To support the fast-growing traffic demand for the next generation mobile communication systems, extensive research efforts have been devoted to exploring various new wireless technologies , such as ultra-dense network, millimeter wave (mmWave) communication, massive multiple-input multiple-output (M-MIMO), non-orthogonal multiple access (NOMA) , and machine-type communication. All these technologies were mainly developed for the terrestrial wireless network with base stations (BSs), relays and access points deployed at fixed locations. Recently, there have been significant interests in using unmanned aerial vehicles (UAVs) as aerial platforms to enable terrestrial communications from the sky . Compared to conventional terrestrial communication, UAV-enabled communication is more swift and flexible to deploy for unexpected or temporary events. Besides, thanks to the UAV’s high altitude, the favorable line-of-sight (LoS) communication links are more likely to be established between UAV and ground users (GUs) , . Thus, UAV-enabled communication has many potential use cases, such as for public safety communication, ground BS offloading, emergency response, and Internet of things (IoT) communication.
Significant research efforts have been devoted to addressing the various new challenges for UAV-enabled communications, such as the UAV-ground channel characterization [4, 5, 6], performance analysis , , and UAV placement optimization [9, 10, 11, 12, 13]
. In particular, the controllable high mobility of UAVs offers a new design degree of freedom to enhance communication performance via trajectory optimization, which has received significant interests recently[14, 15, 16, 17, 18, 19, 20, 21, 22].
In , the authors proposed a general framework via jointly optimizing the transmit power and UAV trajectory to maximize the end-to-end throughput for a UAV-enabled mobile relaying system. Specifically, the transmit power at the source/UAV relay and the UAV trajectory were optimized in an alternating manner iteratively via the technique of block coordinate descent. To tackle the non-convex trajectory optimization in each iteration, the successive convex optimization technique was proposed based on the local lower bound of the rate function. Such techniques have then been applied to various other scenarios in UAV-enabled wireless communications [15, 16, 17, 18, 19, 20, 21]. Note that for all these works employing successive convex optimization and block coordinate descent techniques, the converged results critically depend on the initial UAV trajectory adopted. A straight line based initial trajectory and a circular based initial trajectory were proposed in  and , respectively. Though simple and intuitive, such trajectory initialization schemes do not fully exploit the locations and communication requirements of GUs. This thus gives one of the main motivations of the current work, to devise more sophisticated trajectory initialization schemes for UAV-enabled communications to achieve better converged performance.
, the UAV trajectory was formulated as a mixed integer linear program (MILP) to ensure collision avoidance. In, the receding-horizon path planning approach was applied to demonstrate the capability for a swarm of UAVs to perform autonomous search and localization. Moreover, the authors in  and  investigated the path planning for a single vehicle to collect data from all sensors. Note that the aforementioned works for path planning either focused on other design objectives rather than communication performance, or assumed simplified communication models, such as the disk model in , . In practice, adaptive communication with dynamic power and bandwidth allocation can be exploited along with the UAV trajectory design to achieve enhanced communication performance, as pursued in more recent works such as [14, 15, 16, 17, 18, 19].
In this paper, we study a general UAV-enabled radio access network (RAN) as shown in Fig. 1, where the UAV is employed as an aerial platform supporting multi-mode communications for its served GUs, including data relaying from one GU to another , downlink data transmission to GUs , and uplink data collection from GUs  as special cases. Such a multi-mode aerial communication platform is more practically relevant for a real-life RAN with different traffic demands of the GUs.
For the considered general RAN, two UAV application scenarios of practical interest are further considered. The first one is periodic operation, where the UAV serves the GUs in a periodic manner by following a certain trajectory repeatedly. In this case, our objective is to minimize each periodic flight duration of the UAV for the purpose of minimizing the communication delay of the GUs , while satisfying the average rate requirement of each GU, via jointly optimizing the UAV trajectory, transmit power and bandwidth allocation. The second scenario corresponds to one-time operation, where the UAV serves the GUs with one single fly and then leaves for another mission. This may correspond to practical use cases such as periodic sensing, where the UAV only needs to be dispatched at a given frequency. In this scenario, we aim to minimize the mission completion time for saving UAV time for other missions while satisfying the aggregated throughput requirement of each GU, via jointly optimizing the UAV trajectory and pertinent communication resource allocation. In this case, for the particular data relaying mode, the UAV can only forward to a destination GU the data that has been received from its associated source GU, along its given one-round trajectory, thus resulting in a stringent information-causality constraint ; whereas this constraint can be relaxed in the former periodic operation scenario thanks to the periodic trajectory of the UAV. The main contributions of this paper are summarized as follows.
