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USC Computer Science Technical Reports, no. 796 (2003)

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USC Computer Science Technical Reports, no. 796 (2003)

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Content On the Feasibility of Ad-Hoc Localization Systems
Krishna Kant Chintalapudi Amit Dhariwal Ramesh Govindan
Gaurav Sukhatme
Computer Science Department,
University of Southern California, Los Angeles,
California, USA, 90007.
Abstract
Ad-hoc localization systems enable nodes in a sensor network to fix their positions in a global coordinate system
using a relatively small number of anchor nodes that know their position through external means (e.g., GPS). Because
location information provides context to sensed data, such systems are a critical component of many sensor networks
and have therefore received a fair amount of recent attention in the sensor networks literature. The efficacy of these
systems is a function of the density of deployment and of anchor nodes, as well as the error in distance estimation
(ranging) between nodes.
In this paper, we examine how these factors impact the performance of the system. This examination lays the
groundwork for the main question we consider in this paper: Can the ability to estimate bearing to neighboring
nodes greatly increase the performance of ad-hoc localization systems? We discuss the design of ad-hoc localization
systems that use range together with either bearing or imprecise bearing (such as sectoring) information, and evaluate
these systems using analysis and simulation.
1 Introduction
Sensor network localization has been an active area of research for the last few years. For sensor networks, and more
generally for networks of embedded systems, the ability for nodes to determine their position through automatic means
is recognized as an essential capability. The community has made great strides in ranging technologies, systems for
infrastructured-based localization, and algorithms and techniques for ad-hoc localization (Section 2). This last class is
the subject of this paper.
In an ad-hoc localization system, nodes determine their position in a common coordinate system using a number
of anchor nodes that already know their location (through some external means, such as GPS [6]) in that coordinate
system. These systems assume all nodes possess a ranging capability (the ability to estimate distances to other nodes).
Using their range estimates, nodes use one of several distributed position fixing techniques to determine their positions
in the coordinate system.
There are two characteristics that are highly desirable in a distributed ad-hoc localization system
1
; in fact, we
assume that these are design requirements for such systems. The first requirement is that the performance of such a
system be relatively insensitive to anchor placement, as long as the anchors are not placed in a degenerate configura-
tion. From a sensor network perspective, this is desirable since it may often be difficult to engineer anchor placements
in the environments that these networks are deployed. Another way of saying this is that an ad-hoc localization system
permits unplanned anchor placement. A second requirement is that relatively few anchors be necessary for obtaining
good localization performance. This requirement is motivated by the fact that in some environments it may be difficult
to obtain position estimates through external means (e.g., because GPS signals can be significantly through foliage).
We argue that ad-hoc localization systems should work well with an order of magnitude fewer anchors than nodes.
This rule-of-thumb is motivated by a systems argument; if one in two or three nodes are required to be anchors, it
would significantly constrain the deployment of such systems.
We begin this paper (Section 3) by evaluating the performance of a range of ad-hoc localization techniques pro-
posed in the literature. The performance of ad-hoc localization depends upon several factors: the accuracy of ranging,
1
Our focus in this paper is on distributed ad-hoc localization systems. Henceforth, when we use the term “ad-hoc localization system”, or simply
“localization system”, we mean this class of systems.
1
the density of node placement, the relative density of anchors, as well as the particular position fixing schemes in
use. Using both analysis and extensive simulations, we find that ad-hoc localization systems begin to perform ac-
ceptably only at node densities well beyond the density required for network connectivity. We find this to be rather
pessimistic—being required to deploy more resources to get a component of a system working seems undesirable from
an architectural standpoint. Moreover, we argue that this is a fundamental limitation of ad-hoc localization systems
that use ranging devices only, rather than a shortcoming that can be remedied by designing better localization schemes.
We then consider whether adding the ability to estimate bearing to neighboring nodes can qualitatively improve
the performance of ad-hoc localization schemes (Section 4). We show that there exists a highly accurate position fixing
scheme that uses both range and bearing information in order to localize nodes, at node densities comparable to that
required for network connectivity. This is obviously an idealization, since it is unclear if accurate bearing estimation
devices can be built at the form factors and energy-levels that sensor network nodes require.
What is more feasible, perhaps, is the ability to approximately detect bearing. Guided by this observation, we
examine whether devices that enable nodes to place neighbors within sectors can enable acceptable ad-hoc localization
performance at node densities that are sufficient for connectivity (Section 5). We show that there exists a simple
iterative scheme that can provide attractive ad-hoc localization performance using both range and sector estimates.
We conclude the paper by arguing that, given the importance of node localization in a sensor network, the community
should invest some effort in sector estimation devices that will enable practical ad-hoc localization systems.
2 Related Work
In recent years, there has been significant work in localization for sensor networks and networks of embedded devices.
Localization has also been an active area of research within the robotics community for several years now. We briefly
review the current state of knowledge in localization systems. Our intent is not to be exhaustive in our coverage of the
literature; rather, we list representative pieces of work that help put this paper in context.
Ranging An important focus of the localization literature has been robust techniques for estimating distances be-
tween nodes (ranging). The networking community has focused on two classes of ranging techniques: RF-based
ranging and acoustic ranging. For the purposes of this paper, the exact choice of ranging technique is not important,
but we include a discussion of these techniques because, as we shall see, they do have some bearing on our discussion
of ad-hoc localization.
RF-based ranging, as exemplified by the SpotON [10] and Calamari [21] systems, is based on the premise that
by measuring received signal strength a receiver can determine its distance to a transmitter. This presumes that RF
propagation in an environment can be accurately characterized by a simple path loss model, whose parameters are
known. Using this technique, nodes can estimate distances to all neighbors within radio range. Range errors upwards
of 10% have been reported in the literature [21], usually after a fairly involved calibration step that estimates the path
loss parameters and adjusts for variations in transceiver characteristics.
A second class of ranging schemes is based on measuring the time-of-flight of an acoustic or ultrasound signal [8,
19]. More precisely, these techniques measure the difference in arrival times of simultaneously transmitted radio
and ultrasound signals, then estimate distances knowing the speed of sound. Some approaches in acoustic ranging
use spread spectrum approaches for resilience to multipath, and employ techniques to correct for latencies induced
by other system components [8]. Such techniques provide an order or magnitude better accuracy (1-2% error) over
distances of 3-6 meters (i.e., significantly lower than the nominal radio range of sensor platforms such as the Mica
motes).
Finally, we note that the robotics community has, for some time now, used highly accurate laser range-finders
(which exhibit less than 0.1% ranging error). It is unclear, however, whether such devices can be manufactured in the
form factors and energy-consumption levels required of sensor nodes.
