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Procedía Computer Science 34 (2014) 633 - 638

International Symposium on Emerging Inter-networks, Communication and Mobility

(EICM-2014)

A New RWLS Positioning Method Basing on Distance Information

Hong Dinga*, Lingling Heb, Linhua Zhenga, Jibing Yuana, Liangjun Xianga

aCollege of Electronic Science and Engineering bLibrary National University of Defense Technology, Changsha, Hunan 410073, P.R.China

Abstract

In ultra-wideband (UWB) localization, the ranging precision will determine the localization performance directly. However not all the range measurements(RMs) have the same precision. In this paper, we propose a new weighted least square(WLS) positioning method, which will design the weighted values of each RM according to its reliability. We show the ranging reliability is closely related to the distance between the mobile station(MS) and anchor nodes(ANs). Moreover, if design each weighted value fusing distance with non-line-of-sight(NLOS) identification result, the localization performance will be enhanced. Simulation results show our method performs better than those commonly used WLS methods. © 2014 Publishedby ElsevierB.V.This is anopen access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).

Selection and peer-review under responsibility of Conference Program Chairs Keywords: UWB; localization; WLS

1. Introduction

Impulse radio UWB signal has extremely wide bandwidth, and its pulse duration can be shortened to nanoseconds, even subnanoseconds. The narrow pulse duration provides high time resolution, so ranging methods basing on time, such as time of arrival(TOA) and time difference of arrival(TDOA), can realize high precision and have been widely used1-3. Once the MS gets the necessary amount of RMs, it can perform localization using LS algorithms. However the localization performance is closely related to the ranging accuracy, and not all the RMs have the same reliability. So in order to enhance the LS regression performances, WLS solution can be used4-6. The weighted values designation is a hot researched problem of WLS algorithms. Many methods design the weighted values according to the reliabilities of RMs.

* Corresponding author. fax: +86_731_84518426. E-mail address: susan_dh@sina.com

1877-0509 © 2014 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/3.0/).

Selection and peer-review under responsibility of Conference Program Chairs doi: 10.1016/j.procs.2014.07.087

On the other hand, the TOA estimations under line-of-sight(LOS) environment are relatively precise. However when under NLOS condition, the DP signal penetrates one or more obstacles, which will result in the additive propagation delay. This often introduces positive bias into TOA estimation. Furthermore when under NLOS, the DP signal may not be the strongest of the received multipath signals, which will further increase the DP detection difficulty7. Sometime the DP signal even might be completely blocked. This will introduce larger error to ranging. The RM errors have important influence on the localization precision. So compared with LOS, when under NLOS environment the localization error will increase remarkably8. So we believe the RMs under LOS have higher reliability than those under NLOS. After obtaining the LOS and NLOS RMs, the simplest way is removing the NLOS, and only keeping LOS RMs for localization purpose. But reducing the number of effective nodes will result in positioning accuracy descending, especially when the NLOS identification results are wrong. Moreover, when the number of LOS nodes identified is less than the necessary amount that localization needs, position ambiguity problem will be introduced, so more reliable method is still keeping the NLOS measurements.

In this paper we propose a new range WLS(RWLS) target positioning method which designs the weighted values fusing distance information with NLOS identification results. We believe the RMs under LOS are more reliable than those under NLOS. On the other hand, according to analyze the relation between the signal-to-noise ratio(SNR) of the received signals and the RMs reliability, we believe the RMs with nearer distances are more reliable than those with further distances. When designing the weighted values basing on the distance information, it can also avoid imbalance problems when using distance estimation variance as the weighted value9. Simulation results show that our method has the significantly better positioning performance than those existing WLS methods.

This paper is organized as follows. In section 2 the commonly used LS localization algorithm is provided. In section 3, we propose the RWLS localization algorithm which designs weighted values fusing the distance measurements and NLOS identification results. Simulation results are shown in section 4. Finally, we conclude the paper in section 5.

