On the impact of ranging-error models for ... - Semantic Scholar
On the impact of ranging-error models for simulating indoors location systems Salvador Guardiola
Israel Martin-Escalona
Francisco Barcelo-Arroyo
Department of Telematic Engineering Universitat Politecnica de Catalunya (UPC) Barcelona, Spain [email protected]
Department of Telematic Engineering Universitat Politecnica de Catalunya (UPC) Barcelona, Spain [email protected]
Department of Telematic Engineering Universitat Politecnica de Catalunya (UPC) Barcelona, Spain [email protected]
Abstract— Wireless networks have boosted the interest of users and network operators on location due to the possible synergies between the cellular layout and location techniques. As a result, location methods have been proposed to provide this information in most wireless technologies. Evaluation of the performance of these and further techniques is initially achieved by means of simulation. It is essential to use suitable models for featuring errors that are present in observations. This characterization is especially relevant in time-based techniques such as time-ofarrival (TOA) and time-difference-of-arrival (TDOA). These techniques are significantly impacted by non-line-of-sight conditions; hence their performance is usually evaluated under these conditions. This article compares several ranging-error models adapted to indoors and studies their impact on the performance of time-based techniques (e.g., TOA and TDOA). Results indicate that simple models that are not based on the actual distance between nodes tend to underestimate the positioning error with respect to more complex models. Keywords; ranging, error, TOA, TDOA, NLOS, indoor positioning.
I.
INTRODUCTION
Mobile ad hoc networks (MANETs) have been a focus of the industry, mainly due to their dynamism and selfconfiguring capabilities. The mobility feature of these networks represents a challenge in the design of protocols since nodes can change their position and can associate or leave the network suddenly. Accordingly, many efforts have been addressed to provide MANETs with location solutions that are able to fix the position of all nodes. This knowledge allows the nodes to access a huge range of location-based services. Furthermore, knowing the position allows several operational and maintenance tasks to be optimized. For instance, routing protocols are scalable only if the positions of the nodes are known [1]. Consequently, there is increasing interest in the research of location solutions for MANETs, which aims to provide accurate enough positions for the scenarios they usually are deployed in. Currently, there are several proposals for positioning ad hoc nodes, including fingerprinting [2-3], schemes based on Ultra Wide Band [4-5], and those based on the angle-of-arrival [6]. Time-based techniques seem to be preferred in the ad hoc field,
This research was partially funded by ERDF and the Spanish Government through project TEC2006-09466/TCM.
mainly due to the good trade-off between accuracy, scalability and deployment/maintenance costs. This article is focused on two of these time-based techniques: TOA [7-8] and passiveTDOA [9], which observe the time spent by signals in traveling certain distances and fix the position by means of multilateration algorithms. These techniques have the advantage that they usually do not need specific hardware or off-line calibration steps, which are required in fingerprinting. However, these time measures do not always give a precise estimate of the distance between two nodes, since signals are impacted by several sources of error. These sources of error include dense multi-path propagation, obstacles and noise. Especially relevant in this context is the non-line of sight (NLOS) scenario, in which no direct propagation path is possible between the nodes and consequently, the time observed for nodes separation does not match with the actual separation between nodes (i.e., the rectilinear trajectory). As a result, the actual distance is overestimated. Several proposals have been presented to reproduce the actual impact of NLOS conditions on the time (or distance) measurements. These proposals can be grouped in two classes: those models in which ranging errors depend on the actual range between nodes, and those that do not account for this dependence. All of these proposals have been used in research works indistinctly. However, as long as the distance between nodes increases, finding obstacles in the way is more likely to happen, and hence direct visibility between nodes is less probable. This article aims to evaluate the impact of the ranging-error model on the performance of time-based location techniques. Accordingly, different ranging error models, both accounting and not accounting for the actual distance between nodes, are used under the same scenarios so that their impact on the performance of two time-based techniques (e.g. TOA and passive-TDOA) can be quantified. The rest of the article is organized as follows. Section II presents a brief state of the art on ranging error modeling. Section III proposes several ranging models to compare and describes the scenarios simulated. Section IV shows the results, evaluating the impact of using each ranging model in the location technique under study. Finally, the main conclusions are drawn in section V.
II.
