Cellular Controlled Cooperative Unmanned Aerial Vehicle Networks with Sense-and-Send Protocol
aa r X i v : . [ c s . S Y ] M a y Cellular Controlled Cooperative Unmanned AerialVehicle Networks with Sense-and-Send Protocol
Shuhang Zhang, Hongliang Zhang,
Student Member , IEEE , Boya Di,
Student Member , IEEE ,and Lingyang Song,
Senior Member , IEEE
Abstract —In this paper, we consider a cellular controlled un-manned aerial vehicle (UAV) sensing network in which multipleUAVs cooperatively complete each sensing task. We first proposea sense-and-send protocol where the UAVs collect sensory dataof the tasks and transmit the collected data to the base station.We then formulate a joint trajectory, sensing location, and UAVscheduling optimization problem that minimizes the completiontime for all the sensing tasks in the network. To solve this NP-hard problem efficiently, we decouple it into three sub-problems:trajectory optimization, sensing location optimization, and UAVscheduling. An iterative trajectory, sensing, and scheduling op-timization (ITSSO) algorithm is proposed to solve these sub-problems jointly. The convergence and complexity of the ITSSOalgorithm, together with the system performance are analysed.Simulation results show that the proposed ITSSO algorithmsaves the task completion time by 15% compared to the non-cooperative scheme.
Index Terms —unmanned aerial vehicle, sense-and-send, tra-jectory optimization, cooperative sensing.
I. I
NTRODUCTION
Unmanned aerial vehicle (UAV) is an emerging facilitywhich has been widely applied in military, public, and civilapplications [1]. Among these applications, the use of UAVto perform data sensing has been of particular interest owingto its advantages of on-demand flexible deployment, largerservice coverage compared with the conventional fixed sensornodes, and additional design degrees of freedom by exploitingits high mobility [2]–[4]. Recently, UAVs with cameras orsensors have entered the daily lives to execute various sensingtasks, e.g. air quality index monitoring [5], autonomous targetdetection [6], precision agriculture [7], and water stress quan-tification [8]. Besides, the sensory data collected in such tasksneeds to be immediately transmitted to the base stations (BSs)for further processing in the servers, thereby posing lowlatency requirement on the wireless network.To this end, the cellular network controlled UAV transmis-sion is considered to play an important role in satisfying thelow latency requirement. In the traditional UAV ad hoc sensingnetwork [9], [10], the sensory data is transmitted through UAV-to-UAV and UAV-to-ground communications over unlicensedband, which cannot guarantee the QoS requirements. Recently,the network controlled UAVs are proposed to transmit sensorydata to BSs directly through the cellular network [11]–[13].In [11], the authors analyzed the use of LTE for realizing UAVsensing network, which improves the data rate. In [12], thepotential of UAVs as Internet of Things devices was discussed,
The authors are with the School of Electronics Engineering and Com-puter Science, Peking University, Beijing, China. Email: { shuhangzhang,hongliang.zhang, diboya, lingyang.song } @pku.edu.cn. which can reduce the latency of the network. In [13], a cellularUAV sensing network was proposed to improve the data rate.In this paper, we study a single cell UAV sensing net-work in which UAVs perform sensing tasks and transmit thecollected data to a BS. Note that sensing failure may occurdue to imperfect sensing in practical systems. Therefore, weadvocate UAV cooperation in the sensing networks to furtherimprove the successful sensing probability [14]. To be specific,multiple UAVs are arranged to collect the sensory data for thesame sensing task and to transmit the collected data to theBS separately. In this way, the successful sensing probabilityrequirement for each UAV is loosened [15], [16], and the taskcompletion time of each UAV can be shortened [17].Although UAV cooperation has the advantages in reducingthe sensing failure probability and task completion time, italso involves some challenges. Firstly, as the UAV schedulingwill influence the sensing performance, an efficient schedulingscheme is necessary. Secondly, since each sensing task isperformed by multiple UAVs, the trajectories and sensinglocations of UAVs are coupled with each other. In light ofthese issues, we first propose a sense-and-send protocol tosupport the cooperation and facilitate the scheduling. Then,we optimize the trajectories, sensing locations, and schedulingof these cooperative UAVs to minimize the completion timefor all the tasks. As the problem is NP-hard, we decomposeit into three subproblems, i.e., trajectory optimization, sensinglocation optimization, and UAV scheduling, and solve it by aniterative algorithm with low complexity.Note that in literature, most works focused on either sensingor transmission in UAV networks, instead of joint consideringUAV sensing and transmission. The work in [18] presented aplatform to deal with the cooperation and control of multipleUAVs with sensing and actuation capabilities for load trans-portation and deployment. The estimation problem for both theposition and velocity of a ground moving target was addressedin [19] using a team of cooperative sensing UAVs. In [20],a searching algorithm to make multiple UAVs autonomouslyand cooperatively search roads in the urban environments wasproposed. In [21], the authors considered a multi-UAV enabledwireless communication system, in which the UAVs work asBSs cooperatively to serve the ground users taking the fairnessinto consideration. Multiple UAVs were deployed as wirelessBSs to provide a better communication coverage for groundusers in [22]. The multiple cell scenario is an extension of the single cell scenario, andwill be studied in the future.
The main contributions of this paper can be summarizedbelow.(1) We propose a cooperative UAV sensing network wheremultiple UAVs complete the same sensing task coopera-tively, and transmit the collected data to a BS. A sense-and-send protocol is designed to support such cooperation.(2) We decompose the joint optimization problem intothree sub-problems: trajectory optimization, sensing lo-cation optimization, and UAV scheduling, and proposean iterative trajectory, sensing, and scheduling optimiza-tion (ITSSO) algorithm to jointly solve the sub-problems.The theoretical performance analysis is then studied.(3) Simulation results verify the theoretical analysis, and showthat the proposed cooperative sensing scheme outperformsthe non-cooperative and fixed sensing location schemes.The rest of this paper is organized as follows. In Section II,we describe the system model of the cooperative UAV sensingnetwork. In Section III, we elaborate the sense-and-sendprotocol for the cooperative UAVs. In Section IV, we formulatethe task completion time minimization problem by optimizingthe trajectory, sensing location, and UAV scheduling. TheITSSO algorithm and the algorithm analysis are proposed inSection V. Simulation results are presented in Section VI, andfinally we conclude the paper in Section VII.II. S
YSTEM M ODEL
