# Energy-Efficient UAV Communications With Directional Antennas: Tilting Effect Modeling and Trajectory Optimization

Xinhong Dai, Bin Duo, Xiaojun Yuan, Marco Di Renzo

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Xinhong Dai, Bin Duo, Xiaojun Yuan, Marco Di Renzo. Energy-Efficient UAV Communications With Directional Antennas: Tilting Effect Modeling and Trajectory Optimization. IEEE Transactions on Vehicular Technology, 2025, 74 (7), pp.11194-11206. 10.1109/TVT.2025.3545454. hal-05378141

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# Energy-Efficient UAV Communications With Directional Antennas: Tilting Effect Modeling and Trajectory Optimization

Xinhong Dai<sup>1, ∗</sup>, Bin Duo<sup>1</sup>, Member, IEEE, Xiaojun Yuan<sup>1, †</sup>, Senior Member, IEEE, and Marco Di Renzo<sup>1, ‡</sup>, Fellow, IEEE

**Abstract**—Directional antennas have been widely used in unmanned aerial vehicle (UAV) communications to improve the transmission efficiency. For a UAV mounted with directional antennas, its tilting effect has a crucial impact on the communication performance, which is however ignored in the related studies. In this paper, we first characterize the relationship between the tilting effect of the UAV and its flight status based on force analysis. Then, we investigate a UAV-enabled downlink communication system, where the UAV, equipped with a directional antenna, provides communication services to ground users (GUs). We formulate an energy efficiency (EE) maximization problem that optimizes the UAV trajectory under the consideration of the tilting effect. The formulated problem is challenging to solve due to the complicated expressions of the directional antenna radiation pattern and the adopted propulsion energy consumption model of the UAV. To tackle this, by utilizing the successive convex approximation technique, we propose an approximate antenna radiation pattern that serves as a convex lower bound of the original antenna gain,

The work of Marco Di Renzo was supported in part by the European Union through the Horizon Europe project COVER under Grant Agreement 101086228, the Horizon Europe project UNITE under Grant Agreement 101129618, the Horizon Europe project INSTINCT under Grant Agreement 101139161, and the Horizon Europe project TWIN6G under Grant Agreement 101182794, in part by the Agence Nationale de la Recherche (ANR) through the France 2030 project ANR-PEPR Networks of the Future under Grant Agreement NF-Founds 22-PEFT-0010, and in part by the CHIST-ERA project PASSIONATE under Grant Agreement CHIST-ERA-22-WAI-04 and Grant Agreement ANR-23-CHR4-0003-01. This work was supported in part by Sichuan Science and Technology Program, China, under Grant 2024ZYD0036. The review of this article was coordinated by Dr. Tomaso De Cola. (Corresponding author: Xiaojun Yuan.)

Xinhong Dai was with the National Key Laboratory of Wireless Communications, University of Electronic Science and Technology of China, Chengdu 611731, China. He is now with the Key Laboratory for Information Science of Electromagnetic Waves, Department of Communication Science and Engineering, Fudan University, Shanghai 200433, China (e-mail: xhdai24@m.fudan.edu.cn).

Bin Duo is with the College of Computer Science and Cyber Security, Chengdu University of Technology, Chengdu 610059, China, and also with the National Key Laboratory of Wireless Communications, University of Electronic Science and Technology of China, Chengdu 611731, China (e-mail: duobin@cdut.edu.cn).

Xiaojun Yuan is with the National Key Laboratory of Wireless Communications, University of Electronic Science and Technology of China, Chengdu 611731, China (e-mail: xjyuan@uestc.edu.cn).

Marco Di Renzo is with the Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des Signaux et Systèmes, 91192 Gif-sur-Yvette, France, and also with the King's College London, Centre for Telecommunications Research – Department of Engineering, WC2R 2LS London, U.K. (e-mail: marco.di.renzo@kcl.ac.uk).

and develop an efficient algorithm to obtain a suboptimal solution. Simulation results show that the negative impact of the tilting effect on the EE can be efficiently mitigated and the UAV flight can be made more steady under the tilting effect. In addition, the proposed algorithm allows the UAV to achieve a substantially higher EE compared to the benchmark schemes.

**Index Terms**—Rotary-wing UAV, directional antenna, tilting effect, UAV communications, energy efficiency, trajectory optimization.

## I. INTRODUCTION

T HANKS to several appealing features, such as high agility, swift deployment, and low cost, unmanned aerial vehicles (UAVs) are widely regarded as a promising technology to meet diversified demands in various scenarios [1], [2]. In particular, UAVs can provide line-of-sight (LoS) air-to-ground communications with high probability, which yields a considerable transmission gain in UAV-enabled communications [3]. Thus, UAV-enabled communications have found many applications in wireless networks, e.g., mobile edge computing, secure communications and wireless sensing [4], [5], [6], [7].

To improve the power efficiency of signal transmission and reduce the co-channel interference in neighboring areas [8], directional antennas can be utilized in UAV-aided communications to achieve better communication performance [9]. Different from omni-antennas applied in traditional UAV-enabled communication systems [10], [11], directional antennas can concentrate the radiation power on a relatively small ground area. Based on these features, recent works on UAV communications considered directional antennas to further enhance system performance. The authors of [12] considered a multiuser communication system enabled by a UAV equipped with a directional antenna of adjustable beamwidth, and improved the performance by optimizing the UAV altitude and antenna beamwidth. The authors of [13] considered an angle-dependent radiation pattern for a directional antenna, and optimized the three-dimensional deployment of UAVs to minimize the total average transmit power of the system. In [14], the authors proposed a tractable directional antenna pattern and analyzed the system performance of UAV-assisted networks with a conic directional antenna model, by deriving the network coverage probability via stochastic geometry. In [15], the authors investigated a UAV-enabled wireless power transfer network with a directional array antenna, and maximized the
minimum harvested energy by optimizing the UAV trajectory and the orientation of the directional antenna.

The aforementioned works treat UAVs as aerial stations equipped with directional antennas, and show that typical communication performance metrics, e.g., in terms of the coverage probability and the transmission gain, can be significantly improved. These works mostly focus on optimizing the UAV flight status and the antenna characteristics, e.g., the UAV altitude and trajectory, and the beamwidth of the directional antenna. However, they neglect the impact of the UAV flight status on the directivity by assuming that the UAV flies in an absolutely steady way, which is impractical for UAV communications. For rotary-wing UAVs, their horizontal movement always relies on the horizontal component of the lift, and the tilt of the UAV fuselage can produce this component. This phenomenon is defined as the tilting effect of the UAV flight, which is an important concept in UAV communication systems and also known as the pitch and roll angle variation of the UAV [16], the UAV fluctuation [17] and the rotation effect of the UAV [18]. Since the tilting effect can lead to a deviation of the directional beam and cause the losses in terms of directivity, the impact of the tilting effect cannot be ignored in UAV communications in general, especially when directional antennas are utilized. In addition, the presence of a strong tilting effect usually results in frequent changes in UAV's speed or direction, which causes additional propulsion energy consumption [19]. Therefore, as a key factor affecting the directivity and propulsion power of UAVs equipped with directional antennas, the tilting effect plays a crucial role in the energy-efficient design of UAV-enabled communication systems with directional antennas.

Motivated by these considerations, we analyze a UAV-enabled downlink communication system where the UAV is equipped with a uniform planer array (UPA). We propose an efficient iterative algorithm to maximize the energy efficiency (EE) of the UAV. The main contributions of this paper are summarized as follows:

* We first characterize the relationship between the tilting effect of the UAV and the UAV flight status based on the force analysis. We show that the tilting effect is related to the velocity, acceleration, weight, and other aerodynamic parameters of the UAV, as well as its propulsion energy consumption.

* To maximize the EE of the UAV, we formulate a UAV trajectory optimization problem by considering that the UAV is equipped with a directional antenna and is affected by the tilting effect. The formulated problem is difficult to solve because the expressions of the considered UAV energy consumption model and the antenna radiation pattern are non-convex and complicated. To tackle these difficulties, by using the successive convex approximation (SCA) technique, we propose an approximated radiation pattern that serves as a convex lower bound of the original antenna gain, and develop an efficient iterative algorithm to obtain a suboptimal solution to the formulated problem.

