how the computational cost in evaluating Fig. 14. The average size of bounding-boxes when vmax varies.
candidate samples varies when sd , nd , and vmax increases.
We can see that when sd and vmax varies, the number of
distance computing operations is approximately propor-
tional to the sampling efficiency showed in Fig. 12. From
Fig. 17b, we can see that the computational cost in
evaluating candidate samples in WMCL-A and WMCL-B
increases proportional to nd . However, the cost in these two
algorithms is still always (except in WMCL-A when nd is Fig. 15. When vmax is small, adding can greatly improve the sampling
very large) less than MCL. This is due to the high sampling efficiency.
efficiency in these two algorithms.
Table 1 lists the cost in evaluating candidate samples in
WMCL and VMSL*. We can see that the difference between
the cost in the two algorithms is not very large although the
sampling efficiency in WMCL is much higher than that in
VMSL*. This is because in these two algorithms, the major
source of distance computing is in the computing of pðot jlt Þ
and this is not effected much by the sampling efficiency.

3. Here, we define the distance between two points p and q as kp À qk, Fig. 16. When vmax is large, the improvement in sampling efficiency is
because jp À qj r is equivalent to kp À qk r2 . small.



Fig. 17. Number of distance computing in different algorithms. (a) sd ; (b) nd ; and (c) vmax .

However, WMCL still incurs less computational cost in needs to broadcast its position information and each node
evaluating candidate samples and incurs much less needs to rebroadcast the information it received from its
computational cost in generating candidate samples. beacon neighbors, so the communication cost is Oðsd þ 1Þ.
For WMCL, WMCL-A, and WMCL-B, there are some In WMCL and VMSL*, the communication cost is
additional computational cost incurred in the phase of OðN þ sd þ 1Þ, where N is the number of samples a sensor
bounding-box construction. This is because we need to node reserves in each time unit. This is because in WMCL
compute the intersection points of a circle and a rectangle. and VMSL*, every sensor node also needs to broadcast is
This needs to compute the square roots of some number. estimated position information obtained in last time unit. In
However, the number of this type of operations is only WMCL-A, WMCL-B, and VMSL, the communication cost of
approximately equal to the number of two-Hop beacon each node is Oðsd þ 1 þ kÞ, where k represents the addi-
neighbors and is very small. For the VMSL algorithm the tional information a sensor node broadcasts. In WMCL-A
operation of computing the power of a given number also and WMCL-B, it is the estimated position in last time unit
and ERx and ERy . In WMSL, it is the closeness of the sensor
incurs large computational cost. The number of this
node in last time unit.
operation is proportional to the number of generated
Figs. 18 and 19 shows the communication cost of each
candidate samples in VMSL.
node in different algorithms when sd and nd increase. We
5.3 Communication Cost assume each point is represented with two 4-byte integer
numbers and the closeness is represented with one 4-byte
First we theoretically analyze the communication cost of
float number. We can see that the additional communica-
different algorithms. In MCL and MCB, each beacon node
tion cost incurred in WMCL-A and WMCL-B is very
limited, but is slightly higher than that in VMSL.
TABLE 1 Table 2 lists the communication cost for each node in
Number of Distance Computing in WMCL and VMSL* (Â104 ) WMCL and VMSL* when sd and nd increases. We can find
that the communication cost in these two algorithms is
dominated by the cost in transmitting the whole set of the
samples and is not affected a lot by sd and nd . However,
VMSL* always incurs about 50 percent more communica-
tion cost than WMCL.
5.4 The Impact of Irregular Communications
In this section, we study the performance of the proposed
algorithms under irregular communications. We use the
Quasi Unit Disk Graph (Q-UDG) model presented in [15] to

Fig. 18. Number of bytes transmitted when sd varies. Fig. 19. Number of bytes transmitted when nd varies.



TABLE 2 TABLE 3
Number of Bytes Transmitted in WMCL and VMSL* (Bytes) Cost of Different Operations (50,000 Times)

Fig. 21. Localization accuracy of WMCL-A and Centroid in one randomly
selected network.