Position Fixing A large body of work has examined algorithms for ad-hoc localization schemes. We taxonomize this
literature later (Section 3.1), and only briefly introduce the related work here. Perhaps the earliest pieces of work in the
area of sensor network localization can be attributed to Niculescu et al. [12] and Savvides et al. [20, 19]. In the former,
nodes first estimate their distances to anchors using one of several techniques (DV-hop, DV-distance, and a Euclidean
scheme), then fix their own position using these distances. The latter proposes an elegant N-hop multilateration
scheme which we discuss later. That work also discusses a Kalman filtering based position refinement phase to
2
+
+
1
a
+
2
3
d
d
d d
1
1
2
2
’ ’
r
r
r
2
2
1
1
’ r
’
Figure 1: V oting scheme
+
+
+
+
r
r
r
r
r
r
r
r
r
23
14
1
12
24
34
1 2
4
4
2
3
a
θ
θ
θ
24
14
34
3
Figure 2: Lateration
x
y
Global
Local
x’
y’
θ
θ’
Figure 3: Global and local coordinate
systems
improve position estimates. In more recent work, Savvides et al. discuss the error characteristics and the dependence
on network size and anchor density of their schemes [18]. Savarese et al. [17] propose a three-phase scheme, where
they obtain crude position estimates and use an iterative multi-lateration based scheme to refine the estimates. Each
node uses the locations of its neighbors to multi-laterate and re-estimate its position. Finally, Langendoen et al. [11]
discuss a fairly detailed comparison of all the above schemes in the face of ranging errors, different node densities and
anchor fractions. Our work draws heavily upon theirs.
Somewhat tangentially related to our work is other literature on localization. Examples of such work include: po-
sition inference using specialized beacons [3], optimization methods for centrally estimating locations [5], in-building
localization systems that enable position for context-aware applications [14, 9, 2], systems for estimating orientation
of handheld devices [15], and systems for placing nodes within a relative coordinate system [4]. We do not survey this
literature in detail.
Finally, predating the work in sensor network localization, there exists significant work in robotic node localiza-
tion. This has discovered many tools that are applicable to ad-hoc localization systems: Kalman filter approaches to
cooperative localization [16], refinement of probabilistic estimates of robot positions [7], and maximum-likelihood
estimation of relative robot positions [1]. While not directly relevant to our work, they have influenced the design of
some of our localization schemes.
3 Ad-Hoc Localization Using Ranging
As we have seen in Section 2, most of the distributed ad-hoc localization schemes studied till date leverage the ability
of nodes to estimate distances to other nodes using ranging. Our goal in this section is to fairly extensively evaluate,
through analysis and simulation, a representative sampling of such ad-hoc localization schemes, with the intent of
understanding their error characteristics as a function of node and anchor density. Based on the simulations we gain
some insight into the fundamental limitations (qualitatively and quantitatively) of range-only methods. In the rest of
this section, for conciseness and to distinguish this class of localization approaches from those we consider later in the
paper, we use the term r-only localization to describe these methods.
In some recent work, Langendoen et al. [11] have performed similar analyses. However, our evaluations (presented
in this section) differ from theirs in several respects. In our simulations, we use larger topologies, examine a wider
range of density and anchor ratios, and use a slightly more sophisticated ranging error model. We also introduce some
improved localization schemes of our own into the mix of schemes, and examine more metrics to explore the efficacy
of ad-hoc localization.
3.1 A Taxonomy of Ad-Hoc Localization Schemes
Before describing the schemes we evaluate, we taxonomize r-only localization methods. Following Langendoen et
al. [11], such localization schemes can be divided into three distinct sub-problems or stages: estimating distance
to anchors; getting an initial position estimate; and iteratively refining the position estimate. When only ranging
techniques are available, a sensor needs to estimate its distance to at least three anchors in order to localize itself into a
3
global coordinate system common to those anchors. All r-only localization schemes begin with each sensor trying to
estimate its distance (or some approximation thereof) to anchors — the first sub-problem. Having obtained distances
to three anchors, obtaining an initial position estimate reduces to a simple geometric problem commonly referred to as
lateration. With these estimates, some r-only localization approaches incorporate a refinement scheme that improves
position estimates in the face of ranging error.
In the following subsections, we describe each of these three stages, and discuss the r-only localization schemes
we evaluate.
Estimation of distance to an anchor
Broadly speaking, there are two classes of approaches to estimating distance to anchors: topological and geometric.
Topological techniques are content to roughly estimate distance to anchors using information obtained by message
flooding. Geometric techniques, on the other hand, more carefully compute distances to anchors and generally exhibit
higher accuracy.
One example of a topological approach is what Niculescu et al. [12] the DV-dist approach. In this approach,
each anchor initiates a (possibly constrained) flood. As the flooding message propagates, it accumulates the range
measurement corresponding to each hop. Each node measures its distance to an anchor as the minimum accumulated
distance. This approach consistently over-estimates the distance to anchors and is subject to accumulation of error over
multiple hops. A slight variant of this approach estimates distance using hop-counts to anchors, a technique called
DV-hop [12]. In Section 3.3, we evaluate an ad-hoc localization scheme using the DV-dist approach.
Geometric approaches are more sophisticated, and can be classified into: those that use relative bearing information
when geometrically computing distance to anchors, and those that do not.
An example of the latter is the Euclidean approach proposed by Niculescu et al. [12]. The scheme is illustrated
in Figure 1. Essentially, using its neighbors’ computed distances to an anchor, a node determines various possibilities
for its own distance to the same anchor, and picks the most likely using a “voting” scheme. For example, Node d has
three nodes (1, 2, and 3) which have their estimates of distance (and positions) from anchor node a. Each pair of nodes
gives raise to two possible values of ranges. For example, the pair (1,2) gives raise to the two possible ranges r
1
(for
d
1
) and r
1
(for d
1
), while the pair (2,3) gives rise to r
2
(for d
2
) and r
2
(for d
2
). Under error free conditions, d
1
and d
2
would be identical, while d
2
and d
2
would not and so r
1
would be the true range. With ranging subject to errors, the
true solutions may not overlap, but lie “close” to each other as illustrated by d
1
and d
1
in Figure 1. In the voting based
scheme, each solution votes for solutions in its vicinity and the solution with the maximum votes is chosen as the true
solution. We include this scheme in our evaluations as well. In our implementation of this voting scheme, a solution
from one pair of nodes votes to the closer (in terms of Euclidean distance) of the two solutions from another pair with
a vote whose strength is equal to the negative of the difference between them.
Finally, the last (and most sophisticated) distance estimation technique is a geometric technique that implicitly
computes relative bearing to neighbors (in addition to distance from an anchor) [19]. This technique is based on
relative localization, as illustrated in Figure 2. In this figure, nodes 1, 2 and 3, know their ranges to the anchor node 0
and are within ranging distance of each other (r
12
,r
23
,r
13
can be measured). Node 4 has the three nodes 1, 2 and 3 in
its ranging radius (can measure r
14
,r
24
, r
34
) and hence can calculate its distance to the anchor a. With that information,
node 4 can also determine the relative angles θ
34
, θ
24
, θ
14
. Thus, nodes 1, 2, 3 and 4 can estimate their positions in
a local coordinate system. A set of relatively localized nodes are localized within a local coordinate system, with its
origin as the anchor. In general, this coordinate system may be a translated, rotated and mirrored (relative angles being
measured in the opposite sense) version of the global frame of reference as in Figure 3. The only absolute information
(invariant in the global reference) obtained from relative localization with respect to an anchor are the ranges of a
node from the anchor. Knowing how three or more of its neighbors place themselves in a local coordinate system
with respect to an anchor, a node can place itself in that same relative coordinate system using multi-lateration. In
Section 3.3, we evaluate an ad-hoc localization scheme that uses relative localization.
