# A Generalized Pointing Error Model for
FSO Links with Fixed-Wing UAVs for 6G:
Analysis and Trajectory Optimization

Hyung-Joo Moon, , Chan-Byoung Chae, ,
Kai-Kit Wong, , and Mohamed-Slim Alouini This work was supported in part by
Institute of Information & communications Technology Planning & Evaluation
(IITP) grant funded by the Korea government (MSIT) (2019-0-00685) and in part
by the National Research Foundation of Korea (NRF) grant funded by the Korea
government (MSIT) (2022R1A5A1027646). (Corresponding author: C.-B. Chae.)H.-J.
Moon and C.-B. Chae are with the School of Integrated Technology, Yonsei
University, Seoul 03722, South Korea (e-mail: {moonhj,
cbchae}@yonsei.ac.kr).K.-K. Wong is with the Department of Electronic and
Electrical Engineering, University College London, London WC1E 7JE, UK
(e-mail: kai-kit.wong@ucl.ac.uk). He is also affiliated with Yonsei Frontier
Lab., Yonsei University, Seoul 03722, South Korea.M.-S. Alouini is with the
Computer, Electrical, and Mathematical Science and Engineering Division, King
Abdullah University of Science and Technology, Thuwal 23955, Saudi Arabia
(e-mail: slim.alouini@kaust.edu.sa).

###### Abstract

Free-space optical (FSO) communication is a promising solution to support
wireless backhaul links in emerging 6G non-terrestrial networks. At the link
level, pointing errors in FSO links can significantly impact capacity, making
accurate modeling of these errors essential for both assessing and enhancing
communication performance. In this paper, we introduce a novel model for FSO
pointing errors in unmanned aerial vehicles (UAVs) that incorporates three-
dimensional (3D) jitter, including roll, pitch, and yaw angle jittering. We
derive a probability density function for the pointing error angle based on
the relative position and posture of the UAV to the ground station. This model
is then integrated into a trajectory optimization problem designed to maximize
energy efficiency while meeting constraints on speed, acceleration, and
elevation angle. Our proposed optimization method significantly improves
energy efficiency by adjusting the UAV’s flight trajectory to minimize
exposure to directions highly affected by jitter. The simulation results
emphasize the importance of using UAV-specific 3D jitter models in achieving
accurate performance measurements and effective system optimization in FSO
communication networks. Utilizing our generalized model, the optimized
trajectories achieve up to 11.8$\%$ higher energy efficiency compared to those
derived from conventional Gaussian pointing error models.

###### Index Terms:

6G, free-space optics, fixed-wing UAV, pointing error, energy-efficiency,
trajectory optimization problem.

##  I Introduction

Communication resources are still predominantly concentrated on terrestrial
networks, even with the development of 5G communication networks. To meet the
demands of massive connectivity and increasing data traffic of B5G/6G,
researchers are utilizing the vertical space to create a three-dimensional
(3D) network known as a non-terrestrial network (NTN). This network includes
satellites, high-altitude platforms (HAPs), low-altitude platforms (LAPs), and
ground nodes [1, 2]. Unmanned aerial vehicles (UAVs) are becoming increasingly
popular as a means of expanding the connectivity and capacity of the existing
terrestrial networks [3, 4, 5, 6]. For UAV mobile base stations (BSs), a
reliable wireless backhaul link is essential to connect UAVs to the core
network through a ground gateway. Wireless backhaul systems can be effectively
supported by free-space optical (FSO) communications, a promising solution for
assisting BSs in NTNs.

FSO communications have distinct characteristics compared to radio frequency
(RF)-based wireless communications [7]. They provide long-distance
communication links but only support point-to-point connections. The
transceivers are highly secure against eavesdropping due to the narrow
beamwidth, and the high carrier frequency, in the hundreds of terahertz,
offers significant bandwidth in an unlicensed band. Because of these
advantages, FSO communications are considered suitable for supporting the
connection between radio access networks (or a gateway) and mobile BSs across
the various non-terrestrial layers of the NTN [8, 9, 10]. For instance, relay
UAVs can provide improved data rates to terrestrial FSO networks while
ensuring line-of-sight (LoS) conditions [8]. However, one of the primary
challenges in establishing an FSO backhaul link is overcoming pointing
disturbances caused by atmospheric effects and mechanical jitters. To ensure
signal power at the receiver end when transmitting the narrow optical beam,
transceivers at both ends must precisely align with each other. This
emphasizes the critical role of the pointing, acquisition, and tracking (PAT)
system, which manages link establishment, stabilization, and maintenance [11,
12].

The presence of a PAT system does not guarantee the elimination of pointing
errors. Pointing losses still occur due to residual jitter in the beam
footprint at the receiver plane [13, 14]. The distribution of pointing errors
varies depending on the type of disturbance and can be modeled using different
distributions. Without considering boresight error, the pointing error
distribution is Rayleigh for one degree of freedom (DoF) and Hoyt for two DoFs
[15]. Especially, the Rayleigh distribution is widely applicable because it
generalizes the sum of all jitter factors into two-dimensional (2D) Gaussian
disturbances [16]. On the other hand, the Hoyt distribution is suitable for
modeling pointing errors due to rotary-wing drone jitters. The authors in [17]
separately analyze UAV jitters in the vertical and horizontal directions to
represent the distinctive jitter characteristics of drones.

To the best of our knowledge, this is the first study that investigates the
impact of generalized 3D jitter in UAVs and proposes a method for avoiding
highly disturbed link directions by adjusting the UAVs’ trajectories. Our
results indicate that UAVs should avoid directions significantly affected by
specific jitter orientations. The proposed optimization method enhances energy
efficiency by adjusting trajectories, taking into account both power
consumption and the achievable rate.

###  I-A Related Works

####  I-A1 FSO and Mixed FSO/RF Links

In our system, the UAV can either be a mobile BS or a source node, and data
must be transmitted from the UAV to the ground station (GS) through the
reverse link. The performance of the reverse FSO backhaul link has been
analyzed and optimized in several studies [18, 19, 20]. In [18], the relay
node collects data from multiple users through RF access links and then
delivers the message to the destination node through a parallel FSO/RF link.
Another study explores multi-user data collection at the satellite destination
node in the satellite-air-terrestrial integrated network through HAP relay
nodes [19]. In [20], the authors analyze the mixed FSO/RF relay communications
system and derive various performance metrics and analytical features. Unlike
other studies, the FSO access link from the UAV to ground users is assumed in
[21]. Additionally, the cost-efficient communication between the GS and UAVs
using a modulating retro-reflector is a promising technology for communicating
with UAVs from a fixed GS [7, 22, 21, 23].

####  I-A2 Pointing Error

FSO communication channels are typically modeled as a product of fast channel
fluctuation, atmospheric loss, and pointing loss [16]. Particularly for UAV-
to-GS links, the transmitter on the UAV becomes more susceptible to pointing
errors due to mechanical jitter and payload limitations [24]. The simplest
model for pointing error in the context of a Gaussian beam involves a 2D
independent and identically distributed (iid) Gaussian error. However, this
simple model neglects boresight error and biased jittering. Both of these
factors are considered and generalized into five different models in [15, 25],
which are applicable to various system models, including fixed-point,
shipborne, and UAV-based networks. The distribution of the FSO channel and
performance metrics were derived with different pointing error distributions
in [26] for weak turbulence conditions and in [27] for strong turbulence
conditions. The effect of UAV jittering on the pointing error has also been
studied for mmWave communications [28, 29]. In [28], the authors analyze the
effect of vibrations from UAV transceivers on the mmWave channel model. In
another study [29], the authors explore the impact of roll, pitch, and yaw
jitter of the UAV on the channel training of a 2D antenna array launched at
the UAV. The authors transform the 3D jittering model of drones into the
jittering of the antenna array in the $x$ and $y$ axes.

####  I-A3 UAV Trajectory Optimization

Trajectory optimization has been widely studied as an effective solution for
energy-efficient communications using unmanned aerial vehicles (UAVs).
Research has been conducted on both RF communications [30, 31, 32, 33, 34, 35,
36, 37] and FSO communications [8, 38, 39, 40, 23, 41]. In [30] and [31], the
energy consumption models for rotary-wing and fixed-wing communication UAVs
were proposed, respectively. In both studies, the communication efficiency was
optimized using the successive convex approximation (SCA) method to solve for
the flight trajectory. The SCA method was used in [30, 31, 32, 33] as an
optimization technique for finding effective trajectories from non-convex
problems. In contrast, the block coordinate descent method was used in [34,
35], and online adaptation through deep reinforcement learning was used in
[36, 37]. For FSO communication systems, researchers have proposed various
flight optimization methods with different criteria. In [38], the link
connection time between the GS and the UAV was maximized while considering
speed and acceleration constraints. In [39, 40], the throughput of buffer-
aided mixed FSO/RF relay communications was maximized under flight constraints
and queueing mechanisms. More complex constraints and objectives were defined
in [23, 41] to optimize the multi-UAV system, taking into account the power
consumption of rotary-wing UAVs [30].

###  I-B Contributions and Paper Organization

In light of the previous studies, we introduce a new pointing error model for
fixed-wing UAVs that accounts for the 3D jitter of UAVs. Based on this model,
we present a trajectory optimization algorithm for UAV-assisted FSO
communications. Our results indicate that in the selected scenarios, the
communication performance is significantly influenced by the position and
posture of the UAV relative to the GS. As a result, it is crucial to minimize
the impact of the UAV jitter by selecting the least affected flight path. In
summary, the key contributions of this work are as follows:

  * •

In the context of the UAV-to-GS FSO link, we define the jittering of UAVs in
the yaw, pitch, and roll axes. Using this jittering model, we derive the
pointing error angle as a Hoyt-distributed random variable. The distribution
parameters are determined by the position and posture of the UAV relative to
the GS. We formulate the energy efficiency as the ratio of total link capacity
to power consumption during the mission flight. The link capacity can be
obtained from the link budget of the UAV-to-GS FSO link, which includes the
proposed pointing error model. The flight power consumption and kinetic
characteristics of the UAVs are obtained from the flight model of the fixed-
wing UAVs.

  * •

Using the capacity and power consumption models, we formulate an optimization
problem that maximizes the energy efficiency under the speed, acceleration,
initial and final coordinates, and elevation angle constraints. Finding the
optimal trajectory is a non-trivial problem, primarily due to the highly
nonlinear relationship between the acceleration, posture, and link capacity.
We utilize the SCA method to convert the non-convex problem into a locally-
approximated concave-convex fractional program. During the iteration, the
Dinkelbach method is used to maximize the approximated fractional objective
function under convex constraints.

  * •

The results of the trajectory optimization problem explicitly show the impact
of the generalized 3D jitter of UAVs on the optimal trajectory. Based on these
results, we assert that the proposed jittering model should be considered to
measure the realistic performance of the FSO link.

The remainder of this work is organized as follows. Section II introduces the
UAV flight model, FSO link capacity, and 3D jittering of UAVs with
mathematical expressions. In Section III, we develop the random jittering of
fixed-wing UAVs into the pointing error model. The formulation of the energy-
efficiency maximization problem is described in Section IV. The approximation
of the non-convex objective function and constraints through the selected
optimization techniques is also provided in Section IV, with theorems, lemmas,
and derivations. The numerical results of the generalized pointing error model
and the optimization problem are shown in Section V. Finally, the paper is
concluded in Section VI with meaningful insights.

