I Introduction
The last two decades have witnessed a dramatic advancement in the research and development of wireless sensor network (WSN) for applications in various fields. A WSN typically consists of a large number of sensor nodes (SNs) that are distributed in a wide area of interest. SNs are typically lowcost and lowpower devices, which are able to sense, process, store and transmit information. Although the SNs usually have limited sensing, processing and transmission capabilities individually, their collaborative estimation/detection can be highly efficient and reliable [1, 2].
One typical application of WSN is for the estimation of an unknown parameter (such as pressure, temperature, etc.) in a given field based on noisy observations collected from distributed SNs. Specifically, each SN performs local sensing and signal quantization, then sends the quantized data to a Fusion Center (FC), where the received data from all SNs are jointly processed to produce a final estimate of the unknown parameter. Prior research on distributed estimation in WSN (see, e.g., [1, 2]) has mainly considered the static FC at a fixed location. As a result, SNs may require significantly different transmission power to send their data reliably to the FC due to their nearfar distances from it, which results in inhomogeneous energy consumption rates of the SNs and thus limited lifetime of the WSN.
To overcome this issue, utilizing unmanned aerial vehicle (UAV) as a mobile data collector for WSN has been proposed as a promising solution [3, 4, 5]
. With onboard miniaturized transceivers that enable groundtoair communications, UAVenabled WSN has promising advantages, such as the ease of ondemand and swift deployment, the flexibility with fullycontrollable mobility, and the high probability of having lineofsight (LoS) communication links with the ground SNs. In contrast to fixed FCs, a UAVenabled mobile data collector is able to fly sufficiently close to each SN to collect its sensed data more reliably, thus helping significantly reduce the SNs’ energy consumptions, yet in a more fair manner.
A fundamental problem in UAVenabled WSN for distributed estimation is the design of the UAV’s trajectory (see Fig. 1), which needs to take into account two important considerations. Firstly, for an SN to send its data reliably to the UAV, the UAV needs to fly sufficiently close to the SN (say, within a certain maximum distance assuming an LoS channel between them). Secondly, given a finite flight duration, the UAV’s trajectory should be designed to “cover” (with respect to the given maximum distance) as many SNs as possible to optimize the distributed estimation performance (e.g., minimizing the mean square error (MSE) for the estimated parameter). Notice that in our prior work [5], the SNs’ wakeup schedule and UAV’s trajectory were jointly optimized to minimize the maximum energy consumption of all SNs, while ensuring that the required amount of data is collected reliably from each SN. In contrast to [5] where the UAV needs to collect independent data from all SNs, this paper considers that all SNs’ data contains noisy observations of a common unknown parameter. As a result, their approaches for the UAV trajectory design are also fundamentally different.
UAV trajectory design for optimizing communication performance has received growing interests recently (see. e.g., [6, 9, 7, 10, 8]). In [6], the UAV’s trajectory was jointly optimized with transmission power/rate for throughput maximization in a UAVenabled mobile relaying system, subject to practical mobility constraints of the UAV. The energyefficient UAV communication via optimizing the UAV’s trajectory was studied in [7], which aims to strike an optimal balance between maximizing the communication rate and minimizing the UAV’s propulsion power consumption. The deployment and movement of multiple UAVs, used as aerial base stations to collect data from ground Internet of Things (IoT) devices, was investigated in [8]. The work in [9] maximized the minimum throughput of a multiUAVenabled wireless network by optimizing the multiuser communication scheduling jointly with the UAVs’ trajectory and power control. In [10], the UAV trajectory was designed to minimize the mission completion time for UAVenabled multicasting. Different from the above work, this paper investigates the UAV trajectory design under a new setup for distributed estimation in WSN. The main contributions of this paper are summarized as follows:

First, we show that for distributed estimation in an UAVenabled WSN, minimizing the MSE is equivalent to maximizing the number of SNs whose sensed data are reliably collected by the UAV;

Second, with a given UAV flight duration, we formulate an optimization problem for designing the UAV’s trajectory to maximize the number of covered SNs, subject to the practical constraints on the initial and final locations of the UAV as well as its maximum speed. Although the problem is NPhard, we show that the optimal UAV trajectory consists of connected line segments only;

Third, with the above simplification, an efficient greedy algorithm is proposed to obtain a highquality suboptimal trajectory solution by leveraging the classic traveling salesman problem (TSP) method and applying convex optimization techniques;

Last, numerical results show that the proposed trajectory design achieves significant performance gains in terms of the number of SNs with successful data collection as compared to benchmark schemes.
