Semidefinite Programming Algorithms for 3-D AOA-Based Hybrid Localization

By taking different kinds of measurements at the same time, it may be possible to improve the accuracy of target localization or reduce the number of sensors needed. Time-of-arrival (TOA), time-difference-of-arrival (TDOA), time delay (TD), received signal strength (RSS), and angle-of-arrival (AOA) are the commonly used measurements in wireless localization systems. This article first presents a new weighted least squares (WLS) algorithm for the AOA-only localization problem. Then, a unified solution based on semidefinite programming (SDP) is developed for hybrid AOA/TOA, hybrid AOA/TDOA, hybrid AOA/TD, and hybrid AOA/RSS localization problems. Additionally, thorough simulation results show that the proposed SDP algorithms perform better than state-of-the-art techniques.


I. INTRODUCTION
Target localization has many applications, such as radar, sonar, navigation, and wireless sensor networks [1], [2]. With the rapid progress of electronic technology, different kinds of sensors such as, lidar scanner, phased-array antenna, etc., are integrated into one equipment, such as smartphone [3], [4], [5], [6]. These sensors can gather various types of data to provide a more accurate location estimation. Therefore, fusion-based positioning (also called hybrid localization) with different information has become more prominent in recent years [7], [8], [9], [10].
The loci of TOA/RSS are circles/spheres with the foci at the sensors. For TDOA, the loci are hyperbolae/hyperboloids with the foci at the sensors. For TD, the loci are ellipses/ellipsoids with the foci at the sensors. For AOA, the loci are bearing lines originating from the sensors. As a result, the loci of hybrid AOA/TOA are the intersections of circles/spheres and bearing lines, and the other hybrid cases are similar. However, due to the existence of measurement noise, the loci do not intersect at a single point. Thus, it is necessary to develop algorithms to minimize the effects of noise and provide a single solution for the target position [30].
Modern algorithms for finding the location of a target can generally be put into three groups: iterative solution, linear least squares (LLS), and convex relaxation. The main advantage of an iterative solution is that it is usually simple and direct. However, it has no guarantee of convergence with the global minimum. In other words, it may converge to the local minimum or even diverge. The LLS has the merit of being a closed-form solution and easy to compute. But it is optimal only when the noise level is sufficiently low. In contrast, convex relaxation has the global solution for the relaxed optimization problem, and it also performs well when the noise level is reasonably large [31]. But the disadvantage of convex relaxation is that it needs the CVX toolbox to solve it, and its computation complexity is also large compared to the other two categories.
Recently, some work on hybrid localization algorithms has been proposed. In [32], [33], [34], [35], the hybrid components are TOA and AOA. Only [34] studied the problem of hybrid 3-D AOA/TOA localization, and a least squares (LS) method was proposed. The hybrids in [26], [36] consist of AOA and TDOA. In [36], Cheung et al. proposed a constrained weighted least squares (CWLS) algorithm for the 2-D hybrid AOA and TDOA localization problem. However, it cannot be straightforwardly extended to the case of 3-D localization, because 3-D localization is a much more difficult problem. Actually, 3-D AOA-only localization can also be regarded as a hybrid localization problem, namely, hybrid azimuth angle and elevation angle. Yin et al. [26] proposed a WLS algorithm for the 3-D hybrid AOA and TDOA localization problem. In [37], the hybrid components are AOA and TD, and Amiri et al. presented a WLS algorithm. The hybrids in [30], [38] consist of AOA and RSS. Tomic et al. proposed a generalized trust region sub-problem (GTRS) algorithm in [38], which can be efficiently solved by the bisection method. In [30], Chan et al. considered the more complicated problem that the received power at reference distance is unknown in the RSS measurements, and a two-stage WLS (2SWLS) estimator was proposed. In summary, most state-ofthe-art hybrid localization algorithms can be classified into the category of LLS. Although they have a closed-form solution and are easy to compute, their performances are susceptible to noise levels.
This article focuses on 3-D AOA-based hybrid localization problems, i.e., hybrid AOA/TOA, hybrid AOA/TDOA, hybrid AOA/TD, and hybrid AOA/RSS. First, we present a new WLS algorithm for the 3-D AOA-only localization problem. Next, the corresponding maximum likelihood estimator (MLE) is formulated for each hybrid localization problem. However, these MLE problems are highly nonlinear and nonconvex, which makes optimal solutions hard to obtain. Based on the derivation of WLS for 3-D AOA-only localization problem, these MLE problems are approximated to CWLS problems, which can be effectively solved by SDR techniques. Finally, a unified semidefinite programming (SDP)based solution is developed for these hybrid localization problems.
The main contributions of this article include: r The proposed SDP-based solution has better localization accuracy compared to state-of-the-art LLS-based methods. The rest of this article is organized as follows: Section II describes the measurement model of 3-D AOA localization and derives a WLS algorithm for it. A unified SDP-based solution is formulated in Section III for hybrid AOA/TOA, hybrid AOA/TDOA, hybrid AOA/TD, and hybrid AOA/RSS problems. Section IV provides the performance simulation results.

