A Novel Weighted Localization Method in Wireless Sensor Networks Based on Hybrid RSS/AoA Measurements

A hybrid RSS/AOA indoor localization method based on error variance and measurement noise weighted least squares (ENWLS) is proposed. This method is based on three-dimensional wireless sensor networks, and achieves high-precision indoor positioning without increasing its complexity. We use the first-order Taylor approximation to approximate the linear weighted least square (WLS) error, and use the weighted least squares estimation to roughly estimate the location of the target, then determine the weight matrix by estimating the linear WLS error variance and the measured noise value on the sensor node. Simulation results show that our proposed method is better than other existing hybrid RSS/AOA localization methods.


I. INTRODUCTION
In recent years, localization method plays an increasingly important role in wireless sensor networks (WSNs) [1]- [12]. WSNs are wireless networks composed of sensors. In outdoor environment, thanks to GPS and cellular network, mobile terminal location can achieve high accuracy. However, in indoor environment or with serious shadowing effect, satellite and cellular signals are often interrupted, and localization becomes a problem. This paper introduces an indoor localization method based on WSNs. WSNs are composed of anchors which locations are known and targets which locations are unknown. The location of the targets are determined by the location of the anchors and the measurements of radio signals.
Traditional radio signals include time of arrival (TOA) [13]- [17], time difference of arrival (TDOA) [18], [19], angle of arrival (AOA) [20]- [25] and received signal strength (RSS) [26]- [32]. Generally, the localization method based on independent signal has low accuracy. The hybrid localization methods can be combinations of the four radio signals. It is up to the sensor hardware to decide which kind of measurement method to adopt. Usually, one of RSS, TOA and TDOA is combined with AOA, because RSS, TOA and TDOA are distance related measurements, while AOA is angle related The associate editor coordinating the review of this manuscript and approving it for publication was Prakasam Periasamy . measurement. TOA has good localization accuracy, but it requires clock synchronization between anchors and targets. Obviously, realizing this task is highly challenging and will lead to an increase in the cost and the size of mobile devices. TDOA only needs the clock synchronization of anchors. Generally, the locations of anchors are unchanged, so TDOA can usually be wire and wireless. Compared with TOA, the hardware requirements of TDOA are lower, but more anchors are needed to ensure its accuracy. Since the radio signal travels continuously in the air, AOA can be calculated according to the phase difference of the radio signal reaching different antennas. And using antenna arrays is a common method to measure AOA. In the third-generation mobile telecommunication, antenna arrays were widely used in the base stations. Therefore, the base stations can provide AOA measurements at the mobile terminal and we can use it for positioning. Well, RSS measurements can also be obtained through the base stations. RSS and AOA do not require time synchronization, so they have low requirements for hardware, but they also have defects. Due to the multipath effect, the accuracy of RSS and AOA is greatly reduced in complex terrain, so RSS and AoA are more suitable for open terrain. The goal of this paper is to achieve open indoor positioning, so Non Line of Sight (NLOS) is not considered. Using hybrid RSS/AOA measurements can effectively reduce the impact of measurement noises and greatly improve the localiza-tion accuracy. Due to the strict requirements of TOA and TDOA for precise timing synchronization in measurement, the cost of the hybrid TOA/AOA measurement and the hybrid TDOA/AOA measurements are greatly increased. Therefore, the hybrid RSS/AOA measurements [33]- [36] provides an attractive low-cost solution for indoor localization problems.
In theory, the estimation of the target node location can be completely accurate according to the exact RSS/AOA measurement. However, due to the RSS/AOA measurement error, the localization problem becomes an NP hard problem. Target position estimation based on hybrid RSS/AOA measurement is an optimization problem of non-convex system, and the challenge is to overcome the noise of the measurement. Semi-definite programming (SDP) and second-order cone programming (SOCP) can effectively solve this problem [37]. The SDP/SOCP method proposed by Chang et al. [38] uses the SDP and SOCP methods to transform the non-convex system into a convex system, which has good accuracy. However, the complexity of these methods are too high. The calculation process may delay the positioning. Therefore, it is more practical to use the least square (LS) method to estimate the location after the measurement model is linearized [39]. This method not only has good estimation accuracy, but also greatly reduces the complexity. The weighted least square (WLS) method proposed by Tomic et al. [40] improves the accuracy of estimation without increasing the complexity. But the weight that only changes with the distances between target and anchors is not the best. The error covariance WLS (ECWLS) proposed by Kang et al. [41] changes the weight. First, they use the LS method to estimate the approximate location of the target, and then calculate the approximate error covariance matrix according to the approximate position. The approximate error covariance matrix is used as the weight. Due to the limitation of the number of anchors, the estimation of the variance of measurement noise has large error. This method directly multiplies the estimated of the variance of the measurement noise into the weight, which will increase the error. The two-step error varianceweighted least squares (TELS) proposed by Watanabe is based on AOA measurement [42]. He uses the LS method to estimate the approximate location of the target, and then calculates the variance as the weight. While he only used AOA measurements, that causes a larger error. The above two methods only consider the influence of the noise variance of evaluation function item, but do not consider the influence of the noise value of the measurements. When the noise standard deviation is the same, the one with small noise value should have more weight.
This paper presents an indoor location method with better performance without increasing the complexity, called error variance and noise value WLS (ENWLS). First, the approximate location of the target is estimated by WLS. Then, according to the theoretical derivation and the approximate location of the target, the variance of each anchor and each evaluation function are calculated. Second, the noise values of the measurements are calculated. Finally, the two items are used as the weight of each anchor and each evaluation function. This method called ''ECWLS''. Its performance is better than the existing methods. The simulation results confirm that. Its complexity is the same as the existing method in linear time. In summary, the main contribution of this paper is to propose a hybrid RSS/AOA target indoor localization method witch performance is best.
The rest of this paper is as follows. The Section II introduces the models of the measurements and the related WLS method. The Section III introduces the establishment of the problem and the proposed ENWLS method. The Section IV gives the simulations, the clarifications on comparisons' figures and the complexity analyzes. The Section V summarizes this paper and introduces the future works of the proposed method.

