Complexity Reduction for Hybrid TOA/AOA Localization in UAV-Assisted WSNs

Unmanned aerial vehicles (UAVs)-assisted wireless sensor networks (WSNs) have tremendous potential applications due to their flexibility and rapid deployment. Hybrid time-of-arrival (TOA) and angle-of-arrival (AOA) localization techniques are commonly used due to their high accuracy. The conventional TOA/AOA localization algorithms require both zenith and azimuth angle estimations at each agent. Since a zenith-and-azimuth AOA estimation requires an L-shape antenna array and complicated 2-D signal processing, conventional TOA/AOA algorithms lead to both hardware and computational complexity and high power consumption at each agent. This letter proposes a hybrid TOA/AOA localization algorithm, named <inline-formula><tex-math notation="LaTeX">$\text{T1A}_{a}$</tex-math></inline-formula>, to reduce the agents' complexity. <inline-formula><tex-math notation="LaTeX">$\text{T1A}_{a}$</tex-math></inline-formula> only combines TOA-ranging with the azimuth angle estimation. Thereby, agents only require a 1-D antenna array, less complicated signal processing, and thus, lower power consumption. Such improvements are important for power-limited WSNs and Internet-of-Things (IoT) systems. Simulations are conducted to prove that the proposed <inline-formula><tex-math notation="LaTeX">$\text{T1A}_{a}$</tex-math></inline-formula> technique can achieve similar performance as conventional TOA/AOA methods, while significantly simplifying agents' complexity.


I. INTRODUCTION
Hybrid time-of-arrival (TOA) and angle-of-arrival (AOA) localization is commonly used in wireless sensor networks (WSNs) due to its high accuracy.Conventional hybrid TOA/AOA algorithms require both azimuth and zenith angle estimations at each agent, especially when the agents move in a 3-D space [4].It is shown in [5] and [6] that shadowing will greatly reduce TOA and AOA estimation accuracy.Therefore, using unmanned aerial vehicles (UAVs) as anchors can alleviate the problem of shadowing since there might be less obstruction and the line-of-sight path might be more likely to exist between the elevated drone and the agents [2].To measure both azimuth and zenith angles, the beam of the antenna arrays at the agents must be steered in both horizontal and vertical planes, leading to heavily rotatable mechanical structures such as parabolic antennas, conical antennas, multiple log-periodic antennas [8], or complicated signal processing using L-shaped antennas [3].
To reduce the complexity and prolong the agents' battery working time, in [3], the authors propose a TOA/AOA localization algorithm, referred to as zenith TOA/1AOA (cf., Section II-B2 for more detail), where only TOA and zenith AOA information is used, rather than all TOA, namely azimuth and zenith AOA information, which is referred to as TOA/2AOA algorithm (see Section II-B1).This zenith TOA/1AOA algorithm has been shown to provide equivalent or even better accuracy than the conventional method when sufficient signal bandwidth or a sufficient number of UAVs is used.However, such zenith TOA/1AOA algorithm is relatively inferior compared to the conventional TOA/2AOA algorithm, at a low signal-to-noise ratio range or when the signal bandwidth is limited.This Achilles heel presents even when agents move in a certain horizontal plane, such as the ground.There exist numerous practical applications, such as Internet-of-Things (IoT) for smart agriculture and localization for ground military operations, where the agents travel on a horizontal plane [6].An important observation that could help overcome the drawback of the zenith TOA/1AOA algorithm is that, when agents move on the ground, the altitude of the agent is no longer an important factor.Thus, the measurement of azimuth angles is more important while the zenith AOA measurement is no longer needed.
In this letter, we propose a novel hybrid TOA/AOA localization algorithm, namely, the azimuth TOA/1AOA algorithm, where only azimuth AOA information is used to combine with TOA information for locating the agents.Simulations show that the proposed algorithm not only outperforms the zenith TOA/1AOA algorithm and approaches the performance of TOA/2AOA in many cases, but also has less computational complexity.Cramer-Rao lower bounds (CRLBs) are derived to provide benchmarks for these methods.In addition, the impacts of other parameters of the WSN, such as signal bandwidth, transmit power, UAVs' altitude, and the number of anchors, are evaluated, providing a useful tool for optimizing the complexity and positioning accuracy.
The contribution of this letter is the proposal of a novel azimuth TOA/1AOA localization method that not only provides good accuracy, but also requires simpler hardware and has less computational complexity than the two aforementioned counterparts.

A. System Model
A WSN is deployed as in Fig. 1 including K UAV anchors with known locations at agents denoted as z m = [x m , y m , z m ] T , m = 1, . . ., M. It should be noted that the K UAV anchors here could be understood as either K different UAVs at known locations or K different way points of one UAV along its flight trajectory.For brevity, in this letter, the term K anchors might imply either case.

