www.ietdl.org Published in IET Communications Received on 5th December 2007 Revised on 10th March 2008 doi: 10.1049/iet-com.2007.0603
ISSN 1751-8628
Anchor-free localisation algorithm and performance analysis in wireless sensor networks K. Yu Y.J. Guo Wireless Technologies Laboratory, CSIRO ICT Centre, Marsfield, NSW 2122, Australia E-mail:
[email protected]
Abstract: A hybrid anchor-free localisation scheme for multihop wireless sensor networks is presented. First, a relatively dense group of nodes is selected as a base, which are localised by using the multidimensional scaling method. Secondly, the robust quads (RQ) method is employed to localise other nodes, following which the robust triangle and radio range (RTRR) approach is used to perform the localisation task. The RQ and the RTRR methods are used alternately until no more nodes can be localised by the two approaches. Simulation results demonstrate that the proposed hybrid localisation algorithm performs well in terms of both accuracy and the success rate of localisation. To evaluate the accuracy of anchor-free localisation algorithms, the authors derive two different accuracy measures: the Cramer – Rao lower bound (CRLB) to benchmark the coordinate estimation errors and the approximate lower bound to benchmark the distance errors. Simulation results demonstrate that both the CRLB and the distance error lower bound provide references for the accuracy of the location algorithms.
1
Introduction
Recently, a range of algorithms for localisation in wireless sensor networks (WSNs) have been reported in the literature. Depending on the applications, the algorithms can be either centralised [1, 2] or distributed [3 – 5]. In the centralised scheme, the algorithm is executed in a processing centre or a location server. This centralised scheme requires infrastructure that challenges the ad hoc nature of the network, and the involved long range communications could be time-consuming and energy inefficient. On the other hand, distributed localisation avoids the problems of the centralised scheme by sharing the computational burden among the nodes so that the communications, especially the long range ones, are greatly reduced. In some circumstances, a certain number of nodes are equipped with a global positioning system (GPS) receiver and so their locations are known. Those nodes are often referred to as anchors and their locations are used to determine the positions of other ordinary sensor nodes, which do not have IET Commun., 2009, Vol. 3, Iss. 4, pp. 549– 560 doi: 10.1049/iet-com.2007.0603
GPS receivers. Either distance or connectivity information can be used to locate the ordinary nodes [6–16]. In other circumstances, there is no anchor in the network and hence, no absolute location information is available at any node. To localise sensor nodes in such anchor-free sensor networks, a number of algorithms have been proposed in the literature. In [17], an incremental algorithm, the assumption-based coordinates (ABC) algorithm, is proposed. ABC first selects three in-range nodes and assigns them coordinates to satisfy the inter-node distances, and then it incrementally calculates the coordinates of the nodes using the distances to the three nodes with known locations. This simple incremental scheme would result in error propagation. In [18, 19], every node establishes its local coordinate system by setting itself as the origin. Two other nodes are randomly chosen under the condition that the three nodes do not lie on the same line and can communicate with each other. Then, any other node can be localised if the distance to each of the three nodes can be estimated. After choosing one local coordinate system as the reference, the other local coordinate systems can be adjusted by coordinate transformation. This transformation process 549
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www.ietdl.org would also result in error propagation. In [20], the multidimensional scaling (MDS) technique was employed. This method employs a complete distance matrix and the distance between a pair of non-neighbouring nodes is approximated by the shortest path distance. In [21], five nodes are first selected to establish a global coordinate system. Then, a mass-spring model is exploited and optimisation is performed. Each node position estimate is updated iteratively by using the refined position information of its neighbouring nodes. In [22], a group of four nodes is defined as a robust quad (RQ) if all triangles between these nodes form robust triangles. Whether a triangle is robust or not is determined by a parameter of robustness. The position information of three known nodes in a RQ is used to localise the fourth node in that quad. In [23], a number of ranging-based localisation