Path planning for two unmanned aerial vehicles in passive localization of radio sources

Aerospace Science and Technology 58 (2016) 189–196

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Aerospace Science and Technology www.elsevier.com/locate/aescte

Path planning for two unmanned aerial vehicles in passive localization of radio sources Seyyed Ali Asghar Shahidian, Hadi Soltanizadeh ∗ Department of Electrical and Computer Engineering, Semnan University, Semnan, Iran

a r t i c l e

i n f o

Article history: Received 26 January 2016 Received in revised form 14 March 2016 Accepted 11 August 2016 Available online 16 August 2016 Keywords: Trajectory control Extended Kalman filters Position error covariance Unmanned aerial vehicles Time difference of arrival

a b s t r a c t This paper studies the trajectory control problem for a pair of unmanned aerial vehicles (UAVs) equipped with time of arrival (TOA) sensors to measure the time difference of arrival (TDOA) of the transmitted radio signal to localize the source. The extended Kalman filter (EKF) is applied to estimate the source’s position. The proposed trajectory control strategy encompasses three optimum experimental design criteria based on the position error covariance produced by the EKF. The control strategy steers the UAVs to the positions to minimize the uncertainty about the location of the source. The effectiveness of the proposed approach is illustrated through simulation examples. © 2016 Elsevier Masson SAS. All rights reserved.

1. Introduction Passive localization of radio sources (emitters) has several civilian and military applications in electronic warfare systems, search and rescue scenarios, indoor environments, wireless mobile telecommunication systems and so on. Although, the source could be localized by stationary sensors mounting the sensors on moving vehicles can greatly enhance the localization performance [1]. Aerial localization of radio sources is an appropriate alternative for a large number of applications. The speed, flexible steering ability, and wide vision of the aerial vehicles may improve the localization accuracy and speed. Flying above the ground level reduces the uncertainty about the transmitted signal caused by obstacles and improves the detection of the position related parameters of the radio signal. Optimal trajectory control for a pair of UAVs in radio source localization is tackled in this paper. The localization scenario in this paper estimates the location of a stationary radio emitter recursively based on a sequence of noisy measurements obtained by TOA sensors. To eliminate the uncertainty about the signal transmission time TDOA measurements are applied for source location estimation. According to the measurement equations the estimation accuracy is a function of the relative sensor-source geometry [2,3]. Therefore, in the case that the sensors are mounted on the

*

Corresponding author. E-mail addresses: [email protected] (S.A.A. Shahidian), [email protected] (H. Soltanizadeh). http://dx.doi.org/10.1016/j.ast.2016.08.010 1270-9638/© 2016 Elsevier Masson SAS. All rights reserved.

UAVs the localization performance depends heavily on the UAV trajectories. Sensor platform motion planning has many applications in search and rescue scenarios [4,5], target tracking [6,7], environmental monitoring [8], surveillance systems [9,10], as well as defense applications [11]. The objective of the UAV trajectory optimization is to determine the UAV waypoints in order to maximize the emitter location estimation accuracy. Since the emitter position is unknown, the relative geometry is optimized based on the last emitter position estimation. Consequently, the location estimation accuracy and the trajectory selection effectiveness have mutual impact on each other. A vast amount of work has been performed in the area of optimal UAV trajectory control in radio source localization over the last decade. Do˘gançay [12] has developed online receiver trajectory optimization algorithms for AOA/scan-based emitter localization. Do˘gançay et al. [13] have considered a multisource environment and have allocated a UAV team for localization of each source. In the work presented in [14] the path planning problem for multiple UAVs with heterogeneous payload sensors has been studied. The proposed steering algorithms in [12–14] control the UAV trajectories based on maximizing the determinant of approximated fisher information matrix (FIM). The active target-tracking problem for a team of UAVs equipped with 3-D range-finding sensors has been studied in [15]. A path planning approach has been proposed in [16] for the limited number of UAVs in the RSS based localization of a single source. In the present paper a gradient-based waypoint optimization for a pair of UAVs equipped with TOA sensors applying D-, A- and E-optimality criteria based on the estimated position error covariance produced by the EKF is studied.