* We show that the optimized trajectory based on the proposed algorithm is smoother and more energy-efficient than the trajectories based on other schemes. As a result,

<figure><img src="image_2.png" /><figcaption>Fig. 1. Force analysis for a rotary-wing UAV.</figcaption></figure>

the UAV flight is more steady when the tilting effect is considered at the optimization stage and the negative impact of the tilting effect on the EE can be efficiently mitigated. Numerical results verify that the proposed algorithm allows the UAV to achieve a substantially higher EE compared to benchmark schemes.

The rest of this paper has the following structure: In Section II, we describe the system model, including the tilting effect and the communication model with directional antennas. In Section III, we formulate an optimization problem to maximize the EE of the UAV, and approximate the radiation pattern of the directional antenna to tackle the optimization problem more efficiently. In Section IV, we develop an efficient iterative algorithm to obtain a suboptimal solution to the formulated problem. We discuss the simulation results in Section V and conclude the paper in Section VI.

## II. SYSTEM MODEL

We consider a UAV-enabled downlink communication system, where a rotary-wing UAV equipped with a directional antenna is dispatched to fly at a fixed height $H$, while providing communication services to $K$ ground users (GUs) indexed by the set $K = \{1, \dots, K\}$. The location of each GU $k$ is denoted by $g_k = (x_k, y_k)^T \in \mathbb{R}^{2 \times 1}$. In the following, we characterize the relationship between the tilting effect of the UAV and its flight status, and introduce a discretized propulsion energy consumption model of the UAV. Then, we describe the communication model considering a directional array antenna.

### A. Tilting Effect Modeling

As mentioned above, the horizontal flight of a rotary-wing UAV is commonly impacted by the tilting effect. To characterize the relationship between the tilting effect and the UAV flight status, e.g., the velocity and acceleration of the UAV, we utilize the horizontal force analysis assuming that a rotary-wing UAV flies at a fixed height, as shown in Fig. 1.

From Fig. 1, we denote the thrust of the UAV by $t$, which is perpendicular to the fuselage plane. Also, $t_h$ and $t_v$ represent the horizontal and vertical components of $t$, respectively. From [20],
<table><caption>TABLE I
SIMULATION PARAMETERS</caption><thead><tr><td>Parameter</td><td>Description</td><td>Value</td></tr></thead><tbody><tr><td>T, Δ</td><td>Flight duration, time slot</td><td>150 s, 1 s</td></tr><tr><td>q<sub>I</sub>, q<sub>F</sub></td><td>Initial and final position</td><td>(0, 0)<sup>T</sup>m, (1000, 0)<sup>T</sup>m</td></tr><tr><td>V<sub>max</sub>, a<sub>max</sub>, H</td><td>Maximum flight speed, acceleration, flight height</td><td>40 m/s, 15 m/s<sup>2</sup>, 100 m</td></tr><tr><td>K, g<sub>k</sub></td><td>Number of the GUs and corresponding locations</td><td>6, (200, 200)<sup>T</sup> m, (220, 220)<sup>T</sup> m, (230, 190)<sup>T</sup> m, (600, -40)<sup>T</sup> m, (650, -100)<sup>T</sup> m, (710, -30)<sup>T</sup> m</td></tr><tr><td>ρ<sub>0</sub>, σ<sup>2</sup>, B, P<sub>s</sub></td><td>Communication parameters</td><td>-60 dB, -90 dBm, 1 MHz, 0.1 W</td></tr><tr><td>m, ||g||</td><td>Weight of the UAV and gravitational acceleration</td><td>2 kg, 9.8 m/s<sup>2</sup></td></tr><tr><td>ρ, A, v<sub>0</sub>, d<sub>0</sub>, s, S<sub>FP</sub>, U<sub>tip</sub>, P<sub>0</sub>, P<sub>1</sub></td><td>UAV aerodynamics parameters</td><td>1.225 kg/m<sup>3</sup>, 0.79 m<sup>2</sup>, 7.2, 0.3, 0.1, 0.012 m<sup>2</sup>, 200, 580.65 W, 70.72 W</td></tr><tr><td>ε</td><td>Convergence threshold</td><td>10<sup>-3</sup></td></tr></tbody></table>

<figure><img src="image_3.png" /><figcaption>Fig. 2. Illustration of $\alpha_t$ with different values of velocity and acceleration.</figcaption></figure>

the air drag **d** is defined as

$$ \mathbf{d} = -\frac{1}{2} \rho S_{\text{FP}} \|\mathbf{v}\| \mathbf{v}, \quad (1) $$

where $\rho$ and $S_{\text{FP}}$ denote the air density and the fuselage equivalent flat plate area, respectively. Then, $\mathbf{w} = mg$ denotes the weight of the UAV, where $m$ and $g$ are the mass of the UAV and the gravitational acceleration, respectively.

To model the tilting effect, we first denote the tilt angle by $\alpha_t$, as shown in Fig. 1, and we have $\|\mathbf{t}_h\| = \|\mathbf{t}\| \sin \alpha_t$ and $\|\mathbf{t}_v\| = \|\mathbf{t}\| \cos \alpha_t$. Then, from Newton's second law, we obtain $\mathbf{f} = ma = \mathbf{t}_h + \mathbf{d}$, where $\mathbf{f}$ is the resultant horizontal force, and $\mathbf{a}$ denotes the UAV acceleration of the UAV. Therefore, we have

$$ \tan \alpha_t = \frac{\|\mathbf{t}\| \sin \alpha_t}{\|\mathbf{t}\| \cos \alpha_t} = \frac{\|ma + \frac{1}{2}\rho S_{\text{FP}}\|\mathbf{v}\|\|\mathbf{v}\|}{\|mg\|}. \quad (2) $$

From (2), we see that the tilt angle $\alpha_t$ can be determined by the velocity, acceleration, and other constant parameters of the UAV. To show the relationship between the tilting effect and the

flight status, we briefly analyze the change of $\alpha_t$ with varying $\mathbf{v}$ and $\mathbf{a}$ as shown in Fig. 2, where the simulation parameters are set as detailed in Table I in Section VI. For simplicity, we only analyze the case study when the directions of the UAV velocity and acceleration are parallel, which are denoted by the scalar symbols $v$ and $a$ in Fig. 2, respectively. The negative value of $a$ means that the UAV is under the decelerating status, i.e., the direction of acceleration is opposite to the velocity. From Fig. 2, we see that the tilting effect always exists ($\alpha_t \neq 0$) when the UAV flies horizontally. In addition, the tilting effect becomes more significant (i.e., $\alpha_t$ becomes larger) when the UAV files faster or flies with a dramatic change in speed in both the accelerating and decelerating status. It is worth noting that a flight status with the tilting effect (i.e., for large values of $\mathbf{v}$ and $\mathbf{a}$) causes additional propulsion energy consumption of the UAV. Thus, such developing a relationship between the tilting effect and the UAV flight status can be utilized to design energy-efficient UAV trajectories.

