- Email: [email protected]
Multiple targets localization approach based on virtual fingerprinting in indoor environment
3rd IFAC International Conference on Intelligent Control and Automation Science. September 2-4, 2013. Chengdu, China
Multiple targets localization approach based on virtual fingerprinting in indoor environment Xiansheng Guo ∗,∗∗ Baogen Xu, Zhongchu Rao ∗∗ Hongbing Zhang ∗ ∗
University of Electronic Science and Technology of China Chengdu, 611731 China (e-mail: [email protected]). ∗∗ TONGFANG Electronic Science and Technology Limited Company, Jiujiang, Jiangxi, China, (e-mail: [email protected]) Abstract: The main drawbacks of indoor localization approach based on Received Signal Strength (RSS) is the sensitivity to the fluctuation of measured RSS at a fixed location in difference time. In this paper, a novel multiple target localization approach is proposed by exploiting virtual fingerprinting, which is calculated from spacial relationship between reference nodes and virtual location nodes. The merits of our proposed method is twofold. First, our proposed method does not require the process of RSS fingerprinting constructing, which is a huge burden of localization methods based on RSS. Secondly, our proposed method is more robust than some existing methods, such as TOA-based method and RSS-based method, in big measurement error case. The efficacy of our proposed method are proven by means of some experimental results. Keywords: Virtual fingerprinting, indoor localization, received signal strength(RSS) 1. INTRODUCTION Indoor localization problem aims to determine the physical position of mobile terminals within a network Figuera et al. (2009); Pivato et al. (2011) and is becoming an important potential services in many application scenario. Different approaches to the problem have been proposed recentlyFeng et al. (2012); Guo et al. (2010, 2009), and the most popular being the use of the received signal strength (RSS) at a mobile device from a set of fixed radio devices because the base infrastructure is typically already presented in the form of Wireless Fidelity, or WiFi. As we all known, the severe multipath propagation Guo et al. (2007) in indoor localization is the main reason of the severe fluctuation of RSS for the mobile client even at a fixed location. This kind of fluctuation could result in severe spacial variability of RSS even in the same location in complex indoor environment, which will result in poor performance in location precision and robustness.
fortunately, this technique require a process of estimating parameters using plenty of measurement and the adopted propagation model generally does not describe the actual environment exactly. The latter is based on two stages: Firstly, a database of RSS measurements (also known as fingerprinting) made at a set of known locations can be initially assembled, which is done in an off-line phase, and the resultant database is used as the training set for a statistical learning model Wu et al. (2007); Ouyang et al. (2012). Secondly, in an on-line phase, the learning model is used to estimate a location from a given new set of RSS values. This technique generally overcomes the aforementioned limitations of the propagation model-based techniques, specially in complex indoor environment. However, the main drawback of fingerprinting-based technique is the heavy burden of building the resultant database in off-line phase Bshara et al. (2010).
This paper proposes a novel eliminating approach about spacial variability for indoor localization. By transforming RSS to a new stable space (called dual space), the spacial The existing RSS-based methods can be divided into two variability of RSS can be reduced effectively and the actypes: propagation model-based techniques and fingerprinting-curacy of localization can be improved consequently. The based techniques. The former uses a physical radio propa- merits of our proposed approach is twofold. Firstly, our gation model and information about the building geometry proposed approach does not require the process of building to estimate the location of mobile device. This approach the database so that it is low computational complexity can provide good location results as long as the adopted localization method. Secondly, our proposed approach can propagation model describes the scenario properly. Un- improve the accuracy of localization in complex indoor environment effectively, especially for the big fluctuation of Thanks to the National Natural Science Foundation of China RSS case. The simulation results demonstrate the efficacy under Grant (61201277), China Postdoctoral Science Foundation of our proposed approach. (No.2012M510168), Postdoctoral Science Foundation of Jiangxi Province of China, and the Fundamental Research Funds for the Central Universities (No. ZYGX2010J018) for funding.
978-3-902823-45-8/2013 © IFAC
630
10.3182/20130902-3-CN-3020.00153
IFAC ICONS 2013 September 2-4, 2013. Chengdu, China
2. PROBLEM FORMULATION
(5), nonlinear methods and linear methods can be used to estimate the location of sources. The location estimation of RSS-based method can be given by
2.1 RSS-based approach Consider that there are L anchor nodes and one unknown sensor node in a WSN, where the locations of anchor nodes are respectively denoted by x1 , x2 , · · · , xL , and the location of the unknown sensor node is denoted by x. The dimension of x0 is D × 1 with D = 2 or 3. In this paper, we only consider the case of D = 2 without loss of generality. In this case, the coordinate of ith anchor nodes T is xl = [xl yl ] (l = 1, 2, · · · , L) and the coordinate of T the unknown sensor node x = [x y] is to be estimated. The path loss between the lth anchor and the unknown sensor node can be denoted in term of the following radio propagation path loss model Song (1994) Pr,l = Kl Pt d−α = Kl Pt x − xl −α (1) 2 , l where Pt is a transmitted power, Pr,l is a received power at the lth sensor node, dl is the distance between the unknown sensor node and the lth anchor node. And Kl accounts for all other factors that affect the received power, including the antenna height and antenna gain, while α is the pass loss constant. Depending on the propagation environment, α can vary from 2 to 5.
