An Improved Reference Selection Method in ... - Semantic Scholar
An Improved Reference Selection Method in Linear Least Squares Localization for LOS and NLOS Shixun Wu, Jiping Li, Shouyin Liu The Department of Electronics and Information Engineering Huazhong Normal University, Wuhan 430079, China E-mail: [email protected] Abstract—Linear least squares (LLS) estimation is a low complexity but sub-optimum method for estimating the location of a mobile terminal (MT) from some measured distances. It requires selecting one of the known fixed terminals (FTs) as a reference FT for obtaining a linear set of expressions. In this paper, a new method for selecting the reference FT is proposed, which selects the reference FT based on the minimum residual rather than the smallest measured distance and improves the localization accuracy significantly in Line of sight (LOS) environment. In Non-line of sight (NLOS) environment, a new residual weighting algorithm is proposed, which is based on the proposed LOS algorithm. Moreover, by making use of the interior-point optimization method which can obtain the estimation of NLOS error, a new algorithm which is also based on the proposed LOS algorithm is proposed. Simulation results show that the proposed methods improve the positioning accuracy considerably comparing with other existing methods. Keywords- Reference Selection;Least-Square(LS) Estimation; Wireless Location.
I.
INTRODUCTION
Wireless location technologies, which are designated to estimate the position of a mobile terminal (MT), have drawn a lot of attention over the past few decades. Different types of Location-Based Services (LBSs) have been proposed and studied, including the emergency subscriber safety services [1], the location-based billing, the navigation system, and applications for the Intelligent Transportation System (ITS) [2]. Due to the wide application for LBSs, it is required to provide the location estimation of a MT with high accuracy. Mobile positioning using radiolocation techniques usually involves time of arrival (TOA), time difference of arrival (TDOA), angle of arrival (AOA), signal strength (SS) measurements, or some combination of these methods. Performance limits for the localization accuracy based on these measurements are evaluated by Cramer-Rao lower bound (CRLB). However, CRLB is obtained through the known position of the MT and variance of each measured distance. Hence, CRLB is difficult to achieve in practical environment. In Line of sight (LOS) environment, two iteration methods are proposed in [3][4], one is the two-step maximum likelihood (ML) technique [3], the other is the approximate ML technique [4], which can approach to the CRLB by transforming the nonlinear expressions into matrix form and
performing some manipulation. However, they achieve CRLB at the expense of high complexity. In [5], a linear programming (LP) method is proposed to improve localization accuracy. This LP method is shown to perform as well as the LS estimation when only LOS measured distances are provided, and better than the LS estimation when a mixture of LOS and Non-line of sight (NLOS) measured distances are provided. In [6], the authors analyze the factors affecting the positioning accuracy, which concludes that one method is better or worse than the other depending on the topology and the LOS/NLOS scenarios. The work in [7] develops adaptive LS and least median squares (LMS) position estimation algorithms, which are robust in different environments. In [8], the authors propose an efficient Geometry-constrained Location Estimation (GLE) algorithm, which improves the two-step LS algorithm by imposing additional geometric constraints. The work in [9] proposes three-stage location estimation algorithm, and the estimation of the NLOS bias is obtained by the interior-point optimization (IPO) method. However, it needs a good initial estimation. The linear leastsquares (LLS) algorithm proposed in [10] transforms the nonlinear terms into a linear form and has a low complexity as opposed to a nonlinear least square (NLS) estimation at the expense of some performance loss. Lately, the work in [11] analyzes the localization accuracy of the LLS algorithm both in theory and via simulation, and two closed-form expressions for the mean square errors (MSE) localization in LOS and NLOS scenarios are given. In LOS environment, a method of selecting the fixed terminal (FT) with the minimum measured distance as the reference FT is proposed in [12]. In NLOS environment, it proposes to choose a LOS FT with the minimum measured distance as the reference FT. However, it isn’t known in advance whether the FT is LOS or NLOS. Recently, the work in [13] proposes a dual weighted average algorithm based on LLS to mitigate the effects of NLOS bias and achieves high accuracy in practical scenarios. This paper proposes an improved LLS algorithm in LOS environment. Moreover, we combine the improved LLS algorithm with two existing methods to form two new algorithms in NLOS environment. In our improved LLS algorithm, the reference FT is selected based on the minimum residual rather than the smallest measured distance. For simple expression, we name the improved LLS algorithm as the reference selection algorithm based on the minimum residual
978-1-4244-8327-3/11/$26.00 ©2011 IEEE
(RS-MR). In NLOS environment, it should be better to only choose the LOS FTs to estimate the position. But it is difficult to know whether the FT is LOS or NLOS. The advantage of our proposed RS-MR is to simultaneously obtain the location estimation of MT and the corresponding normalized residual. In fact, the residual contains the information of NLOS errors. Therefore, the RS-MR is introduced to NLOS environment and a new residual weighting algorithm is formed. In addition, we first use the IPO method [9] to get the estimation of the NLOS bias. Then subtract it from the measured distances as the corrected measurements. Finally we choose RS-MR to get the location estimation of MT. This method is noted as the reference FT selection algorithm based on the interior-point optimization (IPO-RS).
