This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2009 proceedings Mapping Techniques for UWB Positioning Eva Arias-de-Reyna Umberto Mengali Department of Signal Theory and Communications University of Seville Camino de los Descubrimientos s/n, 41092 Seville, Spain Email: earias@us.es Department of Information Engineering University of Pisa Via Caruso 16, 56122 Pisa, Italy Email: umberto.mengali@iet.unipi.it Abstract—This paper deals with a wireless indoor positioning problem in which the location of a tag is estimated from range measurements taken by fixed beacons. The measurements may be affected by non-line-of-sight (NLOS) errors that must be mitigated. We discuss a maximum likelihood (ML) positioning technique that assumes a realistic model for the range errors and a signature database providing information on the propagation conditions at every hypothesized tag spot. The database can be gathered from knowledge of the service area infrastructure and through pre-measurements. It is given in the form of a map indicating, at any node of a close-mesh grid, the nature (LOS/NLOS) of the link between that node and each beacon. The performance of the positioning algorithm is assessed by simulation and is compared with other methods available in literature. The results show that the proposed technique provides significant improvements and is robust against mismatches between true and assumed values of the parameters in the range error model. Comparisons with the Cramer-Rao Bound are made. I. I NTRODUCTION In the last few years wireless indoor geolocation has become an important technology for use in a variety of civil and military applications. In the commercial world, for example, there is an increasing need for identifying the location of specific items in warehouses and cargo ships, or for tracking people with special needs in residential environments, hospitals and nursing homes [1]–[3]. As recognized in the IEEE standard 802.15.4a, ultra-wideband (UWB) impulse radio can play a significant role in this context as it allows centimeter accuracy in ranging and low-power and low-cost communications. In this paper we concentrate on indoor geolocation with UWB techniques. In particular we consider a network of beacons deployed inside a building and emitting UWB signals. Their location is known while the unknown coordinates of a tag must be determined from a set of time-of-arrival (TOA) measurements. We envisage a scenario in which the tag can move in a specific service area. For example we want to estimate the position of a robot as is moves inside a laboratory. For simplicity we concentrate on two-dimensional positioning. As is known, three beacons would be sufficient to accurately solve the problem if the links between tag and beacons were This work has been supported by the Ministerio de Educación y Ciencia (Spanish Government) and the European Union (FEDER) under grant TEC2006-13514-C02-02/TCM, by the Ministerio de Ciencia e Innovación (Spanish Government) through the Project COMONSENS of the ConsoliderIngenio 2010 Program, and by the MINERVA Project 2C/039. all in line-of-sight (LOS) [4]. Unfortunately such a condition is seldom met in a typical indoor multipath environment where the propagation is likely to be non-LOS (NLOS), meaning that the direct path (DP) is totally obstructed by large objects or attenuated to such a degree that its energy falls below the threshold detection. When this happens a ray arriving from a reflection can be erroneously declared as the DP, so giving rise to a range measurement with a positive bias [2], [5]. The simple least squares (LS) location estimator suffers severe degradations in the presence of NLOS propagation [6]- [7] and several strategies have been proposed to protect location estimates from NLOS errors. They take advantage of prior information that may be available in various forms: (a) knowledge on the range error statistics; (b) knowledge of the service area layout; (c) location fingerprints of the service area [1], [8]. Point (a) refers to the probability density functions (PDFs) of the range errors at any given spot under LOS and NLOS conditions and to the prior probabilities of LOS/NLOS occurrence. Mathematical models for the range errors in a typical UWB indoor environment are given in [9], [10], [11]. Point (b) refers to the geometry of the service area (perimeter, inside walls, etc.) and to the presence of metallic objects in the building (elevator