, K, and M unknown
Assuming that all anchors and agents are in the communication ranges so that agents can perform TOA and AOA estimations from the beacons of all anchors.From TOA information, the estimated distance to the kth anchor is expressed as dk 2 is the real distance from the agent to the kth anchor, and n k is the estimation error, which is assumed to be an independently and identically distributed (i.i.d.) Gaussian random variable with variance σ 2 d , i.e., n k ∼ N (0, σ 2 d ).Assuming the agent is equipped with a uniformly linear array (ULA), with N antenna elements separated by λ/2 from each other, where λ is the wavelength.The ULA is placed in parallel with the y-axis of the Cartesian coordinate system, thus its broadside is along the x-axis.From AOA estimation, azimuth and zenith angles, denoted as φk and θk , respectively, can be obtained as where ϕ k and θ k are the true azimuth and zenith angles, respectively, which are calculated by and ; and m k and v k are the azimuth and zenith angle estimation errors, respectively.m k and v k are also modeled as i.i.d.zero-mean Gaussian random variables with variances

B. Conventional TOA/2AOA and Zenith TOA/1AOA Localization
1) Conventional TOA/2AOA Localization: Conventionally, hybrid TOA and AOA localization uses the estimated range from TOA measurement combined with zenith and elevation angles to calculate the agent's position.From Fig. 1, the coordinates of the mth agent can be found by solving the following system of linear equations [9]: (3) In the matrix form, (3) can be expressed as where H I = I 3 ⊗ 1 K where I 3 is the 3 × 3 identity matrix, ⊗ denotes Kronecker product, 1 K is a column vector of all ones, and b I = [x 1 + d1 sin φ1 cos θ1 , . . ., x K + dK sin φK cos θK , y 1 + d1 sin φ1 sin θ1 , . . ., y K + dK sin φK sin θK , z 1 − d1 cos θ1 , • • • , z K − dK cos θK ] T .Equation ( 4) can be solved by a WLS solution [7] as as a weighting matrix to apply a higher weight on the shorter links, which gives more accurate estimation results as proved in [7].This solution is denoted as T2A hereafter.
2) Zenith TOA/1AOA Localization:To reduce both hardware and computational complexities of each agent's terminal, in [3], TOA estimation is only combined with the zenith angle so that a singledimensional antenna array is needed.As shown in [3], the WLS solution for this case is obtained as where The solution in (5) using TOA and AOA zenith angle is denoted as T1A z .

C. Proposed Azimuth TOA/1AOA Localization
In many applications, when agents move on a flat surface or the agents' altitude is unimportant, we propose a new method to combine TOA and azimuth AOA estimations as follows.Assuming agents are on the ground, i.e., z m = 0, m = 1, . . ., M, the location of the mth agent is determined by where Using the WLS method to solve (8), we get the of the agent as The solution in ( 9) is referred to as T1A a .From (7), it is clear that Unlike (6) where the accuracy of the agent's estimated coordinates is affected by both distance and zenith angle measurement errors, (10) shows that the accuracy in the azimuth TOA/1AOA algorithm does not depend on the zenith angle measurement errors.As a result, the accuracy of the proposed azimuth TOA/1AOA algorithm will be higher Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
than that of the zenith TOA/1AOA algorithm.This observation will be confirmed later in our simulations.

III. CRAMER-RAO LOWER BOUND (CRLB)
The CRLB is a common benchmark to compare different localization algorithms.CRLBs for T2A, zenith T1A z , and the proposed azimuth T1A a algorithms are derived as follows.
According to [1], the TOA estimation error is determined by the received signal-to-noise ratio (SNR), number of antenna elements N, and mean effective bandwidth B of the transmitted signal, i.e., where c is the speed of light, while the lower bounds of σ ϕ k and σ θ k are The CRLB of an estimator is derived from a Fisher information matrix (FIM), which can be calculated from the log-likelihood function of the estimated vector.The likelihood function of âT2A , conditioned on z, denoted as f (â T2A ; z), is Assuming that the estimations of d k , ϕ k , and θ k are unbiased, i.e., E { dk } = d k , E { φk } = ϕ k , and E { θk } = θ k , following the same steps shown in [10], the CRLB of the T2A estimator is CRLB T2A = Tr(F −1 T2A ), where and

B. Zenith TOA/1AOA for Ground Agents
As this letter focuses on the applications where agents move on a certain horizontal plane where their z-coordinates are constant, the measurement of the parameter z m is no longer needed.As a result, we consider the vector of unknown parameter z m = [x m , y m ] T .The FIM of T1A z can be written as formed by taking elements in the first and second columns of the first K rows and the last K rows of

C. Azimuth TOA/1AOA for Ground Agents
The observation vector of azimuth T1A is written as âT1Aa = [ d1 , . . ., dK , φ1 , . . ., φK ] T ∈ R 2K .Following the same steps as aforementioned, the FIM of T1A a is calculated as F T1Aa = E {(D T1Aa ) T R T1Aa D T1Aa }, where the 2K × 2-sized D T1Aa matrix is obtained by taking elements in the first and second columns of the first 2K rows of D T2A , and The CRLB of the azimuth T1A is CRLB T1Aa = Tr(F −1 T1Aa ).