First, we propose a multi-mode UAV communication platform with periodic operation or one-time operation. For both operation scenarios, we formulate the optimization problems to minimize the UAV periodic flight duration and mission completion time, respectively, via jointly optimizing the UAV trajectory, bandwidth and power allocation. Since the formulated problems are difficult to be directly solved, we propose efficient iterative algorithms to find locally optimal solutions based on successive convex optimization and block coordinate descent techniques.
Second, as the converged results of the proposed algorithms critically depend on the initial UAV trajectory assumed, we propose new methods to design the initial trajectory by fully exploiting the location information and communication requirements of the GUs. Specifically, as the UAV typically has better communication link when it is near GUs, the initial UAV trajectory should be designed so as to approach each GU as much as possible. To this end, we propose the trajectory initialization design based on the Traveling Salesman Problem (TSP) solution for the case of periodic operation, and that based on the Pickup-and-Delivery Problem (PDP) solution for the case of one-time operation. Compared to the existing UAV initial trajectory designs such as the straight-line or circular trajectories, the main novelty of the proposed trajectory initialization lies in the optimized waypoints design and their order of visiting based on the number and location distribution of the GUs, their communication requirements as well as the UAV’s practical mobility constraints such as its maximum speed.
The rest of this paper is organized as follows. Section II introduces the system model and presents the problem formulations for the periodic operation and the one-time operation scenarios, respectively. Section III and Section IV present the proposed algorithms based on successive convex optimization and block coordinate descent techniques for the two operation scenarios, respectively. In Section V, we propose two efficient trajectory initialization designs for the two scenarios, respectively. Numerical results are presented in Section VI to evaluate the performance of the proposed designs. Finally, we conclude this paper in Section VII.
In this paper, scalars and vectors are denoted by italic letters and boldface lower-case letters, respectively.denotes the space of -dimensional real-valued vectors. For a vector a, its Euclidean norm is represented by . denotes the logarithm with base 2. For a time-dependent function , represents the first-order derivative with respect to time . For sets and , means the union of the two sets.
Ii System Model and Problem Formulation
Ii-a System Model
As shown in Fig. 1, we consider a general UAV-enabled wireless RAN, where a UAV serves as an aerial platform for a set of
GUs. In general, the communication modes of the UAV-enabled wireless RAN can be classified into three categories as follows:
Ii-A1 Data Collection in Uplink
The UAV is employed as a flying fusion center to collect data from GUs that are data sources on the ground, such as sensors in IoT .
Ii-A2 Data Transmission in Downlink
In this mode, independent information is transmitted from the UAV to GUs. For example, the UAV may act as a data carrier with pre-cached data to transmit to the intended GUs .
Ii-A3 Data Relaying
The UAV functions as a mobile relay to assist in the communications between multiple pairs of GUs. For each pair, the data is firstly received from the source GU in the uplink and then forwarded to the destination GU in the downlink. By exploiting the LoS links between the UAV and GUs, UAV-enabled mobile relaying is a promising solution to overcome the unreliable terrestrial links between widespread GUs. Practical application scenarios include service recovery after natural disasters, emergency response, etc., , .
Accordingly, the GUs can be generally divided into three groups based on their communication modes. Group 1 corresponds to UAV-assisted data collection, which only involves the uplink communication from the GUs in this group to the UAV. Within this group, we assume that in total independent information flows are transmitted from their respective GUs to the UAV. Group 2 corresponds to data transmission from the UAV to the GUs belonging to this group, where only downlink communication is involved and the UAV transmits in total independent information flows to their corresponding GUs. Lastly, Group 3 corresponds to data relaying, which involves both uplink and downlink communications. For this group, in total information flows are firstly transmitted to the UAV from the source GUs in this group and then forwarded by the UAV to their respective destination GUs. For information relaying, we assume that the UAV employs the decode-and-forward (DF) strategy with a data buffer of sufficiently large size. Notice that in practice, we have , since each GU may correspond to multiple information flows. For ease of presentation, we assume that each GU is only associated with one information flow such that ; whereas the developed results in this paper can be easily generalized to the cases with . By letting and , we define a source GU set as , with the first elements corresponding to source GUs from Group 3 (for information relaying) and the rest from Group 1 (for uplink data collection). Similarly, define a destination GU set as with the first GUs corresponding to destination GUs in Group 3 (for information relaying) and the rest from Group 2 (for downlink transmission). Without loss of generality, we assume that the source GU and the destination GU , , correspond to the same pair in Group 3.
