Joint Placement and Communication Optimization of UAV Base Stations in GPS-Denied Environments

I. Introduction

UNMANNED aerial vehicles (UAVs) are being widely used in wireless communication environments across military, commercial, and industrial sectors for their many benefits including mobility, versatility, and flexibility [1]. For example, UAVs are utilized as aerial platforms to establish, enhance, or extend communication networks. Further, UAVbased sensing systems provide improved detection, tracking, and localization performances [2].

Recently, many studies have started focusing on UAV platforms that integrate both communication and sensing functionalities, commonly known as integrated sensing and communication (ISAC) technology [3]. Incorporating ISAC technologies into UAVs enhance their communication and sensing capabilities, including improved UAV deployment, efficient spectrum usage, increased reliability, and better localization accuracy. Many solid studies have already been conducted aiming to simultaneously enhance communications and sensing capabilities of such dual functional UAV platforms. For instance, in [4], the system’s achievable rate is maximized based on UAV trajectory, user association, target sensing selection, and transmit beamforming. Also, in [5], weighted sum-rate throughput is maximized while ensuring the sensing beampattern gain requirements and the authors in [6] optimized the UAV trajectory in terms of relative position and velocity in scenarios where the UAV simultaneously tracks and communicates with an object. Moreover, in [7], ISAC based channel estimation method is utilized to enhance system performance of UAVs.

Particularly, within the context of localization in terms of sensing in ISAC based UAV systems, the work in [8] considered an environment where a single UAV collaborates with ground base stations (BSs) to provide communication and localization services to user equipments (UEs) on ground. The authors jointly optimized UAV position and network resource allocation to maximize the communication rates of UEs under the localization accuracy constraints of the UEs. In [9], a multiple UAV optimal deployment method that minimizes both the number and location of UAVs is proposed, aiming to improve both the localization accuracy and communication rates of UEs. In [10], the UAV location, user association, and UAV transmission power are jointly optimized under the constraints of the localization accuracy of the UEs. Moreover, the authors in [11] studied a joint UAV allocated power and deployment problem that improved the localization accuracy of all potential UEs in the area of interest. Further, in [12], a joint UAV trajectory and resource optimization problem where the UAV is employed to assist UE localization and data transmission was solved.

In terms of localization, previous works on UAV-based ISAC systems focused on scenarios where UAV-BSs are utilized as reference anchor nodes to provide aid to UEs on ground for localization, which we refer to as “UE-localization”. For such localization systems to work, accurate UAV locations are given in existing works assuming that UAV-BSs are placed at precise positions so that UEs can sense the UAV for positioning. Generally, most UAVs rely on global navigation satellite systems (GNSS) for accurate positioning and among viable options, global positioning system (GPS) is the most widely used system. According to some works such as in [7] and [10], it is directly mentioned that locations of UAVs are obtained via GNSS.

However in reality, GPS signals are vulnerable to interference and jamming because of its low power [13]. Especially, it is simpler to attack aerial platforms such as UAVs compared to ground-based platforms because it is easier to achieve necessary line of sight (LoS) with aerial platforms, making them more susceptible to GPS attacks [14], [15]. In [16], a beamforming technology based jamming system was proposed wherein a ground-based jammer emits a signal targeted at a specified region where the UAV may be located considering the possibility of location error. Moreover, the study in [17] dealt with the susceptibility of UAVs to GPS jamming and provided a backup navigation system in which a multi-antenna communication system was utilized allowing the UAV to return-to-home by estimating the angle of arrival of the ground home transmitter. UAV pseudolites can be deployed to enhance or supplement positioning, navigation, and timing services to overcome limitations in scenarios with such restricted GPS availability, thereby improving the accuracy of target nodes [18], [19]. Therefore, these potential threats have resulted in a gradual increase in the research on anti-jamming solutions. However, the UAV network is still in its initial phase of development and research on UAV-specific anti-jamming strategies remains uncommon, leaving the risk of jamming attacks on UAVs a significant concern [20].

In this paper, we present a “UAV-localization” scenario (as opposed to “UE-localization”) where the GPS-jammed UAV-BS position is estimated by receiving sensed location information of UEs on ground that are utilized as reference anchor nodes. We assume a scenario where the UAV-BS undergo positioning disability because of jamming attacks targeted directly at it, whereas UEs on the ground remain unaffected retaining accurate positioning capabilities. Despite such an attack scenario, the UAV-BS needs to be positioned at the nearest optimal location as possible to continue its role as a base station to provide efficient downlink communication service. UEs remain unaffected, and therefore, they are utilized as reference anchor nodes for localization of UAV-BS 3D position. The UAV-BS uses its ISAC signal to provide communication service to UEs while also performing sensing of UEs needed for localization of its position.