##  II System Model

Figure 1: GS-centered coordinate system.

In this section, we assume that a UAV is performing a mission flight while
maintaining an FSO communication link with a fixed GS. The mission flight is
subject to constraints with a fixed operating altitude and specified initial
and final points. We consider one-hop UAV-to-GS transmission using intensity
modulation and direct detection (IM/DD) FSO communications. In order to ensure
LoS transmission, the elevation angle of the UAV from the GS is constrained to
a certain lower bound. A detailed model of the FSO communication-based mission
flight is described in the following subsections. The parameter notations used
throughout this paper are organized in Table I.

###  II-A Discrete-Time Coordinate System

To establish a GS-centered coordinate system, as depicted in Fig. 1, we first
discretize continuous time into a small time interval $\delta$ and time slot
$k$, where $k\in\mathbb{N}$. The 3D location of the UAV at a given time slot
$k$ is expressed in the coordinate system as $\mathbb{s}[k]$, with the $x$-$y$
plane and $z$ axis corresponding to the ground and altitude, respectively. The
posture of the UAV can be defined by the UAV-centered coordinate system, which
is represented by $x^{\prime}$, $y^{\prime}$, and $z^{\prime}$ axes. At a
given time slot $k$, the heading direction of the UAV and the direction
pointed by the left wing are defined as the $x^{\prime}$ and $y^{\prime}$
axes, respectively.

We define the time of interest as $k\in[1,N]$, where the mission starts at
time slot $1$ and ends at $N$. The velocity and acceleration of the UAV are
determined by its position $\mathbb{s}[k]$. Specifically, the velocity is
given by

$\displaystyle\mathbb{v}[k]=\frac{\mathbb{s}[k+1]-\mathbb{s}[k]}{\delta},\quad
1\leq{k}\leq{N-1}.$ (1)

To facilitate our analysis, we extend the definition of the velocity to time
slot $N$ as

$\displaystyle\mathbb{v}[N]=\mathbb{v}[N-1],$ (2)

or, equivalently, $\mathbb{s}[N+1]=2\mathbb{s}[N]-\mathbb{s}[N-1]$. The
acceleration can be computed as

$\displaystyle\mathbb{a}[k]=\frac{\mathbb{v}[k+1]-\mathbb{v}[k]}{\delta},\quad
1\leq{k}\leq{N-1}.$ (3)

Throughout this paper, we assume a fixed altitude of the UAV as follows:

$\displaystyle s_{z}=H,\quad 1\leq{k}\leq{N},$ (4)

where $\mathbb{s}[k]=[s_{x},s_{y},s_{z}]^{T}$.

TABLE I: Parameters Parameter | Description
---|---
$x$, $y$, $z$ | Axes of the GS-centered coordinate system
$x^{\prime}$, $y^{\prime}$, $z^{\prime}$ | Axes of the UAV-centered coordinate system
$\mathbb{s}[k]$, $\mathbb{v}[k]$, $\mathbb{a}[k]$ |  Position, velocity, and acceleration at time slot $k$
$N$ | Number of time slots
$\delta$ | Time interval between slots
$\phi$, $\psi$, $\theta$ | Roll, pitch, and yaw angles of the UAV posture
$\hat{\mathbb{u}}[k]$ | Average UAV-to-GS pointing vector
$\mathbb{u}[k]$ | Instantaneous UAV-to-GS pointing vector
$C_{\mathbb{E}}$ | Ergodic capacity of UAV-to-GS transmission
$P_{\text{T}}$ | Transmit power of the UAV
$P_{\text{F}}$ | Flight power consumption of the UAV
$E_{\text{cost}}$ | Launch cost of the flight mission
$h_{a}$, $h_{\ell}$, $h_{p}$ | Fast fading, atmospheric loss, and pointing loss
$\theta_{p}$ | UAV-to-GS pointing error angle
$\alpha,\beta,\gamma$ | Roll, pitch, and yaw angle jittering of the UAV
$v_{\text{min}}$, $v_{\text{max}}$ | Minimum and maximum speed of the UAV
$a_{\text{max}}$ | Maximum acceleration of the UAV
$\mathbb{l}_{o}$, $\mathbb{l}_{d}$ | Initial and final position of the mission
$\eta$ | Elevation angle of the UAV

To calculate the UAV-to-GS pointing vector in the UAV-centered coordinate
system, we need to establish the relationship between the two coordinate
systems. The roll, pitch, and yaw angles of the UAV in the GS-centered
coordinate system are denoted as $\phi$, $\psi$, and $\theta$, respectively
(see Fig. 2). Assuming a fixed altitude of the UAV, the pitch angle is always
zero [30]. The yaw angle can be determined using the direction vector of
$\mathbb{v}[k]$, which represents the $x^{\prime}$ axis in the UAV-centered
coordinate system. Specifically, we have

$\displaystyle\cos\theta=\frac{v_{x}}{\left\lVert\mathbb{v}[k]\right\rVert},\,\,\sin\theta=\frac{v_{y}}{\left\lVert\mathbb{v}[k]\right\rVert},$
(5)

where $\mathbb{v}[k]=[v_{x},v_{y},v_{z}]^{T}$. According to [30], acceleration
determines the roll angle (or bank angle) of fixed-wing UAVs. Using the
formula $\tan\phi=-{a_{\perp}}/{\textsl{g}}$, we can derive

$\displaystyle\tan\phi=\frac{1}{\left\lVert\mathbb{v}[k]\right\rVert\textsl{g}}({v_{y}}{a_{x}}-{v_{x}}{a_{y}}),$
(6)

where $a_{\perp}$ and g are the acceleration in the $y^{\prime}$ axis and
gravitational acceleration, respectively, and
$\mathbb{a}[k]=[a_{x},a_{y},a_{z}]^{T}$. We can express the UAV-to-GS pointing
vector, $\hat{\mathbb{u}}[k]$, in the UAV-centered coordinate system, using
the position and posture of the UAV in the GS-centered coordinate system. The
3D rotation matrices with respect to the $x$, $y$, and $z$ axes can be
expressed as follows [29]:

$\displaystyle\mathbb{R}_{\text{x}}(\alpha)=\begin{bmatrix}1&0&0\\\
0&\cos\alpha&-\sin\alpha\\\ 0&\sin\alpha&\cos\alpha\end{bmatrix},$ (7)
$\displaystyle\mathbb{R}_{\text{y}}(\beta)=\begin{bmatrix}\cos\beta&0&\sin\beta\\\
0&1&0\\\ -\sin\beta&0&\cos\beta\end{bmatrix},$
$\displaystyle\mathbb{R}_{\text{z}}(\gamma)=\begin{bmatrix}\cos\gamma&-\sin\gamma&0\\\
\sin\gamma&\cos\gamma&0\\\ 0&0&1\end{bmatrix}.$

We then represent $\hat{\mathbb{u}}[k]$ as

$\displaystyle\hat{\mathbb{u}}[k]=$
$\displaystyle-\mathbb{R}_{\text{post}}(\phi,\psi,\theta)\mathbb{s}[k],$ (8)

where
$\mathbb{R}_{\text{post}}(\phi,\psi,\theta)=\mathbb{R}_{\text{x}}(-\phi)\mathbb{R}_{\text{y}}(-\psi)\mathbb{R}_{\text{z}}(-\theta)$.

Figure 2: 3D jittering of the UAV in the UAV-centered coordinate system.

###  II-B Capacity of an FSO Link

The received signal at the GS for IM/DD FSO communications is modeled as in
[42]:

$\displaystyle\bar{y}=h\bar{x}+\bar{n},$ (9)

where $\bar{y}$ is the received signal, $h$ is the channel coefficient,
$\bar{x}\in\mathbb{R}^{+}$ is the intensity modulated signal, and $\bar{n}$ is
the background Gaussian shot noise. The channel $h$ can be modeled as a
product of independent factors that account for fast channel fluctuations,
atmospheric loss, pointing loss, and responsivity. This is expressed as
$h=h_{a}h_{\ell}{h_{p}}R$. The average transmit power is constrained to
$P_{\text{T}}$, and the shot noise follows
$\bar{n}\sim\mathcal{N}(0,\sigma^{2})$. Then, the average received power at
the GS is given by [16]

$\displaystyle P=h_{a}h_{\ell}{h_{p}}RP_{\text{T}}.$ (10)

The average optical signal-to-noise ratio (ASNR) for this signal model is
$P^{2}/\sigma^{2}$. The fast fluctuations of the FSO signal power are mainly
caused by scintillation during atmospheric propagation [7]. We assume that the
channel fluctuation follows a log-normal distribution, given by

$\displaystyle
f_{h_{a}}(x)=\frac{1}{2x\sqrt{2\pi\sigma_{I}^{2}}}\exp\left(-\frac{(\log{x}+2\sigma_{I}^{2})^{2}}{8\sigma_{I}^{2}}\right),$
(11)

where $\sigma_{I}^{2}$ is the log-amplitude variance [11]. The atmospheric
loss follows the Beer-Lambert law as described in

$\displaystyle h_{\ell}=\exp(-\sigma_{\text{B}}{z}),$ (12)

where $\sigma_{\text{B}}$ is the attenuation coefficient and $z$ represents
the propagation length, which is the distance between the UAV and the GS. The
attenuation coefficient, given by

$\displaystyle\sigma_{\text{B}}=\frac{3.91}{V}\left(\frac{\lambda_{\text{FSO}}}{\lambda_{0}}\right)^{-q_{\text{sca}}},$
(13)

depends on the size distribution of the scattering, represented by
$q_{\text{sca}}$, and the wavelength of the FSO link, $\lambda_{\text{FSO}}$
[7]. The value of $q_{\text{sca}}$ is determined by

$q_{\text{sca}}=\begin{cases}\begin{aligned} 1.6\quad\qquad&V\geq
50\,\text{km}\\\ 1.3\quad\qquad&6\,\text{km}\leq{V}<50\,\text{km}\\\
0.585V^{1/3}\quad&V<6\,\text{km}\end{aligned}\end{cases}$ (14)

where $V$ is the visibility range and $\lambda_{0}=550$ nm. Assuming a
Gaussian beam with a 1-$\sigma$ divergence angle of $2\sigma_{\text{div}}$,
the pointing loss can be described by the following formula [16]:

$\displaystyle h_{p}$
$\displaystyle=A_{0}\exp\left(-\frac{\theta_{p}^{2}}{2\sigma_{\text{div}}^{2}}\right),$
(15)

where $\theta_{p}$ represents the pointing error angle and $A_{0}$ is the
maximum gain at the center of the beam. The 1-$\sigma$ diameter of the beam
footprint at the receiver end can be expressed as $w_{b}=2z\theta_{p}$. Let
$a$ denote the aperture diameter of the receiver. Assuming that the beamwidth
is significantly larger than the aperture size ($w_{b}\gg{a}$), the maximum
gain of $h_{p}$ can be approximated as
$A_{0}\approx\frac{a^{2}}{2z\sigma_{\text{div}}}$ [11].

