Gaofeng WU, Xiaoguang GAO, Xiaowei Fu, Kaifang WAN, Ruohai DI

School of Electronics and Information, Northwestern Polytechnical University, Xi'an 710072, China

KEYWORDS Channel estimation;Gradient methods;Motion control;Optimization;Relay;Unmanned aerial vehicle;Wireless networks

Abstract In this paper,a model-based adaptive mobility control method for an Unmanned Aerial Vehicle(UAV)acting as a communication relay is presented,which is intended to improve the network performance in airborne multi-user systems. The mobility control problem is addressed by jointly considering unknown Radio Frequency (RF) channel parameters, unknown multi-user mobility, and non-available Angle of Arrival (AoA) information of the received signal. A Kalman f ilter and a least-square-based estimation algorithm are used to predict the future user positions and estimate the RF channel parameters between the users and the UAV, respectively. Two different relay application cases are considered: end-to-end and multi-user communications. A line search algorithm is proposed for the former, with its stability given and proven, whereas a simplif ied gradient-based algorithm is proposed for the latter to provide a target relay position at each decision time step, decreasing the two-dimensional search to a one-dimensional search. Simulation results show that the proposed mobility control algorithms can drive the UAV to reach or track the optimal relay position movement, as well as improving network performance. The proposed method ref lects the properties of using different metrics as objective network performance functions.

1. Introduction

Systems consisting of multiple airborne vehicles cooperatively learning and adapting in harsh unknown environments to achieve a common goal have shown their magnitude commonality in both military and civilization applications over the past decades.1-3One key requirement of such systems is realizing and optimizing the communication quality for information exchange purposes. However, increasing distances make this requirement diff icult to achieve for users.4,5

To address this problem, communication relays have been deployed to support system signal transmission. Unmanned Aerial Vehicles (UAVs) equipped with wireless transceivers and receivers as communication relays have been considered ideal for this purpose because of their improved relay performance relative to ground-and satellite-based relays.6,7Furthermore,such on-the-f ly communication relays(without humans)can be easily deployed together with high adaptability and survivability advantages, particularly in harsh environments.Driving relay UAVs toward optimal relay positions to continuously provide multi-user systems with the best possible uplink communication quality is a crucial and challenging problem,requiring excellent mobility control of the relay UAVs.

In research on the mobility control of relay UAVs,the users could be static or mobile, and the usually available input includes location coordinates, Received Signal Strength(RSS)and signal Angle of Arrival(AoA).Methods addressing such problems can be generally classif ied into two groups:model-free and model-based.

Model-free mobility control methods are data driven, only requiring sampled data. Dixon and Frew8proposed a decentralized Extremum Seeking Control (ESC) architecture based on perturbation for chain capacity optimization in end-toend communication. ESC is an adaptive control scheme capable of searching for optimal setpoints in real time using measurements of the performance output or its gradient.However,objective functions in gradient estimation can be measured only when the relay UAV arrives at related positions. Thus,a new chain controller was built by taking advantage of a Lyapunov Guidance Vector Field(LGVF)based loitering method to obtain objective functions perturbed at specif ic positions,requiring only the positions of UAVs and sampled objective function values.9However, this method assumed that the RF environment is quasi-static, which is a hard-to-satisfy requirement in realistic applications where users' mobility has been determined by correlated missions.

In model-based mobility control methods, RF channel models or connectivity models are required. Distance-based channel models are the most commonly used in research on driving relay UAVs to positions to maintain Line of Sight(LoS) connectivity or improve network quality in communication-affected missions.10,11These models are actually over-simplif ied, because controllers using exclusively geographic ranges will lead to a degraded communication performance,9,12thus,RF distribution based models were used in researches:

Yuan et al.13proposed a probabilistic router formation and motion-planning approach by integrating a stochastic channel learning framework14with robotic router optimization. Ono et al.15proposed a variable-rate relaying approach to enable communication among ground stations during disastrous events, where the turning radius and the f light altitude of the relay Unmanned Aircraft(UA)were designed.Mozaffari et al.7proposed a framework for the optimized deployment and mobility control of multiple UAVs intended for energyeff icient uplink data collection from ground Internet of Things(IoT) devices. All these works assumed that users were static,which cannot be satisf ied in many applications. Although other publications, such as Refs.16-18, considered mobile users and better channel models, they still assumed that UAV controllers had prior knowledge of channel parameters. Again,these are unrealistic in many situations,particularly in systems with various unknown user types.

