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Improving Source Localization in NLOS Conditions via Ranging Contraction Giuseppe Destino, Student Member, IEEE and Giuseppe Thadeu Freitas de Abreu, Senior Member, IEEE Centre for Wireless Communications University of Oulu, Finland E-mails:
[email protected],
[email protected]
The remainder of the article is as follows. In section II the localization problem is formulated, in section III we provide an excerpt of the ranging contraction theory, in sections IV and V we introduce the contraction mechanism and the localization algorithm, and finally, in sections VI and VII numerical results and conclusion remarks are given.
Abstract—In this paper a novel distance-based source localization algorithm is proposed that is effective in minimizing the error due to biased measurements. In particular, we show how to exploit the knowledge of the feasibility region, constructed via trilateration, to contract the measured distances such that the cost-function of the LS formulation becomes convex and the global optimum is closer to the true location. The proposed ranging contraction source-localization algorithm is shown via simulations to overperform existing alternatives, such as the unconstrained and the constrained LS approach. The results also show that the localization error performance of the proposed technique remains close to the theoretical position error bound.
II. L EAST-S QUARE F ORMULATIONS OF THE S OURCE L OCALZIATION P ROBLEM Consider a wireless network of NA anchors and a target deployed in the η-dimensional space. An anchor is a node whose position is known a priori, while a target is a node whose location is to be determined. Let ai ∈ Rη and x ∈ Rη be row-vectors whose elements are the coordinates of the ith anchor and the target, respectively. The Euclidean distance between the i-th anchor and the target, denoted by di , is given by di = kai − xkF , where k · kF is the Frobenius norm. A measurement sample of di , denoted by d˜i , is given by
I. I NTRODUCTION In the last few years, the raise of new radio technologies, such as ultra-wideband (UWB), RF-ID, and the development of radio-based ranging techniques, allowed locationinformation to be a key-feature of novel wireless communication systems. In fact, several network optimization functionalities and application services are nowadays developed under the assumption of location-awareness. A typical location-aware network consists of few nodes with known locations (anchors), nodes whose location are to be determined (targets) and an server (centralized or distributed) where the localization algorithm runs [1]. This work focuses on the development of a centralized distance-based localization algorithm for a star-like network architecture, and we mainly address the problem of non-line-of-sight (NLOS) channel conditions typically caused by the blockage or partial obstruction of the direct-path (DP) or by delay propagation of the material crossed. [2], [3]. Specifically, we tackle the issue of bias errors in the ranging measurements, which typically causes poor localization accuracy. In contrast to state-of-the-art algorithms, which are based on constrained optimization methods [4]–[7], we propose a novel non-parametric and unconstrained approach that proves more effective and practical. To be specific, we exploit the novel concept of ranging contraction proposed in [8] to compute a new set of distances used in the optimization. The core of the algorithm exploits the existence of a feasibility region where the target lies inside. In addition, by means of the proposed contraction algorithm, we solve many of the issues tackled in other localization techniques. In short, we are able to modify the optimization into a convex problem, which in turn can provide lower localization errors. 978-1-4244-7157-7/10/$26.00 ©2010 IEEE
d˜i = di + νi + ρi ,
(1)
where νi ∼ N (0, σi2 ) and ρi ∼ U(0, ρMAX ) are random variables with Gaussian and Uniform distributions, respectively. The variable νi models small variations of the error due to thermal-noise, while ρi refers to a bias caused by the missdetection of the DP. Hence, ρi = 0, i.e. ρMAX = 0 corresponds to a distance measurement in LOS conditions, while ρi > 0 with ρMAX σi2 indicates the presence of a NLOS channel. ˆ is Given a set of distances {d˜i }, the target’s location x computed via the following least-square optimization problem ˆ = arg minη x ˆ ∈R x