shafts, chambers etc.) that can obstruct the direct connection between tag and beacon. NLOS mitigation strategies may be ranked according to the amount of prior information they exploit. The more information they use, the more efficient they are. A simple NLOS mitigation scheme is proposed in [12]. No information on the range error statistics is assumed. Only the range measurements and the beacon locations are available. The key idea is that, if the NLOS measurements are relatively few, they can be identified and discarded (or downgraded) and the tag can be accurately located. In this spirit the measurements from all the beacons are taken in combinations of three or more and, for each combination, an intermediate LS tag location estimate is computed. Its residual over the range measurements provides an indicator of the estimate reliability: the smaller the residual, the more trustworthy the estimate. A similar NLOS mitigation scheme (but easier to implement) is discussed in [13]. In reference [14] the NLOS measurements are seen as data affected by gross errors, inconsistent with the LOS error distribution. As such, they are regarded as outliers. Several robust algorithms exist that can cope with a certain percentage 978-1-4244-3435-0/09/$25.00 ©2009 IEEE This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2009 proceedings of outliers [15]. Among them the authors choose the least median of squares (LMedS) estimator. In [16] a wideband CDMA cellular system is envisioned in which location measurements are based on time-differenceof-arrivals (TDOAs) and angle-of-arrivals (AOAs). While the scenario exhibits significant differences in error statistics compared with an indoor UWB environment, the methods in [16] are easily adapted to the problem investigated in the present paper. Mapping methods referred to as location fingerprinting offer another way to counteract NLOS measurements. Instances of these techniques are reported in [1], [17]. Assuming that the service area is relatively small, the idea is to perform premeasurements at each possible tag position and to compile a database of range signatures. Each signature consists of a vector providing the measured distances from tag to beacons. A nearest-neighbor algorithm determines the signature in the database with the minimum Euclidean distance to the actual range metrics. Hereafter we discuss an alternative mapping technique in which a signature represents the beacon states at any tag location (not the beacon ranges to the tag). In particular, the nth signature component is either zero or one, depending on whether the link from tag to the nth beacon is LOS or NLOS. The signatures can be extracted from: (a) the infrastructure of the service area, taking into account size and position of the major metallic objects [10]; (b) field measurements when necessary. This substantial reduction in fingerprint information is compensated by providing range error statistics. As opposed to [1], [17] where no error statistics are needed, we assume that the conditional range error PDFs are known from field measurements [9]- [10] and we exploit them, along with the signature database, to implement a maximum likelihood (ML) position estimator. We believe that the major advantage of storing LOS/NLOS conditions rather than TOA-measured distances is in the robustness of the database against perturbations due to mobile objects in the environment. For example, suppose that the DP from the nth beacon is blocked by an elevator shaft, so creating a shadow zone where the tag is only reached through NLOS propagation. Also, assume that some people are moving around the elevator. As long as the tag lies in the shadow zone, the nth component of a LOS/NLOS signature is unity, independently of the presence of people. On the contrary, in a signature containing TOA-measured distances, the nth component may vary, depending on the positions of the people. The main goals of the paper are as follows. First, we compare the performance of the the ML positioning estimator with that of other localization algorithms available in literature. Second, we establish the standing of all these methods with respect to the conventional (non-Bayesian) Cramer-Rao bound (CRB). Third, we assess the robustness of the ML estimator against mismatches between assumed and actual range error statistics. The rest of the paper proceeds as follows. In the next Section we overview the range error model in [9] and we compute the