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The TOA is a sensor-source range based parameter which has been consistently applied for localization of the radio emitters [17–19]. However, a single TOA does not have any sensor-source range information when the signal emission time is unknown. In the case that there are two such measurements subjected to the same signal the TDOA observation is obtained. TDOA eliminates the need to know the emitter emission time that is required for TOA based localization. A noise free TDOA measurement in twodimensional space specifies the location of the emitter on a hyperbola. N TOA sensors generate N − 1 non-redundant TDOA measurements subjected to an identical transmitted signal. In the absence of measurement noise the emitter is located at the intersection of N − 1 hyperbolae. For two-dimensional noise free localization at least three TDOA measurements are required to find the emitter position [20]. Generally, the measurements are noise corrupted and the hyperbolas will no longer intersect exactly at the emitter location and an estimation algorithm should be applied. Applying an additional sensor (UAV) increases the localization accuracy. However, in addition to the cost of the UAVs applying an additional UAV intensify the complexity of path planning and localization processes and suitable low bandwidth communication network provision for transferring the measurements. Therefore, some applications may require to apply the minimum number of UAVs and increase the target location estimation accuracy through geometry optimization; e.g. in the search and rescue scenarios when there are a large number of sources in a large search area which require to appoint a UAV group to each source. For TDOA localization at least two sensors are required to form at least a single TDOA measurement at each time step and estimate the source location based on the sequence of observations. In this paper a trajectory control strategy is proposed for the minimum number of UAVs in TDOA localization. Several works have addressed the problem of localization of radio sources with limited number of sensors. Okello et al. [21–23] have applied TDOA measurements received by two UAVs as they traverse the predefined trajectories to localize a stationary or moving emitter. Three nonlinear filters: a Gaussian mixture measurement integrated track splitting filter, a multiple model filter with unscented Kalman filters, and a multiple-model filter with extended Kalman filters have been compared in [21]. The researchers in [21–23] have studied the TDOA based localization performance applying two UAVs in the case that the UAV trajectories are predefined. There are not enough studies about the optimal trajectory control for limited number of UAVs in radio source localization with the objective of minimizing the estimation uncertainty. Do˘gançay [24] has proposed a trajectory control algorithm for any number of UAVs equipped with AOA sensors. The UAV paths are optimized by minimizing a cost function comprising the mean-squared error (MSE) of predicted target position estimates produced by the EKF. A trajectory control for two UAVs in DRSS localization of multiple stationary RF emitters is proposed in [25]. In the latter research the path optimization is performed based on the determinant of the approximated FIM. In this paper the localization of a stationary emitter by two UAVs measuring the TOA of the radio signal has been investigated. In order to eliminate the uncertainty about the transmission time the TOA observations at each specified time step form a single TDOA measurement. The sequence of measurements coupled with the emitter motion model is essential to estimate the emitter location over time. An updated estimate of the target location is required in each measurement instant. The interest of the present study is not in deriving the position estimators but rather in optimization of the UAV trajectories that could be achieved applying any unbiased estimator. Since the TDOA is a nonlinear measure-

ment the EKF which is a nonlinear filtering technique is applicable for the proposed problem of this paper. In the trajectory control problem the UAV waypoints should be selected to optimize an objective function, i.e. the path length, the localization accuracy, and so on. The objective of this study is to select the waypoints which increase the source location estimation accuracy. The position error covariance produced by the EKF measures the estimation uncertainty and also is related to the localization geometry. Therefore, minimizing some real-valued functions defined on the position error covariance may find the waypoints that increase the localization accuracy. In this research three most popular optimum experimental design criteria, D-, Aand E-optimality, based on the position error covariance produced by the EKF have been applied for UAV trajectory optimization. This paper is organized as follows. The next section provides the source localization problem description and the EKF design for TDOA localization with two TOA sensors. Then the proposed pathplanning approach for two UAVs is described in the presence of geometric path and movement constraints. Three optimum experimental design criteria based on the position error covariance are presented. Extensive simulation examples for the proposed UAV steering algorithms are provided and conclusions are drawn afterwards. 2. Source localization 2.1. Problem description Consider a stationary radio source at unknown three-directional Cartesian position. The emitter is a radio source with omnidirectional propagation. Two UAVs at known Cartesian positions equipped with omnidirectional antennas are tasked to localize the emitter. The location of the UAVs could be achieved by the Global Positioning System (GPS). Both UAVs are equipped with electronic support measures (ESM) to observe the TOA of the propagated signal. The TOA measurements subjected to an identical transmitted signal at the UAVs are estimated at known time steps. The interval between two consecutive time steps (measurements) is T seconds. The signal trip time (TOA minus the time of transmission) is a linear function of the sensor-source range. Furthermore, the altitude of the UAVs and the source are known for the system; the source is located on the flat ground. Accordingly, without loss of any information a two-dimensional model could be applied for the localization and path planning scenarios. However, the observed three-dimensional TOAs could be converted to the two dimensional measurements through Pythagorean Theorem. The unknown two-dimensional Cartesian position of the source and the known Cartesian positions of the UAVs are denoted by x = [x, y ] T and uki = [u ix,k , u iy ,k ] T , i = 1, 2 respectively. The superscript T denotes the matrix transpose operator and k refers to the instants when the measurements are formed. The TOAs and the UAVs’ positions are transmitted to a processing unit to perform the calculations. The processing unit is located in one of the UAVs. Each UAV measures the time of arrival of the transmitted signal with additive zero-mean white Gaussian noise with a standard deviation of σt . The TOA measurement at the UAV i formed at time step k is given by:

tki = tk + rki /c + nki

(1)

where tk is the source transmission time, = x − is the Euclidean direct distance between the emitter and UAV i at time step k, c is the signal propagation speed, and nki ∼ N (n; 0, σt2 ) is the TOA measurement noise where N (n; m, P ) is the probability density function (pdf) of a Gaussian distribution of variable n with mean m and variance P . rki

uki 

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The TOA is a function of emitter–receiver Euclidean distance and the time of emission. In passive localization scenarios the transmission time of the signal (tk ) is unknown. However, subtraction of two TOA measurements subjected to an identical transmitted signal eliminates the transmission time and results in a single TDOA equation:





τk = tk2 − tk1 = rk2 − rk1 /c + nkτ

(2)

191

that the emitter state estimation error is minimized. The trajectories of the UAVs are optimized in the presence of path constraints such as maximum turn rate and collision avoidance. The waypoint updates are synchronized with sensor measurements. Therefore, the time interval between consecutive waypoints is T seconds and the waypoints at time instant k are given by

uki = uki −1 + ski −1 ,

i = 1, 2, k = 0, 1, . . . ,

(10)

where nkτ = nk2 − nk1 ∼ N (n; 0, 2σt2 ) is the TDOA measurement noise. A single noiseless TDOA measurement in two-dimensional space specifies the potential locations of the emitter on a hyperbola and the additive measurement noise adds an uncertainty to the locus. In the case of N TOA sensors, M = N − 1 non-redundant TDOA measurements are formed and the emitter location is localized by intersecting the M ≥ 3 hyperbolae. In this paper the localization problem is restricted to only two moving sensors. Therefore, the TDOA observations in multiple waypoints/time-steps should be used for localization of the source. Multiplying the TDOA observation by the signal propagation speed (c ) forms the range difference of arrival (RDOA) as follows:

where v ki is the cruising speed of the UAV i between waypoints k and k + 1 and ϕ is the maximum turn rate of the fixed wing UAVs in radians. In addition to turn rate and speed limitations two hard constraints which prevent collision and assure the connectivity between UAVs are given by:

zk = c × τk = hk (x) + w k

dmin < uk1 − uk2  < dmax ,

(3)

where

    hk (x) = x − uk2  − x − uk1 

(4)

2 and w k is a white zero-mean Gaussian noise, w k ∼ N ( w ; 0, σ w ), σ w2 = 2c 2 σt2 . The RDOA is more convenient to apply as the observation equation.

2.2. The EKF design The localization algorithm undergoes the nonlinear measurement equation as (3). Using an optimal nonlinear filter propagates the non-Gaussian distributions and evaluates their mean which makes it computationally expensive. A non-optimal alternative to solve the problem is the EKF which linearizes the nonlinear components of a tracking problem using a first order Taylor series expansion [26]. Assume that the target state vector xˆ k−1 = [ˆxk−1 , yˆ k−1 ] and the error covariance matrix for the filtered state estimate Pk−1 are available at scan time k − 1. At time step k the RDOA is observed and the target state and the error covariance are updated as:

Hk =

∂ hk (x) | ∂ x x=ˆxk−1 

(5)

 2 −1

Kk = Pk−1 Hk Hk Pk−1 HkT + σ w



 xˆ k = xˆ k−1 + Kk zk − h(ˆxk−1 )

Pk = (I − Kk Hk )Pk−1

(6) (7) (8)

where Kk is the Kalman gain, xˆ k is the filtered state estimate, and Pk is the error covariance matrix for the filtered state estimate at time k. The matrix Hk is the Jacobian of the nonlinear measurement function hk (xk ), described in (3), evaluated at the emitter state estimate. The EKF recursions are initialized by:

xˆ 0 = E{x} and

P0 = cov{x}

(9)

3. Path optimization

where ski is the steering control vector (UAV waypoint update) for the ith platform which satisfies the norm and turn rate constraints:

 i s  = v i T , k k  i   u −  ui  ≤ ϕ k k −1





(11a) (11b)

k = 0, 1 , . . .