Initial position estimate and refinement
Once a node has estimated its distance to three anchors, multi-lateration seems to be the most commonly used approach
for obtaining the node’s initial position estimate. The performance of multi-lateration depends on the distances and
relative locations of the anchors and nodes. It works well if the anchors are far apart and enclose the nodes within their
convex hull [18].
4
Finally, three different refinement schemes have been proposed in the literature. The first is iterative multi-
lateration, where nodes continuously estimate their positions using the estimated locations and ranges of their neigh-
bors, and hoping to improve their locations [20]. The second is a Kalman filter based approach, which estimates the
Jacobian of the multi-lateration equations in an attempt to obtain a Gaussian representation for the error in the solu-
tion [19]. The third approach is a heuristic proposed in [17] where bounding boxes for the solutions are iteratively
shrunk.
In the following section, we describe a distributed refinement scheme which explicitly attempts to find locations
for the nodes which “best” fits (in a least-mean-squares sense) the set of all measurements made in the sensor network.
This scheme has independent interest, but also lays the groundwork for some localization schemes we introduce later
in the paper.
Iterative Least-Mean-Squared Refinement
Our proposed refinement scheme attempts to find locations for the sensors, which best explains (in terms of weighted
mean squared error), the set of range measurments taken over the entire network. A range measurement r
i j
taken at a
node n
i
to measure range to another node n
j
, gives an equation,
x
i x
j T
x
i x
j r
i j 0 (1)
Here, x
i x
j ℜ
2
are the locations of nodes n
i
and n
j
. For every measurement taken in the network we would then
have one such equation. Since the number of measurements (hence the number of equations) can far exceed the
number of unknown locations, we have an over-determined system. A common way to solve such an over-determined
system is to find locations which “best” fit the set of equations. Since each measurement could potentially have
different error characteristics, some measurements could be deemed more reliable than others. In such a scenario,
preferentially weighting the reliable equations in the equation set could give improved solutions. For this we adopt the
most commonly used method for finding a best fit for locations—we find locations that minimize the weighted mean
square error taken over all the equations.
We label such a scheme LMSR. LMSR basically weights measurements with their observed σ
i j
, since these are
a direct measure of the certainty of the measurements. As we describe below, LMSR is amenable to a distributed
implementation. The theory behind LMSR assumes that ranging errors are Gaussian with known standard deviations
σ
i j
corresponding to a range measurement r
i j
. This assumption is not critical to the scheme but we do find from
measurements (see below) that the Gaussian assumption holds for range errors. Furthermore, in an implemented
system, we would expect to use the nominal standard deviation obtained from a data sheet, standard deviations obtained
by calibrating the devices prior to measurement, or the standard deviations of multiple range measurements.
We now describe the theory behind LMSR. Let G
N
E
be a graph representing the connectivity of the sensor
network, N be the set of all nodes, and E
e
i j
the set of all edges. An edge exists between two nodes which are within
ranging neighborhood of each other. Further, let A
i
represent the set of nodes that node n
i
can range to. Also let L N,
be the set of all anchors, hence, the x
i
s for n
i L are known, and the remaining x
i
s are unknown.
Formally, LMSR attempts to minimize the objective function,
J
∑
e
i j E
σ
2
i j x
i x
j T
x
i x
j r
i j 2
(2)
The conditions for local minima can be written as, the set of N L simultaneous equations,
∑
n
j A
i
σ
2
i j σ
2
ji
σ
2
i j
r
i j
σ
2
ji
r
ji
x
i x
j T
x
i x
j x
i x
j 0
n
i N
L
n
j N (3)
The above equations can be re-written as,
x
i ∑
n
j A
i σ
2
i j σ
2
ji x
j σ
2
i j
r
i j σ
2
ji
r
ji
x
i x
j T
x
i x
j ∑
n
j A
i
σ
2
i j σ
2
ji
n
i N
L
n
j N (4)
5
a d
η
R
+
+
+
Figure 4: 3 Connectedness
0 1 2 3 4 5 6 7 8 9 10
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Standard Deviation (feet)
Distance from Sonar (feet)
Figure 5: Sonar ranging error std as a fuction of range
Note that Equation 4 depends only on the locations estimates of nodes within x
i
’s ranging neighborhood and the
ranges they measure. This allows for a distributed implementation of LMSR, where nodes continuously exchange their
current location estimates, then use Equation 4 to update their estimates until they converge. LMSR will converge to
the nearest local minima in the objective function, hence an initial guess “sufficiently” close to the global minimum
will force the nodes to a least-mean-squared fit of the measurements. This initial guess is, of course, obtained from
the second stage of r-only localization.
3.2 A Simple Analytical Comparison of the Schemes
Having described the various r-only localization schemes, we now attempt to analyze the schemes and try to predict
their performance. We then solidify these discussions with a quantitative evaluation using simulation of some of these
schemes in Section 3.3.
Estimating distances from anchors: The geometric schemes use ranges and locations of three or more nodes to
relatively locate (or determine range to an anchor) a new node. This imposes a requirement that a node have at least
three nodes within its ranging neighborhood and further that these nodes be connected among themselves (Figure 4).
We call this property 3-connectedness to an anchor. Let the set of all nodes 3-connected to an anchor a
N be denoted
by N
3
a
N. Formally, n
i N
3
a
if and only if there exists n
j n
k n
l N
3
a A
i
such that the sub-graph comprising nodes
n
j n
k n
l
is connected.
Intuitively, 3-connectedness is a strong requirement of sensor deployment topology, far stronger than mere con-
nectedness of the overall topology. As we now argue (using a simple geometric model), 3-connectedness may leave
many nodes unable to find an initial position fix even at moderate node densities. Figure 4 shows a typical scenario,
where, in order to obtain a range estimate from a, node d needs 3 nodes in the region of intersection of the two circles
(representing measurement neighborhoods of a and d) shown. Let R be the radius of the ranging neighborhood.
If the node d is at a distance ηR from the anchor a, the area of overlap is:
O
ηR
R
2
cos
1
η
2
η
2 1
η
2
4
(5)
An average of 3 nodes in this area implies an average node density of
p
e
η
3π
π cos
1 η
2 η
2 1
η
2
4 (6)
in the ranging neighborhood. Nodes beyond one hop from the anchor have η 1 and ρ
e
1
7 7. This implies that
any geometric scheme will require a deployment density of least 8 nodes within the ranging neighborhood for them to
start working. Thus, it is expected that at densities below 8 almost none (or a very small fraction) of the nodes will be
6
localized. Notice that our argument implicitly assumes a reasonably low fraction of anchors, as is desirable for ad-hoc
localization systems. For example, if the three nodes in Figure 4 already had obtained their position through external
means (rather than inferring it from anchor a), our argument would not hold.