## B. Discretized Propulsion Energy Consumption Model

We assume that the flight duration of the UAV is $T_0$, and discretize $T_0$ into $N$ equal time slots, indexed by $\mathcal{N} = \{1, \dots, N\}$. Let $\Delta = T_0/N$ denote each time slot, which is set sufficiently small. Denote the horizontal UAV trajectory by $\mathbf{q} = (x, y)^{\text{T}}$, which can be approximated by the $N$-length position sequence $\{\mathbf{q}[n] = (x[n], y[n])^{\text{T}}\}_{n=1}^N$. Then, the UAV velocity and acceleration can be respectively defined as

$$ \mathbf{v}[n] \triangleq \frac{\mathbf{q}[n+1] - \mathbf{q}[n]}{\Delta}, \quad (3) $$

$$ \mathbf{a}[n] \triangleq \frac{\mathbf{q}[n+2] + \mathbf{q}[n] - 2\mathbf{q}[n+1]}{\Delta^2}. \quad (4) $$

Within each slot $n$, we assume that the UAV location remains unchanged, and the distance between the UAV and the $k$-th GU is given by $d_k[n] = \sqrt{H^2 + \|\mathbf{q}[n] - \mathbf{g}_k\|^2}$, where $\|\cdot\|$ denotes the Euclidean norm. In addition, $\mathbf{q}_I$ and $\mathbf{q}_F$ denote the initial and final locations of the UAV, respectively. We assume that the UAV
is limited by a maximum flight speed $V_{max}$, and a maximum flight acceleration $a_{max}$, i.e., the UAV trajectory satisfies the following constraints:

$$ \mathbf{q}[1] = \mathbf{q}_I, \quad \mathbf{q}[N] = \mathbf{q}_F, \qquad (5) $$

$$ \|\mathbf{q}[n+1] - \mathbf{q}[n]\| \le V_{\max}\Delta, \quad \forall n, \qquad (6) $$

$$ \|\mathbf{q}[n+2] + \mathbf{q}[n] - 2\mathbf{q}[n+1]\| \le a_{\max}\Delta^2 \quad \forall n. \qquad (7) $$

To characterize the propulsion energy consumption of the UAV, we adopt the generalized propulsion energy consumption model proposed in [19], which is formulated as a function of the UAV velocity and acceleration:

$$ P[n] = P_{i}\kappa[n] \sqrt{\left( (\kappa[n])^2 + \frac{\|\mathbf{v}[n]\|^4}{4(v_0)^4} \right)^{\frac{1}{2}} - \frac{\|\mathbf{v}[n]\|^2}{2(v_0)^2}} \\ + P_0 \left( 1 + \frac{3\|\mathbf{v}[n]\|^2}{(U_{\text{tip}})^2} \right) + \frac{1}{2} d_0 \rho s A \|\mathbf{v}[n]\|^3, \qquad (8) $$

where $\kappa[n]$ denotes the thrust-to-weight ratio of the UAV, and

$$ \kappa[n] = \sqrt{1 + \frac{4m^2\|\mathbf{a}[n]\|^2 + \rho^2(S_{\text{FP}})^2\|\mathbf{v}[n]\|^4 + \phi[n]}{4\|\mathbf{m}\mathbf{g}\|^2}}, \qquad (9) $$

with $\phi[n] = 4m\rho S_{\text{FP}}\|\mathbf{v}[n]\|\mathbf{a}[n] \cdot \mathbf{v}[n]$. In (8) and (9), $P_0$ and $P_i$ are two constant coefficients defined in [19], $U_{\text{tip}}$ denotes the tip velocity of the rotor blade, $v_0$ is the mean rotor induced velocity when hovering, $d_0$ and $s$ are the fuselage drag ratio and rotor solidity, respectively, and $A$ denotes the rotor disc area. It is worth noting that, from (8), the UAV flight is constrained by a certain maximum propulsion power [21]. That is, the maximum propulsion power constraint is implicitly imposed in the considered propulsion model. Thus, we will consider the constraints (6) and (7) in the considered problem formulation, without explicitly including the maximum propulsion power constraint.

### C. Communication Model With Directional Antennas

As commonly adopted in [15], [22], [23], we consider that a directional antenna is deployed at the bottom of the UAV, which is modeled as a UPA with $M \times M$ array elements and is assumed to be parallel to the fuselage rotation plane. The antenna spacing is designed to be half of the signal wavelength. In this case, the channel gain can be significantly enlarged towards the orientation direction, i.e., the direction along the central axis of the directional beam. Then, we denote by $\alpha_k[n]$ and $\beta_k[n]$ the deviation angle between the k-th UAV-GU link and the central axis of the directional beam and the azimuth angle in the roll direction with respect to the fuselage rotation plane, as shown in Fig. 3, respectively. Hence, from [22], the actual directional antenna gain for the k-th GU in slot n can be expressed as

$$ G_k^{\text{gain}}[n] = G_0 \times G_k[n] \\ = \pi M^2 \times \left( \frac{\sin(M \frac{\pi}{2} \sin(\alpha_k[n]) \cos(\beta_k[n]))}{M \sin(\frac{\pi}{2} \sin(\alpha_k[n]) \cos(\beta_k[n]))} \right)^2 $$

<figure><img src="image_4.png" /><figcaption>Fig. 3. Illustration of the tilting effect.</figcaption></figure>

$$ \times \left( \frac{\sin(M \frac{\pi}{2} \sin(\alpha_k[n]) \sin(\beta_k[n]))}{M \sin(\frac{\pi}{2} \sin(\alpha_k[n]) \sin(\beta_k[n]))} \right)^2, \qquad (10) $$

where $G_0 = \pi M^2$ is the maximum value of the antenna gain, which is obtained when $\alpha_k[n] = \beta_k[n] = 0$, and $G_k[n]$ is the antenna radiation pattern. It is worth noting that a larger number of the antenna elements M leads to a stronger directivity and a greater antenna gain. From [22], [23], given $\alpha_k[n]$, the main lobe of the radiation pattern is almost the same for different values of $\beta_k[n]$, while the sidelobes are slightly changed. Thus, to simplify the analysis, we assume $\beta_k[n] = 0$ and $G_k[n]$ in (10) can be approximated as

$$ G_k[n] = \left( \frac{\sin(M \frac{\pi}{2} \sin(\alpha_k[n]))}{M \sin(\frac{\pi}{2} \sin(\alpha_k[n]))} \right)^2 . \qquad (11) $$

From (11), we can compute the antenna gain for any given $\alpha_k[n]$. Since $\alpha_k[n]$ is difficult to be obtained directly, we compute $\alpha_k[n]$ based on the geometrical relationship in Fig. 3. Specifically, as shown in Fig. 3, we denote $\alpha_{d,k}[n]$ as the elevation angle of the k-th UAV-GU link in slot n and we have

$$ \alpha_{d,k}[n] = \arctan \left( \frac{\|\mathbf{q}[n] - \mathbf{g}_k\|}{H} \right). \qquad (12) $$

Similarly, $\alpha_t[n]$ in (2) can be written as

$$ \alpha_t[n] = \arctan \left( \frac{\|ma[n] + \frac{1}{2}\rho S_{\text{FP}}\|\mathbf{v}[n]\|\mathbf{v}[n]\|}{\|\mathbf{m}\mathbf{g}\|} \right). \qquad (13) $$

Then, we denote by $g_b[n]$ the projection location on the ground of the central beam in slot n. To formulate $g_b[n]$, we express the velocity and acceleration as a vector composed of components along the x and y axis, i.e., $a[n] = (a_x[n], a_y[n])^T$ and $v[n] = (v_x[n], v_y[n])^T$. Similarly, we decompose $\alpha_t[n]$ into two projected components in the xoz and yoz planes, which are respectively denoted by $\alpha_{t,x}[n]$ and $\alpha_{t,y}[n]$ as follows:

$$ \alpha_{t,x}[n] = \arctan \left( \frac{ma_x[n] + \frac{1}{2}\rho S_{\text{FP}}\|\mathbf{v}[n]\|v_x[n]}{\|\mathbf{m}\mathbf{g}\|} \right), \qquad (14) $$
$$ \alpha_{t,y}[n] = \arctan \left( \frac{ma_y[n] + \frac{1}{2} \rho S_{\text{FP}} \|\mathbf{v}[n]\| v_y[n]}{\|m\mathbf{g}\|} \right). \quad (15) $$

Then, we have

$$ \mathbf{g}_b[n] = (x[n] - H \tan(\alpha_{t,x}[n]), y[n] - H \tan(\alpha_{t,y}[n]))^\mathrm{T}. \quad (16) $$

Based on the law of cosines, $\alpha_k[n]$ can be obtained by the equation

$$ \alpha_k[n] = \arccos \left( \sqrt{\frac{d_b^2[n] + d_k^2[n] - \|\mathbf{g}_k - \mathbf{g}_b[n]\|^2}{2d_b[n]d_k[n]}} \right), \quad (17) $$

and

$$ d_b[n] = \frac{H}{\cos(\alpha_t[n])} = \sqrt{\|\mathbf{q}[n] - \mathbf{g}_b[n]\|^2 + H^2}, \quad (18) $$

which represents the distance between the UAV and $\mathbf{g}_b[n]$. Hence, given $\mathbf{q}[n]$ and the flight status of the UAV, we can obtain the value of $\alpha_k[n]$ and the corresponding $G_k[n]$ based on (11)–(17).