We formulate an optimization problem to obtain the estimated UAV-BS position by selecting reference nodes that maximizes localization accuracy, while simultaneously minimizing the UAV transmission power. Thus, we derive an UAV-BS position that jointly minimizes the GPS jammed UAV-BS communication power while maximizing its localization accuracy for different UE distributions. We utilize the time difference of arrival (TDOA) method, which requires at least four reference anchors nodes, for estimation of a 3D location. The deployment of reference nodes affects the localization performance [21], and therefore we propose three different methods to select the optimal reference nodes. Furthermore, different localization accuracy thresholds for UAV-BS are applied, where position dilution of precision (PDOP) is used as the numerical criterion. Our proposed scheme is applied to different UE distribution models: Random point process (RPP) and clustered point process (CPP). In the RPP model, UEs are randomly positioned while the CPP model represents a more realistic, clustered UE distribution environment, where each big cluster may represent units of platoon in the military field. To the best of our knowledge, no previous work has presented such joint placement and communication system in an aerial GPS denied environment.

The contributions of this paper are summarized below:

· We propose an ISAC based UAV-BS system where the UAV-BS, that provides downlink communication service to UEs on ground is under a GPS jamming attack. Unaffected UEs on ground are utilized as reference anchor nodes to perform “UAV localization” to position the UAV-BS to an optimal location. Our system jointly optimizes the UAV-BS’s ISAC signal in terms of transmission power and localization accuracy of estimated position.

· We present three different reference anchor node selection methods, (center, centroid, and circumcenter methods), which select nodes based on the relative positions of the UEs and the different geometric configurations they each form. The concept of convex hull is used to reduce computational complexity.

· We prepare performance analysis of our proposed scheme, wherein the performance of each method is compared in terms of average transmission power, maximum transmission power, and root mean squared error (RMSE) values under PDOP constraints of the UAV-BS. Simulation is applied to RPP and CPP UE distribution models, and the results show that the proposed methods prove efficient especially in the CPP model. Further, the preferred method based on the communication type (unicast or broadcast) is analyzed.

The remainder of this paper is organized as follows. In Section II, the system model is introduced along with the problem formulation. Section III presents the details of the proposed scheme and in Section IV, MATLAB simulation results are provided to evaluate the proposed algorithms. Finally, the conclusion and future works are presented in Section V.

II. System Model and Problem Formulation

A. System Model

As shown in Fig. 1, a downlink wireless communication scenario is considered where a single rotary wing UAV-BS hovers in air to provide service to UEs on ground with its ISAC signals. We assume that the UAV-BS is initially located at its optimal position [TeX:] $$\mathrm{UAV}_{opt }$$, with an altitude of [TeX:] $$\mathrm{h}_{opt }$$, covering a maximum circular region of radius [TeX:] $$\mathrm{R}_{\max }$$. The optimal position implies that theUAVis located at an optimal altitude maximizing the area of the coverage region for a given maximum allowable pathloss [TeX:] $$\mathrm{PL}_{\max }$$ between UAV and UE. N UEs, [TeX:] $$\mathrm{UE}_N=\left\{u e_i\right\}_{i=1}^N$$ are located at fixed positions inside the coverage area, whereas their altitude is not considered, all set to 0. We assume all UEs are perfectly time synchronized with each other through GPS.

Initially positioned at [TeX:] $$\left(0,0, h_{opt}\right)$$, we consider a scenario where the UAV-BS, [TeX:] $$\mathrm{UAV}_{opt }$$, gets exposed to a GPS jammer. This attack is restricted to the aerial domain, wherein the UAV-BS loses its positioning ability, as opposed to the UEs which maintain their positioning accuracy.

TDOA localization method is employed to perform “UAV-localization” for positioning of the UAV-BS to ensure that the UAV-BS can continue its service as an aerial communication BS for UEs in such GPS-denied environments. The GPS-jammed UAV-BS is positioned to an optimal location by using its ISAC signals to receive location information from UEs utilized as reference anchor nodes. Position information from at least four reference anchor nodes is required for this localization scheme to work for a 3D target. Thus, aid from at least four UEs on ground is required to obtain a unique 3D UAV-BS position. The four selected reference anchor nodes among [TeX:] $$\mathrm{UE}_N$$ are denoted as [TeX:] $$r e f_{1 \sim 4},$$ where [TeX:] $$r e f_{1 \sim 4} \in \mathrm{UE}_N.$$ The deployment of these reference anchor nodes significantly affects the accuracy of the target location to be estimated [21]–[23], making the selection of nodes a critical task for high positioning accuracy.

The UAV-BS target position [TeX:] $$\mathrm{UAV}_{tar}$$ is set accordingly. Its 2D coordinates [TeX:] $$\left(x_t, y_t\right)$$ are set based on absolute positional location or relative UE positions depending on proposed methods, and its altitude [TeX:] $$z_t$$ is fixed at target [TeX:] $$\mathrm{h}_{opt }.$$ The UAV-BS estimated position derived as a result of TDOA localization using [TeX:] $$r e f_{1 \sim 4},$$ is represented as [TeX:] $$\mathrm{UAV}_{est }$$ with coordinates [TeX:] $$\left(\hat{x}_t, \hat{y}_t, \hat{z}_t\right) .$$ A summary of related notations and symbols is given in Table I.