We consider a single cell OFDM cellular network as shownin Fig. 1, which consists of one BS, M UAVs, denoted by M = { , , ..., M } , and K orthogonal subcarriers, denotedby K = { , , ..., K } . Within the cell coverage, there are N sensing tasks to be completed, denoted by N = { , , ...N } .The UAVs perform each task in two steps: UAV sensingand UAV transmission, and thus, these two procedures willbe repeated by the UAVs until all the tasks are completed.Note that different types of sensing tasks require UAVs withdifferent sensors [23], [24]. Therefore, the sensory data of eachtask is collected by a predefined UAV group cooperatively,and the UAVs in this group send the collected data to the BSseparately . We denote the UAV group that performs task j by W j , satisfying W j ⊆ M , and |W j | = q . For UAV i , it isrequired to execute a subset of tasks in sequence, denoted by N i = { , , ...N i } , ∀ i ∈ M , with N i ⊆ N . In the following,we first describe the UAV sensing and UAV transmission steps,and then introduce the task completion time of the UAVs inthe network. A. UAV Sensing
In UAV sensing, it is important to design the trajectory ofeach UAV along which it moves towards the locations of asequence of tasks. Without loss of generality, we denote thelocation of the BS by (0 , , H ) , and the location of task n Unlike most of the previous works, which typically treat UAVs as relaysor BSs [25], [26], our work considers the UAV as a flying mobile terminal inthe UAV sensing network. Tasks such as geological detection can be performed with this model,where each UAV is arranged to perform a series of tasks, and the geologicalinformation of each task is sensed by multiple UAVs. (cid:37)(cid:54) (cid:56)(cid:36)(cid:57)(cid:3)(cid:20)(cid:56)(cid:36)(cid:57)(cid:3)(cid:21) (cid:56)(cid:36)(cid:57)(cid:3)(cid:86)(cid:72)(cid:81)(cid:86)(cid:76)(cid:81)(cid:74)(cid:56)(cid:36)(cid:57)(cid:16)(cid:37)(cid:54)(cid:3)(cid:88)(cid:83)(cid:79)(cid:76)(cid:81)(cid:78) (cid:56)(cid:36)(cid:57)(cid:3)(cid:87)(cid:85)(cid:68)(cid:77)(cid:72)(cid:70)(cid:87)(cid:82)(cid:85)(cid:92)(cid:55)(cid:68)(cid:86)(cid:78)(cid:3)(cid:79)(cid:82)(cid:70)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)
Fig. 1. System model for UAV cooperation. by ( x n , y n , . In time slot t , let l i ( t ) = ( x i ( t ) , y i ( t ) , z i ( t )) be the location of UAV i , and v i ( t ) = ( v xi ( t ) , v yi ( t ) , v zi ( t )) be its velocity, with v i ( t ) = l i ( t ) − l i ( t − . Due to thespace and mechanical limitations, the UAVs have a maximumvelocity v max . For safety consideration, we also assume thatthe altitude of the UAVs in this network should be no lessthan a minimum threshold h min .In time slot t , the distance between UAV i and the BS isexpressed as d i,BS ( t ) = p ( x i ( t )) + ( y i ( t )) + ( z i ( t ) − H ) . (1)The distance between UAV i and task j is given by d i,j ( t ) = p ( x i ( t ) − x n ) + ( y i ( t ) − y n ) + ( z i ( t )) . (2)We utilize the probabilistic sensing model as introducedin [27]–[29], where the successful sensing probability is anexponential function of the distance between the sensing UAVand the task location. The successful sensing probability forUAV i to perform sensing task j can be shown as P R ( i, j ) = e − λd i,j ( t ) , (3)where λ is a parameter evaluating the sensing performance.The probability that task j is successfully completed can beexpressed by P R j = 1 − Y i ∈W j (1 − P R ( i, j )) . (4)Define P R th as the probability threshold, and task j can beconsidered to be completed when P R j ≥ P R th . B. UAV Transmission
In UAV transmission, the UAVs transmit the sensory datato the BS over orthogonal subcarriers to avoid severe inter-ference. We adopt the 3GPP channel model for evaluating theurban macro cellular support for UAVs [31].Let P U be the transmit power of each UAV. The receivedpower at the BS from UAV i in time slot t can then beexpressed as P i,BS ( t ) = P U P L a,i ( t ) / , (5) where P L a,i ( t ) is the average air-to-ground pathloss, definedby P L a,i ( t ) = P L,i ( t ) × P L
L,i ( t )+ P N,i ( t ) × P L
N,i ( t ) . Here, P L
L,i ( t ) and P L
N,i ( t ) are the line-of-sight (LoS) and non-line-of-sight (NLoS) pathloss from UAV i to the BS, with P L
L,i ( t ) = 28 + 22 × log( d i,BS ( t )) + 20 × log( f c ) , and P L
N,i ( t ) = − . − × log( z i ( t ))) × log( d i,BS ( t ))+20 × log( π × f c ) , respectively, where f c is the carrier frequency. P L,i ( t ) and P N,i ( t ) are the probability of LoS and NLoS,respectively, with P N,i ( t ) = 1 − P L,i ( t ) . The expression ofLoS probability is given by P L,i ( t )= , d Hi ( t ) ≤ d , d d Hi ( t ) + e (cid:18) − dHi ( t ) p (cid:19)(cid:18) − d dHi ( t ) (cid:19) , d Hi ( t ) >d , (6)where p = 4300 × log( z i ( t )) − , d = max { × log( z i ( t )) − , } , and d Hi ( t ) = p ( x i ( t )) + ( y i ( t )) .Note that the cooperative sensing and transmission processis completed in a long time period, and small scale fading isneglected in the transmission channel model in problem (12).Therefore, the signal-to-noise ratio (SNR) from UAV i tothe BS is given by γ i ( t ) = P i,BS ( t ) σ , (7)where σ is the variance of additive white Gaussian noisewith zero mean. For fairness consideration, each UAV can beassigned to at most one subcarrier. We define a binary UAVscheduling variable ψ i ( t ) for UAV i in time slot t , where ψ i ( t ) = (cid:26) , UAV i is paired with a subcarrier , , otherwise . (8)Therefore, the data rate from UAV i to the BS is given by R i ( t ) = ψ i ( t ) × W B log (1 + γ i ( t )) , (9)where W B is the bandwidth of a subcarrier. C. Task Completion Time
For UAV i , the relation between two consecutive sensingtime slots τ ji and τ j +1 i is given as τ j +1 i X t = τ ji v i ( t ) = l i ( τ j +1 i ) − l i ( τ ji ) , ∀ j ∈ N i , (10)where l i ( τ ji ) and l i ( τ j +1 i ) are the sensing locations of its j thand j + 1 th task. We define the task completion time of UAV i as the number of time slots it costs to complete the sensingand transmission of all its tasks, which can be expressed as T i = τ N i i + T N i tran,i , (11)where τ N i i is the time slot in which it performs the datacollection for its last task, and T N i tran,i is the time that UAV i cost to complete the data transmission for its last task N i . III. S ENSE - AND -S END P ROTOCOL
In this section, we present the sense-and-send protocol forthe UAV cooperation. As illustrated in Fig. 2, the UAVsperform sensing and data transmission for the tasks in asequence of time slots. For each UAV, the time slots canbe classified into three types: sensing time slot, transmissiontime slot, and empty time slot. For convenience, we definethe location that the UAV performs a sensing task as the sensing location . In each sensing time slot, the UAV collectsand transmits data for its current task at the sensing location.In each transmission time slot, the UAV moves along theoptimized trajectory and transmits the collected data to theBS. A UAV is in an empty time slot if it has completed datatransmission and has not reached the next sensing location. Inan empty time slot, the UAV neither collects data nor transmitsdata, and only moves towards the next sensing location. In thefollowing, we will elaborate on these three types of time slots.When performing a task, a UAV is in the sensing time slotfirst, and then switches to the transmission time slot. Afterseveral transmission time slots, a UAV may either be in theempty slot or in the next sensing time slot. The BS optimizesthe trajectory, sensing location, and UAV scheduling for thecooperative UAVs in advance, and responses the requiredinformation to the corresponding UAVs in every time slot.The interaction between the BS and UAV for completing atask is given in Fig. 3.