AX =
1 p 2
(5)
where ⎡ x1 − xr ⎢x −x A=⎢ 2 r ⎢ ⎢ ⎣ xN − xr ( xi , yi ), i = 1, and
, r,
y1 − yr ⎤ y2 − yr ⎥⎥ ⎥ ⎥ yN − yr ⎦ ( N −1)× 2
(6)
, N is the position of the ith FT.
SYSTEM MODEL
II.
⎡ dˆr2 − dˆ12 − kr + k1 ⎤ ⎢ 2 ⎥ ⎢ dˆr − dˆ22 − kr + k2 ⎥ p=⎢ ⎥ ⎢ ⎥ ⎢ dˆ 2 − dˆ 2 − k + k ⎥ N r N ⎦ ( N −1)×1 ⎣ r
There are N FTs in a wireless network. X = [ x y ]T is the T
position of the MT, X i = [ xi yi ] is the position of the ith FT, dˆi is the measured distance between the MT and the ith FT which is modeled as
dˆi = di + bi + ni = cti , i = 1, 2,
(1)
,N
where c is the speed of light, ti is the TOA of the signal to the ith FT, d i is the actual distance between the MT and the ith
ki = xi2 + yi2 , i = 1, , r , , N . The equation (5) has a LLS solution given by 1 Xˆ = ( AT A) −1 AT p 2
FT, ni ∼ N (0, σ 2 ) is the additive white Gaussian noise (AWGN) with the same variance σ 2 , and bi is a non-negative distance bias, which is given by ⎧ 0, if ith FT is LOS bi = ⎨ ⎩ψ i , if ith FT is NLOS
(2)
For NLOS FTs, the bias term ψ i is modeled as exponential distribution [14]. If all the measured distances in (1) are available, and the negative effect of the noise and the NLOS biases are ignored at different FTs, then the yielding circles do not intersect at the same point, which result in the following inconsistent equations ( x − xi ) 2 + ( y − yi )2 = dˆi2 , i = 1, 2,
,N
N
Res ( X ) = ∑ (dˆi − || Xˆ − X i ||)2
(4)
(8)
The second LLS method (noted as LLS-2) obtains linear equations by subtracting each equation from
N × ( N − 1) 2
all other equations [5]. The transformed measurements are employed for position estimation di , j = dˆi2 − dˆ 2j , i, j = 1, 2,
, N,i < j
(9)
The third LLS method (noted as LLS-3) first obtains the average of the measured distance, which is subtracted from all the equations resulting in N linear equations [7]. The LLS-3 method can be expressed as
(3)
If we can get the estimation of the MT Xˆ = [ xˆ yˆ ]T , then we can define the residual as
(7)
1 di = dˆi2 − N
N
∑ dˆ
2 j
, i = 1, 2,
,N
(10)
j =1
In fact, the LLS-2 is equivalent with LLS-3. III.
REFERENCE FT SELECTION FOR LOS AND NLOS ENVIRONMENTS
i =1
There are N equations in (3), we can get other N − 1 equations by fixing the rth equation and subtracting it from the rest equation for i = 1, 2, , N , (i ≠ r ) . Then manipulating the terms, we obtain the following linear equation
A. Reference FT Selection in LOS In [12], the smallest measured distance is selected as the reference FT among all the measured distances. The index of the reference FT which has the smallest measured distance is given by
r = arg min{dˆi }, i = 1, 2,
,N
i
(11)
However, the minimum measured distance acting as reference FT is not the best selection. We know that the LLS solution in (8) is obtained by minimizing the objective 1 1 function J = ( AX − p )T ( AX − p) . When we choose the 2 2 minimum measured distance as the reference FT, the matrix A and the corresponding vector p are fixed. In fact, the matrix A and the corresponding vector p are variable, which motivates us to find the best matrix A and the corresponding vector p to minimize the objective function J . This is why the minimum residual acting as the reference FT is better than the minimum measured distance in LOS environment. The proposed RS-MR is described as follow 1. Each of the N FTs can be set as the reference FT, so we can get N estimations of MT, which is expressed as Xˆ k , k = 1, 2, , N by (8). 2. For each Xˆ , the corresponding residual (4) is k
k
computed, which is denoted as R es , k = 1, 2, , N . The index of the reference FT is given by
3.
k
r = arg min{Res }, k = 1, 2, k
,N .
The final location estimation of MT is got by (8).
4.