II. RELATED WORKS
This section describes the model of hybrid RSS/AOA measurement and the overview of three existing weighted localization method with good performance. Their shortcomings are also described.
In the WLS method, the weight that only relates to distance is not the best. In the ECWLS method, due to the limited number of anchors, the estimation of the variance of measurement noise is not accurate. In the TELS method, it is assumed that the variance of measurement noise is the same, the influence of measurement noise on weight is not considered. And the RSS measurements are not used in the TELS method, that reduced the accuracy.

Consider a wireless sensor networks (WSNs) with N anchors located at s
T is the unknown location of the sensor (target). The distance between the target and the i-th anchor is x − s i , φ i and α i are true azimuth angles and true elevation angles of AOA, respectively. FIGURE 1 shows the illustration of anchor and target in 3D place. The true values of RSS are given by (1). The true values of AOA are given by (2) and (3) where φ i ∈ (−π, π) and α i ∈ (− π 2 , π 2 ). In the case of actual measurement error, the measurements of RSS and AoA are given by (4) where n i , m i , v i are the measurement errors. The measurement errors in reality are very complicated. We use the noise value of positive distribution to simulate the real situations. n i , m i , v i are the independent white zero mean Gaussian noises of received power, azimuth and elevation respectively, and are modeled as

B. WEIGHTED LEAST SQUARES (WLS)
In [40], the WLS method is proposed. By resorting to spherical coordinates, x − s i was expressed as u i T (x − s i ) for i = 1, · · · , N . For simplicity, the unit vector u i was defined by the values of RSS with actual measurement error as Equation (4a), (4b) and (4c) can be transformed intô WLS is used to estimate the value of x aŝ The problem (9) can be written in a vector form as The closed-form solution in (10) is (11) However, the weight that only relates to distance is not the best.

C. ERROR COVARIANCE WLS (ECWLS)
In [41], the ECWLS method is proposed. First, transform the measurement model (4a), (4b), (4c) into (6), (7), (8) respectively. Then the approximate location of the target is roughly calculated by the LS method in [39] as ε 1i , ε 2i and ε 3i in (6), (7), (8) can be expressed as Assuming that the measurement noises are small enough, λ i , sinφ i and cosα i are expanded by second-order Taylor expansion as follows, where γ = ln 10 10γ . Using x LS instead of x and substitute (14) for (13), get the estimates of ε 1i , ε 2i and ε 3i aŝ . The value of P i , φ i and α i are estimated by x LS as P i , φ i and α i as follows, The covariance matrix is composed of the variances and covariances ofε 1i ,ε 2i andε 3i . The variances and covariances ofε 1i ,ε 2i andε 3i are as The specific calculation of C is as where D 11 , D 22 , D 33 , D 13 , D 31 are diagonal matrices, Then, the covariance matrix of these error terms are used as the weight to calculate the final estimate as However, due to the limited number of anchors, the estimation of the variance of measurement noise is not accurate. This causes a large error.

