2015 International Conference on Computing, Networking and Communications, Invited Position Papers
A Cost-Efficiency Method on Beacon Nodes
Placement for Wireless Localization
Zimu Yuan1,2 , Wei Li2 , Junda Zhu3 , Wei Zhao4
of Chinese Academy of Sciences, China
2 Institute of Computing Technology, Chinese Academy of Sciences, China
3 Department of Electrical and Computer Engineering, University of Macau, Macau
4 Department of Computer and Information Science, University of Macau, Macau
{yuanzimu,liwei}@ict.ac.cn, {jdzhu,weizhao}@umac.mo
1 University
Abstract—Beacon node placement is the prerequisite for wireless localization, and the quality of placement scheme significantly
affects localization accuracy. For a localization system, sufficient
coverage and optimal beacon node distribution are required to
provide satisfactory localization accuracy in a target area. Since
the deployment cost grows with the number of beacon nodes, it
is desirable to keep the number low. In this paper, we propose
a cost-effective method for beacon node placement in arbitrarily
shaped areas. The proposed method can guarantee satisfactory
localization accuracy with a reasonable number of beacon nodes.
It has polynomial time complexity and meets the beacon node
distribution requirement. We theoretically analyze the lower and
upper bounds of the number of beacon nodes, and use numerical
analysis to verify its efficiency. Finally, we compare our approach
to the existing placement methods in different scenarios. The
comparison shows that our method can use fewer beacon nodes
to obtain higher localization accuracy.
I. I NTRODUCTION
Localization is a critical enabler for today’s context-aware
applications. Researchers have devised various approaches
to improve the localization accuracy, e.g., adopting more
accurate signal measurement, using advanced techniques to
alleviate random measurement errors, inventing new models
to calculate target locations, etc. For these approaches, beacon
node placement is the prerequisite to performing localization.
With careful beacon node placement, localization accuracy
can be improved, sometimes significantly. Despite its practical
significance, beacon node placement has not been thoroughly
studied in the existing localization literature. Only some of
the existing works [2][4][6] on node placement considered the
problem in the context of localization.
Basically, two factors of beacon node placement have a significant impact on localization accuracy: the degree of signal
coverage for targets and the distribution of beacon nodes. Sufficient coverage and optimal beacon node distribution schemes
can guarantee satisfactory localization accuracy. Increasing
the number of beacon nodes can sometimes improve the
localization accuracy; however, the resulting deployment cost
may be impractically high. Therefore, a cost-effective method,
which can use fewer beacon nodes to achieve satisfactory
localization accuracy, is of great practical importance.
In this paper, we conduct a systematic research on beacon
node placement for localization. Specifically, our goal is to
design a cost-effective method for beacon node placement with
localization accuracy guarantee. By cost-effective, we mean
that the number of beacon nodes are kept minimal. Our major
contributions are summarized as follows:
• We analyze the effect on localization accuracy by the
coverage and distribution of beacon nodes.
978-1-4799-6959-3/15/$31.00 ©2015 IEEE
We present a general method of beacon node placement
that guarantees sufficient coverage and optimal distribution with fewer beacon nodes.
• We analyze the lower and upper bounds of the number
of beacon nodes for our proposed method.
• We perform several experiments for different scenarios
to compare our method with other placement patterns
under the same coverage requirements. The experimental
results show that our proposed method can attain higher
localization accuracy with fewer beacon nodes.
The rest of the paper is organized as follows. Section
II introduces the general requirements of the beacon node
placement, and formulates the problem. Section III presents
our beacon node placement method in arbitrary shaped areas.
Section IV derives the lower and upper bounds of the number
of beacon nodes. Section V presents the comparative experiment results on localization accuracy. Section VI concludes
the paper.
•
II. T HE P ROBLEM AND R ELATED W ORK
For a localization systems to adopt beacon nodes into
its infrastructure, the quality of beacon node placement has
significant impact on the localization accuracy. We consider
that beacon node placement involves two factors related to
localization accuracy: 1) the target’s signal coverage and 2)
the beacon node distribution.