B. Reference FT Selection in NLOS In NLOS environment, the work in [12] proposes that the selection of the reference FT is based on the minimum measured distance among the LOS FTs. However, this method has two disadvantages. First, we do not know the index set for all the LOS FTs in advance. Second, if the number of LOS FTs is less than three, the method is not appropriate. Thus, two methods are proposed to overcome the two disadvantages. 1) Residual Weighting(Rwgh) Algorithm The work in [14] proposes a residual weighting algorithm. We can use RS-MR to form a new Rwgh algorithm which is described as follow 1. Given N ( N > 3) range measurements from N N ⎛N⎞ M = ∑ ⎜ ⎟ range i =3 ⎝ i ⎠ measurement combinations. Each combination is represented by an index set {Sk , k = 1, , M } . For each combination, computing the location estimations of MT Xˆ k , k = 1, , M with RS-MR
different
2.
FTs
to
form
k
3.
and the normalized residual R es , k = 1, , M k R ( Xˆ , S ) R es = es k k size of Sk The final location estimation of MT X is the weighted linear combination of the intermediate
M
estimation from step 2. Xˆ =
∑ Xˆ
k
k
k =1 M
( Res ) −1 .
k
∑ ( Res )−1 k =1
2) Interior Point Optimization (IPO) method The new Rwgh algorithm is possible to reduce the effect of NLOS error, but it has low positioning accuracy in dense NLOS environment. Another method is first to get the estimations of the NLOS bias bˆi , i = 1, 2, , N , then use dˆi − bˆi as the corrected measurements, finally we can choose RS-MR to get the location estimation of MT. In [9], the authors find the optimal location estimation in the presence of NLOS error by three-stage estimation algorithm, which can also get the estimation of the NLOS bias by IPO method. Here this method is introduced briefly. Firstly, converting (1) to vector form r = h( X ) + b + v
(12)
where ⎛ dˆ1 ⎞ ⎛ || X − X 1 || ⎞ ⎛ d1 ⎞ ⎛ n1 ⎞ ⎜ dˆ ⎟ r = ⎜ 2 ⎟ , h( X ) = ⎜ || X − X 2 || ⎟ , b = ⎜ d 2 ⎟ , v = ⎜ n2 ⎟ ⎜⎜ || X − X || ⎟⎟ ⎜⎜ d ⎟⎟ ⎜⎜ n ⎟⎟ ⎜ dˆ ⎟ N ⎠ ⎝ ⎝ N⎠ ⎝ N⎠ ⎝ N⎠
The nonlinear function h( X ) can be linearized in a reference point X 0 with Taylor series. Ignoring high-order terms
h( X ) ≈ h( X 0 ) + H 0 ( X − X 0 ) where
H0
is
⎡ ∂h1 ⎢ ∂x X 0 , H0 = ⎢ ⎢ ∂hN ⎢ ⎢⎣ ∂x
the
Jacobian
matrix
(13) of
h( X )
at
∂h1 ⎤ ∂y ⎥ ⎥ ∂hN ⎥ ⎥ ∂y ⎦⎥ X = X 0
If no NLOS bias is present, we can get the bias-free estimation of the MT X , which is written as X = ( H 0T R −1 H 0 ) −1 H 0T R −1 (r − h( X 0 ) + H 0 X 0 )
(14)
where R = diag (σ 2 , σ 2 , , σ 2 ) . Considering some constrained conditions, IPO method is used to estimate the NLOS bias b , which can be formulated as min J (b) = bT S T Q −1 Sb − 2Z T S T b
⎧ 0 ≤ bi ≤ ui , i = 1, 2 , N ⎪ s.t ⎨ui = min{ri + rj − Li , j }∀j ≠ i ⎪ Li , j =|| X i − X j || ⎩
(15)
where S = I + H 0V , Q = H 0 ( H 0T R −1 H 0 ) −1 H 0T + R ,
70
(16)
The proposed IPO-RS method is described as follow 1. It is known that three-stage location estimation algorithm needs a good initial estimation from (13). Here, LLS-2 or LLS-3 algorithm is proposed to get the location estimation of the MT, which used as the initial estimation of three-stage method. Then the NLOS bias bˆi can be estimated from (12-15). the corrected measurements 2. Compute d i = dˆi − bˆi , i = 1, 2, , N , and then use RS-MR to get the final location estimation of the MT. IV.