(12)

Optimal trajectory control requires the selection of the UAV steering control vectors in order to optimize a cost function that measures the estimation uncertainty. 3.2. Optimality criteria To compare different steering vectors in each time step a quantitative measure related to the expected accuracy of the emitter location estimates should be applied. The estimation accuracy depends not only on the measurement precision, but also on the localization geometry. Since the measurement precision cannot be modified during the localization, the UAV trajectory selectivity could improve the estimation accuracy. The position error covariance matrix produced by the EKF (Pk ) is such a measure. Given the sensor measurements at time step k, the EKF updates the emitter location estimate. The trajectory control unit determines the sensor configuration to increase the source location estimation accuracy. In general, the position error covariance quantifies the amount of information that the observable random measurement zk carries about the unobservable parameter x (true source location). According to (5)–(8) Pk is independent of the TDOA measurements and depends only on the localization geome2 try, target state estimation, and measurement noise variance σ w . Minimization of some real-valued functions defined on Pk may determine the optimal UAV configuration (localization geometry). Various such functions exist in the optimal experimental design literature including D-, E- and A-optimality [27]. The D-optimality (determinant) criterion that is related to the volume of the uncertainty ellipsoid for the estimates is given by:



  D (Uk ) = ln det(Pk ) ,



Uk =

uk1 uk2



(13)

An E-optimum design is related to the length of the largest axis of the uncertainty ellipsoid. The E-optimality criterion is given by:

 E (Uk ) = λmax (Pk )

(14)

3.1. Problem definition

where λmax (·) denotes the maximum eigenvalue of its argument. An A-optimum design suppresses the average variance of the estimates. The A-optimality (trace) criterion is given by:

The UAV path optimization problem involves the determination of the UAV waypoints at discrete time instants k = 0, 1, 2, . . . such

 A (Uk ) = trace(Pk )

(15)

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Fig. 1. The processing flow for the proposed UAV trajectory control approach for a pair of UAVs.

3.3. Trajectory optimization The optimality criteria (Uk ) (D-, E- or A-optimality) should be minimized to determine the next waypoint such that the emitter state estimation error is minimized. The path optimization problem can be solved by gradient-descent waypoint updates:

uki = uki −1 − μki −1

∂(Uk−1 ) ∂ uki −1

,

i = 1 , 2 , k = 0, 1 , . . .

(16)

where μki −1 is the step-size that normalizes the control inputs in order to satisfy the norm constraint in (11) which is given by:

μki =

v ki T

(17)

∂(Uk )/∂ uki

The gradient of (Uk ) may be approximated by first-order finite difference as:

(Uk + δ i ) − (Uk ) ∂(Uk ) |u =u i ≈ , ∂u δ

Uk = [u i ]4×1

(18)

where δ i is a 4 × 1 column vector which equals to a small positive real number δ for the ith entry corresponding to the ith entry of Uk and zero for the remaining entries. The proposed trajectory control procedure for two UAVs is illustrated in Fig. 1. According to the volume of the search area the source state is initialized by (9). The initial locations of the UAVs are given by the GPS. The UAVs form the RDOA measurement (3) and the EKF updates the source state. One of the control criteria (13)–(15) is optimized to update the UAVs’ waypoints (16)–(18). The new waypoints are expected to provide the most informative measurements. However, the movement constraints must be satisfied, otherwise, the nearest values to the optimized waypoints which satisfy the constraints are selected. Afterwards, the UAVs are steered to the new waypoints. The recursion is continued until a predefined stopping condition is achieved, e.g. maximum waypoints number or minimum root mean-squared error (RMSE). 4. Simulation examples In this section several simulation examples are presented to demonstrate the application of the proposed trajectory control approaches and compare the optimality criteria. The proposed algorithm controls the trajectories of two networked UAVs to increase the accuracy of the localization of a stationary radio source. Each UAV estimates the TOA of the transmitted signal in discrete time instants. A single TDOA measurement is formed at each waypoint