ATmega128L microcontroller. The operating system used is
TinyOS 2.1. For each type of operation, we execute that
operation 50,000 times and measure the time used to
complete the execution. The results are listed in Table 3.
We use two different random generators provided in
Fig. 20. Localization accuracy versus communication irregularity. TinyOS 2.1 (implemented in component RandomC and
RandomLfsrC, respectively) to measure the cost of generat-
study the the impact of irregular communications on the ing candidate samples. The letter one is much faster but
localization accuracy of different algorithms. with less randomness. We can see that generating a
In the Q-UDG model, there is a parameter d 2 ½0; 1Š candidate sample is much costlier than computing the
which is used to control the irregularity of the communica- distance between two samples. We also measured the cost
tions. Two nodes are connected by an edge if their distance of computing the distance between two points represented
is at most dÃr where r is the maximum communication using two float numbers. The cost is 0.706 s and is still much
range as defined in the UDG model. If the distance between lower than the cost of generating a candidate samples.
two nodes is greater than r, then no edge exists between The power operation needed in the VMSL algorithm and
them. When the distance is between dÃr and r, we assume the sqrt operation needed in our proposed algorithms is also
that the probability that there is an edge between the two very costly. However, in our proposed algorithms, the
nodes follows a uniform distribution. number of the sqrt operations is very small (usually less
Fig. 20 shows the localization accuracy of different
than sd ) but in VMSL the number of power operations is
algorithms when the value of the parameter d varies from
much larger (proportional to the number of generated
1 to 0.5. We can see that when d is larger than 0.8, the local-
candidate samples). So it is necessary to improve sampling
ization accuracy of the proposed algorithms only degrades
efficiency in SMC-based algorithms because this can
slightly. However, when d is smaller than 0.8, the localization
dramatically reduce the computational cost.
accuracy degrades dramatically. However, the proposed
algorithms always perform better than other algorithms. 6.2 Validate the Convergence of the Proposed
The irregular communication not only degrades the Algorithms
localization accuracy, but also incurs more computational
In the first set of experiments, we validate the convergence
cost. In our simulation, in all the algorithms the computa-
of our proposed SMC-based algorithms in a real deployed
tional cost increases by a factor of at least 2 when d changes
sensor network. We deployed 40 Micaz motes in a 2:2 m Â
from 1 to 0.5.
2:2 m square region on the ground of our laboratory (see
Fig. 22a). Twelve of them were deployed on the perimeter
6 EXPERIMENTS of the region as beacon nodes, and others were randomly
We have implemented our algorithms in real deployed deployed inside the deployment area. We carefully adjusted
sensor networks to validate the obtained performance results. the transmitting power of the Micaz motes such that the
We first measured the cost of different operations in real transmission range is about 60 cm.
hardware to justify our efforts in reducing the computational We run the WMCL-A algorithm on the sensor nodes and
cost. After that, we validate the convergence of the proposed the result is shown in Fig. 21. From Fig. 21, we can see that
algorithms in real environments. At last, we implement five WMCL-A converges fast in a static sensor network. This
SMC-based localization algorithms: MCL, MCB, VMSL, validates the effectiveness of our proposed algorithms. For
WMCL-A, and WMCL-B in a sensor network consists of comparison, we also implemented the Centroid algorithm
23 Micaz motes and compare their localization accuracy. [5] which is often used as a comparison algorithm in real
implementation. We run the two algorithms in five
6.1 Cost of Key Operations randomly generated networks and found that the localiza-
We have measured the cost of different operations on the tion accuracy of WMCL-A was about 30 percent higher than
Micaz platform [1]. The Micaz mote equips with an Atmel that of Centroid.



Fig. 22. The deployment of sensors. (a) The indoor scenario, and (b) the outdoor scenario.

6.3 Real Implementation in a Outdoor Network algorithms on the improvement of localization accuracy and
We have implemented the aforementioned SMC-based reduction of energy consumption.
localization algorithms and evaluated their localization In our simulations, we find that irregular communica-
accuracy in a static sensor network. We deployed 23 Micaz tions will not only degrade the localization accuracy of the
motes in a 6 M Â 6 M square region on a volleyball court proposed algorithms but also will incur much more
(see Fig. 22b). We set the transmitting power of the sensor computational cost. In our implementation of the proposed
nodes to be about À10 dBm so that the transmission range is algorithms in real environment, we find it is difficult to
about 3 m. Six of the twenty-three nodes were placed on the determine a maximum transmission range between two
perimeter of the deployment region and others were nodes. The connectivity between two sensor nodes is very
randomly deployed inside the region. unstable. The connectivity status may vary greatly in
Fig. 23 shows the localization error of every sensor node different times, in different places, or even between
in different algorithms. The average localization error in different sensor nodes. In this case, the performance in real
different algorithms is listed in Table 4. We can see that our implementation will be much worse than the performance
proposed algorithms outperform other algorithms. The obtained from simulations.
improvement factor is from 23 percent to 37 percent.

ACKNOWLEDGMENTS
7 CONCLUSION
The authors would like to thank the anonymous reviewers
In this paper, we present an accurate and energy efficient and the editors for their invaluable suggestions that
range-free localization algorithm for mobile sensor net- improved the quality of this paper. They also thank Chao
works. We measured the cost of key operations in SMC- Yang and Yingpei Zeng for the help in collecting experi-
based localization algorithms on real hardware and found mental data. The authors gratefully acknowledge the code
that a high sampling efficiency is necessary to reduce the sent by Hu and Evans and the code sent by Masoomeh
computational cost. Then, we propose a set of algorithms
Rudafshani. This work has been supported in part by the
which achieve high sampling efficiency and high localiza-
National Natural Science Foundation of China under Grant
tion accuracy. The results from our simulations and
No. 60873026, 60573132, 60673154, 90718031, and 60721002,
experiments validate the effectiveness of our proposed
the National Grand Fundamental Research 973 Program of
China under Grant No. 2006CB303000 and 2009CB320705,
HK RGC under GRF grant PolyU 5102/08E and HK PolyU
under grant 1-BB6C.

REFERENCES
[1] http://www.xbow.com/products/product_pdf_files/wireless_
pdf/6020-0060-01_a_mic az.pdf, 2008.
[2] A. Baggio and K. Langendoen, “Monte-Carlo Localization for
Mobile Wireless Sensor Networks,” Proc. Conf. Mobile Ad-Hoc and
Fig. 23. Localization accuracy of different algorithms in the outdoor Sensor Networks (MSN ’06), pp. 317-328, 2006.
deployment. [3] P. Bergamo and G. Mazzini, “Localization in Sensor Networks
with Fading and Mobility,” Proc. 13th IEEE Int’l Symp. Personal,
TABLE 4 Indoor and Mobile Radio Comm. (PIMRC ’02), pp. 75-754, 2002.
Average Localization Error in Different Algorithms [4] P. Biswas and Y. Ye, “Semidefinite Programming for Ad Hoc
Wireless Sensor Network Localization,” Proc. Third Int’l Symp.
Processing in Sensor Networks (IPSN ’04), 2004.
[5] N. Bulusu, J. Heidemann, and D. Estrin, “GPS-Less Low Cost
Outdoor Localization for Very Small Devices,” IEEE Personal
Comm., vol. 7, no. 5, pp. 28-34, Oct. 2000.


Achieving High Accuracy and Efficiency in Localizing Mobile Sensor Nodes

Achieving High Accuracy and Efficiency in Localizing Mobile Sensor Nodes

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