B. Optimal UAV-BS Deployment

Before the GPS jamming attack scenario, we assume that the UAV is initially deployed at its optimal altitude that maximizes the area of the coverage region for a given maximum allowable path loss threshold. We adopt the air-to-ground (AtG) channel model as proposed in [24] to derive the optimal location because of its simplicity and generality. Signals emitted from UAV-BSs consist of both LoS and non-LoS (NLoS) signals.Aprobabilistic model must be used to calculate the overall mean path loss that can be expressed as

where [TeX:] $$L_{L o S} \text { and } L_{N L o S}$$ represent path loss models for LoS and NLoS signals, respectively. Meanwhile, [TeX:] $$P_{L o S} \text { and } P_{N L o S}$$ denote the probabilities of establishing a LoS and NLoS connection, respectively. Also, h represents the UAV altitude and [TeX:] $$r_i$$ represents the 2D distance between the UAV and [TeX:] $$u e_i.$$ Organizing this equation according to [24] yields

where the parameters A and B are defined as:

Here, f denotes the carrier frequency, c represents the speed of light, and [TeX:] $$a, b, \eta_{\mathrm{LoS}}, \text { and } \eta_{\mathrm{NLoS}}$$ represent the environmental parameters affecting the model. These parameters determine how the LoS and NLoS components contribute to the pathloss, reflecting the environmental influence on signal propagation.

For a given threshold [TeX:] $$\mathrm{PL}_{\max },$$ the coverage region radius can be expressed as [TeX:] $$\mathrm{R}_{\max }=\left.r_i\right|_{\mathrm{PL}_{h, r_i}=\mathrm{PL}_{\max }}.$$ This equation is an implicit function that provides a critical point indicating the maximum circular radius [TeX:] $$r_i$$ where the UAV altitude h is optimal. By establishing a maximum allowable pathloss threshold of [TeX:] $$\mathrm{PL}_{\max },$$ at 100 dB, it is possible to determine the UAV’s maximum coverage radius [TeX:] $$\mathrm{R}_{\max },$$ and the corresponding optimal altitude [TeX:] $$\mathrm{h}_{opt }.$$ These results, along with the environmental parameters, are summarized in Table II by environment.

C. Problem Formulation

We aim to simultaneously enhance communication performance and localization accuracy for the estimated UAV-BS [TeX:] $$\mathrm{UAV}_{e s t}$$ by optimizing the reference anchor node selection subject to localization accuracy thresholds. We assess communication performance using [TeX:] $$\mathrm{P}_{a v g}$$ for unicast communication and [TeX:] $$\mathrm{P}_{\max }$$ for broadcast communication. Depending on the communication required, we minimize either [TeX:] $$\mathrm{P}_{avg } \text { or } \mathrm{P}_{\max }$$ in conjunction with the localization accuracy, which is measured in terms of RMSE between [TeX:] $$\mathrm{UAV}_{tar } \text { and } \mathrm{UAV}_{est } \text {. }$$ This process is carried out repeatedly, up to N times. K represents the maximum number of feasible reference anchor node selections, and therefore the possible combinations of [TeX:] $$r e f_{1 \sim 4}$$ selected from N UEs. Further, PDOP is used as a metric for the accuracy threshold the [TeX:] $$\mathrm{UAV}_{e s t}$$ must satisfy, represented by [TeX:] $$\tau_{\max }.$$ The final minimum PDOP value [TeX:] $$\mathrm{UAV}_{\min },$$ derived as a result of N iterations should satisfy [TeX:] $$\tau_{\max }$$ for it to be valid.

Here, PDOP is a dimensionless concept used to measure the 3-dimensional (latitude, longitude, and altitude) accuracy of a target, which depends on the geometric arrangement formed by positions of the reference anchor nodes and its relationship to the target. A low PDOP value implies good accuracy, and a high value of PDOP means poor accuracy. For mathematical calculation, the pseudo-range between UAV-BS and the reference anchor nodes should first be calculated. The pseudo-range [TeX:] $$d_{r i}$$ between target UAV-BS and ith reference anchor node is calculated as:

Next, design matrix H is constructed using (5), the coordinates of the UAV-BS, and the coordinates of the four reference anchor nodes.

The covariance matrix Q is then calculated as the inverse of the product of the transposed H and H:

Finally, PDOP is computed as:

where [TeX:] $$Q_{11}, Q_{22}, \text { and } Q_{33}$$ are the diagonal elements of Q, corresponding to the variances in the x, y, and z coordinates, respectively.

[TeX:] $$\mathrm{P}_{a v g}\left(\text { or } \mathrm{P}_{\max }\right.) $$ and RMSE are minimized as represented in (9a) under constraints from (9b) to (9f). (9b) and (9c) showthe constraints for UEs. The N UEs, [TeX:] $$\left\{u e_i\right\}_{i=1}^N$$ are placed inside the circular coverage region of radius [TeX:] $$\mathrm{R}_{\max }.$$ [TeX:] $$\mathrm{UAV}_{tar }$$'s 2D coordinates are set to either (0, 0), [TeX:] $$\left(G_x, G_y\right), \text { or }\left(C_x, C_y\right)$$ depending on the method as denoted in (9d). [TeX:] $$\left(G_x, G_y\right)$$ represents the centroid point of triangle formed by the three reference nodes, [TeX:] $$\Delta r e f_1 r e f_2 r e f_3 \text { and }\left(C_x, C_y\right)$$ represents the circumcenter point of [TeX:] $$\Delta r e f_1 r e f_2 r e f_3.$$ A detailed explanation of the derivation of these points is described in the next subsection, proposed scheme.

