Unmanned aerial vehicle localization using distributed sensors

Abstract

The use of unmanned aerial vehicles of varying types and sizes has increased drastically in recent years. Finding the exact geometric location of these unmanned aerial vehicles is important for several reasons. In this work, we investigate a new approach to localizing unmanned aerial vehicles using multiple image sensors, such as cameras connected through sensor networks, so that sensitive areas can be protected. In the proposed approach, image sensors deployed in the ground measure the azimuth angle and angle of elevation of the target vehicle and send that information to a collector node, so it can estimate the location of the target vehicle based on the collected samples. Two estimation methods are proposed to solve the localization problem. The first scheme localizes the projection of the target vehicle on the horizontal plane and estimates the altitude later. The second scheme estimates the location of the target vehicle directly in three-dimensional space. The performance of the proposed schemes is evaluated through simulation.

Introduction

The use of unmanned aerial vehicles (UAVs), also called drones, has increased dramatically in recent years because of tremendous advancements in UAV technology and a drastic decrease in the price of UAV products. This has brought new challenges to the safety, security, and privacy of the public. Current UAVs are capable of carrying weapons, including weapons of mass destruction. They can be used for smuggling, terrorist attacks, espionage, and stealing wireless data. They can also pose a threat to commercial air flights. It is essential to protect many high-risk areas, such as nuclear power plants, jails, industrial facilities, parliaments, government facilities, embassies, stadiums and concert halls, international borders, no-fly zones, and demilitarized zones, among others.

Many incidents of civil offenses and criminal offenses related to UAVs have been reported. ¹ Most of them involve the illicit use of small UAVs, creating a nuisance to the general public, conducting surveillance and reconnaissance (collecting intelligence on a known “enemy” target), creating airspace interference, causing damage to private or public property (with or without armaments), engaging in cross-border or restricted-area smuggling, using a small UAV in a public demonstration against state agencies, hijacking wireless signals, and so on. Even though small UAV operators may not have bad intentions, UAVs pose threats to airplanes, helicopters, and even to other UAVs, and they are becoming a great challenge for law enforcement agencies. These threats are increasing day by day because of the off-the-shelf availability to anyone of diverse types of UAVs developed by many companies at very affordable prices. Thus, it is very important to develop drone detection techniques to protect a given building or geographic location from possible threats by UAVs.

Based on the type of sensor used, there are three main UAV detection techniques. The first is UAV detection by radio frequency (RF). This technique works fine if a UAV is controlled over an RF channel. However, if a drone is preprogrammed to follow a fixed route using global positioning system (GPS) information without RF communications, ² RF detection simply does not work.

A second approach is detection by radar, which is the traditional technology for detecting flying objects. Evidence shows that in some cases, radar fails to detect UAVs. In particular, radar cannot detect an object within a radar shadow behind a building, mountain, any large obstacle, or masking terrain, and cannot detect objects flying at low altitude, where the electromagnetic waves from radar cannot reach. Also, radar has a hard time detecting small, plastic, electric-powered UAVs because that is not what they were designed to detect. ³ It might be possible to design new radar to detect such small drones. However, if it is modified to also detect small objects, then the amount of noise will increase as birds, and all other small flying objects will surface above the threshold. This drawback limits the use of radar for identifying small UAVs. ⁴

A third approach is detection by vision sensors and image processing. In this approach, a camera usually plays the role of vision sensor to detect UAVs. An experiment by Nistér et al. ⁵ proved that vision can function as a promising navigation sensor that provides accurate localization. Modern cameras can see long distances at a usable resolution. An advantage of vision sensors and image processing–based approaches is that motion detection combined with a speeded-up robust features (SURF) algorithm can properly detect small UAVs while successfully ignoring other flying objects, such as birds. ⁴ Our proposed schemes come under this category.