Based on the channel characteristics, the capacity of the IM/DD FSO
communication link can be expressed as follows [43]:

$\displaystyle C=\frac{1}{2}\log_{2}\left(1+\Gamma\right),$ (16)

where $\Gamma=\frac{e{P}^{2}}{2\pi\sigma^{2}}$. Since the channel fluctuation
is much faster than the kinematic changes of UAVs [44], we select the ergodic
capacity as a performance metric. The ergodic capacity of the link is defined
as follows:

$\displaystyle
C_{\mathbb{E}}=\mathbb{E}\left[\frac{1}{2}\log_{2}\left(1+\Gamma\right)\right],$
(17)

where $\mathbb{E}[\cdot]$ represents the expectation of the function. It is
important to note that the term $h_{p}$ in $\Gamma$ depends on the position
and posture of the UAV, which significantly influences the value of
$C_{\mathbb{E}}$.

Figure 3: Probability density function of the pointing error angle.

###  II-C Flight Power Consumption Model

A major challenge in UAV communications is power consumption during flight.
The power consumption is influenced by the UAV’s dynamics, specifically its
velocity and acceleration, which are represented by the vectors $\mathbb{v}$
and $\mathbb{a}$, respectively. The relationship between these factors and
power consumption is given by the following equation [30]:

$\displaystyle P_{\text{F}}(\mathbb{v},\mathbb{a})\approx$
$\displaystyle{c_{1}\left\lVert\mathbb{v}\right\rVert^{3}+\frac{c_{2}}{\mathbb{\left\lVert
v\right\rVert}}\left(1+\frac{\left\lVert\mathbb{a}\right\rVert^{2}}{\textsl{g}^{2}}\right)+\frac{\Delta{E_{K}}}{\delta}},$
(18)

where $c_{1}=\frac{1}{2}\rho{C}_{D_{0}}S$,
$c_{2}=\frac{2W^{2}}{(\pi{e_{0}}\mathcal{A}_{\mathcal{R}})\rho{S}}$, and
$\Delta{E_{K}}$ is the change in kinetic energy during the time interval of
$\delta$. The symbols $\rho$, ${C}_{D_{0}}$, $S$, $W$, $e_{0}$,
$\mathcal{A}_{\mathcal{R}}$ denote the air density, zero-lift drag
coefficient, reference area, weight, Oswald efficiency, and aspect ratio of
the wing, respectively. The total change in kinetic energy can be calculated
from the initial and final velocity of the UAV, as follows:

$\displaystyle\Delta{E}_{K,\text{tot}}=\frac{1}{2}m(\left\lVert\mathbb{v}[N]\right\rVert^{2}-\left\lVert\mathbb{v}[1]\right\rVert^{2}),$
(19)

where $m$ is the mass of the UAV. For a sufficiently long flight time
($\delta{N}\gg{1}$), we can ignore $\Delta{E}_{K,\text{tot}}$, since
$\left\lVert\mathbb{v}[1]\right\rVert$ and
$\left\lVert\mathbb{v}[N]\right\rVert$ are both restricted by the maximum and
minimum velocity constraints. Thus, the power consumption at time slot $k$ is
approximated as follows:

$\displaystyle P_{\text{F}}$
$\displaystyle(\mathbb{v}[k],\mathbb{a}[k])\approx{c_{1}}\left\lVert\mathbb{v}[k]\right\rVert^{3}+\frac{c_{2}}{\left\lVert\mathbb{v}[k]\right\rVert}\left(1+\frac{\left\lVert\mathbb{a}[k]\right\rVert^{2}}{\textsl{g}^{2}}\right).$
(20)

Through trajectory optimization, the power consumption can be economized by
determining the most energy-efficient flight path for operating the mission.

##  III Pointing Error Analysis

In this section, we introduce the 3D jittering model of the UAV, building upon
the system model outlined in Section II-A. To describe the instantaneous UAV-
to-GS pointing vector, which incorporates the impact of residual jittering in
the PAT system, we develop the jittering matrix. We then establish the
pointing error angle as the deviation between the instantaneous pointing
vector and the average pointing vector, excluding the jittering effect.
Finally, we derive the distribution of the pointing error angle.

###  III-A 3D Jittering Model of Fixed-wing Aircraft

By analyzing the flight model and the relationship between the two coordinate
systems introduced in Section II-A, we obtain $\hat{\mathbb{u}}[k]$, a UAV-to-
GS pointing vector when the posture of the UAV is fixed along the
$x^{\prime}$, $y^{\prime}$, and $z^{\prime}$ axes. To account for the effect
of the jitter, we first express it as a transformation matrix. From (7), the
jittering matrix can be expressed as follows:

$\displaystyle\mathbb{R}_{\text{jitter}}(\alpha,\beta,\gamma)=$
$\displaystyle\mathbb{R}_{\text{x}}(\alpha)\mathbb{R}_{\text{y}}(\beta)\mathbb{R}_{\text{z}}(\gamma)$
(21) $\displaystyle\approx$ $\displaystyle\begin{bmatrix}1&-\gamma&\beta\\\
\gamma&1&-\alpha\\\ -\beta&\alpha&1\end{bmatrix},$

where $\alpha$, $\beta$, and $\gamma$ are roll, pitch, and yaw angle jittering
of the UAV, respectively. We apply small angle approximations
($\sin\epsilon\approx\epsilon$ and $\cos\epsilon\approx{1}$) to simplify
calculations. We consider the 3D jittering angles as random variables,
represented by the vector $\mathbb{x}=\left[\alpha,\beta,\gamma\right]^{T}$
that follows a multivariate Gaussian distribution
$\mathbb{x}\sim\mathcal{N}(\mathbb{\mu},\mathbb{\Sigma})$. The mean and
covariance of $\mathbb{x}$ are defined by

$\displaystyle\mathbb{\mu}=[0,0,0]^{T},\,\mathbb{\Sigma}=\begin{bmatrix}\sigma_{\alpha}^{2}&\rho_{\alpha,\beta}{\sigma_{\alpha}}{\sigma_{\beta}}&\rho_{\gamma,\alpha}{\sigma_{\gamma}}{\sigma_{\alpha}}\\\
\rho_{\alpha,\beta}{\sigma_{\alpha}}{\sigma_{\beta}}&\sigma_{\beta}^{2}&\rho_{\beta,\gamma}{\sigma_{\beta}}{\sigma_{\gamma}}\\\
\rho_{\gamma,\alpha}{\sigma_{\gamma}}{\sigma_{\alpha}}&\rho_{\beta,\gamma}{\sigma_{\beta}}{\sigma_{\gamma}}&\sigma_{\gamma}^{2}\end{bmatrix}.$
(22)

The covariance matrix $\mathbb{\Sigma}$ represents the jittering
characteristics of the fixed-wing UAV. Now we develop the instantaneous
pointing vector, including the effect of the 3D jitter, as

$\displaystyle\mathbb{u}[k]=$
$\displaystyle-\mathbb{R}_{\text{jitter}}(\alpha,\beta,\gamma)\mathbb{R}_{\text{post}}(\phi,\psi,\theta)\mathbb{s}[k].$
(23)

It can be equivalently expressed as
$\mathbb{u}[k]=\mathbb{R}_{\text{jitter}}(\alpha,\beta,\gamma)\hat{\mathbb{u}}[k]$.
Finally, the pointing error angle $\theta_{p}$ is given by

$\displaystyle\sin\theta_{\text{p}}=\frac{\lvert\mathbb{u}[k]\times\hat{\mathbb{u}}[k]\rvert}{\lvert\mathbb{u}[k]\rvert\lvert\hat{\mathbb{u}}[k]\rvert},$
(24)

which characterizes the angle difference between the average pointing vector
$\hat{\mathbb{u}}[k]$ and the instantaneous pointing vector $\mathbb{u}[k]$
from the perspective of the UAV.

###  III-B Distribution of the Pointing Error Angle

When the boresight error is neglected, the pointing error angle is defined as
the angle difference between the actual pointing vector $\mathbb{u}[k]$, which
incorporates the effect of the 3D jitter, and the average pointing vector
$\hat{\mathbb{u}}[k]$. We proceed to analyze the characteristics of
$\theta_{p}$, the random variable defined in (24).

TABLE II: Comparison of the pointing error models Model | Parameter | $\lambda_{1}$ | $\lambda_{2}$ |  | $\mathbb{E}[\theta_{p}^{2}]=$
---
$\lambda_{1}+\lambda_{2}$
Ours |  $\theta=0$, $\rho_{\text{jit}}=0$ | $0.9664$ | $0.0522$ | $1.0186$
Ours |  $\theta=0$, $\rho_{\text{jit}}=0.5$ | $0.9202$ | $0.0324$ | $0.9526$
2D Model |  $\theta=0$ [17] | $0.5449$ | $0.2827$ | $0.8276$
Ours |  $\theta=\pi/2$, $\rho_{\text{jit}}=0$ | $0.3797$ | $0.0891$ | $0.4688$
Ours |  $\theta=\pi/2$, $\rho_{\text{jit}}=0.5$ | $0.3723$ | $0.0640$ | $0.4363$
2D Model |  $\theta=\pi/2$ [17] | $0.5449$ | $0.2074$ | $0.7523$

###### Theorem 1

The distribution of $\theta_{p}$ follows the Hoyt distribution as

$\displaystyle\theta_{p}\sim\mathcal{H}(q,\Omega),$ (25)

where $q=\sqrt{\lambda_{1}/\lambda_{2}}$ and $\Omega=\lambda_{1}+\lambda_{2}$
[17]. The symbols $\lambda_{1}$, $\lambda_{2}$ ($\lambda_{1}\geq\lambda_{2}$)
are the nonzero eigenvalues of
$\mathbb{\Sigma}^{1/2}\mathbb{A}\mathbb{\Sigma}^{1/2}$, where $\mathbb{A}$ is
defined as

$\displaystyle\mathbb{A}=\mathbb{I}-\frac{1}{z^{2}}{\hat{\mathbb{u}}[k]}{\hat{\mathbb{u}}[k]}^{T},$
(26)

with the identity matrix $\mathbb{I}$, and
$z={({\hat{\mathbb{u}}[k]}^{T}{\hat{\mathbb{u}}[k]})}^{1/2}$ representing the
link distance.