Chamseddine et al. proposed and proved a guidance law19for the mobility control of a relay UAV in a multi-ground unit system using only RSS and AoA, where the users' mobility requirement was relaxed compared to Dixon's method. This guidance law was based on the idea that the path loss exponent was 2; however, the exponent actually differed from 2-6 according to channel characteristics and environment properties.4,20Moreover, obtaining the AoA of the received signal is quite diff icult if the relay UAV is equipped with a single isotropic antenna.21

None of the above mentioned works, model-based or model-free,directly addressed the relay UAV mobility control problem by jointly considering the unknown RF channel parameters, unknown multi-user mobilities and non-available AoA information of the received signal.The primary contribution of this study is a model-based mobility control method for driving a single relay UAV to an expected location for optimal communication network performance. The method only uses measured GPS and Received Signal Strength (RSS) information by jointly considering the aforementioned three aspects in an airborne multi-user system. The key point of this work is building an optimal relay position seeking approach instead of a combined UAV autopilot system and a high-level decision module model. A guidance law was subsequently employed to complete the framework.

In identifying the optimal relay position for the UAV, a Kalman Filter (KF) was used to predict the future positions of the user nodes. The channel parameters could then be estimated using a Least Square Estimation (LSE) algorithm with little measured RSS information,hence,the RF distribution in the region could be predicted. These prior accomplishments made it possible to solve the relay position seeking nonlinear programming problems which can be classif ied into two groups: end-to-end communication and multi-user communication.A line search algorithm was proposed and proven adequate for the former case.A simplif ied gradient-based algorithm was proposed instead of global search to optimize the global and worst-two-channel network performance for the latter case,where a special situation with equal path loss exponents was discussed.

The remainder of this paper is organized as follows: Section 2 presents the formulated problem and some preliminary models used in this research, which proved highly convenient for the study of relay UAV mobility control in literature. Sections 3 and 4 provide the framework of the proposed mobility control architecture with a model-based wireless channel parameter estimation method given in Section 3.Section 4 presents approaches for seeking optimal relay positions in various scenarios based on the pre-estimated information, together with an LGVF guidance law for driving the relay UAV toward the identif ied position. The performance of this approach was verif ied via simulation, and the results are given in Section 5,followed by concluding remarks in Section 6.

2. System model and problem formulation

In this work, a networked cooperative system consisting of N user vehicles ni∈N =｛n1,n2,...,nN｝ and one relay UAV u are considered. The UAV is a f ixed-wing aircraft and each node is equipped with a communication node.

2.1. UAV kinematic model

The relay UAV is assumed to have an on-board autopilot providing low-level f light control, and each aircraft possesses a fast-inner loop to achieve the required turn rates, altitudes,and airspeed commands. A simplif ied high-level kinematic model is used instead of an exact model of the combined aircraft and the autopilot system. The aircraft is also assumed to f ly at a f ixed,designated altitude with a maintained velocity,leaving only the commanded turn rate for consideration. This is realistic in many cases,9and the UAV acceleration does not impact the focus of the work,which is a decision system determining optimal relay positions of the relay UAV as mentioned in the later part of this paper.

Let pu=[xu,yu]Tbe the position vector of the UAV in the operation region M with a speed vector [˙xu, ˙yu]Tin Cartesian inertial coordinates that evolves according to the standard(Cartesian) bicycle-like kinematic model22

where ψu(yù)∈[0,2π) is the compass heading angle, vuis the ground speed that is assumed constant herein, and ωuis the commanded turn rate.The maximum turn rate ωu,maxis limited to a maximum bank angle φmaxbecause of the vehicle operational performance constraints

where g is the acceleration constant due to gravity.

2.2. KF-based position prediction

Although the relay UAV can be informed of the user positions via a Global Positioning System(GPS),GPS data is inaccurate and cannot provide future positions. A f irst order Auto-Regressive (AR) model is employed to model the user vehicle dynamics and a KF is used by the UAV to predict the user positions at the next time instance.