Assume that all the GUs share the same bandwidth $B$, and the UAV transmits the signal with a fixed power $P_s$. Since $P_s$ is much smaller than $P[n]$ in (8), we ignore it in the analysis of the system energy consumption [6]. Then, we consider that the communication links between the UAV and the users are dominated by LoS links. Thus, the channel power gain at the $k$-th GU in slot $n$ is expressed as $\frac{\rho_0 \pi M^2 G_k[n]}{\sigma^2 d_k^2[n]}$, leading to an achievable rate for the $k$-th GU equal to

$$ R_k[n] = B \log_2 \left( 1 + \frac{P_s \rho_0 \pi M^2 G_k[n]}{\sigma^2 d_k^2[n]} \right), \quad (19) $$

where $\rho_0$ denotes the channel power gain at the reference distance and $\sigma^2$ is the power of the additive white Gaussian noise at the GUs.

### III. PROBLEM FORMULATION AND APPROXIMATION

In this section, we first formulate an optimization problem to maximize the EE of the UAV. Since the considered antenna radiation pattern is difficult to handle in the optimization problem, we propose an approximate antenna pattern to make the problem more tractable.

#### A. Problem Formulation

Since the directivity and the propulsion energy consumption are affected by the tilting effect, we aim to maximize the EE of the UAV-enabled communication system. We introduce an auxiliary variable $R_m$ that is upper bounded by the minimum achievable rate among all GUs<fnref n="1" /> over the flight duration [24], i.e., $R_m \le \min(\sum_{n=1}^N R_k[n])$, $\forall k$. Then, we maximize the ratio of $R_m$ to the UAV energy consumption by jointly optimizing $R_m$

and the UAV trajectory. From this problem definition, the EE of the UAV can be maximized, and at the same time, the minimum achievable rate is also improved. Thus, the optimization problem can be formulated as

$$ \max_{\mathbf{q}, R_m} \frac{R_m}{\sum_{n=1}^N P[n]} \quad (20a) $$

$$ \begin{array}{ll} \text{s.t.} & R_m \le \sum_{n=1}^{N} R_k[n], \quad \forall k, \\ & (5), (6), (7), (11)-(17), \quad (20b) \end{array} $$

Note that constraint (20b) must hold with equality to obtain the optimal solution to the formulated problem. That is, the optimal $R_m$ must be equal to the minimum achievable rate among the GUs, since otherwise $R_m$ can be increased to further increase the objective value. In this case, the objective function and (20b) ensure that the UAV serves all the GUs fairly. Clearly, (20b) is a non-convex constraint because the trigonometric computations in (11)–(17) and $P[n]$ are non-convex with respect to $\mathbf{q}[n]$. As a result, problem (20) is a non-convex fractional program, which is difficult to solve in general. Thus, we next introduce some approximations to tackle the problem efficiently.

#### B. Approximate Radiation Pattern

The antenna pattern given in (11) has sine functions in both the numerator and denominator, which makes it not analytically tractable when considering problem (20). To this end, it is highly beneficial to consider an approximation of the directional antenna pattern in (11), which provides both computation accuracy and tractability. Based on the form of (11), we know that the antenna pattern $G_k[n]$ serves as a multimodal function similar to the Sinc function defined as $f(x) = \frac{1}{x}$ [25], and the envelope of the Sinc function can be expressed as $f(x) = \frac{1}{x}$. Then, it is easy to verify that the envelope function of $G_k[n]$ can be approximated by a function as $f(x) = \frac{b}{x^a}$ similar to the Sinc function, where $a$ and $b$ are tuning parameters. In addition, since $G_k[n]$ equals 1 when $\alpha_k[n] = 0$ and the width of the main lobe is determined by $M$, we propose an approximate form of the $G_k[n]$ as

$$ G_k[n] \approx \frac{b}{(M\alpha_k[n])^a + b}. \quad (21) $$

Furthermore, we aim to establish a simpler numerical relationship between $G_k[n]$ and the trajectory $\mathbf{q}[n]$ (i.e., the UAV flight status). From (12) and (13), $\alpha_{d,k}[n]$ and $\alpha_t[n]$ are both computed by the tangent function of UAV trajectory $\mathbf{q}[n]$. Besides, the UAV velocity and acceleration are numerically equivalent to $\mathbf{q}[n]$ from (3) and (4), and we commonly analyze the accuracy of the approximation based on the half-power beamwidth of the pattern. Since $\alpha_k[n]$ is determined by these two angles from the geometric relations in Fig. 3, and the value of $\alpha_k[n]$ is relatively small and satisfies $\alpha_k[n] \approx \tan(\alpha_k[n])$. Thus, we consider reformulating the antenna pattern with the tangent function as

$$ G_k[n] = \frac{b}{(M \tan(\alpha_k[n]))^a + b}. \quad (22) $$

<footnote n="1">¹We note that $R_m$ is an optimization variable in the considered optimization problem, which can be defined as a constant threshold for different GU's communication demands in some application-driven scenarios, where the optimized UAV trajectory is highly related to $R_m$.</footnote>
<figure><img src="image_5.png" /><figcaption>Fig. 4. Comparison between the approximate pattern (22) and the original pattern (11) with varying M.</figcaption></figure>

Note that we can apply existing function fitting methods (e.g., the least square method) based on (22) to determine a and b and approximate $G_k[n]$ in (11). Fig. 4 shows the pattern comparison between the original $G_k[n]$ in (11) and the approximated $G_k[n]$ in (22). Specifically, we see that by appropriately choosing the fitting parameters, the approximate patterns via (22) fit the corresponding original pattern, especially within the half-power beamwidth. Thus, (22) approximatively characterizes the functional relationship between $G_k[n]$ and $\alpha_k[n]$ with a more simplified mathematical form, which is easier to tackle in the following approximations and optimizations.

### C. Problem Approximation

As detailed in the previous section, $G_k[n]$ is now a function of $\tan(\alpha_k[n])$. To facilitate the solution to problem (20), we introduce some additional approximations as follows. According to the geometrical relationship among $\alpha_k[n]$, $\alpha_t[n]$, and $\alpha_{d,k}[n]$ in Fig. 3, we introduce the following upper bound for $\alpha_k[n]$

$$ \alpha_k^{\text{ub}}[n] = \alpha_t[n] + \alpha_{d,k}[n] \ge \alpha_k[n]. \quad (23) $$

In general, $\alpha_t[n]$ is smaller than $\alpha_{d,k}[n]$ for the UAV flight when the UAV is away from the GUs, and both $\alpha_t[n]$ and $\alpha_{d,k}[n]$ are relatively small when the UAV is flying over the GUs. Moreover, from (12) and (13), the values of the tangent function of $\alpha_t[n]$ and $\alpha_{d,k}[n]$ can be directly obtained based on trajectory $\mathbf{q}[n]$. Thus, we can utilize the approximation $\tan(\alpha_t[n] + \alpha_{d,k}[n]) \approx \tan(\alpha_t[n]) + \tan(\alpha_{d,k}[n])$ to establish a more direct relationship between $G_k[n]$ in (22) and the trajectory $\mathbf{q}[n]$. Then, we introduce an auxiliary variable set $\Xi = \{\xi_k[n], \forall k, n\}$, in which $\xi_k[n]$ satisfies

$$ \begin{aligned} \xi_k[n] &= \frac{\|\mathbf{q}[n] - \mathbf{g}_k\|}{H} + \frac{\|m\mathbf{a}[n]\| + \frac{1}{2}\rho S_{\text{FP}}\|\mathbf{v}[n]\|^2}{\|m\mathbf{g}\|}, \\ &\ge \tan(\alpha_{d,k}[n]) + \tan(\alpha_t[n]) \approx \tan(\alpha_k^{\text{ub}}[n]), \\ &\ge \tan(\alpha_k[n]), \end{aligned} \quad (24) $$

and $\xi_k[n]$ is a convex function with respect to $\mathbf{q}[n]$ because of the convexity preservation of the norm operation. From (23) and (24), a lower bound of $G_k[n]$ in (22) can be written as $G_k^{lb,1}[n] = \frac{b}{(M\xi_k[n])^a+b}$, which leads to the following lower bound of $R_k[n]$:

$$ R_k^{\text{lb},1}[n] = B \log_2 \left( 1 + \frac{P_s \rho_0 \pi M^2 G_k^{\text{lb},1}[n]}{\sigma^2 d_k^2[n]} \right). \quad (25) $$

Therefore, problem (20) can be recast as

$$ \max_{\mathbf{q}, \Xi, R_m} \frac{R_m}{\sum_{n=1}^{N} P[n]} \quad (26a) $$

$$ \begin{array}{l} \text{s.t.} \quad R_m \le \sum_{n=1}^{N} R_k^{\text{lb},1}[n], \quad \forall k, \\ (5), (6), (7), (24). \end{array} \quad (26b) $$

The solution to the approximated problem (26) serves as a lower bound for the value of (20a). Moreover, problem (26) is still difficult to solve due to the non-convexity of $P[n]$ with respect to $\mathbf{q}[n]$ and the non-convex constraint (26b). To tackle this problem, we propose an efficient iterative algorithm based on the SCA technique.