STATE OF THE ART ON RANGING MODELS
Several approaches have been followed to model the ranging error. All these models provide an expression to produce the errors on the ranging estimation as ,
(1)
where d is the actual distance between two nodes, is the estimation for such distance and er is the error on the distance estimation. Note that Equation 1 can be easily applied to time measurements if it is divided by the propagation speed (which is usually the speed of light). One of the simplest approaches consists of considering ranging-errors to be uniformly distributed, as proposed by Chan et al.[10]. However, this assumption seems to be far from reality. Accordingly, Alavi et al. [11] propose to model the ranging error according to the visibility conditions, which is a usual assumption. It consists of a sum of a Gaussian and exponential random variables in order to model both line-ofsight (LOS) and non-line-of-sight conditions. Specifically, the ranging error in LOS scenarios is modeled as a Gaussian distribution with a mean of 0 and a standard deviation of σ. In NLOS scenarios, the same error is modeled as a weighted sum of two random variables: a Gaussian component for the zeromean errors (mainly due to the measurement system), and an exponential variable which is responsible for the bias introduced by NLOS conditions. However, modeling NLOS errors as an exponential variable has a major disadvantage: it produces the “zero-effect,” which means that the probability of the direct path is very high. However, it is not likely to happen in the NLOS scenario, which the exponential errors are mainly addressed for. Alavi et al. and Xu et al. [12, 13] propose using a Gaussian distribution instead of an exponential to overcome this effect. All of these works are based on the common assumption that ranging error is a stationary random variable that does not depend on time or distance. However, recent works [12-16] show that more realistic error figures are achieved if the actual distance between nodes is considered in the computation of the ranging error, especially under NLOS conditions. Xu et al. [13] show that the dependence on the distance is modulated by the variance of the Gaussian component of the ranging model, i.e., the one associated with errors under LOS conditions. Marco et al. [14] evaluate a location technique using a ranging model in which NLOS is modeled as an exponential random variable whose mean value is proportional to the delay spread. The delay spread is represented as a lognormal variable whose median increases with the actual distance between nodes, following the model presented in [17]: .
(2)
T1 is the median delay spread at a distance of 1 km, d is the distance in kilometers and y is a lognormal variable with zero median and standard deviation σy. Thus, the further the nodes are from each other, the larger the ranging errors are. The same applies to [15], in which a fixed value for the delay spread is used (i.e., y random variable in Equation 2 is removed). A different approach is followed in [16], in which three scenarios are proposed for indoor environments: LOS, NLOS1
and NLOS2. NLOS1 and NLOS2 represent, respectively, light and hard signal propagation conditions for the NLOS environment. All of them are then adjusted according to empirical measures. Finally, the ranging model consists of a weighted sum of the three scenarios in which the weights depend on the probability of being in each of them. The dependence on the distance between nodes is set in these weights instead of each component. Thus, nodes becoming more separated involve a higher probability of being in a NLOS scenario, even though the NLOS component stays the same. A further step is presented in [12], where authors propose two scenarios for indoors: LOS and NLOS. The model associated with the LOS component presents a logarithmic dependence on the distance, whereas the model for the NLOS component is range-independent. The range error is a weighted combination of these two scenarios in which the probability of being in each scenario is again range-dependent. Accordingly, several solutions for the range-error modeling are proposed, but the impact of using different range-error models in the position computation is not explored. This work aims to study the differences in the accuracy simulated for two time-based techniques using different range-error models. The following sections provide further details on these models, the simulation conditions, and the results obtained. III.
SIMULATION AND SCENARIOS
Simulation has been chosen for evaluating the range models. Two location techniques (2-way TOA and passiveTDOA) are accounted for, fixing the positions [9]. The simulated scenario consists of four access points located at the corners of a squared area, which is a usual assumption. Accordingly, errors due to geometric dilution of precision are out of the scope of this work. Two terminals are emplaced in the simulation area, each of them using one of the techniques implemented (i.e., 2-way TOA and passive-TDOA). For both techniques, the Levenberg-Marquardt [18] least squares algorithm is used for fixing positions. Three ranging-error models are tested in this study: exponential range-independent, exponential range-dependent, and Gaussian range-dependent. Errors with the first model are computed as 0,
,
(3)
where 0, is a Gaussian random variable with zero-mean and standard deviation of σ; Exp(λ) is an exponential random variable with a mean of λ, and w1 and w2 are the weights applied to each component. Errors produced by the exponential range-dependent model are computed as 0,
,
(4)
where the average value β is computed as .
(5)
Finally, the Gaussian range-dependent ranging errors are computed as 0,
,
,
(6)
RMS of position in meters
RMS of position in meters
Range-independent Exponential Gaussian
RMS of the ranging errorr in meters b)
TOA accuracy with LM algorithm
RMS of position in meters
TOA accuracy with LLS algorrithm
RMS of position in meters
a)
RMS of the rangging error in meters
RMS of the ranging errorr in meters c)
RMS of the rangging error in meters
TDOA accuracy with LLS algoorithm
d)
TDOA accuracyy with LM algorithm
Figure 1. Accuracy achieved by TOA and passive TDO OA techniques with linear and non-linear least squares algorithms
where ρ is the coefficient of variation, i.e., the ratio between the standard deviation and the mean values. As it can be seen, all of these models differ only in the compponent that models the NLOS conditions. Therefore, the weightts w1 and w2 have been set to 0.26 and 0.74, respectively, givinng more relevance to the NLOS conditions [9]. Table I presennts the rest of the parameters of the ranging-error models sim mulated, where σy stands for the standard deviation of the lognoormal variable y in Equation 2. These values have been borroweed from [9, 12, 15] and adapted to indoor conditions (one of thee most constrained scenarios). Only the and parameters havve been modulated in order to change the RMS for the ranging error (through the u to 2 meters are T1 parameter). RMSs for the ranging error up considered, which are figures usually prooposed for UWB location systems working indoors. Monteecarlo simulations were run to compute the accuracy of TOA and a passive-TDOA using the ranging-models presented above. The T procedure used for those simulations is the same as in [9]. Simulations were carried out using different simulation areaas from 10 to 50 meters. Due to the length restrictions of the paper, p only results for scenarios with landmarks separated 20 are a included as an example of medium node coverage scenario.