Let si,k=[xi,k,yi,k, ˙xi,k, ˙yi,k]Tdenote the state of user niat time instance tk,where ˙xi,kand ˙yi,kdenote the velocities of user niin the x and y directions,respectively. According to the AR model, the state transition equation from time instance tkto tk+1is given by

where F is the state transition matrix ref lecting the transition between states si,k-1and si,k, Δt represents the time length of tkand tk-1, and ξi,k-1represents the zero-mean Gaussian variable with a covariance matrix Qi,k-1=ref lecting the process noise, where I4is a 4×4 identity matrix. The user position information is informed by the users and the GPS output contains errors and delay, leaving it subject to inaccuracy. The noisy observation of the position of user niat time instance tkcan be given as where T represents the observation matrix, and νi,krepresents the zero-mean Gaussian variable with covariance matrix Ri,k=σ2νI2ref lecting the observation noise, where I2is a 2×2 identity matrix.The standard KF implementation procedure is presented as follows:

Initialization

Prediction

State measurement and covariance matrix

2.3. Optimal relay position seeking problem formulation

The relay UAV mobility is controlled to drive the UAV to an expected location for optimal network performance. Let pi=[xi,yi], i=1,2,...,N denote the position of user ni. The UAV is assumed to operate at a constant height and speed,hence, the only controllable variable of the UAV is its turn rate, ˙ψu(yù). Thus, this work considers solving the following optimization problem

where J is the objective function ref lecting network performance.

Solving the problem in Eq.(13)is quite diff icult because the objective function is only directly affected by the relay position, which is coupled with the autopilot system. Thus, the decision-making system is formulated by determining the optimal relay position.A guidance law is utilized to drive the UAV to this position. The main focus of this work concentrates on the former problem, which can be derived from Eq. (13) and is given as

2.4. Assumptions

The following assumptions are made throughout the remainder of this paper:

(1) The f ixed-wing UAV is assumed to have a constant speed and altitude during the relay mission, and the motion of each user is determined only by its own mission, inf luenced neither by the network nor the UAV states.

(2) The UAV could be informed of the user positions with negligible channel performance requirements, whereas the channel parameters between the UAV and each user could be various and unknown to the UAV beforehand.Therefore, they should be estimated by the UAV with online sampling progress.

(3) Users are airborne vehicles operating at relatively high altitudes such that no obstacles exist between any of them and the UAV, allowing line of sight communications to be assumed.

3. Online estimation of wireless channel parameters

A f ixed-wing relay UAV is deployed to improve the communication performance of a multi-vehicle system in which vehicles are executing missions in a region or an environment.The two main aspects of this problem are communication and mobility control. The problem is approached by controlling the relay UAV mobility to optimize the network communication performance. The regional RF distribution should be exclusively conf igured prior to optimization using small numbers of online measured positions and RSS samples.

3.1. Received signal model

Both the relay UAV u and N user nodes ni∈N(i=1,2,···,N)are assumed to be equipped with omnidirectional antenna in the region M. Airborne vehicles are assumed to operate at relatively high altitudes, such that an air-to-air path loss model is considered as follows

where Spu,piis the received signal strength at the receiver of the relay UAV located at position puwith transmitted powerfrom the transmitter of user nilocated at position pi,is a gain based on equipment characteristics, Gi=αiis the path loss exponent varying from 2 to 6, d0is a reference distance for the antenna far f ield,and diis the Euclidean distance between the transmitter and the receiver:

3.2. Estimation of the channel parameters

To estimate regional RF distribution, parameters of the channel between the relay and each user should f irst be characterized. Mostof i, et al. built a framework to estimate the channel parameters14,23by modeling the communication channel as a multi-scale system with three major dynamics (pathloss, shadowing, and multipath fading) and stipulating that the user is static and the channel parameters are constant,because shadowing and multipath components are strongly related to the user positions in the region due to the existence of terrain,buildings,and obstacles.Instead,this work builds a different channel estimation framework considering the path loss model, and def ining the other effects as Gaussian distributed variables, because: (A) the communication network is assumed airborne with no obstacles, thus the main effects rely on path loss; (B) the lognormal distributed random variable models the shadowing effects well acceptable, despite the good multipath fading match provided by the Nakagami distribution.