Moreover, [TeX:] $$\mathrm{UAV}_{tar }$$'s altitude, [TeX:] $$z_t,$$ is set to [TeX:] $$\mathrm{h}_{opt }$$ represented by constraint (8e). [TeX:] $$\tau_{\max }$$ depicts the PDOP threshold that [TeX:] $$\mathrm{UAV}_{est }$$ needs to satisfy and (8f) denotes that the minimum PDOP value [TeX:] $$\mathrm{pdop}_{min }$$ out of [TeX:] $$\mathrm{pdop}_k (k=1, \cdots, K)$$ from K iterations of four reference node pair selection, should not exceed the maximum threshold of [TeX:] $$\tau_{\max }$$.

subject to:

The transmission power from UAV-BS to [TeX:] $$u e_i, \mathrm{P}_t(i),$$ can be calculated using (10a), where [TeX:] $$\mathrm{P}_{\min}$$ represents the minimum required transmission power, and [TeX:] $$\mathrm{PL}_{h, r_i}$$ represents the path loss between [TeX:] $$\mathrm{UAV}_{est} \text{ and } ue_i.$$ [TeX:] $$\mathrm{P}_{avg } \text { and } \mathrm{P}_{\max }$$ can be derived using (10b) and (10c), whereas RMSE between [TeX:] $$\operatorname{UAV}_{tar }\left(\left(x_t, y_t, z_t\right)\right) \text { and } \operatorname{UAV}_{e s t}\left(\left(\hat{x}_t, \hat{y}_t, \hat{z}_t\right)\right)$$ can be calculated with (10d).

III. Proposed Scheme

Our proposed solution for the formulated problem involves selecting reference nodes (Phase 1) and then conduction of localization based on the priorly selected nodes (Phase 2). This process is applied across three different methods, which primarily differ in how the reference nodes are selected. Selecting the reference nodes appropriately is critical because their geographical placement significantly influences localization accuracy. The three methods are the center, centroid, and circumcenter methods. We assess each method by comparing key evaluation metrics, such as communication efficiency and localization accuracy. Further, we determine the most effective method based on the communication scenario, whether unicast or broadcast.

The flow chart of our entire proposed scheme is shown in Fig. 2. Our proposed scheme first obtains the UAV-BS initial position [TeX:] $$\mathrm{h}_{opt }$$ by environment, as organized in Table II. Next, initialization is performed by setting [TeX:] $$\mathrm{pdop}_0$$ to an infinite number and k = 1. Phase 1 and Phase 2 are repeated for K cases, where K represents the total number of possible cases for selecting four reference nodes from N UEs. If [TeX:] $$\mathrm{pdop}_k$$ obtained from the kth iteration satisfies [TeX:] $$\mathrm{pdop}_k \lt \mathrm{pdop}_{k-1},$$ the reference anchor nodes from this iteration are selected as [TeX:] $$r e f_{1 \sim 4},$$ and [TeX:] $$\mathrm{pdop}_k$$ is assigned to [TeX:] $$\mathrm{pdop}_{\min}$$. The previous selected reference nodes are discarded. This algorithm iterates K times updating [TeX:] $$r e f_{1 \sim 4}$$ and [TeX:] $$\mathrm{pdop}_{\min}$$ when conditions are met. After K iterations, the final [TeX:] $$\mathrm{pdop}_{\min}$$ value is compared with the PDOP constraint [TeX:] $$\tau_{\max }$$ for validity check. If satisfied, thus if [TeX:] $$\operatorname{pdop}_{\min } \leq \tau_{\max },$$ it is regarded as a valid case.

A. Phase 1: Selection of Reference Nodes

In order to estimate the 3D position of UAV-BS, position information from at least four reference nodes is required to perform TDOA localization. The basic concept behind the proposed reference node selection is picking nodes as symmetric and spread-out as possible based on triangular method [21].

Selecting four UEs as reference anchor nodes from N UEs is computationally complex with K resulting in [TeX:] $${ }_N C_4$$ cases. Mathematically, [TeX:] $${ }_N C_4$$ represents the number of ways to choose 4 nodes from a set of N UEs without regard to the order. This number is calculated using the combination formula [TeX:] $${ }_n C_r=n!/(r!(n-r)!).$$ In order to reduce complexity, we utilize the concept of a convex hull, which is a well known concept in the field of computational geometry in mathematics [25]. A convex hull represents the smallest convex shape enclosing all set of points in Euclidean space.