At a density of p, the expected distance μ between two nodes is:
μ
p
π
p
R (7)
This means that the expected distance of the closest node to a beyond the ranging neighborhood of a is:
χ
p
R
μ
p
1
μ
p
(8)
We wish to determine, the critical node density (p
c
), which roughly guarantees that almost all nodes in the network
will be 3-connected. This will allow localization by geometric schemes at low anchor ratios. Suppose the node density
is such that the closest node beyond the ranging neighborhood is 3-connected to the anchor. Then, this node can
determine its relative position (or distance) and act as an anchor for the closest node beyond its ranging neighborhod
and so on. Such a density will then will allow localization of almost all nodes at very low anchor ratios. This condition
is given by the equation,
p
e
χ
p
c R
p
c
(9)
Solving equation 9 gives ρ
c 14 9845. Hence, 3-connectedness becomes highly likely for almost all nodes at densities
around 14 to 15 and one would expect almost all the nodes to be localized (with very low error) at these densities and
beyond. Furthermore, at low densities, those nodes that do obtain an initial position estimate get quite accurate
estimates using geometric schemes. Our simulations in Section 3.3 bear this out.
By contrast, the topological schemes do not require 3-connectedness and the existence of a single network path to
the anchor is sufficient to obtain distances to anchors. Using a similar analysis, the critical density can be calculated
as,
1
3
p
e
χ
p
c R
p
c (10)
since, the overlap region now requires only one node rather than three. The solution to equation 10 is ρ
c 5 9956.
Hence, it follows that connectedness will be highly likely at densities of 5-6 nodes per ranging neighborhood and
beyond. One expects almost all nodes to be localized at and beyond these densities at low anchor fractions. Note that
the above arguments do not depend on actual values of radio range or total number of nodes in the network, but only
depend on the average number of nodes in the ranging neighborhood.
While the density requirements for topological approaches seem to be more relaxed, this comes at the cost of
increased localization error. The ranges calculated beyond one hop are consistent over-estimates, sometimes by a very
large margin. We quantify this tradeoff in Section 3.3.
Effect of refinement on the schemes: The geometric approaches, provide fairly accurate estimates for the range
from anchors. This, coupled with such range estimates from several anchors provides a fairly accurate initial guess to
work with. The unlocalized nodes do not provide an initial guess to work with and hence cannot be refined. As pointed
out by Langendoen et al. [11], refinement does not improve performance significantly for geometric approaches.
On the other hand, the topological schemes for estimating distances to anchors leave enough room for improvement
by refinement. For example, in the DV-dist approach at least, the estimates of ranges to anchors is incorrect often
resulting in highly inaccurate initial guesses.
3.3 Evaluating Ad-Hoc Localization Systems
In the previous sections, we taxonomized the various r-only ad-hoc localization schemes known in the literature, and
examined analytically their performance as a function of node density. In this section, we use simulation to discuss
the performance of r-only localization schemes.
7
The Schemes According to our taxonomy (Section 3.1), r-only localization can be classified into three phases. For
the second of these phases, obtaining an initial position fix, almost all published schemes use multi-lateration. So do
we.
For the last of these phases, refinement, we use the LMSR scheme described in Section 3.1. This choice is
somewhat different from that published in the literature, but we are fairly confident that LMSR is representative of
the benefits that refinement can provide. For geometric schemes, refinement provides very little benefit anyway.
We found that, for topological schemes (DV-dist), the performance of our refinement scheme “closely” matched the
results reported by [11]. For these reasons, we do not believe that evaluating other refinements schemes will change
the conclusions we make below.
The first, and most crucial, phase in range-only ad-hoc localization is estimating a node’s distance to anchors. This
phase presents the most choices. In our evaluations, we choose three schemes, representative of three different classes
of our taxonomy: the Euclidean [12] geometric scheme that does not estimate relative bearing to anchors; the relative
localization technique that implicitly computes relative bearing to anchors; and the DV-dist topological scheme. We
believe that this set adequately represents the state of the art in range-only ad-hoc localization mechanisms.
Simulation Methodology All our simulations were carried out in a home-grown simulator. We carefully imple-
mented many of the schemes described in the literature, and our results are in agreement with published results. Since
our approach used an independent implementation, it represents a validation of most of the prior work in the field.
All our simulations use a field of 1000 randomly deployed nodes (node locations being drawn from a uniform dis-
tribution over the region). We chose the ranging distance to be 10m, and we also assume that this is the communication
radius.
The density of the nodes (the number of nodes in a ranging neighborhood) was controlled by increasing the area
of deployment while keeping the number of nodes 1000. For example, an 252m 252m region results in a density of
5, while a 200m 200m region results in a density of 8 nodes. A fraction of nodes (the anchor fraction) were chosen
to be anchor nodes.
Most of these assumptions are consistent with the literature. One place where we deviate from the literature is
in the use of a different model for ranging error. Most published work, at least in sensor network localization, has
assumed that ranging error is independent of the ranging distance. The error model we use is a Gaussian with its
standard deviation varying linearly with range, and given by σ
r
αr. Consider a uniform error model which uses
a fixed standard deviation σ. In comparing this with our distance dependent error model where the standard deviation
at the maximum distance was also σ, we see that our error model is more generous (or less pessimal) towards ad-hoc
localization schemes; for smaller ranges, it assigns less ranging error. Our conclusions are not affected by this choice
of error model; we have verified that using a uniform error model gives qualitatively the same result.
To validate our error model, we conducted measurements using a sonar ranging device on an off-the-shelf robot.
Although the robot is significantly more capable that a sensor node, we anticipate that the error characteristics of
the sonar will be similar to the ultra-sound ranging devices in consideration for sensor networks [8]. In fact, our
error measurements in this device (about 1% over 3-4m) agrees qualitatively with ultrasound ranging error estimates
measured elsewhere [8, 20]. Figure 5 shows the variation of standard deviation as a function of distance. Error variance
increases with distance, and for the range we consider, a linear fit works reasonable well with an α of approximately
0.7.
Furthermore, the Gaussian assumption is, to a large extent, borne out by measurements. For example, we took
measurements on a sonar ranging device at a 5 foot distance. The distribution of samples at this distance is shown in
Figure 6 and is visually Gaussian with a slightly longer right tail. With increasing distance, this tail becomes slightly
more pronounced.
Metrics Most of the literature focuses on either RMS error or mean error as measures of the accuracy of ad-hoc
localization. Both measures define the error of localization as the distance between the computed location and the
actual (ground truth) location. RMS error can be skewed by nodes with large localization error, and often gives
pessimistic estimates of localization error. Mean error weighs all values of localization errors equally and hence
provides a more realistic estimate of the expected value of error. Mean error is defined as,
e
μ 1
n
n
∑
i 1
x
i ˆ x
i T
x
i ˆ x
i (11)
8
5.1 5.15 5.2 5.25 5.3 5.35
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
range (feet)
fraction of smaples
Figure 6: Sonar Ranging Measurement
Distribution
5 6 7 8 9 10 11 12 13 14 15
0
10
20
30
40
50
60
70
80
90
100
Node denisty
20% Euc
20%, Rel
20%,DV
Localization extent (%)
Figure 7: Localization extent
5 6 7 8 9 10 11 12 13 14 15
0
5
10
15
20
25
30
35
40
45
Node density
20% Euc
20%, Rel
20%,DV
Localization error (% of R)
Figure 8: Localization Error
where ˆ x
i
are the estimates of x
i
found by the algorithm. In this paper, we use mean error as a measure of localization
accuracy, both because it is less skewed than RMS error by large error values, but also because it is more easily
understood as an error measure. We compute mean error only on nodes that were able to obtain a position estimate at
all.