D. TWO-STEP ERROR VARIANCE-WEIGHTED LEAST SQUARES (TELS)
In [42], the TELS method is proposed. Unlike WLS and ECWLS, TELS only uses AOA measurements for localization. The closed form solution of the target is obtained by the LS method asx where By using the geometric relationship of trigonometric functions, ε 2i , ε 3i in (4) can be transformed into (22) where d i = x LS − s i . Define that the estimations of ε 2i and ε 3i areε 2i = −r i m i andε 3i = −d i v i . The variances ofε 2i andε 3i are calculated as Assuming that σ m i = σ v i . Replace σ m i and σ v i with σ . Then, (23) can be written as The weight matrix relates to (24) as Then, these error terms is used as the weight to calculate the final estimate as The advantage of the TELS method is that, it doesn't need previous environmental information. Although environmental information is uncertain, it always changes in a certain range. RSS measurements are very easy to get. Therefore, the measurement accuracy will be reduced if the RSS measurements are not used. And the influence of measurement noise on weight is not considered in this method.
In order to improve the accuracy, we add the RSS measurements into the method with appropriate weight, which is related to error variance and the measured noise value on the sensor nodes.

III. PROBLEM FORMULATION AND THE PROPOSED METHOD
This section gives the ML estimate which is difficult to solve and the ENWLS method is described. First, calculate the approximate value of x WLS with the WLS method. Then calculate weight matrix C and S with x WLS . Finally, calculate the approximate value of x ENWLS , with the weight, CS.  Table 1.

A. PROBLEM FORMULATION
Based on the Gaussian noise measurement models (1), (2), (3) and (4), we can formulate the ML estimator of target location x as follows, However, this problem is non-convex and difficult to solve.

B. COMPUTATION OF WEIGHT MATRIX C
The weight matrix C is related to the variances of each anchor and each evaluation function item. Assuming that |n i |, |m i | and |v i | 1. Using a first-order Taylor approximation, ε 1i , ε 2i , ε 3i in (6), (7), (8) are written as (28), (29), (30) The variances of (28), (29), (30) are calculated as Due to the limited number of anchor nodes, the estimations of σ n i , σ m i , σ v i are not accurate. So, assume that σ n i = σ m i = VOLUME 9, 2021 σ v i = σ . The influence of different standard deviation and noise value of each anchor and each evaluation function item will be reflected in weight matrix S. (31) can be written in a equivalent form as The weights of each anchor and each evaluation function item are inversely proportional to their variance, so the weight matrix C can be written as It does not affect the final estimate that multiply each term of the weights in C by σ 2 . Replace x with the estimate of x, x WLS . So, (33) can be written as The matrix C consists of C 1 , C 2 and C 3

C. COMPUTATION OF WEIGHT MATRIX S
The weight matrix S is related to the values of n i , m i and v i . n i , m i and v i can be estimated byx WLS aŝ The weight matrix S can be written as (37) and (38) The matrix S consists of S 1 , S 2 and S 3

D. THE ENWLS ESTIMATE
First, calculate the approximate value of x with the WLS method in (11). Then, calculate weight matrix C and S withx WLS . The ENWLS estimator for the target location, x ENWLS , is derived as follows By expressing (39) in matrix form, the estimated target position becomes a closed-form solution aŝ

