Elsevier

Signal Processing

Volume 108, March 2015, Pages 498-508
Signal Processing

Accuracy bounds of non-Gaussian Bayesian tracking in a NLOS environment

https://doi.org/10.1016/j.sigpro.2014.10.025Get rights and content

Highlights

  • Complex signal modeling that utilizes a number of assumptions is described.

  • Localization algorithm for nanoLOC technology is proposed.

  • Approximation of an empirical probability function for measured distance and RSS is described thoroughly.

  • The knowledge of non-line of sight case for a given measured distance based on RSS value increases localization accuracy.

Abstract

With a growth of the popularity of wireless sensor networks it has became obvious that Bayesian filter is most commonly used method for the sensor localization. The multiple sensors in one device allow to build different variations of the filter through the definition of the components of Bayesian rule. This paper presents a localization algorithm that is based on Bayesian filter and an attempt to evaluate the accuracy of the algorithm for nanoLOC technology. Empirical probability density functions for two Non-Line of Sight (NLOS) cases and different mobile object movement models have been used in the algorithm. The usage of extreme variants of the movement model allows to estimate the lower and upper bounds of the localization algorithm accuracy. Any movement sensor data incorporated into the algorithm produces the algorithm variant where accuracy is within the bounds. Also, the paper presents a technique for recovering the probability density function of distance overestimates from biased measurements and proposes relative accuracy estimate based on the least squares method. The estimate can be used instead of Cramer–Rao lower bound when there is no analytical probability density function.

Introduction

The last decade has shown the growth in popularity of sensor networks in its ability to locate sensors in different circumstances. This interest is fueled by multiple applications of sensor networks in different human activities [1].

One of the central problems in sensor networks is the localization of unknown positions of nodes. This paper presents centralized localization algorithm based on known distributions of sensor measurements.

Many different approaches have been proposed to locate unknown node positions. The multiplicity of these approaches could be explained by different physical features of the sensor networks and by the environments where the networks are deployed. All approaches can be grouped as follows:

  • anchor nodes presence (nodes with known position);

  • location technique: connectivity information, Received Signal Strength (RSS), Angle of Arrival (AoA), Time of Flight (ToF) (Time of Arrival (ToA), Round-trip Time of Flight (RToF), Time Difference of Arrival (TDoA) [2];

  • environment properties that cause signal distortion: Line of Sight (LOS), Non-line of Sight (NLOS), etc;

  • localization algorithm implementation: distributed, centralized [3];

  • method to account the measurement error: global optimization, error distribution accounting [4], [5], [6], [7], [1].

Numerous investigations have focused on the prior knowledge of measurement error distributions to build a localization method. In [7] Savvides et al. perform detailed analysis of error impact on localization accuracy using Gaussian probability density function (PDF) for distance measurements. Although Gaussian PDF is a natural assumption for approximating measurements and have been widely used in literature [7], [8], [5], [9], [10], real distributions not always comply with normal distribution. In [5], authors eliminate this imprecision in an error PDF model by adding mixture distribution with a weight to Gaussian. However, this adapted model probably will not be fit in different environments and sensor hardware. The reason to use any kind of analytical model for PDF is due to the limited sensor network capacity. Distributed algorithms commonly transfer only parameters of a model that are sufficient to reproduce the distribution at the nodes. Centralized algorithms do not suffer from this limitation.

Also, analytical model can be used to estimate Cramer–Rao Lower Bound (CRLB). However, there are methods to estimate the relative localization accuracy without the requirement to have an analytical model. One of such methods is proposed in this paper.

A number of researchers have shown improvements in the location accuracy when multiple measurements based on different techniques are fused together [11], [12], [13], [14], [15]. Design of localization systems where multiple techniques are used is complex enough. Hence, we suggest that a common protocol between the system components is required to reuse different localization algorithms and to provide flexibility of fusing products of different techniques. The most common format for localization algorithm׳s product presentation is a PDF based on two- or three-dimensional Euclidean space. Bayesian filter is a good model for location estimate when there are data from multiple sensors – it fuses PDFs of different sensors in a natural manner.

