Abstract
In this paper, we address the target localization problem based on hybrid range and angle measurements with unknown transmit power in both noncooperative and cooperative scenarios, respectively. By analyzing the approximate expressions of the noise terms in the measurement models, an original nonconvex localization problem is formulated according to the least square criterion. This problem is transformed into a mixed semi-definite programing/second-order cone programing by using convex relaxation techniques. Computer simulation results verify that the proposed algorithm can effectively solve the localization problem with good performance in the unknown transmit power case.
1. Introduction
Nowadays, target location has been an increasingly important information due to its wide applications in navigation, tracking, monitoring, etc. [1]. However, the global positioning system (GPS) cannot provide accurate location information in indoor or other harsh environments due to the disability of GPS signals to penetrate in-building materials [2, 3]. Wireless sensor networks attract more attention as they can provide localization information in such harsh environments. Based on the localization measurements related to distance, the methods can be divided into time of arrival (TOA) [4, 5], time difference of arrival (TDOA) [6], received signal strength (RSS) [7–9], angle of arrival (AOA) [9, 10], and their combination [11–13]. Compared with TDOA and TOA, the RSS-based localization method is attractive due to its low cost and easy tractability, and does not cause substantial power consumption. However, the performance of the single measurement localization method is limited, especially because the accurate model parameters, including transmit power and path loss exponent, are difficult to be determined in the actual environment. Currently, many methods based on hybrid RSS–AOA measurements have been proposed to improve the performance [14–17]. In the case of noncooperative localization, Yu [14] proposed a WLS estimator based on hybrid RSS–AOA measurements when the transmit power is known and provided a closed-form solution without considering any constraint conditions, causing low estimation accuracy. Qi et al. [15] proposed a WLS estimator when the transmit power is known, which is converted into RLS-semi-definite programing (SDP) problem by using the semi-definite relaxation (SDR) technique. The authors also extended the results to the case when the transmit power is unknown. In the case of cooperative localization, Tomic et al. [16] studied the localization problem when the transmit power is known. They fuzed RSS and AOA measurements by using spherical characteristics, proposed a least square (LS) estimator through a linearized measurement model, and provided a closed-form solution. Tomic et al. [17] also studied the problem when the transmit power is unknown and formulated an original LS problem, which is transformed into an SDP problem by using SDR technology.
In this paper, we address the noncooperative and cooperative hybrid RSS–AOA-based localization problem with unknown transmit power. We jointly estimate the target location and transmit power through the following three steps. First, we derive the approximation expressions of noise terms using first-order Taylor expansion for measurement models. Next, we formulate a nonconvex localization problem based on the LS criterion. Finally, we transform the nonconvex problems into convex ones by introducing some auxiliary variables and using some relaxation techniques.
Notation: In this paper, uppercase bold letters and lowercase bold letters, respectively, denote matrices and vectors, lowercase regular letters denote scalars. denotes the norm of a vector, denotes the transpose of a vector or matrix, denotes dimensional identity matrix, denotes dimensional vector, matrix denotes that matrix is a semi-definite matrix.
The reminder of the paper is arranged as follows: Section 2 describes the localization model and the derivation of the algorithm in detail, Section 3 analyses the computational complexity of the proposed and other discussed algorithms, Section 4 discusses Cramer–Rao Lower Bound (CRLB) of the proposed algorithm, Section 5 shows simulation results and analyze in detail.
2. Problem Formulation
A sensor network consists of targets with unknown locations and anchors with known locations in a 3D localization scenario, as shown in Figure 1.
Let and denote target/anchor and target/target communication links, respectively, where denotes effective communication range of sensors.
In this paper, hybrid range and angle measurements are utilized to model our localization problem.
First, RSS measurement can be modeled by using path loss model, the relationship between RSS measurement and path loss is , where denotes transmit power, and denote path loss value and receive power between the two nodes and , respectively. Then, the RSS measurement can be modeled as follows:where denotes the reference distance, denotes the reference path loss value at denotes path loss exponent, and denote the real distance between anchor/target and target/target, respectively, RSS measurement noises and can be modeled as and , respectively.
Second, according to the geometric relationship, the AOA azimuth and elevation measurements can be modeled, respectively, as follows:where the azimuth and elevation measurement noises and are also modeled as and .
Next, we perform first-order Taylor expansion and omit the second-order and above noise terms when the noise is small.
In this case, the expressions of noise terms of (1)–(6) have the following approximations:where , , , , , , , , .
The expressions in (9) and (10) we have , .
Similarly, from (11) and (12), we have , .
We define a vector () as a column vector including all target locations, then , where denotes the ith column of the M-dimensional identity matrix, and denotes Kronecker product.
According to LS criterion, the joint estimation of unknown parameters of and can be formulated as follows:
2.1. Noncooperative Localization
In the noncooperative localization scenario, there is no communication link between target and target, so the set B in models (1)–(6) is empty. For ease of understanding, we just consider one target, i.e., . In this case, (13) can be rewritten as follows:
Since (14) is a nonconvex problem, and it is difficult to obtain its solution directly. In the following section, we develop methods to transform this problem into a convex optimization problem by applying some relaxation techniques. For this purpose, we introduce auxiliary variable , and , where , then (14) can be expressed as follows:
Furthermore, we introduce auxiliary variables and , where and , then problem (15) is equivalent to the following problem:
Next, we relax into , which can be transformed into a linear matrix inequality, i.e., by Schur complement [18], is transform into , additionally, is transformed into by second-order cone relaxation (SOCR) technique.