ˆ = arg min (rRSS − fRSS (x))T (rRSS − fRSS (x)). (10) x x
2.2 TOA-based localization approach Without loss of generality, we assume that the source emits a signal at time 0 and the lth sensor receives it at time tl ; dl , l = 1, 2, · · · , L, (11) c where c 3 × 108 ms−1 is the speeds of light. The range measurement of TOA is modeled as tl =
rTOA,l = dl + nTOA,l ,
where nTOA,l is the range error in rTOA,l , which results from the TOA disturbance, and the vector form of Equation (12) can be written as rTOA = fTOA + nTOA ,
rRSS,l = 10logPr,l − 10logKl − 10logPt , The RSS signal model is simplified to
(3)
(4) rRSS,l = −10αlogdl + nRSS,l . The vector form of Equation (4) can be expressed as rRSS = fRSS (x) + nRSS ,
(5)
where T
rRSS = [rRSS,1 rRSS,2 · · · , rRSS,L ] ,
(6)
T
nRSS = [nRSS,1 nRSS,2 · · · , nRSS,L ] ,
(7)
(13)
where T
Field trials have validated that the disturbance in RSS is lognormal distributed. Accordingly, the lognormal path loss model can be expressed as 10logPr,l = 10logKl + 10logPt − 10αlogdl + nRSS,l . (2) Now, the unit on left side of (2) becomes dBm and the disturbance nRSS,l , l = 1, 2, · · · , L are zero-mean uncorrelated 2 }. Let Gaussian processes with variances {σRSS,l
(12)
rTOA = [rTOA,1 rTOA,2 , · · · , rTOA,L ] , T
(14)
nTOA = [nTOA,1 nTOA,2 , · · · , nTOA,L ] ,
(15)
fTOA (x) = d.
(16)
and
Assume that {nTOA,l } are zero-mean uncorrelated Gaus2 }, the PDF of TOA sian processes with variances {σTOA,l can be expressed as
exp − 12 (rTOA − d)T C−1 TOA (rTOA − d) p (rTOA ) = (17) , L/2 (2π) |CTOA |1/2 2 2 2 , σTOA,2 , · · · , σTOA,L where CTOA = diag σTOA,1 is the covariance matrix of nTOA . Based on Equation (13), nonlinear methods and linear methods can be used to estimate the location of sources. The location estimation of RSS-based method can be given by ˆ = arg min (rTOA − fTOA (x))T (rTOA − fTOA (x)).(18) x
and
x
(8) fRSS (x) = p = −10αlog (d) . The distance vector d has the following expression: ⎤ ⎡ 2 2 (x − x1 ) + (y − y1 ) ⎥ ⎢ ⎥ ⎢ ⎢ (x − x2 )2 + (y − y2 )2 ⎥ T ⎥ .(9) ⎢ d = [d1 , d2 , · · · , dL ] = ⎢ ⎥ .. ⎥ ⎢ . ⎦ ⎣ 2
2
(x − xL ) + (y − yL )
The source location problem based on RSS measurement is to estimate x given {rRSS,l } or rRSS .Based on Equation 631
From Equation (18), three commonly used local search methods, namely, Newton-Raphson, Gauss-Newton, and steepest descent methods can be used to estimate the locations of multiple target. The iterative Newton-Raphson procedure for x ˆ is k k x ˆk+1 = x ˆ ∇ JNLS,TOA x ˆ (19) , ˆk − H−1 JNLS,TOA x k k ˆ and ∇ JNLS,TOA x ˆ are the where H JNLS,TOA x corresponding Hessian matrix and gradient vector computed at the kth iterative estimation, namely, x ˆk , and they have the forms of
IFAC ICONS 2013 September 2-4, 2013. Chengdu, China
H (JNLS,TOA (x)) = =
∂ 2 JNLS,TOA (x) ∂x∂xT
∂2 J (x) ∂ 2 J NLS,TOA ∂x2
(20) NLS,TOA (x)
∂x∂y
∂ 2 JNLS,TOA (x) ∂ 2 JNLS,TOA (x) ∂y∂x ∂y 2
,
and ∇ (JNLS,TOA (x)) =
∂JNLS,TOA (x) ∂x ∂JNLS,TOA (x) ∂y
where m = 1, 2, · · · , L,n = m + 1, m + 2, · · · , L, and k = 1, 2, · · · , L (L − 1) /2. Hence the dimension of D is (L (L − 1) /2) × N . The constructed dual fingerprinting can alleviate the spacial variability of RSS by comparing the differences of RSS values between each pair of sensors. And the measure function g (·) in dual fingerprinting case is defined as L(L−1)/2
.
(21)
g (fDUAL (r) , D) = min i
Note that RSS-based techniques and TOA-based techniques are very sensitive to the range measurement error so that the performances of them are poor in complex indoor environment. In the next section, we present a novel location approach to improve the performance of them.
|fDUAL (r (k)) − di (k)|, (26)
k=1
where |·| is absolute operator and the location estimation ˆ is given by of target x ˆ= x
N
xi · g (fDUAL (r) , D).