SIMULATION RESULTS
Monte-Carlo simulations are performed to compare the proposed algorithms with the existing methods. Assume there are four FTs which are positioned at X 1 = [−1, 20] , X 2 = [10,10] , X 3 = [4, −6] , X 4 = [−5, −5] . The location of MT is X = [2, 2] . To compare the performance of different algorithms, we need to analyze their mean square error (MSE). The definition of MSE is MSE = E || X − Xˆ ||2
The mean square error(MSE)
Xˆ = X + Vbˆ
The minimum distance as the RS The minimum residual as the RS CRLB
60
50
40
30
20
10
0
0
5
10
15
20
25
30
35
40
The variance of measurement error
Fig.1 The comparison of the minimum measured distance as the reference selection (RS) and the minimum residual as the RS 60
The mean square error(MSE)
Z = r − h( X 0 ) + H 0 ( X 0 − X ) , V = −( H 0T R −1 H 0 ) −1 H 0T R −1 . Solving the optimization problem in (15), we can obtain the estimation of the NLOS bias bˆ . Finally, an optimal estimation of the MT position Xˆ can be got as follow
The minimum distance as the RS(one nlos) The Rwgh algorithm(one nlos) The minimum distance as the RS(two nlos) The Rwgh algorithm(two nlos) The minimum distance as the RS(three nlos) The Rwgh algorithm(three nlos) CRLB
50
40
30
20
10
(17) 0
1
1.5
2
2.5
3
3.5
4
The mean of NLOS error
Fig.2 MSE VS the mean of NLOS error when σ 2 = 1 . 60
The minimum distance as the RS(one nlos) The Rwgh algorithm(one nlos) The minimum distance as the RS(two nlos) The Rwgh algorithm(two nlos) The minimum distance as the RS(three nlos) The Rwgh algorithm(three nlos) CRLB
55
The mean square error(MSE)
where Xˆ is the location estimation of the MT X . All the simulation results are obtained based on 10,000 independent runs. In LOS environment, Fig.1 demonstrates that the proposed algorithm (RS-MR) is better than the minimum measured distance acting as the reference FT, and the performance improves significantly with the increasing of LOS variance σ 2 . In NLOS environment, we assume that the NLOS error is exponentially distributed [14] with mean λ . Fig.2 compares new Rwgh with the LOS algorithm for different LOS/NLOS conditions. We observe that the new Rwgh indeed mitigates the effect of the NLOS error, especially when there is only one NLOS FT. Fig.3 demonstrates the effect of LOS variance σ 2 for different LOS/NLOS conditions. As shown in Fig.3, the Rwgh algorithm improves the performance when the LOS variance is large. Fig.4-Fig.6 are shown to compare three different algorithms when λ and σ 2 are different, and the initial estimation of the three-stage algorithm is obtained by LLS-2 or LLS-3 algorithm. We find that the proposed IPO-RS algorithm is always better than two other algorithms for different parameters, and the mean of NLOS error has great effect on the three-stage algorithm.
50 45 40 35 30 25 20 15 10 5 2
4
6
8
10
12
14
16
The variance of measurement error
Fig.3 MSE VS the variance of measurement error when λ = 2 .
V.
The mean square error(MSE)
25
In this paper, three algorithms are presented in LOS and NLOS environments. Simulation results demonstrate that: 1) in LOS environment, the proposed RS-MR method is better than the minimum distance acting as the reference FT algorithm; 2) in NLOS environment, the new Rwgh method can indeed mitigate the effect of NLOS error, and the algorithm only has better performance when the number of LOS FTs is more than two; 3) the proposed IPO-RS method has better performance than the minimum distance (IPO-RS) method, and the performance of the three-stage algorithm decreases significantly in dense NLOS environment.
three-stage algorithm the minimum distance(IPO-RS) the minimum residual(IPO-RS) CRLB
20
CONCLUSION
15
10
5
REFERENCES [1]
0 3LOS,1NLOS
2LOS,2NLOS
1LOS,3NLOS
0LOS,4NLOS
LOS/NLOS condition of the measurements
Fig.4 MSE VS different LOS/NLOS conditions when λ = 4, σ 2 = 1 .
[2]
[3]
The mean square error(MSE)
25
20
three-stage algorithm the minimum distance(IPO-RS) the minimum residual(IPO-RS) CRLB
[4]
[5]
15
[6]
10
[7]
5 3LOS,1NLOS
2LOS,2NLOS
1LOS,3NLOS
0LOS,4NLOS
LOS/NLOS condition of the measurements
Fig.5 MSE VS different LOS/NLOS conditions when λ = 4, σ 2 = 4 .
[8]
[9]
The mean square error(MSE)
60
50
three-stage algorithm the minimum distance(IPO-RS) the minimum residual(IPO-RS) CRLB
[10]
[11]
40
30
[12] 20
[13]
10
0 3LOS1NLOS
2LOS2NLOS
1LOS3NLOS
LOS/NLOS condition of the measurements
Fig.6 MSE VS different LOS/NLOS conditions when λ = 6, σ 2 = 4 .