Then, the final optimal problem can be written as follows:
The above is the proposed mixed SDP/second-order cone programing (SOCP) algorithm for noncooperative localization. This mixed SDP/SOCP algorithm is referred to as SDP/SOCP1.
2.2. Cooperative Localization
In the cooperative localization, we introduce variables , then (13) is equivalent to the following:
For the sake of ease, we stack the variables into vector . And we introduce variables , where . Then (18) is equivalent to the following:
The following optimization problems are formed by using techniques similar to the problem formation of (17), we relax into , and into .
The algorithm is referred to as SDP/SOCP2. SDP/SOCP1 and SDP/SOCP2 can be solved by CVX [24] in MATLAB.
3. Complexity Analysis
There exists a tradeoff between algorithm estimation accuracy and computational complexity, which is one of the most important indicators for evaluating the realizability of the algorithm. We analyze the worst-case complexity, i.e., we assume that the nodes in the network are interconnected, the total number of the communication link is , where . The formulation of hybrid SDP/SOCP in worst-case is shown as follows [19]:where denotes the number of equation constraints, and denote the dimensions of semidefinite cone (SDC) and second-order cone (SOC), respectively. and denote the number of constraints of SDC and SOC, respectively, is the so-called barrier parameter. Since we investigate the worst-case complexity of the algorithm, the total number of communication links of the network is , where .
Based on (21), we calculate the complexity of the proposed and the existing algorithm in noncooperative and cooperative localization.
From Tables 1 and 2, we observe that the complexity of the algorithm is related to the number of sensors. Compared with noncooperative localization, the complexity of the cooperative localization is greatly higher.
4. CRLB
The CRLB is an unbiased estimation that can achieve the optimal estimation accuracy theoretically and can be used as a criterion for evaluating the performance of the localization algorithm. In the study of Kay [22], CRLB is expressed by the trace of , i.e.where denotes Fisher Information Matrix (FIM), in cooperative localization, we define a variable , the expression of FIM is shown as follows:where is joint probability density function of the unknown parameter. is as follows:where is a square matrix whose elements were given in [23], F2 and F4 are given as follows:
, and .
5. Results and Discussion
In this section, we compare the performance of the proposed algorithm with existing methods for both noncooperative and cooperative localization.
The normalized root mean square error (NRMSE) is introduced to evaluate localization performance, which is defined as follows:andwhere denotes Monte Carlo (Mc) runs, denotes the number of the targets, and , respectively, denote true location and estimated location of the target in the MC run. Particularly, when , NRMSE is just RMSE (root mean square error).
We deploy anchors and targets randomly in a cube space in each Mc run. The reference distance is , the path loss exponent (PLE) is . However, in practice, PLE is an experience value, which is closely related to the hardware condition and the wireless signal propagation environment, so PLE is assumed to follow the uniform distribution on an interval , i.e., .
5.1. Noncooperative Localization
In noncooperative WSN, targets communicate with anchors, exclusively. We compare the proposed algorithm with the previous algorithm, “WLS1,” proposed by Yu [14], and “GTRS1,” proposed by Tomic et al. [17]. For the sake of generality, we assume . In this part of simulation, .
Figure 2 shows the RMSE comparison of the proposed algorithm with the other methods versus the number of anchors when . Figure 2 shows that the localization performance becomes better as the number of anchors increases. This is due to the fact that more measurement information is utilized to estimate the target location. Furthermore, the estimation error of the proposed SDP/SOCP1 algorithm is smaller than that of and the proposed algorithm is closer to CRLB.
Figure 3 is the RMSE versus standard deviation of RSS measurement noise on target location estimation. Figure 3 shows that the localization performance of all the discussed algorithms becomes worse as the quality of measurement weakens, as expected. Furthermore, the proposed algorithm provides the best performance among the discussed algorithms.
Figures 4 and 5 are the RMSE versus standard deviation of azimuth and elevation measurement noise on target location estimation, respectively. We get the similar conclusion as in Figure 3.
Figure 6 shows the RMSE comparison of the proposed algorithm with the others versus the number of anchors when . Figure 6 shows that the RMSE of the transmit power estimation is decrease as the number of anchors increases. Figure 6 also shows that the proposed algorithm provides the best estimation performance among the two discussed algorithms.
5.2. Cooperative Localization
Figure 7 shows the RMSE comparison of the methods versus the number of anchors when . In the simulation, . Figure 7 shows that the NRMSE decreases as the number of nodes increases since more measurement information in target/target communication links can be captured to locate the target. Though the performance of the proposed algorithm is slightly better compared to SDP, it confirms that the localization performance of combined measurement in a hybrid system is much better than single measurement in a traditional system.
6. Conclusions
In the paper, we have addressed the localization problem based on hybrid RSS–AOA measurement in noncooperative and cooperative scenarios when transmit power is unknown. Based on measurement model, we formulate original optimal problems according to LS criterion and transform the nonconvex problem into mixed SDP/SOCP problems by some relaxation techniques, by means of which the unknown parameters and are jointly estimated.
We investigate the influence of a number of anchors and standard deviation of different types of measurement noise on localization accuracy in noncooperative localization scenarios; the results demonstrate the advantage of the proposed algorithm. Also, we investigate the influence of the number of targets on localization in a cooperative localization scenario; the results demonstrate that the performance of hybrid measurement is obviously better than a single measurement.
Data Availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would gratefully acknowledge the grants from the International Cooperation Pro-ject of the Ministry of Science and Technology under grant 2018YFE0206500, by the National Natural Science Foundation of China under grant 61571250, in part by the Zhejiang Natural Science Foundation under grant LY22F010018.