I Introduction
Recently, unmanned aerial vehicles (UAVs) communication is an emerging example to assist nextgeneration remote users with reliable connectivity [1]. The UAV communication system is less expensive than the terrestrial base station (BS) platform due to swift, dynamic, ondemand, flexible, and reconfigurable features. Moreover, the UAV relay is controllable. Due to its higher altitude, it often experiences a significant line of sight (LOS) communication links.
The UAVs can be loosely classified
[2] based on operation, such as aerial BS, relay, and collecting information. When the BS is malfunctioning, UAV is deployed to serve as aerial BS [2]. Moreover, the UAVs also stay as quasistationary on the serving area to support the nodes. The scenarios, such as the (BS) offloading in hot spots and BS hardware limitations, require a fast service recovery, and the UAV is an excellent choice [3], [4]. If the BS is not available due to expensive installation costs in physically unreachable areas, the UAVs are deployed as relays to increase the BS capacity. Thus, the UAV relays are responsible for providing wireless connectivity for remote users in adversarial environments, such as natural disaster recovery, military operation, and rescue operation, etc. Moreover, UAV relays can be deployed to collect or disseminate information [5], [6]. Collecting or disseminating information is vital in various domains, such as periodic sensing or smart cities application. This is because sensors’ operational power is reduced if the UAVs fly over them to communicate, which results in a more extended network lifetime.The UAV relay has two categories, mobile and static. Like mobile UAV, the static UAV has a better chance of LOS, it is not dynamic and consumes more propulsion energy. Moreover, the static UAV serves the only region, such as the football stadium, where UAV does not need to move [7]. However, mobile relaying has more advantages than static relaying, such as costeffective, swift deployment, serving an ondemand basis [8]. Due to the mobility of relay, it provides opportunities for improving the wireless network performance by adjusting the relay location. In the recent few years, research has been conducted on UAV relaying because of its range extension capability [9] – [12]. Moreover, the application of the UAV relaying increases the overall system performance. Unfortunately, due to the limited mobility of nodes and backhaul techniques, most of the conducted researches on the UAV relaying is static.
The authors in [13] studied the throughput analysis for mobile UAV relay. They achieved the optimal BS and UAV transmit power while designing the trajectory. However, their proposed model is limited due to 1) considering one known adversary, and 2) not considering the UAV energy consumption. Though their algorithm optimized the UAV power to design trajectory, it did not design the optimal energysaving trajectory. This is because UAV transmit power (up to few Watts) has a negligible effect compared to propulsion energy consumption (PEC). UAV PEC is typically up to a few hundred KWatts.
We design mobile relaying communication to make the problem practical. Unfortunately, there are new challenges for UAV relaying communications. Specifically, onboard power consumption during the finite time limits the UAV relay performance because of its fixed size. The energyefficient (EE) UAV communication, defining total communicated information bits normalized by UAV PEC [14, 15], is an essential paramount feature. Moreover, the UAV is required forwarding information from BS to users by ensuring the physical layer security. Additionally, the UAV has broadcast nature communication links, which may lead to substantial physical layer security concerns for uncertain adversaries.
Researchers are working on designing EE UAV networks broadly. However, it needs more attention to secure the network. For example, the authors in [16] – [17] designed EE UAV communication. However, they did not consider the UAV onboard energy consumption and physical layer security. Moreover, they considered straightforward UAV path planning. A robust resource allocation to maximize the secrecy is studied in [18] in the presence of adversaries. However, the authors did not consider UAV energy minimization. In our previous work in [19], we optimize the UAV worstcase secrecy rate (WCSR) in adversarial networks via resource allocation. We proposed an algorithm that considers informationcausalityconstraint (ICC) while maximizing WCSR.
