NLOS Error Mitigation for Mobile Location Estimation in Wireless ...

NLOS Error Mitigation for Mobile Location Estimation in Wireless Networks Kegen Yu

Y. Jay Guo

Wireless Technologies Laboratory CSIRO ICT Centre Marsfield NSW 2122, Australia Email: [email protected]

Wireless Technologies Laboratory CSIRO ICT Centre Marsfield NSW 2122, Australia Email: [email protected]

Abstract—Most radio positioning methods are based on the measurements of distance between different wireless nodes. Owing to the existence of non-line-of-sight (NLOS) radio propagation, unfortunately, not all the measured distances are reliable. One way to tackle the problem of positioning is therefore to take two-steps: (i) identifying the NLOS measurements; (ii) smart signal processing of the mixed LOS and NLOS measurements. This paper is focused on the second issue. Under the assumption that the NLOS measurements have been identified, we first propose a simple method to suppress the effect of the NLOS error. Simulation results demonstrate that the proposed method achieves similar or better accuracy than several other known methods and the computational complexity is reduced considerably. We also present an optimal location estimator under the assumption of Gaussian distributed measurement noise and Rayleigh distributed NLOS error. Although it is difficult to achieve the optimal performance in practice due to modeling uncertainties, the optimal estimator provides a performance benchmark. Keywords---mobile location; NLOS error mitigation; Taylorseries LS estimator; optimal estimator; wireless networks.

I. INTRODUCTION Position location and tracking is an important issue for many applications and services. Police and other personnel need the location information to provide emergency assistance. When a vehicle is stolen or even a pet is lost, it can be readily found if its location can be determined. In healthcare, doctors and nurses need to know where their patients are so that services can be given in time. In factories, construction fields, hospitals and offices, it is often necessary to know the location of the tools, material and assets in general. One main error source in radio positioning is the NLOS signal propagation, which makes a signal travels an extra time/distance from a transmitter to a receiver. Complete removal of the NLOS impact may be impractical; however, various methods have been proposed to mitigate the impact. The residual based approach [1]-[4] basically relies on a large number of measurements which are grouped into subsets. Location estimates from each subset of measurements are evaluated by its residual. The final location estimate is obtained either by weighting the different results or by only using some selected measurements. The constrained optimization method [5-7] exploits optimization techniques [8] and geometrical constraints. The propagation model based method [9]-[12]

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either directly employs the existing propagation models or empirically develops a model based on experimental results. Then, statistical estimation/detection theory is applied. Similarly, the error statistics based method [13]-[16] assumes that the NLOS error has certain distribution. On the other hand, the database based method [9], [17], [18] makes use of a signature database established a priori through an extensive survey. When the MS is to be located, some measurements are performed and results are forwarded to the location server in which the MS location is estimated by comparing the measured data with the recorded fingerprints. In this paper we first propose an efficient method to reduce the NLOS effect in the event that both the LOS and NLOS measurements are required to produce a position estimate. By assuming the knowledge of the NLOS measurements, a simple Taylor series (TS) based least-square (LS) algorithm is developed. Compared to the constrained optimization method, the proposed TS-LS algorithm has a substantially reduced complexity [19] even accounting for the extra complexity incurred due to the NLOS identification, and achieves a better location accuracy on average. Although the NLOS identification is not dealt with in the paper, several existing techniques [1], [2], [4] may be employed. Additionally, the optimal location estimator is derived when the measurement noise is Gaussian and the NLOS error is Rayleigh distributed. The optimal estimator produces an excellent performance reference. The remainder of the paper is organized as follows. In section II the measurement model is defined. In section III the proposed TS-LS algorithm is described. In section IV the maximum log likelihood estimator is studied, which serves as a performance benchmark. In section V some simulation results are presented for performance evaluation and comparisons.

II.

MEASUREMENT MODEL

The network considered consists of N fixed stations (FSs) whose locations are known as ( xi , yi , zi ), i = 1, 2, ..., N , and a number of mobile stations (MSs) whose locations are to be determined. The unknown three dimensional coordinates of the MS of interest are denoted by ( x, y , z ) . The Euclidean distance between FS i and the MS is given by

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d i ( x, y, z) = ( x − xi ) 2 + ( y − yi ) 2 + ( z − z i ) 2 , i = 1, 2, ..., N .