Ii System Model and Problem Formulation
As shown in Fig. 1, we consider a WSN consisting of SNs arbitrarily located on the ground, denoted by . The horizontal coordinate of SN is denoted by , . Each SN can observe, quantize and transmit its observation for an unknown parameter to the FC, which estimates the parameter based on the received information.
Iia Distributed Estimation
Each SN makes a noisy observation on a deterministic parameter (e.g., temperature). The realvalue observation by SN is modeled as
(1) 
where
is the observation noise that is assumed to be spatially uncorrelated for different SNs with zero mean and variance
. We further assume that the noise variances for all SNs are identical, i.e., . Denote by the signal range that the sensors can observe, where is a known constant that is typically determined by the sensor’s dynamic range. In other words, .The local processing at SN consists of the following: (i) an uniform quantizer with quantization levels, where denotes the number of quantization bits and represents the quantization step size; (ii) a modulator, which maps the quantization bits into a number of symbols based on certain modulation scheme, such as binary phase shift keying (BPSK); and (iii) transmission of the modulated symbols to the FC. It is shown in [11] that with uniform quantizer, the quantization noise variance for can be obtained as . For a homogeneous sensor network with equal observation noise power for all SNs, we assume that all SNs generate the same number of quantization bits, i.e., [1]. The FC then performs the linear estimation based on the received data from all SNs to recover
using the Quasi Best Linear Unbiased Estimators (QuasiBLUE)
[2], and the corresponding MSE can be obtained as(2) 
where is the number of SNs whose sensed data are reliably collected.
The expression in (2) shows that for the considered distributed estimation, the MSE is inversely proportional to the number of SNs whose data are reliably collected. Therefore, in order to minimize MSE for the distributed estimation, the FC should successfully collect data from as many SNs as possible.
IiB UAV Data Collection
For the UAVenabled WSN, a UAV is employed as a flying data collector/FC for a given time horizon , which collects the quantized information from SNs and jointly estimates the parameter . It is assumed that the UAV flies at a fixed altitude of in meter (m) and the maximum speed is denoted as in meter/second (m/s). The initial and final UAV horizontal locations are predetermined and denoted as , respectively, where so that there exists at least one feasible trajectory for the UAV to fly from to in a straight line within . The UAV’s flying trajectory projected on the ground is denoted as .
We assume that the transmit power for each SN is given (but can be different among SNs in general, depending on each SN’s energy availability). Thus, in order to satisfy the minimum required signaltonoise ratio (SNR) at the UAV for reliable data collection from each SN , the UAV location projected on the ground should lie within its communication range, which is denoted by . For each SN , define the coverage area . In general, an SN with smaller transmit power has a smaller given the same for all SNs. As a result, the UAV can collect the data reliably from as long as it is within , as shown in Fig. 1. In the following, we refer to the event that the UAV enters into as UAV visits . For example, in Fig. 1, the UAV has visited SNs , , and . Since the number of quantization bits is typically small for practical applications (e.g., bits) [1], the required transmission time for the quantized information can be neglected compared to the UAV flight time. In other words, as long as the UAV visits , we assume that the sensed data by can be reliably collected by the UAV.
IiC Problem Formulation
Define the indicator function and indicator variable as follows,
(3)  
(4) 
where indicates whether UAV is within or not at each time instant , and indicates whether UAV visits (at least once) during the time horizon . We assume that all the SNs’ locations as well as their communication ranges are known to the UAV. The UAV trajectory design problem to maximize the number of visited SNs for distributed estimation is thus formulated as,
(P1):  
s.t.  (6)  
In (P1), constraint (5) corresponds to the maximum UAV speed constraint, with denoting the timederivative of , and constraints (6) specify the initial/final locations for the UAV.