For the issue of coverage, it indicates the density of beacon
nodes in a given area. Basically, denser coverage of beacon
nodes will bring redundant measurements, which can improve
localization accuracy by alleviate random measurement errors.
Fingerprint-based [14][23] and proximity-based [8][24] methods usually require at least one coverage for every point in
the given area. For higher localization accuracy, more beacon
nodes such as 2-coverage should be provided. For methods
utilizing triangulation and trilateration [7][13][15][17][22], 3coverage or more coverage should be satisfied considering
the solvability of the system of equations. For other methods
such as SDP [5], MDS [19][20] and hop-based methods
[11][12][16], few or even no beacon nodes are needed, namely,
0-coverage is required.
For the issue of distribution, specific distribution of beacon
nodes are also required for different localization methods.
Some localization methods [8][14][23][24] consider every
point within a given area to be equally important. Thus, the
beacon nodes should be placed uniformly. For methods such
as [7][13][15][17][22], the beacon node distribution with a
good value of Geometric Dilution of Precision (GDoP) [21]
will help improve the localization accuracy. For these methods,
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2015 International Conference on Computing, Networking and Communications, Invited Position Papers
the beacon nodes should be placed such that the GDoP values
are optimal.
From the above discussion, the beacon node placement
requirement of localization methods can be stated as:
req(A) = (k, d, r)
(1)
Here, A indicates a given area and r is the localization
radius of beacon nodes. k is the coverage requirement, whose
value indicates how many beacon nodes are required to cover
every point in A. d represents the distribution requirement,
whose type can be Uniform, GDoP Optimal, Random or other
value specified. We think that the definition in equation (1) is
valuable to designers of localization systems. In other words,
by specifying req(A) and finding a placement, the accuracy
for localization methods in given areas can be guaranteed.
III. B EACON N ODES P LACEMENT WITH O PTIMAL
D ISTRIBUTION
In this section, we present a cost-effective method of beacon
node placement satisfying the requirement req(A) = (k, d, r)
using as few beacon nodes as possible. This method works in
three steps: 1) The classical triangle lattice pattern is applied
to achieve 1-coverage placement; 2) Optimal distribution is
calculated for localization methods; 3) The k-coverage pattern
is searched over a set of sample points and fitted to an arbitrary
shaped area.
A. Obtaining 1-Coverage Placement
B. Calculating Optimal Distributions
Consider the placement requirement (k, d, r). After applying the triangle lattice pattern, additional beacon nodes (or
additional hexagonal lattices) need to be placed to meet the
k-coverage requirement if k > 1. These beacon nodes should
also be placed following the distribution requirement d at the
same time. However, the distribution requirement d differs
with different localization methods, deployment environments
and planning decisions. In other words, a different distribution
requirement d determines a different optimal beacon nodes
placement pattern, e.g., the k-optimal pattern represents the
k-coverage with optimal distribution. To evaluate a pattern,
we define an evaluation function f (d, P ) for the specific
distribution requirement d, where the set P consists of k
points. Here, a better pattern has a smaller value of f (d, P ),
and vice versa.
Next, we explain the design of function f (d, P ) for different
localization methods with different distribution requirements
d. For clarity, we take two common distribution requirements
as examples, which are d1 = Uniform and d2 = GDoP Optimal.
A hexagonal lattice hl is sampled for evaluating f (d, P ).
1) Uniform Placement: With the k-coverage requirement
satisfied, every point of the hexagonal lattice hl is within the
localization radius r of at least k placed beacon nodes. The
area of lattice hl can be viewed as being covered by some
overlapped k-coverage sub areas. Under the requirement of
d1 = Uniform, the k-coverage sub-areas should be guaranteed
to be as equal in size as possible. Let a(A) denote the size of
area A. Suppose that there are q such sub-areas, A1 , A2 , ..., Aq ,
for the set of selected points P = {oi , p1 , p2 , ..., pk−1 }. We
can define the function f (d, P ) as:
f (d, P ) =
q
(a(Ai ) − E(a(A1 ), a(A2 ), ..., a(Aq )))2
(2)
i=1
(a) Equilateral triangles
where E(a(A1 ), a(A2 ), ..., a(Aq )) = (a(A1 ) + a(A2 ) + ... +
a(Aq ))/q. For equation (2), approximately equally sized subareas suggests a good pattern, and f (d, P ) has a small value.