2. The LS localization algorithm

The LS algorithm is a common localization method. When the RM of each AN has been known, the LS estimation of the target position can be shown as9

where z and z,- are the position of MS and the i th AN separately. d t is RM of the l th AN. Re(z) is the position estimation residual error. The weighted value ft can be used to represent the reliability of each measurement, so in fact it is a WLS method. If the reliability information can't be obtained, ft can all be set to 1. Solving equation (1) commonly requires numerical optimization methods, such as the steepest descent method10, Gauss-Newton method11. These methods need large computational costs. Furthermore in order to ensure the target function not converging to local minimum, these methods need sufficiently accurate initial parameter estimation.

3. The RWLS localization method

3.1. The idea weighting according to the distance information

Usually in LS localization algorithm, it assumes the reliability of each distance measurement is equal. That is to say in equation (1), it sets the same weighted value ft for the different AN. However under different environment, even the same under LOS or NLOS situation, the reliability of each measurement is obviously different. In this paper, we suggest designing the weighted value following next two steps. Firstly identify the LOS or NLOS RMs. Secondly design the weighted value for each measurement under LOS or NLOS according to its distance separately.

According to the electromagnetic wave propagation property, usually the mean received power can be written as the function about frequency f (Hz) and distance d (m). It is often modeled as

Px (d, f ) = PaGa ( f )VttGrx (f )Vn

do A d j

Ptx is the transmitted power. Gtx and Grx are transmitted and received antenna gains respectively. nx and qrx are transmitted and received antenna efficiencies respectively. c is the electromagnetic wave propagation velocity. d0 is reference distance generally equal to 1 meter. Ne is pathloss exponent.

For UWB signal, as its relative bandwidth is very large, the frequency f in equation (2) can't be seen as the constant like in narrow band signal. But the change of received signal power caused by the frequency dependence will not be considered in this paper.

Usually we can assume measuring precision of each AN is highly consistent. So each RM noise will approximately have the same probability density distribution function, and the distribution parameters are approximately the same. Under this assumption, SNR of the received signal will have important influence on the final RM error. That is to say, when a certain AN has higher SNR, its RM will have higher reliability. Therefore the weighted value of WLS algorithm can be designed as the function about SNR. From equation (2) we can know that the mean received signal power is inversely proportional to the distance with an exponential law. It means the SNR of the received signal is inversely proportional to the distance with an exponential law. That is

Px ( d, f)

x d -n

According to above and considering the calculation convenience, we can design the weighted value in equation (1)

P, - ko -d

where k0 is a constant. It can be designed according to the NLOS identification result.

3.2. The RWLS algorithm weighting basing on the distance information and NLOS identification results

In the RWLS algorithm, we can process the nonlinear WLS problem utilizing linearization technique. Without losing generality, set zr as reference point, then the linearized equations can be written as the follow matrix form

WAz = Wp

x1 - xr tt - yr

XN-1 Xr yN-1 yr

d2 - i/f + x2 - x2r + y2 - y2 d2r - d22 + x22 - x2 + y22 - y,2

d2r - d+ xN- x,2 + y2N_! - y2r

In equation (5), each element of the diagonal matrix W corresponds to the RWLS weighted value. According to the weighted value calculation method given by equation (4), the weighted matrix can be written as:

W = k0diag( , d1 d

'd2 UN-1

Reference to the LS solving formula of the linear equations, the solution for equation (5) is

z = 1( AT WA ) A T W2 p

Formulae (9) and (10) provide the general solution forms for RWLS target localization which designs the weighted value basing on distance information. It hasn't distinguished whether each AN is under LOS or NLOS condition. In reality, if the NLOS information can be obtained, each value in W can be further optimized. So according to the NLOS identification results the RMs will be divided into two classifications. Then we can set the weighted value range for each class separately. For example we can set

i = \,2,...,Nnlos {Pt>k2, i = NNLOS+\,...,N

In the class of NLOS RMs, the weighted value of the node whose distance is the minimum can be set to k\. The other Nnlos -1 nodes can design their weighted values according to distance square reciprocal form. That is

NLOS min

Similarly in the class of LOS, the weighted value of the node whose distance is the maximum can be set to k2. The other Nlos -1 nodes can design their weighted values according to distance square reciprocal form. That is

n _ U dLOSm ax

P, - k2—sr-

The values of k1 and k2 can be selected experienced according to the actual application environment.