approaches are evaluated and analysed under a number of ranging models. In this paper, we investigate GPS/anchor-free node localisation in WSNs. It is assumed that distance measurements are already made hop-by-hop between neighbouring sensor nodes. There are a number of technologies and techniques for performing the distance measurement. The simplest way is to measure the received signal strength and then apply a path loss model, such as the log path loss model [24] or the Walfisch–Ikegami model [25], to calculate the distance. The distance can also be determined based on the time-of-arrival measurements and the measurements of round-trip time-of-flight of a radio signal [26, 27]. In particular, with the ultra wideband technology, sub-metre or decimetre accuracy of measurements can be produced [28–31]. When both a radio signal and an ultra sound signal are employed, an extremely accurate time-offlight estimate and hence distance estimate can be obtained [32]. Using the distance measurements, a hybrid localisation scheme is proposed by combining the MDS and the RQ methods [20, 22], and making use of the robust triangle and radio range (RTRR) technique as well. The advantage of the MDS method is that each node can be localised if the one- or multiple-hop distance between the node and any other node can be estimated. However, the accuracy of the MDS method may not be satisfactory because of the use of multiple-hop distances, which usually have large errors, especially when the distance is over many hops. On the other hand, the RQ method can achieve good accuracy by decreasing the probability of flipping, but the number of nodes that cannot be localised because of the lack of RQ may be large. The essence of the hybrid algorithm is to exploit the advantages of both the MDS and the RQ methods, and to compensate for their drawbacks. More specifically, the purpose is to localise more nodes than the RQ method, and to achieve higher accuracy than the MDS method and even the RQ method. It works as follows: (1) first, a group of nodes are localised by using the MDS approach to form a base. (2) Then, the other nodes are localised starting from the base and by using the RQ and the RTRR approaches 550 & The Institution of Engineering and Technology 2009
alternately until no more nodes can be localised by the two methods. When a location estimate is obtained, residual checking is applied to ensure the reliability of the estimate and, thus, to reduce the effect of abnormal errors and error propagation. In the presence of multiple location estimates of the same node because of the node being in multiple RQ, averaging is used to produce an improved location estimate. Thus, compared with the existing methods, the proposed algorithm aims to achieve better localisation accuracy. An important issue in anchorfree localisation is the accuracy evaluation of the algorithms and the fundamental performance limits. We derive the Cramer – Rao lower bound (CRLB) which serves as the position coordinate estimation accuracy limit. Also, an approximate lower bound is derived to measure the distance error, which is defined as the difference between the true distance and that of two estimated locations of a pair of nodes. To evaluate fairly the accuracy of the coordinate estimates, we propose an efficient approach to transform the location estimates to minimise the difference between the assigned coordinates and the estimated ones. To our knowledge, this is the first time the accuracy of the anchor-free location algorithm is evaluated against the CRLB and the approximate lower bound. The remainder of the paper is organised as follows: Section 2 provides a detailed description of the hybrid node localisation method; Section 3 derives the CRLB when the distance measurement error is Gaussian distributed; Section 4 presents a method to transform the location estimates for the determination of the coordinate estimation accuracy; Section 5 derives an approximate lower bound to measure the distance errors; Section 6 shows some simulation results; and Section 7 concludes the paper.
2
Hybrid node localisation
It is assumed that distance measurements between each pair of nodes that are within the radio range are available, that is d^i, j ¼ di, j þ eij ,
1 i, j N ,
i=j
(1)
where there are N nodes in the network, eij is the distance estimation error, and di, j
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ (xi xj )2 þ (yi yj )2
(2)
where (xi , yi) and (xj , yj) are the coordinates of nodes i and j, respectively. To begin with, we need to select a group of nodes in the network that can communicate with each other and the number of the nodes in the group should be as large as possible. The reason, we only choose nodes within the radio range of each other is that the MDS algorithm performs well when using one-hop distance estimates. IET Commun., 2009, Vol. 3, Iss. 4, pp. 549– 560 doi: 10.1049/iet-com.2007.0603