In this article, we propose new methods to localize UAVs, overcoming the limitations of radar detection. In the proposed schemes, the radar shadow problem is resolved by deploying a sufficiently large number of image sensors, such as cameras connected through sensor networks, around the region needing protection. Each image sensor measures the azimuth angle and angle of elevation of the target UAV and sends that angle information to the data collector (sink) node through sensor networks. ⁶ Then, the data collector node estimates the location of the target UAV based on the collected angle samples. It is necessary to detect a UAV in the image in order to measure the angles.

The remainder of this article is organized as follows. Section “Literature review” reviews the related literature. In section “UAV localization scheme,” we describe our proposed aerial vehicle localization scheme. In section “Analysis results,” we compare the estimation accuracy of different schemes for low-altitude target UAVs and high-altitude target UAVs, and we present analysis results. In section “Conclusion and future work,” we summarize the work in this article and describe possible future research that builds upon this study.

Literature review

Localization is the process of finding the physical location of a UAV in accordance with some real or virtual coordinate system. Localization is an important task when direct measurement of the UAV location is not available. Generally, system performance with localization is evaluated based on the accuracy of the estimated location information at a given time.

In the literature, vision-based UAV detection has been investigated intensively in order to avoid collisions between UAVs.^7–9 Most of those vision-based approaches focused only on the detection of other UAVs, but not on localization.

Although distance estimation has been considered, ⁷ the research used a training data set for a specific UAV, and thus, the estimation is not likely to work reliably for any arbitrary UAV.

Boddhu et al. ² considered detection and tracking of hostile drones based on the paradigm of human-as-sensor. In this scheme, each user with a specific smartphone application contributes to small UAV detection by sending small UAV-related information (e.g. flight direction, user position) manually into the sensor cloud. Then, the sensor cloud notifies other users involved in the project and attempts to track the UAV.

Although this scheme is similar to our scheme, in that it uses multiple sensors, and the detailed localization method is very different, since localization is done based on users’ manual feedback.

At variance to our purpose, there is a lot of work in the literature on localizing ground-based objects when imaged from UAVs.^10–15 In these methods, the target is basically localized using its pixel location in an image, with measurement of UAV position and attitude and camera pose angle. These vision-based localization techniques in unknown environments were classified and reviewed by Ben-Afia et al. ¹⁶

To the best of our knowledge, the issue of localizing a UAV with multiple distributed sensors has not yet been discussed intensively, because most of UAVs specifically designed for civil and commercial applications were used to localize targets on the ground. In this work, we investigate the issue of localizing a UAV with multiple ground sensors in detail. Unlike most of the UAV localization schemes in the literature that focus on localizing objects in the ground with image sensors mounted on UAVs,^10–15 in the proposed schemes image sensors are deployed in the ground, that is, the region needing protection. Figure 1(a) shows the deployment of image sensors for UAV localization.

Figure 1.

(a) Deployment of image sensors for UAV localization and (b) measurement of azimuth angle (θ) and angle of elevation (φ) of a UAV by sensor nodes.

When we have two sensors monitoring a selected target UAV, if we know the angle of the target UAV measured by each sensor and the distance between two sensors, then the location of the target UAV can be determined by triangulation. ² However, if there is error in the measured angles, the accuracy of the triangulation method is significantly low. This issue is discussed in section “Analysis results.”

In this work, we investigate two new methods to improve localization accuracy by incorporating multiple sensors and solving an optimization problem based on the measurement results from multiple sensors while overcoming the limitations of a simple triangulation method. We analyze the proposed scheme by numerical analysis, and we kept a comparison of the scheme with experimental results for our future work.

UAV localization scheme

In this section, we discuss detailed methods to estimate the location of a UAV using the angle measurement values provided by image sensors deployed over a selected range of a geographic region. N denotes the total number of sensors in the selected region, and (x_i, y_i, z_i) represents the position of ith sensor node S_i, with $(x_U,y_U,z_U)$ as the position of target UAV node U at time t. Each sensor measures the azimuth angle (θ_i) and the angle of elevation (φ_i) of the target UAV at time t, as shown in Figure 1(b) and sends that information to the centralized collector (sink) node through the sensor network. Then, the sink node estimates the location of the UAV based on the collected angle values. The location estimation methods are described below.