IV.
ANCE ASSESSMENTS PERFORMA
This section aims to evaluuate the impact of the rangingerror model on the accuracy coomputed in common range-based techniques (e.g., TOA and TDOA). Figure 1 shows the evolution of the RMS of the poositioning error with the RMS of the ranging error. Exponential and Gaussian references in this figure stand for the distance-ddependent ranging-error models based on those random variabbles. Accuracy is computed by means of the Levenberg-Marqquardt (LM) non-linear squares algorithm, which is fed with a coarse position computed by means of a linear least squaress (LLS) algorithm. Both figures are included as references. Data in Fig. 1 show that foor the TOA technique, the range model used constrains the resuults on the accuracy. Confidence intervals at 95% for positioningg results are displayed as dashed lines for the TOA technique. Inn the case of TDOA, figures for the confidence interval are neegligible (less than 0.1% of the metric presented), and are hencce removed from pictures. When TABLE I. σ 0.0129 m
SIMU ULATION PARAMETERS
λ 0.1185 – 0.5561 m
T1 1 1.57 – 23.41 ns
ε 0.3
σy 4.0
ρ 0.94
Filtering probability Filtering probability
Range-independent Exponential Gaussian
RMS of the ranging error in meters m Figure 2. Filtering rate using LM algorithm in the passivve-TDOA
using TOA, the range-independent model leads l to optimistic results, i.e., it produces errors lower than thoose produced with range-dependent models, which are assumedd to produce range errors closer to reality [12-16]. Furthermoree, the higher RMS for the ranging error the larger difference between rangeindependent and dependent models. Only sligght differences can be found between Gaussian and exponentiaal range-dependent models, probably because these two modeels present similar statistics as they have the same mean andd nearly the same deviation (all of them are proportional to dε). ) Consequently, it can be stated that accuracy measured in term ms of the RMS of the error is not sensitive to the distributiion of the NLOS component of the range model. The same does not apply to passive-TDOA technique, in which the range-dependency seems not to impact on the accuracy results. In the case of the TDOA technique, onlyy slight differences can be found between range-dependent and range-independent models. This result is due mainly to the fact that passiveTDOA uses noisier data to fix the position, since the position of one of the landmark involves certain errorr. This error masks the impact of the ranging-error mode in the position A to be almost computation and leads to passive-TDOA insensitive to the model used. Results in the accuracy of the 2-way TOA algorithm using the LM algorithm are very close to those achhieved by means of the regular linear least squares algorithm. This similarity is TABLE II. Probability 66% (TOA) 66% (TDOA) 95% (TOA) 95% (TDOA)
LINEAR REGRESSION FOR RANGING ER RROR PERCENTILES Error Model Range-independent Exponential Gaussian Range-independent Exponential Gaussian Range-independent Exponential Gaussian Range-independent Exponential Gaussian
C Coefficients b0
- 0,0789 - 0,0757 - 0,0766 - 0,0306 - 0,0146 - 0,0202 - 0,1381 - 0,1598 - 0,0699 - 0,0788 - 0,0334 - 0,1752
b1
0,8771 0,9027 0,9821 0,8839 0,8916 0,9546 1,6871 1,9425 1,7544 1,7663 1,8206 1,9877
R²
0,9900 0,9933 0,9948 0,9975 0,9984 0,9990 0,9951 0,9958 0,9986 0,9927 0,9959 0,9835
because the observables are good enough to provide good results in both algorithms, soo that no clear differences are expected between them. The same does not apply to the passive-TDOA, in which positiions using LM algorithm are less stable than in the case of linear estimation. The main reason for that is that LM is fed with lineear estimations, which involve a considerable error. Accordinglyy, the LM algorithm is often not able to converge to the optimum solution. All positions involving an error larger thhan the distance between two landmarks (i.e., 20 meters) arre removed from the results in order to remove aberrant errorrs (i.e., outliers). This threshold corresponds to twice the errror of cell identification and accordingly it can be conssidered to be a conservative assumption. Fig. 2 shows the percentage of TDOA solutions T this percentage is zero for that have been rejected (with TOA all scenarios and all error models). The filtering rate is similar wever, the exponential rangewith the three models. How dependent model tends to diverrge more frequently as the RMS of the ranging error grows, whilst w the Gaussian and rangeindependent models grow smooother. The inclusion of range dependence does not seem to t impact dramatically on the stability of the positioning algorrithm. Two figures usually accoounted for location technique evaluation are the 66th and 95 9 th percentiles. Data regarding those percentiles seem to folloow a linear dependence with the RMS of the ranging error. Accordingly, A a linear regression over the data gathered was perfformed as ,