Sensing Time Slot : The UAV hovers on the sensing locationand performs data collection and transmission in sensing timeslot. The UAV first sends UAV beacon to the BS over controlchannel, which contains the information of its location, theongoing sensing task, the location of the next sensing task,and transmission request. The BS then informs the UAV ofthe subcarrier allocation result and sensing location of its nexttask. Afterwards, the UAV performs data collection until theend of the time slot. The UAV performs data transmission tothe BS simultaneously if it is allocated to a subcarrier. For task j , the UAV hovers on the sensing location to collect data foronly one time slot, with a data collection rate R js . If the UAVhas not finished data transmission in the sensing time slot, itswitches to transmission time slot, otherwise, it switches to anempty time slot. Transmission Time Slot : When the UAV requires datatransmission to the BS, it will operate in the transmission timeslot. In the transmission time slot, the UAV first sends UAVbeacon to the BS over control channel, which contains theinformation of transmission request, UAV location, data lengthto transmit, and the location of the next sensing location. TheBS then informs the UAV of its trajectory and UAV schedulingsolutions in this time slot. Afterwards, each UAV moves alongthe optimized trajectory. In the meanwhile, the UAV performsdata transmission if it is allocated to a subcarrier. Otherwise,the UAV cannot transmit data in the current time slot and willsend data transmission request again in the next transmissiontime slot. The collected data with respect to task j should beuploaded to the BS before UAV i starts the next sensing task,i.e. P τ j +1 i t = τ ji +1 R i ( t ) ≥ R js , where τ ji is the sensing time slotof UAV i for its j th task. 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Sense-and-send protocol. 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Fig. 3. UAV-BS interaction process in the sense-and-send protocol.
If a UAV does not complete its data transmission in thecurrent time slot, it will occupy another transmission time slot,and request data transmission to the BS again in the next timeslot. When a UAV completes data transmission for the currenttask, it switches to sensing time slot if it has arrived at thesensing location of the next task. Otherwise, it switches to theempty time slot.
Empty Time Slot : A UAV is in the empty time slot whenit has completed the data transmission for the current task,and has not arrived at the next sensing location. In emptytime slot, the UAV sends UAV beacon that contains its currentlocation and its next sensing location to the BS over control channel. The BS responses the corresponding trajectory to theUAV. The UAV then moves along the optimized trajectory withneither sensing nor transmission in such a time slot. The UAVwill switch to sensing time slot when it arrives at the sensinglocation of the next task.
Remark 1:
In each time slot, at most K UAVs can performdata transmission to the BS. When more than K UAVs sendtransmission request to the BS in one time slot, the UAVshave to share the subcarriers in a time division multiplexingmanner.To describe the signaling cost over the control channelsfor the proposed protocol, we assume that the UAV beaconcontains no more than κ messages, and the trajectory, sensinglocation, and UAV scheduling responce contains at most ι messages. Therefore, the maximum signaling cost of thenetwork is M × ( κ + ι ) in each time slot. The maximumsignaling cost is restricted by the number of UAVs, and thesignaling of each user costs no more than hundreds of bits [30].Thus, the signaling cost of the network is tolerable.IV. P ROBLEM F ORMULATION
In this section, we first formulate the task completiontime minimization problem. Afterwards, we decompose it intothree sub-problems, and introduce the proposed algorithm thatsolves the three sub-problems iteratively.
A. Problem Formulation
Note that the time for completing all the tasks in thisnetwork is determined by the maximum task completion timeof the UAVs. Let T max be the maximum task completion timeof the UAVs in this network, i.e. T max = max { T i } , ∀ i ∈ M .To complete all the tasks efficiently, our objective is tominimize the maximum task completion time of the UAVs byoptimizing UAV trajectory that consists of speed and direction of the UAV, UAV sensing location, and UAV scheduling. Thus,the problem can be formulated by min { v i ( t ) , l i ( τ ji ) , ψ i ( t ) } T max , (12a) s.t. z i ( t ) ≥ h min , ∀ i ∈ M , (12b) k v i ( t ) k ≤ v max , ∀ i ∈ M , (12c) k v i ( τ ji ) k = 0 , ∀ i ∈ M , j ∈ N i , (12d) P R j ≥ P R th , ∀ j ∈ N , (12e) τ j +1 i X t = τ ji +1 R i ( t ) ≥ R js , ∀ i ∈ M , j ∈ N i , (12f) M X i =1 ψ i ( t ) ≤ K, ≤ t ≤ T max , (12g) ψ i ( t ) = { , } . (12h)The altitude and velocity constraints are given by (12b)and (12c), respectively. (12d) shows that the UAV’s velocityis zero when performing sensing, and (12e) implies that thesuccessful sensing probability for each task should be no lessthan the given threshold P R th . Constraint (12f) explains thatthe data transmission is required to be completed before thenext sensing task, and (12g) is the UAV scheduling constraint. B. Problem Decomposition
Problem (12) contains both continuous variables v i ( t ) and l i ( τ ji ) , and binary variable ψ i ( t ) , which is NP-hard. To solvethis problem efficiently, we propose an ITSSO algorithm, bysolving its three sub-problems: trajectory optimization, sensinglocation optimization, and UAV scheduling iteratively.In the trajectory optimization sub-problem, given the sens-ing locations l i ( τ ji ) , ∀ i ∈ M , j ∈ N i and the UAV scheduling ψ i ( t ) , ∀ i ∈ M , we can observe that different UAVs are inde-pendent, and different tasks of a UAV are irrelevant. Therefore,in this sub-problem, the trajectory for a single UAV betweentwo successive tasks can be solved in parallel. Without loss ofgenerality, we study the trajectory for UAV i between its j thand j + 1 th task in the rest of this subsection, and the UAVtrajectory optimization sub-problem can be written as min v i ( t ) ( τ j +1 i − τ ji ) , (13a) s.t. z i ( t ) ≥ h min , ∀ i ∈ M , (13b) k v i ( t ) k ≤ v max , (13c) k v i ( τ ji ) k = 0 , (13d) τ j +1 i X t = τ ji +1 R i ( t ) ≥ R js . (13e)In the sensing location optimization sub-problem, given theUAV scheduling result and trajectory optimization method, thesensing location optimization sub-problem can be written as min { l i ( τ ji ) } T max , (14a) s.t. k v i ( t ) k ≤ v max , ∀ i ∈ M , (14b) k v i ( τ ji ) k = 0 , ∀ i ∈ M , j ∈ N i , (14c) P R j ≥ P R th , ∀ j ∈ N , (14d) τ j +1 i X t = τ ji +1 R i ( t ) ≥ R js , ∀ i ∈ M , j ∈ N i . (14e)In the UAV scheduling sub-problem, given the trajectoryoptimization and sensing location optimization of each UAV,the UAV scheduling sub-problem can be written as min { ψ i ( t ) } T max , (15a) s.t. τ j +1 i X t = τ ji +1 R i ( t ) ≥ R js , ∀ i ∈ M , j ∈ N i , (15b) M X i =1 ψ i ( t ) ≤ K, ≤ t ≤ T max , (15c) ψ i ( t ) = { , } . (15d) C. Iterative Algorithm Description
In this subsection, we introduce the proposed ITSSO al-gorithm to solve problem (12), where trajectory optimiza-tion, sensing location optimization, and UAV scheduling sub-problems are solved iteratively. We firstly find an initial feasi-ble solution of problem (12) that satisfies all its constraints. Inthe initial solution, the trajectory, sensing location, and UAVscheduling of this solution are denoted by { v i ( t ) } , { l i ( τ ji ) } ,and { ψ i ( t ) } , respectively. We set the initial sensing locationof task j as ( x n , y n , h min ) for all the UAVs in W j . The initialtrajectory for UAV i between task j and j + 1 is set as theline segment between the two sensing locations, with the UAVspeed being v that satisfies v l i ( τ j +1 i ) − l i ( τ ji ) ≪ R js R , where R is the average transmission rate of UAV i for task j . In theinitial UAV scheduling the subcarriers are randomly allocatedto K UAVs in each time slot.We then perform iterations of trajectory optimization, sens-ing location optimization, and UAV scheduling until thecompletion time for all the tasks converges. In each iter-ation, the trajectory optimization given in Section V-A isperformed firstly with the sensing location optimization andUAV scheduling results given in the last iteration, and thetrajectory variables are updated. Next, the sensing locationoptimization is performed as shown in Section V-B, withthe UAV scheduling obtained in the last iteration, and thetrajectory optimization results. Afterwards, we perform UAVscheduling as described in Section V-C, given the trajectoryoptimization and sensing location optimization results. Whenan iteration is completed, we compare the completion timefor all the tasks obtained in the last two iterations. If thecompletion time for all the tasks does not decrease withthe last iteration, the algorithm terminates and the result isobtained. Otherwise, the ITSSO algorithm continues to thenext iteration.