To simplify the problem, we make the following assumptions. (a) We assume that each sensor node has the ability to detect an aerial moving object and extract the azimuth angle and angle of elevation of the target UAV using image-processing techniques on a picture it took.^7–9 (b) The clocks of the sensor nodes are synchronized, and thus, all of the working sensor nodes can take a picture at the same time. (c) The entire set of sensor nodes shares a common coordinate system, as shown in Figure 1(b), and they face in the same direction, that is, the direction of the positive y-axis. The collector node knows the position of each sensor, for example, the coordinates (x_i, y_i, z_i) for the ith sensor, S_i.

The location of the UAV needs to be determined in three-dimensional (3D) space, that is, in the reference coordinate system. We attempt to solve this problem in two different ways. In the first approach, we localize the projection of the target UAV on the horizontal plane and estimate the location of the UAV in 3D space by calculating its altitude afterwards. In the second approach, the estimation is done directly in 3D space.

Localization of the target UAV through the projection image on the horizontal plane

In this subsection, we attempt to find the location of the target UAV in two steps. In the first step, we estimate the location of the projection of the target UAV on the horizontal two-dimensional (2D) space using azimuth angles measured by sensors. In the second step, we estimate the altitude of the target UAV using the measured angles of elevation.

Figure 1(b) shows two sensors, S_i and S_j, and the target UAV U, along with their coordinates. Let us consider the projection of the points S_i, S_j, and U on the horizontal plane, which is defined by the equation z = 0. By projection, S_i, S_j, and U are mapped to $S{\prime }_i=(x_i,y_i,0)$ , $S{\prime }_j=(x_j,y_j,0)$ , and $U\prime =(x_U,y_U,0)$ , respectively. From azimuth angle θ_i measured by S_i and the coordinates of $S{\prime }_i$ , the equation of the line (l_i) emanating from $S{\prime }_j$ toward U′, which is referred to as the target-pointing line (TPL) of $S{\prime }_j$ in this article, can easily be obtained as

$$a_{i}x+b_{i}y+c_i=0$$ (1)

where $a_i=\sin {\theta }_i$ , $b_i=-\cos {\theta }_i$ , and $c_i=-\sin {\theta }_i{x}_i+\cos {\theta }_i{y}_i$ . The equation of TPL l_j passing $S{\prime }_j$ can be obtained in a similar manner from θ_j and the coordinates of $S{\prime }_j$ . Then, the coordinates of U′, $(x_U,y_U,0)$ , can be calculated from the intersection of l_i and l_j. Thus, the x and y coordinates of target U can be obtained from the intersection of two TPLs on the horizontal plane.

From φ_i and the relative position between S_i and U in Figure 1(b), we obtain

$$\tan ({\varphi }_i)=\frac{z_U-z_i}{\sqrt{{(x_i-x_U)}^2+{(y_i-y_U)}^2}}$$ (2)

By solving the above equation in terms of $z_U$ , the altitude of U, that is, $z_U$ , is obtained as

$$z_U=z_i+\tan (j_i)\sqrt{{(x_i-x_U)}^2+{(y_i-y_U)}^2}$$

Thus, we find that the location of target UAV U can be determined by the above method if two distinct sensors can provide accurate information about the azimuth angle (θ) and angle of elevation (φ). However, the angle measurement of each sensor node may not be accurate under practical situations due to limited resolution of the images, distortion of the images by an unclean lens, and so on. Thus, we investigate localization of the UAV under conditions such as noisy measurement in more detail hereafter.

As we have seen in Figure 1(b), when the location of U′ (i.e. the projection of U on the horizontal plane) is determined reliably, the calculation of altitude $(z_U)$ becomes trivial with equation (2). Thus, we concentrate on estimation of the location of U′ based on possibly noisy angle measurement samples from many sensor nodes.