Let [0,T] denote the time interval of the relay mission, and the channel parameters be updated at time instance tk, 0 ＜k ≤L, which also represents the decision time instances. We consider that the wireless channels are sparsely sampled at user positions (subscript tkdropped hereafter)Pi=｛｝?M, i=1,2,···,N, during time[tk-1,tk) with a given environment. These channel measurements can be executed by the UAV along its trajectory Pu=｛｝?M,i=1,2,···,N. Let Di=10[lg(｜｜｜｜),lg(｜｜｜｜),...,lg(｜｜｜｜)]Tand yi=[y1,y2,...,yκ]Tdenote the corresponding distance vector between user niand the relay UAV,and the vector of all signal power measurements, respectively, in dB. This yields

where Iκdenotes the vector of ones with length κ,θi=[Gi,dB,αi,PL]Tis the vector of the path loss parameters in d B, and Xi=[]Twith χi=χ(｜｜pu-pi｜｜). As the Gaussian distribution model is a good match for χi, and the sampling procedure is completely independent,thus,X is taken as an uncorrelated zero-mean Gaussian random vector with the covariance vector E XXT=σ2i Iκ×κ, where Iκ×κis a κ×κ identity matrix. The main purpose of this channel estimation work is f inding the optimal estimations of θiand.

With the assumption of independent fading variables, the possibility density function (pdf) of yican be written as:

A Least Square Estimation (LSE) algorithm23can then be employed to estimate θiandas

Once the underlying communication channel parameters are estimated, the RSS at any position pe∈M could be estimated as

where ^pirepresents the predicted position of user niusing the aforementioned Kalman f ilter.

Neither the channel model presented here nor that proposed by Mostof i can be directly applied to moving users with continuously changing channel parameters(such as a relay for multi-users in complicated terrain regions), because these dynamics change within short sampling periods. Estimating the wireless channel parameters in such complicated regions for moving users requires further discussion.

4. Optimal relay position seeking and guidance law

This subsection proposes and discusses adaptive optimal relay position seeking algorithms for end-to-end and multi-user communication relay missions.

4.1. Optimal relay position-seeking for end-to-end communication

In end-to-end communication, a lead node is deployed to accomplish tasks in a far-f ield region. An airborne relay UAV is used to satisfy the LoS requirement and improve the communication performance between the lead node and the control station.24Information exchange should occur only between two distinguished users, and chain performance is normally limited by the uplink channel with the smaller received signal strength, hence, the optimal relay positionseeking problem can be expressed as

Maximizing the continuous objective function J=min｛Spu,p1,Spu,p2｝ is equivalent to f inding p*usuch that

The use of the min(·) function in Eq. (22) means that J is non-smooth, thus the derivation of J needs to be understood in terms of the least-norm element of the generalized gradient.

Proposition 1. Seeking the optimal relay positionis equivalent to seeking a position puwhere (1) Spu,p1=Spu,p2and (2)pu=p1+λ(p2-p1)(or pu=p2+(1-λ)(p1-p2)), where.0 ＜λ ＜1.

Proof. Condition (2) should be considered to mean that position puis located in segment [p1,p2] def ined by p1and p2. This can be proven in two steps:

First, in a suff iciency proof: if p=p*, then S1(p)=S2(p)and p=p1+λ(p2-p1). Assuming S1＞S2, according to Eq.(24), ?J=?S2, then ?S2=0, which is only obtained when d2=0, while Siis inversely proportional to distance diand S1＞S2.Thus d1＜0,which is impossible and the assumption cannot hold. Similarly, S1■S2. Thus, S1(p)=S2(p). Next,assume that?[p1,p2]. Def initely, there exists one position p′∈[p1,p2] satisfying Sp′,p1=Sp′,p2, then Spu,p1=Spu,p2≥Sp′,p1=Sp′,p2. This yields dpu,p1≤dp′,p1and dpu,p2≤dp′,p2, resulting in dpu,p1+dpu,p2≤dp′,p1+dp′,p2=dp2,p1,which is also impossible because of the triangle properties.The assumption cannot hold, hence, theremust be p*u∈[p1,p2],which completes the suff iciency proof.