In our study, given a set of UEs, [TeX:] $$\mathrm{UE}_N=\left\{u e_i\right\}_{i=1}^N,$$ the convex hull is the smallest convex polygon that encloses all N nodes. Mathematically, the convex hull [TeX:] $$\operatorname{Conv}\left(\mathrm{UE}_N\right)$$ consists of points formed by the convex combinations of [TeX:] $$\mathrm{UE}_N.$$

where [TeX:] $$\lambda_i$$ is the ith scalar coefficient that forms a convex combination of the points in [TeX:] $$\mathrm{UE}_N.$$ A convex combination is a specific type of linear combination where each coefficient [TeX:] $$\lambda_i$$ is non-negative, and the sum of all coefficients equals 1. These conditions ensure that the resulting point of this combination lies within the convex region formed by the set [TeX:] $$\mathrm{UE}_N.$$ Essentially, each point within the convex hull can be represented as a convex combination of the vertices of the hull.

The first three reference nodes, [TeX:] $$r e f_1, r e f_2 \text {, and } r e f_3$$ are UEs that are spatially equidistant from the target. The convex hull implementation is applied when selecting these initial three reference nodes. The selection of [TeX:] $$r e f_1, r e f_2 \text {, and } r e f_3$$ is identical for all three methods. The last node [TeX:] $$r e f_4,$$ which is selected based on proximity to the desired central point differs

for all three methods depending on how the desired central point is defined. The center method selects [TeX:] $$r e f_4$$ using a location-based approach while the center and circumcenter methods utilize a UE-centric approach. The description of the process of selecting [TeX:] $$r e f_1, r e f_2, r e f_3 \text {, and } r e f_4$$ is shown in Algorithm 1.

In detail, the first three reference nodes [TeX:] $$r e f_1, r e f_2 \text {, and } r e f_3$$ are selected based on the nodes that form the triangle with the maximum area among all possible triangles formed from the set [TeX:] $$\mathrm{UE}_N$$. This selection method reflects the concept of choosing three nodes that are as spatially spread out as possible within the given set of UEs. It must be proven that the three vertices of the triangle with the maximum area are included in the convex hull [TeX:] $$\operatorname{Conv}\left(\mathrm{UE}_N\right)$$. We provide this proof using a contradiction approach.

Proposition: Given a set of points [TeX:] $$\mathrm{UE}_N$$, the vertices of the triangle with the maximum area are contained within the convex hull [TeX:] $$\operatorname{Conv}\left(\mathrm{UE}_N\right)$$.

Proof: Assume, for the sake of contradiction, that there exists a triangle with maximum area with vertices p, q, and r from [TeX:] $$\mathrm{UE}_N,$$ where at least one vertex, say p, is not on [TeX:] $$\operatorname{Conv}\left(\mathrm{UE}_N\right)$$. Since p is not a node included in [TeX:] $$\operatorname{Conv}\left(\mathrm{UE}_N\right)$$, there exists a point [TeX:] $$p^{\prime}$$, on the boundary of [TeX:] $$\operatorname{Conv}\left(\mathrm{UE}_N\right)$$ such that the triangle formed by [TeX:] $$p^{\prime}$$, q, amd r has a larger area than the triangle formed by p, q, and r. This follows from the fact that moving p towards [TeX:] $$p^{\prime}$$ along the direction increasing the distance to the line segment [q, r] will increase the area, contradicting our initial assumption of maximality. Hence, all vertices of the triangle with maximum area must be on [TeX:] $$\operatorname{Conv}\left(\mathrm{UE}_N\right)$$.

The area of a triangle formed by the three points from [TeX:] $$\operatorname{Conv}\left(\mathrm{UE}_N\right)$$ can be calculated using the determinant of a matrix. Let the three points be [TeX:] $$r e f_1\left(x_{r 1}, y_{r 1}\right), r e f_2\left(x_{r 2}, y_{r 2}\right),$$ and [TeX:] $$r e f_3\left(x_{r 3}, y_{r 3}\right).$$ (The altitude term, [TeX:] $$z_{r i}$$ is ignored here for simplicity since it is fixed at 0.) Then, the area of the triangle [TeX:] $$\Delta r e f_1 r e f_2 r e f_3$$ is given by (12).

The fourth reference node [TeX:] $$r e f_4$$ is selected based on its proximity to a specific point of interest. For the center method, [TeX:] $$r e f_4$$ is selected as the UE with the smallest Euclidean distance to the origin, (0,0). For centroid and circumcenter methods, [TeX:] $$r e f_4$$ is selected as the nearest UE to the centroid and circumcenter points of [TeX:] $$\Delta r e f_1 r e f_2 r e f_3,$$ respectively. The centroid point [TeX:] $$\left(G_x, G_y\right)$$ and circumcenter point [TeX:] $$\left(C_x, C_y\right)$$ can be calculated using (13) and (14), where [TeX:] $$\theta_1, \theta_2 \text {, and } \theta_3$$ represent the angles at the vertices [TeX:] $$r e f_1, r e f_2 \text {, and } r e f_3$$ of the triangle, respectively.