A limited theoretical analysis of UAV relaying security was studied in [20, 21]. In [20], the authors investigated the joint BS/UAV power and trajectory optimization. The system considers adversaries, which are partially known by the UAV. In their investigation, the UAV is considered as the aerial base station. However, the authors proposed model is limited to serving close users. Moreover, they did not consider UAV PEC, which is an important paradigm to design trajectory. The authors in [21] investigated the UAV security and their proposed networks had UAV relay, BS, user, and one adversary. Their proposed model is limited to the perfect location of adversaries to the UAV. This assumption is limited application in realworld scenarios. They considered neither robust resource allocation nor UAV EE. The authors in [20, 21] analyzed the security aspect to maximize the throughput. However, their algorithms did not have an optimal trajectory and UAV EE for longdistance users since they did not investigate 1) UAV relaying security for unknown adversaries 2) designing an optimal trajectory for EE UAV. We explained the differences in the revised manuscript. We hope that the reviewer would be satisfied with our explanation and revised paper.
The system considers adversaries, which are partially known by the UAV. In their investigation, the UAV is considered as the aerial base station. However, the authors proposed model is limited to serving closedistance users. Moreover, they did not consider UAV PEC, which is a crucial paradigm to design trajectory. The authors in investigated the UAV security, and their proposed networks had UAV relay, BS, user, and one adversary. Their proposed model is limited to the perfect location of adversaries to the UAV. This assumption is the limited application in realworld scenarios. They considered neither robust resource allocation nor UAV EC. However, their algorithms did not have an optimal trajectory and UAV EE for the longdistance user since they did not investigate 1) UAV relaying security for unknown adversaries, 2) designing an optimal trajectory for EE UAV.
In [22], UAV relaying communication is studied, which helps to forward independent data to different users. The authors maximized the data volume and relay trajectory by using a simple algorithm. In [23], the authors optimized the UAV flying path at a fixed altitude. The authors in [24] investigated the optimal UAV trajectory, considering the UAV onboard energy for the energyaware coverage path. Their study considered a quadrotor UAV measurementbased energy model, which was applied to aerial imaging. Mobile UAV communication is studied in [25] by assuming that the relay moves randomly, which follows a specific mobility model. They maximized the UAV mobile relay statistical characteristics via throughput. The authors investigated throughput for UAV relaying networks in [26]. They achieved the minimum UAV transmit power and trajectory.
All of the above works consider UAV trajectory optimization. There is still scope for research to design EE UAV communication and ICC, while the adversaries try to hide in the wireless networks. Adversaries often use artificial noise, which increases the wireless network noise level. This nature helps them hide their presence even when the user is close to relay. Moreover, the adversaries collaborate, making their presence protect from legitimate nodes. Moreover, most of the aircraft track optimization investigations are not studied for wireless networking purposes.
The above proposals and models aim to achieve optimal solutions on simplified algorithms. Thus, we focus on developing a more realworld model and achieving a suboptimal EE UAV relaying networks using an iterative algorithm. We design EE UAV mobile relaying via optimizing the UAV and the BS robust resource allocation, which considers the joint WCSR and UAV PEC. Naturally, the best channel modeling is achieved for the maximum throughput, if the UAV mobile relay stays fixed to the possible nearest location from the user. However, this scenario results in the inefficient UAV PEC modeling due to the UAV hovering at zero speed [27]. Thus, there must be an optimal tradeoff between maximizing the average WCSR and also optimizing PEC. Our main contributions in this paper are described as follows:

We consider a scenario, where the user receives data from the BS. Due to the longer distance, there is no direct link between them. Thus, UAV relaying is a promising solution to forward data to the user. The system has uncertain adversaries who try to intercept the UAVuser link. We achieve the optimal achieve the average WCSR via optimizing joint UAV/BS transmit power, and the UAV trajectory. To achieve the EE, we employ the fixedwing UAV PEC, which is the function of speed and acceleration. Based on this model, we formulate the EE UAV relay.

We investigate EE UAV relaying maximization problem subject to ICC, UAV trajectory, speed, acceleration, UAV/BS power constraints. ICC [28] is applied to capture the UAV broadcast communication from the BS.

The optimization problem is nonconvex. Moreover, it is also fractional. We apply an iterative algorithm, which considers successive convex approximation (SCA) [29], Dinkelbach [30] and Taylor series expansion [31]. This iterative algorithm achieves the solution. However, the solutions are suboptimal. Initially, we find optimal UAV/BS transit power control and speed under given UAV path planning and acceleration. Next, we minimize trajectory and acceleration using optimal power control and speed. The algorithm repeats until convergence.