Otherwise, f (d, P ) has a large value for a bad pattern for
which the sub-areas differ in size.
2) GDoP Optimal Placement: With the GDoP requirement
d2 = GDoP Optimal, the function f (d, P ) should have a small
value for a good GDoP distribution [21] with the selected point
set P . We can define the function f (d, P ) as:
f (d, P ) =
GDoP (P )dP
(3)
(b) Regular hexagonal lattices
hl
Fig. 1.
The triangle lattice pattern [10]
Triangle lattice pattern has been widely used to cover an
area, which was originally proposed in [10]. Such a pattern
is tessellated by equilateral triangles as shown in figure 1(a).
By
√ setting the length of these equilateral triangle sides as
3r (with r being the localization radius), and placing beacon
nodes in the center of the blue triangles, 1-coverage can be
readily achieved. It can be seen from figure 1(b) that the
area is covered by circles with radius of r. Each circle has
a beacon node placed in its center. By applying the triangle
lattice pattern, any point p in the area is covered by at least
one circle, i.e., with the signal coverage of at least one beacon
node. Using Voronoi tessellation [3], we can divide the entire
area into regular hexagonal lattices. Each of these hexagonal
lattices encloses all points whose distance to its center is less
than or equal to the distance to the centers of other hexagonal
lattices.
where GDoP (P ) denotes the distribution value of GDoP. We
assign infinity to the points with undefined GDoP values.
Equation (3) performs the integral of GDoP (P ) over the
area of hl. A good GDoP distribution has a small value in
GDoP (P ), and thus, a small value in f (d, P ).
C. k-Covering An Arbitrarily Shaped Area
After applying the triangle lattice pattern, 1-coverage
can be achieved for the target area. Let the set HL =
{hl1 , hl2 , ..., hln } denote all the hexagonal lattices placed by
the triangle lattice pattern. Assume that n beacon nodes have
been placed at locations o1 , o2 , ..., on , where oi is the center
of the i-th hexagonal lattice.
To find the k-optimal pattern, we perform an exhaustive
search over sampled points. First, we randomly select a
hexagonal lattice hli centered at oi from HL, and sample a set
of m points in this lattice. To meet the k-coverage requirement,
each time we select k − 1 points, p1 , p2 , ..., pk−1 , out of
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2015 International Conference on Computing, Networking and Communications, Invited Position Papers
Fig. 2.
Fig. 3.
Node placement in an arbitrary shaped area
the set of m points, and translate a copy of the pattern HL
simultaneously with the movement of the point oi . As the k−1
copies of the point oi moves to each of these k − 1 selected
points, k − 1 copies of HL are applied to the target area.
After this process, every point of the target area is covered by
at least k hexagonal lattices. Then we place beacon nodes at
the center of new generated hexagonal lattices. The function
f (d, P ), P = {oi , p1 , p2 , ..., pk−1 }, can be used to evaluate
the pattern of beacon nodes placement. By traversing all k − 1
combinations of points out of the set of m points, we can find
the best pattern that minimizes the function f (d, P ), which
is the k-coverage pattern with optimal distribution (k-optimal
pattern). 1
After finding the k-coverage pattern with optimal distribution, we should fit it to a given area A. When carefully fitted,
fewer beacon nodes may be used to achieve the placement
requirement req(A) = (k, d, r). For a given area A that could
be arbitrarily shaped, the search method is adopted similar to
the above. We randomly select a hexagonal lattice hli centered
at the point oi , and sample a set of t points, p1 , p2 , ..., pt , in
hli . Then, randomly choose a point ps in the given area A,
and translate the entire pattern together with the movement
of each of these t points. By counting the number of beacon
nodes placed when each of these t points, p1 , p2 , ..., pt , is
moved onto the point ps , we can find the point that fits the
pattern to the area A with the fewest beacon nodes out of these
t points.