and the target state is updated recursively through the EKF. The waypoints of the UAVs are selected based on the real-valued functions, D-, E-, and A-optimality, defined on the position error covariance produced by the EKF. The initial position of the UAVs and the true location of the source are specified randomly by the standard uniform distribution in the simulations. The location of the emitter is unknown for the system, however, it is restricted to a circle shaped area with the center on [9, 5] T and radius of 3 km. The initial position of the UAVs are restricted to a ring shaped area with the center on [9, 5] T km, the smaller radius of 5 km and the larger radius of 9 km. The EKF is initialized to xˆ 0 = [9, 5] T and P0 = diag(3, 3). Both UAVs have the same cruising speed of v = 150 km/h with a measurement time interval of T = 10 s. The maximum turn rate for the UAVs is ϕ = 30◦ (i.e. 3◦ /s). The RMSE for emitter location estimates is given by

RMSEk =



tr(Pk )

(19)

where Pk is the filtered estimation covariance of the EKF at time step k. Fig. 2 illustrates the sampled trajectories and the evolution of the RMSE for different optimization criteria of the proposed UAV trajectory control approach. The reference sensor measurement error for TDOA is σ w = 100 m. The localization mission is stopped when the RMSE reaches to 40 m. In the next simulation the three-dimensional trajectory control is presented. The UAVs are initially located at:

u10 = [4, 6, 0.1] T km u20 = [18, 4, 0.1] T km

(20)

The true location of the source is at x = [9, 3, 0] T km. Fig. 3 illustrates the D-optimum UAV paths for target localization after 30 waypoint updates, 60 waypoint updates, and 90 waypoint updates. To compare the application of the three optimality criteria more accurately the following simulations are represented. The results in Fig. 4 are the average of the 1000 Monte Carlo runs for different reference sensor measurement errors. Each run is stopped when the RMSE reaches to 40 m. A simulation is deemed to be diverged if the RMSE does not reach to 40 m at most in 200 waypoints or the RMSE exceeds 5 km in one of the time steps. The percentage of the diverged runs in all 1000 runs is called the divergence rate. The average of the number of waypoints and the divergence rate are provided in Fig. 4. The diverged runs are not included in the statistics of Fig. 4. The simulation is repeated for different standard deviation σ w of the measurement error.

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Fig. 2. The proposed trajectories of two UAVs based on (a) D-optimality, (b) E-optimality, (c) A-optimality. (Final UAV locations are marked with ∗.) (d) Evolution of the RMSE for different criteria.

The average of the RMSE of converged Monte Carlo runs at each time step is given by:

RMSEk =

1

1000

N iter

RMSEki

(21)

i =1

where N iter is the number of converged runs and RMSEki equals zero if the ith run is diverged. The average of the RMSE of 1000 Monte Carlo runs of all 50 waypoints is represented in Fig. 5,

RMSE =

50 1

50

RMSEk

(22)

k =1

In the next simulation the number of waypoints is restricted to 50. The results in Fig. 5 are the average of the converged runs of 1000 Monte Carlo simulations. If the RMSE in only one time step exceeds 5 km or the RMSE of the waypoint 50 exceeds 100 m the run is regarded diverged. Fig. 4 illustrates the average of the number of time steps and the divergence rate of 1000 Monte Carlo runs for different TDOA measurement variances. Each run is assumed to be diverged if the RMSE does not reach to 40 m at most in 200 waypoints or the RMSE exceed 5 km in only one waypoint. The average of the number of waypoints of each UAV for converged runs are represented in Fig. 4(a). Accordingly, the target localization based on the Doptimality criterion trajectory control converges to RMSE = 40 m faster than the other criteria. However, the A-optimality requires lower waypoints than E-optimality criterion. Fig. 4(b) shows the divergence rate of different measurement variances. For all the