4. Simulation Results

Fig. 1. The simulation scene arrangement diagram

The simulation scene is a rectangular area with the length 60m and width 40m, as shown in Fig. 1. The dashed box area in the middle is a rectangle with length 40m and width 20m. The MS is in the center of the area, with coordination z=[x,y]T=[0,0]T m. The fixed targets(FTs) that are the ANs(FT-1 to FT-6) are spread in the district, with z1=[-26,-10]Tm, z2=[-25.5,11]Tm, z3=[24,-9]T m, z4=[25,9.5]T m, z5=[0.5,-11]T m, z6=[-0.5,10]T m separately.

In the simulation CM3 and CM4 of IEEE 802.15.4a model are adopted as LOS and NLOS channel models separately. Not losing generality, we suppose that all the ANs have the same noise measurement power and the received power satisfies the change relationship given by (3). TOA estimation is realized using GML algorithm. The positioning performance evaluation is according to the RMSE of the position estimation.

RMSE = {e {||z - z||2 J}'2 = {trace [ E {(z - z)(z - z)T j]}^ (14)

25 30 35

SNR(dB)

SNR(dB)

(a) FT-1 is under NLOS

(b) FT-1 and FT-2 is under NLOS

30 SNR(dB)

— LS-I WLS-V WLS-R CRLB

30 SNR(dB)

--►-LS-I WLS-V WLS-R CRLB

(c) FT-1 to FT-3 is under NLOS (d) FT-1 to FT-4 is under NLOS

Fig. 2 The localization performance with different number of NLOS nodes

Simulation results are shown as Fig.2. They illustrate the target position estimation RMSE changing with the SNR. The horizontal axis shows SNR(dB), and vertical axis shows RMSE of the target position estimation(meter). In order to observing larger dynamic range of the RMSE variation, the vertical axis illustrates using logarithmic scale, not linear scale. The SNR defines according to the nearest measurement node, and the SNR of other measurement nodes will be calculated in accordance with inversely proportional to the square of the distance, with range from 20dB to 40dB. The calculation interval is 2dB. According to each SNR, we adopt Monte-Carlo simulation, and randomly select 50 channel models to acquire 50 noise realization sample values. In the simulation we compare the RMSE of out RWLS method with the commonly used WLS method using unit weighted matrix W. We also compare with the method using variance weighted matrix9. The three methods are denoted as WLS-R, LS-I and WLS-V separately. We also provide the CRLB of RMSE estimation for comparison. In the figure the changing curves of LS-I, WLS-V and WLS-R are illustrated as "O", "□" and "A" separately. CRLB is illustrated as solid line. We give four simulation results assuming with 1, 2, 3 or 4 NLOS nodes. In the simulation, we assume that the NLOS situations have been successfully identified. Out method designs weighted values according to formula (12) and (13) with k1=0.1, k2=1.

The results in Fig. 2 show comparing with LS-I, WLS-R has less RMSE in the given SNR range. It indicates our method provides more reasonable judgement for the reliability of the RMs under different SNR. When SNR is low WLS-V has the largest RMSE value in the three, and its localization performance is the worst. When SNR is high enough the distance measurement variance will relatively more stable, so the weighting effect will more obvious, thus its RMSE will lower than LS-I.

5. Conclusions

In this paper we study the UWB localization problems using WLS algorithms. We propose a new RWLS positioning method. It utilizes the distance between each AN and the MS designing the WLS optimal weight, and it also fuses the NLOS identification results with the distance weighted matrix design in order to further improve the localization precision. Because it uses the RM reliability of each AN more reasonably, our method can perform better localization performance.

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