Also motivated by our discussion in Section 3.2, we introduce a new metric that measures the extent of localization.
This is essentially a quantile metric and is defined as the fraction of non-anchor nodes that are localized to within 2
meters of their ground-truth position. In our simulations, 2m represents a fairly conservative estimate (about 20%
of radio range). The extent metric serves to differentiate geometric schemes (which rely on 3-connectedness) and
topological schemes (which do not). Much of the literature has ignored this metric, choosing to evaluate localization
error only in regimes with high localization extent. We argue in Section 3.3 that this metric brings out a more complete
picture of the dynamic performance range of ad-hoc localization schemes.
Simulation Results We have evaluated our three representative schemes using simulation for three different ranging
errors (0.1%, 1% and 5%) and three different anchor fractions (5%, 10% and 20%). Given space constraints, we present
results from one combination (20% anchors, 1% error) that represents the state of the art ranging error using ultrasound
ranging, and a 20% anchor fraction that represents a fairly generous sprinkling of anchors. Other combinations of
errors and anchor densities do not affect our conclusions of Section 3.4.
Figure 7 shows the localization extent for our three schemes. Qualitatively, there seem to be two classes of
performance: the DV-dist approach (a topological scheme) has higher localization extent at lower node densities than
the Euclidean and the relative localization schemes (the geometric schemes), but those two catch up at about a density
of 12. Clearly, the 3-connectedness requirement greatly affects the performance of the second class; only 20 to 30%
of the nodes are able to get a position fix at all at low densities. In the DV-dist approach, even a low densities, more
than 95% of the nodes are able to get a position fix (as predicted by our analysis), but their estimates are very poor in
quality (a greater than 20% error in position). Finally, notice that the simulations are consistent with our analysis: at
densities of 14 or 15, the network becomes 3-connected.
The same two classes are visible when we examine the localization error (Figure 8). The geometric schemes exhibit
low mean error (and also low RMS error – we have computed this metric but do not include it in our evaluations)
across the entire range of densities because if a node gets a position estimate, it gets a good one. On the other hand,
the topological schemes start with high error at low densities and achieve good performance only at densities of
9 or 10. There is some performance difference between the two geometric schemes (Euclidean outperforms relative
localization), but we suspect that their performance could be made comparable by applying well-known optimizations.
We did not do this since our intent was not to compare similar schemes, but understand classes of performance).
The qualitative conclusion we draw from this is that r-only localization schemes require 11-12 nodes within the
ranging neighborhood in order to achieve 90% localization and 5% accuracy.
3.4 The Feasibility of Ad-Hoc Localization Systems
Our conclusion above represents a somewhat subjective choice (90% extent, 5% mean error). However, we believe
these thresholds to be fairly generous from the perspective of a deployed system. Also our evaluations are fairly
extensive and borne out by analyses. We argue that this result is pessimistic, because in order for r-only localization
9
schemes to work, almost twice as many nodes are required as for mere connectivity (it is well known that a radio
network with uniformly distributed nodes is connected, with high likelihood, if it has an expected number of 5-6
neighbors per node). This is an architectural oddity, since it isn’t clear one should have to deploy twice as many
resources for only one component (albeit a very important one) of a system.
We should note that the literature has long hinted at this pessimism, preferring to evaluate their schemes at relatively
high densities. However, we could not find an explicit statement of this pessimism, rather a general acceptance of the
state of affairs. We do not mean to disparage existing pieces of work; many of the localization schemes we have
referred to have very elegant and intricate algorithms, and we could not (despite trying) improve any of them in
any significant way. Rather, we think our observation is fundamental of r-only localization and optimizations and
improved refinement schemes cannot change this. Euclidean schemes require 3-connectedness, which occurs only
at high densities. Topological schemes on the other hand rely on the congruency between topological and physical
distance for low error and this sets in only at high densities.
There are three counter-arguments to our position, which we now discuss.
Argument 1: Why not just deploy more anchors? This is a reasonable position, and when it is feasible to make 30%
or more of the nodes as anchors, then with high likelihood r-only localization will begin to work well. We have seen
this in our simulations. However, this goes against the basic rationale for ad-hoc localization systems (Section 1).
Argument 2: What’s wrong with deploying more resources: we need them anyway for increased network lifetimes?
On the surface, this is an attractive argument. From an architectural perspective, however, we believe that requiring
deployments to be redundant so that ad-hoc localization schemes work is undesirable.
But, the question of higher density deployments is a bit more subtle than just that. Note that in our simulations
(and this is consistent with current practice in the literature), when we say “density” we mean the number of nodes
within ranging radius. In drawing our conclusions of the previous section, we implicitly equated ranging radius and
communication radius. While this assumption is true of RF-based ranging, the ranging errors for this technique are
quite high (10% even with significant calibration [21]). The more accurate ultrasound ranging devices have a reported
range of 3-6m [20, 8], while the current radios on the Mica motes already have a range of tens of meters and the
new Chipcon radios are reported to have a greater range. This means that r-only schemes might, in theory, require
impossibly high density of nodes within the communication radius. This is not a fundamental limitation, however, and
there are many possible approaches to circumvent this: controlling the transmit power on the radios (which might be
necessary if node deployments need to be fairly dense anyway for application reasons), using UWB localization (the
current generation UWB localization methods are reputed to exhibit high error), or finding low-power ranging devices
which can range at higher distances (this may be difficult: a fairly good off-the-shelf sonar, the Polaroid 6500, which
can range up to 35 ft. requires a 2A current draw when taking measurements).
Argument 3: Random anchor placement makes no sense, so we should perhaps consider more constrained place-
ments? While more constrained placements, like placement of anchors along the periphery of the network can im-
prove localization error, we do not believe they can increase localization extent at low densities. For the geometric
schemes, recall that the 3-connectedness argument does not require a certain number or placement of anchors (as long
as the fraction of anchors is low); rather it is a more fundamental property of node density. Topological schemes rely
on the congruency of topology and physical distance to get reasonable error, and again this cannot be improved by
preferentially placing more anchors. We have also verified this using our simulations, and the graphs are omitted for
brevity.
Having painted this rather pessimistic picture, we now ask: Is there an alternative solution?
4 Using Bearing Information for Higher Localization Extent
We have argued that r-only localization schemes exhibit fairly poor performance. In this section, we discuss whether
ad-hoc localization schemes could exhibit improved localization performance (low mean error and high localization
extent even at low node densities) if each node had the ability to estimate bearing to its neighboring nodes, in addition
to range. To our knowledge, this question has not been considered in the literature before.
In this section, we first describe a distributed algorithm for range-and-bearing based ad-hoc localization—we use
the term r
θ localization to describe this class of ad-hoc localization systems. We end this section with a discussion
of the performance and practical feasibility of r
θ localization.