## IV. PROPOSED ALGORITHM

In this section, to tackle the non-convexity of problem (26), we relax the objective function in (26) as the ratio of the minimum achievable rate $R_m$ of each GU to the upper bound of $P[n]$ based on slack variables and corresponding convex constraints. Then, we propose an iterative algorithm to obtain a suboptimal solution to problem (26) as well as problem (20).

### A. Relaxation on $P[n]$

We first deal with $P[n]$. Note that $P[n]$ in (8) is a non-convex function of the UAV velocity and acceleration, which is challenging to handle. Moreover, the last two terms of $P[n]$ in (8) are convex with respect to $\mathbf{v}[n]$ because of the convexity preservation of the norm operation. Thus, we need to focus on tackling the non-convexity of the first term in (8). We see that the value of $P[n]$ increases monotonically as $\kappa[n]$ increases. Thus, we first obtain a strict upper bound of $\kappa[n]$ (denoted as $\kappa^{\text{ub}}[n]$) based on Cauchy-Schwarz inequality (i.e., $\mathbf{a}[n] \cdot \mathbf{v}[n] \le \|\mathbf{a}[n]\|\|\mathbf{v}[n]\|$), and introduce a slack variable set $\mathcal{L} = \{\ell[n]\}_{n=1}^N$ satisfying

$$ \ell[n] \ge \kappa^{\text{ub}}[n] = \sqrt{1 + \frac{(2m\|\mathbf{a}[n]\| + \rho S_{\text{FP}}\|\mathbf{v}[n]\|^2)^2}{\|4m\mathbf{g}\|^2}} \ge \kappa[n]. \quad (27) $$

Then, we aim to obtain an upper bound of the first term of $P[n]$ in (8). Similarly, we introduce a slack variable set $\mathcal{W} = \{\omega[n]\}_{n=1}^N$ such that

$$ \omega[n] \ge \ell[n] \sqrt{\left( (\ell[n])^2 + \frac{\|\mathbf{v}[n]\|^4}{4(v_0)^4} \right)^{\frac{1}{2}} - \frac{\|\mathbf{v}[n]\|^2}{2(v_0)^2}}, \quad (28) $$
and thus we have a strict upper bound of $P[n]$ as

$$P^{\text{ub}}[n] = P_{\text{i}}\omega[n] + P_{0}\left(1 + \frac{3\|\mathbf{v}[n]\|^2}{(U_{\text{tip}})^2}\right) + \frac{1}{2}d_{0}\rho s A \|\mathbf{v}[n]\|^3. \quad (29)$$

To deal with the non-convexity of (27) and (28), we have the following steps:

1) We introduce a slack variable set $\mathcal{J} = \{j[n]\}_{n=1}^N$ such that $j[n] \ge (\ell^2[n] + \frac{\|\mathbf{v}[n]\|^4}{4(v_0)^4})^{\frac{1}{2}} - \frac{\|\mathbf{v}[n]\|^2}{2(v_0)^2}$. Based on some straightforward transformations, we have the following inequalities:

$$1 + \left( \frac{\|\mathbf{a}[n]\|}{\|\mathbf{g}\|} + \frac{\rho S_{\text{FP}} \|\mathbf{v}[n]\|^2}{\|2m\mathbf{g}\|} \right)^2 \le \ell^2[n], \quad (30)$$

$\ell^2[n] \le j^2[n] + j[n] \frac{\|\mathbf{v}[n]\|^2}{(v_0)^2}$, and $\omega^2[n] \ge \ell^2[n]j[n]$.

2) To make these inequalities tractable in the optimization, we utilize the fractional bivariate function $f(x, y) = \frac{x^2}{y}$ to establish the convex functions of the slack variables. This function is jointly convex for $x > 0$ and $y > 0$ [26]. Then, we have the following constraints with convex forms in both sides of inequalities:

$$\frac{\ell^2[n]}{j[n]} \le j[n] + \frac{\|\mathbf{v}[n]\|^2}{(v_0)^2}, \quad (31)$$

$$\ell^2[n] \le \frac{\omega^2[n]}{j[n]}. \quad (32)$$

3) For constraints (30)–(32), we apply the first-order Taylor expansion at given local points denoted by $\mathbf{q}^t = \{\mathbf{q}^t[n]\}_{n=1}^N$, $\mathcal{W}^t = \{\omega^t[n]\}_{n=1}^N$, $\mathcal{L}^t = \{\ell^t[n]\}_{n=1}^N$, and $\mathcal{J}^t = \{j^t[n]\}_{n=1}^N$ in the t-th iteration, respectively, which results in the following convex constraints:

$$1 + f_{\mathbf{a},\mathbf{v}}[n] \le (\ell^t[n])^2 + 2\ell^t[n](\ell[n] - \ell^t[n]), \quad (33)$$

$$\frac{\ell^2[n]}{j[n]} \le j[n] - \frac{\|\mathbf{v}^t[n]\|^2}{v_0^2} + 2 \frac{(\mathbf{v}^t[n])^\text{T} \mathbf{v}[n]}{(v_0)^2}, \quad (34)$$

$$\ell^2[n] \le \frac{(\omega^t[n])^2}{j^t[n]} + \frac{2\omega^t[n](\omega[n] - \omega^t[n])}{j^t[n]} - \left(\frac{\omega^t[n]}{j^t[n]}\right)^2 (j[n] - j^t[n]), \quad (35)$$

where $f_{\mathbf{a},\mathbf{v}}[n] = (\frac{\|\mathbf{a}[n]\|}{\|\mathbf{g}\|} + \frac{\rho S_{\text{FP}} \|\mathbf{v}[n]\|^2}{\|2m\mathbf{g}\|})^2$ is convex with respect to $\mathbf{q}[n]$ and $\mathbf{v}^t[n] = \frac{\mathbf{q}^t[n+1] - \mathbf{q}^t[n]}{\Delta}$.

## B. Reformulated Problem

We next tackle the non-convexity of (26b). We introduce two slack variable sets $\mathcal{D} = \{D_k[n], \forall k, n\}$ and $\mathcal{S} = \{S_k[n], \forall k, n\}$, which satisfy $D_k[n] \le G_k^{\text{lb},1}[n]$ and $\frac{d_k[n]}{D_k[n]} \le S_k[n]$, respectively. From (25), we have $B \log_2(1 + \frac{\gamma_0}{S_k[n]}) \le R_k^{\text{lb},1}[n]$, where $\gamma_0 = \frac{\pi M^2 \rho_0 P_s}{\sigma^2}$. Note that for $D_k[n] \le G_k^{\text{lb},1}[n] = \frac{b}{(M\xi_k[n])^{a+b}}$, the right-hand side of this inequality is non-convex with respect to the variable $\xi_k[n]$. However, it is observed that, there is a unique tangent point of the function $G_k^{\text{lb},1}[n]$ (denoted as $\xi_o[n]$)

<figure><img src="image_6.png" /><figcaption>Fig. 5. Illustration of the connected function ($M = 2, a = 2.3, b = 1.3$).</figcaption></figure>

such that the corresponding tangent line past the point (0,1). When $\xi_k[n] \ge \xi_o[n]$, this function is convex as shown in Fig. 5.