(1)

Without loss of generality, let us assume that there are N los LOS FSs, each of which has a direct LOS signal propagation path to the mobile, while the remaining N − N los do not have due to blockage. In the presence of measurement noise and possible NLOS error, the distance measurement equations are modeled as

ri = d i ( x, y , z ) + ε i , i = 1, 2, ..., N ,

(2)

where



εi = 

ni ,

i = 1, 2 , ..., N,

(3)

Li + ni , i = N los + 1, ..., N,

where Li is the extra distance in addition to the LOS distance that the signal travels from FS i to the MS or vice versa, due to the blockage of the LOS path, and ni is the measurement noise. The NLOS distance error Li is modeled as Rayleigh distributed random variable1. On the other hand, the measurement noise is generally modeled as a zero-mean Gaussian random variable. III. TAYLOR SERIES LEAST-SQUARE ALGORITHM To begin with, we need to obtain an initial mobile position estimate. Linearizing the distance measurement equations [21] produces

2 xi x + 2 yi y + 2 z i z − R = Ri − (ri − ε i ) , 2

θˆ = ( A T WA ) −1 A T Wb ,

(8)

where W is a diagonal weighting matrix. Intuitively, the LOS measurements should be weighted more than the NLOS terms. Each LOS term may be given a different weight if the measurement reliability can be judged. When the mobile location estimate is available, we can recalculate the distance between the mobile and each of the FSs. It would be expected that the re-calculated distance might be more accurate than the NLOS corrupted distance measurement especially when the NLOS error is large. For this reason, we replace the NLOS corrupted measurements with the recalculated values in obtaining a more accurate location estimate. Taylor series (TS) expansion based LS algorithm [22] is often considered for position estimation in LOS situations due to its good trade-off between accuracy and complexity. Let us exploit it for our 3-D positioning. Giving the initial position estimates xˆ , yˆ , and zˆ , produced by (8), we have,

x = xˆ + δ x , y = yˆ + δ y , z = zˆ + δ z , where

δ x , δ y , and δ z

are the coordinate errors/increments to

be determined. Expanding d i ( x, y, z ) in a Taylor series around the position estimate and retaining the first two terms, we obtain

ai ,1δ x + ai , 2δ y + ai , 3δ z ≈ ri − dˆi , i = 1, 2, ..., N los ,

(4)

(5)

where in the second equation in (9) the NLOS corrupted measurements are cancelled by the estimated distance so that both of them disappear and

dˆi = ( xˆ − xi ) 2 + ( yˆ − yi ) 2 + ( zˆ − z i ) 2 ,

θ = [ x y z R]T ,

ai ,1 =

2 z1 2z2 ... 2zN

− 1 − 1 ∈ ...   − 1

N ×4

N ×1

,

(6)

Aδ ≈ h, .

(10)

Equation (9) can be rewritten as (11)

where

 a1,1 a 2 ,1 A=  ...  a N ,1

Then, (4) can be written in a compact form:as:

Aθ ≈ b.

xˆ − xi yˆ − yi zˆ − zi , ai , 2 = , ai ,3 = . dˆi dˆi dˆi

(7)

1

When positioning in a cellular network, the distance error is also widely modeled as an exponentially distributed random variable [20].

a1, 2 a2 , 2 ... aN ,2

δ = [δ x δ y δ z ]T ,

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a1, 3  a2,3  ∈ ...   a N ,3 

R

2 y1 2 y2 ... 2 yN

R

 2 x1  2x A= 2  ...  2 x N

R

Define

b = [ R1 − r12 R2 − r22 ... RN − rN2 ]T ∈

(9)

ai ,1δ x + ai , 2δ y + ai , 3δ z ≈ 0, i = N los + 1, ..., N ,

where

R = x 2 + y 2 + z 2 , Ri = xi2 + yi2 + zi2 .

Applying the weighted least-square (LS) estimator, we obtain the estimate of the mobile location:

N ×3

,

R

h = [r1 − dˆ1 ... rNlos − dˆ Nlos 0]T ∈

N ×1

The weighted least-square estimator for (11) produces

δ ≈ ( A T WA) −1 A T Wh,

(12)

Recall that measurements ri , i = 1, 2, ..., N los , are LOS measurements, while the remainders are NLOS measurements. Assuming that all the measurements are independent each other, we have the joint conditional density function: N los

where W is a weighting matrix. Then update the position estimate according to

xˆ = xˆ + δ x , yˆ = yˆ + δ y , zˆ = zˆ + δ z .

p (r | ϑ ) = ∏ i =1

(13)

  (ri − d i ) 2  1 −  + (17) exp ∏ 2 2   σ 2 i = Nlos +1  2π σ si / σ ni n i      (ri − d i ) 2   ri − d i − Q − ri − d i   exp 3 2  σ si / σ i 2σ si   σ si σ ni / σ i    N

Continually refine the position estimate until a pre-defined criterion is satisfied, for instance, a threshold for δ or for the number of iterations is crossed. To summarize, the algorithm works in the following manner: first generate an initial position guess ( xˆ , yˆ , zˆ ) , using (8); then, compute the coordinate increments δ with (12); and update the position estimate using (13); finally, repeat the process until the threshold is crossed.