www.ietdl.org On the other hand, the MDS algorithm tends to produce large errors when a few multihop/shortest path distance estimates are used. Then, we choose three nodes in the group to set up a local coordinate system. Without loss of generality, the three nodes are numbered 1, 2 and 3, respectively. It is required that the three chosen nodes do not lie in a straight line. One of the three (say node 1) is set at the origin (x1 , y1 ) ¼ (0, 0) and the other (say node 2) is on the positive x-axis (x2 , y2) ¼ (d12 , 0), where d12 is the distance between nodes 1 and 2. The third node (say node 3) has a positive y-axis coordinate, that is 2 2 x22 þ d13 d23 , 2x2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 x23 y3 ¼ d13
x3 ¼
Once the three nodes are chosen and their coordinates are assigned, the coordinates of other nodes are uniquely established. Note that since the distance estimates are available instead of the true values, we can only obtain the estimates of the coordinates x2 , x3 and y3 . We apply the MDS algorithm to localise the group of n selected nodes whose position matrix is defined as [33] P¼
x1 y1
x2 y2
. . . xn . . . yn
T
Since the position estimates P^ are produced purely based on distance measurements, the resulting coordinate system can be significantly different from the established one. Hence, if the coordinate system is already established such as defined by nodes 1, 2 and 3, coordinate transformation is required. In the case that there is no coordinate system developed, the generated coordinate system from the MDS may be adopted. In this case, the origin of the coordinate system is not set at the location of a node in general. The advantage of the adopted coordinate system is that it avoids the errors caused by coordinate transformation. The position estimation accuracy may be enhanced by using optimisation techniques [34– 38]. This requires a node in the group to have the computational power to run an iterative minimisation algorithm. After the n selected nodes are localised, we make use of the RQ approach to localise the other nodes in the network. An unknown node can be localised if it and three other nodes can form a RQ [22]. A quad is robust if all the four triangles formed by the nodes are robust. The RQ method defines a parameter of robustness g which may be set to 3s, where s is the standard deviation (STD) of the distance error. When the smallest angle us and the shortest edge ds of a triangle satisfy
(3) ds sin2 us . g
and the MDS algorithm can be described as follows: 1. Construct the squared-distance matrix D whose elements are [D]i, j ¼ d^i,2 j
5. The positions of the n nodes are estimated as the first two pffiffiffiffiffi column components of the matrix V ¼ U 0 L0 , that is, P^ ¼ V (1 : n, 1 : 2).
(6)
the triangle is defined to be robust. Fig. 1 shows an example of a RQ , whereas Fig. 2 shows an example of a non-RQ. When the positions of the three nodes of a RQ are known, the fourth node can be robustly localised.
(4)
where d^i, j is the estimate of distance di,j . 2. Compute the matrix B ¼ 2(1/2) CDC, where D is the squared-distance matrix whose elements are defined by (4) and 1 C ¼ I eeT n
(5)
where I is the identity matrix of dimensions of n n and e is a column vector of length n, whose elements are all ones. 3. Decompose B as B ¼ U LU T , where L ¼ diag{l1 , l2 , . . . , ln } is a diagonal matrix of eigenvalues of B, and U is the matrix whose column vectors are the corresponding eigenvectors. 4. Normalise the eigenvectors to unit eigenvectors, sort the eigenvalues in a non-increasing order to form L0 , and rearrange the unit vector matrix accordingly, resulting in U 0 . IET Commun., 2009, Vol. 3, Iss. 4, pp. 549– 560 doi: 10.1049/iet-com.2007.0603
Figure 1 Example of a RQ 551
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www.ietdl.org Apparently, choosing two out of the three known nodes results in three different combinations and, hence, three different solutions would be produced. Then, the position estimate can be set to the average of the solutions. In the event that a node is among more than one RQ , the position estimate may be further improved by averaging all the available results. Note that to reduce the effect of the abnormal errors, before averaging, we check each position estimate by comparing the estimated distance (based on the position estimate) with the original distance measurement. If the distance difference is larger than a threshold, we drop the corresponding position estimate. This RQ-based localisation is performed successively throughout the network until all connected RQs are localised.