0LOS4NLOS
[14]
Federal Communications Commission, “Revision of the Commissions Rules to Insure Compatibility with Enhanced 911 Emergency Calling Systems,” 1996. S.Feng and C.L.Law, “Assisted GPS and Its Impact on Navigation in Intelligent Transportation Systems,” IEEE Intelligent Transportation Systems, 2002, pp.926-931. Y.T.Chan and K.C.Ho, “A Simple and efficient estimator for hyperbolic location,” IEEE Trans. Signal Processing, vol.42, no.8, pp.1905-1915. Y.T.Chan, H.Y.C.Hang, and P.C.Ching, “Exact and approximate maximum likelihood localization algorithms,” IEEE Trans. Vehicular Technology, vol. 55, no.1, pp.10-16.Jan. 2006. S.Venkatesh and R.M Buehrer, “A linear programming approach to NLOS error mitigation in sensor networks,” in Proc. IEEE Int. Conf. on Information Processing in Sensor Networks (IPSN), Nashville, TN, Apr. 2006, pp. 301-308. V.Dizdarevic and K.Witrisal, “On impact of topology and cost function on LSE position determination in wireless networks,” in Proc. Workshop on Positioning, Navigation , and Commun. (WPNC), Hannover, Germany, Mar.2006, pp.129-138. Z.Li, W.Trappe, Y.Zhang, and B.Nath, “Robust statistical methods for securing wireless localization in sensor networks,” in Proc. IEEE Int. Symp. Information Processing in Sensor Networks(IPSN). Los Angles, CA, Apr.2005, pp.91-98. C.L.Chen and K.T.Feng, “An efficient geometry-constrained location estimation algorithm for NLOS environments,” in Proc. IEEE Int. Conf. Wireless Networks, Commun, Mobile Computing, Hawaii, USA, June 2005, pp.244-249. W.Kim, J.G.Lee, and G..I.Jee, “The interior-point method for an optimal treatment of bias in trilateration location,” IEEE Trans. Vehic, Technol, vol.55, no.4, pp. 1291-1301,July 2006. J.J.Caffery, “A new approach to the geometry of TOA location,” in Proc. IEEE Vehic. Technol.Conf.(VTC), vol.4, Boston, MA, Sep.2000, pp.1943-1949. I.Guvenc, C.C.Chong, and F.Watanabe, “Analysis of a linear leastsquares location technique in LOS and NLOS environments,” in Proc, IEEE Vehic, Technol. Conf. (VTC), Dublin, Ireland, Apr.2007, pp.1886-1890. I.Guvenc, S. Gezici, F.Watanabe, and H..Inamura, “Enhancements to linear least squares localization through reference selection and ML estimation,” in Proc. IEEE Wireless Commun. Networking Conf. (WCNC), Las Vegas, NV, Apr.2008, pp.284-289. N.Bhagwat, K.Liu, and B.Jabbari,”Robust bias mitigation algorithm for localization in wireless netorks,” in Proc.IEEE International conference on communication (ICC), Cape Town, May,2010, pp.1-5. P.C.Chen, “A non-line-of-sight error mitigation algorithm in location estimation,” in Proc. IEEE Int. Conf. wireless Commun. Networking (WCNC), vol.1, New Orleans, LA, Sept. 1999, pp.316-320.
I.
INTRODUCTION
Wireless location technologies, which are designated to estimate the position of a mobile terminal (MT), have drawn a lot of attention over the past few decades. Different types of Location-Based Services (LBSs) have been proposed and studied, including the emergency subscriber safety services [1], the location-based billing, the navigation system, and applications for the Intelligent Transportation System (ITS) [2]. Due to the wide application for LBSs, it is required to provide the location estimation of a MT with high accuracy. Mobile positioning using radiolocation techniques usually involves time of arrival (TOA), time difference of arrival (TDOA), angle of arrival (AOA), signal strength (SS) measurements, or some combination of these methods. Performance limits for the localization accuracy based on these measurements are evaluated by Cramer-Rao lower bound (CRLB). However, CRLB is obtained through the known position of the MT and variance of each measured distance. Hence, CRLB is difficult to achieve in practical environment. In Line of sight (LOS) environment, two iteration methods are proposed in [3][4], one is the two-step maximum likelihood (ML) technique [3], the other is the approximate ML technique [4], which can approach to the CRLB by transforming the nonlinear expressions into matrix form and
performing some manipulation. However, they achieve CRLB at the expense of high complexity. In [5], a linear programming (LP) method is proposed to improve localization accuracy. This LP method is shown to perform as well as the LS estimation when only LOS measured distances are provided, and better than the LS estimation when a mixture of LOS and Non-line of sight (NLOS) measured distances are provided. In [6], the authors analyze the factors affecting the positioning accuracy, which concludes that one method is better or worse than the other depending on the topology and the LOS/NLOS scenarios. The work in [7] develops adaptive LS and least median squares (LMS) position estimation algorithms, which are robust in different environments. In [8], the authors propose an efficient Geometry-constrained Location Estimation (GLE) algorithm, which improves the two-step LS algorithm by imposing additional geometric constraints. The work in [9] proposes three-stage location estimation algorithm, and the estimation of the NLOS bias is obtained by the interior-point optimization (IPO) method. However, it needs a good initial estimation. The linear leastsquares (LLS) algorithm proposed in [10] transforms the nonlinear terms into a linear form and has a low complexity as opposed to a nonlinear least square (NLS) estimation at the expense of some performance loss. Lately, the work in [11] analyzes the localization accuracy of the LLS algorithm both in theory and via simulation, and two closed-form expressions for the mean square errors (MSE) localization in LOS and NLOS scenarios are given. In LOS environment, a method of selecting the fixed terminal (FT) with the minimum measured distance as the reference FT is proposed in [12]. In NLOS environment, it proposes to choose a LOS FT with the minimum measured distance as the reference FT. However, it isn’t known in advance whether the FT is LOS or NLOS. Recently, the work in [13] proposes a dual weighted average algorithm based on LLS to mitigate the effects of NLOS bias and achieves high accuracy in practical scenarios. This paper proposes an improved LLS algorithm in LOS environment. Moreover, we combine the improved LLS algorithm with two existing methods to form two new algorithms in NLOS environment. In our improved LLS algorithm, the reference FT is selected based on the minimum residual rather than the smallest measured distance. For simple expression, we name the improved LLS algorithm as the reference selection algorithm based on the minimum residual
978-1-4244-8327-3/11/$26.00 ©2011 IEEE
(RS-MR). In NLOS environment, it should be better to only choose the LOS FTs to estimate the position. But it is difficult to know whether the FT is LOS or NLOS. The advantage of our proposed RS-MR is to simultaneously obtain the location estimation of MT and the corresponding normalized residual. In fact, the residual contains the information of NLOS errors. Therefore, the RS-MR is introduced to NLOS environment and a new residual weighting algorithm is formed. In addition, we first use the IPO method [9] to get the estimation of the NLOS bias. Then subtract it from the measured distances as the corrected measurements. Finally we choose RS-MR to get the location estimation of MT. This method is noted as the reference FT selection algorithm based on the interior-point optimization (IPO-RS).