The time complexity of calculating f (d, P ) and recording
the placed location of beacon nodes are O(mk−1 ) and O(t),
respectively. It is an exhaustive search of the k-coverage
pattern with optimal distribution. For a special distribution
requirement d, better methods could be designed according
to the spatial property of f (d, P ). However, it is impossible
to design such methods for a general beacon node placement
method without knowledge of the spatial property. Although
as an exhaustive search, searching the k-coverage pattern with
optimal distribution is with an acceptable time complexity of
O(mk−1 ).
IV. A NALYSIS ON THE N UMBER OF B EACON N ODES
A. The Lower Bound
Let LB(nk , A) denote the minimal number of beacon nodes
used to achieve k-coverage in an arbitrarily shaped area A.
1 It
is a discrete method to sample a set of points for searching the k-optimal
pattern. The reason we do not use the continuous method here is that f (d, P )
can be defined as a discontinuous function. Also, it is easier to implement a
discrete method, which meets our goal of designing an effective and easily
realized placement method.
Node placement in the rectangle area
Consider an arbitrarily shaped area A with a maximal width
of w, as shown in figure 2. Assume there are total C columns
in A.
Theorem 1. Given an arbitrary shaped area A, the lower
bound
C
hi
√ (4)
LB(nk , A) = k
3r
i=1
Proof: The lower bound to achieve 1-coverage in area
A can be calculated by summing up the counts of hexagonal
lattices at least required in each column, that is, LB(n1 , A) =
C
√hi
i=1 3r . After applying the triangle lattice pattern, k ×
LB(n1 , A) is the minimal number of beacon nodes for all
possible distribution requirement d. Thus, we have the lower
bound of equation (4) for the k-optimal pattern.
B. The Upper Bound
Let U B(nk , A) denote the upper bound of beacon nodes
used to achieve k-coverage in an arbitrarily shaped area A.
Let the rectangle v0 v1 v2 v3 as shown in figure 3 denote the
rectangle area Ar . Apply the triangle lattice pattern to this
rectangle area. Focusing on the left and bottom corner of
v0 v1 v2 v3 , we can find that if the whole hexagonal lattices (or
the triangle lattice pattern) move right or upward slightly, extra
lattices (or extra beacon nodes) will need to be placed. So the
locations, where the hexagonal lattices are placed in figure 3,
achieve minimal waste in the left and bottom boundaries. Since
the length of v0 v1 and v0 v3 can be arbitrary, there are several
different cases that should be considered in the right and top
boundaries. The point v1 is between line l0 and l1 (H <
|v0 v1 |
v1 |
√
≤ H +0.5) or line l1 and l2 (H +0.5 < |v√03r
≤ H +1),
3r
H ∈ N, denoted as case l0 l1 and case l1 l2 , respectively. The
0 v3 |
≤ W + 1/3),
point v1 is between line l3 and l4 (W < |v3r
|v0 v3 |
line l4 and l5 (W + 1/3 < 3r ≤ W + 5/6) or line l5 and
0 v3 |
l6 (W + 5/6 < |v3r
≤ W + 1), W ∈ N, denoted as case
l3 l4 , case l4 l5 and case l5 l6 , respectively.
We have the Theorem 2 for U B(n1 , Ar ) and Theorem 3 for
U B(nk , Ar ). The proofs are omitted due to space limitations.