Figure 2 Example of a non-RQ Assume that the known coordinates of the three nodes are (^xi , y^ i ), (^xj , y^ j ) and (^xk , y^k ), respectively. For notational convenience, we will drop the hats of the coordinate estimates. The simplest way to determine the position of the fourth node, (x‘ , y‘ ), is to use two known nodes (say i and j) to produce two solutions of the unknown position, and then to use the third known node (say k) to get rid of one solution. More specifically, defining xij ¼ xi xj ,
yij ¼ yi yj
g ¼ 0:5(di‘2 dj‘2 þ x2j þ yj2 (x2i þ yi2 ))
(7)
the two solutions are given by 8 (1) y ¼ y‘(2) ¼ g=yji , > > < ‘ x(1) ‘ ¼ xi þ dx , > > : (2) x‘ ¼ xi dx 8 > x(1) ¼ x(2) > ‘ ¼ g=xji , > < ‘ y‘(1) ¼ yi þ dy , > > > : y(2) ¼ y d i y ‘
xi ¼ xj , yi = yj qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx ¼ di‘2 (y‘(1) yi )2
(8)
xi = xj , yi ¼ yj qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 dy ¼ di‘2 (x(1) ‘ xi )
(9)
In the event that there are nodes whose locations cannot be determined by the RQ method because of the absence of RQ , we propose the RTRR approach to localise the remainder of the nodes, which can be described using Fig. 3. Node a is the neighbour of both nodes b and c which have been localised, but not necessarily in connection to each other. Node d has been localised and it is the neighbour of node c, but not the neighbour of a. In the case that the triangle abc is robust, we can obtain two solutions for the position estimate of node a, that is, the two locations indicated by a0 and a00 . Node e is the neighbour of a and a0 , whereas node d is the neighbour of a00 , but not the neighbour of a. We can eliminate the solution a00 since a0 is not the neighbour of node d, but the neighbour of e, whereas a00 is the neighbour of d, but not the neighbour of e. Thus, we can make use of the neighbours of nodes a, namely b and c, which have been already localised to help localise node a. More similar information would result in a more accurate judgement. Also, multiple robust triangles may produce multiple position estimates that can be further processed such as by
or 8 (1) > y‘ ¼ b=a þ c, > > > > > a ¼ 1 þ (yji =xji )2 , > > > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > < c ¼ (b=a)2 c1 =a, > > y‘(1) ¼ b=a c > > > > (1) > > x(1) > ‘ ¼ ( g yji y‘ )=xji > > (2) : x‘ ¼ ( g yji y‘(2) )=xji
xi = xj ,
yi = yj
b ¼ yi þ ( g=xji xi )yji =xji c1 ¼ yi2 di‘2 þ ( g=xji xi )2
(10) 552 & The Institution of Engineering and Technology 2009
Figure 3 Example of localising a node based on the RTRR approach IET Commun., 2009, Vol. 3, Iss. 4, pp. 549– 560 doi: 10.1049/iet-com.2007.0603
www.ietdl.org and the measurement vector as r^ ¼ [d^1,2 , . . . , d^1,N , d^2,3 , . . . , d^2,N , . . . , d^N 1,N ]T [ RNr 1 (12)
Figure 4 Block diagram of the hybrid localisation scheme averaging to obtain more accurate position information. The RTRR method is applied to each unlocalised node, and then the RQ approach is applied if there are still nodes to be localised. The RQ and the RTRR algorithms are performed alternately until no more nodes can be localised by the two methods. Fig. 4 shows the block diagram of the proposed hybrid localisation scheme. It is non-trivial to point out that once a node is localised, its position estimate can be refined by making use of the position information and distance measurements of all its localised neighbours, and by running a minimisation algorithm, provided that the node has the computational power to do so. It is also worth mentioning that for applications, which require the knowledge of global positions of the nodes, at least three nodes must be equipped with GPS receivers and they do not lie in a straight line. With the global position information of these nodes, one can readily obtain the global locations of other nodes that are not equipped with GPS receivers by performing coordinate transformation. Further, it is worth pointing out that one of the important issues in WSN is related to the resource constraints. Typically, sensor nodes are powered by batteries that have a limited power source. Also, sensor nodes have rather limited memory space. One may envision that there are two types of cheap nodes in the network: the very cheap node performing basic calculus and with a limited time of connection with the other surrounding nodes, and the node with more power and memory, which will be able to run the relatively complicated algorithms such as the MDS.