NA X i=1
2 wi d˜i − dˆi ,
(2)
ˆ kF and wi is a weighing factor that relates where dˆi , kai − x to the reliability of d˜i [9], [10] (in this paper wi = 1 ∀i). In the presence of severe ranging errors, especially due to NLOS, the global minimum of equation (2) is largely shifted from the true point, thus the above problem formulation leads to large localization errors in the presence of bias. In order to improve the performance of a LS-based localization algorithm, constrained optimizations can be utilized as those proposed in [6], [11], [12]. For instance, the method described in [6] uses a priori information on LOS/NLOS channel conditions to formulate the following constrained optimization approach 56
2
ˆ x
=
arg minη ˆ ∈R x
NA X i=1
2 hi d˜i − dˆi ,
Consider a network with NA = 3 and a target, and assume x enclosed. All anchor-target links are assumed in NLOS channel conditions, therefore ρi > 0 ∀i. In figure 1 we illustrate the contour levels of f (ˆ x) and we indicate with dots, the points where the Hessian matrix ∇2xˆ f (ˆ x) 0. We consider different conditions in order to exemplify the aforementioned theory. The first study, illustrated in figure 1(a), consists of the evaluation of LS objective function with exact distance, i.e. d˜i = di . We recognize that f (ˆ x) is generally convex in the whole domain but around the anchors. Next, we evaluate the LS objective function with the original distance measurements d˜i ’s, and as shown in figure 1(b), noisy measurements cause multiple minima and, in addition, modify the convex area, which is significantly reduced. The intuition is that concave areas expand with d˜i ≥ di . From figure 1(c), this phenomenon is more evident. Indeed, in contrast to figure 1(b), the assumption of contracted distances causes a reduction of concave areas and, subsequently an expansion of the convex regions. Theorem 1 and Corollary 2, in fact, prove this observation. Finally, figure 1(d) shows that by means of negative contracted distances, f (ˆ x) is convex in the whole domain, as proved in Theorem 2. IV. R ANGING C ONTRACTION A LGORITHM From the previous section, we established the advantage of utilizing a set of contracted distances d¯i ’s to improve convexity of the objective function inside the convex-hull. In the sequel, we will propose a mechanism that allows for the computation of the contracted distances given a set of measurements {d˜i }. To begin with, consider the following definition and Lemma. Definition 4 (Feasibility region): Given a set of distance measurements d˜i ’s, the feasibility region I of the LS sourcelocalization problem is defined as I , {ˆ x|dˆi ≤ d˜i ∀i}. (7)
(3)
s.t. dˆi ≤ d˜i , ∀i | hi = 1, where hi = 1 if the i-th link is in LOS condition or hi = 0 otherwise. Alternatively, considering ρi ’s as variables, the LS-problem in equation (2) can be reformulated as [12] ˆ x
=
arg
min
NA X
ˆ ∈Rη ,ˆ x bi ∈R+ i=1
2 ¯ iˆbi , d˜i − dˆi − h
(4)
s.t. dˆi ≤ d˜i , ∀i. ˆbi ≤ min {d˜i + d˜j − di,j }, j=1,...,NA
ˆbi ≥ 0 where di,j is the distance between the i-th and the j-th anchor ¯ i is the complement of hi , i.e. h ¯ i = 0 if hi = 1 and and h vice-versa. Our approach, instead, relies only on the simple information that ρi > 0 and the feasibility region, defined in the section IV, exists. These information will be sufficient to design a mechanism that adjusts the distance measurements (distance contraction algorithm), and yields a robust source-localization algorithm based on an unconstrained optimization. III. T HEORY OF R ANGING C ONTRACTION The theory of ranging contraction was presented for the first time in [8] and in what follows, only an excerpt will be provided. For the proofs of Theorems, Lemmas and Corollaries we refer to the original paper [8]. ˆ such that Lemma 1 (Convexity): Let d˜i = di , ∀i. For all x NA P ˆ di −di ≥ 0 then ∇2xˆ f (ˆ x) 0, where f (ˆ x) denotes the di i=1
objective function in equation (2). Definition 1 (Convex-hull): Let C({ai }) denote the convexhull defined by the anchors. Definition 2 (Enclosed point): A point x ∈ C({ai }) is referred to as enclosed. Corollary 1 (Convexity with Enclosure): Let B ⊆ C({ai }) be a non-empty convex set. If x ∈ C({ai }) and d˜i = di , ∀i then, ∃B | ∇2xˆ f (ˆ x) 0, ∀ˆ x ∈ B.