Second, in the necessity proof: if Spu,p1=Spu,p2and pu∈[p1,p2], then pu=Assume that position p′satisf ies Sp′,p1=Sp′,p2and p′=p1+λ(p2-p1), while p′is not the optimal relay position. Sinceis the optimal relay position,this produces Sp′,p1=Sp′,p2＞Spu,p1=Spu,p2, and thus dp′,p1＜dpu,p1. dp2,p1is constant, thus, dp′,p2＞dpu,p2, resulting in Sp′,p2＜Spu,p2. This contradicts Sp′,p2＞Spu,p2, thus, p′=must hold. Proposition 1 is proven.

With Proposition 1,the objective function depends only on parameter λ. A line search approach can thus be employed to decrease the optimal relay position search space in airborne end-to-end communications. The adaptive optimal relay position-seeking algorithm is given in Table 1.

Note that in some situations, the relay UAV cannot converge to the optimal relay position as time increases. For these, the algorithm convergence is studied:

Proposition 2. To drive the relay UAV to converge with the optimal position, the velocity of the UAV must be faster than the maximum velocity of the users, namely v ＞max(v1,v2).

Proof. Let vp*be the velocity of the optimal relay position. If the relay UAV is desired to converge to p*,v ＞vp*must be satisf ied.As proven in Proposition 1,p*=p1+λ(p2-p1),resulting in v*=v1+λ(v2-v1) by derivation, where 0 ＜λ ＜1. As v= ｜｜˙p*｜｜＜max ｜｜v1｜｜, ｜｜v2｜｜( ), if v ＞max(v1,v2), then,v ＞ ｜｜v*｜｜must hold, where ｜｜v*｜｜is another form of vp*. This proves Proposition 2.

Proposition 2 shows that UAVs with operation velocities faster than the maximum possible velocities of user vehicles are preferred for relays in airborne end-to-end communication situations, ensuring the stability of a relay task.

Algorithm 1 contains k decision steps, none of which consists loop operations. The complexity of KF is no more than O(2×12×43) as the computational complexities of matrix multiplication and inversion approximately equal to O(p3),where O(p3)is the matrix dimension.The complexity of LSE is no more than O(4×κ3), where κ denotes the number of samples. The complexity of one dimensional line search algorithm is O(z), where O(z) is the iterations. Since κ is normally much bigger than 4, the complexity of each step is mainly caused by LSE algorithm. The complexity of Algorithm 1 is O(kκ3).

Table 1 Adaptive optimal relay position-seeking algorithm(Algorithm 1).

4.2. Optimal relay position seeking for multi-user communication

Each user of an airborne multi-user system may need to communicate with any other users; however, the uplink performances are different. The min(·) function described in Section 4.1 is thus inappropriate for the evaluation of the performance of the multi-user communication network. Two types of objective functions are typically used to ref lect the network uplink performance: the global network performance and the worst-two-channel performance.25

4.2.1. Optimizing the global network performance

As in Ref.26, the objective function given in Eq. (25) is employed to ref lect the global network performance and drive the relay UAV to the optimal relay position p*u:

First, consider the special case where the path loss exponents of all user vehicles are def ined by α1=α2=···=αN=2. Solving Eqs. (23) and (25) yields

A similar adaptive algorithm described in Section 4.1 can then be applied to seek the optimal relay position, where step 5 from Table 1 could be directly calculated and replaced by Eq.(26).

However,many situations do not fulf ill this assumption.To f ind the global optimal solution to Eq. (25), a multidimensional search method should be executed, which is diff icult.This work proposes a method of decreasing compatibility to solve this problem. The authors of Ref.9proposed a gradient based method, in which a position in the positive gradient direction was selected at each decision time instance and used to drive the UAV to this position using the guidance law. The relay UAV could also converge to the global optimal position.A loiter method was used to obtain the signal strength data for the estimation of the gradient direction in Ref.9, whereas the channel parameters were already estimated in this work as per Section 2. Thus, a new method is proposed to calculate the gradient direction of Eq. (25) with less computation consumption.