We consider a three-dimensional (3D) Cartesian coordinate system, where the locations of each source GU and destination GU are denoted as , , and , , respectively. We assume that the UAV flies at a given constant altitude . Furthermore, for a given time horizon of duration , denote the UAV trajectory projected on the ground as , . Let be the maximum UAV speed in meter/second (m/s). We then have the following constraint , The time-varying distance between the UAV and the GUs can be written as
We further assume that both the uplink and downlink channels are dominated by LoS links. Thereby, the channel power gains follow the free-space path loss model given by
where denotes the channel power gain at the reference distance of m.
Let the total available bandwidth be denoted as . The UAV is assumed to employ the frequency division multiple access (FDMA) scheme with dynamic bandwidth allocation among all GUs. Specifically, at time instant , denote as the fraction of the total bandwidth that is allocated for the source GU , and as that allocated for the destination GU . We then have the following constraints:
Note that the above dynamic FDMA scheme includes both conventional time division multiple access (TDMA) with dynamic user time scheduling and FDMA with fixed user bandwidth allocation as special cases. In particular, when all and are set as binary values 1 or 0, we have the dynamic TDMA scheme. On the other hand, when , and , , we have the non-dynamic FDMA scheme.
Denote by the transmit power for the source GU if , which is assumed to be constant. The instantaneous normalized achievable rate in bits/second/Hertz (bps/Hz) for this GU can be expressed as
where represents the additive white Gaussian noise (AWGN) power spectral density in watts/Hz, and is the time-dependent channel-to-noise power ratio, and denotes the reference signal-to-noise ratio (SNR) at the reference distance of m.
Similarly, let denote the UAV’s transmit power for the destination GU at time . The instantaneous achievable rate in bps/Hz for this GU is thus expressed as
where is the channel-to-noise power ratio from the UAV to the destination GU . Let denote the maximum total transmit power by the UAV. For the downlink transmission from the UAV to the destination GUs, we then have the following power constraint .
Ii-B Problem Formulation
Generally speaking, a UAV serving as a multi-mode aerial platform may have two operation scenarios in practice: periodic operation versus one-time operation, explained as follows.
Ii-B1 Periodic Operation
With periodic operation, the UAV needs to remain airborne to serve the GUs periodically, where the GUs keep generating service requests to the UAV. We assume that the average rate requirements in bps for uplink and downlink communication corresponding to the different flows are , , and , , respectively. In particular, for the data relaying service in Group 3, the uplink and downlink average rate requirements for each pair should be balanced, i.e., , . Without loss of generality, we assume that the UAV flies above the GUs following a periodic trajectory with period , where is a design variable. Note that in practice, it is desirable to minimize in order to avoid large communication delay of GUs .
For notational convenience, define , and . Our objective is to minimize the UAV flight period , via jointly optimizing the UAV’s trajectory , the downlink transmit power , as well as the bandwidth allocation , while satisfying the average rate requirements by the GUs. The problem can be formulated as
where the constraint (8i) ensures that the UAV returns to the initial location at the end of each period.
Different from the prior work  which focuses on maximizing the minimum throughput over all GUs in downlink communication with given , we here study the flight period minimization problem in a more general setup, where uplink communication, downlink communication and data relaying modes are all taken into account and is also a design variable.
Ii-B2 One-Time Operation
In the second scenario, the UAV only needs to serve the GUs once by one single fly mission. This corresponds to many practical scenarios where the service requests by the GUs are intermittent. In this case, the UAV mission is regarded as completed once the throughput in bits (instead of the average rate as in periodic operation) for each information flow meets the target requirement of the GUs. Denote the uplink and downlink throughput requirements corresponding to different information flows as bits, , and bits, , respectively. Similar to the periodic operation scenario, for the particular data relaying service, the uplink and downlink throughput requirements should be balanced for each source-destination pair, namely , . Further, denote by the flight duration (or mission completion time) required by the UAV to meet the throughput requirements of all the information flows.