We denote the optimization objective function after the r thiteration by T max (cid:16) { v i ( t ) } r , { l i ( τ ji ) } r , { ψ i ( t ) } r (cid:17) . In iteration r , the trajectory optimization variables { v i ( t ) } , the sensinglocation optimization variables { l i ( τ ji ) } , and the UAV schedul-ing variables { ψ i ( t ) } are denoted by { v i ( t ) } r , { l i ( τ ji ) } r , and { ψ i ( t ) } r , respectively. The ITSSO algorithm is summarizedin detail as shown in Algorithm 1. Algorithm 1:
Iterative Trajectory, Sensing, and SchedulingOptimization Algorithm.
Initialization:
Set r = 0 , find an initial solution ofproblem (12) that satisfies all its constraints, denote thecurrent trajectory, sensing location, and UAV schedulingby { v i ( t ) } , { l i ( τ ji ) } , and { ψ i ( t ) } , respectively; while T max (cid:16) { v i ( t ) } r − , { l i ( τ ji ) } r − , { ψ i ( t ) } r − (cid:17) − T max (cid:16) { v i ( t ) } r , { l i ( τ ji ) } r , { ψ i ( t ) } r (cid:17) > do r = r + 1 ;Solve the trajectory optimization sub-problem, given { l i ( τ ji ) } r − and { ψ i ( t ) } r − ;Solve the sensing location optimization sub-problem,given { v i ( t ) } r and { ψ i ( t ) } r − ;Solve the UAV scheduling sub-problem, given { v i ( t ) } r and { l i ( τ ji ) } r ; endOutput: { v i ( t ) } r , { l i ( τ ji ) } r , { ψ i ( t ) } r ;V. I TERATIVE T RAJECTORY , S
ENSING , AND S CHEDULING O PTIMIZATION A LGORITHM
In this section, we first introduce the algorithms that solvethe three subproblems (13), (14), and (15), respectively. Af-terwards, we analyse the performance of the proposed ITSSOalgorithm.
A. Trajectory Optimization
In this subsection, we provide a detailed description ofthe UAV trajectory optimization algorithm (13). Note thatwe utilize the standard aerial vehicular channel fading modelas proposed in [31], which makes the expression of con-straint (13e) very complicated. Therefore, problem (13) cannot be solved with the existing optimization methods. In thefollowing, we optimize the speed and moving direction of theUAVs with a novel algorithm utilizing geometry theorems andextremum principles.
1) UAV Speed Optimization:
Assume that the transmissiondistance from a UAV to the BS is much larger than theUAV velocity, i.e. d i,BS ( t ) ≫ v max , ∀ i ∈ M , we have thefollowing theorem on the optimization of UAV speed. Theorem 1:
The optimal solution can be achieved when thespeed of the UAV is v max between τ ji and τ j +1 i , i.e. k v i ( t ) k = v max , τ ji ≤ t ≤ τ j +1 i , ∀ i ∈ M , j ∈ N i . Proof.
See Appendix A.According to the proof of Theorem 1, we have the followingremark. (cid:37)(cid:54) (cid:54)(cid:72)(cid:81)(cid:86)(cid:76)(cid:81)(cid:74)(cid:3)(cid:79)(cid:82)(cid:70)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3) (cid:73)(cid:82)(cid:85)(cid:3)(cid:87)(cid:68)(cid:86)(cid:78) j (cid:54)(cid:72)(cid:81)(cid:86)(cid:76)(cid:81)(cid:74)(cid:3)(cid:79)(cid:82)(cid:70)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:3) (cid:73)(cid:82)(cid:85)(cid:3)(cid:87)(cid:68)(cid:86)(cid:78) j+1 (cid:55)(cid:88)(cid:85)(cid:81)(cid:76)(cid:81)(cid:74)(cid:3)(cid:83)(cid:82)(cid:76)(cid:81)(cid:87) (cid:54)(cid:72)(cid:81)(cid:71)(cid:76)(cid:81)(cid:74)(cid:16)(cid:51)(cid:85)(cid:76)(cid:82)(cid:85)(cid:76)(cid:87)(cid:92)(cid:3)(cid:71)(cid:72)(cid:87)(cid:82)(cid:88)(cid:85)(cid:54)(cid:72)(cid:81)(cid:86)(cid:76)(cid:81)(cid:74)(cid:16)(cid:51)(cid:85)(cid:76)(cid:82)(cid:85)(cid:76)(cid:87)(cid:92)(cid:3) (cid:53)(cid:82)(cid:88)(cid:87)(cid:72) Fig. 4. Moving direction optimization.
Remark 2:
With t ′ being the optimal solution, a trajectorywith the length of t ′ × v max can be given.Therefore, we set the UAV speed as v max in the followingparts.