(7)
where RMSEp and RMSEr standd for the RMS of positioning and ranging errors respectively. Taable II shows the parameters for the linear regression as well as the coefficient of determination o the variability on the data is R2. As it can be seen, most of explained by the linear approoximation, which enforces the linear assumption taken for thee percentiles. As it can be seen, this linear behavior is not alterred by the ranging model used. However, differences betw ween models are detected. Specifically, range-dependent models provide higher figures for both percentiles than rangee-independent models, i.e., they provide more conservative values than the latter. This statement agrees with the resullts presented above for the RMS of the positioning error. Furthhermore, it is observed that the Gaussian model provides higheer figures for the 66th percentile in both techniques than the achhieved by the exponential model. On the other hand, figures for the 95th percentile indicate that the Gaussian model provides larger errors on the passivee model does the same TDOA technique whereas the exponential in the case of TOA technique. It can be said that in the case of the 95th percentile, the results are close enough to not exhibit significant differences. Finally, Fig. 3 shows an estimation of the probability density function of the positioning error for the three error t the value gathered. All three models simulated, according to models tend to follow the samee shape in the probability density function. However, differencess appear if range-dependent and independent models are compaared. Range independent model tend to concentrate the posittioning error on lower values, especially in the case of TOA, with w noticeable variations on the probability. The same does not apply to range-dependent
Empiric density function
Empiric density function
Range-independent Exponential Gaussian
Positioning errorr
Positioningg error
Figure 3. Empirical density distribution of the position errror in the 2-wayTOA (left) and passive-TDOA (right) techniques
models, whose probability function is smoothher and distributes the positioning error with a longer queue. Thhis result reinforces what has been already observed: the range-independent model leads to more optimistic results than range-dependent ones and consequently, its use is discouraged for perfoormance evaluation purposes in favor of those accounting forr the actual range between nodes. V.
[5]
[6]
[7]
CONCLUSIONS
This article presents the comparison between different ranging error models applied in the siimulation of two positioning techniques based in time measurements (2-way a tested, focusing TOA and passive-TDOA). Several models are the study on the difference between range-dependent and range-independent models. Models have been adapted to indoor environments, and Montecarlo simulaations were run in order to gather positions fixed through TOA T and passiveTDOA techniques. Results show that possitioning errors in range-independent models tend to be optim mistic if compared with those obtained by means of range-ddependent models, leading to a performance better than what it should have been. This result is especially noticeable in the casee of positions fixed by means of TOA techniques; positions compputed by means of passive-TDOA do not seem to be as sensitive to the ranging model. Given that range-dependence is noot only a intuitive assumption, but is also supported by field studies, it can be concluded that range-dependent models woulld make it possible to evaluate positioning algorithms more reealistically and to avoid an overestimation of their performancee, at the expense of a minor increase of the computational cost.
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
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[3]
[4]
I. Stojmenovic, “Position-based routing in ad hoc networks”, IEEE Communications Magazine, vol. 40, no. 7. pp. 1288-134, July 2002. M. Brunato, R. Battiti, “Statistical learning theory for location fingerprinting in wireless LANs”, Elsevier Compuuter Networks, vol. 47, issue 6, pp. 825-845, November 2004. Widyawan, M. Klepal, D. Pesch, “Influence of Prredicted and Measured Fingerprint on the Accuracy of RSSI-based Indooor Location Systems”, 4th Workshop on Positioning, Navigation and Com mmunication (WPNC), pp. 145-151, March 2007. A. Hatami, K. Pahlavan, “Performance Compariison of RSS and TOA Indoor Geolocation Based on UWB Measuurement of Channel Characteristics”, IEEE 17th International Sym mposium on Personal, Indoor and Mobile Radio Communications, pp. 1-6, September 2006.