**Lemma 1:** Given a continuous and closed interval $x \in [a, b]$, and a point $x = c \in (a, b)$ in this interval. Two functions $f(x)$ and $g(x)$, where $f(x)$ is convex for $x \in [a, c)$, and $g(x)$ is convex for $x \in [c, b]$, are connected together at the point $x = c$ to construct a new function $h(x)$ as

$$h(x) = \begin{cases} f(x), & \text{for } x \in [a, c), \\ g(x), & \text{for } x \in [c, b]. \end{cases}$$

If $h(x)$ is continuous and differentiable at $x = c$, then the connected function $h(x)$ is also convex in $[a, b]$.

**Proof:** Since $h(x)$ is continuous and differentiable at $x = c$, the left derivative of $h(c)$ is equal to its right derivative. Then, from the convexity of $f(x)$ and $g(x)$, it is clear that the derivative of $h(x)$ is monotonically non-decreasing over the interval $[a, b]$ [26]. Therefore, $h(x)$ is also convex in $[a, b]$.

Then, denote by $C(\xi_o[n])$ the unique tangent point. With **Lemma 1**, we can verify that the function connected by $C(\xi_o[n])$ with $\xi_k[n] \le \xi_o[n]$ and the part of $G_k^{\text{lb},1}[n]$ with $\xi_k[n] \ge \xi_o[n]$, is still convex. Furthermore, the function value of this connected function serves as a strict lower bound for the value of $G_k^{\text{lb},1}[n]$. When $\xi_k[n] \le \xi_o[n]$, the connected function is linear, and its Taylor expansion is the function itself. Thus, we can apply the first-order Taylor expansion to the connected function at given local points denoted by $\Xi^t = \{\xi_k^t[n], \forall k, n\}$ in t-th iteration. Then, we have

$$G_k^{\text{lb},1}[n] \ge G_k^{\text{lb},2}[n] = \begin{cases} C(\xi_o[n]), & \xi_k[n] \le \xi_o[n], \\ f_{\xi_k}[n], & \xi_k[n] \ge \xi_o[n], \end{cases} \quad (36)$$

where

$$f_{\xi_k}[n] = - \frac{abM^a(\xi_k^t[n])^{a-1}}{((M\xi_k^t[n])^a + b)^2} (\xi_k[n] - \xi_k^t[n]) + \frac{b}{(M\xi_k^t[n])^a + b}. \quad (37)$$
**Algorithm 1: Proposed Algorithm Based on SCA.**

**Require** $g_k, N$, number of array elements $M^2$, convergence threshold $\epsilon$.

**1: Initialization:**

Determine $a$ and $b$ in (22) based on the given $M$;

Let $q^0, R_m^0$ and other initial values of slack variables be feasible for problem (39);

Denote by $t$ the iteration index of SCA and set $t = 0$;

Denote by EE$^0$ the initially computed objective value of problem (39).

**2: repeat**

3: In $t$-th iteration, solve problem (39) with given local points $q^{t-1}$ and other obtained values of slack variables in the last iteration, and denote by EE$^t$ the obtained objective value;

4: Update $t = t + 1$.

5: **until** The objective value EE$^t$ converges within convergence threshold $\epsilon > 0$.

**Ensure** The optimal trajectory $q$, the actual achievable rate, propulsion energy consumption and EE based on $q$ and original directional gain in (10).

Similarly, given the local points denoted by $S^t = \{S_k^t[n], \forall k, n\}$ in $t$-th iteration, we can express a lower bound of $R_k^{\text{lb},1}[n]$ via the first-order Taylor expansion:

$$ R_k^{\text{lb},2}[n] = B \left( \log_2 \left( 1 + \frac{\gamma_0}{S_k^t[n]} \right) - \frac{\gamma_0(S_k[n] - S_k^t[n])}{\ln 2 S_k^t[n] (S_k^t[n] + \gamma_0)} \right) \\ \le B \log_2 \left( 1 + \frac{\gamma_0}{S_k^t[n]} \right) \le R_k^{\text{lb},1}[n]. \quad (38) $$

Therefore, problem (26) can be recast as

$$ \max_{q, \Xi, W, L, J, D, S, R_m} \frac{R_m}{\sum_{n=1}^{N} P^{\text{ub}}[n]} \quad (39a) $$

$$ \text{s.t.} \quad R_m \le \sum_{n=1}^{N} R_k^{\text{lb},2}[n], \quad \forall k, \quad (39b) $$

$$ D_k[n] \le G_k^{\text{lb},2}[n], \quad \forall k, n, \quad (39c) $$

$$ \frac{d_k^2[n]}{D_k[n]} \le S_k[n], \quad \forall k, n, \quad (5), (6), (7), (24), (33), (34), (35). \quad (39d) $$

Problem (39) is a quasi-convex optimization problem since (39a) includes a linear numerator and a convex denominator, and all constraints are convex. As such, it can be solved optimally and efficiently via fractional programming techniques, e.g., the Dinkelbach algorithm [27].

### C. Overall Algorithm and Computational Complexity

The proposed algorithm is summarized in Algorithm 1. Clearly, problem (39) serves as a lower bound for the optimal value of problem (26) and also for the optimal value of problem (20). Then, in each $t$-th iteration, the obtained objective value

EE$^t$ is non-decreasing and is upper bounded by a finite value, which guarantees the convergence of the proposed algorithm based on the SCA technique [28]. Thus, by using the proposed algorithm, problem (39) can be solved optimally and the optimization process ends when the value of (39a) falls below a pre-specified precision $\epsilon > 0$. Since a larger number of time slots incurs higher computational complexity, in practice, we set an appropriate number of time slots to avoid excessively high complexity. Problem (39) is solved based on the standard interior-point method [29], the required computational complexity of the proposed algorithm is thus $O((3KN + 5N)^{3.5} \log(1/\epsilon))$.

## V. NUMERICAL RESULTS

In this section, numerical simulations are illustrated to analyze the performance of the proposed algorithm which maximizes the EE considering the tilting effect at the optimization stage. The proposed solution is denoted as MEE-TE.

### A. Simulation Settings

To better verify the effectiveness of the proposed scheme, we consider three benchmark schemes: 1) Maximizing the EE by ignoring the tilting effect (denoted as MEE-ITE), which corresponds to solving problem (39) based on the proposed algorithm with $\alpha_t[n] = 0$; 2) Maximizing the achievable rate by considering the tilting effect in the optimization (denoted as MAR-TE) 3) Maximizing the achievable rate by ignoring the tilting effect of the UAV (denoted as MAR-ITE). For completeness, we note that the optimization problems for the MAR-TE and MAR-ITE schemes can be formulated as

$$ \max_{q, \Xi, D, S, R_m} R_m $$

s.t. Similar constraints in (39).

In practice, the GUs are usually distributed in clusters of varying densities rather than uniformly [30]. Moreover, the effective communication coverage area of the UAV is determined by the directional antenna (e.g., the number of array elements) [13], [14]. Thus, to better investigate the system performance considering that the UAV is equipped with a directional antenna array, we assume $K = 6$ GUs which are divided into two clusters with different separation distances. The simulation parameters are summarized in Table I. Also, the aerodynamics parameters of the UAV are set as in [6], [19], [31]. Note that the approximate antenna radiation pattern in (22) is used in the proposed algorithm for simplicity. To ensure the validity of the simulations, the following results are computed with the original directional gain in (10). Moreover, all simulations are conducted by using CVX on a computer with a 3.9 GHz CPU and 16 GB RAM memory. The CVX solver is Mosek and the error tolerance for the variable-step solver is $1e^{-3}$.