II. 3-D AOA LOCALIZATION
Consider a wireless sensor network of M sensor nodes with s i = [x i , y i , z i ] T being the known coordinates of the ith node. And u = [x, y, z] T be the unknown coordinates of the target.
The azimuth and elevation angle measurements in 3-D space are expressed as where θ 0 i and φ 0 i are the unknown true values of the azimuth and elevation angles, respectively. v i and w i are measurement noise. For easy of analysis, the noise v i and w i are assumed to be independent identically distributed, respectively. They obey zero-mean Gaussian distribution, and their covariance matrices are Q a and Q e , whose diagonal elements are σ 2 a and σ 2 e , respectively. The true azimuth angle θ 0 i and the true elevation angle φ 0 i can be expressed as [22] θ 0 From the true azimuth angle θ 0 i , we have and from the true elevation angle φ 0 i , we have (3) and combining (4b) yield Next, using measurements θ i and φ i to approximate θ 0 i and φ 0 i yields As a result, (5) can be written as:

and
Stacking the M equations for i = 1, . . . , M from (7) gives: and B 11 , B 12 , B 21 , B 22 , and B 3 are diagonal matrices, whose diagonal elements are Finally, a CWLS estimator for g is formulated as where As we know, a CWLS problem with one quadratic constraint can be effectively and efficiently solved by the Lagrange multiplier method [39]. However, from (13), it can be seen that the number of quadratic constraints is greater than one, which leads to the CWLS problem in (13) not being effectively solved by the Lagrange multiplier method.
Ignoring the constraints in (13b), the CWLS problem is degraded to a WLS problem, which is a quadratic and convex function with respect to g. By letting the gradient of the cost function in (13a) with respect to g be zero, it results in a WLS solution and target position can be obtained by u WLS = g WLS (1 : 3) . 1 Remark 1: The above WLS algorithm needs to know the target position u to calculate the weighted matrices W 1 , W 2 , and W 3 . Here, we adopt the following approach. First, let to find a rough estimate of u via (15). Then the rough estimate is used to calculate W 1 , W 2 , and W 3 , which are subsequently used to calculate (15) again to get the final result. This process is repeated only once, because one iteration is sufficient to give an accurate result [13], [26].

III. UNIFIED 3-D AOA-BASED HYBRID LOCALIZATION
In this section, a unified solution for four kinds of 3-D AOAbased hybrid localization problems is developed.

A. HYBRID AOA AND TOA LOCALIZATION
The TOA measurement with ith sensor can be expressed as: where r i is the distance measurement (distance equals TOA multiplied by propagation speed), and n i is the measurement noise. The noise n i obeys a zero-mean Gaussian distribution; its covariance matrix is Q t , which is a diagonal matrix, whose diagonal elements are σ 2 t . The MLE of target position u can be written as: 1 It should be noted that constraints existing in g are not used in the scenario of AOA-only. Because the unconstrained WLS already has good performance for AOA-only localization problem, which is validated by the numerical simulation in Section IV. In the following hybrid localization problems, the constraints need to be utilized because they can improve the estimation accuracy.
Although the above MLE is optimal, it is still hard to address directly due to its high nonlinearity and nonconvexity.
Here, we use the formulated CWLS estimator in (13) to approximate the first part of MLE in (18), i.e., is replaced by Then the MLE in (18) can be approximated to: As mentioned earlier, CWLS with multiple quadratic constraints cannot be effectively solved by the Lagrange multiplier method. Nevertheless, the SDP method is still suitable for this case.
Next, we will resort to semidefinite relaxation (SDR) techniques to solve the above CWLS problem. First, let G = gg T , then the objective function in (21a) is equivalent to where the constant terms are discarded since they do not affect the optimization results. The constraints in (21b) can be written as: Besides, the non-convex constraint G = gg T can be relaxed to: Finally, an SDP based algorithm is formulated as: s. t. (23), (24). (25b)

Remark 2:
The above SDP algorithm also needs to know the target position u to calculate the weighted matrices W 1 , W 2 , W 3 . First, we adopt the LS method to find a rough estimate of u Then the rough estimate is used to calculate W 1 , W 2 , and W 3 , which are subsequently used to calculate (25) to get the final result.