2) UAV Moving Direction Optimization:
Since the speed ofthe UAV has been obtained by Theorem 1, we then proposean efficient method to solve the moving direction of UAV i between τ ji and τ j +1 i .For simplicity, we denote the time between UAV i ’s j th and j + 1 th task by δ ji = τ j +1 i − τ ji − . Let [ x ] be the minimuminteger that is no smaller than x . In problem (13), the lowerbound of δ ji can be expressed as δ j,lbi = [ l i ( τ j +1 i ) − l i ( τ ji ) v max ] , (16)which corresponds to a line segment trajectory, with l i ( τ j +1 i ) − l i ( τ ji ) being its moving direction. This direction is the solutionif constraint (13e) can be satisfied, i.e. P τ ji + δ j,lbi t = τ ji +1 R i ( t ) ≥ R js .Otherwise, the UAV has to make a detour to approach the BSfor a larger transmission rate, which also leads to a larger taskcompletion time.As illustrated in Fig. 4, the moving direction of the detouredtrajectory contains two parts, namely sending-priority detourand sensing-priority route. In the sending-priority detour part,the UAV detours to the BS for a larger transmission rate, andin the sensing-priority route part, the UAV moves toward itsnext sensing location with the shortest time. • Sending-Priority Detour:
In the sending-priority detourpart, we maximize the transmission rate of the UAV, sothat constraint (13e) can be satisfied with minimum timeslots. To achieve the maximum achievable rate, the UAVmoves along the direction with the maximum rate ascentvelocity, i.e. the gradient of the transmission rate R i ( t ) ,which can be expressed as ∇ R i ( t ) = ( ∂R i ( t ) ∂x , ∂R i ( t ) ∂y , ∂R i ( t ) ∂z ) , ( l i ( t ) > h min ) , (17)where the expression of R i ( t ) can be derived by equa-tions (1), (2), (5)-(9). In time slot t , if the altitude of the trajectory is below the minimum threshold h min , theUAV has to adjust its moving direction to ∇ R i ( t ) = ( ∂R i ( t ) ∂x , ∂R i ( t ) ∂y , . (18) • Sensing-Priority Route:
We define the endpoint of thesending-priority detour as the turning point , denoted by l tri ( τ ji ) . In the sensing-priority route part, the UAV movesfrom the turning point to the sensing location of the nexttask. To minimize the task completion time, the trajectoryof the this part is optimized as a line segment, with l i ( τ j +1 i ) − l tri ( τ ji ) being its moving direction.We denote the time duration of the sending-priority detourand the sensing-priority route by δ j, i and δ j, i , respectively.Our target is to find the minimum δ j, i + δ j, i that satisfiesconstraint (13e). We can observe that a larger δ j, i impliesa larger δ j, i since δ j, i is positively related with the detourdistance to the BS. Therefore, the minimum δ j, i + δ j, i isachieved with the minimum feasible δ j, i . The solution ofthe moving direction optimization problem is summarized inAlgorithm 2. Algorithm 2:
Moving Direction Optimization.
Initialization:
Set moving direction as l i ( τ j +1 i ) − l i ( τ ji ) ,and δ j, i = 0 ; while Constraint (13e) is not satisfied do δ j, i = δ j, i + 1 ;Optimize the moving direction of sending-prioritydetour;Find the location of the turning point ;Optimize the moving direction of sensing-priorityroute; end Set the current moving direction as the final solution;
B. Sensing Location Optimization
In this subsection, we propose a method to solve the sensinglocation optimization subproblem (14). It is known that con-straint (14e) is non-convex, and thus problem (14) can not besolved directly. Since the trajectory between two consecutivesensing locations has been optimized in Section V-A, theUAV trajectory is one-to-one correspondence with the sensinglocations. Note that constraints (14b), (14c), and (14e) can besatisfied with the trajectory optimization method proposed inSection V-A when a sensing location optimization method isgiven. In the following, we first analyse the properties of thesensing location, and then solve this sub-problem with a localsearch method.
Theorem 2:
For each UAV, the optimal sensing locationis collinear with the corresponding turning point and the tasklocation . Proof.
To satisfy constraint (14d), we assume that the distancebetween the sensing location a UAV and the task location If the UAV trajectory does not detour to the BS, the start point can beconsidered as the turning point . (cid:55)(cid:88)(cid:85)(cid:81)(cid:76)(cid:81)(cid:74)(cid:3)(cid:83)(cid:82)(cid:76)(cid:81)(cid:87) (cid:54)(cid:72)(cid:81)(cid:86)(cid:76)(cid:81)(cid:74)(cid:3)(cid:79)(cid:82)(cid:70)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:55)(cid:68)(cid:86)(cid:78)(cid:3) (cid:79)(cid:82)(cid:70)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81) Fig. 5. Illustration of Theorem 2. should be no more than d ′ while considering the sensing loca-tions of other UAVs are fixed. As shown in Fig. 5, the feasiblesolutions of the sensing location is a hemispheroid, with thetask location being the center and d ′ being the radius. We canobserve that the sensing location with the shortest trajectoryis on the the intersection of the hemispheroid and the linesegment from the turning point to the task location. Therefore,when the task completion time of a UAV is minimized, thesensing location is collinear with the turning point and thetask location.In the following, we derive the upper bound and lowerbound of δ ji , ∀ i ∈ M , j ∈ N i . Theorem 3:
The lower bound of δ ji , denoted by δ j,lbi , isachieved when the sensing location and the turning point are overlapped. The upper bound of δ ji , denoted by δ j,ubi ,is achieved when the the sensing location of task j + 1 is ( x j +1 , y j +1 , h min ) . Proof.
When the sensing location and the turning point areoverlapped, the UAV trajectory is the gradient of the trans-mission rate, which corresponds to the maximum achievabletransmission rate. Therefore, constraint (14e) can be satisfiedwith minimum number of transmission time slots. When thesensing location of task j + 1 is ( x j +1 , y j +1 , h min ) , the dis-tance between sensing location and task location is minimized,and the successful sensing probability is maximized. On thiscondition, the length of the trajectory is maximized, whichcorresponds to the maximum δ ji .It is shown that task completion time T i and successfulsensing probability P R ( i, j ) are negatively correlated. There-fore, a trade-off between the task completion time of UAV i and successful sensing probability is required to minimize thecompletion time for all the tasks while guaranteeing the thesuccessful sensing probability constraint (14d).When solving the sensing location optimization sub-problem, we propose a local search method to reduce thecomputational complexity. We first give an initial solution thatsatisfies all the constraints of problem (14), and set it as thecurrent solution. The local search method contains iterationsof sensing location adjustment. In each iteration, the algorithmis performed task by task. For task j , we first reduce themaximum task completion time of the UAVs in set W j byone time slot, and then adjust the sensing location of other UAVs in set W j to keep constraint (14d) satisfied. The localsearch method terminates when the completion time for all thetasks can not be reduced by the adjustment of any task.The detailed process of the sensing location optimizationmethod is elaborated in Algorithm 3. When performing thesensing location adjustment for task j , without loss of gener-ality, we assume that the maximum task completion time ofthe UAVs in W j is T i . If δ ji > δ j,lbi , we reduce δ ji by one timeslot by adjusting its sensing location. Denote U by the set ofUAVs in W j that satisfies δ jm ≤ δ j,ubm , and T m ≤ T i − ,so that the maximum task completion time of the UAVs in W j dose not increase. We adjust the sensing locations ofthe UAVs in set U to decrease the distances between theirsensing locations and the task location until constraint (14d)is satisfied. Denote the task completion time increment of UAV m by ∆ t m , its sensing location after adjustment is given as l i ( τ jm ) = l i ( τ jm ) + ∆ t m · v max · l m ( τ jm ) − l trm ( τ j − m ) | l m ( τ jm ) − l trm ( τ j − m ) | , ∀ m ∈ W j .If constraint (14d) can not be satisfied until U is empty, themaximum task completion time T i can not be reduced byadjusting the sensing locations of UAVs in W j . Algorithm 3:
Sensing Location Optimization.
Initialization:
Give a set of initial sensing locations { l i ( τ ji ) } ; while The maximum task completion time of any task isreduced in the last iteration dofor j = 1 : N do Find the maximum task completion time T i , ∀ i ∈ W j ; if δ ji > δ j,lbi then δ ji = δ ji − ;Adjust UAV i ’s sensing location l i ( τ ji ) ; if Constraint (14d) is satisfied then
Continue; elsewhile
Constraint (14d) is not satisfied doif U = ∅ then Break; endfor m ∈ U do δ jm = δ jm + 1 ; l i ( τ jm ) = l i ( τ jm )+ v max · l m ( τ jm ) − l tri ( τ j − m ) | l m ( τ jm ) − l trm ( τ j − m ) | ; endendendif Constraint (14d) cannot be satisfied then
Recover the adjusted data and continue; endendendendTheorem 4:
The iterative sensing location optimizationmethod is convergent.