Utilizing such reference node selection method decreases computational complexity, K, from [TeX:] $${ }_N C_4 \text { to }{ }_L C_3,$$ where L represents the total number of nodes in [TeX:] $$\operatorname{Conv}\left(\mathrm{UE}_N\right)$$. This is because upon selection of the first three reference nodes, [TeX:] $$r e f_1, r e f_2 \text {, and } r e f_3$$, the final node [TeX:] $$r e f_4$$ is decided automatically for each method. Accordingly, the time complexity expression of generating such combinations, derived in the form of big-O notation is decreased from [TeX:] $$O\left(2^N\right) \text { to } O\left(2^L\right) .$$ Since L is reduced significantly from N, the time complexity efficiency is improved notably.

Examples of UE distribution and its corresponding convex hull, selected reference nodes, centroid point, and circumcenter point based for RPP UE distribution model is shown in Fig. 3(a), 3(b), and 3(c) for N = 10, 50, 90. In Fig. 3(d), 3(e), and 3(f) examples generated for CPP model when N = 100 is presented with CF values of CF = 1, 2, 3 respectively.

B. Phase 2: Localization Based on Selected Nodes

Based on the selected reference anchor nodes from the first phase, UAV-BS localization is performed using TDOA localization method. To estimate the targetUAV-BS 3D location, measurements from at least four nodes are required, whose positions are known, stationary, and timely synchronized with one another. Before conducting localization, the target UAV-BS position [TeX:] $$\mathrm{UAV}_{tar }\left(x_t, y_t, z_t\right)$$ needs to be set, which is set differently based on node selection methods. For the center method, [TeX:] $$\mathrm{UAV}_{tar }$$ is [TeX:] $$\left(0,0, h_{o p t}\right),$$ which is identical to the UAV-BS’s initial optimal location. For the centroid and circumcenter methods, [TeX:] $$\mathrm{UAV}_{tar }$$ equals [TeX:] $$\left(G_x, G_y, h_{opt }\right) \text { and }\left(C_x, C_y, h_{opt }\right) \text {, }$$ respectively. A summary of selection of [TeX:] $$r e f_4$$ from Phase 1 and the decision of [TeX:] $$\mathrm{UAV}_{tar }$$ from Phase 2 by each of the three proposed methods is summarized in Table III.

The range [TeX:] $$d_{ri }$$ between the ith reference node and target UAV-BS can be obtained with (5). Next the range difference between the target UAV-BS with two difference reference nodes, [TeX:] $$r e f_i \text { and } r e f_j, D_{i, j},$$ can be expressed as

where c represents the speed of light and [TeX:] $$t_i$$ represents the time the signal arrives from [TeX:] $$r e f_i.$$ [TeX:] $$\tau_{i,j}$$ depicts the TDOA between [TeX:] $$r e f_i \text { and } r e f_j \text {. }$$ Thus,

and

Since i = 1, 2, 3, 4, we obtain four equations like given in (17). Solving these four equations leads to a unique 3D target location. The solution is solved through the linearization of [TeX:] $$d_{ri}$$ from (17) with Taylor series expansion upon the initial estimation of the target UAV-BS position, [TeX:] $$\mathrm{UAV}_0=\left(x_{t_0}, y_{t_0}, z_{t_0}\right)$$ [26].

The linear approximation of the distance from each reference point to the target is:

where [TeX:] $$d_{ri, 0}$$ is the initial distance from [TeX:] $$\mathrm{UAV}_0 \text{ to } r e f_i,$$ and the partial derivatives represent the components of the direction vector from [TeX:] $$\mathrm{UAV}_0 \text{ to } r e f_i.$$

These linearized equations form a system:

where [TeX:] $$\mathbf{A}$$ is a matrix derived from the gradients of the distances, [TeX:] $$\mathbf{b}$$ is a vector representing the differences in squared distances derived from the measured TDOAs, and [TeX:] $$\delta \mathbf{p}$$ is the change in position as [TeX:] $$\delta \mathbf{p}=\left(x_t-x_{t_0}, y_t-y_{t_0}, z_t-z_{t_0}\right).$$ [TeX:] $$\mathbf{A}$$ can be expressed with the following equation.

We solve this system using the least squares method:

and update the position of the UAV-BS iteratively:

IV. Performance Analysis

A. Simulation Setup

We evaluate the UAV-BS joint placement and communication system across two different UE distribution models: RPP and CPP model. [TeX:] $$\mathrm{P}_{a v g}, \mathrm{P}_{\max },$$ and RMSE values are compared as performance metrics for each reference node selection method under RPP and CPP models. [TeX:] $$\mathrm{P}_{a v g}$$ is a used for evaluating unicast communication, whereas [TeX:] $$\mathrm{P}_{\max }$$ is used for broadcast communication. In addition, for the CPP distribution scenario, the percentage of valid cases that satisfy different PDOP threshold value are also analyzed.

As described in Table IV, each simulation run is repeated 20, 000 times and the ensemble averages of the performance metrics are driven. PDOP constraints are set to 6 and 20 in which the former indicates requirement for “good” accuracy while the latter is a threshold to be met for “fair” accuracy. The ratings for these DOP values are suggested in [27]. Further, we assume a suburban environment in which the UAV-BS is initially placed at its optimal location of [TeX:] $$\mathrm{h}_{opt }=403.9 \mathrm{~m} \text { and } \mathrm{R}_{\max }=1089.8 \mathrm{~m}$$ as shown in Table II. These values were derived using the pathloss model from [24] under maximum allowable pathloss threshold [TeX:] $$\mathrm{PL}_{\max }$$ of 100 dB.