Furthermore, for data relaying in one-time operation scenario, we need to impose the stringent information-causality constraints, i.e., at any time instant , the UAV can only forward the data that has already been previously received from the source GU in Group 3. Note that such information constraints do not need to be explicitly imposed for the periodic operation scenario since the UAV may forward the information received from the previous period, as long as the total information bits received from the source equal to that forwarded to the corresponding destination at each period to ensure the long-term balance. The information-causality constraints for data relaying in one-time operation can be expressed as 
Note that the left-hand side (LHS) of (9) is the aggregated information bits that have been forwarded by the UAV to the destination GU at time , and the right-hand side (RHS) represents those which have been received from the source GU at the same time. For one-time operation, we aim to minimize the mission completion time via a joint trajectory, spectrum and power allocation design. In practice, minimizing the completion time is of high practical interest since it helps save more time/energy for the UAV to serve other missions. The problem can be formulated as
Note that in (P2), no constraints on the UAV’s initial and final locations are imposed, i.e., they can be freely designed for performance optimization. The developed results can be easily extended to include such constraints similarly as in .
Besides, it should also be noted that in the prior work , the special case of UAV-enabled relaying with one pair of source and destination GUs has been studied, where the end-to-end throughput is maximized with a pre-determined time horizon . In (P2), we study the completion time minimization problem in the general setup with multiple GUs and modes, where the results in  cannot be directly applied.
Iii Proposed Solution for Periodic Operation
In this section, we consider the flight period minimization problem (P1) for the periodic operation. Problem (P1) is challenging to solve for two reasons. First, the problem requires to optimize continuous functions , and , which essentially involve an infinite number of optimization variables that are closely coupled with each other. Secondly, the integral in the LHS of (8a) and (8b) involve the optimization variable as the upper bound of the integration interval, which lack closed-form expressions. To tackle these issues, we first introduce the following optimization problem for any given period :
Problem (P1.1) aims to maximize the minimum ratio between the achievable average rate and the target rate requirement of each GU. For any given flight period , denote the optimal value of (P1.1) as . It is not difficult to see that for any given , the target rate requirements of all GUs are achievable if and only if . Therefore, problem (P1) is equivalent to
The optimal value of problem (P1.1) is non-decreasing with .
Please refer to Appendix A. ∎
By applying Lemma 1, problem (P1.2) can be solved by applying a bisection search over until the equality in (12) holds. Thus, the main task of solving (P1) is to find an efficient algorithm for (P1.1) for any given .
To obtain a more tractable form of (P1.1), we apply a discrete state-space approximation. Specifically, the time horizon is equally divided into time slots, i.e., , , with representing the time step which is sufficiently small such that the distance between the UAV and the GUs can be assumed to be approximately constant within each time slot. Therefore, the UAV’s trajectory over can be specified by , . As a result, the UAV speed constraints (8h) can be represented as
where denotes the maximum distance that the UAV can travel within each time slot. The bandwidth and transmit power allocation can be similarly discretized as , , , . Then, the achievable rate between the GUs and the UAV at time slot can be expressed as
Besides, , and are rewritten as , and , respectively. As a result, problem (P1.1) is reformulated as
where constraints (18a)-(18i) represent the discrete-time equivalents of (11a), (11b), (8c)-(8i), respectively. As constraints (18a) and (18b) are non-convex with respect to variables , and , problem (P1.3) is difficult to be directly solved in general. In the following, we propose an efficient suboptimal solution to (P1.3) based on successive convex optimization and block coordinate descent techniques, similarly as in . The main idea is to solve the two sub-problems of (P1.3) iteratively, namely the power and bandwidth optimization with fixed trajectory, and trajectory optimization with fixed power and bandwidth allocation. Then, the block coordinate descent method is employed to optimize the two sets of variables in an alternating manner until the objective value converges within a prescribed accuracy.
Iii-a Power and Bandwidth Optimization with Fixed Trajectory
First, we consider the sub-problem to optimize the UAV transmit power and bandwidth allocation , for any given feasible UAV trajectory . In this case, the time-varying variables in (16) and (17) are also determined. This sub-problem of (P1.3) is given by
It can be shown that the LHS of (19a) is concave with respect to the bandwidth allocation , and the LHS of (19b) is jointly concave with respect to the bandwidth allocation and the transmit power , and all other constraints are convex. Therefore, (P1.4) is a convex optimization problem, which can be efficiently solved via existing software such as CVX  or applying the Lagrange duality , for which the details are omitted for brevity.
Iii-B Trajectory optimization with Fixed Power and Bandwidth Allocation
In this subsection, we consider the other sub-problem to optimize the UAV trajectory by assuming that the transmit power and bandwidth allocation are given. However, even with fixed power and bandwidth allocation, the trajectory optimization in (P1.3) is still a non-convex problem due to non-convex constraints (18a) and (18b). To tackle such non-convexity, the successive convex optimization technique similar to that used in  and  can be applied, for which a lower bound of the original problem is sequentially maximized by optimizing the trajectory at each iteration. To this end, we need the following result.