Our prior work in [14] covered part of the throughput maximization considering EE UAV relaying. We proposed an iterative algorithm by considering throughput maximization based on LOS communication links and propulsion energy minimization. On the other hand, this paper considers EE UAV relaying communication in the presence of uncertain adversaries.
Ii System Model
Iia Description of parameters
Fig. 1 shows the information transmitted from the BS to single user via the UAV relay in the presence of uncertain adversaries. We also consider a few obstacles in rural or remote areas. Both the user and adversaries are located in this region. No direct link is established by BS with the user and the uncertain adversaries due to long distance. Since the user resides in rural or remote areas, far from BS. Thus, due to the higher altitude of UAV and fewer obstacles, we safely assume only the LOS link and the system has a negligible effect on the Gaussian Additive channel, Rayleigh fading channel, or Rician fading channel. However, it is in our interest to investigate the model in the dense urban area with more obstacles and both LOS/NLOS links. We have left the extension for future effort.
Each node has a single antenna. The UAV works as a relay that helps to communicate the BS to the user. Thus, UAV forwards the received information from the BS to the legitimate user on the ground. The UAV has fixed flying altitude, while both user and BS locations are known. The UAV does not require to change its altitude since we consider the rural environment, with a less number of higher obstacles. Let’s say the UAV changes its altitude and tries to reach an optimal altitude, which requires frequent ascending/descending. Thus, the frequent ascending/descending will significantly consume propulsion energy, which results that the UAV will not be able to fly sufficiently long time due to its onboard power limitation. The uncertain adversaries may intercept the UAVtouser link. In our proposed system, the UAV does not have the adversary location information. The UAV only knows the region where the adversaries are located. We define the uncertain adversaries set as =. As the UAV has a better LOS advantages, we neglect the shadowing and multipath effects in our proposed model. In the few subsections, we explain the proposed system model in detail.
IiB UAV flight time and node locations
UAV provides service to the single user in time horizon, where is seconds. Thus, is continuous. In the paper, is discretized into number of equal slots, having slot size and . Moreover, we consider that each time slot is static and equal.
We apply the BS/UAV/user/adversaries positions in a 2D coordinate system. Each node is static except for the timevarying UAV positions. The time dependent UAV location is . is defined as the UAV fixed altitude, which can avoid tall obstacles. The users’ static location is and the static BS location is . The initial UAV location is due to ICC explained in Section IIE. On the other hand the UAV final position is .
IiC Uncertain adversaries
The UAV has both higher LOS chance of communication links and broadcast communication nature because of higher UAV altitude. Thus, this link is used to send information from UAV to the user. Unfortunately, due to the broadcast communication nature, the adversaries may take advantage to intercept the legitimate information. Moreover, the adversaries have always the hiding nature from the source and destination. Thus, it is not easy for the UAV to know the actual adversarial locations.
To tackle the adversary location issue, we assume UAV has the circular region of the adversary’s residence information. The actual location of uncertain adversary , where , is calculated from the circular region as follows:
(1) 
where is the actual adversary location .
defines the estimated location of adversary
. The approximated errors from actual uncertain adversary location is defined as , where is set of possible errors of the uncertain adversary . The following needs to be satisfied if the uncertain adversary resides on the circular region.(2) 
where is the radius of the circular region.
IiD Various links
We use UAV flight time and node location in Section IIB, we can calculate the various channel gains and data rates for free space. For example, we calculate the distance between the UAV location and BS location to achieve its corresponding channel gain. Channel gain of BStoUAV is:
(3) 
where is the channel power gain calculated when = 1 m (the reference distance) [35].
At this point, we apply the Shannon capacity to achieve the rate from the formulated channel gain. The BStoUAV data rate is:
(4) 
where is AWGN noise power. The BS power in time slot is . The SNR in (4) is calculated using the channel gain between the BS and the UAV.
