Power Minimization of UAV-Assisted Full-Duplex Mobile Relaying via Hybrid Trajectory Design

—This paper solved a propulsion power minimization problem subject to a total data-bits constraint of an unmanned aerial vehicle (UAV) enabled full-duplex mobile relaying system, where a full-duplex UAV relay is dispatched as a mobile relay to assist data transfer from a source to a destination by using three trajectory flying modes, namely, the UAV first flies in a circle above the source, next flies to the destination in a straight line, and finally flies in a circle above the destination until all the data bits has been transferred. Since the propulsion power minimization problem is a non-convex mixed integer programming problem and its closed-form solution is hard to obtain, it is transformed to three sub-problems so as to simplify its solution. After solving the three sub-problems, an iterative algorithm is proposed to achieve a sub-optimal solution to the propulsion power minimization problem, leading to a new hybrid circular/straight trajectory (HCST) design. Computer simulations are conducted and the results validated the proposed HCST design. It is shown that compared to the straight or circular flight trajectory design, the HCST performs well in terms of energy saving for the long distance and big data communication cases.


Introduction
The use of unmanned aerial vehicles (UAVs) for assisting wireless communications has attracted increasing interest recently [1]. Compared to the traditional terrestrial communications, UAV-assisted communications are more cost-effective by enabling on-demand operations, more flexible and swift for deployment due to the high mobility of UAVs. In addition, UAV-ground communication links are more likely to have line-of-sight (LoS) channels which offers higher capacity than that provided by terrestrial links. Thus, UAVs shall play an extremely important role in future wireless communications with applications such as temporary traffic offloading for cellular base stations [2]- [4], cooperative communications for mobile relaying [5]- [8], information dissemination and data collection for Internet of Things [9], etc.
In this paper, we considered a full-duplex mobile relaying system consisting of a fixed-wing UAV, a source and a destination. The source would like to transfer data to the destination assisted by the UAV which acts as a decode-and-forward (DF) relay. The UAV relay has three trajectory flying modes, namely, the UAV first flies in a circle above the source, next flies to the destination in a straight line, and finally flies in a circle above the destination until all the data bits has been transferred. It is noted in [10] that an UAV's communication-related power is usually much smaller than its propulsion power. Therefore, the main goal of this paper is to minimize the total propulsion power of the UAV. We formulate the trajectory design as a non-convex mixed integer programming problem. To solve the problem, it is dived into a few sub-problems, by solving which a sub-optimal solution is acquired. Then, an iterative algorithm is proposed to obtain a hybrid circular/straight trajectory (HCST) design which can approach to the optimal trajectory minimizing the total propulsion power.
The rest of this paper is organized as follows: Section 2 describes the related works in literature. Section 3 introduces the system model under investigation and formulates a general optimization problem corresponding to the propulsion power minimization. By dividing the optimization problem into three sub-problems, the power minimization problem is solved in Section 4, leading to a new HCST design. Computer simulations of the proposed HCST design with performance comparisons are presented in Section 5. Section 6 concludes the paper.