10
Iterative Least-Mean-Squares Refinement using Range and Bearing The key intuition for the efficacy of r
θ
localization can be seen from a simple geometric argument. Suppose that a node n
i
has computed its position x
i
(note
that, per our notation in Section 3.1 x
i
is a vector that represents the node’s position). Now, if n
i
is able to measure its
range (r
i j
) and its bearing (θ
i j
) to another nearby node n
j
, then the coordinates of n
j
would given by
x
j x
i r
i j
cos
θ
i j r
i j
sin
θ
i j (12)
Thus, a node needs only one neighbor that has localized itself to estimate its own position. Contrast this with the 3-
connected requirement of r-only schemes that use geometric methods for estimating distances to anchors (Section 3.2).
Furthermore, it follows the discussion in Section 3.2 that one would expect r
θ localization schemes to be effective
even at densities of 5-6 nodes in the ranging neighborhood. We use simulation to verify if this is true.
But, in the presence of range and bearing measurement errors, this simple approach could propagate these errors
throughout the network. How, then, can we accurately estimate position in a distributed fashion?
The approach we use (Least-Mean-Squares refinement using Range and Bearing, or LMSRB) is very similar to
the iterative refinement scheme described for r-only localization in Section 3.1. Each r
θ measurement leads to the
equation,
x
i x
j Δx
i j 0
(13)
here Δx
i j
is given by the second term on the right hand side of Equation 12.
As with LMSR, LMSRB finds the locations which “best” fit the set of all readings. The basic idea is to cast the po-
sition estimation problem within an optimization framework, and then observe that minimizing the objective function
is amenable to re-formulation in a manner that would permit distributed, iterative, computation (see Appendix A). The
key difference here is that, because of the particular form of the objective function, there is one and only one solution.
This means that, unlike LMSR, LMSRB does not require a “good” initial guess.
LMSRB also weights r
θ measurements based on their level of certainty. For this we assume the r
θ measure-
ments to have zero mean Gaussian error. However, we expect that our algorithms will not be significantly perturbed by
any slight deviation from the Gaussian, since the Gaussian assumption merely provides a weighting mechanism. Let
the standard deviations in measurements, r
i j
and θ
i j
be σ
r
i j
and σ
θ
i j
respectively. Note, that each measurement could
potentially have a different standard deviation. This allows for the use of sensors with different characteristics and
variation of standard deviation with range. Assuming relatively small errors (up to about 10 cones in bearing errors)
and ignoring third and higher order moments, Δx
i j
can be approximated by a Gaussian random vector with covariance
matrix:
ΔM
i j σ
2
r
i j
cos
2
θ
i j σ
2
θ
i j
r
2
i j
sin
2
θ
i j cos
θ
i j sin
θ
i j σ
2
r
i j
r
2
i j
σ
2
θ
i j
cos
θ
i j sin
θ
i j σ
2
r r
2
i j
σ
2
θ
i j
σ
2
r
i j
sin
2
θ
i j σ
2
θ
i j
r
2
i j
cos
2
θ
i j (14)
One can use the covariance matrix (see Appendix A) as a weighting function to formulate the localization problem as
a weighted least-squares optimization.
It turns out that the conditions for local minima in this formulation can be written as:
x
i ∑
n
j A
i
ΔM
1
i j
x
j Δx
i j ΔM
1
ji
x
j Δx
ji ∑
n
j A
i
ΔM
1
i j ΔM
1
ji
1
i
j n
i N
L
n
j N (15)
In this equation, x
i
depends only on the measured ranges and angles from its neighbors. Solving these equations using
the Gauss Seidel iterative technique corresponds to solving the equations from an initial guess successively.
Thus, LMSRB would simply involve each node broadcasting its current position estimate locally to its neighbors
(together with its own estimates of its distances and bearings to them). Each node would then use Equation 15 to
update its current estimate and repeat the process until the estimate converges.
Results We evaluated the r
θ scheme for different node densities, three different anchor densities (5, 10 and 20%)
and three different error values (0.1%, 1% and 5%). We used the same error model for ranging errors as used in
evaluating r-only schemes. We evaluated the r
θ scheme for three different kinds of bearing error: low (a 2 cone),
moderate (a 4 cone), and high (an 8 cone). The low values were motivated by laser rangefinders (which are known
to provide millimeter level accuracies at 10m, within a 2 cone). The higher values were motivated by a sonar ranging
device or a less expensive laser device for bearing.
11
5 6 7 8 9 10 11 12 13 14 15
80
82
84
86
88
90
92
94
96
98
100
Node density
Localization extent (%)
5%
10%
20%
Figure 9: Localization Extent
5 6 7 8 9 10 11 12 13 14 15
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Node density
Localization error (% of R)
5%
10%
20%
Figure 10: Localization Error
5 6 7 8 9 10 11 12 13 14 15
0
0.5
1
1.5
2
2.5
Localization error (% of R)
0.1%,0.5deg
1%,1.0deg
5%,2.5deg
Node density
Figure 11: Localization Error for Higher
Bearing Error
We use the same simulation framework as for the r-only localization schemes (Section 3.3). The only new addition
is the error model for bearing which we assume to be a zero mean Gaussian. A bearing measurement having a Gaussian
error with standard deviation σ
θ
, results in a cone of 4σ
θ
. LMSRB scheme usually stabilizes within 20-30 iterations,
depending on anchor ratios; however, we ran the experiments to 100 iterations to ensure asymptotic behavior. All
the results are averaged over 50 simulations and the 99% confidence interval for localization extent was 0.5% and for
localization error 5cm.
As seen from Figure 9, even with an anchor fraction of 5% and node density of 5 the localization extent is more
than 90% and reaches 98% by node density 6 for moderate error conditions. This follows from (and validates) the
analytical discussion in Section 3.2. Furthermore, localization extent only improves with higher anchor fractions and
node densities. Figure 10 depicts the localization error as a function of density at moderate error levels (4 cone). Even
at an anchor fraction of 5% and node density of 5, mean localization error is less than 1% of ranging neighborhood
(10cm in this case). As expected, performance only improves with increase in node densities and higher anchor
fractions.
Figure 11 captures how errors in range and bearing effect the algorithm at 10% anchor fraction. An error of 2.5 translates to an average position error of
2R
3
sin2 5=30cm(3%),since the expected range of a node is
2R
3
and an angular
deviation of θ at this distance implies a physical deviation of
2R
3
sinθ. Morover, a σ
θ
of 2.5 implies an 8 cone.
Also, considering that error might accumulate over multiple hops, it actually extremely encouraging that at high error,
and a node density of 5, the mean error stays below 2.5% (25cm). The explanation for this is that the LMSRB is a
global minimization and takes into account all the measurements in the sensor network. We did not, however, find any
significant difference in localization extent with much higher anchor densities.
In conclusion, LMSRB provides a high localization extent and low localization errors even at low node densities
and low anchor fractions.
Discussion Clearly, then, LMSRB shows that it is possible to achieve very good localization performance even at
low densities (where, as before, by density we mean the number of nodes within ranging radius). Our LMSRB scheme
is a contribution to the literature; to our knowledge no other work has attempted to use range and bearing information
to obtain location (although a recent piece of work [13] attempts to use only bearing information to estimate position
and requires the 3-connectedness property to work).