corresponding CRB in positioning estimation. Section III describes the ML estimator and introduces a specific service area that is used as a study case to assess estimation performance. Section IV summarizes the positioning schemes in [12], [14], [16] while Section V provides simulation results and comparisons. Finally, Section VI offers some conclusions. II. R ANGE ERROR MODEL , LIKELIHOOD FUNCTION AND CRB The following notations are adopted throughout the paper. The coordinates of the tag are indicated as z = (x, y)T . There are N beacons located at known positions z n = (xn , yn )T (n = 1, 2, · · · , N ) and the nth beacon provides a range measurement rn which is modelled as rn = dn + n where dn is the true distance from the tag, i.e.,  dn = (x − xn )2 + (y − yn )2 (1) (2) while n is the range error. Several statistical models have been suggested for n . In this paper we concentrate on the one reported in [9] but the discussion is easily extended to the more complex model in [10]. A. Range error model The authors in [9] express n as the sum of two terms1 n = n,LOS + ξn n,N LOS (3) where n,LOS is caused by multipath propagation and is always present in an indoor environment. In case of LOS propagation it is the only source of error considered. Instead n,N LOS corresponds to a situation in which the direct path is undetected (NLOS). Finally ξn is a binary variable taking value 0 with LOS and 1 with NLOS propagation. The term n,LOS is modelled as n,LOS = γ log(1 + dn ), where γ is a Gaussian random variable with mean mLOS and standard deviation σLOS . It follows that n,LOS is Gaussian with mean (4) mn,LOS = mLOS log(1 + dn ) and variance 2 2 = σLOS [log(1 + dn )]2 σn,LOS (5) Similarly, n,N LOS is modelled as a Gaussian RV (independent of n,LOS ) but, as opposed to n,LOS , its mean and variance do not depend on dn mn,N LOS = mN LOS 2 σn,N LOS = 2 σN LOS (6) (7) Putting these facts together and bearing in mind (3), we conclude that n is Gaussian with mean mn = mLOS log(1 + dn ) + ξn mN LOS (8) 1 For convenience we have renamed the original parameters in [9] and have suppressed a subscript W indicating dependence on the signal bandwidth. This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2009 proceedings and variance 2 2 σn2 = σLOS [log(1 + dn )]2 + ξn σN LOS (9) Empirical values of mLOS , mN LOS , σLOS and σN LOS are given in Table I of [9] as a function of the signal bandwidth. From the above it follows that the PDF of the range measurement rn can be written as   1 (rn − dn − mn )2 p(rn ; z) = √ (10) exp − 2σn2 2πσn Also, as the range measurements are all independent and Gaussian distributed, the PDF of r = (r1 , r2 , · · · , rN )T is given by   N  (rn − dn − mn )2 1 √ (11) exp − p(r; z) = 2σn2 2πσn n=1 from which the log-likelihood function Λ(r; z)  ln p(r; z) is computed as Λ(r; z) = −N ln √ 2π − N  ln σn n=1 − N  (rn − dn − mn )2 2σn2 n=1 (12) B. Cramer-Rao Bound The CRB for the estimate ẑ = (x̂, ŷ)T of the tag location is computed from the Fisher information matrix   ∂Λ ∂Λ 2 E{( ∂Λ ∂x ) } E{ ∂x ∂y } (13) J ∂Λ ∂Λ 2 E{ ∂Λ ∂x ∂y } E{( ∂y ) } where the expectations are taken with respect to p(r; z) and the z value corresponds to the true tag location. The CRB is expressed in terms of the diagonal elements of J−1 as CRB(z)  [J−1 ]11 + [J−1 ]22 (14) The explicit expression of CRB(z) is easily derived from (14) but is not given here for space limitations. Two remarks are in order. The first is that (14) gives a bound to the variance of any unbiased estimator when the tag occupies the specific location z. In practice, however, we are interested in an average bound that accounts for all the possible tag locations. In the simulations shown later this average is computed as the arithmetic mean of CRB(z) over the random tag locations involved in the simulation. The second remark stems from the observation that the computation of CRB(z) ignores the plan of the service area, i.e., its perimeter and possible internal obstacles. For example, in the simulations we assume a rectangular service area limited by external walls and containing two obstacles that the tag cannot go through. Now, suppose the tag is close to an obstacle. While this fact is irrelevant in the calculation of CRB(z), it can be taken into account by a location estimator. In particular, some of the algorithms considered later involve the minimization of a cost function. As the minimization is performed over the useful service area, it follows that