3
where d^i, j is the distance measurement between nodes i and j as given in (1) when they are within the radio range, and Nr N (N 1)=2. When the distance measurement error ei, j is a Gaussian random variable (RV) of zero mean and variance s2d^ , we can obtain the following conditional i, j
probability density function as N 1 Y N Y pffiffiffiffiffiffi s1 exp p(^r ju) ¼ ( 2p)Nr d^
9 8 > 1 X N (d^ d )2 > = < NX i, j i, j 2 > > 2sd^ ; : i¼1 j¼iþ1 i, j
The CRLB is widely used to evaluate the accuracy of any unbiased estimators and provides a performance benchmark [39]. In the case of positioning in cellular networks and anchor-based localisation in wireless sensor networks, the CRLB has already been derived such as in [37, 38, 40]. The standard CRLB is determined as follows. Define the parameter vector as
di, j R0
where R0 is the radio range. Let u^ be the estimate of u. Then, the standard CRLB for the ith parameter of the position estimation vector u^ , that is, u^ i , is defined as h i CRLB(u^ i ) ¼ F 1 (u^ )
u ¼ [x1 , x2 , . . . , xN , y1 , y2 , . . . , yN ] [ R IET Commun., 2009, Vol. 3, Iss. 4, pp. 549– 560 doi: 10.1049/iet-com.2007.0603
(14)
i,i
where F (u^ ) is the Fisher information matrix (FIM) whose elements are defined by h
" # 2 @ ln p(^ r j u ) F (u^ ) ¼ E , j,k @uj @uk i
j, k ¼ 1, 2, . . . , N
(15)
where uk is the kth element of u and the log likelihood function ln p(^r ju) is given by
ln p(^r ju) ¼
N 1 X i¼1
F ^ F (u) ¼ F xx yx
N (d^ d )2 X i, j i, j
2s2d^
j¼iþ1 di, j R0
(16)
i, j
(11)
F xy F yy
[ R2Nr 2Nr
(17)
where by defining
ui ¼ 2N 1
(13)
The FIM can be written in a block matrix form as
Cramer– Rao lower bound
T
i, j
j¼iþ1 di, j R0
i¼1
N X (d^i, j di, j )2 j¼1 j=i,di, j R0
2s2d^
(18)
i, j
553
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www.ietdl.org In the event that u^ is an unbiased estimate of u, H is given as
the components of the FIM can be derived as "
# @2 u i [F xx ]i,i ¼ E ¼ @xi2
N X (xi xj )2 j¼1 j=i,di, j R0
H¼
@uT 2 @r1 6 @x1 6 6 @r 6 2 6 @x ¼6 6 1 6 .. 6 . 6 4 @rNr @x1
s2d^ di,2 j i, j
"
[F xx ]i, j
# (xi xj )2 @2 u i ¼E ¼ 2 2 , j = i, di, j R0 @xi @xj s ^ di, j
# @2 u i [F xy ]i,i ¼ E ¼ @xi @yi
di, j
"
N X (xi xj )(yi yj )
s2d^ di,2 j i, j
j¼1 j=i,di, j R0
"
[F xy ]i, j
# (xi xj )(yi yj ) @2 ui ¼E , ¼ @xi @yj s2^ di,2 j
j=i,di, j R0
[F yy ]i, j ¼ E
2
.
...
3 @r1 @yN 7 7 @r2 7 7 ... @yN 7 7 [ RNr 2N 7 .. 7 .. . 7 . 7 @rNr 5 ... @yN ...
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2N u 1 X 1 t trace(C b ) [C b ]ii ¼ CRLBav ¼ 2N i¼1 2N
2
(yi yj ) @ ui ¼ 2 2 , j = i, di, j R0 @yi @yj s ^ di, j di, j
It can be seen that the standard CRLB defined in (14) does not exist for anchor-free localisation since the FIM is singular. The rank of the FIM can be up to 2N 2 3, instead of 2N. This is because of the fact that the node configuration remains the same after translation, rotation or reflection. To handle the complication of the singular FIM, we make use of the results in [41], that is, giving
[ RNr 1 , di, j R0
..
@r1 @y1 @r2 @y1 .. . @rNr @y1
In the simulation, we will evaluate the accuracy of the proposed algorithm in terms of the root-mean-square error (RMSE) of the location estimates. To benchmark the accuracy of the localisation algorithms, we use the averaged CRLB which is defined as
i, j
r ¼ [d1,2 , . . . , d1,N , d2,3 , . . . , d2,N , . . . , dN 1,N ]T
...
@r1 @xN @r2 @xN .. . @rNr @xN
(25)
j = i, di, j R0 " # N X (yi yj )2 @2 u i [F yy ]i,i ¼ E ¼ @yi2 s2d^ di,2 j j¼1 #
...