AX =
1 p 2
(5)
where ⎡ x1 − xr ⎢x −x A=⎢ 2 r ⎢ ⎢ ⎣ xN − xr ( xi , yi ), i = 1, and
, r,
y1 − yr ⎤ y2 − yr ⎥⎥ ⎥ ⎥ yN − yr ⎦ ( N −1)× 2
(6)
, N is the position of the ith FT.
SYSTEM MODEL
II.
⎡ dˆr2 − dˆ12 − kr + k1 ⎤ ⎢ 2 ⎥ ⎢ dˆr − dˆ22 − kr + k2 ⎥ p=⎢ ⎥ ⎢ ⎥ ⎢ dˆ 2 − dˆ 2 − k + k ⎥ N r N ⎦ ( N −1)×1 ⎣ r
There are N FTs in a wireless network. X = [ x y ]T is the T
position of the MT, X i = [ xi yi ] is the position of the ith FT, dˆi is the measured distance between the MT and the ith FT which is modeled as
dˆi = di + bi + ni = cti , i = 1, 2,
(1)
,N
where c is the speed of light, ti is the TOA of the signal to the ith FT, d i is the actual distance between the MT and the ith
ki = xi2 + yi2 , i = 1, , r , , N . The equation (5) has a LLS solution given by 1 Xˆ = ( AT A) −1 AT p 2
FT, ni ∼ N (0, σ 2 ) is the additive white Gaussian noise (AWGN) with the same variance σ 2 , and bi is a non-negative distance bias, which is given by ⎧ 0, if ith FT is LOS bi = ⎨ ⎩ψ i , if ith FT is NLOS
(2)
For NLOS FTs, the bias term ψ i is modeled as exponential distribution [14]. If all the measured distances in (1) are available, and the negative effect of the noise and the NLOS biases are ignored at different FTs, then the yielding circles do not intersect at the same point, which result in the following inconsistent equations ( x − xi ) 2 + ( y − yi )2 = dˆi2 , i = 1, 2,
,N
N
Res ( X ) = ∑ (dˆi − || Xˆ − X i ||)2
(4)
(8)
The second LLS method (noted as LLS-2) obtains linear equations by subtracting each equation from
N × ( N − 1) 2
all other equations [5]. The transformed measurements are employed for position estimation di , j = dˆi2 − dˆ 2j , i, j = 1, 2,
, N,i < j
(9)
The third LLS method (noted as LLS-3) first obtains the average of the measured distance, which is subtracted from all the equations resulting in N linear equations [7]. The LLS-3 method can be expressed as
(3)
If we can get the estimation of the MT Xˆ = [ xˆ yˆ ]T , then we can define the residual as
(7)
1 di = dˆi2 − N
N
∑ dˆ
2 j
, i = 1, 2,
,N
(10)
j =1
In fact, the LLS-2 is equivalent with LLS-3. III.
REFERENCE FT SELECTION FOR LOS AND NLOS ENVIRONMENTS
i =1
There are N equations in (3), we can get other N − 1 equations by fixing the rth equation and subtracting it from the rest equation for i = 1, 2, , N , (i ≠ r ) . Then manipulating the terms, we obtain the following linear equation
A. Reference FT Selection in LOS In [12], the smallest measured distance is selected as the reference FT among all the measured distances. The index of the reference FT which has the smallest measured distance is given by
r = arg min{dˆi }, i = 1, 2,
,N
i
(11)
However, the minimum measured distance acting as reference FT is not the best selection. We know that the LLS solution in (8) is obtained by minimizing the objective 1 1 function J = ( AX − p )T ( AX − p) . When we choose the 2 2 minimum measured distance as the reference FT, the matrix A and the corresponding vector p are fixed. In fact, the matrix A and the corresponding vector p are variable, which motivates us to find the best matrix A and the corresponding vector p to minimize the objective function J . This is why the minimum residual acting as the reference FT is better than the minimum measured distance in LOS environment. The proposed RS-MR is described as follow 1. Each of the N FTs can be set as the reference FT, so we can get N estimations of MT, which is expressed as Xˆ k , k = 1, 2, , N by (8). 2. For each Xˆ , the corresponding residual (4) is k
k
computed, which is denoted as R es , k = 1, 2, , N . The index of the reference FT is given by
3.
k
r = arg min{Res }, k = 1, 2, k
,N .