Proof.
For task j , the maximum task completion time of the UAVs in W j may reduce or remain unchanged in eachiteration. Therefore, the time for completing all the tasks doesnot increase with the iterations. It is known that the timefor completing all the tasks has a lower bound. Therefore,the maximum task completion time can not reduce infinitely,and the iterative sensing location optimization method isconvergent. Theorem 5:
The complexity of the iterative sensing locationoptimization method is O ( N × q × N i × M ) . Proof.
In each iteration of the sensing location optimizationmethod, each of the N tasks is visited for one time, and thetotal number of UAV sensing for the N tasks is N × q . There-fore, the complexity of each iteration is N × q . The numberof iterations is determined by the number of task completiontime reduction when performing the local search algorithm.It is known that each UAV has a lower bound of its taskcompletion time, and the task completion time reduction of aUAV is in direct proportion to its task number N i . Therefore,the total number of task completion time reduction is no morethan N i × M , and the number of iteration is no more than N i × M . In conclusion, the complexity of the iterative sensinglocation optimization method is O ( N × q × N i × M ) . C. UAV Scheduling
In this subsection, we introduce the UAV scheduling methodthat solves sub-problem (15). Problem (15) is non-convexsince ψ i ( t ) is a discrete variable, which can not be solveddirectly. In the following, we propose an efficient method thatperforms UAV scheduling time slot by time slot.In each time slot, if the BS receives no more than K trans-mission requests, each of the request UAV will be allocatedwith one subcarrier. In the time slot that the BS receives morethan K transmission requests, the BS allocates the subcarriersto the K requesting UAVs with maximum task completiontime. The task completion time of each UAV is then updatedwith the change of UAV scheduling in this time slot, and thenthe BS continues the UAV scheduling of the next time slot.The process of UAV scheduling is shown as Algorithm 4. Algorithm 4:
UAV Scheduling Method. for t = 1 : T max doif Transmission request number < K then
Allocate a subcarrier to each of the request UAV; else
Allocate a subcarrier to the K requesting UAVswith maximum task completion time; end Update the task completion time of each UAV withthe new UAV scheduling; end
D. Performance Analysis
In this subsection, we first analyse the performance ofthe proposed ITSSO algorithm, including its convergence and complexity, and then analyse the system performance of thenetwork.
1) Convergence:
Theorem 6:
The proposed ITSSO algorithm is convergent.
Proof.
As shown in Section V-B, given the trajectory opti-mization method, the time for completing all the tasks does notincrease with the sensing location optimization. In the UAVscheduling, the time for completing all the tasks decreaseseach time the BS rearranges the subcarriers. Therefore, thetime for completing all the tasks does not increase with theiterations of the ITSSO algorithm. It is known that the time forcompleting all the tasks has a lower bound in such a network,and the objective function can not decrease infinitely. The timefor completing all the tasks will converge to a stable valueafter limited iterations, i.e. the proposed ITSSO algorithm isconvergent.
2) Complexity:
Theorem 7:
The complexity of the proposed ITSSO algo-rithm is O ( N × q × N i × M ) . Proof.
The proposed ITSSO algorithm consists iterations oftrajectory optimization, sensing location optimization, andUAV scheduling. In each iteration, the complexity of trajectoryoptimization is O ( M ) , and the complexity of UAV schedulingis O ( K × T max ) = O ( K × N i ) . The complexity of sensinglocation optimization is O ( N × q × N i × M ) , which is provedin Theorem 5. The number of ITSSO algorithm iterationsis relevant to the reduction of the time for completing allthe tasks, which is in direct proportion to the number oftasks N . Therefore, the complexity of the proposed ITSSOalgorithm is O ( N × ( M + K × N i + N × q × N i × M )) = O ( N × q × N i × M ) .
3) System Performance Analysis:
In this part, we analysethe impact of the cooperate UAV number q and the sensingprobability threshold P R th on the task completion time of theUAVs in the network. Theorem 8:
The average rate of change of UAV i ’s taskcompletion time T i to the cooperative UAV number q is ∆ T max ∆ q = (1 − P R th ) /q ln(1 − P R th ) λ (1 − (1 − P R th ) /q ) q × N i v max . (19) Proof.
See Appendix B.
Theorem 9:
The average rate of change of UAV i ’s taskcompletion time T i to the sensing probability threshold P R th is ∆ T max ∆ P R th = (1 − P R th ) /q − λq (1 − (1 − P R th ) /q ) × N i v max . (20) Proof.
See Appendix C.In the following, we analyse the dominated factor on thecompletion time for all the tasks.
Theorem 10:
The transmission resource is a dominatedfactor on the completion time for all the tasks when thenetwork is crowded.
Proof.
We denote the possibility that a UAV in transmissiontime slot is allocated with a subcarrier by p t , with p t ∝ K/M . Therefore, the average time that a UAV finishes data trans-mission for a task is η × MK , where η is a proportionalitycoefficient. The average time that a UAV cost to finish a taskcan be given as max { η × MK , ¯ δ lb } , where ¯ δ lb is the averagelower bound of the time that a UAV cost to finish a task.Therefore, the transmission resource is the dominated factoron the completion time for all the tasks when K satisfies η × MK > ¯ δ lb , i.e., K < η × M ¯ δ lb .We then discuss the impact of sensing task size R js onthe completion time for all the tasks in different transmissionresource schemes.1) High Transmission Resource:
In high transmission re-source scheme, the most of the UAVs in transmission timeslot are allocated with a subcarrier. The impact of sensingtask size R js on the completion time for all the tasks isnot significant when R js is at a low level, since most ofthe data transmission tasks can be completed without adetour trajectory. When R js is at a high level, the UAVsare more likely to detour to the BS for data transmission,and R js becomes a dominated factor on the completiontime for all the tasks.2) Low Transmission Resource:
In low transmission re-source scheme, the subcarriers are occupied by the UAVsin most of the time slots. On this condition, the taskcompletion time of the UAVs are extended, and mostof the UAVs detour to the BS for data transmission.A larger sensing task size R js corresponds to a longersensing-priority detour to the BS i.e. ∂T max ∂R js > . Withthe increment of sensing-priority detour, the UAV movescloser to the BS, which improves the average data rate.Therefore, we have ∂ T max ∂ ( R js ) < , and the sensing tasksize R js has a more significant impact on the completiontime for all the tasks T max when it is at a low level.The dominated factor on the completion time for all thetasks is concluded as Table I. The transmission resource K is adominated factor on the completion time for all the tasks whenit is at a low level. The sensing data size R js is a dominatedfactor of the completion time for all the tasks when R js and K are both at a high level or low level. TABLE ID
OMINATED FACTOR ON THE COMPLETION TIME FOR ALL THE TASKS .Transmission Resource K High LowSensing Task Size R js High R js K Low Neither R js & K VI. S
IMULATION R ESULTS
In this section, we evaluate the performance of the proposedITSSO algorithm. The selection of the simulation parame-ters are based on the existing 3GPP specifications [31] andworks [13]. For comparison, the following schemes are alsoperformed: • Non-Cooperative (NC) Scheme:
In NC scheme, eachtask is required to be completed with only one UAV, i.e. TABLE IIS
IMULATION P ARAMETERS
Parameter Value
BS height H
25 mCarrier frequency f c K W B R s σ -96 dBmUAV transmit power P U v max
50 m/sMinimum UAV altitude h min
10 mSensing performance parameter λ P R th q = 1 , and the number of tasks in the network is thesame with the proposed ITSSO scheme. The proposedtrajectory optimization, sensing location optimization,and UAV scheduling methods are also performed in NCscheme. • Fixed Sensing Location (FSL) Scheme:
The FSL isgiven as mentioned in [32]. In FSL scheme, the sensinglocations of the UAVs are given as the location rightover the locations of the corresponding tasks, with fixedheight H F SL = 50 m, and the sensing probabilityconstraint (12e) is not considered in this scheme. Thetask arrangement for each UAV in FSL scheme is thesame with the proposed ITSSO scheme, and the proposedtrajectory optimization and UAV scheduling methods areutilized in FSL scheme.For simulation setup, the initial location of the UAVs are ran-domly and uniformly distributed in an 3-dimension area of 500m ×
500 m ×
100 m, and the tasks are uniformly distributedon the ground of this area. We assume that the number of tasksfor different UAVs are equal, i.e. N i = N j = N × qM , ∀ i, j ∈ M ,and the task arrangement for each UAV is given randomly.The data collection rate for different tasks are fixed, denotedby R js = R s , ∀ j ∈ N . The values of N , q , and m are given ineach figure. All curves are generated by over 10000 instancesof the proposed algorithm. The simulation parameters arelisted in Table II.Fig. 6 depicts the completion time for all the tasks T max v.s.the number of cooperative UAVs q for each task. The numberof tasks in the network is set as 20, and each UAV is arrangedto complete 4 tasks. The number of UAVs M varies withvariable q , satisfying M × N i = N × q . It is shown that thecompletion time for all the tasks decreases with the incrementof cooperative UAVs for a task. The reason is that the averagedistance between the UAV and the task when performing datacollection increases with a larger cooperative UAV number, Number of cooperative UAVs for a task (q) C o m p l e t i on t i m e f o r a ll t he t a sks ( t i m e s l o t ) PR th =0.9PR th =0.8PR th =0.7 Fig. 6. Number of cooperative UAVs for a task vs. Completion time for allthe tasks ( N =20, N i =4). which corresponds to a shorter moving distance. The slopesof the curves decrease with the increment of q , which satisfiesthe theoretical results given in Theorem 8. The completiontime for all the tasks decreases for about 8% when we changethe sensing probability threshold from 0.9 to 0.8, and it furtherdecreases about 6% if the threshold is reduced to 0.7. Number of task for a UAV (N i ) C o m p l e t i on t i m e f o r a ll t he t a sks ( t i m e s l o t ) ISTSONCTAW
Fig. 7. Number of tasks for a UAV vs. Completion time for all the tasks( M =20, q =4). Fig. 7 shows the completion time for all the tasks T max v.s. the number of tasks for a UAV N i . In the ITSSO andFSL schemes, we set the number of UAVs as 20, and thenumber of cooperative UAVs for a task as 4. In the NC scheme,the number of UAVs is also set as 20, and each task is onlyperformed with one UAV. The completion time for all the tasksincreases linearly with the number of tasks for a UAV. Theslope of the ITSSO scheme is about 15% lower than that ofthe NC scheme due to a shorter average UAV moving distance.The completion time for all the tasks of the FSL scheme isover 50% larger than the ITSSO scheme due to the lack ofsensing location optimization.In Fig. 8, we plot the relation between the sensing prob-ability threshold P R th and the completion time for all thetasks T max . Here, we set the number of tasks as 20, andthe number of tasks for a UAV as 4. In the ITSSO and FSL Sensing probability threshold (PR th ) C o m p l e t i on t i m e f o r a ll t he t a sks ( t i m e s l o t ) ISTSONCTAW
Fig. 8. Sensing probability threshold vs. Completion time for all the tasks( N =20, N i =4, q =4). schemes, the number of cooperative UAVs for a task is setas 4. In the NC scheme, each task is performed by onlyone UAV. We can observe that the completion time for allthe tasks of the ITSSO and NC scheme increases with thesensing probability threshold due to the change of the sensinglocations. The completion time for all the tasks in the ITSSOscheme increases from 22 to 29 when the sensing probabilitythreshold increases from 0.5 to 0.9, and the completion timefor all the tasks in the NC scheme increases from 27 to 32 inthe same range of sensing probability threshold. The slope isconsistent with the theoretical results given in Theorem 9. Thecompletion time for all the tasks in the FSL scheme is around42, since the sensing locations are fixed in this scheme. Sensing task size (R s )
10 15 20 25 30 35 40 45 50 C o m p l e t i on t i m e f o r a ll t he t a sks ( t i m e s l o t ) ISTSONCTAW
Fig. 9. Sensing task size vs. Completion time for all the tasks ( N =20, N i =4, q =4). Fig. 9 shows the completion time for all the tasks as afunction of the sensing task size R s . We set the number oftasks as 20, and the number of UAVs as 10. In the ITSSO andFSL scheme, the number of cooperative UAVs for a task isset as 4. In the NC scheme, each task is performed by onlyone UAV. For the ITSSO scheme, the completion time for allthe tasks is around 28-30 when R s ≤ Mbps, where mostof the data transmission tasks can be completed without adetour trajectory. The completion time for all the tasks starts to increase significantly when R s > Mbps, since the UAVsare more likely to detour to the BS for data transmission.Note that the increment of the completion time for all thetasks is mainly caused by the trajectory detouring. Therefore,the completion time for all the tasks in all the three schemesincreases with the sensing task size, and the difference amongthe three schemes decreases with a larger R s . Sensing probability threshold PR th1-10 -1 -2 -3 -4 -5 -6 M i n i m u m c oope r a t i v e U AV f o r a t a sk q h min =10, Simulationh min =10, Theoreticalh min =20, Simulationh min =20,Theoretical Fig. 10. Sensing probability threshold vs. Minimum number of cooperativeUAVs for a task.
In Fig. 10, we study the minimum number of cooperativeUAVs required for completing a task with different sensingprobability thresholds. Given a high sensing probability re-quirement, UAV cooperation is necessary for the UAVs tocomplete a sensing task. We can observe that the minimumnumber of UAVs required for a task is logarithmically relatedto the sensing probability threshold, which is consistent withthe theoretical results in (3) and (4). The difference betweenthe simulation result and the theoretical result is less than 0.2within the simulation range. When we set
P R th = 1 − − ,the minimum number of UAVs required for a task is 1. Whenthe sensing probability threshold is set as P R th = 1 − − ,at least 6 UAVs are required to complete a task. The requirednumber of cooperative UAVs increases for about 50% whenwe adjust the minimum UAV altitude h min from 10 m to 20 m. Number of subcarriers (K) C o m p l e t i on t i m e f o r a ll t he t a sks ( t i m e s l o t ) R s =10MbpsR s =20MbpsR s =30MbpsR s =40MbpsR s =50Mbps Fig. 11. Number of subcarriers vs. Completion time for all the tasks ( q = 4 , M = 20 ). Fig. 11 illustrates the impact of subcarrier number K andtask size R s on the completion time for all the tasks. Thenumber of UAV is set as M = 20 , and number of cooperativeUAV for a task is set as q = 4 . It is shown that the completiontime for all the tasks is strongly affected by the task size R s with R s ≤ Mbps, and the marginal impact of R s decreaseswhen R s > Mbps. Given a fixed R s , the completion timefor all the tasks decreases rapidly with the number of subcar-riers with K ≤ . The ratio decreases until convergence withthe increment of the number of subcarriers. The simulationcurves satisfies the analysis of the dominated factor of thecompletion time for all the tasks given in Section V-D.VII. C ONCLUSIONS
In this paper, we studied a single cell UAV sensing net-work where multiple cooperative UAVs perform sensing andtransmission. We first proposed a sense-and-send protocol tofacilitate the cooperation, and formulated a joint trajectory,sensing location, and UAV scheduling optimization problem tominimize the completion time for all the tasks. To solve theNP-hard problem, we decoupled it into three sub-problems:trajectory optimization, sensing location optimization, andUAV scheduling, and proposed an iterative algorithm to solveit. We then analyzed the system performance, from which wecan infer that the UAV cooperation reduces the completiontime for all the tasks, and the marginal gain becomes smallerwith the increment of cooperative UAV number. Simulationresults showed that the completion time for all the tasks inthe proposed ITSSO scheme is 15% less than the NC scheme,and over 50% less than the FSL scheme. The transmissionresource is a dominated factor on the completion time for allthe tasks when it is at a low level. The sensing data size is adominated factor of the completion time for all the tasks whenthe size of the sensing task and the transmission resource areboth at a high level, or both at a low level.A
PPENDIX AP ROOF OF T HEOREM Proof.