(19)
di, j
"
@r
(20)
the covariance matrix of u^ h i C ¼ E (u^ E[u^ ])(u^ E[u^ ])T
(21)
C Cb ¼ H F yH T
(22)
satisfies
4
(26)
Accuracy of location estimates
The CRLB is the lower bound that any unbiased estimators cannot surpass. The accuracy of a specific localisation algorithm is often evaluated by calculating the RMSE of the estimated location coordinates and compared with the CRLB. It is noted that when the coordinate system is established by using either the three selected nodes or the base nodes, the true positions of the nodes are given. Owing to estimation errors, the true positions of the second node and the third node as discussed in Section 2 may not be the same as the estimated ones. The difference between the true positions of the chosen three nodes or the base nodes and the estimated ones means that even when the relative positions of all other nodes can be calculated precisely, the RMSE could still be significant. Since we are dealing with anchor-free sensor networks, the effect of the coordinate system shift must be removed by using a coordinate transformation in order to access fairly the accuracy of the location estimates. Here, we propose an efficient method to perform the coordinate mapping as follows. Let
where
H¼
@(E[u^ ] u) @u
T
þ
@r T
@u
(23)
and F y is the Moore– Penrose pseudo-inverse of the FIM F , which is defined as F y ¼ (F T F )1 F T 554 & The Institution of Engineering and Technology 2009
(24)
pi ¼ [xi yi ]T , i ¼ 1, 2, . . . , N
(27)
be the pre-defined node locations and p^ i ¼ [^xi y^i ]T , i ¼ 1, 2, . . . , N
(28)
be the location estimate of pi . The centroid of the pre-defined IET Commun., 2009, Vol. 3, Iss. 4, pp. 549– 560 doi: 10.1049/iet-com.2007.0603
www.ietdl.org error-free locations of all the localised nodes is defined as N 1X c ¼ p N i¼1 i o
(29)
and the centroid of the estimated locations is given as ce ¼
N 1X p^ N i¼1 i
(30) t21
p^ 0i ¼ p^ i þ (co c e )
(31)
Then, the estimated locations are transformed according to
t12 t22
[ R22
(33)
is the transform matrix which is determined by T ¼ arg min T
( N X
) jjT p^ 0i pi jj2
(34)
i¼1
where jj jj is the vector norm. The problem is to determine the matrix T that minimises N X
jjT p^ 0i pi jj2 ¼
i¼1
N X
{(t11 x0i þ t12 yi0 xi )2
i¼1
(35)
þ (t21 x0i þ t22 yi0 yi )2 } Differentiating the right-hand side of (35) with respect to t11 , t12 , t21 and t22 , respectively, and then setting them to zero, we obtain 8 N N N X X X > 02 0 0 > > x þ t x y ¼ xi x0i t > 12 i i i < 11 i¼1
i¼1
i¼1
N N N > X X X > > > x0i yi0 þ t12 yi02 ¼ xi yi0 : t11 i¼1
i¼1
! N N N N X X X 1 X 02 0 0 0 0 ¼ x (x y ) (xi xi ) (xj yj ) D i¼1 i j¼1 j j i¼1 j¼1 ! N N N N X X X 1 X 0 02 0 0 0 ¼ (x y ) y (xi yi ) (yj yj ) D i¼1 i i j¼1 j i¼1 j¼1 ! N N N N X X X 1 X ¼ x02 (y0 y ) (x0i yi0 ) (x0j yj ) D i¼1 i j¼1 j j i¼1 j¼1
(36)
i¼1
8 N N N X X X > 02 0 0 > > t x þ t x y ¼ x0i yi > 21 22 i i i < i¼1
i¼1
N N N > X X X > 0 0 02 > > t x y þ t y ¼ yi0 yi : 21 22 i i i i¼1
i¼1
(38)
where
D¼
N X
x02 i
N X j¼1
yj02
N X
!2 (x0i yi0 )
(39)
i¼1
Therefore after the mapping, the accuracy of the coordinate estimates can be fairly evaluated with respect to the predefined node locations. The RMSE of the location estimates is employed to measure the accuracy of the algorithm, which is defined as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u 1 X t ((^x00 xi )2 þ (^yi00 yi )2 ) 2N i¼1 i
5 Distance error based accuracy measure Another performance measure that can be used to evaluate the accuracy of anchor-free localisation algorithms is the RMS of the difference between the true distance and the estimated one of each pair of nodes, that is vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N 1 N u1 X X t (d~ di, j )2 N1 i¼1 j¼iþ1 i, j
i¼1
IET Commun., 2009, Vol. 3, Iss. 4, pp. 549– 560 doi: 10.1049/iet-com.2007.0603
(41)
where di,j is the true distance between nodes i and j as defined by (1), and d~ i, j is the distance between the estimated locations of nodes i and j, which is defined as d~ i, j ¼
(37)
(40)
where x^ 00i and y^ i00 are the components of the vector p^ 00i that is defined in (32).
and
i¼1
!