The final location estimation of MT is got by (8).
4.
B. Reference FT Selection in NLOS In NLOS environment, the work in [12] proposes that the selection of the reference FT is based on the minimum measured distance among the LOS FTs. However, this method has two disadvantages. First, we do not know the index set for all the LOS FTs in advance. Second, if the number of LOS FTs is less than three, the method is not appropriate. Thus, two methods are proposed to overcome the two disadvantages. 1) Residual Weighting(Rwgh) Algorithm The work in [14] proposes a residual weighting algorithm. We can use RS-MR to form a new Rwgh algorithm which is described as follow 1. Given N ( N > 3) range measurements from N N ⎛N⎞ M = ∑ ⎜ ⎟ range i =3 ⎝ i ⎠ measurement combinations. Each combination is represented by an index set {Sk , k = 1, , M } . For each combination, computing the location estimations of MT Xˆ k , k = 1, , M with RS-MR
different
2.
FTs
to
form
k
3.
and the normalized residual R es , k = 1, , M k R ( Xˆ , S ) R es = es k k size of Sk The final location estimation of MT X is the weighted linear combination of the intermediate
M
estimation from step 2. Xˆ =
∑ Xˆ
k
k
k =1 M
( Res ) −1 .
k
∑ ( Res )−1 k =1
2) Interior Point Optimization (IPO) method The new Rwgh algorithm is possible to reduce the effect of NLOS error, but it has low positioning accuracy in dense NLOS environment. Another method is first to get the estimations of the NLOS bias bˆi , i = 1, 2, , N , then use dˆi − bˆi as the corrected measurements, finally we can choose RS-MR to get the location estimation of MT. In [9], the authors find the optimal location estimation in the presence of NLOS error by three-stage estimation algorithm, which can also get the estimation of the NLOS bias by IPO method. Here this method is introduced briefly. Firstly, converting (1) to vector form r = h( X ) + b + v
(12)
where ⎛ dˆ1 ⎞ ⎛ || X − X 1 || ⎞ ⎛ d1 ⎞ ⎛ n1 ⎞ ⎜ dˆ ⎟ r = ⎜ 2 ⎟ , h( X ) = ⎜ || X − X 2 || ⎟ , b = ⎜ d 2 ⎟ , v = ⎜ n2 ⎟ ⎜⎜ || X − X || ⎟⎟ ⎜⎜ d ⎟⎟ ⎜⎜ n ⎟⎟ ⎜ dˆ ⎟ N ⎠ ⎝ ⎝ N⎠ ⎝ N⎠ ⎝ N⎠
The nonlinear function h( X ) can be linearized in a reference point X 0 with Taylor series. Ignoring high-order terms
h( X ) ≈ h( X 0 ) + H 0 ( X − X 0 ) where
H0
is
⎡ ∂h1 ⎢ ∂x X 0 , H0 = ⎢ ⎢ ∂hN ⎢ ⎢⎣ ∂x
the
Jacobian
matrix
(13) of
h( X )
at
∂h1 ⎤ ∂y ⎥ ⎥ ∂hN ⎥ ⎥ ∂y ⎦⎥ X = X 0
If no NLOS bias is present, we can get the bias-free estimation of the MT X , which is written as X = ( H 0T R −1 H 0 ) −1 H 0T R −1 (r − h( X 0 ) + H 0 X 0 )
(14)
where R = diag (σ 2 , σ 2 , , σ 2 ) . Considering some constrained conditions, IPO method is used to estimate the NLOS bias b , which can be formulated as min J (b) = bT S T Q −1 Sb − 2Z T S T b
⎧ 0 ≤ bi ≤ ui , i = 1, 2 , N ⎪ s.t ⎨ui = min{ri + rj − Li , j }∀j ≠ i ⎪ Li , j =|| X i − X j || ⎩
(15)
where S = I + H 0V , Q = H 0 ( H 0T R −1 H 0 ) −1 H 0T + R ,
70
(16)
The proposed IPO-RS method is described as follow 1. It is known that three-stage location estimation algorithm needs a good initial estimation from (13). Here, LLS-2 or LLS-3 algorithm is proposed to get the location estimation of the MT, which used as the initial estimation of three-stage method. Then the NLOS bias bˆi can be estimated from (12-15). the corrected measurements 2. Compute d i = dˆi − bˆi , i = 1, 2, , N , and then use RS-MR to get the final location estimation of the MT. IV.