(27)
i=1
3. THE PROPOSED LOCATION APPROACH The fundamental objective in a fingerprinting location ˆ from the reference locations system is to estimate x stored in a database. This database can also be called as fingerprinting, which is formulated as D = fFP (R) ,
(22)
and the location estimation can be given by ˆ= x
N
xi · g (fFP (r) , D),
(23)
The merits of our proposed location approach is twofold: Firstly, the dimension of original measured RSS fingerprinting is L×N (L is the number of anchor nodes and N is the number of reference locations), while the dimension of transformed dual fingerprinting becomes (L (L − 1) /2) × N . It is obvious that the dimension of transformed dual fingerprinting is far more than that of original fingerprinting if enough anchor nodes are available. Secondly, the transformed dual fingerprinting can utilize the structure of RSS efficiently because the differences between RSS values are used. 4. SIMULATION RESULTS
i=1
where xi is the location of ith known reference grids. And R = [r1 , r2 , · · · , rN ] is a range matrix with dimension L × N , which is computed from N known reference locations. In other words, we do not need to build RSS fingerprinting in offline phase, which usually waste plenty of time in practice, or we can call R virtual fingerprinting. D = [d1 , d2 , · · · , dN ] , i = 1, 2, · · · , N is a constructed fingerprinting with di being the transformed fingerprinting of the jth reference location, and fFP (·) represents a transform function between the measured RSS and the constructed fingerprinting. The measure g (·) is a function that measures the correlation between measurements and locations and r is the online measured RSS. Many previous research has shown that it is feasible to perform positioning in an alternative transformed space, where the location information is more efficiently utilized. Several transformed methods have been discussed, including MDA, PCA, etc,. In this paper, we present a novel transform function fDUAL who transforms the measured RSS to a dual space to reduce the variability of RSS. The dual space fingerprinting can be expressed as DDUAL = fDUAL (R) ,
(24)
where fDUAL(·) is the transform function between virtual fingerprinting and the constructed fingerprinting. The elements of dual fingerprinting are 1 or -1 depending on the difference between the RSS values in each column of measured RSS, i.e., R. Mathematically, it can be formulated as 1, ri (m) − ri (n) ≥ 0 di (k) = fDUAL (ri ) = , (25) −1, ri (m) − ri (n) < 0 632
4.1 The diamond geometry of five reference nodes Consider a 2-D geometry of L = 5 reference nodes with known coordinates at (0, 2),(2, 4),(2, 2), (2, 0), (4, 2), is shown in Fig. 1, which is a diamond with a central node. The unknown source position is distributed randomly around (x, y) = (2, 3), the range error variance, σl2 , is assigned proportional to d2l with SNR = d2l /σl2 . We compare the cumulative probability density (CDF) of error of listed algorithms with our proposed approach and the results are based on 1000 independent runs. Some existing algorithms are listed for comparing the location performance. They include RSS-based method and two TOA-based methods. The RSS-based method uses Equation (10) as cost function. The two TOA-based methods: linear least square (LLS) method and NewtonRaphson iterative method. The Newton-Raphson iterative method is based on Equation (18). The initial target location is set to [3, 2] and the iteration number is 50. (See So (2012) for more details). So if we want to obtain a better performance in general case, a efficient combination method can be used. To show the performance of our approach in different SNRs, we plot the RMSE of location error for different methods with SNR=10dB in Fig. 2. To show the performance of four listed methods, we also show the RMSE of location error for different methods with SNR=-10dB in Fig. 3. From the two figures, it is easily find that our approach is more robust to measured range error than other three methods. The superiority is remarkable as SNR decreases because our approach makes full use of the structure information of measured RSS. This superiority is very useful for complex indoor location environment because severe multi-path is everywhere.
IFAC ICONS 2013 September 2-4, 2013. Chengdu, China
located on the four corner and the central of rectangle, which is depicted in Fig. 4. The unknown source position is distributed randomly around (x, y) = (2, 3), the range error variance, σl2 , is assigned proportional to d2l with SNR = d2l /σl2 . We also compare the cumulative probability density (CDF) of error of listed algorithms with our proposed approach and the results are based on 1000 independent runs. The initial target location is set to [3, 2] and the iteration number is 50. To compare the location performance of listed approaches, we plot the RMSE of location error of different approaches in Fig. 5 with SNR being 10 dB. We are also interested in how sensitive is the performance of the proposed method with respect to the value of low SNR, we illustrate the RMSE of location error of different approaches in Fig. 6. As can be seen from Figs. 5 and 6, our method is very robust to the change of SNR. Generally speaking, the lower SNR is, the more robust of our method. The interpretation is that our method obtains the locations of multiple target by means of comparing the difference between two adjacent data of RSS fingerprint, which is efficient in dealing with the spacial fluctuation of RSS fingerprinting. From the results of two experiments, we can find that the different geometries of reference nodes give different location performance under the same SNR. But the performance of our proposed approach is always outperform the other listed algorithms at low SNR, which can be seen from Figs. 3 and 6. For higher SNR case, the performance of LLS, Newton-Raphson, and RSS methods is promoted, which can be seen from Figs. 2 and 5. If we need higher localization performance, a efficient combination method of these algorithm can be used to determine the final precision of location, such as some well known resampling methodGuo et al. (2010).