### B. Analysis of Optimized Trajectories

Fig. 6 shows the optimized trajectories of the UAV based on the maximization of the achievable rate when the directional antenna has $2 \times 2$ array elements. Specifically, see the left half
<figure><img src="image_7.png" /><figcaption>Fig. 6. Trajectories based on the achievable rate maximization ($M = 2$).</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 7. Trajectories based on the EE maximization ($M = 2$).</figcaption></figure>

of Fig. 6, let us consider the MAR-ITE scheme. The UAV hovers over each GU for a while, and flies to the center of the cluster to hover for a period of time to achieve a higher achievable rate. However, considering the tilting effect in the optimization, the UAV based on the MAR-TE scheme chooses to hover over the center of the cluster instead of flying among the GUs. This is because the GUs in this cluster are distributed compactly and can be covered efficiently by the coverage area of the main lobe of the directional beam when $M = 2$. Moreover, the UAV hovering within a small area can avoid frequent velocity changes and attenuate the tilting effect compared to the MAR-ITE scheme. In the right half of Fig. 6, the separation distances among the GUs in the cluster are larger than those in another cluster, and the GUs cannot be covered simultaneously by the coverage area of the directional beam. In this case, the UAV has to hover over each GU instead of the center of the cluster for better directivity. It is noted that the UAV flies among the GUs at a higher speed based on the MAR-ITE scheme, but lowers its speed based on the MAR-TE scheme to mitigate the negative impact of the tilting effect. Thus, with the MAR-TE scheme, the directional beam of the antenna can be better directed toward the GUs so that the UAV can achieve a higher rate. Fig. 7 shows the optimized

<figure><img src="image_9.png" /><figcaption>Fig. 8. Trajectories based on the achievable rate maximization ($M = 4$).</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 9. Trajectories based on the EE maximization ($M = 4$).</figcaption></figure>

trajectories of the UAV based on the EE maximization, where the UAV flies in a smoother way (i.e., circling over the cluster) to avoid dramatic changes in velocity and to save more propulsion energy. Similar to Fig. 6, the UAV in the MEE-TE scheme flies more steadily at a lower speed to attenuate the tilting effect compared to the MEE-ITE scheme.

Figs. 8 and 9 show similar trajectories when $M = 4$, where the directional antenna has a stronger directivity and the coverage area of the main lobe of the directional beam becomes narrower. In this case, the UAV chooses to fly over each GU one by one instead of hovering over the center of the cluster. The optimized trajectories in Fig. 8 are similar to the trajectories in the right half of Fig. 6. From Fig. 9, we see that the UAV circles within a small area over each GU instead of hovering over each GU to save more propulsion energy. For rotary-wing UAVs, hovering is not the most energy-saving solution for the flight, and this conclusion is detailed as follows. Besides, when the tilting effect is considered in the optimization, the UAV flies over each GU more steadily at a lower speed compared to the MEE-ITE scheme, which makes the directional beam better directed toward GUs so that the UAV can achieve a higher EE.
<figure><img src="image_11.png" /><figcaption>Fig. 10. Illustration of the average value of $\alpha_k$ based on the EE maximization ($M = 4$).</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 11. Illustration of $\alpha_k$ for GU 4, 5 and 6 based on the EE maximization ($M = 4$).</figcaption></figure>

### C. Analysis of Tilt Angle and Propulsion Power

Figs. 6–9 show that considering the tilting effect in the optimization can lead to a more steady trajectory and a better directivity of the antenna. To better understand these results, we further analyze the changes in the angular deviation (i.e., $\alpha_k[n]$) as shown in Figs. 10 and 11. Fig. 10 shows the change of the average value of $\alpha_k[n]$ based on the EE maximization. Specifically, we randomly generate 300 sets of samples to represent different location distributions for the GU's and conduct 300 trajectory optimization experiments.<fnref n="2" /> For each GU, we calculate the average value of the tilt angle $\alpha_k$ over 20 slots when the UAV is closest to the GU. Then, the average values of the tilt angles for all the GU's are averaged across the 300 experiments to statistically present the deviation of the directional beams towards each GU. From Fig. 10, it is observed that the average values

<figure><img src="image_13.png" /><figcaption>Fig. 12. Instantaneous power comparison ($M = 4$)</figcaption></figure>

of the tilt angles obtained with the proposed MEE-TE scheme are significantly lower than those obtained with the MEE-ITE scheme. This validates that considering the tilting effect then optimizing the trajectory can significantly mitigate its negative impact on the UAV communication from a statistical perspective. Moreover, with the MEE-TE scheme, the directional beam of the antenna can be better aligned with each GU and thus the communication efficiency of the UAV is significantly enhanced. Furthermore, Fig. 11 shows the change of $\alpha_k[n]$ for GU 4, 5 and 6 in the later 80 slots of the flight duration based on a single simulation. Since the UAV chooses not to hover over the GU's shown in Fig. 9, $\alpha_k[n]$ in Fig. 11 is almost greater than zero. Particularly, analyzing the circled part in Fig. 11, we evince that $\alpha_k[n]$ based on the MEE-TE scheme is significantly lower than that based on the MEE-ITE scheme, and is close to zero when the UAV flies over the corresponding GU. From a statistical point of view and a specific trajectory point of view, the negative impact of the tilting effect is effectively mitigated and the UAV can achieve a better antenna directivity with the proposed MEE-TE scheme.

Fig. 12 shows the instantaneous power comparison based on the proposed four schemes during the whole flight duration. Specifically, we see that the instantaneous power with the MAR-ITE scheme is significantly higher than other schemes within some slots.<fnref n="3" /> This is because the UAV flies at a very high acceleration when leaving or approaching each GU as shown in Fig. 6 and Fig. 8, which causes significantly additional propulsion energy consumption. Moreover, comparing the MAR-TE scheme with the MAR-ITE scheme, we can draw a conclusion that a more energy-efficient trajectory design can be obtained by considering the tilting effect in the optimization. This is because the attenuation of the tilting effect leads to a more steady flight as analyzed before, and efficiently avoids the additional propulsion energy consumption, which is similar to the optimized trajectory

<footnote n="2">²Each sample still includes two GU clusters with different densities, and each cluster contains three GU's with varying inter-cluster separation distances. Based on these samples, we then conduct 300 trajectory optimization experiments to obtain the corresponding trajectory samples with different schemes.</footnote>

<footnote n="3">³Note that the actual UAV flight cannot follow the corresponding optimized trajectory with so dramatic velocity change. Thus, the high values of instantaneous power obtained with the MAR-ITE scheme are just computed theoretically, and do not correspond to the actual UAV flight.</footnote>
<figure><img src="image_14.png" /><figcaption>Fig. 13. Function value comparison based on the MEE-TE scheme ($M = 4$). (a) GU4. (b) GU5. (c) GU6.</figcaption></figure>

based on the EE maximization. Particularly, we can find that MEE-ITE is the most energy-saving scheme, especially when the UAV flies over the GU5. It is worth noting that the instantaneous power based on the MAR-TE and MAR-ITE schemes is higher than that on the other schemes when the UAV flies over the GU5, which means flying at a relatively low speed is more energy-saving than hovering. This conclusion is similarly verified in [32] and [19], and also explains why the UAV chooses not to hover over the GU5 for energy saving.

As analyzed in Section III, to make the antenna pattern more tractable in the optimization, we reformulate the antenna pattern as (22), which is a function of $tan(\alpha_k)$. Then, we introduce the upper bound $tan(\alpha_k^{ub})$ and the auxiliary variable $\xi_k$ based on the approximation that their values are approximately equal. To verify the effectiveness of the approximation and investigate the relationship among $tan(\alpha_k)$, $tan(\alpha_k^{ub})$ and $\xi_k$, Fig. 13 shows the changes in their values for GU 4, 5 and 6 in the last 80 slots of the flight duration. Specifically, from the three subfigures of Fig. 13, we see that the value of $tan(\alpha_k^{ub})$ always serves as the upper bound of the value of $tan(\alpha_k)$, and the value of $\xi_k$ is also greater than the value of $tan(\alpha_k)$. In addition, the value of $\xi_k$ nearly equals the value of $tan(\alpha_k^{ub})$ in most slots especially when the UAV flies over the GU5 steadily so that the tilting effect is not obvious and their values are smaller. Thus, the approximation $tan(\alpha_k^{ub}) \approx \xi_k$ is accurate in the proposed algorithm and $R_k^{lb,1}$ in (25) serves as a lower bound of $R_k$. From the circled parts in the three subfigures, we can find that, when the UAV flies over each GU, the corresponding values of $tan(\alpha_k)$, $tan(\alpha_k^{ub})$ and $\xi_k$ are nearly equal and close to zero. This demonstrates that the proposed approximation is accurate and the tilting effect is significantly attenuated as analyzed in Figs. 9 and 11.