When the number of sensor nodes is three or more, and the azimuth angle measurement is not accurate due to noise, there is not likely to be a single point where all the TPLs from the sensor nodes meet. In such a case, we estimate the location of U′ as point $U^{*}$ , which minimizes summation of the square of the distance between the selected point and the TPL from each sensor node, and this can be mathematically formulated as follows. The distance between a point, β = (t₁, t₂) ^T and TPL l_i given in equation (1), d((t₁, t₂), l_i), can be expressed as

$$\begin{array}{c}\multicolumn{1}{c}{d((t_1,t_2),l_i)=\frac{|a_i{t}_1+b_i{t}_2+c_i|}{\sqrt{a_i^2+b_i^2}}}\\ \multicolumn{1}{c}{=|a_i{t}_1+b_i{t}_2+c_i|}\end{array}$$

where the second equality is valid because $a_i^2+b_i^2={\sin }^2{\theta }_i+{\cos }^2{\theta }_i=1$ by the definition of a_i and b_i in equation (1). Let S(U) denote the set of sensors that detected specific target UAV U. Let us assume that |S(U)| = M, and the sensors belonging to S(U) are numbered sequentially from 1 to M. Then, $U^{*}$ , which estimates U′, can be expressed as

$$\begin{array}{c}\multicolumn{1}{c}{U^{*}=\underset{\beta ={(t_1,t_2)}^T}{\mathrm{arg\; min}}\sum _{i\in S\left(U\right)}d{((t_1,t_2),l_i)}^2}\\ \multicolumn{1}{c}{=\underset{\beta ={(t_1,t_2)}^T}{\mathrm{arg\; min}}\sum _{i=1}^M{|a_i{t}_1+b_i{t}_2+c_i|}^2}\end{array}$$

If we define C = (–c₁,–c₂, …, –c_M) ^T and

$$\left[\begin{array}{cc}{a}_1\hfill & b_1\hfill \\ \multicolumn{1}{c}{a_2}& b_2\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill \\ \multicolumn{1}{c}{a_{\mathrm{M}}}& a_{\mathrm{M}}\hfill \end{array}\right]$$ (4)

then $U^{*}$ from equation (3) can be expressed in terms of C and X as

$$U^{*}=\underset{\beta ={(t_1,t_2)}^T}{\mathrm{arg\; min}}\Vert C-X\beta {\Vert }^2$$

This is a well-known linear least square optimization problem, and the optimal solution for this problem can be obtained as follows ¹⁷

$$U^{*}=(X^{T}X{)}^{-1}{X}^{T}C$$ (5)

From the definition of X in equation (4), X^TX becomes a 2 × 2 matrix, and thus, it is not complicated to calculate the inverse of X^TX in equation (5).

Localization of the target UAV in 3D space

As a second approach to UAV localization, we solve an optimization problem directly in 3D space. A line that passes sensor S_i and has the azimuth angle and the elevation angle measured by S_i is called a 3D-TPL and is denoted as L_i. An arbitrary point, $\overrightarrow{v}$ , on the 3D-TPL Li can be represented as

$$\overrightarrow{v}=\left[\begin{array}{c}{x}_i+\cos ({\varphi }_i)\cos ({\theta }_i)t\hfill \\ \multicolumn{1}{c}{y_i+\cos ({\varphi }_i)\sin ({\theta }_i)t}\\ \multicolumn{1}{c}{z_i+\sin ({\varphi }_i)t}\end{array}\right]$$

where t is a real number corresponding to each point. The distance between point γ = (t₁, t₂, t₃) ^T and the 3D-TPL Li, d((t₁, t₂, t₃),L_i), can be expressed as follows ¹⁸

$$\begin{array}{c}d((t_1,t_2,t_3),L_i)=(t_1-x_i{)}^2+(t_2-y_i{)}^2+(t_3-z_i{)}^2\hfill \\ -((x_i-t_1)\cos ({\varphi }_i)\cos ({\theta }_i)+(y_i-t_2)\cos ({\varphi }_i)\sin ({\theta }_i)\hfill \\ +(z_i-t_3)\sin ({\varphi }_i){)}^2\hfill \end{array}$$ (6)