The key question is: Is r
θ localization practical? That is, can accurate bearing estimation devices be built at the
form and energy factors that sensor nodes require? We do not know the answer to this question. Certainly, at larger
form-factors, laser range finders can also estimate bearing, but they are also power-hungry and expensive.
One possible way to estimate bearing cheaply would be to use a ring of charge-coupled devices (CCDs) that detects
light at certain frequencies. A node can emit a light pulse, and the CCD array on a receiving node can estimate bearing
to the transmitter using little energy. However, while this seems plausible to us, we have not investigated how such a
device might actually be engineered.
12
5 Range-Sector Based Localization Systems
Given that the feasibility of building accurate bearing estimation devices is an open question, we ask: if nodes could
estimate imprecise bearing to each other, how would the resulting ad-hoc localization scheme perform? More pre-
cisely, if we allowed nodes to have sector estimation devices (those that could place a neighbor within, say a 45 sector), would the performance of r-sector localization schemes be closer to r
θ scheme or r-only schemes?
We now describe an r-sector distributed ad-hoc localization scheme, and evaluate its performance.
Iterative Least-Mean-Square Refinement with Range and Sector LMSRB cannot be directly extended to perform
r-sector localization, for two reasons. First, sectoring devices are more likely to exhibit uniform, rather than Gaussian
(or “close” to Gaussian) errors. Second, the first order approximation that we used to derive the covariance matrix in
Equation 14 does not hold since the higher order terms will have a significant contribution beyond cones of 10 .
Instead, since the only relatively precise information available is the range information, our LMSRS (Iterative
Least-Mean-Squares refinement with Range and Sector) algorithm essentially uses the LMSR algorithm described
in Section 3.1. LMSR requires a good initial guess for the locations of the nodes, and for this we use the sector
information. A node gets an initial guess using Equation 13 and the center of the sector as an estimate for bearing to
the neighbor.
It is possible, with a little communication, to obtain a better initial guess; this is not crucial for LMSRS but can
speed up its convergence. Consider Figure 14 where A has an initial position estimate, and node B is trying to get a
position for itself using Equation 13. A and B announce to each other the sector that each thinks the other lies in (e.g.,
B might say “A lies in my 45 to 90 sector”). Let CAD represents the sector which A thinks B is in. Let EBF be the
sector which B thinks A is in. Suppose BF were the true bearing of A measured from B, then B would lie on the line
AF’. Similarly if BE were the true bearing of A measured from B, then B would lie on somewhere on the line AE’.
This means that using the sector information of B, B must lie in the sector F’AE’. The intersection of sectors CAD and
F’AE’ represents, a tighter bound on the region where B lies relative to A. Hence, now, B can initialize itself along the
angle bisector of F’AD (AB’) at a range
r
AB
σ 1
AB
r
BA
σ 1
BA
σ 1
AB
σ 1
BA
, as depicted in Figure 14. Here, r
AB
and r
BA
(and respectively
the σs) are ranges and standard deviations measured at A to B and vice versa. This approach provides a better initial
guess only if the orientations of A and B are not identical, which is highly likely even in loosely planned deployments.
Our simulation setup for LMSRS is identical to that for LMSRB. In general, LMSRS takes about 40-50 iterations
(recall that both LMSRB and LMSRS schemes are distributed schemes; when we say iterations we mean the number
of message exchanges required to converge). All our simulations actually run over 100 iterations and reflect asymptotic
results. Our results are averaged over 50 simulations to ensure that the 99% confidence intervals for localization extent
are less than 1% and for localization error are less than 5cm.
Results In our simulations, we found that that sensitivity of LMSRS to ranging error is relatively small at all anchor
fractions and node densities. Accordingly (given the limited space), we discuss the how LMSRS performs for three
different sector angles (30, 45 and 60) at three different anchor ratios (5%, 10% and 20%). In Figure 12 and Figure 13
we plot the variation of localization extent and error for these combinations. The most notable results are that at even
as large 30 sectors, 10% anchor ratio and a density of 5 nodes, more than 90% nodes are localized within 2mts and
the mean error is around 5.5% (55cm). The corresponding values at 20% anchor ratio are 95% and 3% (30cm). The
performance improves at higher node densities and is expected to increase at higher anchor ratios. While these are
extremely encouraging results, we examine the effect increased sector angle has on the performance. At 60 sectors
and 20% anchor ratio and node density of 5, the localization extent is around 90% and the localization error is about
6% (60cm). This implies that even with sectors as large as 60 one could acheive “satisfactory” localization if 20%
anchors were deployed.
Discussion These results imply that an ad-hoc localization system that uses range and sector estimates can approach
the performance offered by our r
θ scheme. More importantly, the localization extent and error are acceptable even
at low densities.
As with r
θ localization, the question is: Are sector estimation devices really practical, particularly for sensor
networks? We don’t know, given that we know of no devices explicitly built to estimate sectors (directional antennas
might, for example, be able to estimate sector bearing, but are not explicitly designed for that purpose). Of course,
there hasn’t been, till date, a pressing need to develop such a device. We believe our finding indicates that there is now
13
5 6 7 8 9 10 11 12 13 14 15
60
65
70
75
80
85
90
95
100
Node density
Localization extent (%)
5%,30
10%,30
20%,30
5%,45
10%,45
20%,45
5%,60
10%,60
20%,60
Figure 12: Localization extent for LMSRS
5 6 7 8 9 10 11 12 13 14 15
0
2
4
6
8
10
12
14
16
18
Node density
5%,30
10%,30
20%,30
5%,45
10%,45
20%,45
5%,60
10%,60
20%,60
Localization error (% of R)
Figure 13: Localization error for LMSRS
A
B
C
D
E
F
F’
E’
B’
Figure 14: Obtaining an initial guess
a need; r-sector localization will enable ad-hoc localization systems at reasonable deployment densities, and ad-hoc
localization seems to be an essential component of wireless sensor networks.
Certainly, it is more reasonable to expect that sector estimation devices will be easier to engineer, and perhaps
cheaper, than bearing estimation devices. As before, a simple CCD array might actually be quite feasible, and we
intend to examine the likelihood of manufacturing such a device.
6 Conclusions and Future Work
This paper makes the argument that the class of ad-hoc localization schemes considered in the literature so far fun-
damentally require high node densities in order to get acceptable performance. By contrast, new schemes that can
use range and bearing estimates, or even range and sector estimates, can give good performance even at densities that
ensure node connectivity.
Clearly much work is needed before ad-hoc localization systems that use ranging and sectoring devices become a
reality. Sector estimation devices have to be built, our algorithms have to be validated by implementation and actual
deployment in realistic environments. We are continuing to pursue these directions.
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A The Theory Behind our r
θ Localization Scheme
Continuing with the terminology defined in Section 4, our approach seeks to minimize the objective function,
J
∑
e
i j
E
x
i x
j Δx
i j T
ΔM 1
i j
x
i x
j Δx
i j (A-1)
The objective function in A-1 is a commonly used weighted least mean square formulation, based on the Mahalanobis norm.