We assume that in the optimal trajectory, there exists atime slot t , in which the speed of the UAV is k v i ( t ) k = v ′ ,with v ′ < v max . In the following, we will prove that thereexists a solution with k v i ( t ) k = v max , whose performanceis no worse than the one with k v i ( t ) k = v ′ .Given the location of l i ( t − and l i ( t + 1) , the possiblelocation of l i ( t ) is shown as the two red points in Fig. 12(a).When we set k v i ( t ) k = v max , the possible location of theUAV in time slot t moves to the blue points. If the BS ison the right side of the polyline, at least one blue point isnearer to the BS than both of the red points. Therefore, setting k v i ( t ) k = v max can improve the transmission rate in timeslot t when the BS is on the right side of the polyline.Similarly, we analyse the possible location of l i ( t − in Fig. 12(b). Given the location of l i ( t − and l i ( t ) ,the possible location of l i ( t − is shown as the two redpoints with k v i ( t ) k = v ′ . When we set k v i ( t ) k = v max ,the possible location of the UAV in time slot t − moves tothe blue points. Similarly, we get a polyline, on the left side (cid:95)(cid:95) (cid:89) (cid:76) (cid:11)(cid:87) (cid:19) (cid:14)(cid:20)(cid:12)(cid:95)(cid:95)(cid:95)(cid:95) (cid:89) (cid:76) (cid:11)(cid:87) (cid:19) (cid:12)(cid:95)(cid:95)(cid:32)(cid:89) (cid:255) (cid:95)(cid:95) (cid:89) (cid:76) (cid:11)(cid:87) (cid:19) (cid:12)(cid:95)(cid:95)(cid:32)(cid:89) (cid:80)(cid:68)(cid:91) (cid:79) (cid:76) (cid:11)(cid:87) (cid:19) (cid:14)(cid:20)(cid:12) (cid:79) (cid:76) (cid:11)(cid:87) (cid:19)(cid:20) (cid:16)(cid:20)(cid:12) (a) (cid:95)(cid:95) (cid:89) (cid:76) (cid:11)(cid:87) (cid:19) (cid:14)(cid:20)(cid:12)(cid:95)(cid:95) (cid:79) (cid:76) (cid:11)(cid:87) (cid:19) (cid:14)(cid:20)(cid:12) (cid:79) (cid:76) (cid:11)(cid:87) (cid:19) (cid:12) (cid:79) (cid:76) (cid:11)(cid:87) (cid:19) (cid:16)(cid:21)(cid:12) (cid:95)(cid:95) (cid:89) (cid:76) (cid:11)(cid:87) (cid:19) (cid:16)(cid:20)(cid:12)(cid:95)(cid:95) (b)Fig. 12. Proof of Theorem 1. of which at least one blue point is nearer to the BS than bothof the red points. The transmission rate in time slot t − can be improved when the BS is on the left side of the newpolyline.It can be easily proved that the two polylines are not parallel,and a quadrangle is obtained with the two polylines. Since thetransmission distance is much larger than the UAV velocity,i.e. d i,BS ( t ) ≫ v max , the BS is outside of the quadrangle.Therefore, the transmission rate of at least one time slot canbe improved when k v i ( t ) k is changed from v ′ to v max . Alarger transmission rate reduces the number of transmissiontime slots, and thus, the task completion time is no more thanthe solution with k v i ( t ) k = v ′ .A PPENDIX BP ROOF OF T HEOREM Proof.
According to equations (3) and (4), a larger cooperativeUAV number q requires lower successful sensing probabilityfor each UAV. Given the sensing probability threshold P R th ,when a UAV performs data collection, the distance between theUAV and the sensing task can be longer with a larger q . Whenconsidering the average change of rate, we assume that thedistance between every UAV and the sensing task are the samewhen performing data collection, and the sensing probabilityof each task equals the sensing probability threshold P R th ,i.e. d i,j ( τ ji ) = d , ∀ i, m ∈ W j , (21) P R j = P R th , ∀ j ∈ N . (22)When substituting (21) and (22) into (3) and (4), we have P R th = 1 − (1 − e − λd ) q . (23) The average rate of change of sensing distance to the cooperateUAV number can be achieved with the derivation of q inequation (23), which is shown as ∆ d ∆ q = − (1 − P R th ) /q ln(1 − P R th ) λ (1 − (1 − P R th ) /q ) q . (24)For each UAV, the increment of sensing distance equals to thedecrement of moving distance. Given the UAV speed v max , theaverage rate of change of time for each task to the cooperateUAV number is ∆ δ ji ∆ q = − ∆ d v max ∆ q . For UAV i , the averagerate of change of its task completion time to the cooperateUAV number is ∆ T i ∆ q = − N i ∆ d v max ∆ q = (1 − P R th ) /q ln(1 − P R th ) λ (1 − (1 − P R th ) /q ) q × N i v max . A PPENDIX CP ROOF OF T HEOREM Proof.
Similar with the proof of Theorem 8, we assume thatthe distance between every UAV and the sensing task arethe same when performing data collection, and the sensingprobability of each task equals the sensing probability thresh-old
P R th . Therefore, equations (21), (22), and (23) are alsosatisfied in the prove of Theorem 9. The average rate of changeof sensing distance for one task to the sensing probabilitythreshold can be achieved with the derivation of P R th inequation (23), which is shown as ∆ d ∆ P R th = − (1 − P R th ) /q − λq (1 − (1 − P R th ) /q ) . (25)For each UAV, the increment of sensing distance equals to thedecrement of moving distance. Given the UAV speed v max ,the average rate of change of time for each task to the sensingprobability threshold is ∆ δ ji ∆ P R th = − ∆ d v max ∆ P R th . For UAV i ,the average rate of change of its task completion time to thesensing probability threshold is ∆ T i ∆ P R th = − N i ∆ d v max ∆ P R th = (1 − P R th ) /q − λq (1 − (1 − P R th ) /q ) × N i v max .R EFERENCES[1] M. Erdelj, E. Natalizio, K. R. Chowdhury, and I. F. Akyildiz, “Help fromthe sky: leveraging UAVs for disaster management,”
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