the locations inside the obstacles or outside the service area’s perimeter are never tested nor selected as final estimates. In conclusion, the CRB must be viewed as a performance bound for estimators that ignore restrictions on z due to the building layout. It need not be a bound for estimators that take such restrictions into account. It is worth mentioning that knowledge on z could be incorporated by modelling z as a random variable with a PDF p(z) that accounts for the constraints imposed by the service area layout. This would pave the way to a number of Bayesian bounds, including the Bayesian CRB [18]. Unfortunately the computation of such bounds involves restrictive conditions on p(r;z) that are not satisfied when z is constrained within finite regions. III. ML LOCATION ESTIMATOR We propose an ML location estimator that exploits knowledge of the propagation conditions (LOS/NLOS) for every beacon at every possible tag position. The specific setting taken as a study case is a rectangular area of size 20 m × 10 m, with two internal square metallic objects of side 4 m. The obstacles completely block the UWB signals, thus generating shadow zones where NLOS conditions take place. We assume that such conditions only occur when the direct path is obstructed by the metallic obstacles2 . Thus, since geometry and location of the obstacles are known, for each beacon it can be established whether the propagation is LOS or NLOS at any possible tag spot. Collecting these data together generates a signature database for use in the estimation process. Figure 1 shows the service area layout, with red squares representing the metallic obstacles and the dark blue area indicating the shadow zone for a beacon lying in the origin of the coordinates. The ML estimator maximizes the log-likelihood function Λ(r; z) in (12) over all the possible tag locations z. Actually the maximization is carried out over the nodes of a grid with a mesh size of 10 cm. For each hypothesized tag location the state (LOS/NLOS) of the beacons is provided by the signature database. Maximization is performed over the useful service area. Sub-areas the tag cannot reach are excluded. In particular, the maximum of Λ(r; z) is looked for within the perimeter of the service area and outside the metallic obstacles. IV. OTHER POSITIONING SCHEMES In this section we overview other localization methods with the aim of comparing them with the ML algorithm. A. Method in [12] All the possible combinations of three or more range meaN surements are considered. Their number is K = i=3 Ni . Each combination is represented by the set of indices Sk 2 The scenario is readily extended to the case in which the NLOS conditions are caused by a large distance from tag to beacon, such that the DP is attenuated below the detection threshold. This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2009 proceedings C. Method in [16] One of the methods in [16] assumes a PDF as in (10) but does not exploit any precise knowledge of the state (LOS/NLOS) of the tag-to-beacon links. The only informa(N LOS) (z) the tion available is of statistical nature. Call Pn probability that the signal from the nth beacon undergoes NLOS propagation at the tag location z. Obviously, the LOS (LOS) (N LOS) (z) = 1 − Pn (z). Then, the probability is Pn likelihood function of z is 10 shadow zone y (m) 8 6 4 metallic obstacle metallic obstacle 2 p(r; z) = 0 0 Fig. 1. 5 10 x (m) 15 20 +p(rn ; z)|ξn =1 Pn(N LOS) (z) (19) (k = 1, 2, · · · , K) of the corresponding beacons. For each set an intermediate LS estimate of z is computed as z (15) where Res(z; Sk ) is the sum of the squared differences between the range measurements and the distances of the beacons to the hypothesized position z  2 [rn − z − z n ] (16) Res(z; Sk ) = n∈Sk Res(z; Sk ) is calculated at z = ẑ k and is normalized to the number of elements in Sk (ẑ k ; Sk )  Res(ẑ k ; Sk ) Size of Sk (17) The inverse of (ẑ k ; Sk ) is viewed as a measure of the reliability of the estimate ẑ k and the final estimate of z is taken as a weighted combination of the type ẑ = K −1 k=1 ẑ k ((ẑ k ; Sk )) K −1 k=1 ((ẑ k ; Sk )) p(rn ; z)|ξn =0 Pn(LOS) (z) n=1 Service area and shadow zone for a beacon lying at (0,0). ẑ k = arg min{Res(z; Sk )} N  (18) B. Method in [14] The method in [14] is similar to that in [12] except that only combinations of three beacons at a time are considered and, instead of minimizing the sum of the squared residues, the median of the