B. UE Distribution Model Setup

In the RPP model, UEs are randomly positioned inside the coverage region. On the other hand, the CPP model provides a more realistic representation of clustered node distributions, which is relevant in environments like military battlefield scenarios where UEs are organized into clustered formations such as battalions, squadrons, or platoons. For both models, the UEs distributed on ground are fixed at known 2D positions and their altitude is neglected and thus, it is set to 0.

1) Random point process (RPP) model: N UE nodes are randomly distributed according to Poisson point processes (PPPs) within the circular coverage region of radius [TeX:] $$\mathrm{R}_{\max }$$ centered at the origin. Each node [TeX:] $$u e_i=\left(x_i, y_i\right)$$ is defined by its coordinates where [TeX:] $$x_i \text{ and } y_i$$ represent independent random variables uniformly distributed over the interval [TeX:] $$\left[-\mathrm{R}_{\max }, \mathrm{R}_{\max }\right]$$ subject to the constraint [TeX:] $$x_i^2+y_i^2 \leq \mathrm{R}_{\max }^2 .$$ The density of the point process is λ, where [TeX:] $$\lambda=N /\left(\pi \mathrm{R}_{\max }^2\right) .$$ The total number of UEs generated N, is varied.

2) Clustered point process (CPP) model: In the clustered point process (CPP) model, the total number of UEs generated is fixed at N. However, the degree of spatial clustering among these UEs is modulated by the clustering factor (CF), which quantitatively influences the density of UE distribution. A higher CF indicates a stronger clustering tendency. This model accommodates two scenarios: PPP when CF = 1 and CPP when CF > 1. When CF = 1, UEs are distributed according to a standard PPP, characterized by a random spatial distribution without additional clustering. Thus, UEs are distributed according to PPP from the RPP model.

For CF > 1, each UE node initially positioned at [TeX:] $$\left(x_i, y_i\right)$$ is treated as the center of a potential cluster. The number of additional UEs to be clustered around each center, [TeX:] $$M_i,$$ follows a Poisson distribution: [TeX:] $$M_i \sim \operatorname{Pois}((\mathrm{CF}-1) \times 10) .$$ Thus, for each cluster center, additional points are generated by iterating from 1 to [TeX:] $$M_i.$$ This approach enables the density of nearby UEs to increase with CF value, enhancing the clustering effect around each initial UE position.

The coordinates of the additional points [TeX:] $$\left(x_{i j}, y_{i j}\right)$$ for [TeX:] $$j=1, \cdots, M_i,$$ are determined by introducing offsets [TeX:] $$\left(\delta_x, \delta_y\right)$$ defined by normally distributed random variables: [TeX:] $$\delta_x \sim \mathcal{N}(0,(0.05 \cdot \left.\left.\mathrm{R}_{\max }\right)^2\right) \text { and } \delta_y \sim \mathcal{N}\left(0,\left(0.05 \cdot \mathrm{R}_{\max }\right)^2\right) .$$ The generated points are accepted if they lie within the boundary of a circular region with radius [TeX:] $$\mathrm{R}_{\max }.$$ This is checked by computing the squared distance from the cluster center to the point, [TeX:] $$\Delta=\left(x_i+\delta_x\right)^2+\left(y_i+\delta_y\right)^2,$$ and comparing it with [TeX:] $$\mathrm{R}_{\max }^2$$ (22). Points that fall outside of this region are discarded, and the generation process is repeated. This iterative process continues until the total number of generated UEs reaches the predefined target value, N. This ensures a clustered distribution where the density of nodes around each center reflects the specified clustering factor.

where [TeX:] $$\Delta=\left(x_i+\delta_x\right)^2+\left(y_i+\delta_y\right)^2 .$$

This approach maintains the spatial integrity by ensuring that all points fall within the predefined circular boundary. As the clustering factor increases, so does the concentration of UEs around each cluster center, enhancing the model’s ability to represent various degrees of spatial clustering.

C. Simulation Results for RPP UE Distribution Model

For the RPP UE distribution model, to assess our proposed scheme, simulations were performed with varying number of UEs. The range was selected to represent sparse to dense networks, from 10 to 90 UEs, incrementing by 20 UEs. Our scheme’s efficiency within the RPP UE distribution model is shown through an analysis utilizing center, centroid, and circumcenter reference node selection methods.

Fig. 4(a) illustrates the relationship between the average transmission power [TeX:] $$\mathrm{P}_{a v g}$$ and number of UEs. At a lower UE count, such as N = 10, [TeX:] $$\mathrm{P}_{a v g}$$ does not follow a discernible trend, suggesting an initial phase of adjustment. However, a stabilization in [TeX:] $$\mathrm{P}_{a v g}$$ is observed across all methods regardless of PDOP constraints with an increase in the number of UEs. The difference in power levels for PDOP thresholds of 6 and 20 becomes negligible when the number of UEs generated surpasses 30. This indicates that the UAV-BS can sustain positioning accuracy beyond a certain UE density without escalating power consumption.