4 3.5
y(meter)
3 2.5 2 1.5 1 0.5 0 0
0.5
1
1.5
2 2.5 x(meter)
3
3.5
4
Fig. 1. The geometry of reference nodes. 1
Cumulative probability density
0.9 0.8 0.7 0.6 0.5 0.4 0.3
LLS Our approach Newton−Raphson RSS
0.2 0.1 0 0
0.5
1
RMSE(m)
1.5
2
4 3.5
Fig. 2. The root mean square error (RMSE) of location error of different algorithms, SNR=10dB. y(meter)
1 LLS Our approach Newton−Raphson RSS
Cumulative probability density
0.9 0.8 0.7
3 2.5 2 1.5 1
0.6
0.5
0.5 0 0
0.4 0.3 0.2
0.5
1
1.5
2 2.5 x(meter)
3
3.5
4
Fig. 4. The rectangle geometry of five reference nodes.
0.1 0 0
0.5
1
RMSE(m)
1.5
5. CONCLUSIONS
2
Fig. 3. The root mean square error (RMSE) of location error of different algorithms, SNR=-10dB. 4.2 The rectangle geometry of five reference nodes To test the influence of geometry of reference nodes to our algorithm, we consider that the five reference nodes are 633
In this paper, we proposed a novel approach to eliminate space variability of RSS in indoor localization,namely, a novel multiple target localization approach is proposed by exploiting virtual fingerprinting, which is calculated from spacial relationship between reference nodes and virtual location nodes. The merits of our proposed method is twofold. Firstly, our proposed method does not require the process of RSS fingerprinting constructing, which is
IFAC ICONS 2013 September 2-4, 2013. Chengdu, China
1
Cumulative probability density
0.9 0.8
LLS Virtual Newton−Raphson RSS
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.5
1
RMSE(m)
1.5
2
Fig. 5. The root mean square error (RMSE) of location error of different algorithms, SNR=10dB. 1
Cumulative probability density
0.9 0.8
LLS Virtual Newton−Raphson RSS
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.5
1
RMSE(m)
1.5
2
Fig. 6. The root mean square error (RMSE) of location error of different algorithms, SNR=-10dB. a huge burden of localization methods based on RSS. The dual fingerprinting is calculated from the ranges between virtual reference locations and reference nodes. Secondly, our proposed method is more robust than some existing methods, such as TOA-based method and RSSbased method, in more high measurement error case. The simulation results have demonstrated the efficacy of our proposed algorithms. ACKNOWLEDGEMENTS This work is supported by the National Natural Science Foundation of China under Grant (61201277), China Postdoctoral Science Foundation (No.2012M510168), Postdoctoral Science Foundation of Jiangxi Province of China, and the Fundamental Research Funds for the Central Universities (No. ZYGX2010J018) for funding. REFERENCES Bshara, M., Orguner, U., Gustafsson, F., and Van Biesen, L. (2010). Fingerprinting localization in wireless networks based on received-signal-strength measurements: 634
a case study on wimax networks. Vehicular Technology, IEEE Transactions on, 59(1), 283–294. Feng, C., Au, W., Valaee, S., and Tan, Z. (2012). Received signal strength based indoor positioning using compressive sensing. Mobile Computing, IEEE Transactions on, 11(12), 1983–1993. Figuera, C., Mora-Jim´enez, I., Guerrero-Curieses, A., ´ Rojo-Alvarez, J., Everss, E., Wilby, M., and RamosLopez, J. (2009). Nonparametric model comparison and uncertainty evaluation for signal strength indoor location. Mobile Computing, IEEE Transactions on, 8(9), 1250–1264. Guo, X., Wan, Q., Chang, C., and Lam, E.Y. (2010). Source localization using a sparse representation framework to achieve superresolution. Multidimensional Systems and Signal Processing, 21(4), 391–402. Guo, X., Wan, Q., Yang, W., and Lei, X. (2009). Lowcomplexity 2d coherently distributed sources decoupled doas estimation method. Science in China Series F: Information Sciences, 52(5), 835–842. Guo, X., Wan, Q., Wu, B., and Yang, W. (2007). Parameters localisation of coherently distributed sources based on sparse signal representation. IET Radar, Sonar & Navigation, 1(4), 261–265. Ouyang, R.W., Wong, A.K.S., Lea, C.T., and Chiang, M. (2012). Indoor location estimation with reduced calibration exploiting unlabeled data via hybrid generative/discriminative learning. Mobile Computing, IEEE Transactions on, 11(11), 1613–1626. Pivato, P., Palopoli, L., and Petri, D. (2011). Accuracy of rss-based centroid localization algorithms in an indoor environment. Instrumentation and Measurement, IEEE Transactions on, 60(10), 3451–3460. So, H.C. (2012). Source localization: Algorithms and analysis. Handbook of Position Location, 25–66. Song, H.L. (1994). Automatic vehicle location in cellular communications systems. Vehicular Technology, IEEE Transactions on, 43(4), 902–908. Wu, Z., Li, C., Ng, J.K.Y., and Leung, K.R.P.H. (2007). Location estimation via support vector regression. Mobile Computing, IEEE Transactions on, 6(3), 311–321.