Related works
Currently, there are many works in literature that concern the problem of maximizing information throughput or spectrum efficiency of UAV-assisted mobile relaying systems [11]- [16]. It is known that UAVs have limited on-board energy, and thus energy saving has been recognized as a more important factor for UAV communications.
The authors of [17] investigated an energy-efficiency problem by optimizing the UAV's speed and load factors in circular trajectory. In [18], an energy-efficiency maximization problem was solved by jointly studying beamforming, power allocation, circular radius and flight speed of an UAV-assisted half-duplex mobile relaying system. In [19], a power consumption trade-off between ground terminal andU A V sw a sde r iv edf o rci r c u l ara nd straight trajectories, respectively. The authors of [10] developed a load-carry-and-delivery scheme for a full-duplex UAV relaying network with the aim of enhancing the energy efficiency of the network. In [20], a rotary-wing UAV was used to communicate with multiple ground nodes, and an efficient algorithm was proposed to optimize the hovering locations and duration, as well as the flying trajectory so as to minimize the system energy consumption. The authors of [21] investigated an energy-efficiency problem with the aim of maximizing the data bits transferred per the UAV's propulsion energy via the trajectory design. It should be noted that most of the existing works considered the circular or straight trajectory design separately. It is noted in [21] that there exists an upper bound of information throughput if a straight trajectory is adopted. It means that straight trajectory is not suitable for the case where a huge amount of data need to be transferred. In case that a circular trajectory is employed, the system information throughput will be poor if the source and destination nodes are located far away [17]. In view of the shortcomings of circular and straight trajectories, this paper investigates a HCST design for power minimization. Fig. 1 gives the system model, where an UAV denoted by R is dispatched as a mobile relay to assist data transfer from the source S to the destination D. Suppose that S and D are located far away, and thus no direct link exists between S and D due to high path loss caused by long distance and/or obstacles. Moreover, it is assumed that S would like to transfer a huge amount of data to D. respectively, and the UAV flies at a fixed altitude H which corresponds to the minimum altitude required for terrain or building avoidance without frequent aircraft ascending or descending [10]. Denote by T the mission completion time. Throughout the whole mission time, the UAV needs to collect Q bits data from S and then forwards them to D. There exists three trajectory modes described as the following. Suppose that the initial location of the UAV is   1 ,0,   rH . First, the UAV flies n laps according to mode 1.

Method
During the whole mission time, a full-duplex relaying fashion is employed, namely, the UAV collects data from S and forwards them to D simultaneously. Note that for simplicity, the UAV's take-off and landing phases are ignored. It is worth-mentioning that the UAV's trajectory should change between different modes smoothly, and the UAV's speed cannot be switched instantly between v1, v2 and v3. However, since the total power consumption of an UAV is very large, the change of the UAV's kinetic energy between speeds v1, v2 and v3 and the difference between the practical trajectory and the trajectory in Fig. 1 Here, it is assumed that the UAV R is equipped with a data-buffer of sufficiently large size and the communication links from R to S and D are dominated by the Line of Sight channels [11]. Moreover, the Doppler effect due to the UAV mobility is assumed to be perfectly compensated [11]. Then, the time-varying channel between S and R follows the free-space path loss model and can be written as Likewise, the time-varying channel between R and D can be expressed as In (4) and (5), β0 denotes the channel power at the reference distance d0 = 1m. Assume that R collects and forwards data using different frequency bands, and thus there is no loop-interference at the UAV [11]. Then, the corresponding signals received by R and D can be, respectively, written as In (6) and (7) where σ 2 denotes the noise power at the UAV R and D, and . It is worth-mentioning that in the paper, data bits are normalized by bandwidth, and thus without specific instructions, bit means bit/Hz. In order to ensure Q bits data transfer during mission completion time T , (10) must be satisfied.
Moreover, the UAV can only forward data that has already been collected from S, and thus the following information-causality constraint must be satisfied [20]     It is well known that the total power consumption of an UAV consists of two parts, i.e., communication-related and propulsion power consumption. Here, only the UAV's propulsion power consumption is considered as the communication-related power consumption is usually much smaller than the propulsion power consumption [10].
According to [19], propulsion power consumption of a fixed-wing UAV with circular and straight trajectories can be modeled as   where min V and max V are the UAV's the minimum and maximum flight velocity constraints, and (12b) and (12c) are the practical physical constraints.

Solution of Power Minimization Problem
In this section, the power minimization problem (12) is solved, leading to a HCST design. As a first step, the following lemma is presented so as to simplify the optimization problem. Since problem (12) is a non-convex mixed integer programming problem, it cannot be directly solved with standard convex optimization techniques. In the following, problem (12) is divided into three sub-problems by solving which a sub-optimal solution to the power minimization problem is acquired.
Recall that S and D are located far away and the UAV relay assists the data transfer between S and D by adopting three trajectory modes. In mode 1, the UAV is much closer to S than to D, and thus it has dominant traffic volume in collecting data from the source S. On the contrary, in mode 3 the traffic of the link from the UAV to D is dominant. Therefore, in order to simplify (12), two sub-problems corresponding to the power minimization in modes 1 and 3 are addressed first, where only the link having the dominant traffic volume is considered. Then, the two sub-problems can be formulated as (18) Of note is that in (18)   v are the optimal solutions of (19).
However, the solution of problem (22) is still hard to achieve due to the integer constraint n .T h e n ,t h e following two lemmas are proposed so as to simplify problem (22).