Similarly, the UAV and user channel gain is expressed as follows:
(5) 
The UAVtouser data rate is:
(6) 
where the UAV power in time slot is . The SNR in (6) is calculated using the channel gain between the UAV and the user.
Similarly, channel gain of UAVtoadversary is:
(7) 
The UAVtoadversary data rate is:
(8) 
The SNR in (8) is calculated using the channel gain between the UAV and the adversaries.
IiE Informationcausalityconstraint (ICC)
Legitimate data is sent by BS to UAV in time slot, . After that, UAV forwards that data to the user. ICC states that UAV forwards the received data to user during other time slots, i.e. [21] is:
(9) 
(10) 
UAV does not forward legitimate information to user when n=1. On the other hand, there is no transmission by the BS to the UAV when n=N. Thus, and .
IiF UAV propulsion energy consumption (PEC)
UAV propulsion energy is the total energy consumed for the time horizon. It has a considerable effect on EE UAV relaying system performance. There is other energy consumption that occurs due to signal processing, radiation, and electronics circuit, etc. This amount of energy is trivial compared to UAV propulsion energy [2]. When the UAV has fixed wings with no abnormality, such as the backward thrust generation against the forwarding speed, the UAV hovering path becomes the propulsion energy function. We aim to design EE UAV relaying communication via designing the optimal path planning, velocity, acceleration, and transmit power control. Moreover, the UAV hovering path requires an optimal tradeoff to balance WCSR maximization and energy minimization. The UAV PEC [2] is expressed as follows:
(11) 
where both , are constant. Their values depend on many factors, such as relay weight/wing size, etc. is the speed and is acceleration. is gravitational constant. Moreover, (11) neglects the UAV transmit power due to the meager amount of power compared to the UAV PEC. is kinetic energy, which is , and the mass of UAV is .
IiG Transmission protocol
Fig. 2 shows a data transmission protocol for the proposed model. The BS sends data to the user via UAV using the optimal EE trajectory from our proposed algorithm (more detail of the proposed algorithm is explained in Section III). Prior that UAV verifies data by ICC and detects the uncertain adversaries from the known circular region. Finally, the UAV designs the optimal EE trajectory so that it can communicate between user and BS during the data transmission and UAV flight time. Note that designing an optimal path for UAV considers the joint optimization of WCSR and PEC. Throughout the process, the BS does not establish the user and adversaries links due to long distance from user and adversaries.
Iii Optimal EE UAV relay
We design EE UAV by considering WCSR and UAV PEC. Now we formulate EE UAV problem for UAV flight time slot, which combines (6), (8), and (11). Thus, optimization problem can be formulated with related constraints.
(12a)  
(12b)  
(12c)  
(12d)  
(12e)  
(12f)  
(12g)  
(12h)  
(12i)  
where
(13) 
where , , and are expressed in (6), (8) and (11), respectively. Moreover, (12a) illustrates WCSR [36] and UAV PEC. (12b) and (12c) define the UAV flying speed limit. (12d) is the acceleration limit. UAV mobility expression is in (12e). The BS peak power constraint is defined in (12f), where is the highest BS transmit power. Average power control of the BS and the UAV are expressed in (12g) and (12h), respectively, where and are the UAV and BS average power. However, (12) is not easy to solve optimally due to 1) not being a convex, and 2) uncertain infinite possible error numbers to find the actual adversaries locations. Thus, we propose an suboptimal approach to solve (12).
To solve (12) suboptimally: first, we fix the and and solve , , and . In the second step, we apply relay path and acceleration to achieve the optimal solution both BS/UAV power and speed.