_Proof:_ See Appendix A. $\blacksquare$

During the proof of _Theorem_ 1, we obtain an expression for the expected
value of $\theta_{p}^{2}$:

$\displaystyle\mathbb{E}[\theta_{p}^{2}]=\lambda_{1}+\lambda_{2},$ (27)

which can also be computed using
$\text{Tr}(\mathbb{\Sigma}^{1/2}\mathbb{A}\mathbb{\Sigma}^{1/2})=\lambda_{1}+\lambda_{2}$.
Fig. 3 presents a numerical comparison of the probability density function of
$\theta_{p}$ for our generalized model, which incorporates the 3D jittering
characteristics, and the pointing error model from [17], that considers only
two DoFs of drone jitter. In each case illustrated in Fig. 3, the angles
$\psi=-10^{\circ}$ and $\phi=0$ are held constant. We simulate different
values of $\theta$ and the correlation coefficients
$\rho_{\alpha,\beta}=\rho_{\beta,\gamma}=\rho_{\gamma,\alpha}=\rho_{\text{jit}}$.
The standard deviations for the jittering along each axis are set to
$\sigma_{\alpha}=1$ mrad for roll, $\sigma_{\beta}=0.3$ mrad for pitch, and
$\sigma_{\gamma}=0.1$ mrad for yaw. In the 2D jitter model, where
$\sigma_{\alpha}$ and $\sigma_{\beta}$ are not separately defined, we use
$\sigma_{\text{x-y}}=\sqrt{(\sigma_{\alpha}^{2}+\sigma_{\beta}^{2})/2}=0.738$
mrad and $\sigma_{\gamma}=0.1$ mrad to best represent the actual jitter
characteristics using only two DoFs. The positions of the GS and the UAV are
assumed to be $[0,0,0]$ m and $[50,550,600]$ m, respectively. We evaluate the
pointing error angle under two different heading directions, $\theta=0$ and
$\theta=\pi/2$. The results depicted in Fig. 3 and Table II highlight
significant differences in the distribution between the proposed generalized
pointing error model and the model proposed in [17]. It emphasizes the
importance of considering the UAV-specific 3D jittering characteristics in
link performance assessments.

##  IV Trajectory Optimization for Mission Flight

In this section, we first define the trajectory optimization problem aimed at
maximizing energy efficiency. Since the problem is highly non-convex, we
iteratively solve using the SCA method. Each iteration of the SCA method
involves approximating the non-convex functions, and we provide detailed
descriptions of these approximations. Subsequently, we utilize the Dinkelbach
method to convert the problem, which originally has a fractional objective
function, into an equivalent convex problem that can be addressed iteratively.

###  IV-A Problem Formulation

Based on the system model introduced in Section II, we aim to optimize the
trajectory of the UAV over time to maximize the energy efficiency. Throughout
this section, we determine $\mathbb{s}[k]$, the position of the UAV in the GS-
centered coordinate system at each time slot $1\leq{k}\leq{N}$. We formulate
this problem as follows:

$\displaystyle\text{(P1)}:\,\,\begin{split}\max_{\mathbb{s}[k]}\quad&\frac{\sum_{k=1}^{N}C_{\mathbb{E}}[k]}{\sum_{k=1}^{N-1}P_{\text{F}}(\mathbb{v}[k],\mathbb{a}[k])+NP_{\text{T}}+{E_{\text{cost}}}/{\delta}}\end{split}$
(28a)
$\displaystyle\begin{split}\textrm{s.t.}\quad&v_{\text{min}}\leq\left\lVert\mathbb{v}[k]\right\rVert\leq{{v}_{\text{max}}},\quad
1\leq{k}\leq{N}\end{split}$ (28b)
$\displaystyle\begin{split}&\left\lVert\mathbb{a}[k]\right\rVert\leq{a_{\text{max}}},\quad
1\leq{k}\leq{N-1}\end{split}$ (28c)
$\displaystyle\begin{split}&(\ref{velocity}),(\ref{velocityout}),(\ref{acceleration}),(\ref{altitude})\end{split}$
(28d)
$\displaystyle\begin{split}&\mathbb{s}[1]=\mathbb{l}_{o},\,\mathbb{s}[N]=\mathbb{l}_{d}\end{split}$
(28e)
$\displaystyle\begin{split}&\frac{\pi}{4}\leq\eta\leq\frac{\pi}{2},{\quad}1\leq{k}\leq{N}\end{split}$
(28f)

where $E_{\text{cost}}$ is the launch cost of the mission, $v_{\text{min}}$
and $v_{\text{max}}$ represent the maximum and minimum speed, respectively,
and $a_{\text{max}}$ is the maximum acceleration. The parameters
$\mathbb{l}_{o}$ and $\mathbb{l}_{d}$ represent the initial and final
position, respectively. The elevation angle $\eta$ is defined as
$\eta=\arctan({s_{z}}/{\sqrt{s_{x}^{2}+s_{y}^{2}}})$ where
$\mathbb{s}[k]=[s_{x},s_{y},s_{z}]^{T}$. The constraints in (P1) include (28b)
for the minimum and maximum speed of the UAV, (LABEL:pc2) for the maximum
acceleration, (LABEL:pc00) for the relationship between position, velocity,
and acceleration, (LABEL:pc3) for the initial and final points of the mission.
We also assume the elevation angle constraint in (LABEL:pc5), which guarantees
the LoS condition. While the constraints (LABEL:pc00), (LABEL:pc3), and
(LABEL:pc5) are convex, (28a), (28b), and (LABEL:pc2) are non-convex. To solve
this highly non-convex problem, we utilize the SCA method.

###  IV-B Successive Convex Approximation

In this subsection, we first approximate the non-convex objective function
(28a) to a concave-convex fractional function. Specifically, we approximate
the numerator to a concave function using the logarithmic approximation and by
introducing auxiliary variables, while the denominator is approximated to a
convex function by introducing an auxiliary variable. Then, (28b) and
(LABEL:pc2) are approximated to convex constraints. During the SCA method, we
utilize the optimized variables from the $p$-th iteration as known parameters
in the $(p+1)$-th iteration.

To convert (16) into a more tractable function, we apply logarithmic
approximation, which provides a tight lower bound of the original function
[45]. The parameters of the logarithmic approximation can be obtained during
the iterations of the SCA method. We can approximate the capacity function for
the $(p+1)$-th iteration utilizing the optimization results of the $p$-th
iteration. This approximation offers the lower bound as follows:

$\displaystyle\log_{2}\left(1+\Gamma\right)\geq\frac{1}{\log
2}(\nabla_{\text{L}}^{(p)}\log\Gamma+\delta_{\text{L}}^{(p)}),$ (29)

where the parameters can be obtained using $\Gamma_{\text{L}}^{(p)}$ for the
tightness at $\Gamma=\Gamma_{\text{L}}^{(p)}$ as

$\displaystyle\nabla_{\text{L}}^{(p)}=\frac{\Gamma_{\text{L}}^{(p)}}{1+\Gamma_{\text{L}}^{(p)}},\,\delta_{\text{L}}^{(p)}=\log(1+\Gamma_{\text{L}}^{(p)})-\frac{\Gamma_{\text{L}}^{(p)}}{1+\Gamma_{\text{L}}^{(p)}}\log\Gamma_{\text{L}}^{(p)}.$
(30)

We introduce a new variable $\Gamma_{\text{L}}^{(p)}$ to avoid confusion with
$\Gamma^{(p)}$, which can be interpreted as a random value of $\Gamma$ for the
given variables in the $p$-th iteration. From (16) and (29), the approximated
ergodic capacity during the iteration of the SCA method is derived as follows:

$\displaystyle C_{\mathbb{E}}=\frac{1}{2\log
2}(\nabla_{\text{L}}^{(p)}\mathbb{E}[\log\Gamma]+\delta_{\text{L}}^{(p)}).$
(31)

###### Lemma 1

Using $\Gamma=\frac{e{P}^{2}}{2\pi\sigma^{2}}$, we can express
$\mathbb{E}[\log\Gamma]$ in (31) as follows:

$\displaystyle\mathbb{E}[\log\Gamma]=$ $\displaystyle
c_{3}-2\sigma_{\text{B}}{z}-2\log{z}-\frac{\lambda_{1}+\lambda_{2}}{z^{2}\sigma_{\text{div}}^{2}},$
(32)

where
$c_{3}=\log\left(\frac{e{R}^{2}{P_{\text{T}}}^{2}a^{4}}{8\pi\sigma^{2}\sigma_{\text{div}}^{2}}\right)-4\sigma_{I}^{2}$
and $z$, $\lambda_{1}$, $\lambda_{2}$ are the functions of
$\hat{\mathbb{u}}[k]$ which are defined as in _Theorem_ 1.

_Proof:_ See Appendix B. $\blacksquare$

Algorithm 1 Algorithm for Solving (P1) Using the SCA and Dinkelbach Methods.

1:Let the set of original variables as
$\mathbb{V}=\\{\mathbb{s},\hat{\mathbb{u}},\mathbb{v},\mathbb{a},\bm{\kappa}\\}$,
and the set of auxiliary variables as
$\mathbb{A}=\\{S,\check{U},\check{V},\check{P},Q,R\\}$.

2:Let
$\mathbb{V}^{(p)}=\\{\mathbb{s}^{(p)},\hat{\mathbb{u}}^{(p)},\mathbb{v}^{(p)},\mathbb{a}^{(p)},\bm{\kappa}^{(p)}\\}$
and
$\mathbb{A}^{(p)}=\\{S^{(p)},\check{U}^{(p)},\check{V}^{(p)},\check{P}^{(p)},Q^{(p)},R^{(p)}\\}$.

3:For $k\in\mathbb{N},\,1\leq{k}\leq{N}$, initialize $\mathbb{s}^{(0)}[k]$,
$\hat{\mathbb{u}}^{(0)}[k]$, $\mathbb{v}^{(0)}[k]$, $\mathbb{a}^{(0)}[k]$,
$\bm{\kappa}^{(0)}[k]$, $S_{k}^{(0)}$, $\check{U}_{k}^{(0)}$,
$\check{V}_{k}^{(0)}$, and for $k\in\mathbb{N},\,1\leq{k}\leq{N-1}$,
initialize $\check{P}_{k}^{(0)}$, $Q_{k}^{(0)}$, $R_{k}^{(0)}$.

4:Set threshold $\tau_{v}$.

5:$p=0$.

6:repeat

7: $\forall k$, obtain $\Gamma_{\text{L},k}^{(p)}$,
$\nabla_{\text{L},k}^{(p)}$, and $\delta_{\text{L},k}^{(p)}$ through (34) and
(30).

8: Set $\lambda_{\text{min}}$, $\lambda_{\text{max}}$, and
$\lambda^{(0)}=(\lambda_{\text{min}}+\lambda_{\text{max}})/2$.

9: Set threshold $\tau_{f}$.

10: $i=0$.

11: repeat

12: Solve (P3), which is a convex problem, and find the optimal value, $F$.

13: if $F>0$ then

14: $\lambda^{(i+1)}=(\lambda^{(i)}+\lambda_{\text{min}})/2$.

15: else

16: $\lambda^{(i+1)}=(\lambda^{(i)}+\lambda_{\text{max}})/2$.

17: $i\leftarrow{i+1}$.

18: until $|F|\leq{\tau_{f}}$.

19: For $\mathbb{V}$ and $\mathbb{A}$, the set of the final solution,
$\mathbb{V}^{(p+1)}=\mathbb{V}$ and $\mathbb{A}^{(p+1)}=\mathbb{A}$.

20: $p\leftarrow{p+1}$.

21:until
$\left\lVert\mathbb{V}^{(p)}-\mathbb{V}^{(p-1)}\right\rVert+\left\lVert\mathbb{A}^{(p)}-\mathbb{A}^{(p-1)}\right\rVert<\tau_{v}$.