## D. Analysis of Achievable Rate and EE

The tilting effect can influence both the communication performance and the propulsion energy consumption of the UAV equipped with a directional antenna, and the number of array elements $M$ is a key factor determining the directivity of the antenna. Thus, Figs. 14 and 15 show the achievable rate and the EE for all schemes with different values of $M$, respectively. From Fig. 14, we see that the achievable rate for all schemes increases as the number of $M$ increases. The reason is that the maximum directional gain of the antenna becomes larger as $M$ increases, and the UAV can achieve a higher rate. For any given

<figure><img src="image_15.png" /><figcaption>Fig. 14. Achievable rate comparison with varying $M$.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 15. EE comparison with varying $M$.</figcaption></figure>

$M$, the MAR-TE scheme presents the best performance and the schemes considering the tilting effect in the optimization outperform the other schemes in terms of achievable rate. This result shows the negative impact of the tilting effect on the directivity of the antenna. Furthermore, we can find that the achievable rate with the MEE-TE scheme is closer to the MAR-TE scheme as the number of $M$ increases. Because the tilting effect is not severe when the UAV flies steadily and slowly over the GU5,
and the UAV can achieve a similar performance compared to hovering over the GU's.

In terms of EE, from Fig. 15, we see that the EE for all schemes shows an upward trend as the number of *M* increases, and the EE obtained with the MAR-ITE scheme goes down at first. In addition, we can find that the proposed MEE-TE scheme is most energy-efficient compared to the other benchmarks, and the EE based on the EE maximization is higher than the EE based on the achievable rate maximization for any given *M*. Because the trajectory based on the EE maximization is smoother, the negative impact of the tilting effect on the EE is efficiently mitigated. It is worth mentioning that, even though the EE based on the MAR-TE scheme is lower than the schemes based on the EE maximization (i.e., the MEE-TE and MEE-ITE schemes), it significantly outperforms the MAR-ITE scheme in terms of EE. Thus, this phenomenon demonstrates that mitigating the tilting effect can avoid additional propulsion energy consumption of the UAV as shown in Fig. 12, i.e., we can obtain an energy-efficient UAV trajectory design by considering the tilting effect in the optimization, not just from the energy-saving perspective. To sum up, the obtained results show the effectiveness of the proposed MEE-TE scheme, and verify the importance of considering the tilting effect in UAV communications with directional antennas.

## VI. CONCLUSION

In this paper, we investigated the tilting effect on energy-efficient UAV-enabled communications. We first characterized the relationship between the tilting effect of the UAV and the UAV flight status. Then, we considered a UAV-enabled communication system with a directional array antenna and formulated an optimization problem to maximize the EE of the UAV. To tackle this problem, we proposed an approximate radiation pattern of the antenna, and developed an efficient iterative algorithm to solve the problem. Numerical results showed that the optimized trajectory based on EE maximization is smoother than that on the achievable rate maximization, where the tilting effect is attenuated. Besides, the UAV flight based on the proposed algorithm becomes more steady and the negative impact of the tilting effect on the EE can be efficiently mitigated. The proposed algorithm allows the UAV to achieve a substantially higher EE than the benchmark schemes.

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<figure><img src="image_17.png" /></figure>

Xinhong Dai received the B.S. degree in communication engineering from Chongqing University, Chongqing, China, in 2021, and the M.S. degree in information and communication engineering from the University of Electronic Science and Technology of China, Chengdu, China, in 2024. He is currently working toward the Ph.D. degree with the Department of Communication Science and Engineering, Fudan University, Shanghai, China. His research interests include UAV communications, mobile edge computing, and integrated sensing and communication.

<figure><img src="image_18.png" /></figure>

Bin Duo (Member, IEEE) received the Ph.D. degrees in information and communication engineering from the Harbin Institute of Technology, Harbin, China, in 2014, and from The University of Sydney, Sydney, NSW, Australia, in 2016, respectively. From 2018 to 2019, he was with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore. Since 2018, he has been with the National Key Laboratory of Wireless Communications, University of Electronic Science and Technology of China, as a Research Fellow. He is currently

a Professor with the College of Computer Science and Cyber Security, Chengdu University of Technology, Chengdu, China. His research interests include convex optimization theory, UAV communications for 6G, and IoT security.

<figure><img src="image_19.png" /></figure>

Xiaojun Yuan (Senior Member, IEEE) received the Ph.D. degree in electrical engineering from the City University of Hong Kong, Hong Kong, in 2009. From 2009 to 2011, he was a Research Fellow with the Department of Electronic Engineering, City University of Hong Kong. He was also a Visiting Scholar with the Department of Electrical Engineering, University of Hawaii at Manoa, Honolulu, HI, USA, in spring and summer 2009, as well as in the same period of 2010. From 2011 to 2014, he was a Research Assistant Professor with the Institute of Network Coding, The

Chinese University of Hong Kong. From 2014 to 2017, he was an Assistant Professor with the School of Information Science and Technology, ShanghaiTech University, Shanghai, China. He is currently a Professor with the National Key Laboratory of Wireless Communications, University of Electronic Science and Technology of China, Chengdu, China. He has authored or coauthored more than 280 peer-reviewed research papers in the leading international journals and conferences in the related areas. His research interests include signal processing, machine learning, wireless communications, intelligent communications, structured signal reconstruction, Bayesian approximate inference, distributed and learning. He has served on several technical programs for international conferences. He was an Editor of IEEE leading journals, including IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS and IEEE TRANSACTIONS ON COMMUNICATIONS. He was a co-recipient of the Best Paper Award of IEEE International Conference on Communications 2014, co-recipient of the Best Journal Paper Award of IEEE Technical Committee on Green Communications and Computing 2017, and co-recipient of IEEE Heinrich Hertz Award for Best Communication Letter 2022.

<figure><img src="image_20.png" /></figure>

Marco Di Renzo (Fellow, IEEE) received the Laurea (cum laude) and Ph.D. degrees in electrical engineering from the University of L'Aquila, L'Aquila, Italy, in 2003 and 2007, respectively, and the Habilitation à Diriger des Recherches (Doctor of Science) degree from Paris-Saclay University (formerly University Paris-Sud), Gif-sur-Yvette, France, in 2013. He is currently a CNRS Research Director (Professor) and the Head of the Intelligent Physical Communications group with the Laboratory of Signals and Systems, CNRS & CentraleSupélec, Paris-Saclay University, Paris, France, and also a Chair Professor in telecommunications engineering with the Centre for Telecommunications Research, Department of Engineering, King's College London, London, U.K. He was a France-Nokia Chair of Excellence in ICT with the University of Oulu, Oulu, Finland, a Tan Chin Tuan Exchange Fellow in engineering with Nanyang Technological University, Singapore, a Fulbright Fellow with The City University of New York, New York City, NY, USA, a Nokia Foundation Visiting Professor with Aalto University, Espoo, Finland, and a Royal Academy of Engineering Distinguished Visiting Fellow with Queen's University Belfast, Belfast, U.K. He is a Fellow of the IET, EURASIP, and AAIA, an Academician of AIIA, an Ordinary Member of the European Academy of Sciences and Arts, an Ordinary Member of the Academia Europaea, an Ambassador of the European Association on Antennas and Propagation, and a Highly Cited Researcher. He was the recipient of Michel Monpetit Prize conferred by the French Academy of Sciences, IEEE Communications Society Heinrich Hertz Award, and IEEE Communications Society Marconi Prize Paper Award in Wireless Communications. From 2019 to 2023, he was the Editor-in-Chief of IEEE COMMUNICATIONS LETTERS. His current main roles within the IEEE Communications Society include serving as a Voting Member of the Fellow Evaluation Standing Committee, as the Chair of the Publications Misconduct Ad Hoc Committee, and as the Director of Journals.