Please refer to Appendix B. ∎
Proposition 1 shows that for any given local trajectory , and are respectively lower-bounded by and , which are both concave functions with respect to . As a result, for any given local trajectory , a lower bound of the optimal value of the original problem (P1.3) with fixed power and bandwidth allocation can be obtained by solving the following problem
Note that due to the lower bound given in Proposition 1, if (22a) and (22b) are satisfied, then the constraints (18a) and (18b) with the same power and bandwidth allocation are guaranteed to be satisfied as well, but the reverse is not true. Therefore, the feasible region of (P1.5) is in general a subset of that of (P1.3), and its optimal solution serves as a lower bound to that for (P1.3) with fixed power and bandwidth allocation. (P1.5) is a convex optimization problem, which can be efficiently solved with the standard convex optimization techniques or existing solvers such as CVX .
Iii-C Iterative Power, Bandwidth and Trajectory Optimization
Based on the results obtained above, we propose an iterative algorithm for (P1.3) based on the block coordinate descent technique. The details are summarized in Algorithm 1.
Since in each iteration of Algorithm 1, (P1.5) is optimally solved with given local trajectory , whose objective value is non-decreasing over iterations and upper-bounded by a finite value, Algorithm 1 is guaranteed to converge to at least a locally optimal solution. Note that for step 4 of Algorithm 1, an alternative way is to successively optimize the trajectory multiple times until convergence. The resulted objective value is also non-decreasing over iterations, thus its convergence is also guaranteed.
Iv Proposed Solution for One-Time Operation
In this section, we study the optimization problem (P2) for the one-time operation. Similar to (P1), in order to solve (P2), we first consider the following problem for any given UAV operation time :
Problem (P2.1) aims to maximize the minimum ratio between the achievable throughout and the target requirement. For any given operation time , let the optimal solution to (P2.1) be denoted as . Then it is not difficult to see that all throughput requirements of (P2) are achievable if and only if . Therefore, (P2) is equivalent to finding the minimum such that . Furthermore, as the time only appears in the upper limit of the integral in (23a) and (23b) (no normalization by as in (P1.1)), it is quite obvious that the LHS of (23a) and (23b) are non-decreasing with . Thus, is also non-decreasing with . Therefore, (P2) can be solved by solving (P2.1) and applying a bisection search over the completion time .
Similar to Section III, for any given , problem (P2.1) can be recast in a discrete equivalent form as
where (24) represents the discrete-time equivalent of the information-causality constraints in (9). As constraints (24a)-(24) are non-convex, problem (P2.2) is difficult to be optimally solved. Similar to Section III, we apply the successive convex optimization and block coordinate descent techniques to (P2.2) by iteratively solving the two sub-problems, namely the power and bandwidth optimization with fixed trajectory, and trajectory optimization with fixed power and bandwidth allocation, as detailed in the next.
Iv-a Power and Bandwidth Optimization with Fixed Trajectory
With the given UAV trajectory , problem (P2.2) reduces to optimizing the UAV transmit power and bandwidth allocation . By introducing slack variables , , problem (P2.2) can be equivalently transformed to
Note that if at the optimal solution to (P2.3), there exists one constraint in (25) that is satisfied with strict inequality, we are always able to decrease the corresponding transmit power and/or the bandwidth allocation to make the constraint active. This implies that there always exists an optimal solution to (P2.3) at which all constraints in (25) are active, and thus (P2.3) is equivalent to (P2.2) for any given trajectory. Furthermore, it can be verified that all constraints of (P2.3) are convex, thus (P2.3) is a convex optimization problem, which can be efficiently solved via standard convex optimization software such as CVX .
Iv-B Trajectory optimization with Fixed Power and Bandwidth Allocation
Next, we consider the other sub-problem to optimize the UAV trajectory for any given transmit power and bandwidth allocation . To deal with non-convex constraints (24a)-(24), the successive convex optimization is employed based on the lower bounds given in Proposition 1. Specifically, for any given local trajectory, by introducing slack variables , the resulted problem is given by
With the above two sub-problems solved, (P2.2) can be solved by iteratively optimizing the power and bandwidth allocation and the trajectory with similar steps as in Algorithm 1. Furthermore, the completion time minimization problem in (P2) can be solved via a bisection search over while solving (P2.2) in each iteration. The details are omitted for brevity.