Iiia Suboptimal solution 1
We first formulate the optimization problem to achieve , ,and for given and . Using (12), the reformulated the suboptimal problem is:
(14)  
The standard optimization toolbox, such as cannot find the solution of (14) due to the nonconvexity of (10), (12b), (12c), and (14). First, we reformulate ICC in (10) as follows:
(15) 
(16) 
where is newly added variable. We reformulate (14) as follows:
(17)  
We replace by , which is a new variable in (17). However, (17) is still nonconvex because of and (12c). We apply a variable, in . Thus, the reformulated problem is expressed as follows:
(18a)  
(18b)  
(18c)  
where
(19) 
To find the solution, (18) is required to satisfy all of its constraints as a convex problem. However, (18a) has still uncertain and infinite numbers of the actual locations errors of the adversaries. To tackle the WCSR in (18a), following is expressed:
(20) 
can be defined (1) as follows:
(21) 
can be rewritten, using (1).
(22) 
Still (21) is not convex and thus, not tractable due to . Thus, using (2) in (22), we achieve the expression.
(23) 
where
(24) 
However, is very small. However, from (23), the UAVadversary distance is larger than adversary region . Distance between the UAVadversary distance is:
(25) 
Following conditions can happen, such as the UAVadversary distance is either greater/equal or less than the radius of the circular region. For example, if , then is written using (23) as follows:
(26) 
On the other hand, if , then is written using (23) as follows:
(27) 
We tackle the WCSR, UAV PEC, and the constraints as a convex function. However, EE UAV maximization problem is not yet soluble because it is still the fractional in (18a). Due to the fractional problem, we cannot apply the optimization toolbox to achieve the solution. Thus, we employ Dinkelbach method [32] to tackle the fractional nature of the objective function. Fortunately, this approach confirms convergence with local optima.
(28)  
where is numerical number. Moreover, is updated in iterative fashion as . Now, (28) is convex. it can be solved via convex optimization toolbox, such as [34].
IiiB Suboptimal solution 2
In the subsection, we apply the solution, achieved in Section IIIA, to achieve the solution of the rest of the optimizing variables in our proposed model. Using the optimal , , and , we reformulate the optimization problem from (12) as follows:
(29)  
where
(30) 
(31) 
where
(32) 
However, (29) is nonconvex problem because of the fractional objective function and ICC in (10). Due to the infinite number of possible multiple adversaries locations errors, (29) is challenging to solve suboptimally in the polynomialtime series. We tackle nonconvexity of (29) by applying the slack variables and . Thus, the newly formulated optimization is:
(33a)  
(33b)  
(33c)  
(33d)  
where , , and .
Thus, the similar suboptimal solution of (29) shares the similar solution of (33). We focus to solve (33) to find the suboptimal solution of trajectory and acceleration.
Proof. Refer to Appendix B.
Still (33a) is still a nonconvex problem due to infinite possible errors from the real location in (33c). Thus, we apply (1)  (2) in (33c) as follows:
(34) 
(35) 
We apply the  mathematical approach, which can tackle the infinite number of possible uncertain location errors of adversary . Thus, a feasible point exists, for example , such that
(36) 
The following implication also need to be held.
(37) 
if and only if exists such that
(38) 
where . Thus, (38) and (33c) are equivalent. Now, (33) is:
(39a)  
(39b)  
(39c)  
is slack variables, where , where . Moreover, due to the nonconvexity of (39a) and (10), (39) is not tractable. On the other hand, is now convex in nature. Thus, (38) is non linear function as it contains . Thus, nonconvexity and nonlinearity of (39) make it difficult solving suboptimally. We apply an iterative algorithm, which can tackle (39). The algorithm achieves the approximate solution of (39). It can be expressed as follows:
The feasible sets of are , , and , respectively. These feasible points are also feasible to (39). Due to nonconvexity of , we apply the first order Taylor expansion series at as follows:
(40) 
Similarly, we apply the first order Taylor expansion series at for as follows:
(41) 
We tackle the nonlinearity of by applying the Taylor series expansion at the feasible points.
(42) 
(43) 
We can transform (39) using (40)  (43) as follows:
(45a)  
(45b)  
(45c)  
(45d)  
where
(46) 
(47) 
where and are derived from (40) and (41), respectively. We reformulate ICC to (15)  (16) from (10).
(48) 
(49) 
We tackle the nonconvexity of (48)  (49) by adding the variable. Thus, (48) is:
(50) 
(51) 
where is introduced variable and
(52) 
Now, (39) becomes:
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