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K. Yu, I. Oppermann, “Performaance of UWB position estimation based on time-of-arrival measurementts”, International Workshop on Ultra Wideband Systems 2004, pp. 4000-404, May 2004. S. Venkatraman, J.Jr. Caffery,, “Hybrid TOA/AOA techniques for mobile location in non-line-off-sight environments”, IEEE Wireless Communications and Networkingg Conference (WCNC 2004), vol. 1, pp. 274-278, March 2004. M. Ciurana, F. Barcelo-Arroyo, F. Izquierdo, “A ranging system with IEEE 802.11 data frames”, Wireeless Communications and Networking Conference (WCNC), pp. 2092-22096, March 2007. D. Kang, Y. Namgoong, S. Yang, Y S. Choi, Y. Shin, “A simple asynchronous UWB position loccation algorithm based on single roundtrip transmission”, International Conference on Advanced ACT). vol. 3, pp. 20-22, 2006. Communication Technology (ICA I. Martin-Escalona, Francisco Barcelo-Arroyo, “A new time-based algorithm for positioning mobbile terminals in wireless networks”, EURASIP Journal on Advances in Signal Processing, vol. 2008, pp. 110, 2008. Yiu-Tong Chan, Wing-Yue Tsuui, Hing-Cheung So, Pak-chung Ching, “Time-of-Arrival Based Localizzation Under NLOS Conditions”, IEEE Transactions On Vehicular Technnology, vol. 55, no. 1, January 2006 Bardia Alavi, Kaveh Pahlavan, “Modeling of the Distance Error for Indoor Geolocation”, IEEE Wireeless Communications and Networking Conference (WCNC), vol. 1, pp. 668-672, March 2003. Bardia Alavi, Kaveh Pahlavan, “Modeling of the TOA-based Distance B Indoor Radio Measurements”, IEEE Measurement Error Using UWB Communications Letters, vol. 10, No. 4, April 2006. Jun Xu, Maode Ma, Choi Lookk Law, “Theoretical Lower Bound for UWB TDOA Positioning”, IEEE Global Telecommunications Conference (Globecom), pp. 41001-4105, Nov. 2007. A. Marco, R. Casas, A. Asensio, V. Coarasa, R. Blasco, A. Ibarz, “Least Median of Squares for Non-Liine-of-Sight Error Mitigation in GSM Localization”, IEEE Personnal, Indoor and Mobile Radio Communications (PIMRC), pp. 1-5, 1 Sept. 2008. Jens Schroeder, Stefan Galler, Kyandoghere K Kyamakya, Klaus Jobmann ,“NLOS detection algorithmss for Ultra-Wideband localization”, Positioning, Navigation and Communication, 4th Workshop on Positioning, Navigation and Coommunication (WPNC ), pp. 159-166, March 2007. B. Denis, N. Daniele,“NLOS Raanging Error Mitigation in a Distributed Positioning Algorithm for Indooor UWB Ad-Hoc Networks” , IEEE International Workshop on Wireeless Ad-Hoc Networks, pp. 356- 360, June 2004. Larry J. Greenstein, Vinko Ercegg, Yu Shuan Yeh, Martin V. Clark, “A New Path-Gain/Delay-Spread Propagation P Model for Digital Cellular Channels”, IEEE Transactions On O Vehicular Technology, vol. 46, no. 2, May 1997. D.W. Marquardt, “An Algoritthm for Least-Squares Estimation of Nonlinear Parameters”, SIAM Joournal on Applied Mathematics, vol. 11, no. 2. pp. 431-441, 1963.
Israel Martin-Escalona
Francisco Barcelo-Arroyo
Department of Telematic Engineering Universitat Politecnica de Catalunya (UPC) Barcelona, Spain [email protected]
Department of Telematic Engineering Universitat Politecnica de Catalunya (UPC) Barcelona, Spain [email protected]
Department of Telematic Engineering Universitat Politecnica de Catalunya (UPC) Barcelona, Spain [email protected]
Abstract— Wireless networks have boosted the interest of users and network operators on location due to the possible synergies between the cellular layout and location techniques. As a result, location methods have been proposed to provide this information in most wireless technologies. Evaluation of the performance of these and further techniques is initially achieved by means of simulation. It is essential to use suitable models for featuring errors that are present in observations. This characterization is especially relevant in time-based techniques such as time-ofarrival (TOA) and time-difference-of-arrival (TDOA). These techniques are significantly impacted by non-line-of-sight conditions; hence their performance is usually evaluated under these conditions. This article compares several ranging-error models adapted to indoors and studies their impact on the performance of time-based techniques (e.g., TOA and TDOA). Results indicate that simple models that are not based on the actual distance between nodes tend to underestimate the positioning error with respect to more complex models. Keywords; ranging, error, TOA, TDOA, NLOS, indoor positioning.
I.
INTRODUCTION
Mobile ad hoc networks (MANETs) have been a focus of the industry, mainly due to their dynamism and selfconfiguring capabilities. The mobility feature of these networks represents a challenge in the design of protocols since nodes can change their position and can associate or leave the network suddenly. Accordingly, many efforts have been addressed to provide MANETs with location solutions that are able to fix the position of all nodes. This knowledge allows the nodes to access a huge range of location-based services. Furthermore, knowing the position allows several operational and maintenance tasks to be optimized. For instance, routing protocols are scalable only if the positions of the nodes are known [1]. Consequently, there is increasing interest in the research of location solutions for MANETs, which aims to provide accurate enough positions for the scenarios they usually are deployed in. Currently, there are several proposals for positioning ad hoc nodes, including fingerprinting [2-3], schemes based on Ultra Wide Band [4-5], and those based on the angle-of-arrival [6]. Time-based techniques seem to be preferred in the ad hoc field,
This research was partially funded by ERDF and the Spanish Government through project TEC2006-09466/TCM.