V Initial Trajectory Design
The proposed algorithms for both periodic and one-time operation scenarios require the UAV initial trajectory to be specified, and their converged results via the successive convex optimization and block coordinate descent techniques depend on the UAV trajectory initialization in general. In this section, we propose new trajectory initialization schemes for the periodic and one-time operation scenarios, respectively. Note that due to the additional information-causality constraints for the one-time operation scenario, these two operation scenarios generally require different trajectory initializations.
Intuitively, the UAV trajectory should be designed so that when the UAV is scheduled to communicate with a particular GU, it should be as close to the GU as possible. One intuitive approach for trajectory initialization is to minimize the UAV traveling time among different GUs, so that when the given value of is sufficiently large, the UAV will be able to reach the top of all GUs to enjoy the best communication channel. Given the maximum speed , the problem of minimizing the traveling time is thus equivalent to minimizing the total traveling distance. The above approach will be used in the following designs, as detailed later.
V-a Initial Trajectory Design for Periodic Operation
In this subsection, we design the UAV initial trajectory for Algorithm 1 in the case of periodic operation for any given value of . For notational convenience, let denote the set containing all GUs, i.e., , where GUs correspond to source GUs in while GUs correspond to destination GUs in . The locations of all GUs in , and , are compactly denoted as , where represents the index of the GU in .
For given , we first consider the problem of minimizing the traveling distance/time for the UAV to visit all GUs by determining their optimal visiting orders, which is essentially the classic TSP . Although TSP is NP-hard, various algorithms have been proposed to find high-quality approximate solutions within a reasonable computational complexity . After solving the TSP, we obtain the minimum traveling time required, denoted as , as well as the permutation order , with representing the index of the th GU to be visited. In the following, for any given flight period , the UAV initial trajectory is designed by distinguishing two cases, depending on whether is no smaller than , as follows.
Case 1: . In this case, is sufficiently large so that the UAV is able to reach the top of each GU within each flight period. The remaining time can be spent by the UAV to hover above the GUs. To obtain an effective method for determining the hovering time allocation among the GUs, let denote the time required for the UAV to satisfy the average rate requirement for GU , by assuming that the UAV only communicates with it when hovering on its top. We thus have
Then the total hovering time can be proportionally divided among the GUs as
Following the visiting order and the hovering time allocation in (28) for each GU, the initial trajectory in the case of can be constructed accordingly.
Case 2: . In this case, the given time is insufficient for the UAV to reach the top of all GUs. To design a feasible initial trajectory, we first specify a disk-shaped region for each GU in , which is centered at the corresponding GU with radius . As illustrated in Fig. 2, the main idea is to minimize the UAV traveling distance by properly designing the UAV trajectory and radius , so that the UAV is able to reach each disk region. The problem can be formulated as
where constraints (29a) ensure that for each GU in , there exists at least one time instant such that the distance between the UAV and the GU is no larger than . This guarantees that all disks are traversed by the UAV.
For any given radius , denote the optimal value of (P3) as . It is not difficult to see that is non-increasing with . Thus, the optimal solution to (P3) can be obtained by solving the corresponding problem with fixed , and then applying a bisection search to find the optimal radius . In the following, we focus on solving (P3) with any given radius .
The optimal trajectory to (P3) should only contain connected line segments.
Similar to the proof of Theorem 1 in , Lemma 2 can be shown by contradiction. Suppose on the contrary that at the optimal solution , there exists at least one curved portion along the trajectory. Then we can always construct an alternative trajectory composed of line segments only, that achieves less traveling time . To this end, it is first noted that at the optimal solution to (P3), the UAV should always travel with the maximum speed , i.e., the constraint (29b) should be satisfied with equality.
For each node with , denote by as the earliest time instance when the UAV reaches its disk region with trajectory , i.e., . Then the trajectory of the UAV can be partitioned into portions, with the th portion specified by time interval , , where , . For the th time interval, we may replace the original trajectory portion with a line segment directly connecting and . Obviously, this replacement not only ensures the feasibility of (29a), but also reduces the traveling distance for the UAV. Therefore, if the optimal solution trajectory contains a curved portion, we are always able to construct an alternative trajectory by sequentially connecting that achieves . Thus, any trajectory with curved portion cannot be the optimal trajectory to (P3). This completes the proof. ∎
With Lemma 2, for any given , problem (P3) is recast to optimizing a set of waypoints inside the disks, which are the starting and ending points of the line segments, and finding the optimal permutation order to visit these waypoints. Let the waypoint inside the disk associated with GU be denoted as , . The traveling time required can be expressed as