This paper presents a localization algorithm of one of the modules of RealTrac™localization system that performs calculation of PDF of the mobile node location in two-dimensional Euclidean space using prior PDF of measurements error [16]. RealTrac™is based on nanoLOC hardware technology that provides distance and RSS measurements. The measurements are processed between anchor nodes with known locations and mobile nodes using RToF technique [17]. The algorithm deals with RSS measurements to make a decision – whether there are NLOS or LOS cases. For NLOS and LOS cases, different table-based PDF of anchor-to-node measurements are used.

NLOS cases have a number of gradations that depend on the wall/floor type. In [5], authors also apply different distributions for plasterboard and concrete walls. It was found that only two gradations – NLOS and LOS – are sufficient to noticeably improve the localization accuracy. Adding more gradations does not improve the accuracy as much as the first two. Therefore, this paper concentrates only on NLOS and LOS (NLOS-cases). The novelty of the proposed algorithm is that the algorithm takes into account RSS values to define the probability of a given NLOS-case.

Depending on different mobile node movement models a localization algorithm can produce location estimates of a different accuracy. The lower and upper bounds for a localization accuracy are derived in this paper for the localization algorithms that are based on Bayesian rule.

Section 2 defines the localization problem in terms of probability theory. Section 3 describes a number of assumptions that can be admitted to simplify the localization algorithm. PDF approximation technique is presented in Section 4. After that, the mobile motion model is derived in Section 5 that provides localization error bounds for the algorithm. Section 6 describes a numerical solution for location calculation using theoretical formulas. 7 Least-squares localization approach, 8 Relative localization accuracy estimate recall for least-squares method and shows how it can be used to relatively estimate a localization accuracy of any Bayesian-based algorithm. Finally, the experimental data description and results of localization accuracy estimation are provided in Section 9 with corresponding discussion. Conclusions are presented in the last section.

Section snippets

Localization problem and Bayesian filter

The common localization problem can be defined as follows. Any localization system as a dynamic system can be presented as Fig. 1. The system input here is unknown (mobile) node position X=(x,y)T and the system output is a number of measurements Y=(Y1,Y2,,Yi,Ym) on m anchor points. Yi=(y1,y2,,yk)T is a complex measured signal in i-th anchor point; k is a number of different sensors. yj components can be a distance, time-of-flight (ToF), a radio signal strength (RSS), an angle of arrival or

NLOS incorporation technique

In RealTrac™system anchor points measure a received signal strength (RSS) rss and distance d^ to a mobile node [16]. Hence, Y can be defined as Y=(rss,d^)T. The direct way to calculate f(Y|X) in (1) is to create an empirical PDF for all possible X in a full localization area LX. This can take a lot of human resources and requires to reestimate f(Y|X) for each new environment. Therefore, this method is not efficient, but is the most precise in a set of methods based on Bayesian formula (1). We

PDF approximation technique

The functions fη(Δd) and f(rss,d) are approximated with empirical functions f^η(Δd) and f^(rss,d) that are based on collected data (see Fig. 2, Fig. 4, Fig. 5). Due to known access point positions (si) and mobile node positions, d and Δd can be calculated and utilized to form these functions. More details on how to derive fη(Δd) are provided in [20].

Empirical function f^(rss,d) is derived as follows. A number of intervals I={I1,I2,,Ik,} are constructed from RSS sample values rssij:rssmin=mini,

Mobile node motion model

f(X) in (1) defines a prior known motion model of a mobile node. If location measurements are obtained in a sequence, then f(X) can be based on the product of location estimation on a previous step. Knowing f(X|Yj1) values for each point XAj1R2 of a localization area and speed model fv(ΔX)=deffv:R2R,f(X) on j-th step can be defined asfj(X)=(fvf(·|Yj1))(X),where ⁎ is a convolution operator(fg)(X)=defAjf(XX)g(X)dX.Here, the speed model is defined as a function fv(ΔX) that is