mainly due to the good trade-off between accuracy, scalability and deployment/maintenance costs. This article is focused on two of these time-based techniques: TOA [7-8] and passiveTDOA [9], which observe the time spent by signals in traveling certain distances and fix the position by means of multilateration algorithms. These techniques have the advantage that they usually do not need specific hardware or off-line calibration steps, which are required in fingerprinting. However, these time measures do not always give a precise estimate of the distance between two nodes, since signals are impacted by several sources of error. These sources of error include dense multi-path propagation, obstacles and noise. Especially relevant in this context is the non-line of sight (NLOS) scenario, in which no direct propagation path is possible between the nodes and consequently, the time observed for nodes separation does not match with the actual separation between nodes (i.e., the rectilinear trajectory). As a result, the actual distance is overestimated. Several proposals have been presented to reproduce the actual impact of NLOS conditions on the time (or distance) measurements. These proposals can be grouped in two classes: those models in which ranging errors depend on the actual range between nodes, and those that do not account for this dependence. All of these proposals have been used in research works indistinctly. However, as long as the distance between nodes increases, finding obstacles in the way is more likely to happen, and hence direct visibility between nodes is less probable. This article aims to evaluate the impact of the ranging-error model on the performance of time-based location techniques. Accordingly, different ranging error models, both accounting and not accounting for the actual distance between nodes, are used under the same scenarios so that their impact on the performance of two time-based techniques (e.g. TOA and passive-TDOA) can be quantified. The rest of the article is organized as follows. Section II presents a brief state of the art on ranging error modeling. Section III proposes several ranging models to compare and describes the scenarios simulated. Section IV shows the results, evaluating the impact of using each ranging model in the location technique under study. Finally, the main conclusions are drawn in section V.
II.
STATE OF THE ART ON RANGING MODELS
Several approaches have been followed to model the ranging error. All these models provide an expression to produce the errors on the ranging estimation as ,
(1)
where d is the actual distance between two nodes, is the estimation for such distance and er is the error on the distance estimation. Note that Equation 1 can be easily applied to time measurements if it is divided by the propagation speed (which is usually the speed of light). One of the simplest approaches consists of considering ranging-errors to be uniformly distributed, as proposed by Chan et al.[10]. However, this assumption seems to be far from reality. Accordingly, Alavi et al. [11] propose to model the ranging error according to the visibility conditions, which is a usual assumption. It consists of a sum of a Gaussian and exponential random variables in order to model both line-ofsight (LOS) and non-line-of-sight conditions. Specifically, the ranging error in LOS scenarios is modeled as a Gaussian distribution with a mean of 0 and a standard deviation of σ. In NLOS scenarios, the same error is modeled as a weighted sum of two random variables: a Gaussian component for the zeromean errors (mainly due to the measurement system), and an exponential variable which is responsible for the bias introduced by NLOS conditions. However, modeling NLOS errors as an exponential variable has a major disadvantage: it produces the “zero-effect,” which means that the probability of the direct path is very high. However, it is not likely to happen in the NLOS scenario, which the exponential errors are mainly addressed for. Alavi et al. and Xu et al. [12, 13] propose using a Gaussian distribution instead of an exponential to overcome this effect. All of these works are based on the common assumption that ranging error is a stationary random variable that does not depend on time or distance. However, recent works [12-16] show that more realistic error figures are achieved if the actual distance between nodes is considered in the computation of the ranging error, especially under NLOS conditions. Xu et al. [13] show that the dependence on the distance is modulated by the variance of the Gaussian component of the ranging model, i.e., the one associated with errors under LOS conditions. Marco et al. [14] evaluate a location technique using a ranging model in which NLOS is modeled as an exponential random variable whose mean value is proportional to the delay spread. The delay spread is represented as a lognormal variable whose median increases with the actual distance between nodes, following the model presented in [17]: .
(2)
T1 is the median delay spread at a distance of 1 km, d is the distance in kilometers and y is a lognormal variable with zero median and standard deviation σy. Thus, the further the nodes are from each other, the larger the ranging errors are. The same applies to [15], in which a fixed value for the delay spread is used (i.e., y random variable in Equation 2 is removed). A different approach is followed in [16], in which three scenarios are proposed for indoor environments: LOS, NLOS1
and NLOS2. NLOS1 and NLOS2 represent, respectively, light and hard signal propagation conditions for the NLOS environment. All of them are then adjusted according to empirical measures. Finally, the ranging model consists of a weighted sum of the three scenarios in which the weights depend on the probability of being in each of them. The dependence on the distance between nodes is set in these weights instead of each component. Thus, nodes becoming more separated involve a higher probability of being in a NLOS scenario, even though the NLOS component stays the same. A further step is presented in [12], where authors propose two scenarios for indoors: LOS and NLOS. The model associated with the LOS component presents a logarithmic dependence on the distance, whereas the model for the NLOS component is range-independent. The range error is a weighted combination of these two scenarios in which the probability of being in each scenario is again range-dependent. Accordingly, several solutions for the range-error modeling are proposed, but the impact of using different range-error models in the position computation is not explored. This work aims to study the differences in the accuracy simulated for two time-based techniques using different range-error models. The following sections provide further details on these models, the simulation conditions, and the results obtained. III.