Since L_i is determined from the measurement of S_i, we define f_i(t₁, t₂, t₃) as f_i(t₁, t₂, t₃) = d((t₁, t₂, t₃),L_i). We estimate the location of target UAV U as $\hat{U}$ , defined as

$$\hat{U}=\underset{\gamma ={(t_1,t_2,t_3)}^T}{\mathrm{arg\; min}}F(t_1,t_2,t_3)$$ (7)

where $F(t_1,t_2,t_3)=\sum _{1\le i\le M}{f}_i(t_1,t_2,t_3)$ , and M is the number of sensors that detected U. In other words, U is estimated as the point $\hat{U}$ that minimizes summation of the squares of the distance between the selected point and the 3D-TPL from each sensor in 3D space.

The eigenvalues of the Hessian matrix of $f_i(t_1,t_2,t_3)$ can be obtained by solving the following equation

$$\begin{array}{c}0=|{\nabla }^2{f}_i(t_1,t_2,t_3)-\lambda I|\hfill \\ =\left|\begin{array}{ccc}(2-\lambda )-2{\cos }^2{\varphi }_i{\cos }^2{\theta }_i\hfill & -2{\cos }^2{\varphi }_i\cos {\theta }_i\sin {\theta }_i\hfill & -2\cos {\varphi }_i\sin {\varphi }_i\cos {\theta }_i\hfill \\ \multicolumn{1}{c}{-2{\cos }^2{\varphi }_i\cos {\theta }_i\sin {\theta }_i}& (2-\lambda )-2{\cos }^2{\varphi }_i{\sin }^2{\theta }_i\hfill & -2\cos {\varphi }_i\sin {\varphi }_i\sin {\theta }_i\hfill \\ \multicolumn{1}{c}{-2\cos {\varphi }_i\sin {\varphi }_i\cos {\theta }_i}& -2\cos {\varphi }_i\sin {\varphi }_i\sin {\theta }_i\hfill & (2-\lambda )-2{\sin }^2{\varphi }_i\hfill \end{array}\right|\hfill \\ =\lambda (\lambda -2{)}^2\hfill \end{array}$$

where I is a 3 × 3 identity matrix. Since the minimum eigenvalue is zero, the Hessian matrix of $f_i(t_1,t_2,t_3)$ is positive semidefinite for any $t_1,t_2,t_3$ , and thus, $f_i(t_1,t_2,t_3)$ defined with equation (6) is convex. ¹⁹ Since f_i is convex, the summation of $f{\prime }_{i}s$ , that is, F in equation (7), becomes convex again. Since F is quadratic and convex, F has a global minimum at the point γ = (t₁, t₂, t₃) ^T , where the following equations are simultaneously satisfied

$$\begin{array}{c}\frac{\partial F}{\partial t_1}=2\sum _{i=1}^M\{(t_1-x_i)+((x_i-t_1)\cos {\varphi }_i\cos {\theta }_i+(y_i-t_2)\cos {\varphi }_i\sin {\theta }_i+(z_i-t_3)\sin {\varphi }_i)\times \cos {\varphi }_i\cos {\theta }_i\}\hfill \\ =0\hfill \\ \frac{\partial F}{\partial t_2}=2\sum _{i=1}^M\{(t_2-y_i)+((x_i-t_1)\cos {\varphi }_i\cos {\theta }_i+(y_i-t_2)\cos {\varphi }_i\sin {\theta }_i+(z_i-t_3)\sin {\varphi }_i)\times \cos {\varphi }_i\cos {\theta }_i\}\hfill \\ =0\hfill \\ \frac{\partial F}{\partial t_3}=2\sum _{i=1}^M\{(t_3-z_i)+((x_i-t_1)\cos {\varphi }_i\cos {\theta }_i+(y_i-t_2)\cos {\varphi }_i\sin {\theta }_i+(z_i-t_3)\sin {\varphi }_i)\times \sin {\varphi }_i\}\hfill \\ =0\hfill \end{array}$$