22:end

Now that we place the form of $\log_{2}\left(1+\Gamma\right)$ with the form of
$\log\Gamma$, we can directly substitute $(\ref{capacitytheo})$ into
$(\ref{finalergodic})$. We apply the above logarithmic approximation and
introduce auxiliary variables to transform the numerator and denominator of
the objective function into concave and convex functions, respectively. This
transformation allows us to apply the Dinkelbach method, assuming that the
numerator and denominator of (28a) are always positive, which is a natural
assumption. The superscript $(p)$ denotes the result parameter value obtained
from the $p$-th problem. Note that (P2) is the problem of the $(p+1)$-th
iteration of the SCA method, and the objective is to find the $(p+1)$-th
optimized variables.

$\displaystyle\text{(P2)}:$
$\displaystyle\begin{split}\max_{\mathbb{s}[k]}\quad&\frac{\sum_{k=1}^{N}\nabla_{\text{L},k}^{(p)}(c_{3}-c_{4}\left\lVert\mathbb{s}[k]\right\rVert-
c_{5}\check{U}_{k}^{2}-c_{6}\check{V}_{k})+\delta_{\text{L},k}^{(p)}}{\sum_{k=1}^{N-1}\check{P}_{k}+NP_{\text{T}}+{E_{\text{cost}}}/{\delta}}\end{split}$
(33a)
$\displaystyle\begin{split}\textrm{s.t.}\quad&v_{\text{min}}\leq\left\lVert\mathbb{v}[k]\right\rVert\leq{{v}_{\text{max}}},\quad
1\leq{k}\leq{N}\end{split}$ (33b)
$\displaystyle\begin{split}&\left\lVert\mathbb{a}[k]\right\rVert\leq{a_{\text{max}}},\quad
1\leq{k}\leq{N-1}\end{split}$ (33c)
$\displaystyle\begin{split}&(\ref{velocity}),(\ref{velocityout}),(\ref{acceleration}),(\ref{altitude})\end{split}$
(33d)
$\displaystyle\begin{split}&S_{k}^{2}\leq\left\lVert\mathbb{s}[k]\right\rVert^{2},\quad\forall{k}\end{split}$
(33e)
$\displaystyle\begin{split}&S_{k}\check{U}_{k}\geq\sqrt{{\hat{\mathbb{u}}[k]}^{T}\mathbb{D}\hat{\mathbb{u}}[k]},\quad\forall{k}\end{split}$
(33f)
$\displaystyle\begin{split}&\check{V}_{k}\geq\log(\left\lVert\mathbb{s}[k]\right\rVert),\quad\forall{k}\end{split}$
(33g)
$\displaystyle\begin{split}&\mathbb{s}[1]=\mathbb{l}_{o},\,\mathbb{s}[N]=\mathbb{l}_{d}\end{split}$
(33h)
$\displaystyle\begin{split}&\check{P}_{k}\geq{c_{1}}\left\lVert\mathbb{v}[k]\right\rVert^{3}+c_{2}Q_{k},\quad
1\leq{k}\leq{N-1}\end{split}$ (33i)
$\displaystyle\begin{split}&R_{k}^{2}\leq\left\lVert\mathbb{v}[k]\right\rVert^{2},\quad
1\leq{k}\leq{N-1}\end{split}$ (33j) $\displaystyle\begin{split}&Q_{k}R_{k}\geq
1+\frac{\left\lVert\mathbb{a}[k]\right\rVert^{2}}{\textsl{g}},\quad
1\leq{k}\leq{N-1}\end{split}$ (33k)
$\displaystyle\begin{split}&\frac{\pi}{4}\leq\eta\leq\frac{\pi}{2},\quad\forall{k}\end{split}$
(33l)

The parameters $\nabla_{\text{L},k}^{(p)}$ and $\delta_{\text{L},k}^{(p)}$ can
be obtained using (30). We can adjust $\Gamma_{\text{L},k}^{(p)}$
heuristically, by setting it to

$\displaystyle\Gamma_{\text{L},k}^{(p)}=\exp($ $\displaystyle
c_{3}-c_{4}\left\lVert\mathbb{s}^{(p)}[k]\right\rVert-
c_{5}\check{U}_{k}^{(p)2}-c_{6}\check{V}^{(p)}),$ (34)

which is obtained by
$\Gamma_{\text{L},k}^{(p)}=e^{\mathbb{E}[\log\Gamma_{k}^{(p)}]}$, where
$\Gamma_{k}^{(p)}$ is a random value of $\Gamma$ at time slot $k$ in the
$p$-th iteration. The symbols $c_{4}$, $c_{5}$, and $c_{6}$ are constants with
values of $c_{4}=2{\sigma_{\text{B}}}$, $c_{5}={1}/{\sigma_{\text{div}}^{2}}$,
and $c_{6}=2$.

TABLE III: Simulation parameters Parameter | Value
---|---
Altitude ($H$) | $600\,\,$\mathrm{m}$$
Transmit power ($P_{\text{T}}$) | $10\,\,$\mathrm{mW}$$
$P_{\text{T}}/\sigma$ | $30\,\,\mathrm{dB}$
Responsivity ($R$) | $0.5\,\,\mathrm{A}/\mathrm{W}$
Aperture diameter ($a$) | $20\,\,$\mathrm{cm}$$
Log-amplitude standard deviation ($\sigma_{I}$) | $0.3$
Visibility range ($V$) | $3\,\,$\mathrm{km}$$
Attenuation coefficient ($\sigma_{\text{B}}$) | $5.44\times{10^{-4}}\,\,\mathrm{dB}/\mathrm{km}$
Time interval between slots ($\delta$) | $0.2\,\,$\mathrm{s}$$
Minimum speed ($v_{\text{min}}$) | $3\,\,\mathrm{m}/\mathrm{s}$
Maximum speed ($v_{\text{max}}$) | $100\,\,\mathrm{m}/\mathrm{s}$
Maximum acceleration ($a_{\text{max}}$) | $5\,\,\mathrm{m}/\mathrm{s}^{2}$
Gravitational acceleration ($g$) | $9.8\,\,\mathrm{m}/\mathrm{s}^{2}$

In the formulation process of (P2), the total capacity
$\sum_{k=1}^{N}C_{\mathbb{E}}[k]$ in (P1) is approximated by (31) and (32).
Also, we introduce the auxiliary variables $\check{U}_{k}$ and
$\check{V}_{k}$, which are constrained by (LABEL:nc2) and (LABEL:nc3),
respectively. Here, (LABEL:nc2) is derived from
$\frac{\lambda_{1}+\lambda_{2}}{z^{2}}=\frac{{\hat{\mathbb{u}}[k]}^{T}\mathbb{D}\hat{\mathbb{u}}[k]}{\left\lVert\mathbb{s}[k]\right\rVert^{2}}$,
where
$\mathbb{D}=\mathbb{\text{diag}}(\sigma_{\beta}^{2}+\sigma_{\gamma}^{2},\sigma_{\gamma}^{2}+\sigma_{\alpha}^{2},\sigma_{\alpha}^{2}+\sigma_{\beta}^{2})$.
Due to the introduced variables, the numerator of (33a) becomes concave. To
make the denominator convex, we introduce $\check{P}_{k}$ which is constrained
by (LABEL:nc5). Still, (33b), (LABEL:nc1), (LABEL:nc2), (LABEL:nc3),
(LABEL:nc6), and (LABEL:nc7) are non-convex constraints. Each should be
replaced with the approximated convex constraints to apply the SCA method.

(a)  (b)  (c)

Figure 4: The energy-efficient trajectories for the mission where the initial
and final points are different (moving mission). We vary the parameter values
by (a) $\sigma_{\text{div}}=1.5$ mrad, (b) $\sigma_{\text{div}}=2.0$ mrad, and
(c) $\sigma_{\text{div}}=2.5$ mrad to compare the impact of the pointing error
on the trajectory.

(a)  (b)  (c)

Figure 5: The energy-efficient trajectories for the mission where the initial
and final points are the same (hovering mission). The parameter is set to
$\sigma_{\text{div}}=1.5$ mrad, and each of the subfigures varies by the
dominant jittering direction.

The constraints (33b), (LABEL:nc1), (LABEL:nc6), and (LABEL:nc7) can be
approximated by the following inequalities [33]. The non-convex constraint
(33b) is restricted to

$\displaystyle\left\lVert\mathbb{v}[k]\right\rVert\leq{{v}_{\text{max}}},\quad
1\leq{k}\leq{N-1},$ (35) $\displaystyle
2{\mathbb{v}^{(p)}[k]}^{T}\mathbb{v}[k]-\left\lVert\mathbb{v}^{(p)}[k]\right\rVert^{2}\geq{v_{\text{min}}^{2}},\quad
1\leq{k}\leq{N-1},$ (36)

which is tight for $\mathbb{v}[k]=\mathbb{v}^{(p)}[k]$. The non-convex
constraint (LABEL:nc1) is restricted to

$\displaystyle
2{\mathbb{s}^{(p)}[k]}^{T}\mathbb{s}[k]-\left\lVert\mathbb{s}^{(p)}[k]\right\rVert^{2}\geq{S_{k}^{2}},\quad\forall{k},$
(37)

which is tight for $\mathbb{s}[k]=\mathbb{s}^{(p)}[k]$. The non-convex
constraint (LABEL:nc6) is restricted to

$\displaystyle
2{\mathbb{v}^{(p)}[k]}^{T}\mathbb{v}[k]-\left\lVert\mathbb{v}^{(p)}[k]\right\rVert^{2}\geq{R_{k}^{2}},\quad
1\leq{k}\leq{N-1},$ (38)

which is tight for $\mathbb{v}[k]=\mathbb{v}^{(p)}[k]$. The non-convex
constraint (LABEL:nc7) is restricted to

$\displaystyle\left\lVert
Q_{k}-R_{k},Q_{k}^{(p)}+R_{k}^{(p)},L_{k}-1,2,\frac{2\mathbb{a}[k]}{\textsl{g}}\right\rVert$
$\displaystyle\leq{L_{k}+1},$ (39) $\displaystyle{\quad}1\leq{k}$
$\displaystyle\leq{N-1},$

where $L_{k}={(Q_{k}^{(p)}+R_{k}^{(p)})(Q_{k}+R_{k})}/{2}$. The approximated
constraint (39) is tight for $Q_{k}=Q_{k}^{(p)}$ and $R_{k}=R_{k}^{(p)}$. The
non-convex constraint (LABEL:nc2) is approximated as the following theorem.