control criteria zero simulations have been diverged for σ w < 50 m which is a large amount of standard deviation of the noise for real situations. Therefore, the proposed trajectory control strategy applying each criterion effectively improves the target localization. However, to compare the optimality criteria the divergence rate for σ w > 50 m is included in Fig. 4(b). For 50 m < σ w < 80 m applying A-optimality resulted in the lowest divergence rate and for 160 m < σ w < 200 m the largest divergence rate has achieved by the E-optimality based trajectory control. The divergence rate for D-optimality criterion is approximately three percent for σ w > 80 m. Fig. 5 represents the average of the RMSE of all 50 waypoints of 1000 Monte Carlo simulations (RMSE) and the divergence rate for different measurement variances. Each run is continued to 50th time step and in the case that the RMSE exceeds 100 m at 50th waypoint or exceeds 5 km at only one waypoint the simulation is deemed to be diverged. Applying each optimality criteria results in approximately the similar RMSE; however, E-optimality achieves slightly higher RMSE compared to the other criteria. In terms of the divergence rate the optimality criteria are sorted as D-, A- and E-optimality from the lowest rate to the highest rate. Although, for some measurement variances the optimality criteria have underperformed. The present paper have proposed a trajectory control approach for only two UAVs based on the position error covariance produced by the EKF. The FIM is related to the estimation error covariance and is applied as an approximate optimization criterion in [12–14]. In the absence of process noise, the FIM inversely equals to the covariance recursions of the EKF with the difference that the Jacobian of the measurements for the FIM is evaluated at the

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Fig. 3. The proposed three-dimensional trajectories of two UAVs based on D-optimality criterion after (a) 30 waypoint updates, (b) 60 waypoint updates, and (c) 90 waypoint updates; (d) evolution of the RMSE.

Fig. 4. (a) The average of the number of waypoints and (b) the divergence rate for 1000 Monte Carlo runs applying the proposed control approach based on the different optimality criteria.

true state [28]. Since the true location of the target is unknown the FIM is approximated by replacing the true emitter state with the current emitter state estimate [12–14]. Therefore, the approximated FIM does not provide further information compared to the covariance matrix of the EKF and only imposes a further computational burden on the system. Furthermore, the trajectory control strategies proposed in [12–14] apply the FIM of the current mea-

surements; the prior FIM has been ignored. In the case that there are only one TDOA measurement the FIM of each observation is singular. Consequently the proposed approaches in [12–14] require at least two measurements at each time step and are not compatible to the TDOA localization applying a pair of UAVs. The path planning for the minimum number of UAVs in TDOA localization has not been tackled by the researchers. Applying a pair of UAVs

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Fig. 5. (a) The average of the RMSE of 1000 Monte Carlo runs of all 50 waypoints and (b) the average of the divergence rate for 1000 Monte Carlo runs applying the proposed control approach based on the different optimality criteria. The number of waypoints are restricted to 50 for each run.

in source localization is an important topic since some applications are constrained to apply the minimum number of UAVs; e.g. in the search and rescue scenarios when there are a large number of sources in a large search area which require to appoint a UAV group to localize each source. Therefore, it is essential to be able to apply a minimum number of UAVs and increase the source location estimation accuracy through geometry optimization. 5. Conclusions In this paper a trajectory control algorithm for two UAVs has been developed in passive localization of a radio source. The target has been assumed to be a stationary source with omnidirectional propagation. The control strategy encompasses three optimum experimental design criteria based on the position error covariance produced by the EKF. The application of the mentioned criteria has been compared in multiple simulations. The various simulations illustrated the characteristics of each criterion. It is concluded that the proposed trajectory control approach is efficient for the real situations. Applying all the optimality criteria resulted in acceptable low divergence rate for large enough measurement noise variances. The proposed control approach localizes the source with predefined accuracy in acceptable low number of waypoints. Although, D-optimality requires the least and the Eoptimality requires the most number of waypoints in the group. In the case that the UAVs are required to pass a predefined number of waypoints the D-optimality demonstrated the best performance and E-optimality represented the worst performance in terms of divergence rate. According to the results it could be concluded that D- excels A- and A- excels E-optimality in terms of divergence rate, number of waypoints and the RMSE. Conflict of interest statement There is no conflict of interest. References [1] Y. Oshman, P. Davidson, Optimization of observer trajectories for bearingsonly target localization, IEEE Trans. Aerosp. Electron. Syst. 35 (3) (1999) 892–902. [2] A.N. Bishop, B. Fidan, B. Anderson, K. Do˘gançay, P.N. Pathirana, Optimal range-difference-based localization considering geometrical constraints, IEEE J. Ocean. Eng. 33 (3) (2008) 289–301.

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