Multiple targets localization approach based on virtual fingerprinting in indoor environment Xiansheng Guo ∗,∗∗ Baogen Xu, Zhongchu Rao ∗∗ Hongbing Zhang ∗ ∗
University of Electronic Science and Technology of China Chengdu, 611731 China (e-mail: [email protected]). ∗∗ TONGFANG Electronic Science and Technology Limited Company, Jiujiang, Jiangxi, China, (e-mail: [email protected]) Abstract: The main drawbacks of indoor localization approach based on Received Signal Strength (RSS) is the sensitivity to the fluctuation of measured RSS at a fixed location in difference time. In this paper, a novel multiple target localization approach is proposed by exploiting virtual fingerprinting, which is calculated from spacial relationship between reference nodes and virtual location nodes. The merits of our proposed method is twofold. First, our proposed method does not require the process of RSS fingerprinting constructing, which is a huge burden of localization methods based on RSS. Secondly, our proposed method is more robust than some existing methods, such as TOA-based method and RSS-based method, in big measurement error case. The efficacy of our proposed method are proven by means of some experimental results. Keywords: Virtual fingerprinting, indoor localization, received signal strength(RSS) 1. INTRODUCTION Indoor localization problem aims to determine the physical position of mobile terminals within a network Figuera et al. (2009); Pivato et al. (2011) and is becoming an important potential services in many application scenario. Different approaches to the problem have been proposed recentlyFeng et al. (2012); Guo et al. (2010, 2009), and the most popular being the use of the received signal strength (RSS) at a mobile device from a set of fixed radio devices because the base infrastructure is typically already presented in the form of Wireless Fidelity, or WiFi. As we all known, the severe multipath propagation Guo et al. (2007) in indoor localization is the main reason of the severe fluctuation of RSS for the mobile client even at a fixed location. This kind of fluctuation could result in severe spacial variability of RSS even in the same location in complex indoor environment, which will result in poor performance in location precision and robustness.
fortunately, this technique require a process of estimating parameters using plenty of measurement and the adopted propagation model generally does not describe the actual environment exactly. The latter is based on two stages: Firstly, a database of RSS measurements (also known as fingerprinting) made at a set of known locations can be initially assembled, which is done in an off-line phase, and the resultant database is used as the training set for a statistical learning model Wu et al. (2007); Ouyang et al. (2012). Secondly, in an on-line phase, the learning model is used to estimate a location from a given new set of RSS values. This technique generally overcomes the aforementioned limitations of the propagation model-based techniques, specially in complex indoor environment. However, the main drawback of fingerprinting-based technique is the heavy burden of building the resultant database in off-line phase Bshara et al. (2010).
This paper proposes a novel eliminating approach about spacial variability for indoor localization. By transforming RSS to a new stable space (called dual space), the spacial The existing RSS-based methods can be divided into two variability of RSS can be reduced effectively and the actypes: propagation model-based techniques and fingerprinting-curacy of localization can be improved consequently. The based techniques. The former uses a physical radio propa- merits of our proposed approach is twofold. Firstly, our gation model and information about the building geometry proposed approach does not require the process of building to estimate the location of mobile device. This approach the database so that it is low computational complexity can provide good location results as long as the adopted localization method. Secondly, our proposed approach can propagation model describes the scenario properly. Un- improve the accuracy of localization in complex indoor environment effectively, especially for the big fluctuation of Thanks to the National Natural Science Foundation of China RSS case. The simulation results demonstrate the efficacy under Grant (61201277), China Postdoctoral Science Foundation of our proposed approach. (No.2012M510168), Postdoctoral Science Foundation of Jiangxi Province of China, and the Fundamental Research Funds for the Central Universities (No. ZYGX2010J018) for funding.
978-3-902823-45-8/2013 © IFAC
630
10.3182/20130902-3-CN-3020.00153
IFAC ICONS 2013 September 2-4, 2013. Chengdu, China
2. PROBLEM FORMULATION
(5), nonlinear methods and linear methods can be used to estimate the location of sources. The location estimation of RSS-based method can be given by
2.1 RSS-based approach Consider that there are L anchor nodes and one unknown sensor node in a WSN, where the locations of anchor nodes are respectively denoted by x1 , x2 , · · · , xL , and the location of the unknown sensor node is denoted by x. The dimension of x0 is D × 1 with D = 2 or 3. In this paper, we only consider the case of D = 2 without loss of generality. In this case, the coordinate of ith anchor nodes T is xl = [xl yl ] (l = 1, 2, · · · , L) and the coordinate of T the unknown sensor node x = [x y] is to be estimated. The path loss between the lth anchor and the unknown sensor node can be denoted in term of the following radio propagation path loss model Song (1994) Pr,l = Kl Pt d−α = Kl Pt x − xl −α (1) 2 , l where Pt is a transmitted power, Pr,l is a received power at the lth sensor node, dl is the distance between the unknown sensor node and the lth anchor node. And Kl accounts for all other factors that affect the received power, including the antenna height and antenna gain, while α is the pass loss constant. Depending on the propagation environment, α can vary from 2 to 5.