###### Theorem 2

The non-convex constraint (LABEL:nc2) can be approximated into the convex
constraint as

$\displaystyle S_{k}^{(p)}$ $\displaystyle
U_{k}+U_{k}^{(p)}S_{k}-S_{k}^{(p)}U_{k}^{(p)}\geq\left({\hat{\mathbb{u}}^{(p)}[k]}^{T}\mathbb{D}\hat{\mathbb{u}}^{(p)}[k]\right)^{\frac{1}{2}}$
(40)
$\displaystyle+\left({\hat{\mathbb{u}}^{(p)}[k]}^{T}\mathbb{D}\hat{\mathbb{u}}^{(p)}[k]\right)^{-\frac{1}{2}}\hat{\mathbb{u}}^{(p)}[k]\mathbb{D}\Delta\hat{\mathbb{u}}[k],\quad\forall{k},$

where
$\mathbb{D}=\mathbb{\text{diag}}(\sigma_{\beta}^{2}+\sigma_{\gamma}^{2},\sigma_{\gamma}^{2}+\sigma_{\alpha}^{2},\sigma_{\alpha}^{2}+\sigma_{\beta}^{2})$.
The symbols $\hat{\mathbb{u}}^{(p)}[k]$ and $\Delta\hat{\mathbb{u}}[k]$ are
expressed as (41) and (42), respectively, where

$\displaystyle\hat{\mathbb{u}}^{(p)}[k]=\begin{bmatrix}s_{x}^{(p)}\cos\theta^{(p)}+s_{y}^{(p)}\sin\theta^{(p)}\\\\[2.0pt]
-s_{x}^{(p)}\cos\phi^{(p)}\sin\theta^{(p)}+s_{y}^{(p)}\cos\phi^{(p)}\cos\theta^{(p)}+s_{z}^{(p)}\sin\phi^{(p)}\\\\[2.0pt]
-s_{x}^{(p)}\sin\phi^{(p)}\sin\theta^{(p)}+s_{y}^{(p)}\sin\phi^{(p)}\cos\theta^{(p)}+s_{z}^{(p)}\cos\phi^{(p)}\end{bmatrix}$
(41)

$\displaystyle\Delta\hat{\mathbb{u}}[k]=\begin{bmatrix}s_{x}^{(p)}\Delta\cos\theta+\cos\theta^{(p)}\Delta{s_{x}}+s_{y}^{(p)}\Delta\sin\theta+\sin\theta^{(p)}\Delta{s_{y}}\\\\[3.0pt]
-s_{x}^{(p)}\cos\phi^{(p)}\Delta\sin\theta-
s_{x}^{(p)}\sin\theta^{(p)}\Delta\cos\phi-\cos\phi^{(p)}\sin\theta^{(p)}\Delta{s_{x}}+s_{y}^{(p)}\cos\phi^{(p)}\Delta\cos\theta\\\
+s_{y}^{(p)}\cos\theta^{(p)}\Delta\cos\phi+\cos\phi^{(p)}\cos\theta^{(p)}\Delta{s_{y}}+s_{z}^{(p)}\Delta\sin\phi+\sin\phi^{(p)}\Delta{s_{z}}\\\\[3.0pt]
-s_{x}^{(p)}\sin\phi^{(p)}\Delta\sin\theta-
s_{x}^{(p)}\sin\theta^{(p)}\Delta\sin\phi-\sin\phi^{(p)}\sin\theta^{(p)}\Delta{s_{x}}+s_{y}^{(p)}\sin\phi^{(p)}\Delta\cos\theta\\\
+s_{y}^{(p)}\cos\theta^{(p)}\Delta\sin\phi+\sin\phi^{(p)}\cos\theta^{(p)}\Delta{s_{y}}+s_{z}^{(p)}\Delta\cos\phi+\cos\phi^{(p)}\Delta{s_{z}}\end{bmatrix}$
(42)

$\displaystyle\Delta\sin\phi=(a_{0}^{2}+1)^{-3/2}\left\lVert\mathbb{v}^{(p)}[k]\right\rVert^{-3}\\{(v_{y}^{(p)2}a_{y}^{(p)}-v_{x}^{(p)}v_{y}^{(p)}a_{x}^{(p)})\Delta{v_{x}}+(v_{x}^{(p)2}a_{x}^{(p)}-v_{x}^{(p)}v_{y}^{(p)}a_{y}^{(p)})\Delta{v_{x}}$
(43)
$\displaystyle+v_{x}^{(p)}(v_{x}^{(p)2}+v_{y}^{(p)2})\Delta{a_{y}}-v_{y}^{(p)}(v_{x}^{(p)2}+v_{y}^{(p)2})\Delta{a_{x}}\\},$
$\displaystyle\Delta\cos\phi=a_{0}(a_{0}^{2}+1)^{-3/2}\left\lVert\mathbb{v}^{(p)}[k]\right\rVert^{-3}\\{(v_{y}^{(p)2}a_{y}^{(p)}-v_{x}^{(p)}v_{y}^{(p)}a_{x}^{(p)})\Delta{v_{x}}+(v_{x}^{(p)2}a_{x}^{(p)}-v_{x}^{(p)}v_{y}^{(p)}a_{y}^{(p)})\Delta{v_{x}}$
$\displaystyle+v_{x}^{(p)}(v_{x}^{(p)2}+v_{y}^{(p)2})\Delta{a_{y}}-v_{y}^{(p)}(v_{x}^{(p)2}+v_{y}^{(p)2})\Delta{a_{x}}\\}$

$\displaystyle\cos\theta^{(p)}=\frac{v_{x}^{(p)}}{\left\lVert\mathbb{v}^{(p)}[k]\right\rVert},\,\,\sin\theta^{(p)}=\frac{v_{y}^{(p)}}{\left\lVert\mathbb{v}^{(p)}[k]\right\rVert}$
(44)
$\displaystyle\Delta\sin\theta=\left\lVert\mathbb{v}^{(p)}[k]\right\rVert^{-3}(v_{y}^{(p)2}\Delta{v_{x}}-v_{x}^{(p)}v_{y}^{(p)}\Delta{v_{y}}),$
(45)
$\displaystyle\Delta\cos\theta=\left\lVert\mathbb{v}^{(p)}[k]\right\rVert^{-3}(v_{x}^{(p)2}\Delta{v_{y}}-v_{x}^{(p)}v_{y}^{(p)}\Delta{v_{x}}),$
$\displaystyle\sin\phi^{(p)}=\frac{a_{0}}{\sqrt{a_{0}^{2}+1}},\,\,\cos\phi^{(p)}=\frac{1}{\sqrt{a_{0}^{2}+1}},$
(46)

and (43). The symbol $a_{0}$ is expressed as follows:

$\displaystyle
a_{0}=\frac{v_{y}^{(p)}a_{x}^{(p)}-v_{x}^{(p)}a_{y}^{(p)}}{\left\lVert\mathbb{v}[k]\right\rVert\textsl{g}}.$
(47)

In (42), $\Delta\hat{\mathbb{u}}[k]$ is approximated into the linear
combination of
${\Delta}s_{x},{\Delta}s_{y},{\Delta}v_{x},{\Delta}v_{y},{\Delta}a_{x},{\Delta}a_{y}$,
where ${\Delta}s_{x}=s_{x}-s_{x}^{(p)}$, ${\Delta}s_{y}=s_{y}-s_{y}^{(p)}$,
${\Delta}v_{x}=v_{x}-v_{x}^{(p)}$, ${\Delta}v_{y}=v_{y}-v_{y}^{(p)}$,
${\Delta}a_{x}=a_{x}-a_{x}^{(p)}$, ${\Delta}a_{y}=a_{y}-a_{y}^{(p)}$. The
approximated constraint (40) is also tight for the variable values of the
$p$-th iteration.

_Proof:_ See Appendix C. $\blacksquare$

###### Lemma 2

The constraint (LABEL:nc3) is convex under (LABEL:elev2).

_Proof:_ See Appendix D. $\blacksquare$

In addition, although (LABEL:nc3) is a non-convex constraint, it is not
necessary to transform it into a convex constraint according to _Lemma_ 2.

###  IV-C Dinkelbach Method

In Section IV-B, the optimization problem with the concave-convex fractional
objective function is formulated as (P2), and the non-convex constraints are
approximated to (35)-(40). We formulate the final problem to iteratively solve
(P2) by the Dinkelbach method [46, 47, 48]. The total capacity and power
consumption are denoted as $C_{\text{tot}}$ and $P_{\text{tot}}$,
respectively, and can be represented as

$\displaystyle{C}_{\text{tot}}$
$\displaystyle=\sum_{k=1}^{N}\nabla_{\text{L},k}^{(p)}(c_{3}-c_{4}\left\lVert\mathbb{s}[k]\right\rVert-
c_{5}\check{U}_{k}^{2}-c_{6}\check{V}_{k})+\delta_{\text{L},k}^{(p)},$ (48)
$\displaystyle{P}_{\text{tot}}$
$\displaystyle=\sum_{k=1}^{N-1}\check{P}_{k}+NP_{\text{T}}+{E_{\text{cost}}}/{\delta},$

derived from (33a). Then, we define (P3) as follows:

$\displaystyle\text{(P3)}:$
$\displaystyle\begin{split}\min_{\mathbb{s}[k]}\quad&{F}=-{C}_{\text{tot}}+\lambda{P}_{\text{tot}}\end{split}$
(49a)
$\displaystyle\begin{split}\textrm{s.t.}\quad&(\ref{convexc}),(\ref{convexd}),(\ref{nc2})-(\ref{nc5}),(\ref{elev2}),(\ref{ca9})-(\ref{ca6})\end{split}$
(49b)

Algorithm 1 provides the suboptimal trajectory of the fixed-wing UAV that
improves the energy efficiency. First, the original and auxiliary variables
are initialized to formulate the approximated problem and conduct the SCA
method. The feasible set of (P2) can be approximated into convex. However,
(33a) is still non-convex. Therefore, we apply the Dinkelbach method to
iteratively solve (P3) and obtain the solution for the $p$-th iteration of the
SCA method.

###  IV-D Convergence and Complexity Analysis

The SCA method guarantees convergence only when the feasible set of the
approximated problem is both convex and included in the feasible set of the
original problem [49]. However, (40) is convex but not included in the
solution set of (LABEL:nc2). Nevertheless, the right-hand side (RHS) of
(LABEL:nc2) is a smooth nonlinear function (compared to other constraints)
with respect to $\mathbb{s}[k]$, $\mathbb{v}[k]$, and $\mathbb{a}[k]$, and it
is non-negative, meaning that the equality of (LABEL:nc2) always holds since
$\check{U}_{k}$ must be minimized. As a result, although there is no general
convex restriction for (LABEL:nc2), the proposed approximation into the convex
constraint (40) closely approximates the value of $\check{U}_{k}^{2}$ in (33a)
and does not compromise the physical feasibility. In summary, although
Algorithm 1 may not result in a converged solution, it only results in a
slight oscillation of the objective value without resulting in infeasible
solutions. Moreover, since all constraints in (P3), including (40), are tight
for the variable values of the $p$-th iteration, the algorithm converges in
most cases.

The computational complexity of our proposed algorithm can be evaluated based
on the complexity of solving the convex optimization problem (P3). The number
of iterations required can vary depending on factors such as the type of
objective and constraint functions, threshold values, and learning rates. For
a convex optimization problem, the interior-point method provides a
generalized computational complexity of
$\mathcal{O}(\max\\{n_{\text{v}}^{3},n_{\text{v}}^{2}n_{\text{c}},F\\})$,
where $n_{v}$ represents the number of variables, $n_{c}$ represents the
number of constraints, and $F$ represents the cost of evaluating the first and
second derivatives of the objective function and constraints [39, 50]. Despite
the complicated formulation of the constraints, particularly for (40), $F$
follows $\mathcal{O}(n_{\text{v}})$, suggesting it can generally be ignored.
The optimization variables are the $x$ and $y$ components of the vectors
$\mathbb{s}[k]$, so $n_{\text{v}}=2N$. The number of constraints related to
$(\ref{convexc})$, $(\ref{convexd})$, $(\ref{nc2})-(\ref{nc5})$,
$(\ref{elev2})$, and $(\ref{ca9})-(\ref{ca6})$ are $N-1$, $2N-1$, $N$, $N$,
$4$, $N-1$, $N$, $N-1$, $N-1$, $N$, $N-1$, $N-1$, and $N$ respectively.
Consequently, the total number of constraints is $n_{\text{c}}=13N-3$.
Therefore, each iteration of our optimization process has a polynomial
computational complexity of $\mathcal{O}(N^{3})$.