For each user ni,i=1,2,...,N,the received signal strength Sp′u(β),^piat position p′u(β)=pu+[R cosβ,R sinβ]T, 0 ≤β ＜2π from user niat position^pican be predicted based on the former wireless channel parameter estimation model(Eq.(21)),where puis the present position of the relay UAV,R is a constant distance away from pu, and ^piis the predicted position of user niusing a Kalman f ilter. The objective function J can now be reformulated as J(β) with J(β)=0 ≤β ＜2π.

The positive gradient direction β*can thus be estimated as:

J(β)depends only on one parameter,β,hence,a line search method is suff icient to solve Eq. (27), where β*is the fastest direction to improve the network performance. The optimalt

arget relay position of the UAV during the next time step is given as Eq. (28) based on the gradient climbing method:where γ is a pre-def ined dimensionless quantity, and(cosβ*,sinβ*) is the unit direction vector of the optimal target position. The adaptive algorithm to solve the general optimal relay position seeking problem is given in Table 2.

Although this is a generalized method that could be applied to optimal relay position seeking in end-to-end communication,Algorithm 1 requires only one dimensional search with less calculation,while Algorithm 2 is not suggested for these situations.

In Algorithm 2, there exist N users. Taking advantage of the complexity results from Algorithm 1, we can easily f ind that the complexity of Algorithm 2 is O(kNκ3).

4.2.2. Optimizing the worst-two-channel network performance

The global network performance optimization approach uses measured signals to drive the relay UAV to an optimal position. However, this leads to cases with extraordinarily poor performance in some channels, preventing information exchange through these channels.If a mission requires all users to be served, another optimal relay position seeking method can be employed by optimizing the worst-two-channel performance, which differs from end-to-end communication. In the latter situation, the messages are exchanged only between two users,whereas it is unknown where messages will be transmitted in the former. As a result, a new objective function is presented as a network performance metric.

Consider ni1and ni2to be the two users with the smallest signal strengths, Spu,pi1and Spu,pi2, respectively, at the receiver of the relay UAV.The optimal relay position seeking problem can now be expressed as

As a similar optimization procedure exists in the global network performance situation, a line search method can then be employed to f ind the optimal target position in each time step,and the optimization algorithm is not replicated herein. Theobjective function in Eq. (29) can replace that in Eq. (25) in step 5 of Algorithm 2 to implement the adaptive optimal relay position seeking method by optimizing the worst-two-channel performance as described above.

Table 2 Adaptive algorithm to solve the general optimal relay position seeking problem (Algorithm 2).

4.3. Guidance law based on Lyapunov guidance vector f ield

The mobility controller of the relay UAV iteratively estimates the optimal position, called the control point and written as(subscript tkdropped for this section). A guidance law is required to drive the f ixed-wing UAV to this position. A Lyapunov Guidance Vector Field (LGVF) controller,presented by Frew et al.27,28was used in this work. Since the UAV ground speed is assumed constant and a f irst order kinematic model is considered, the LGVF guidance law aims to generate the command parameter ˙ψu(yù), representing the vehicle turning rate determined at time instance tk. The relay UAV then f lies with this turning rate during time steps tkand tk+1.

Let r=pu-=[xr,yr]Tbe the relative position vectors of the present UAV position relative to the control point. The kinematic model of the UAV can then be rewritten as follows:

where vr=is the relative speed, ηiis the relative course angle, and kηis a gain with

The command parameter ˙ψu(yù)is generated by minimizing the Lyapunov function:

A Guidance Vector Field (GVF) function f(r) was used to drive the UAV to the desired orbit with a radius rd,k:

The desired relative heading ηdand feedforward of the desired relative heading rate ˙ηdis then be calculated as

A Heading Tracking Controller (HTC) is used to regulate the aircraft onto in the guidance vector f ield. Combining ηdand ˙ηdto yield the turn rate command:

where Klis a gain, -π ＜〈ηu-ηd〉≤π represents the unwrapped difference with considering UAV kinematic constraints:

It should be f igured out that the UAV acceleration is not considered in this guidance law; in other words, the guidance law should be modif ied if the acceleration has to be considered in the application so as to drive the UAV to maintain tracking the optimal relay position.