Numerical location calculation

Formula (1) is a general form of the localization problem in terms of probability theory. Eqs. (11), (14) are theoretically derived functions with a number of assumptions accepted. If fη(Δd) and fη(rss,d) in (11) are defined analytically, then the objective function (2) can be directly solved. Alternatively, in practice numerical analysis allows to find the optimum without directly solving the objective function. Numerical algorithm iterates over all possible solutions X and compares values of

Least-squares localization approach

Least-squares approach applied to the localization problem minimizes next objective function:z=i=1m(d^idi)2=iΔdi2where d^i is a measurement of a true distance di.

If a localization area is bounded by m circles, then, in most of the cases, z has the minimum value on the border of the localization area due to the nonobservance of one of the required conditions of the original LS-approach. According to Gauss–Markov theorem in a linear regression model the errors must have expectation zero and be

Relative localization accuracy estimate

To compare different localization algorithms it is required to comply with equal conditions. Ideally, all algorithms should use equal traces and measurement sets. Naturally, equal sets have been used for comparing different variants of the algorithm. However, to compare the results with results in other papers, a standardized estimate is needed. One of such estimates is Cramer–Rao Lower Bound (CRLB). Authors often use CRLB if PDFs are defined analytically. Alternatively, we propose a relative

Experimental data

Sample data was collected on the second floor of the IT-park building (Russia, Petrozavodsk, Lenina st., 31) using RealTrac™ localization system [16]. Data was sampled at 49 points (see black dots in Fig. 12) and consists of measurements from 9 anchor points (see squares with “X” inside in Fig. 12). 100–200 measurement sets were collected for each point. Each set consists of measurements from 3–7 access point. Each access point measurement consists of a pair – RSS and distance to mobile node

Conclusion

Bayes formula (1) allows to produce different custom localization algorithms through specifying its two components f(Y|X) and f(X). For nanoLOC technology this paper describes a localization algorithm that is based on (1) and a number of the independence of anchor measurements, the independence of a distance overestimates on a true distance and that LOS-case depends only on a signal pathloss. Also, it has been defined the way to measure the algorithm accuracy bounds for different mobile object

Acknowledgment

The research work described in this publication was ordered by the RTL-Service JSC, and was supported by the Petrozavodsk State University (within the program of the Strategic Development Program of PetrSU) and the Ministry for Education and Science of the Russian Federation (Contract 14.574.21.0059) and the Ministry for Economic Development of the Republic of Karelia (Russian Federation).

References (22)

  • I. Akyildiz et al.

    Wireless sensor networksa survey

    Comput. Netw.

    (2002)
  • M. Vemula et al.

    Sensor self-localization with beacon position uncertainty

    Signal Process.

    (2009)
  • C.-Y. Chong et al.

    Sensor networksevolution, opportunities, and challenges

    Proc. IEEE

    (2003)
  • M. Vossiek et al.

    Wireless local positioning

    IEEE Microw. Mag.

    (2003)
  • A. Savvides, H. Park, M.B. Srivastava, The bits and flops of the n-hop multilateration primitive for node localization...
  • B. Denis, N. Daniele, Nlos ranging error mitigation in a distributed positioning algorithm for indoor uwb ad-hoc...
  • D. Meger, D. Marinakis, I. Rekleitis, G. Dudek, Inferring a probability distribution function for the pose of a sensor...
  • A. Savvides et al.

    An analysis of error inducing parameters in multihop sensor node localization

    IEEE Trans. Mobile Comput.

    (2005)
  • B. Alavi, K. Pahlavan, Modeling of the distance error for indoor geolocation, in: Wireless Communications and...
  • B. Alavi, K. Pahlavan, Bandwidth effect on distance error modeling for indoor geolocation, in: 14th IEEE Proceedings on...
  • J. Parviainen, J. Kantola, J. Collin, Differential barometry in personal navigation, in: Position, Location and...
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