##  V Numerical Results

Figure 6: Convergence of the SCA method (moving mission). Figure 7:
Convergence of the SCA method (hovering mission). Figure 8: Spectral
efficiency of the trajectories over time (hovering mission). Figure 9: Power
consumption of the trajectories over time (hovering mission).

In this section, we present numerical results that validate the significance
of the proposed pointing error model and the algorithm for designing an
energy-efficient trajectory. The simulation parameters are specified in Table
III. The altitude of the UAV is set to $600$ m and $400$ m, and the transmit
power is $10$ mW with $P_{\text{T}}/\sigma=30$ dB [23, 38]. The flight time is
$20$ s and $80$ s for two different flight missions, and is discretized into
time slots with a $0.2$ s interval. The minimum and maximum speeds of the UAV
are $3$ and $100$ $\mathrm{m/s}$, respectively, and the maximum acceleration
is $5$ $\mathrm{m/s^{2}}$. The acceleration constraint also limits the bank
angle during flight to $|{\phi}|\leq{27.03}\cdots^{\circ}$ [30]. To account
for the fixed cost of the flight mission, we introduce the parameter
$E_{\text{cost}}$. Increasing $E_{\text{cost}}$ to a larger value induces more
variations in the trajectory. We set the value to $10^{5}$J and $4\times
10^{5}$J for two different missions, respectively. We assume that the GS is
located at $[0,0,0]$ m in the GS-centered coordinate system.

###  V-A Comparison Between Different Jitter Characteristics

We begin by comparing the energy-efficient trajectory obtained from Algorithm
1 for different types of UAV jitter. Fig. 4 and Fig. 5 illustrate the
trajectory for dominant roll jitter, pitch jitter, yaw jitter, and symmetric
jitter which are represented by
$[\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma}]=[1,0.1,0.1],$ $[0.1,1,0.1],$
$[0.1,0.1,1],$ $[0.583,0.583,0.583]$ $\mathrm{mrad}$, respectively. In all
cases, the total magnitude of jittering remains constant at
$\sqrt{\text{Tr}(\Sigma)}=1.01$ $\mathrm{mrad}$. The covariance between the
jittering directions is assumed to be zero
($\rho_{\alpha,\beta}=\rho_{\beta,\gamma}=\rho_{\gamma,\alpha}=0$). In Fig. 4,
the trajectories of the moving mission are displayed, where the initial and
final points of the mission are set at $[54,200,600]$ m and $[450,200,600]$ m,
respectively. For the moving mission, we initialize the parameters with a
uniform linear motion from the initial to the final point. Fig. 5, the
hovering mission, sets the initial and final points to $[0,0,H]$ m, where
$H=400,\,600$ m. For the hovering mission, we initialize the trajectory with a
clockwise circular path centered at $[0,-60,H]$ m. The flight duration is $20$
s and $80$ s for the two different missions, respectively.

Fig. 4a exhibits largest difference between trajectories, since $\theta_{p}$
has a more significant impact on performance when $\sigma_{\text{div}}$ is
small, as expressed in (15). For the symmetric jitter case, the UAV aims to
approach the GS while reducing energy consumption through smooth flight. A
high amount of pitch jitter significantly affects the energy-efficient
trajectory. The UAV should keep the $y^{\prime}$ axis parallel to the link
direction to avoid the pitch jitter. Consequently, at the beginning of the
flight, the UAV turns in a different direction compared to the other cases.
The UAV also chooses to fly further later in the flight to utilize the bank
angle caused by acceleration. When the roll jitter dominates, the UAV tends to
align the $x^{\prime}$ axis with the line connecting the GS and the final
point. Finally, for the yaw jitter case, the UAV accelerates at the end of the
flight to minimize the pointing error. Similar differences are also shown in
Fig. 4b and Fig. 4c. However, as $\sigma_{\text{div}}$ increases, trajectories
optimized for asymmetric jitter characteristics begin to converge towards
those optimized for symmetric jitter characteristics. We also present the
simulation results of the hovering trajectory of the UAVs in Fig. 5. We
compare the energy-efficient trajectory for different types of jittering
features when the initial and final points are the same.

Figure 10: Average spectral efficiency of trajectories derived using different
jitter models.

In Fig. 6 and Fig. 7, we present the convergence of the SCA method for the two
different missions: the moving mission and the hovering mission. For each
iteration, we calculate the total energy efficiency, represented as
$C_{\text{tot}}/P_{\text{tot}}$. Iteration stops when convergence is detected
by the algorithm. Generally, the proposed algorithm converges effectively;
however, slight oscillations are observed in some cases with dominant yaw
jitter. This occurs because the feasible set of (40) is not included in the
original feasible set. Nevertheless, as discussed in Section IV-C, the
iterative solutions do not violate the physical feasibility defined in the
system model, even during the oscillation.

The spectral efficiency presented in Fig. 8 over the trajectories of the
hovering mission illustrates the strong impact of the posture of the UAV on
the pointing error. Especially for the pitch jitter case, the corresponding
trajectory depicted in Fig. 5b demonstrates that the $y^{\prime}$ axis of the
UAV is aligned with the ground station (GS) due to the acceleration during the
large circular flight, except for the beginning and end of the flight. As a
result, although the UAV is closest to the GS, the spectral efficiency is
lower at both ends of the flight. Finally, in Fig. 9, we display the value of
$P_{\text{F}}(\mathbb{v}[k],\mathbb{a}[k])$ for the moving mission. In our
results, the UAV generally has a higher speed and acceleration for the pitch
jitter case. However, the highest power consumption was found in the yaw
jitter case, which is in $10$ s and $70$ s regions with high acceleration and
relatively low speed.

Figure 11: Average power consumption of trajectories derived using different
jitter models. Figure 12: Relative energy efficiency of trajectories under
varying transmit power and altitude conditions.

###  V-B Comparison with Conventional Jitter Models

To illustrate the significance of integrating generalized jitter
characteristics into trajectory design, we evaluate the performance of our
trajectory optimization algorithm using various models of pointing error. In
this simulation, the actual UAV jitter is characterized by
$[\sigma_{\alpha},\sigma_{\beta},\sigma_{\gamma}]=[0.1,1,0.1]$ mrad. The 3D
jitter model proposed in this paper fully utilizes the knowledge of
$\sigma_{\alpha}$, $\sigma_{\beta}$, and $\sigma_{\gamma}$. However,
conventional models do not account for all three DoFs, thus missing
information about the actual jittering characteristics. For the two DoF jitter
model, the jitter characteristics are best represented by
$\sigma_{\alpha}=\sigma_{\beta}=0.711$ mrad and $\sigma_{\gamma}=0.1$ mrad
[17]. For a DoF of 1, implying that $\mathbb{\Sigma}$ is a scaled identity
matrix, the jitter is best represented by
$\sigma_{\alpha}=\sigma_{\beta}=\sigma_{\gamma}=0.583$ mrad. We implement our
algorithm to derive the trajectories optimized to pointing error models with
one, two, and three DoFs, and then compare their performances under the actual
three DoF jitter model. Additionally, a clockwise circular trajectory centered
at $[0,-60,H]$ m is used in our simulations as a baseline method. It is
important to note that this study is the first to incorporate a jitter model
with three DoFs for trajectory design of fixed-wing aircraft.

Our algorithm is designed to maximize energy efficiency, defined as the
achievable rate divided by the energy consumption. In Fig. 10, we compare the
average achievable rate of optimized trajectories under varying transmit power
levels and different DoF of jitter characteristics. The results indicate that
the average spectral efficiency is primarily dependent on the transmit power.
Only a slight performance improvement is observed from 1D to 3D jitter models
in terms of spectral efficiency. This suggests that the algorithm prioritizes
reducing energy consumption over significantly enhancing spectral efficiency.
As illustrated in Fig. 11, the reduction in power consumption is more
pronounced, particularly in regions of low transmit power where the algorithm
effectively exploits detailed jitter characteristics to minimize power usage.
Although using a circular trajectory without optimization might seem
intuitive, it consistently shows the lowest performance on both metrics.
Furthermore, at lower altitudes, the algorithm tends to focus more on power
saving within the power-rate tradeoff. This emphasis on energy conservation is
due to the higher average received power at lower altitudes, which lowers the
gradient of the rate with respect to changes in trajectory.

In Fig. 12, we evaluate different jitter models from the perspective of energy
efficiency. Across most power and altitude conditions, incorporating multiple
DoFs into the jitter model results in a more effectively optimized trajectory
design. We set the energy efficiency of the 3D jitter case as the benchmark at
100$\%$ and present the relative values of energy efficiency under different
jitter models. Compared to the conventional 1D symmetric jitter model,
incorporating a 3D jitter model yields approximately a 10$\%$ performance
improvement, especially noticeable in the low transmit power region.

##  VI Conclusion

In this paper, we first proposed a generalized pointing error model for fixed-
wing UAVs in FSO communications, considering the 3D jitter of UAVs. Based on
this, we developed a trajectory optimization algorithm to maximize energy
efficiency while meeting various constraints such as speed, acceleration, and
elevation angles. To solve the complex non-convex problem, we employed the
iterative methods and derive an approximation for the constraint associated
with the pointing error. Our results emphasize the importance of incorporating
the UAV-specific 3D jittering model for accurate performance measurement and
system optimization in NTN scenarios. In our simulation, strategically
adjusting the flight trajectory significantly reduces the impact of UAV jitter
on communication performance, achieving up to a 11.8$\%$ improvement in energy
efficiency. This work provides valuable insights into FSO communications
performed by fixed-wing UAVs in aerial networks and establishes a basis for
further research investigating its applicability in various communication
scenarios.