5. Simulation and results

In this section, simulations are provided to demonstrate the proposed mobility control algorithms and their performances.Mobile users in simulations were constrained to the smooth turn mobility model29,and the f ixed-wing UAV is constrained by its kinematic characteristics.The velocities and headings of mobile users at each time step are not available to the UAV beforehand, but are estimated using the KF algorithm as described in Section 2.2. The wireless channel parameters are not known to the UAV beforehand, either, but are estimated based on Eq. (32) given in Section 3.2

5.1. Relay for end-to-end communication

Because the methods of Dixon9and Chamseddine19cannot be applied to applications with mobile users and unknown AoA information, as discussed herein, comparisons are drawn among the proposed work, theoretical results and two other methods: (A) optimal position seeking using a distance based model, where the optimal position is selected as the coordinated center of the users; (B) a different channel parameter estimation method based on Maximum Likehood Estimation9(MLE) provided by Malmichegini and Mostof i is used to replace the LSE estimator in Algorithm 1, which is denoted as the ‘‘Mostof i-MLE” algorithm.

5.1.1. Two static users

In the f irst simulation,two static users are randomly placed in the area, and Algorithm 1 is applied to drive the relay UAV from its randomly initialized position to seek and move to the optimal relay position. The user channel parameters are=100 mW,=100, α1=2.8 and=100 mW,G′2 =200, α2=3.0, the UAV velocity 40 m/s, the bank angle limited to 40°, and the loiter radius is 200 m. The initial positions are p1=[370,2348], p2=[7701,2194] and pu=[3210,6626]. Figs. 1 and 2 show the results.

The square and blue points in Fig.1 represent the two static users,and the UAV f lies from the dark-square point to seek its optimal position using Algorithm 1. The f lying path of the UAV is denoted by the red line, whereas the theoretical optimal path(fastest path for the improvement of network performance) considering the kinematic constraints of the UAV is denoted by the blue line.The green line denotes the path using the distance based optimal position seeking method, and the cyan line denotes the f lying path using the Mostof i-MLE algorithm.The theoretical relay position is[4629,2258]and the loiter center of the UAV is[4679.2305],which are notably close to each other.

The network performance is evaluated using Eq.(22)and is shown in Fig. 2, where the red line shows the network performance achieved by the UAV,the blue line the theoretical optimal network performance, the green line the network performance achieved using the distance based method, and the cyan line indicates the network performance achieved using the Mostof i-MLE algorithm.Oscillations emerge in the curves because the f ixed wing UAV could not hover at a single point.

These two f igures show that:(A)Although the speed of the network performance improvements achieved by Algorithm 1 is lower than the theoretical path, it is acceptable, and the UAV gradually loiters close to the optimal relay position with the network performance converging to the theoretical maximum network performance. (B) The performance of the distance based optimal relay position seeking algorithm is rather poor, which is, therefore, better not be used in applications. The Mostof i-MLE algorithm converges faster than the proposed algorithm. However, the complexity of this algorithm (＞O((z′(15+κ)+12)×κ3), where z′is the iterations for seeking the optimal value of the correlation distance) is much higher than the proposed LSE algorithm. In many realistic airborne relay missions, the computational capabilities of the UAVs are strictly limited, and more importantly, the performance of the proposed algorithm is acceptable because the UAV path converges to the theoretical optimal path and the network performance also converges. Thus, the proposed algorithm shows its value in realistic applications where computation capabilities are limited.

5.1.2. Two mobile users

This simulation considers two mobile users,where the channel parameters between them and the UAV are=100 mW,

The UAV velocity is 40 m/s and limited to a 40°maximum bank angle. The user velocities are 15 m/s and 20 m/s,respectively.

Fig. 1 Comparisons of UAV paths in relaying for end-to-end communications between two static users.

Fig. 2 Comparisons of network performance variations in relaying for end-to-end communications between two static users.

Fig. 3 User, relay UAV, and optimal relay position paths in relay for end-to-end communication between two mobile users.