Lemma 2 (Property of the feasibility region): If I is not an empty set, then i) I is a convex set, ii) x ∈ I. Proof: See [6], [13], [14]. The Lemma above provides fundamental information about the location of the target. Indeed, by force of property ii), we can ensure that by confining the search within I, the true target’s location can be found. This result, in fact, is one of the fundaments of all algorithms based on tri-lateration mechanism such those given in equations (3) and (4). In what follows, instead, we will show that the knowledge about the feasibility region I can also be used to correct the distance measurements. To this end, consider the following Theorem and Corollary. Theorem 3 (Unique minimum-distance projection): Let A ⊆ Rn be a closed convex set and let z ∈ / A. Then P · z ∈ A is the unique projection of z on A if and only if
(5)
Definition 3 (Contracted distance): Let d¯i ≤ di , then d¯i is referred as contracted distance. ˆ∈ Theorem 1 (Convexity with contracted distances): Let x B. If d˜i = d¯i and d¯i ≥ 0 ∀i, then ∇2xˆ f (ˆ x) 0 ∀ˆ x ∈ B. Corollary 2 (Wider convexity): Let B 0 be a non-empty convex set such that B ⊆ B 0 ⊆ C({ai }). If x ∈ C({ai }) and d˜i = d¯i , d¯i ≥ 0 ∀i then, ∃B 0 | ∇2xˆ f (ˆ x) 0, ∀ˆ x ∈ B0 .
(6)
Theorem 2 (Full convexity with negative contraction): If d˜i = d¯i and d¯i ≤ 0 ∀i, then ∇2xˆ f (ˆ x) 0 ∀ˆ x ∈ Rη . In the sequel, we illustrate the ranging contraction theory with a typical source-localization study.
hz − P · z, y − P · zi ≤ 0 ∀y ∈ A,
(8)
where P ∈ Rn×n is the projection matrix and h, i is the inner product. 57
3
Convex area of the LS Cost-Function NA = 3, NT = 1, d˜i = di
Convex area of the LS Cost-Function NA = 3, NT = 1, d˜i
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(a) Exact distances
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Convex area of the LS Cost-Function NA = 3, NT = 1, d˜i = d¯i , d¯i ≥ 0
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Convex area of the LS Cost-Function NA = 3, NT = 1, d˜i = d¯i , d¯i ≤ 0
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(c) Contracted distances Fig. 1.
z.
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(d) Negative contracted distances
Illustrative example of the distance contraction theory.
Invoking Theorem 3, P · ai can be found by an exhaustive search within I until equation (8) is satisfied. This approach, however, is not practical due to the large number of trials to be required. An alternative method to obtain d¯i is proposed in [15], [16, thm. 2.3]. In this case the minimum distance problem is posed as a maximization problem using hyperplanes. For the sake of completion, we provide the following definitions [17]. Definition 5 (Supporting hyperplane): Let A be a set in Rn . The supporting hyperplane H has form
Proof: See [15] and reference thereby. Corollary 3 (Nearest point): P · z is the point in A nearest
Proof: See [15]. From the above, we recognize that ∀ ai such that ai ∈ / I, the minimum distance separating ai to I can be computed by equation (8). By construction, such a distance is shorter than di and therefore, it is a contracted distance. Thus, the desired d¯i can be computed as d¯i = kai − P · ai kF ,
H , {z|cT · z = cT · z0 },
(9)
where c 6= 0, cT · z ≤ cT · z0 , the superscript z0 ∈ bd(A) and bd() is the boundary.
where P · ai is the unique projection of ai on I. 58
(10) T
is transpose,
4
Next, we use equation (9) to compute d¯i . Finally, we minimize equation (2) where d˜i ’s are replaced by d¯i ’s. In figure 1, we illustrate the aforementioned idea with an example. Reconsider the network shown in figure with NA = 3 anchors and a target. All links are in NLOS conditions and we set ρMAX = 0.2 and σ = 0.05. We indicate in dash-lines the circular traces computed from the measurements d˜i > di . Using equation (17) we verify the existence of I, which is indicated in bold-line. By means of ˆ 0 . As mentioned above, x ˆ 0 is used equation (18), we compute x to initialize the constrained optimization given in equation (14). Next, using equation (14) for all anchor-target links, we obtain the set of contracted distances d¯i ’s, which will replace d˜i ’s in equation (2). By construction, the shortest distance from the i-th anchor to I corresponds to a circular trace that is tangent to the feasibility region. Given the convexity of I the shortest distance is also smaller than the exact one, thus each d¯i is a contracted distance. Finally, using equation (2) ˆ. the target’s location estimate is computed and indicated by x
Definition 6 (Separating hyperplane): Let A and D two disjoint convex set in Rn . The separating hyperplane H is defined as H , {z|cT · z = b}, (11) such that cT · z ≤ b, z ∈ A and cT · z ≥ b, z ∈ D. (12) In light of the definitions above, the minimum distance from z∈ / A to A can be found by solving the following optimization problem [15] [16] kz − P · zkF
=
s.t.