(32)
where t11 t21
t22
N N N N X X X 1 X ¼ (xi x0i ) yj02 (xi yi0 ) (x0j yj0 ) D i¼1 j¼1 i¼1 j¼1
i¼1
p^ 00i ¼ T p^ 0i
t11
t12
The first step in the mapping is to translate the centroid of the estimated locations to the centroid of the pre-defined node locations. That is, the estimated locations are changed as
T¼
From (36) and (37), we can readily obtain
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (^xi x^ j )2 þ (^yi y^ j )2
(42)
Note that the distance measurement, d^i, j , is only available when the two nodes are within the radio range of each other, whereas the distance d~ i, j can be computed for any 555
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www.ietdl.org pair of nodes that are localised. It is obvious that for a given node configuration, the distance between a pair of nodes remains the same when any coordinate transformation is performed on the node locations. This is the advantage of using the distance error as an accuracy measure. As the CRLB described in Section 3 is used as a lower bound on the accuracy of the coordinate estimates, we are motivated to derive an approximate lower bound on the estimated distance errors. Expanding the estimated distance d~ i, j in a Taylor series at the true distance and retaining the first two terms, we obtain d~ i, j ’ di, j þ
@di, j @di, j @di, j @di, j Dxi þ Dxj þ Dyi þ Dyj @xi @xj @yi @yj (43)
where Dxi ¼ x^ i xi , Dyi ¼ y^ i yi , 1 i N @di, j @di, j 1 1 ¼ (x xj ), ¼ (y yj ), @xi @yi di, j i di, j i
(44)
i ¼ 1, . . . , N 1, j ¼ i þ 1, . . . , N Assuming that Dxi , Dxj , Dyi independent, we obtain
s2d~ i, j
@di, j ’ @xi
2
s2x^ i
@di, j þ @xj
!2
s2x^ j
and Dyj
are mutually
!2 @di, j 2 @di, j þ sy^i þ s2y^j @yi @yj
(45) Replacing the variances of the coordinate estimates in the above equation by their corresponding CRLBs defined in (22), we obtain the approximate lower bound on the estimated distance as
LBd~ i, j
@di, j ’ @xi
2
@di, j [C b ]i,i þ @xj
6
Simulation results
In this section, we evaluate the proposed hybrid localisation method through simulation, and examine the accuracy measures described in Sections 3 and 5. We consider a network in which the nodes are randomly deployed in a square area with dimensions of 1 km 1 km. The radio/ communication range of each node is set to 300 m. The distance measurement error is modelled as a Gaussian distributed RV of zero mean and a STD proportional to the true distance.
6.1 Impact of distance measurement accuracy and network size First, let us show that the transformation described in Section 4 is necessary. Fig. 5 shows the localisation results based on the proposed algorithm under a one-network configuration which has 120 nodes. The dots denote the true node configuration under a pre-defined coordinate system and the circles denote the configuration of the estimated node locations. The two locations of the same node are connected by a solid line. Fig. 6 shows the corresponding results after the location estimates, which are already shown in Fig. 5, are transformed to better match the pre-defined node locations. Based on the length of the connection lines, we can roughly judge that the estimated locations in Fig. 6 are more accurate than those in Fig. 5. More precisely, it is found that the RMSE of the coordinate estimation is 10.9 and 26.7 m with and without transformation, respectively. Clearly, it is crucial to transform the estimated locations before calculating the coordinate estimation errors; otherwise, the location accuracy would be underestimated. Since the distance between each pair of estimated locations does not change with the transformation, the RMSE of the distance estimates remains the same and equal to 8.3 m. Thus, the RMSE of the distance errors might provide a more accurate measure of the anchor-free location algorithms.
!2 [C b ]j, j
!2 @di, j @di, j [C b ]N þi, N þi þ þ [C b ]N þj, N þj @yi @yj
(46) and hence
s2d~ LBd~ i, j i, j
(47)
Accordingly, we can obtain the averaged lower bound as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N 1 X N X u 1 LB ~ LBd~ ¼ t N (N 1) i¼1 j¼iþ1 d i, j 556 & The Institution of Engineering and Technology 2009
(48)
Figure 5 Position estimation results under one-node configuration True and estimated positions are denoted by dot and circle, respectively, and connected by a solid line
IET Commun., 2009, Vol. 3, Iss. 4, pp. 549– 560 doi: 10.1049/iet-com.2007.0603
www.ietdl.org Fig. 8 compares the RMSE of the coordinate estimation errors under three node populations. As observed the location accuracy improves as the node population increases. This is because of the fact that a higher node density results in more neighbouring nodes such that more information can be employed to localise the unknown nodes.