Fig.3 shows the path of the relay UAV (red line), paths of the two mobile users(blue lines),the theoretical relay position set(green line),the path of the relay UAV using a model-based optimal position seeking method(yellow line),and the path of the relay UAV using the Mostof i-MLE algorithm (cyan line).The red line demonstrates that the UAV eventually converged to the optimal relay position and stably tracked the changing relay position. Fig. 4 also proves this conclusion, because the realistic network performance using the proposed mobility control method improved and converged with the theoretical maximum network performance.

Fig.4 Comparison between the network performances achieved by Algorithm 1 and theoretical maximum in relay for end-to-end communication between two mobile users.

Comparison between the red line and cyan line shows that the network performance provided by replacing LSE in Algorithm 1 with the Mostof i-MLE algorithm will be improved to a limited degree. However, as aforementioned,the computational cost also improves.

The abovementioned simulation demonstrates that Algorithm 1 is effective in controlling the mobility of the relay UAV even in scenarios without prior knowledge of the channel parameters or the two mobile users,whereas the method proposed by Dixon9,12cannot be applied to scenarios with mobile users.

5.2. Relay for multi- user communication

As the distance based optimal relay position seeking algorithm has shown really poor performance and the Mostof i-MLE algorithm requires much more computational capability, this section concentrates on comparisons between Algorithm 2 and the theoretically optimal path.

This section considers a scenario with 6 mobile users following random moving paths denoted by blue lines in Fig. 5.The relay UAV has a velocity of 30 m/s, a maximum bank angle of 40°, and an initial position indicated by the red and square points in Fig. 5. For the channel parameters, the user velocities are randomly set between 10 m/s and 20 m/s, their transmitting powers varying from 100-250 mW, the channel gains from 100-200, and the path loss exponents from 2.0 to 4.0. Considering the different metric functions of the network performances, namely, the global and the worst-two-channel network performances (Eqs. (27) and (29), respectively), the realistic UAV movement and theoretically optimal paths are studied,with results shown in Fig.5.The theoretically optimal paths are generated by solving the nonlinear programming problem under the assumption that all required knowledge is available, which is practically impossible in realistic applications.

The separated green and cyan lines in Fig. 5 show that the theoretically optimal relay positions differed between the two different network performance metric functions. Similarly,Algorithm 2 provides different UAV f light paths(red and dark lines in Fig. 5) for the different network performance metric functions.

Fig. 5 Paths of relay UAV and optimal relay position reached by optimizing global and worst-two-channel network performances.

Fig. 6 Global network performance variation curve.

Fig.7 Worst two channel network performance variation curve.

The changes in the network performances related to the paths in Fig.5 are studied.Fig.6 shows that the global network performances generated by Algorithm 2 through respectively optimizing the global and worst-two-channel network performances demonstrate that:(A)the global network performance(red line in Fig.6)is improved by the UAV and eventually converged with the theoretical maximum(green line in Fig.6);(B)the global network performance generated by the worst-twochannel optimal path is worse than that of the global optimal path,because the red line is lower than the blue line.

Fig. 7 shows the results of the worst-two-channel network performance with the same setup. In this case, however, the worst-two-channel network performance achieved on the path optimizing the global network performance is worse than that achieved by the path optimizing the worst-two-channel network performance.

The simulations in this section demonstrate that Algorithm 2 is effective in controlling the relay UAV mobility in scenarios with multiple and mobile users without using AoA information or assuming that the path loss exponents equal 2(as compared to the algorithm proposed by Chamseddine19).

6. Conclusions

(1) This study considered the problem of using a UAV as a relay for supporting airborne multi-user communication by jointly considering unknown Radio Frequency (RF)channel parameters, user mobility and non-available Angle of Arrival (AoA) information provided by the receiver.

(2) A line search algorithm was proposed for end-to-end communication relay to drive the UAV toward or track the optimal relay position. Its effectiveness and stability were proven.

(3) Different network performance functions were considered for multiple mobile user communication relays,and a simplif ied gradient based controlling algorithm was proposed to generate the optimal target point at each time step based on the estimated channel parameters and the predicted user positions.

(4) Both the proposed mobility control algorithms need only a one-dimensional search, requiring no AoA information or RF parameters,and could be used in applications with mobile users.

Acknowledgement

This study was supported by the National Natural Science Foundation of China (No. 61573285).