##  Appendix A Proof of Theorem 1

First, we apply small angle approximation ($\theta_{p}\approx\sin\theta_{p}$)
to (24). Then, we obtain

$\displaystyle\theta_{p}^{2}$
$\displaystyle=\mathbb{x}^{T}\mathbb{A}\mathbb{x},$ (50)

with $\mathbb{A}$ defined as (26). The random vector $\mathbb{x}$ can be
substituted by
$\mathbb{x}=\mathbb{\Sigma}^{1/2}(\mathbb{\Sigma}^{-1/2}\mathbb{x})$, where
$\mathbb{\Sigma}^{-1/2}\mathbb{x}$ is a random vector with iid unit Gaussian
distributed elements. We conduct eigenvalue decomposition as
$\mathbb{\Sigma}^{1/2}\mathbb{A}\mathbb{\Sigma}^{1/2}=\mathbb{P}^{T}\mathbb{\Lambda}\mathbb{P}$,
where
$\mathbb{\Lambda}=\mathbb{\text{diag}}(\lambda_{1},\lambda_{2},\lambda_{3})$.
Since ${rank}(\mathbb{A})=2$ and
$\mathbb{\Sigma}^{1/2}\mathbb{A}\mathbb{\Sigma}^{1/2}$ is positive-
semidefinite, we can assume that
$\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}=0$. So far, we obtain

$\displaystyle\theta_{p}^{2}$
$\displaystyle=(\mathbb{\Sigma}^{-1/2}\mathbb{x})^{T}\mathbb{P}^{T}{\mathbb{\Lambda}}\mathbb{P}(\mathbb{\Sigma}^{-1/2}\mathbb{x}).$
(51)

Then, $\mathbb{P}\mathbb{\Sigma}^{-1/2}\mathbb{x}$ can be simplified into
$\mathbb{P}\mathbb{\Sigma}^{-1/2}\mathbb{x}=\mathbb{z}^{\prime}$ where
$\mathbb{z}^{\prime}=\left[\zeta_{1},\zeta_{2},\zeta_{3}\right]^{T}$ is an iid
Gaussian random vector. Finally, from

$\displaystyle\theta_{p}^{2}=\mathbb{z}^{\prime
T}\mathbb{\Lambda}\mathbb{z}^{\prime},$ (52)

we obtain (25) by the definition of the Hoyt distribution [51].

##  Appendix B Proof of Lemma 1

The term $\mathbb{E}[\log\Gamma]$ can be divided into the sum of each of the
channel terms. Since $\Gamma=\frac{e{P}^{2}}{2\pi\sigma^{2}}$, it is expressed
as

$\displaystyle\mathbb{E}[\log\Gamma]=$
$\displaystyle\log\left(\frac{e{h_{\ell}}^{2}{R}^{2}{P_{\text{T}}}^{2}}{2\pi\sigma^{2}}\right)$
(53)
$\displaystyle+2\mathbb{E}[\log{h_{a}}]+2\log{h_{\ell}}+2\mathbb{E}[\log{h_{p}}],$

where $h_{a}$ and $h_{p}$ are the random variables. We obtain

$\displaystyle\mathbb{E}[\log{h_{a}}]=-2\sigma_{I}^{2},$ (54)

from (11) and the property of the log-normal distribution, and

$\displaystyle\log{h_{\ell}}=-\sigma_{\text{B}}z,$ (55)

from (12). From (15) and (27),

$\displaystyle\mathbb{E}[\log{h_{p}}]=$
$\displaystyle\log\left({\frac{a^{2}}{2\sigma_{\text{div}}}}\right)-\log{z}-\frac{\lambda_{1}+\lambda_{2}}{2{z^{2}}\sigma_{\text{div}}^{2}},$
(56)

By substituting (54), (55), (56) into (53), we obtain (32).

##  Appendix C Proof of Theorem 2

We apply first order Taylor approximation to the both sides of (LABEL:nc2).
The LHS of the (40) is achieved by substituting
$S_{k}=S_{k}^{(p)}+(S_{k}-S_{k}^{(p)})$ and
$U_{k}=U_{k}^{(p)}+(U_{k}-U_{k}^{(p)})$. The RHS is a function of
$\mathbb{s}[k]$, $\mathbb{v}[k]$, and $\mathbb{a}[k]$ since
$\hat{\mathbb{u}}[k]$ can be represented as

$\displaystyle\hat{\mathbb{u}}[k]=\begin{bmatrix}s_{x}\cos\theta+s_{y}\sin\theta\\\
-s_{x}\cos\phi\sin\theta+s_{y}\cos\phi\cos\theta+s_{z}\sin\phi\\\
-s_{x}\sin\phi\sin\theta+s_{y}\sin\phi\cos\theta+s_{z}\cos\phi\end{bmatrix},$
(57)

and $\theta$, $\phi$ are the function of $\mathbb{v}[k]$ and $\mathbb{a}[k]$.
Through the chain rule, we obtain the first order Taylor approximation
$\hat{\mathbb{u}}[k]\approx\hat{\mathbb{u}}^{(p)}[k]+\Delta\hat{\mathbb{u}}[k]$,
which is a function of
${\Delta}s_{x},{\Delta}s_{y},{\Delta}v_{x},{\Delta}v_{y},{\Delta}a_{x},{\Delta}a_{y}$,
where (41)-(43). Also through the chain rule, the linear approximation below
is possible:

$\displaystyle\sqrt{{\hat{\mathbb{u}}}^{T}\mathbb{D}\hat{\mathbb{u}}}\approx\left({\hat{\mathbb{u}}^{(p)}[k]}^{T}\mathbb{D}\hat{\mathbb{u}}^{(p)}[k]\right)^{\frac{1}{2}}$
(58)
$\displaystyle\quad+\left({\hat{\mathbb{u}}^{(p)}[k]}^{T}\mathbb{D}\hat{\mathbb{u}}^{(p)}[k]\right)^{-\frac{1}{2}}\hat{\mathbb{u}}^{(p)}[k]\mathbb{D}\Delta\hat{\mathbb{u}}[k].$

##  Appendix D Proof of Lemma 2

Let $\mathcal{F}$ be a function defined as follows:

$\displaystyle\mathcal{F}(\mathbb{x})=\log(\left\lVert\mathbb{x}\right\rVert+H^{2})$
(59)

The Hessian of $\mathcal{F}$ is

$\displaystyle\mathbb{H}_{\mathcal{F}}(\mathbb{x})=\frac{1}{2(\mathbb{x}^{T}\mathbb{x}+H^{2})^{2}}\left[(\mathbb{x}^{T}\mathbb{x}+H^{2})\mathbb{I}-2\mathbb{x}\mathbb{x}^{T}\right],$
(60)

where $\mathbb{I}$ is the identity matrix. Let us prove that
$\pi/4\leq\arctan(H/\sqrt{s_{x}^{2}+s_{y}^{2}})\leq\pi/2$ is a sufficient
condition to the convexity of $\log({\sqrt{s_{x}^{2}+s_{y}^{2}+H^{2}}})$. Let
$\mathbb{s}^{\prime}=[s_{x},s_{y}]^{T}$. The condition can be equivalently
represented as $\mathbb{s}^{T}\mathbb{s}\leq{H^{2}}$. The eigenvalues of
$\mathbb{H}_{\mathcal{F}}(\mathbb{s^{\prime}})$ are as follows:

$\displaystyle\lambda_{\mathcal{F},1}=\frac{1}{2({\mathbb{s^{\prime}}}^{T}\mathbb{s^{\prime}}+H^{2})},\,\lambda_{\mathcal{F},2}=\frac{-\mathbb{s^{\prime}}^{T}\mathbb{s^{\prime}}+H^{2}}{2(\mathbb{s^{\prime}}^{T}\mathbb{s^{\prime}}+H^{2})^{2}}.$
(61)

Under the condition of $\mathbb{s}^{T}\mathbb{s}\leq{H^{2}}$,
$\mathbb{H}_{\mathcal{F}}(\mathbb{s^{\prime}})$ is a positive semidefinite
matrix. Therefore, $\frac{1}{2}\mathcal{F}(\mathbb{s})$ is convex, which
guarantees convexity of the constraint (LABEL:nc3).

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|  Hyung-Joo Moon (Student Member, IEEE) received the B.S. degree from the
School of Integrated Technology, Yonsei University, South Korea, in 2019,
where he is currently pursuing the Ph.D. degree. His research interests
include performance analysis and system optimization for emerging technologies
in 6G non-terrestrial network (NTN).
---|---
|  Chan-Byoung Chae (Fellow, IEEE) received the Ph.D. degree in electrical and
computer engineering from The University of Texas at Austin (UT), USA in 2008.
Prior to joining UT, he was a Research Engineer at the Telecommunications
Research and Development Center, Samsung Electronics, Suwon, South Korea, from
2001 to 2005. He is currently an Underwood Distinguished Professor with the
School of Integrated Technology, Yonsei University, South Korea. Before
joining Yonsei University, he was with Bell Labs, Alcatel-Lucent, Murray Hill,
NJ, USA, from 2009 to 2011, as a Member of Technical Staff, and Harvard
University, Cambridge, MA, USA, from 2008 to 2009, as a Post-Doctoral Research
Fellow. Dr. Chae was a recipient/co-recipient of the Ministry of Education
Award in 2024, the KICS Haedong Scholar Award in 2023, the CES Innovation
Award in 2023, the IEEE ICC Best Demo Award in 2022, the IEEE WCNC Best Demo
Award in 2020, the Best Young Engineer Award from the National Academy of
Engineering of Korea (NAEK) in 2019, the IEEE DySPAN Best Demo Award in 2018,
the IEEE/KICS Journal of Communications and Networks Best Paper Award in 2018,
the IEEE INFOCOM Best Demo Award in 2015, the IEIE/IEEE Joint Award for Young
IT Engineer of the Year in 2014, the KICS Haedong Young Scholar Award in 2013,
the IEEE Signal Processing Magazine Best Paper Award in 2013, the IEEE ComSoc
AP Outstanding Young Researcher Award in 2012, and the IEEE VTS Dan. E. Noble
Fellowship Award in 2008. Dr. Chae has held several editorial positions,
including Editor-in-Chief of the IEEE Transactions on Molecular, Biological,
and Multi-Scale Communications, Senior Editor of the IEEE Wireless
Communications Letters, and Editor for the IEEE Communications Magazine, IEEE
Transactions on Wireless Communications, and IEEE Wireless Communications
Letters. He was an IEEE ComSoc Distinguished Lecturer from 2020 to 2023. He is
an IEEE Fellow and NAEK Fellow.
---|---
|  (Kit) Kai-Kit Wong (Fellow, IEEE) received the BEng, the MPhil, and the PhD
degrees, all in electrical and electronic engineering, from the Hong Kong
University of Science and Technology, Hong Kong, in 1996, 1998, and 2001,
respectively. After graduation, he took up academic and research positions at
the University of Hong Kong, Lucent Technologies, Bell-Labs, Holmdel, the
Smart Antennas Research Group of Stanford University, and the University of
Hull, UK. He is Chair in Wireless Communications at the Department of
Electronic and Electrical Engineering, University College London, UK. His
current research centers around 5G and beyond mobile communications. He is a
co-recipient of the 2013 IEEE Signal Processing Letters Best Paper Award and
the 2000 IEEE VTS Japan Chapter Award at the IEEE Vehicular Technology
Conference in Japan in 2000, and a few other international best paper awards.
He is Fellow of IEEE and IET and is also on the editorial board of several
international journals. He served as the Editor-in-Chief for IEEE Wireless
Communications Letters between 2020 and 2023.
---|---
|  Mohamed-Slim Alouini (Fellow, IEEE) was born in Tunis, Tunisia. He received
the Ph.D. degree in Electrical Engineering from the California Institute of
Technology (Caltech), Pasadena, CA, USA, in 1998. He served as a faculty
member in the University of Minnesota, Minneapolis, MN, USA, then in the Texas
A&M University at Qatar, Education City, Doha, Qatar before joining King
Abdullah University of Science and Technology (KAUST), Thuwal, Makkah
Province, Saudi Arabia as a Professor of Electrical Engineering in 2009. His
current research interests include the modeling, design, and performance
analysis of wireless communication systems.
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