The above set of simultaneous equations can be summarized as

$$(I-Y^{T}Y)\gamma =G$$ (8)

where Y and G are M × 3 and 3 × 1 matrices, respectively, given by

$$\begin{array}{c}Y=\left[\begin{array}{ccc}\cos ({\varphi }_1)\cos ({\theta }_1)\hfill & \cos ({\varphi }_1)\sin ({\theta }_1)\hfill & \sin ({\varphi }_1)\hfill \\ \multicolumn{1}{c}{\cos ({\varphi }_2)\cos ({\theta }_2)}& \cos ({\varphi }_2)\sin ({\theta }_2)\hfill & \sin ({\varphi }_2)\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & ⋮\hfill \\ \multicolumn{1}{c}{\cos ({\varphi }_M)\cos ({\theta }_M)}& \cos ({\varphi }_M)\sin ({\theta }_M)\hfill & \sin ({\varphi }_M)\hfill \end{array}\right]\hfill \\ G=\left[\begin{array}{c}\sum _{i=1}^M\left\{\begin{array}{c}{x}_i(1-\mathrm{co}{\mathrm{s}}^2({\varphi }_i)\mathrm{co}{\mathrm{s}}^2({\theta }_i))\hfill \\ \multicolumn{1}{c}{-y_i\mathrm{co}{\mathrm{s}}^2({\varphi }_i)\cos ({\theta }_i)\sin ({\theta }_i)}\\ \multicolumn{1}{c}{-z_i\cos ({\varphi }_i)\sin ({\varphi }_i)\cos ({\theta }_i)}\end{array}\right\}\hfill \\ \multicolumn{1}{c}{\sum _{i=1}^M\left\{\begin{array}{c}-x_i\mathrm{co}{\mathrm{s}}^2({\varphi }_i)\cos ({\theta }_i)\sin ({\theta }_i)\hfill \\ \multicolumn{1}{c}{+y_i(1-\mathrm{co}{\mathrm{s}}^2({\varphi }_i)\mathrm{si}{\mathrm{n}}^2({\theta }_i))}\\ \multicolumn{1}{c}{-z_i\cos ({\varphi }_i)\sin ({\varphi }_i)\sin ({\theta }_i)}\end{array}\right\}}\\ \multicolumn{1}{c}{\sum _{i=1}^M\left\{\begin{array}{c}-x_i\cos ({\varphi }_i)\sin ({\varphi }_i)\cos ({\theta }_i)\hfill \\ \multicolumn{1}{c}{-y_i\cos ({\varphi }_i)\sin ({\varphi }_i)\sin ({\theta }_i)}\\ \multicolumn{1}{c}{+z_i(1-\mathrm{si}{\mathrm{n}}^2({\varphi }_i))}\end{array}\right\}}\end{array}\right]\hfill \end{array}$$

Since the solution of equation (8) can be obtained by multiplying $(I-Y^{T}Y{)}^{-1}$ on both sides of equation (8), and combining equations (7) and (8) yields

$$\hat{U}=(I-Y^{T}Y{)}^{-1}G$$

Analysis results

In this section, we evaluate the accuracy of our proposed UAV localization schemes through simulation. Our proposed UAV localization scheme discussed in subsection “Localization of the target UAV through the projection image on the horizontal plane” will be referred to as 2D-TPL scheme, since the least square estimation is based on the TPL on the horizontal plane. Since the estimation is based on the 3D-TPL, the second localization scheme, discussed in subsection “Localization of the target UAV in 3D space,” will be referred to as the 3D-TPL scheme. For comparison purposes, we consider one more scheme, referred to as the centroid scheme in this article. The centroid scheme first finds the intersection of two TPLs for every possible pair of TPLs on the horizontal plane. Then, the projection of the target UAV, U′, is estimated by the centroid, that is, arithmetic mean position, of all intersection points. The altitude is estimated with equation (2) in the same manner as the 2D-TPL scheme.