SIMULATION AND SCENARIOS
Simulation has been chosen for evaluating the range models. Two location techniques (2-way TOA and passiveTDOA) are accounted for, fixing the positions [9]. The simulated scenario consists of four access points located at the corners of a squared area, which is a usual assumption. Accordingly, errors due to geometric dilution of precision are out of the scope of this work. Two terminals are emplaced in the simulation area, each of them using one of the techniques implemented (i.e., 2-way TOA and passive-TDOA). For both techniques, the Levenberg-Marquardt [18] least squares algorithm is used for fixing positions. Three ranging-error models are tested in this study: exponential range-independent, exponential range-dependent, and Gaussian range-dependent. Errors with the first model are computed as 0,
,
(3)
where 0, is a Gaussian random variable with zero-mean and standard deviation of σ; Exp(λ) is an exponential random variable with a mean of λ, and w1 and w2 are the weights applied to each component. Errors produced by the exponential range-dependent model are computed as 0,
,
(4)
where the average value β is computed as .
(5)
Finally, the Gaussian range-dependent ranging errors are computed as 0,
,
,
(6)
RMS of position in meters
RMS of position in meters
Range-independent Exponential Gaussian
RMS of the ranging errorr in meters b)
TOA accuracy with LM algorithm
RMS of position in meters
TOA accuracy with LLS algorrithm
RMS of position in meters
a)
RMS of the rangging error in meters
RMS of the ranging errorr in meters c)
RMS of the rangging error in meters
TDOA accuracy with LLS algoorithm
d)
TDOA accuracyy with LM algorithm
Figure 1. Accuracy achieved by TOA and passive TDO OA techniques with linear and non-linear least squares algorithms
where ρ is the coefficient of variation, i.e., the ratio between the standard deviation and the mean values. As it can be seen, all of these models differ only in the compponent that models the NLOS conditions. Therefore, the weightts w1 and w2 have been set to 0.26 and 0.74, respectively, givinng more relevance to the NLOS conditions [9]. Table I presennts the rest of the parameters of the ranging-error models sim mulated, where σy stands for the standard deviation of the lognoormal variable y in Equation 2. These values have been borroweed from [9, 12, 15] and adapted to indoor conditions (one of thee most constrained scenarios). Only the and parameters havve been modulated in order to change the RMS for the ranging error (through the u to 2 meters are T1 parameter). RMSs for the ranging error up considered, which are figures usually prooposed for UWB location systems working indoors. Monteecarlo simulations were run to compute the accuracy of TOA and a passive-TDOA using the ranging-models presented above. The T procedure used for those simulations is the same as in [9]. Simulations were carried out using different simulation areaas from 10 to 50 meters. Due to the length restrictions of the paper, p only results for scenarios with landmarks separated 20 are a included as an example of medium node coverage scenario.
IV.
ANCE ASSESSMENTS PERFORMA
This section aims to evaluuate the impact of the rangingerror model on the accuracy coomputed in common range-based techniques (e.g., TOA and TDOA). Figure 1 shows the evolution of the RMS of the poositioning error with the RMS of the ranging error. Exponential and Gaussian references in this figure stand for the distance-ddependent ranging-error models based on those random variabbles. Accuracy is computed by means of the Levenberg-Marqquardt (LM) non-linear squares algorithm, which is fed with a coarse position computed by means of a linear least squaress (LLS) algorithm. Both figures are included as references. Data in Fig. 1 show that foor the TOA technique, the range model used constrains the resuults on the accuracy. Confidence intervals at 95% for positioningg results are displayed as dashed lines for the TOA technique. Inn the case of TDOA, figures for the confidence interval are neegligible (less than 0.1% of the metric presented), and are hencce removed from pictures. When TABLE I. σ 0.0129 m
SIMU ULATION PARAMETERS
λ 0.1185 – 0.5561 m
T1 1 1.57 – 23.41 ns
ε 0.3
σy 4.0
ρ 0.94
Filtering probability Filtering probability
Range-independent Exponential Gaussian
RMS of the ranging error in meters m Figure 2. Filtering rate using LM algorithm in the passivve-TDOA
using TOA, the range-independent model leads l to optimistic results, i.e., it produces errors lower than thoose produced with range-dependent models, which are assumedd to produce range errors closer to reality [12-16]. Furthermoree, the higher RMS for the ranging error the larger difference between rangeindependent and dependent models. Only sligght differences can be found between Gaussian and exponentiaal range-dependent models, probably because these two modeels present similar statistics as they have the same mean andd nearly the same deviation (all of them are proportional to dε). ) Consequently, it can be stated that accuracy measured in term ms of the RMS of the error is not sensitive to the distributiion of the NLOS component of the range model. The same does not apply to passive-TDOA technique, in which the range-dependency seems not to impact on the accuracy results. In the case of the TDOA technique, onlyy slight differences can be found between range-dependent and range-independent models. This result is due mainly to the fact that passiveTDOA uses noisier data to fix the position, since the position of one of the landmark involves certain errorr. This error masks the impact of the ranging-error mode in the position A to be almost computation and leads to passive-TDOA insensitive to the model used. Results in the accuracy of the 2-way TOA algorithm using the LM algorithm are very close to those achhieved by means of the regular linear least squares algorithm. This similarity is TABLE II. Probability 66% (TOA) 66% (TDOA) 95% (TOA) 95% (TDOA)