Fig. 4(b) shows a gradual and consistent escalation in maximum transmission power [TeX:] $$\mathrm{P}_{\max}$$ with the increase in number of UEs. A near-constant [TeX:] $$\mathrm{P}_{\max}$$ is maintained across all three methods as the number of UEs generated increases, implying that enforcing stricter PDOP constraints does not necessarily result in higher maximum power requirements.

Fig. 5 shows a decrease in RMSE with an increasing number of UEs, enhancing the UAV-BS positioning accuracy, possibly due to selection from more various data points leading to a better estimation. This reflects a slight trade-off between power consumption and localization accuracy within the RPP UE distribution framework. It suggests that our system maintains a steady communication performance across different accuracy demands with localization accuracy improvements. This robustness ensures that communicational effectiveness is preserved with enhanced localization accuracy, even when high localization accuracy is sought.

D. Simulation Results for CPP UE Distribution Model

For CPP model, CF among UEs was varied to analyze performance under different degrees of UE clustering intensity. The CF values were selected to represent a spectrum of clustering scenarios, from low (CF = 1, representing PPP) to high (CF = 3, indicating high clustering). The performance of our scheme within the CPP UE distribution model is analyzed for the three reference node selection methods: Center, centroid, and circumcenter method.

As shown in Figs. 6(a) and 6(b), for all three methods, center, centroid, and circumcenter, both [TeX:] $$\mathrm{P}_{avg} \text{ and } \mathrm{P}_{\max}$$ required by the UAV-BS exhibit a decreasing trend as CF intensifies. This proves the efficiency of our scheme in terms of communication. In addition, a distinction in performance based onPDOPthreshold [TeX:] $$\tau_{\max }$$, is observed; with lower [TeX:] $$\mathrm{P}_{avg} \text{ and } \mathrm{P}_{\max}$$ needed at the stricter [TeX:] $$\tau_{\max }$$ of 6. This underlines our scheme’s efficiency in simultaneously optimizing communication power and localization performance in CPP UE distribution models.

In terms of specific methodologies, the centroid approach demands the least [TeX:] $$\mathrm{P}_{avg}$$ as CF rises, whereas the circumcenter method proves to be more power-efficient in terms of [TeX:] $$\mathrm{P}_{\max}$$ with an increase in CF. The rationale behind this can be illustrated by Fig. 7, which compares the average distances to UEs and to the origin point (0, 0) from each reference point, [TeX:] $$\left(G_x, G_y\right) \text{ and } \left(C_x, C_y\right)$$ of each method. With a higher CF, the centroid point [TeX:] $$\left(G_x, G_y\right)$$ is formed closer to the UEs compared to from [TeX:] $$\left(C_x, C_y\right)$$, suggesting its advantage for unicast communications that utilize average power. Conversely, the circumcenter point [TeX:] $$\left(C_x, C_y\right)$$, being closer to the origin compared to [TeX:] $$\left(G_x, G_y\right)$$, is better suited for broadcast communications.

Furthermore, Fig. 8 evaluates RMSE in relation to CF. While RMSE escalates with an increase in CF, indicating a potential trade-off with power, the magnitude of change is not substantial across CF values, suggesting this trade-off is minimal. Further, the RMSE for the center method surges more significantly than for the centroid and circumcenter methods as CF escalates, thereby affirming the superior performance of the centroid and circumcenter methods over the center method in terms of localization accuracy.

In Fig. 9, we observe the proportion of valid cases in relation to PDOP constraints for clustered UE distributions, analyzed across CF values of 1, 2, and 3. Thus, the evaluation shows the number of simulations that meet the predetermined PDOP constraint out of 20,000 simulation runs for each environment and method. This figure distinctly illustrates that with the elevation of [TeX:] $$\tau_{\max }$$ values, there is a corresponding increase in the percentage of valid cases. This trend indicates that as the demands for UAV-BS localization accuracy become less stringent, the ability of our system to deliver valid results improves, eventually approaching a 100% convergence rate. This pattern holds true across all reference node selection methods and clustering factors, underscoring the adaptability of the model to varying precision requirements in real-world scenarios.

V. Conclusions and Future Work

We formulated a joint placement and communication model in aerial GPS-denied environments, where UEs on ground positioned at accurate locations are utilized as reference anchor nodes to position the UAV-BS to a near-optimal location. The communication efficiency was evaluated in terms of transmission power and simultaneously, we aimed to minimize the localization accuracy of the estimated UAV-BS position by employing different methods for selecting reference nodes. Simulation was conducted under RPP and CPP UE distributed environments for our three proposed methods (center, centroid, and circumcenter methods). In RPP UE distribution model, the UAV-BS maintains its localization accuracy without a significant increase in transmission power with an increase in the number of UEs, even under stringent PDOP constraints. Further investigation into the CPP model revealed that higher UE clustering leads to reduced power requirements even under strict PDOP constraints. For future works, we plan focus on how to increases validity for strict PDOP thresholds. Further, we plan to develop our idea to non-static UE environments, aiming to solve problems with reinforcement learning for future works.