ˆ = arg min (rRSS − fRSS (x))T (rRSS − fRSS (x)). (10) x x
2.2 TOA-based localization approach Without loss of generality, we assume that the source emits a signal at time 0 and the lth sensor receives it at time tl ; dl , l = 1, 2, · · · , L, (11) c where c 3 × 108 ms−1 is the speeds of light. The range measurement of TOA is modeled as tl =
rTOA,l = dl + nTOA,l ,
where nTOA,l is the range error in rTOA,l , which results from the TOA disturbance, and the vector form of Equation (12) can be written as rTOA = fTOA + nTOA ,
rRSS,l = 10logPr,l − 10logKl − 10logPt , The RSS signal model is simplified to
(3)
(4) rRSS,l = −10αlogdl + nRSS,l . The vector form of Equation (4) can be expressed as rRSS = fRSS (x) + nRSS ,
(5)
where T
rRSS = [rRSS,1 rRSS,2 · · · , rRSS,L ] ,
(6)
T
nRSS = [nRSS,1 nRSS,2 · · · , nRSS,L ] ,
(7)
(13)
where T
Field trials have validated that the disturbance in RSS is lognormal distributed. Accordingly, the lognormal path loss model can be expressed as 10logPr,l = 10logKl + 10logPt − 10αlogdl + nRSS,l . (2) Now, the unit on left side of (2) becomes dBm and the disturbance nRSS,l , l = 1, 2, · · · , L are zero-mean uncorrelated 2 }. Let Gaussian processes with variances {σRSS,l
(12)
rTOA = [rTOA,1 rTOA,2 , · · · , rTOA,L ] , T
(14)
nTOA = [nTOA,1 nTOA,2 , · · · , nTOA,L ] ,
(15)
fTOA (x) = d.
(16)
and
Assume that {nTOA,l } are zero-mean uncorrelated Gaus2 }, the PDF of TOA sian processes with variances {σTOA,l can be expressed as
exp − 12 (rTOA − d)T C−1 TOA (rTOA − d) p (rTOA ) = (17) , L/2 (2π) |CTOA |1/2 2 2 2 , σTOA,2 , · · · , σTOA,L where CTOA = diag σTOA,1 is the covariance matrix of nTOA . Based on Equation (13), nonlinear methods and linear methods can be used to estimate the location of sources. The location estimation of RSS-based method can be given by ˆ = arg min (rTOA − fTOA (x))T (rTOA − fTOA (x)).(18) x
and
x
(8) fRSS (x) = p = −10αlog (d) . The distance vector d has the following expression: ⎤ ⎡ 2 2 (x − x1 ) + (y − y1 ) ⎥ ⎢ ⎥ ⎢ ⎢ (x − x2 )2 + (y − y2 )2 ⎥ T ⎥ .(9) ⎢ d = [d1 , d2 , · · · , dL ] = ⎢ ⎥ .. ⎥ ⎢ . ⎦ ⎣ 2
2
(x − xL ) + (y − yL )
The source location problem based on RSS measurement is to estimate x given {rRSS,l } or rRSS .Based on Equation 631
From Equation (18), three commonly used local search methods, namely, Newton-Raphson, Gauss-Newton, and steepest descent methods can be used to estimate the locations of multiple target. The iterative Newton-Raphson procedure for x ˆ is k k x ˆk+1 = x ˆ ∇ JNLS,TOA x ˆ (19) , ˆk − H−1 JNLS,TOA x k k ˆ and ∇ JNLS,TOA x ˆ are the where H JNLS,TOA x corresponding Hessian matrix and gradient vector computed at the kth iterative estimation, namely, x ˆk , and they have the forms of
IFAC ICONS 2013 September 2-4, 2013. Chengdu, China
H (JNLS,TOA (x)) = =
∂ 2 JNLS,TOA (x) ∂x∂xT
∂2 J (x) ∂ 2 J NLS,TOA ∂x2
(20) NLS,TOA (x)
∂x∂y
∂ 2 JNLS,TOA (x) ∂ 2 JNLS,TOA (x) ∂y∂x ∂y 2
,
and ∇ (JNLS,TOA (x)) =
∂JNLS,TOA (x) ∂x ∂JNLS,TOA (x) ∂y
where m = 1, 2, · · · , L,n = m + 1, m + 2, · · · , L, and k = 1, 2, · · · , L (L − 1) /2. Hence the dimension of D is (L (L − 1) /2) × N . The constructed dual fingerprinting can alleviate the spacial variability of RSS by comparing the differences of RSS values between each pair of sensors. And the measure function g (·) in dual fingerprinting case is defined as L(L−1)/2
.
(21)
g (fDUAL (r) , D) = min i
Note that RSS-based techniques and TOA-based techniques are very sensitive to the range measurement error so that the performances of them are poor in complex indoor environment. In the next section, we present a novel location approach to improve the performance of them.
|fDUAL (r (k)) − di (k)|, (26)
k=1
where |·| is absolute operator and the location estimation ˆ is given by of target x ˆ= x
N
xi · g (fDUAL (r) , D).