LINEAR REGRESSION FOR RANGING ER RROR PERCENTILES Error Model Range-independent Exponential Gaussian Range-independent Exponential Gaussian Range-independent Exponential Gaussian Range-independent Exponential Gaussian
C Coefficients b0
- 0,0789 - 0,0757 - 0,0766 - 0,0306 - 0,0146 - 0,0202 - 0,1381 - 0,1598 - 0,0699 - 0,0788 - 0,0334 - 0,1752
b1
0,8771 0,9027 0,9821 0,8839 0,8916 0,9546 1,6871 1,9425 1,7544 1,7663 1,8206 1,9877
R²
0,9900 0,9933 0,9948 0,9975 0,9984 0,9990 0,9951 0,9958 0,9986 0,9927 0,9959 0,9835
because the observables are good enough to provide good results in both algorithms, soo that no clear differences are expected between them. The same does not apply to the passive-TDOA, in which positiions using LM algorithm are less stable than in the case of linear estimation. The main reason for that is that LM is fed with lineear estimations, which involve a considerable error. Accordinglyy, the LM algorithm is often not able to converge to the optimum solution. All positions involving an error larger thhan the distance between two landmarks (i.e., 20 meters) arre removed from the results in order to remove aberrant errorrs (i.e., outliers). This threshold corresponds to twice the errror of cell identification and accordingly it can be conssidered to be a conservative assumption. Fig. 2 shows the percentage of TDOA solutions T this percentage is zero for that have been rejected (with TOA all scenarios and all error models). The filtering rate is similar wever, the exponential rangewith the three models. How dependent model tends to diverrge more frequently as the RMS of the ranging error grows, whilst w the Gaussian and rangeindependent models grow smooother. The inclusion of range dependence does not seem to t impact dramatically on the stability of the positioning algorrithm. Two figures usually accoounted for location technique evaluation are the 66th and 95 9 th percentiles. Data regarding those percentiles seem to folloow a linear dependence with the RMS of the ranging error. Accordingly, A a linear regression over the data gathered was perfformed as ,
(7)
where RMSEp and RMSEr standd for the RMS of positioning and ranging errors respectively. Taable II shows the parameters for the linear regression as well as the coefficient of determination o the variability on the data is R2. As it can be seen, most of explained by the linear approoximation, which enforces the linear assumption taken for thee percentiles. As it can be seen, this linear behavior is not alterred by the ranging model used. However, differences betw ween models are detected. Specifically, range-dependent models provide higher figures for both percentiles than rangee-independent models, i.e., they provide more conservative values than the latter. This statement agrees with the resullts presented above for the RMS of the positioning error. Furthhermore, it is observed that the Gaussian model provides higheer figures for the 66th percentile in both techniques than the achhieved by the exponential model. On the other hand, figures for the 95th percentile indicate that the Gaussian model provides larger errors on the passivee model does the same TDOA technique whereas the exponential in the case of TOA technique. It can be said that in the case of the 95th percentile, the results are close enough to not exhibit significant differences. Finally, Fig. 3 shows an estimation of the probability density function of the positioning error for the three error t the value gathered. All three models simulated, according to models tend to follow the samee shape in the probability density function. However, differencess appear if range-dependent and independent models are compaared. Range independent model tend to concentrate the posittioning error on lower values, especially in the case of TOA, with w noticeable variations on the probability. The same does not apply to range-dependent
Empiric density function
Empiric density function
Range-independent Exponential Gaussian
Positioning errorr
Positioningg error
Figure 3. Empirical density distribution of the position errror in the 2-wayTOA (left) and passive-TDOA (right) techniques
models, whose probability function is smoothher and distributes the positioning error with a longer queue. Thhis result reinforces what has been already observed: the range-independent model leads to more optimistic results than range-dependent ones and consequently, its use is discouraged for perfoormance evaluation purposes in favor of those accounting forr the actual range between nodes. V.
[5]
[6]
[7]
CONCLUSIONS
This article presents the comparison between different ranging error models applied in the siimulation of two positioning techniques based in time measurements (2-way a tested, focusing TOA and passive-TDOA). Several models are the study on the difference between range-dependent and range-independent models. Models have been adapted to indoor environments, and Montecarlo simulaations were run in order to gather positions fixed through TOA T and passiveTDOA techniques. Results show that possitioning errors in range-independent models tend to be optim mistic if compared with those obtained by means of range-ddependent models, leading to a performance better than what it should have been. This result is especially noticeable in the casee of positions fixed by means of TOA techniques; positions compputed by means of passive-TDOA do not seem to be as sensitive to the ranging model. Given that range-dependence is noot only a intuitive assumption, but is also supported by field studies, it can be concluded that range-dependent models woulld make it possible to evaluate positioning algorithms more reealistically and to avoid an overestimation of their performancee, at the expense of a minor increase of the computational cost.
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
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