The conditions for local minima are given by the set of N L simultaneous equations,
∂J
∂x
i
∑
n
j
A
i
ΔM 1
i j
x
i x
j Δx
i j ΔM 1
ji
x
i x
j Δx
ji 0 i j n
i N
L n
j N
(A-2)
This gives a set of N L simulataneous equations,
∑
n
j
A
i
ΔM 1
i j
ΔM 1
ji
x
i ∑
n
j
A
i
ΔM 1
i j
ΔM 1
ji
x
j ∑
n
j
A
i
ΔM 1
i j
Δx
i j ΔM 1
ji
Δx
ji i j n
i N
L n
j N
(A-3)
All these equations can be consicely written in a matrix form as BX
f and X can be found as B 1
f . However, B 1
cannot be
calculated in a distributed manner, moreover the size of B can be o N , making this computationally infeasible, since there can be
potentially thousands or even millions of nodes in the sensor field. The key to finding a distributed solution lies in re-writing these
equations as,
x
i
∑
n
j
A
i
ΔM 1
i j
x
j Δx
i j ΔM 1
ji
x
j Δx
ji ∑
n
j
A
i
ΔM 1
i j
ΔM 1
ji
1
i j n
i N
L n
j N
(A-4)
Since, the objective function is quadratic, there can only be one unique solution to this equation. As iterations proceed the
scheme converges to the globally unique solution. It can be shown that this scheme is guranteed to converge from any initial state.
A formal proof of convergence is given in Appendix B.
15
B Proof convergence for r
θ adhoc localization
Claim I : A necessary and sufficient condition for convergence of Gauss Seidal on BX
f is ρ M 1, where B
I
C. Here,
ρ C is the spectral norm of C.
Proof : Let X
θ
be the initial guess, and let X
t
be the estimate after the t
th
iteration. Then,
X
t I
i t
1
∑
i 1
C
i
f
C
t
X
0 (B-1)
As t ∞ X
t
I
C 1
f iff ρ C 1,
since I
∑
i ∞
i 1
C
i I
C 1
. Hence proved.
C is a symmetric matrix of the form,
0 H
21 H N
L
1
H
21
0
0
0 H N
L
N
L
1
H N
L
1 H N
L
N
L
1
0
.
Here 0 is 2 2 zero matrix and H
i j
are 2 2 matrices.
Claim II : ρ C 1 if ρ
∑
l n
l 1
H
il , ρ
∑
l n
l 1
H
li 1 and H
i j
are all positive definite or 0 n
i N
L and atleast one H
i j
is
positive definite.
Proof :
ρ C max CX
X
(B-2)
max X
T
C
T
CX
X
T
X
X
ℜ
2 N
L
(B-3)
Let K
C
T
C, and K
i j
∑
l n
l 1
H
il
H
l j
.
ρ
K
i j ρ
l n
∑
l 1
H
il
H
l j ρ
l n
∑
l 1
H
il
H
l j l
1
n
∑
l
1
1
l
2
n
∑
l
2
1
H
il
1
H
l
2
j (B-4)
since multiplication two positive definite or (0 matrix) matrices results in a positive definite matrix or (0 matrix) and addition of
positive definite matrices or (0 matrix) results in a positive definite matrix with a larger (or equal) spectral norm.
ρ
n
∑
l 1
H
il
H
l j n
∑
l
1
1
n
∑
l
2
1
H
il
1
H
l
2
j ρ
l n
∑
l 1
H
il
l n
∑
l 1
H
li ρ
i n
∑
i 1
H
il ρ
i n
∑
i 1
H
il 1 (B-5)
If atleast one H
i j
is positive definite then, atleast one ρ K
i j
1.
Let X
x
1
x N
L
, x
i ℜ
2
.
X
T
C
T
CX
X
T
KX
∑∑ x
T
i
K
i j
x
j
∑∑
x
T
i
K
i j
x
j
∑∑
x
T
i
x
T
j
K
i j
∑∑
x
T
i
x
T
j
∑∑ x
T
i
x
T
j
X
T
X
Hence,
X
T
C
T
CX
X
T
X
1, i.e. ρ C 1.
Claim III : H
i j
are all positive definite or 0.
Proof : Let L be the set of all anchors. If n
j L then H
i j
0, else
H
i j
∑
j
Ad j i
ΔM 1
i j
ΔM 1
i j
1
ΔM 1
i j
ΔM 1
ji
(B-6)
16
Now, ΔM
i j
are all covariance matrices for gaussian random variables, and hence are positive definite. Sum,inverse and product
of positive definite matrices are positive definite. Hence, H
i j
are all positive definite.
Claim IV : ρ
∑
l n
l 1
H
il , ρ
∑
l n
l 1
H
li 1.
Proof :
∑
l
H
il
∑
n
j
A
i
ΔM 1
i j
δM 1
ji
1
∑
j
n
j L
ΔM 1
i j
ΔM 1
ji
(B-7)
Let ∑
n
j L
ΔM 1
i j
ΔM 1
ji
a
i
and ∑
n
j
L
ΔM 1
i j
ΔM 1
ji
b
i
, then ∑
l n
l 1
H
il
a
i b
i
1
a
i
.
Since, ΔM
i j
are positive definite symmetric matrices, so are a
i
and b
i
. This means there exists a rotated coordinate system
(unitary transform) which diagonalizes a
i
without altering its eigen values. Let a
i
λ
1
0
0 λ
2 and b
i
α γ
γ β
. Here,
αβ γ
2
and λ
1
,λ
2
0.
a
i b
i
1
a
i
1
α
λ
1
β
λ
2
γ
2
β
λ
2
λ
1 γλ
2
γλ
1
α
λ
1
λ
2 (B-8)
The largest eigen value than is given by,
2λ
1
λ
2 αλ
2 βλ
1
2λ
1
λ
2 αλ
2 βλ
1
2
4λ
1
λ
2
α
λ
1
β
λ
2
γ
2 2 α
λ
1
β
λ
2
γ
2 .
It is then straight forward to that the eigen value is less than 1 iff 0 αβ
γ
2
, which is true since b
i
is positive definite.
The same argument can be used to prove that ρ ∑ H
li
1. Hence the claim is proved.
Claims I,II,III and IV when combined imply that, the algorithm is guaranteed to converge monotonically to a globally unique
solution.
Note that, if not even one H
i j
is non-zero, it means that, all nodes in the network are isolated i.e. have no neighbor and the
localization is impossible.
17

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Description

Krishna Kant Chintalapudi, Amit Dhariwal, Ramesh Govindan, Gaurav Sukhatme. "On the feasibilty of ad-hoc localization systems." Computer Science Technical Reports (Los Angeles, California, USA: University of Southern California. Department of Computer Science) no. 796 (2003).

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Creator Chintalapudi, Krishna Kant (author), Dhariwal, Amit (author), Govindan (), Ramesh (author), Sukhatme, Gaurav (author)

Core Title USC Computer Science Technical Reports, no. 796 (2003)

Alternative Title On the feasibilty of ad-hoc localization systems (title)

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Repository Name USC Viterbi School of Engineering Department of Computer Science

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Publisher Department of Computer Science,USC Viterbi School of Engineering, University of Southern California, 3650 McClintock Avenue, Los Angeles, California, 90089, USA (publisher)