Theorem 2. For the rectangle area Ar , the upper bound
U B(n1 , Ar ) = (p1 + p2 )p3 + p4
where
548
p1 = H + 1, p2 = H + 1
p1 = H + 2, p2 = H + 1
Case l0 l1
Case l1 l2
(5)
(6)
2015 International Conference on Computing, Networking and Communications, Invited Position Papers
⎧
⎨p3 = W, p4 = p1
p3 = W, p4 = p1 + p2
⎩
p3 = W, p4 = 2p1 + p2
Case l3 l4
Case l4 l5
Case l5 l6
(7)
Theorem 3. For the rectangle area Ar , the upper bound
U B(nk , Ar ) = U B(n1 , Ar ) + (k − 1)[(p1 + p2 )p3 + p4 ] (8)
where
p1 = p1 + 1, p2 = p2 (or p1 = p1 , p2 = p2 + 1) Case l0 l1
p1 = p 1 , p 2 = p 2 + 1
Case l1 l2
(9)
⎧ ⎪
Case l3 l4
⎨p3 = p3 , p4 = p4 + p2
p3 = p3 , p4 = p4 + p1 (or p4 = p4 + p2 )
Case l4 l5
⎪
⎩ p = p , p = p + p
Case
l 5 l6
3
4
3
4
2
(10)
Based on Theorem 2 and Theorem 3, we have:
0 v3 |
≤ 1/3 for a rectangle
Corollary 1. If we have 0 < |v3r
area Ar , the upper bound
(H + 1) + (k − 1)(2H + 3) Case l0 l1
U B(0, 13 ] (nk , Ar ) =
(H + 2) + (k − 1)(2H + 4) Case l1 l2
(11)
0 v3 |
Corollary 2. If we have 0 < |v3r
≤ 1/3 for a rectangle
area Ar and another area As with U B(0, 13 ] (n1 , Ar ) =
U B(n1 , As ), the upper bound
U B(0, 13 ] (nk , Ar ) ≥ U B(nk , As )
(12)
Consider the worst case of using maximum extra lattices (or
extra beacon nodes) by the result of Corollary 2. Following
the parameter setting in Section IV-A, we have the following
theorem:
Theorem 4. Given an arbitrary shaped area A, the upper
bound
C
U B(0, 13 ] (nk , Ai )
(13)
U B(nk , A) =
i=1
V. E XPERIMENTS ON VARIOUS S CENARIOS
In this section, we compare our method with two other
node placement methods, RKC [9] and CERACC [1], which
also provide k-coverage for a given area. RKC randomly
selects beacon nodes for achieving k-coverage of a given
area. CERACC deterministically selects beacon nodes from
lenses of slices.2 For performance comparison in localization
accuracy, three scenarios are considered.
A. Scenario 1: Uniform Placement with 3-Coverage Satisfied
in Rectangle Area
For this scenario, LANDMARC [14] is used to perform
localization in a 500 m × 500 m rectangle area. The placement
requirement of LANDMARC is to uniformly place beacon
nodes with 3-coverage satisfied. In figure 4 and 7, the localization radius r ranges from 10m to 50m. For each value of
r, both the result of RKC and CERACC are averaged from
repeated construction of beacon nodes placement of 20 times.
The nodes placement pattern, 3-optimal pattern, constructed by
2 Both the RKC and CERACC method take a set of candidate nodes as input.
Beacon nodes are selected from these candidate nodes. In the experiment, we
r
as candidate nodes set
sample a set of points with the interval distance of 10
for the RKC and CERACC method.
our method has better localization accuracy with less beacon
nodes used for the placement requirement. In figure 4, the
localization result of 3-optimal pattern has on average 19.2%
and 51.7% lower localization error than the result of beacon
nodes placement pattern by CERACC and RKC, respectively.
In figure 7, the number of beacon nodes placed by 3-optimal
pattern is on average 60.2% and 33.9% lower than the number
of nodes placed by CERACC and RKC, respectively.
B. Scenario 2: GDoP Optimal Placement with 3-Coverage
Satisfied in Rectangle Area
For scenario 2, the trilateration method [15] is used to
perform localization within a 500 m × 500 m rectangle area.