Lemma 2:
For problem (22), the optimal solution of the number of the laps in mode 1 can be expressed as 11  Q the optimal data bits received by the UAV in modes 1 and 2, respectively, which can be obtained by solving problem (22) and then substituting the obtained solutions into (15)   According to lemma 2 and lemma 3, an iterative algorithm is proposed to solve problem (22), leading to a sub-optimal solution to the propulsion power minimization problem (12). First, 1  r , 1  v , 2  r and 3  v are acquired by solving (18) and (20)  RR   QQand repeat the first two steps until a convergent solution is found. Then, the algorithm of the sub-optimal solution for the propulsion power minimization problem can be given as follows.
Algorithm: iteration algorithm for sub-optimal propulsion power minimization problem.
Step 2: Set the value of  n according to lemma 2.
Step 6: Output the sub-optimal radius 1  r ,velocity 1  v and duration 1  T in mode 1; output the sub-optimal velocity 2  v and duration 2  T in mode 2; output the sub-optimal radius 2  r ,velocity 3  v and duration 3  T in mode 3. Then calculate the total power consumption E according to (12a).
It is worth-mentioning that according to lemma 2 ,  n has two possible solutions, i.e., 11  into the iteration algorithm. Actually, the optimal solution of the algorithm is the one having the smaller power consumption.

Simulation Results and Discussions
In this section, computer simulations are conducted to validate the proposed HCST design. In order to make  The sub-optimal results are achieved by the iteration algorithm mentioned above. While a search algorithm is used to solve the problem (12) which leads to the optimal results. It is shown in Fig. 1 that the UAV's trajectory and total propulsion power consumption of sub-optimal and optimal are closed.
In Fig. 2, the distance between S and D is set to 5000m  L , the transmit power of S and UAV relay is set to SR 1W  PP . As shown in Fig. 2 (a)  . The reason is a larger number of lap in mode 1 can achieve a smaller total power consumption with the increasing of Q. Then, the radius 1 r and speed 1 v can be derived by updating the number of lap in mode 1 which causing the jump phenomenon. And the number of lap in mode 1 is given in Table 1. Meanwhile, the circular radius 2 r and speed 3 v are almost not changed. Since there is no lap constraint in mode 3, and the energy-efficiency trajectory has the potential to achieve the minimum power consumption. propulsion power consumption. It is shown in Fig. 3 that the HCS perform welli nb i gd a t aa n dl o n gd i s t a n c e compared to straight and circular trajectory.
In Fig. 3 (a), the distance between S and D is set to 5000m  L , the transmit power of S and UAV relay is set to SR 1W  PP . Fig. 3 (a) present the total propulsion power consumption of the HCS and straight flight trajectory, showing that the HCS can saving more energy compare to the straight flight trajectory when the total data bits Q is large. For instance, Fig. 3 (a) shows that the HCS can achieve about 18.4kWe n e r g ys a v i n g s compared to the straight flight trajectory when Q=200bit. In Fig. 3 (b), the total data bits is set to 300bit  Q ,the transmit power of S and UAV relay is set to SR 1W   PP . Fig. 3 (b) present the total propulsion power consumption of the HCS and circular flight trajectory, showing that the HCS can saving more energy compare to the circular flight trajectory when the distance L is large. For instance, Fig. 3 (b) shows that the HCS can achieve about 2.7kW energy savings compared to the circular flight trajectory when L=1000m.

Conclusions
This paper proposes a new HCS trajectory referring to the case of long distance and big data communication.
And investigates the unmanned aerial vehicle's (UAV) propulsion power minimization problem subject to the total data bits constraint of a UAV-enabled full-duplex one-way mobile relaying system. Considering a sub-optimal problem, and the corresponding sub-optimal solutions are obtained by an iterative algorithm. The results show that the HCS performs better than straight or circular flight trajectory in terms of energy saving when long distance and big data communication is considered.

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