B. HYBRID AOA AND TDOA LOCALIZATION
The TDOA measurement between ith sensor and the reference sensor (without loss of generality, the first sensor is selected as the reference sensor) can be expressed as: where r i1 is the distance difference measurement, which equals TDOA multiplied by the signal speed, and n i1 is the measurement noise. The noise n i1 obeys a zero-mean Gaussian distribution, its covariance Q d = σ 2 d R, where R is a symmetric matrix of order M − 1, whose diagonal elements are 1 and the rest of the elements are 0.5 [40].
The MLE of target position u can be written as: Similar to the derivations in (21), a CWLS problem approximated to the MLE problem in (28) can be formulated as: The above CWLS problem can also be solved by SDR techniques. The derivations for it are omitted. The reader can refer to Section III-A.

C. HYBRID AOA AND TD LOCALIZATION
In the scenario of distributed multiple input and multiple output (MIMO) radar systems [16], the TD measurement between ith transmitter and jth receiver can be expressed as: where r i j is the bistatic range measurement, and n i j is the measurement noise. The noise n i j obeys a zero-mean Gaussian distribution, its covariance matrix is Q m , which is a diagonal matrix, whose diagonal elements are σ 2 m . The MLE of target position u can be written as: Let Using the results in (13), an approximation to the MLE in (31) can be formulated as: where the constant terms are discarded. The constraints in (32) can be written as: The non-convex constraint G e = g e g T e can be relaxed to: 1 g e T g e G e 0.

Remark 3:
The above SDP algorithm also needs to know the target position u to calculate the weighted matrices W 1 , W 2 , W 3 . First, we adopt LS method to find a rough estimate of u Then the rough estimate is used to calculate W 1 , W 2 , and W 3 , which are subsequently used to calculate (37) to get the final result.

D. HYBRID AOA AND RSS LOCALIZATION 1) p 0 IS KNOWN
According to the path-loss model [20], the received average power (in dBm) at ith sensor can be expressed as: where p 0 is the received power at a reference distance d 0 , which is related to the transmit power [41], γ is the path-loss exponent, d i = u − s i represents the distance between the ith sensor and target node. The noise n i obeys a zero-mean Gaussian distribution, its covariance matrix is Q r , which is a diagonal matrix, whose diagonal elements are σ 2 r . Without loss of generality, d 0 = 1 m.
The MLE of target position u can be written as: Next, we take some approximations for the RSS equation. From (39), we have Let h i = 10 (41) can be written as  (13) and combining the approximation in (42), an approximation to the MLE in (40) can be formulated as: The above CWLS problem can also be solved by SDR techniques. The derivations for it are also omitted. The reader can refer to Section III-A.

2) p 0 IS UNKNOWN
When p 0 is unknown, the RSS equation can be recast as Let α = 10 p 0 10γ , and f i = 10 Let Similar to the derivations in (43), a CWLS problem can be formulated as: Again, the above CWLS problem can be solved by the SDR techniques, and the derivations for it are also omitted. Finally, an SDP algorithm is formulated as: Remark 4: The above SDP algorithm needs to know the target position u to calculate the weighted matrices W 1 , W 2 , and W 3 . Besides, the computation of the weighted matrix W p also needs to know α. First, we adopt the following LS method   to find a rough estimate of u Then the rough estimate is used to calculate W 1 , W 2 , and W 3 . Second, let α = 1 to obtain a rough W p , then with the calculated W 1 , W 2 , and W 3 to calculate (47). Third, using the result ofα =ĝ p (M + 4) to updateŴ p , and calculate (47) again to obtain the final estimate.