residues is minimized. The algorithm proceeds as follows. The number of beacon combinations is now K = N3 . For a given combination Sk , an intermediate estimate of z is obtained as in (15). Next the 2 set of squared residues [rn − z − z n ] (n = 1, 2, · · · , N ) is computed and its median Mk is calculated, so that at the end of the process we have a set of K intermediate estimates ẑ k and the corresponding medians Mk . The final estimate is taken as the intermediate estimate with the minimum median. The authors in [16] look for the maximum of a modified (N LOS) (LOS) (z) and Pn (z) are version of (19) in which Pn (N LOS) (LOS) and Pn independent of replaced by quantities Pn z. One difficulty in going further is to assign sensible values (N LOS) (LOS) and Pn . We explored two routes. In one to Pn (LOS) case we put Pn equal to the fraction of the service area in LOS connection with the nth beacon. In the other we set (LOS) (N LOS) = Pn = 1/2. As experimental evidence shows Pn minimal difference in the results, in the following we only report on the second case. V. S IMULATION RESULTS Simulations have been run with the specific service area introduced in Section III. Here, 2000 tag positions are randomly chosen and, for each position, 10 different realizations of the range measurements are generated. The ML estimator (henceforth referred to as ML with signature database, shortly MLSD) is compared with the algorithms in [12], [14] and [16]. The parameters of the range error model are taken from Table I of [9] for a signal bandwidth of 500 MHz and have the following values: mLOS = 0.21 m, mN LOS = 1.62 m, σLOS = 26.9 cm and σN LOS = 80.9 cm. The performance of the algorithms is expressed as the root mean square error (RMSE) in the position estimates. The number of beacons N varies from 3 to 8. The first four are placed at the corners of the service area’s perimeter. The 5th and 6th are at the mid points of the longer walls. The 7th and 8th are at the mid points of the shorter walls. The CRB is computed as the square root of the arithmetic average of CRB(z) over the set of random tag locations tested in the simulation. All the algorithms exploit knowledge of the service area plan in the way mentioned in Section II-B. Figure 2 illustrates performance comparisons. As expected, MLSD and the algorithm in [16] have superior performance as they exploit range error statistics. On the other hand MLSD outperforms the algorithm in [16] because it takes advantage of the signature database and, in consequence, has precise information on the propagation conditions. Finally, MLSD is occasionally better than the CRB, in particular when the number of beacons is small. This confirms that CRB is a bound only for estimators with no knowledge of the service area. This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2009 proceedings 3.5 Ref. [12] 3 Ref. [14] Ref. [16] RMSE (m) 2.5 MLSD CRB 2 1.5 1 0.5 0 3 4 5 6 7 8 N Fig. 2. RMSE of the positioning error for different location algorithms as a function of the number of beacons. 2.5 Ref. [16] no mismatch Ref. [16] with mismatch 2 MLSD no mismatch RMSE (m) MLSD with mismatch CRB 1.5 1 0.5 0 3 4 5 6 7 8 N Fig. 3. RMSE of the positioning error in the presence of a mismatch between true and assumed parameters. As perfect knowledge of the range error statistics is unrealistic, the MLSD performance has been checked in the presence of a mismatch between true and assumed values of the parameters in the range error model. Figure 3 compares MLSD with the method in [16] with and without mismatch. The mismatch consists of taking mN LOS and σN LOS 30% greater than their true values. As is seen, the impact of the mismatch is marginal. VI. C ONCLUSIONS We have investigated a TOA-based geolocation algorithm that exploits information on the range error statistics and the layout of the service area. In particular it makes use of a signature database where the the propagation conditions at every possible tag spot are recorded. The database is in the spirit of the standard location fingerprinting approach but is View publication stats simpler to implement and store. Each signature provides in fact just the states (LOS/NLOS) of the beacons, not the detailed range measurements. The performance of the proposed algorithm has been compared with that of other methods available in literature. As expected, a tradeoff exists between performance and complexity. 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