1 max aT · z − γA (a), τ a kak ≤ τ
(13)
where γA (a) is the supporting function of A and τ is an arbitrary positive value [15]. Considering our objective, that is to find the minimum distance from ai to I, this approach can be difficult to apply because of the unknown supporting function γI (a). Therefore, we propose an alternative optimization problem. ¯ i ∈ I to ai is Theorem 4: Let ai ∈ / I. The nearest point x 2 given by ¯ i = arg max d˜i − dˆ , x (14)
VI. S IMULATION R ESULTS
ˆ ∈I x
In this section, the performance of the proposed distance contraction (DC) localization algorithm will be evaluated and compared to state-of-the-art algorithms, namely, the unconstrained linear-global distance continuation (L-GDC) [18], the constrained non-linear LS (C-NLS) [13] and the sequential quadratic programming (SQP) [12] approaches that solve equations (2), (3) and (4), respectively. In addition, the results are compared to the position error bound (PEB) [3] given by
s.t. d˜i − dˆ ≥ 0, ∀i. ¯ i ∈ I to ai is Proof: By definition the nearest point x ¯ ¯ i , arg min d. x (15) ¯ ∈I x
In order to obtain the formulation in equation (14), do ¯i , x =
¯ arg min d¯ = arg max(−d), ¯ ∈I x
¯ ∈I x
(16)
¯ = arg max(d˜i − d) ¯ 2, arg max(d˜i − d) ¯ ∈I x
¯ ∈I x
PEB ,
where the first equality holds due to a sign change, the second ¯ ∀¯ one since d˜i ≥ 0 and the last one becasue d˜i ≥ d, x ∈ I. Using the Theorem above, P · ai can be easily computed from equation (14) and, by means of equation (9) we obtain the desired contracted distance d¯i .
η = 2, NA = 3, NT = 1
1.2
This section will provide a concise description of the operations to be executed by the localization algorithm utilizing contracted distances. First, we verify the existence of the feasibility region I. To this end, it is sufficient that for any η or more anchor-to-target links the following inequalities hold
1
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y-axis
(17)
If I does not exist, meaning there is no intersection, then the measurements d˜i ≤ di ∀i, thus d˜i ’s are already contracted. In this case, the following steps will not be considered and equation (2) can be minimized. On the other hand, if I exists, then we use equation (14) for all i-th anchor-target links. In so doing, we initialize the optimization with a point x0 ∈ I which, for instance, can be computed as NA 2 X ˆ 0 = arg minη x max 0, dˆi − d˜i , (18) ˆ ∈R x
0.4
0.2
Anchor Target ˆ CN LS x ˆ SQP x ˆ DC x
0
−0.2 −0.2
i=1
where the cost-function is convex and 0 in I.
(19)
Example of Localization with Distance Contraction
V. L OCALIZATION A LGORITHM
di,j ≤ d˜i + d˜j .
GDOP2 , A
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x-axis Fig. 2.
59
Illustration of the contraction algorithm
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η = 2, NA = 4, NT = 1, ρMAX = 0.2, σ = 0.05
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Fig. 5. Comparison of the localization MSE performance as function of the number of anchors.
Fig. 3. Comparison of the localization MSE performance as function of the maximum bias.
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Fig. 4.
Localization Error
Comparison of the CDF localization error.
Fig. 6.
where A=
1 √ σπ 2ρMAX
where the indexes p and l indicate the l-th realization of the p-th network configuration and, the CDF of ζ is computed as CDF , Pr{ζ ≤ ξ},
(24)
where ξ is the localization accuracy. We perform two studies. In the first one, we evaluate the performances of the algorithms as a function of the ranging error, and mainly, of the maximum bias ρMAX . In the second one, instead, we evaluate the impact of the number of anchors on the localization error. Starting from the first case of study, the typical network consists of NA = 4 anchors and a target, which are randomly
i=1
(21) where θi is the angle between the target and the i-th anchor measured with respect to the horizon. The performances of each algorithm are evaluated utilizing the mean-square-error (MSE) and the cumulative density function (CDF) of the localization error ζ ˆ kF . ζ , kx − x
(23)
l=1
−∞
i=1
Comparison of the CDF localization error.