The placement requirement of trilateration is to 3-cover the
area with better GDoP distribution of beacon nodes. For
this evaluation, the localization radius r changes from 10m
to 50m. For each value of r, both the result of RKC and
CERACC are averaged from 20 times construction of beacon
nodes placement. The 3-optimal pattern has better localization
accuracy with fewer beacon nodes used. In figure 5, the
localization result of trilateration by 3-optimal pattern has on
average 12.4% and 30.5% lower localization error than the
result of the pattern by CERACC and RKC, respectively. The
comparison result of the number of beacon nodes used is the
same with the result in figure 7.
C. Scenario 3: Uniform Placement with 3-Coverage Satisfied
in MetroFi
In this scenario, the MetroFi dataset [18] is used. MetroFi
involves 72 access points, and samples over more than
200, 000 locations with the received signal strength in a citywide area. These 200, 000 locations are taken as candidate
locations to be placed with beacon nodes in the evaluation.
The LANDMARC method is applied to locate these 72
access points. The placement requirement is to uniformly
place beacon nodes with 3-coverage satisfied the same as the
requirement in scenario 1 3 . When applying the 3-optimal
pattern in the evaluation, the selected locations of beacon
nodes to be placed may not be the sample points in the dataset
of MetroFi. In this case, for each selected location unlocated
at the sample points, we replace it with the location of its
nearest point over these 200, 000 sample points in 3-optimal
pattern. Also, the same process is applied to the placement
pattern of the CERACC and RKC method in their beacon
nodes selection.
As can be seen from figure 6 and 8, the localization radius
r changes from 63m to 80m in the evaluation. The 3-optimal
pattern has on average 55.7% less localization error (figure
6) with 37.0% less beacon nodes placed (figure 8) than the
result of the RKC method. The placement pattern by the
CERACC method has average 8.0% less localization error
than the 3-optimal pattern but with more than twice beacon
nodes placed (2.28 times). The 3-optimal pattern has a slight
performance decrease when selected locations unlocated at the
sample points are replaced with the locations of their nearest
sample points, but needs much less beacon nodes to be placed
than the placement pattern by the CERACC method.
VI. C ONCLUSIONS AND D ISCUSSION
In this paper, we represent the problem of the beacon nodes
placement as a triple, req(A) = (k, d, r), for a given arbitrarily
3 Since the measurement of signal strength could be with large error, it
would be inappropriate to use trilateration method that performs badly with
the measurement of large error. So we do not consider trilateration method
and its localization requirement in scenario 2 here.
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2015 International Conference on Computing, Networking and Communications, Invited Position Papers
15
10
60
4
2
5
20
30
40
Localization Radii (m)
The comparison of errors in scenario 1
Fig. 5.
20
30
40
Localization Radii (m)
Fig. 6.
65
70
75
Localization Radii (m)
80
The comparison of errors in scenario 3
x 10
3íOptimal
CERACC
RKC
10
6000
4000
2000
Fig. 7.
0
50
The comparison of errors in scenario 2
3íOptimal
CERACC
RKC
8000
0
10
40
4
10000
Num of Nodes
0
10
50
3íOptimal
CERACC
RKC
20
Num of Nodes
0
10
Fig. 4.
80
3íOptimal
CERACC
RKC
6
Errors (m)
20
Errors (m)
8
3íOptimal
CERACC
RKC
Errors (m)
25
8
6
4
2
20
30
40
Localization Radii (m)
0
50
The comparison of beacon nodes placed in scenario 1 and 2
shaped area A. For this triple, we design a k-optimal pattern
that can be easily implemented. In the theoretical analysis, we
derive the lower and upper bounds of the number of beacon
nodes, as required by the designed pattern. In the experiment,
we compare our designed pattern with two other placement
methods that can also provide k-coverage of an area. Both
the results in theoretical analysis and experiment show the
efficiency of our placement method.
ACKNOWLEDGMENT
This work is supported in part by the Macau Science
and Technology Development Fund, under grant number
FDCT/023/2013/A1, the Hi-Tech Research and Development
(863) Program of China (Grant No. 2013AA01A212 and
2013AA01A209), International Science and Technology Cooperation Program of China (Grant No. 2013DFA10690),
and the introduction of innovative R&D team program of
Guangdong Province (NO.201001D0104726115).
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