IV. SIMULATION RESULTS AND ANALYSES
Simulations are conducted in MATLAB to evaluate the performance of the proposed azimuth TOA/1AOA and compare it to that of the conventional TOA/2AOA and zenith TOA/1AOA.For brevity, in the following figures, we denote the three algorithms as T1A a , T2A, and T1A z , respectively.Four UAVs are randomly deployed at the altitude h = 300 m transmitting beacons to ten agents traveling in an area of 1000 m × 1000 m.Path loss model follows the one in [3].The noise spectral density is −174 dBm/Hz.The beacons with bandwidth B are transmitted over 2-GHz carrier frequency and the power P T dBm.Each agent is equipped with an antenna array of ten elements, spacing at half the wavelength for performing TOA and AOA estimations.Using (11) and ( 12), the standard deviations of estimation noises σ d k , σ ϕ k , and σ θ k can be obtained.Since the agents are assumed to travel in a 2-D plane, the root mean square error (RMSE) to compare different localization algorithms is calculated as where E {.} stands for expectation over Monte Carlo simulations.Fig. 2 presents the RMSE of the proposed T1A a algorithm compared to those of T1A z and T2A and the CRLBs of these methods accordingly.It is obvious that the proposed T1A a performs much better than T1A z and approaches T2A's performance, especially when the transmit power is higher than 20 dBm.In the low transmission power condition, T1A a achieves a significant improvement compared to T1A z , with the RMSE of the former being only half of that of the latter when P T = 5 dBm.We then evaluate the mentioned methods under different transmit signal bandwidths.Fig. 3 shows the proposed T1A a algorithm's performance compared with that of T1A z and T2A counterparts over a range of signal bandwidth from 500 kHz to 6 MHz with the transmit power P T = 10 dBm.The figure confirms that the proposed T1A a approaches the T2A algorithm and outperforms T1A z algorithm at a low-to-medium signal bandwidth.
It can also be seen from Fig. 3 that, with the increase of the signal bandwidth, the accuracy of T1A a and T2A first tends to improve and then reduce, while that of T1A z improves.To explain this phenomenon, the dependence of σ d k and σ ϕ k on the signal bandwidth are plotted in Fig. 4. The transmitted power P T = 10 dBm is considered in this simulation.Clearly, when the bandwidth increases, σ d k reduces exponentially fast, while σ ϕ k increases exponentially.This is because σ d k is inversely proportional to both signal bandwidth and SNR [cf., (11)].Meanwhile, σ ϕ k only depends on SNR [cf., (12)], which reduces due to increasing noise over a wider bandwidth.Therefore, when the bandwidth increases, σ ϕ k becomes higher, leading to less accurate T1A a and T2A.This confirms that the signal bandwidth of the WSN with the proposed T1A a can be reduced while maintaining the accuracy performance at an acceptable level.
The impacts of the number of anchors and their height on the performance of the proposed algorithm are examined.Fig. 5 shows the performance of the three algorithms when 4-10 anchors are used under two cases of UAV's altitudes of 300 and 500 m, respectively.We can see that the anchors' altitude has a significant impact on the performance of both T1A a and T1A z algorithms, while it is much less significant in the case of T2A.In addition, when UAVs are flying at a certain altitude above the ground, having more UAVs improves the performance of all algorithms.However, while the number of anchors has a significant impact on the accuracy of T1A z , it has less significance for the cases of T1A a and T2A.Hence, with the proposed T1A a , the number of anchors can be reduced compared to T1A z .This will translate into less complex signal processing, lower localization latency, and more power saving.Finally, Table 1 compares the average elapsed time (in seconds) of the three algorithms.Clearly, the computational complexity of the proposed azimuth TOA/1AOA is the best among the three algorithms.

V. CONCLUSION
This letter evaluates different approaches using hybrid TOA/AOA localization algorithms.A novel way of combining TOA ranging information and azimuth AOA estimation has been proposed.We prove that when the altitudes of agents remain unchanged or on the ground plane, combining TOA and azimuth AOA estimations can achieve almost the same level of accuracy as conventional hybrid schemes where TOA is combined with both azimuth and zenith AOA estimations.The proposed method also outperforms our previous findings which use TOA and zenith AOA estimations, especially when the transmit power or the signal bandwidth is limited.The proposed method, therefore, significantly reduces the complexity of agents and power consumption of WSNs with assisted UAVs.

Fig. 3 .
Fig. 3. RMSE (m) versus signal bandwidth (MHz) with the UAV altitude of 300 m and transmit power of 10 dBm.

Fig. 5 .
Fig. 5. Localization accuracy versus number of anchors with the UAV altitudes of 300 and 500 m, transmit power of 10 dBm, and bandwidth of 500 kHz.

TABLE 1 .
Comparison of Computational Complexity