SIMULATION RESULTS
Monte-Carlo simulations are performed to compare the proposed algorithms with the existing methods. Assume there are four FTs which are positioned at X 1 = [−1, 20] , X 2 = [10,10] , X 3 = [4, −6] , X 4 = [−5, −5] . The location of MT is X = [2, 2] . To compare the performance of different algorithms, we need to analyze their mean square error (MSE). The definition of MSE is MSE = E || X − Xˆ ||2
The mean square error(MSE)
Xˆ = X + Vbˆ
The minimum distance as the RS The minimum residual as the RS CRLB
60
50
40
30
20
10
0
0
5
10
15
20
25
30
35
40
The variance of measurement error
Fig.1 The comparison of the minimum measured distance as the reference selection (RS) and the minimum residual as the RS 60
The mean square error(MSE)
Z = r − h( X 0 ) + H 0 ( X 0 − X ) , V = −( H 0T R −1 H 0 ) −1 H 0T R −1 . Solving the optimization problem in (15), we can obtain the estimation of the NLOS bias bˆ . Finally, an optimal estimation of the MT position Xˆ can be got as follow
The minimum distance as the RS(one nlos) The Rwgh algorithm(one nlos) The minimum distance as the RS(two nlos) The Rwgh algorithm(two nlos) The minimum distance as the RS(three nlos) The Rwgh algorithm(three nlos) CRLB
50
40
30
20
10
(17) 0
1
1.5
2
2.5
3
3.5
4
The mean of NLOS error
Fig.2 MSE VS the mean of NLOS error when σ 2 = 1 . 60
The minimum distance as the RS(one nlos) The Rwgh algorithm(one nlos) The minimum distance as the RS(two nlos) The Rwgh algorithm(two nlos) The minimum distance as the RS(three nlos) The Rwgh algorithm(three nlos) CRLB
55
The mean square error(MSE)
where Xˆ is the location estimation of the MT X . All the simulation results are obtained based on 10,000 independent runs. In LOS environment, Fig.1 demonstrates that the proposed algorithm (RS-MR) is better than the minimum measured distance acting as the reference FT, and the performance improves significantly with the increasing of LOS variance σ 2 . In NLOS environment, we assume that the NLOS error is exponentially distributed [14] with mean λ . Fig.2 compares new Rwgh with the LOS algorithm for different LOS/NLOS conditions. We observe that the new Rwgh indeed mitigates the effect of the NLOS error, especially when there is only one NLOS FT. Fig.3 demonstrates the effect of LOS variance σ 2 for different LOS/NLOS conditions. As shown in Fig.3, the Rwgh algorithm improves the performance when the LOS variance is large. Fig.4-Fig.6 are shown to compare three different algorithms when λ and σ 2 are different, and the initial estimation of the three-stage algorithm is obtained by LLS-2 or LLS-3 algorithm. We find that the proposed IPO-RS algorithm is always better than two other algorithms for different parameters, and the mean of NLOS error has great effect on the three-stage algorithm.
50 45 40 35 30 25 20 15 10 5 2
4
6
8
10
12
14
16
The variance of measurement error
Fig.3 MSE VS the variance of measurement error when λ = 2 .
V.
The mean square error(MSE)
25
In this paper, three algorithms are presented in LOS and NLOS environments. Simulation results demonstrate that: 1) in LOS environment, the proposed RS-MR method is better than the minimum distance acting as the reference FT algorithm; 2) in NLOS environment, the new Rwgh method can indeed mitigate the effect of NLOS error, and the algorithm only has better performance when the number of LOS FTs is more than two; 3) the proposed IPO-RS method has better performance than the minimum distance (IPO-RS) method, and the performance of the three-stage algorithm decreases significantly in dense NLOS environment.
three-stage algorithm the minimum distance(IPO-RS) the minimum residual(IPO-RS) CRLB
20
CONCLUSION
15
10
5
REFERENCES [1]
0 3LOS,1NLOS
2LOS,2NLOS
1LOS,3NLOS
0LOS,4NLOS
LOS/NLOS condition of the measurements
Fig.4 MSE VS different LOS/NLOS conditions when λ = 4, σ 2 = 1 .
[2]
[3]
The mean square error(MSE)
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three-stage algorithm the minimum distance(IPO-RS) the minimum residual(IPO-RS) CRLB
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2LOS,2NLOS
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0LOS,4NLOS
LOS/NLOS condition of the measurements
Fig.5 MSE VS different LOS/NLOS conditions when λ = 4, σ 2 = 4 .
[8]
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The mean square error(MSE)
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three-stage algorithm the minimum distance(IPO-RS) the minimum residual(IPO-RS) CRLB
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0 3LOS1NLOS
2LOS2NLOS
1LOS3NLOS
LOS/NLOS condition of the measurements
Fig.6 MSE VS different LOS/NLOS conditions when λ = 6, σ 2 = 4 .
0LOS4NLOS
[14]
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