6.2 Performance comparison
Figure 6 Results after transforming the estimated node locations
Fig. 7 shows the RMSE of the coordinate estimation errors and that of the distance errors with respect to the STD of the distance measurement errors. Also shown are the CRLB and the approximate lower bound. When the STD is at 4% of the true distance, the RMSE of both the coordinate estimation errors and the distance errors are less than 6 m when the population is 150. As expected, the location accuracy degrades as the STD of the distance measurement error increases. The gap between the accuracy of the location estimates for the CRLB and the approximate lower bound is relatively large, especially when the distance measurement error is large. One possible way to reduce the gap is to perform the global optimisation over all the location estimates. However, this will increase the computational complexity tremendously, and it is not a solution for a distributed network. Another way to narrow down the gap is to exploit the error statistics of the distance measurement. We will continue the work to see how close the accuracy of the algorithm will approach the bounds.
Figure 7 Lower bounds and the accuracy of the proposed algorithm in the event of 150 nodes in the network: ‘dissimu’ denotes the RMSE of the distance errors calculated by (41); ‘coor-simu’ denotes the RMSE of the coordinate estimation errors computed by (40); ‘dis-ana’ denotes the approximate lower bound determined by (48); ‘coor-ana’ denotes the CRLB determined by (26) IET Commun., 2009, Vol. 3, Iss. 4, pp. 549– 560 doi: 10.1049/iet-com.2007.0603
In this section, we compare the performance of the proposed algorithm with that of the MDS method [20] and the RQ algorithm [22]. Fig. 9 shows the cumulative distribution of the absolute values of the distance errors when there are 150 and 90 nodes in the network, respectively. The accuracies of the three algorithms, that is, the proposed algorithm, the RQ algorithm and the MDS method are plotted for comparison. For each node population (150 and 90), 100 different node deployment realisations are examined. As expected, more nodes in the network produce better position estimation accuracy as already
Figure 8 STD of the coordinate estimation errors of the proposed algorithm when there are 90, 120 and 150 nodes in the network, respectively
Figure 9 Cumulative distribution probability of the distance error of the proposed algorithm, the RQ method and the MDS method when there are 150 and 90 nodes in the network, respectively ‘proposed (150)’ denotes results for the proposed algorithm with 150 nodes in the network
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www.ietdl.org Table 1 Success rate of localisation of the proposed algorithm and the RQ method when there are either 150 or 90 nodes in the network Proposed method, % RQ method, % 150 nodes
99.47
95.24
90 nodes
96.48
53.13
observed earlier. It can be seen that the accuracy of the proposed algorithm is considerably higher than that of the MDS method and the RQ method. Compared with the RQ method, the accuracy improvement of the proposed algorithm comes from the good accuracy of the location estimates of the base nodes, the residual checking to avoid abnormal errors, and the averaging of estimates from multiple RQ. Table 1 shows the average success rates of localisation of the proposed method and the RQ method. The success rate of localisation is defined as the ratio of the number of nodes that have been localised to the number of all nodes in the network. The unlocalised nodes result from the fact that the condition(s) of localising a node is (are) not satisfied. For instance, a node cannot communicate with at least two other nodes and, hence, its location cannot be determined based on distance measurements. When a node is within the radio range of another node, its location is constrained to a circular region with a radius equal to the radio range. Another example is that a node cannot form a RQ with other three nodes and hence, its location cannot be determined by only using the RQ method. Clearly, the success rate of the proposed algorithm is higher than that of the RQ method, especially when the node density is relatively low. This is because of the fact that even when the RQ algorithm stops because of the lack of RQ, the RTRR approach can resume the localisation process. This is attributed to the fact that the conditions of the RTRR approach can usually be satisfied more easily than those of the RQ method. Note that all nodes can be localised by using the MDS method provided that any pair of nodes is connected via a one-hop or multihop path. The distance between two nodes that are out of the radio range is computed based on the shortest path distance in the MDS algorithm [20].
7
Conclusions
In this paper, we investigated anchor-free node localisation in multihop WSNs. A hybrid localisation scheme was proposed and numerically evaluated. The proposed approach combines the MDS, RQ and RTRR approaches to achieve both good accuracy and success rate of localisation. The effectiveness of the proposed algorithm has been demonstrated through simulation. Also, we investigated the fundamental performance limits of anchor-free localisation in multihop WSNs. Both the CRLB and the approximate lower bound were 558 & The Institution of Engineering and Technology 2009
derived to serve as accuracy references. The accuracy of the algorithms can be evaluated based on both the coordinate estimation errors and the distance estimation errors.
8
Acknowledgment
The authors would like to thank the anonymous reviewers for their valuable comments and suggestions.
9
References
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