(27)
i=1
3. THE PROPOSED LOCATION APPROACH The fundamental objective in a fingerprinting location ˆ from the reference locations system is to estimate x stored in a database. This database can also be called as fingerprinting, which is formulated as D = fFP (R) ,
(22)
and the location estimation can be given by ˆ= x
N
xi · g (fFP (r) , D),
(23)
The merits of our proposed location approach is twofold: Firstly, the dimension of original measured RSS fingerprinting is L×N (L is the number of anchor nodes and N is the number of reference locations), while the dimension of transformed dual fingerprinting becomes (L (L − 1) /2) × N . It is obvious that the dimension of transformed dual fingerprinting is far more than that of original fingerprinting if enough anchor nodes are available. Secondly, the transformed dual fingerprinting can utilize the structure of RSS efficiently because the differences between RSS values are used. 4. SIMULATION RESULTS
i=1
where xi is the location of ith known reference grids. And R = [r1 , r2 , · · · , rN ] is a range matrix with dimension L × N , which is computed from N known reference locations. In other words, we do not need to build RSS fingerprinting in offline phase, which usually waste plenty of time in practice, or we can call R virtual fingerprinting. D = [d1 , d2 , · · · , dN ] , i = 1, 2, · · · , N is a constructed fingerprinting with di being the transformed fingerprinting of the jth reference location, and fFP (·) represents a transform function between the measured RSS and the constructed fingerprinting. The measure g (·) is a function that measures the correlation between measurements and locations and r is the online measured RSS. Many previous research has shown that it is feasible to perform positioning in an alternative transformed space, where the location information is more efficiently utilized. Several transformed methods have been discussed, including MDA, PCA, etc,. In this paper, we present a novel transform function fDUAL who transforms the measured RSS to a dual space to reduce the variability of RSS. The dual space fingerprinting can be expressed as DDUAL = fDUAL (R) ,
(24)
where fDUAL(·) is the transform function between virtual fingerprinting and the constructed fingerprinting. The elements of dual fingerprinting are 1 or -1 depending on the difference between the RSS values in each column of measured RSS, i.e., R. Mathematically, it can be formulated as 1, ri (m) − ri (n) ≥ 0 di (k) = fDUAL (ri ) = , (25) −1, ri (m) − ri (n) < 0 632
4.1 The diamond geometry of five reference nodes Consider a 2-D geometry of L = 5 reference nodes with known coordinates at (0, 2),(2, 4),(2, 2), (2, 0), (4, 2), is shown in Fig. 1, which is a diamond with a central node. The unknown source position is distributed randomly around (x, y) = (2, 3), the range error variance, σl2 , is assigned proportional to d2l with SNR = d2l /σl2 . We compare the cumulative probability density (CDF) of error of listed algorithms with our proposed approach and the results are based on 1000 independent runs. Some existing algorithms are listed for comparing the location performance. They include RSS-based method and two TOA-based methods. The RSS-based method uses Equation (10) as cost function. The two TOA-based methods: linear least square (LLS) method and NewtonRaphson iterative method. The Newton-Raphson iterative method is based on Equation (18). The initial target location is set to [3, 2] and the iteration number is 50. (See So (2012) for more details). So if we want to obtain a better performance in general case, a efficient combination method can be used. To show the performance of our approach in different SNRs, we plot the RMSE of location error for different methods with SNR=10dB in Fig. 2. To show the performance of four listed methods, we also show the RMSE of location error for different methods with SNR=-10dB in Fig. 3. From the two figures, it is easily find that our approach is more robust to measured range error than other three methods. The superiority is remarkable as SNR decreases because our approach makes full use of the structure information of measured RSS. This superiority is very useful for complex indoor location environment because severe multi-path is everywhere.
IFAC ICONS 2013 September 2-4, 2013. Chengdu, China
located on the four corner and the central of rectangle, which is depicted in Fig. 4. The unknown source position is distributed randomly around (x, y) = (2, 3), the range error variance, σl2 , is assigned proportional to d2l with SNR = d2l /σl2 . We also compare the cumulative probability density (CDF) of error of listed algorithms with our proposed approach and the results are based on 1000 independent runs. The initial target location is set to [3, 2] and the iteration number is 50. To compare the location performance of listed approaches, we plot the RMSE of location error of different approaches in Fig. 5 with SNR being 10 dB. We are also interested in how sensitive is the performance of the proposed method with respect to the value of low SNR, we illustrate the RMSE of location error of different approaches in Fig. 6. As can be seen from Figs. 5 and 6, our method is very robust to the change of SNR. Generally speaking, the lower SNR is, the more robust of our method. The interpretation is that our method obtains the locations of multiple target by means of comparing the difference between two adjacent data of RSS fingerprint, which is efficient in dealing with the spacial fluctuation of RSS fingerprinting. From the results of two experiments, we can find that the different geometries of reference nodes give different location performance under the same SNR. But the performance of our proposed approach is always outperform the other listed algorithms at low SNR, which can be seen from Figs. 3 and 6. For higher SNR case, the performance of LLS, Newton-Raphson, and RSS methods is promoted, which can be seen from Figs. 2 and 5. If we need higher localization performance, a efficient combination method of these algorithm can be used to determine the final precision of location, such as some well known resampling methodGuo et al. (2010).