IV. SIMULATION RESULTS
This section presents several numerical simulations to show the performance of the proposed SDP algorithms and compare it with state-of-the-art methods, namely, [22] (label as 'WLS', for AOA-only localization), [34] (label as 'LS-Yu', for AOA/TOA localization), [26] (label as 'WLS-Yin', for AOA/TDOA localization), [37] (label as 'WLS-Amiri', for AOA/TD localization), [38] (label as 'LC-GTRS-Tomic', for AOA/RSS localization with known p 0 ), [30] (label as '2SWLS-Chan', for AOA/RSS localization with unknown p 0 ), and the Cramé r-Rao lower bound (CRLB). The proposed SDP algorithms are implemented by the CVX toolbox using SDPT3 as a solver and with the best precision [42]. We consider a 3-D network with five sensors and one target. The positions of sensors are fixed, and the position of target is randomly deployed. More specifically, a total of 10 random target positions were used to obtain the average CRLB and RMSE, and 1000 Monte Carlo realizations were generated for each target position.
In the scenarios of AOA-only, hybrid AOA/TOA, hybrid AOA/TDOA, and hybrid AOA/TD, the sensors' positions are listed in Table 1   All the measurement noises are independent and identically distributed, and they follow a zero-mean Gaussian distribution. The covariance matrices are set as: In Fig. 1, we compare the proposed WLS algorithm with another WLS algorithm given in [22] for the scenario of AOA-only localization. It can be seen that these two WLS algorithms have same performance when σ a is smaller than −15 dB rad, and the proposed WLS has slight superiority when σ a = −10 dB rad (656 m versus 714 m).
Figs. 2 and 3 are drawn from the scenario of AOA/TOA localization. From these two figures, it can be seen that the CRLB with hybrid AOA and TOA measurements is lower than that of AOA measurement or TOA measurement alone. This observation validates the advantage of hybrid localization, i.e., hybrid localization can provide higher positioning accuracy. More importantly, the proposed SDP algorithm attains the  CRLB of hybrid measurements and is superior to the LS-Yu algorithm. In Fig. 2, the standard variance of TOA σ d is fixed to 1 m, and the RMSEs are varying with the standard variance of AOA σ a . In Fig. 3, the standard variance of AOA σ a is fixed to 0.5 • , and the RMSEs are varying with the standard variance of TOA σ d . It is observed that the LS-Yu method cannot attain the CRLB of hybrid localization even when σ a = −40 dB rad in Fig. 2. This is because σ d = 1 m is not sufficiently small for the LS-Yu algorithm. A similar result is observed in Fig. 3.
Figs. 4 and 5 are plotted with the scenario of AOA/TDOA localization. In Fig. 4, it is obvious that the proposed algorithm is better than the WLS-Yin method when σ a > −20 dB rad. This result validates that the proposed SDP algorithm has a higher threshold compared with the WLS method. Besides, it is observed that the WLS-Yin method cannot attain the CRLB of hybrid localization even when σ d = −20 dB m in Fig. 5. This is because σ a = 0.5 • is not sufficiently small.
Figs. 6 and 7 are plotted with the scenario of AOA/TD localization. From Fig. 6, it can be seen that the proposed SDP algorithm has a higher threshold compared with the WLS-Amiri method. This result validates the superiority of the proposed SDP algorithm again. Besides, in Figs. 3, 5, and 7, it is observed that the RMSEs of the proposed SDP   algorithms are slightly above CRLB when σ d = −20 dB m. The reasonable explanation is that the CVX toolbox cannot provide a high-precision solution when the variance of measurement noise is very small. It should be noted that this odd phenomenon has been reported [31].
The scenario of AOA/RSS localization is plotted from Figs. 8-11. The first two figures are drawn with a known  p 0 , and the last two figures are drawn with an unknown p 0 . Comparisons between the proposed SDP algorithms and two other state-of-the-art methods show that the proposed SDP algorithms have better accuracy.
In Fig. 12, we examine the effect of the number of sensors varying from 5 to 9 in the scenario of AOA/TOA   we can see that the three CRLBs and two RMSEs decrease as the number of sensors increases, and the proposed algorithm attains the CRLB of hybrid measurements, while the LS-Yu method is above the CRLB.
Finally, the average running time of different algorithms is given in Table 3. It can be seen that the average running time of the proposed SDP algorithms is great than the LLS-based algorithms. This is due to the fact that the SDP-based algorithm needs the CVX toolbox to solve it, whereas, the LLSbased algorithm just needs some simple matrix calculations.

V. CONCLUSION
In this article, we develop a new WLS algorithm for the 3D AOA localization problem. Then, a unified solution based on SDP is developed for AOA-based hybrid localization problems. Although the MLEs for hybrid localization problems are optimal, they are hard to address. Based on the derivations of WLS for AOA localization problem, the MLE problems are approximated to CWLS problems, which can be effectively addressed by SDR techniques. Finally, SDP-based algorithms are developed for hybrid localization problems. Simulation results also show that the proposed algorithms have superior estimation accuracy compared to the state-of-the-art methods at the cost of more running time.