IV. ANALYSIS OF RESULTS
In this section, we verify the performance of the proposed method via computer simulations. All observations are generated by using (4). The target and anchors are randomly deployed inside a box with an edge length B = 15 m for each Monte Carlo run (M c ). As considered in most existing projects, the reference distance is set to d 0 = 1 m, the reference path loss to P 0 = −10 dBm. The PLE changes according to the environmental conditions, so perfect knowledge of the PLE is virtually impossible to obtain in practice.
Here, we assume that PLE is a random value in interval [2.2, 2.8] for each Monte Carlo run. Performance is evaluated by calculating the root mean square error (RMSE), defined where M c is the Number of runs, x i is the true location of target in the i-th run,x i is the estimated location of target in the i-th run. M c is set to 50000. Obviously, the lower RMSE, the better performance.  The variables of FIGURE 3 to 7 are given in TABLE.1, where σ n i represents the standard deviation of received power noise, σ m i represents the standard deviation of azimuth angle noise, σ v i represents the standard deviation of elevation angle noise, N represents the number of anchor nodes. We compare the performance of our method with the SR-WLS method in [9], the LS method in [39], the WLS method in [40], the ECWLS method in [41], the TELS method in [42] and the SDP/SOCP method in [38].
A. DIFFERENT STANDARD DEVIATIONS OF AZIMUTH ANGLE, ELEVATION ANGLE AND RECEIVED POWER FIGURE 3 shows the relationship between RMSE and the standard deviation of elevation angle noise, σ v i . Set the standard deviation of azimuth angle noise σ m i to 10 deg, the standard deviation of received power noise σ n i to 6 dBm and the numbers of anchors N to 10. As shown in FIGURE 3, RMSE of all considered methods increases with the increase of σ v i .
The weight matrix C of the proposed method ENWLS considers the influence of the different noise values of the different measurements. Due to the rationality of the weight of the proposed method, it is the best among all considered methods. FIGURE 4 shows the relationship between RMSE and the standard deviation of azimuth noise, σ m i . Set the standard deviation of azimuth angle noise σ v i to 10 deg, the standard deviation of received power noise σ n i to 6 dBm and the numbers of anchors N to 10. As shown in FIGURE 4, since the weight of the proposed method relates to the noise values, its performance is better than that of the others. FIGURE 5 shows the relationship between RMSE and the standard deviation of RSS noise, σ n i . Set the standard deviation of azimuth angle noise σ m i to 10 deg, the standard deviation of received power noise σ v i to 10 deg and the numbers of anchors N to 10. As shown in FIGURE 5, RMSE of the LS method and the SR-WLS method increase rapidly  with the increase of σ n i . This is because the weight of RSS measurement is too large. Due to the weight matrix S, the RMSE of the proposed method does not increase significantly with the increase of σ n i . That is because when σ n i rises, the weight matrix S will reduce the weight of the function term corresponding to σ n i to avoid larger error. This is also the reason why the WLS method and the TELS method are also relatively stable. The proposed method is stable and more accurate. FIGURE 6 shows the relationship between RMSE and the number of anchors, N , when the measured standard deviation is large. Set the standard deviation of azimuth angle noise σ m i to 10 deg, the standard deviation of elevation angle noise σ v i to 10 deg, the standard deviation of received power noise σ n i to 6 dBm. As shown in FIGURE 6, when the measured standard deviation is large, the RMSE of the proposed method is the best among all methods except when N = 3. When N = 3, the RMSE of the WLS method is a little better than the proposed method. That is because the number of anchor nodes is too small, resulting in inaccurate estimation of noise value, so there is a large error in the weight matrix S of the proposed method ENWLS. That causes a negative impact on the RMSE. This is also the reason why the ECWLS method has the worst performance when N = 3. However, in practice, the number of anchor nodes is generally more than 3, so the proposed method still has the best performance in practice.   FIGURE 7 shows the relationship between RMSE and the number of anchors, N , when the measured standard deviation is small. Set the standard deviation of azimuth angle noise σ m i to 4 deg, the standard deviation of elevation angle noise σ v i to 4 deg, the standard deviation of received power noise σ n i to 2dBm. Due to the rational use of RSS data, the performance of ENWLS is better than that of TELS. As shown in FIGURE 7, when the measured standard deviation is small, the RMSE of the proposed method is the best among all considered methods. However, the complexity of the SR-WLS method and the SDP/SOCP method are O (KN ) and O N 3.5 , respectively. Due to using SDP and SOCP methods, the complexity of the SDP/SOCP method is the highest, which may delay the positioning.

V. CONCLUSION
This paper proposes an indoor target localization method, ENWLS, based on hybrid RSS/AOA measurement in 3D wireless sensor networks. This method uses the approximate WLS method, its weight matrix is related to error variance and measurement noise. Its complexity is the same as the existing method in linear time. Simulation results show that this method has better performance than the existing hybrid RSS/AOA position method.
As the future work on this method, the system model will be considered closer to the actual situation. We will consider multi-user situation. Users can also receive RSS measurements from each other, which will make the positioning more accurate. Multipath effect will be considered. We will try to reduce the impact of multipath effect through rigorous mathematical derivation. In the actual situation, the interior structure may be complicated. Therefore, we will add the impact of NOLS, so that our method can be applied in more actual scenarios.