4 3.5
y(meter)
3 2.5 2 1.5 1 0.5 0 0
0.5
1
1.5
2 2.5 x(meter)
3
3.5
4
Fig. 1. The geometry of reference nodes. 1
Cumulative probability density
0.9 0.8 0.7 0.6 0.5 0.4 0.3
LLS Our approach Newton−Raphson RSS
0.2 0.1 0 0
0.5
1
RMSE(m)
1.5
2
4 3.5
Fig. 2. The root mean square error (RMSE) of location error of different algorithms, SNR=10dB. y(meter)
1 LLS Our approach Newton−Raphson RSS
Cumulative probability density
0.9 0.8 0.7
3 2.5 2 1.5 1
0.6
0.5
0.5 0 0
0.4 0.3 0.2
0.5
1
1.5
2 2.5 x(meter)
3
3.5
4
Fig. 4. The rectangle geometry of five reference nodes.
0.1 0 0
0.5
1
RMSE(m)
1.5
5. CONCLUSIONS
2
Fig. 3. The root mean square error (RMSE) of location error of different algorithms, SNR=-10dB. 4.2 The rectangle geometry of five reference nodes To test the influence of geometry of reference nodes to our algorithm, we consider that the five reference nodes are 633
In this paper, we proposed a novel approach to eliminate space variability of RSS in indoor localization,namely, a novel multiple target localization approach is proposed by exploiting virtual fingerprinting, which is calculated from spacial relationship between reference nodes and virtual location nodes. The merits of our proposed method is twofold. Firstly, our proposed method does not require the process of RSS fingerprinting constructing, which is
IFAC ICONS 2013 September 2-4, 2013. Chengdu, China
1
Cumulative probability density
0.9 0.8
LLS Virtual Newton−Raphson RSS
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.5
1
RMSE(m)
1.5
2
Fig. 5. The root mean square error (RMSE) of location error of different algorithms, SNR=10dB. 1
Cumulative probability density
0.9 0.8
LLS Virtual Newton−Raphson RSS
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.5
1
RMSE(m)
1.5
2
Fig. 6. The root mean square error (RMSE) of location error of different algorithms, SNR=-10dB. a huge burden of localization methods based on RSS. The dual fingerprinting is calculated from the ranges between virtual reference locations and reference nodes. Secondly, our proposed method is more robust than some existing methods, such as TOA-based method and RSSbased method, in more high measurement error case. The simulation results have demonstrated the efficacy of our proposed algorithms. ACKNOWLEDGEMENTS This work is supported by the National Natural Science Foundation of China under Grant (61201277), China Postdoctoral Science Foundation (No.2012M510168), Postdoctoral Science Foundation of Jiangxi Province of China, and the Fundamental Research Funds for the Central Universities (No. ZYGX2010J018) for funding. REFERENCES Bshara, M., Orguner, U., Gustafsson, F., and Van Biesen, L. (2010). Fingerprinting localization in wireless networks based on received-signal-strength measurements: 634
a case study on wimax networks. Vehicular Technology, IEEE Transactions on, 59(1), 283–294. Feng, C., Au, W., Valaee, S., and Tan, Z. (2012). Received signal strength based indoor positioning using compressive sensing. Mobile Computing, IEEE Transactions on, 11(12), 1983–1993. Figuera, C., Mora-Jim´enez, I., Guerrero-Curieses, A., ´ Rojo-Alvarez, J., Everss, E., Wilby, M., and RamosLopez, J. (2009). Nonparametric model comparison and uncertainty evaluation for signal strength indoor location. Mobile Computing, IEEE Transactions on, 8(9), 1250–1264. Guo, X., Wan, Q., Chang, C., and Lam, E.Y. (2010). Source localization using a sparse representation framework to achieve superresolution. Multidimensional Systems and Signal Processing, 21(4), 391–402. Guo, X., Wan, Q., Yang, W., and Lei, X. (2009). Lowcomplexity 2d coherently distributed sources decoupled doas estimation method. Science in China Series F: Information Sciences, 52(5), 835–842. Guo, X., Wan, Q., Wu, B., and Yang, W. (2007). Parameters localisation of coherently distributed sources based on sparse signal representation. IET Radar, Sonar & Navigation, 1(4), 261–265. Ouyang, R.W., Wong, A.K.S., Lea, C.T., and Chiang, M. (2012). Indoor location estimation with reduced calibration exploiting unlabeled data via hybrid generative/discriminative learning. Mobile Computing, IEEE Transactions on, 11(11), 1613–1626. Pivato, P., Palopoli, L., and Petri, D. (2011). Accuracy of rss-based centroid localization algorithms in an indoor environment. Instrumentation and Measurement, IEEE Transactions on, 60(10), 3451–3460. So, H.C. (2012). Source localization: Algorithms and analysis. Handbook of Position Location, 25–66. Song, H.L. (1994). Automatic vehicle location in cellular communications systems. Vehicular Technology, IEEE Transactions on, 43(4), 902–908. Wu, Z., Li, C., Ng, J.K.Y., and Leung, K.R.P.H. (2007). Location estimation via support vector regression. Mobile Computing, IEEE Transactions on, 6(3), 311–321.