Specifically, the MSE is given by P L 1 XX 2 MSE , ζ(l,p) , LP p=1
+∞ 2 Z √ exp −y + ρσMAX − exp −y 2 √ √ dy. Q( 2y) − Q( 2y + ρMAX σ )
(20) and GDOP, that states for Geometric Dilution of Precision, is NA GDOP , N 2 N N PA PA PA 2 2 cos(θi ) sin(θi ) − cos(θi ) sin(θi ) i=1
L-GDC CNLS SQP DC
(22) 60
6
deployed. All anchor-target links are considered in NLOS conditions. The results are shown in figures 3 and 4. In both figures, the proposed DC localization algorithm outperforms the alternatives, and more relevantly, performs close to the PEB. This result is very remarkable because the proposed ranging contraction algorithm has not been designed to compensate/estimate the bias, as it turns out, it provides a seem-less compensation in a non-paramtric fashion. In figure 3, we also notice that the MSE achieved with the DC algorithm are slightly below the PEB. The reason is still under investigation, but the main intuition is the fact that by using a contraction algorithm the statistics of d¯i ’s are no longer Gaussian. Thus, the PEB computed in [3] does not hold anymore. In the second case of study, instead, we consider a network whose number of anchors varies from 3 to 10. As in the previous case, anchors and target are randomly located, all anchor-target links are considered in NLOS conditions, the noise standard deviation σ = 0.05, and ρMAX = 0.2. The results of this test are shown in figures 5 and 6. Also in this case the DC algorithm is the best performing method and, in particular from figure 3, it is shown that only little advantage is ripped by increasing the number of anchors.
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This result indicates that utilizing the proposed DC algorithm, a localization system can provide accurate location estimates with very few anchors, and by product, it will result energy and computational cost efficient.
VII. C ONCLUSION In this paper, we proposed a robust distance-based source localization algorithm that relies on the novel concept of ranging contraction. It is shown that by means of contracted distances, we can improve the convexity of the LS costfunction and, in turn, we can decrease the localization error due biased distance measurements. In the paper, we described in detail both a ranging contraction algorithm and the localization method. We compared the performance of the DC technique to state-of-the-art algorithms and, we showed that the novel method outperforms the alternatives. R EFERENCES [1] G. Mao, B. Fidan, and B. D. O. Anderson, “Wireless sensor network localization techniques,” Computer Networks: The Intern. J. of Comp. and Telecomm. Networking, vol. 51, no. 10, pp. 2529–2553, July 2007. [2] C. Gentile and A. Kik, “An evaluation of ultra wideband technology for indoor ranging,” in Proc. IEEE Global Telecommunications Conference (GLOBECOM), 27 Nov.-Dec. 1 2006, pp. 1–6. [3] D. Jourdan, D. Dardari, and M. Win, “Position error bound for UWB localization in dense cluttered environments,” in Proc. IEEE International Conference on Communcations, vol. 8, June 2006, pp. 3705–3710. [4] A. J. Weiss and J. S. Picard, “Network localization with biased range measurements,” IEEE Trans. Wireless Commun., vol. 7, no. 1, pp. 298– 304, January 2008. [5] B. Denis and N. Daniele, “NLOS ranging error mitigation in a distributed positioning algorithm for indoor UWB ad-hoc networks,” in Proc. IEEE Intern. Workshop on Wireless Ad-Hoc Netw., 2004, pp. 356–360. [6] S. Venkatesh and R. M. Buehrer, “NLOS mitigation using linear programming in ultrawideband location-aware networks,” IEEE Trans. Veh. Technol., vol. 56, no. 5, Part 2, pp. 3182 – 3198, September 2007. [7] K. Yu and Y. Guo, “Improved positioning algorithms for nonline-of-sight environments,” IEEE Transactions on Vehicular Technology, vol. 57, no. 4, pp. 2342–2353, July 2008. [8] G. Destino and G. Abreu, “Reformulating the least-square source localization problem with contracted distances,” in Proc. IEEE 43th Asilomar Conference on Signals, Systems and Computers, 2009. [9] G. Destino and A. G., “Weighing strategy for network localization under scarce ranging information,” IEEE Trans. Wireless Commun., 2009.
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