Taking logarithm on both sides of (17) and ignoring the irrelevant constants, we obtain the following log likelihood:

IV. OPTIMAL LOCATION ESTIMATION When the measurement noise is Gaussian distributed and the NLOS error is Rayleigh distributed, respectively, as:

pn = pL =

1 2π σ n L

σ2

e

−

e

−

(14)

i

 ε i2  −  exp  2σ n2  2π σ s2i / σ ni i    ε2    ε εi , + 3 i exp − i 2 Q −  2σ s   σ s σ n / σ i  σ si / σ i i  i i   

1

(15)

r = [r1 r2 ... rN ]T .

vector

and

measurement

i

i

The mobile location estimate is produced by maximizing the above log likelihood. Equivalently, the optimal location estimate is achieved by:

min{− Λ(r | ϑ )} x, y,z

subject to : ( x L , y L , z L ) ≤ ( x, y, z ) ≤ ( xU , y U , z U ),

(19)

where the bounds of the unknown coordinates which come from the dimensions of the location area are applied to enhance the performance.

V.

i

ϑ = [ x y z ]T ,

i

It is worthy to mention that the performance of the maximum log likelihood estimator produces a performance reference, which may be difficult to achieve due to uncertainties in noise and error distributions and related parameters.

σ s = σ n2 + σ i2 . Define the position respectively as

(18)

 (r − d ) 2   r − d i  σi (ri − d i ) exp − i 2i Q − i   σs 2σ s   σ s σ n / σ i  

,

where Q(.) is the standard Q-function and i

 (ri − d i ) 2   σ ni − + ln exp  ∑ 2   2 σ i = Nlos +1 ni  2π  

,

we can achieve an optimal performance of the location problem by applying the maximum likelihood (ML) estimator, provided that the parameters of the distributions are known. Assuming that the measurement noise and the NLOS error are independent random variables, the joint distribution of the two random variables can be derived as

pε i =

(ri − d i ) 2 Λ (r | ϑ ) = ∑ + 2σ n2i i =1 N

L2 2σ 2

Nlos

n2 2σ n2

 (ri − d i ) 2  × exp − 2   σ 2 2π σ ni ni   1

vector

(16)

SOME SIMULATION RESULTS

In this section we present some simulation results to evaluate the accuracy of the proposed location algorithms. Two other algorithms are also selected for performance comparisons. One is in [6] where the location estimation is first changed into a quadratic programming problem through two steps. The corresponding results in the plots are denoted by "two-step". The other is the constrained optimization approach in [5] which may be defined as:

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N

min ∑ wi (ri − d i ) 2

1 Cumulative Distribution (100%)

x, y , z

(20)

i =1

subject to : d i ≤ ri and the related results are denoted by "consOpt" in the plots. Also in the plots, "optimal" stands for the optimal (maximum log likelihood) location estimator, while "TS-LS” denotes results from the TS-LS estimator. The dimensions of the location area are 30m × 30m × 30m. and the six fixed stations are located at (0, 0, 0), (30, 0, 0), (0, 30, 0), (0, 0, 30), (30, 30, 30), and (30, 30, 0) (unit in meter), respectively. The target mobile location is randomly selected in the area and a number of 5000 different locations are examined. The unknown coordinates are upper bounded by U

U

noise is Gaussian with zero mean and standard deviation (STD) 42 cm. The NLOS error is Rayleigh distributed with the parameter σ set at three meters such that the mean is 3.76 meters and the STD is 1.97 meters. All the weighting matrices are simply chosen to be an identity matrix. Figures 1 and 2 show the cumulative distribution probability of the location errors when there are three and five NLOS FSs, respectively. Clearly, the proposed TS-LS algorithm has a similar or even better performance than the other two methods which did not exploit the LOS/NLOS information. Also we can observe that the optimal location estimator substantially outperforms the other three algorithms.

Cumulative Distribution (100%)

1 0.8

0.4 optimal TS-LS consOpt two-step

0.2

1

2 3 4 Location Error (m)

5

6

Figure 2: Cumulative distribution probability of location error when there are five NLOS FSs. All the legends have the same meanings as in Figure 1.

VI.

CONCLUSIONS

In this paper we have investigated position estimation in an NLOS environment and two location algorithms were derived. Making use of the LOS/NLOS measurement information, the derived TS-LS algorithm achieves a similar or even better accuracy than other two methods reported in the literature. Comparatively, the method also has a lower computational complexity. Under the assumption that the measurement noise is Gaussian distributed and the NLOS error is Rayleigh distributed, we derived a maximum log likelihood (optimal) location estimator, which dramatically outperforms the other three algorithms. REFERENCES

0.6 [1]

0.4 optimal TS-LS consOpt two-step

0.2 0 0

0.6

0 0

U

( x , y , z ) = (30, 30, 30) and lower bounded by ( x L , y L , z L ) = (0, 0, 0) , respectively. The measurement

0.8

1

2 3 4 Location Error (m)

5

[2]

[3]

6

Figure 1: Cumulative distribution probability of location error when there are three NLOS FSs. Optimal: results produced by (19). TS-LS: results yielded using the proposed TS-LS algorithm. two-step: results produced by the method in [6]. consOpt: results obtained using (20).

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