Simulation environment and parameters

We evaluated the proposed schemes in the following simulation environments. In our simulation, we classify the target UAV based on its altitude, since the sensors are assumed to be deployed over a sufficiently wide range of the geographic region. We consider two types of UAV: a high-altitude UAV, where the altitude is 550 m, and a low-altitude UAV, where the altitude is 50 m. The number of sensors, M, changes from 4 to 30, and when the number of sensors is selected, the position of each sensor is uniformly selected from the 3D range (0, 1000) × (0, 100) × (0, 10). However, the position of the target UAV is fixed at (500, 200, 50) for the low-altitude UAV and fixed at (500, 200, 550) for the high-altitude UAV. The relative position between the target UAV and each sensor is randomized by random selection of sensor positions. As described in section “UAV localization scheme,” all the sensors are directed toward the positive y-axis.

E_h and E_v are random variables representing the measurement error for azimuth angle and elevation angle, respectively. Both of them are modeled as a Gaussian random variable with the same zero mean and a standard deviation of σ_e degrees. We consider two values, 1.0 and 5.0, for σ_e.

Evaluation results

Figure 2(a) compares the accuracies of the 2D-TPL, 3D-TPL, and centroid schemes for a low-altitude target UAV and various numbers of sensors. We ran 20 simulations for each number of sensors, and the average error is shown in the figure. The centroid scheme exhibits significantly worse performance than the proposed schemes, and we found the same trend in other scenarios. Thus, we concentrate on a comparison of 2T-TPL and 3D-TPL hereafter. Figure 2 shows that the average error in 2D-TPL and 3D-TPL tends to decrease as the number of sensors increases. The larger the value of σ_e, the larger the estimation error for both proposed schemes. 2D-TPL and 3D-TPL exhibit similar performance for a diverse number of sensors when the altitude of the target UAV is low.

Figure 2.

Comparison of estimation accuracies for the 2D-TPL, 3D-TPL, and centroid schemes for a low-altitude target UAV at target location (500, 200, 50): (a) comparison of the 2D-TPL, 3D-TPL, and centroid schemes (σ_e = 1.0) and (b) comparison of the 2D-TPL and 3D-TPL schemes (σ_e = 5.0).

Figure 3 compares the accuracies of 2D-TPL and 3D-TPL for a high-altitude target UAV. The 2D-TPL and 3D-TPL schemes show similar performance when σ_e is small. However, 3D-TPL outperforms 2D-TPL with a lower average error when σ_e is large and the altitude of the target UAV is high. 2D-TPL attempts to concentrate on accurate localization of the target UAV projected on the horizontal plane, sacrificing accuracy in altitude estimation. This attempt might be successful when the absolute value of the target altitude is small, compared to the magnitude of the horizontal distance. However, it may not be successful when the altitude estimation error is not small due to a high altitude of the target, as shown in Figure 3(b). When the measurement error in the angle of elevation is the same, the magnitude of the altitude error is larger for a higher altitude.

Figure 3.

Comparison of estimation accuracies of the 2D-TPL and 3D-TPL schemes for a high-altitude target UAV at target location (500, 200, 550): (a) σ_e = 1.0 and (b) σ_e = 5.0.

Conclusion and future work

In this work, we proposed two schemes for localizing UAVs based on measurement of azimuth angle and angle of elevation by many image sensors distributed over a wide geographic region. The first scheme (2D-TPL) attempts to localize the projected image of the target UAV on the horizontal plane and finds the altitude of the target UAV as the next step. The second scheme (3D-TPL) attempts to directly localize the target UAV in 3D space. The simulation results show that both schemes exhibit very similar performance in terms of localization error when the altitude of the target UAV is low. However, 3D-TPL outperforms 2D-TPL, especially when the altitude of the target UAV is high and the measurement error in the angle of elevation is large. Comparison of the simulation results with experimental results is left for future work.