Iii Proposed Solution
(P1) is a nonconvex optimization problem, since the objective function is a nonconcave function, which involves timedependent indicator functions in terms of the UAV trajectory. In the following, we first show the structure of the optimal UAV trajectory solution to (P1).
Iiia Optimal Trajectory Structure and Problem Reformulation
Theorem 1.
Without loss of optimality to problem (P1), the UAV trajectory can be assumed to consist of connected line segments only.
Proof.
Theorem 1 is proved by showing that for any given feasible trajectory of (P1), which contains curved path, we can always construct another feasible trajectory consisting of only connected line segments, which satisfies the conditions in (5), (6), and achieves the same objective value. Specifically, for any given , the indicator variables can be obtained based on (3) and (4), and the objective value of (P1) (i.e., the number of visited SNs) can be obtained as . Without loss of generality, we assume that the visited SNs are , where is the index of the visited SNs in , . Let be the waypoint that the UAV enters into for the first time, , then , where . We rearrange with the increasing order of and obtain a sequence of ordered waypoins , where is a permutation of . Let and . Then we have , where denotes the flying time between waypoints and along the given trajectory . We can then replace any curved trajectory path between waypoints and , with a line segment and obtain the alternative trajectory . Thus, with the same flying time , the required flying speed can be reduced since line segment between any given pair of waypoints and yields the minimum distance. Therefore, satisfies the constraints (5) and (6), and yet achieves the same objective value for (P1). This concludes the proof of Theorem 1. ∎
Based on Theorem 1 and its proof, (P1) can be solved by determining the optimal subset of SNs that are visited, denoted as with cardinality , their optimal visiting order , and the optimal waypoints each for an SN , such that the data from can be received when the UAV is at and the total distance of the resulting path is no greater than . Therefore, (P1) can be reformulated as
s.t.  (9)  
Consider a special case of (P2) with , (i.e., the UAV can collect data reliably from the SN only when it is directly above the SN), then only and the visiting order need to be determined to maximize the number of visited SNs within duration . This problem is essentially equivalent to the selective TSP problem (or orienteering problem), which is known to be NPhard [12]. Therefore, problem (P2) with is also NPhard and more difficult to solve than TSP in general.
IiiB Proposed Algorithm for (P2)
One straightforward approach for solving (P2) is via exhaustively searching all possible subsets and the visiting order of each , and then determining whether the minimum distance of path that visits with order is no greater than . However, searching all possible subsets of has an exponential complexity of , which is infeasible for large values of . Therefore, we propose an efficient suboptimal solution to (P2) by a greedy iterative algorithm.
The key idea of our proposed solution is to maintain a working set containing SNs that the UAV needs to visit, and add only one additional SN in at each iteration. Initially, is set as empty, and we make a greedy choice to select from the complement set , which leads to the minimum traveling distance to visit . The above process iterates until or when the required visiting time is greater than . Let and . The proposed greedy algorithm for (P2) is summarized in Algorithm 1. In Algorithm 1, is the flying distance of the UAV that flies over each SN following the increasing index of all the SNs in , which is an upper bound of the minimum flying distance to visit all SNs in .
Note that in step 7 of Algorithm 1, the UAV trajectory is designed with a given SN set to minimize the UAV traveling distance. The problem is formulated as
s.t.  (10) 
In Algorithm 1, after executing the inner iteration from step 5 to step 11, if adding any additional SN in the complementary set leads to a traveling distance greater than , then step 9 will not be executed and remains equal to as initialized in step 4, and the outer iteration in step 3 terminates. Otherwise, step 9 will be executed and one additional SN will be added into with , and the outer iteration continues. Therefore, the size of increases over the iterations until either or adding any additional SN will lead to a traveling distance greater than ; thus, Algorithm 1 is guaranteed to converge. Furthermore, Algorithm 1 requires at maximum iterations, which is significantly less than required by exhaustive search.
Thus, the remaining task for Algorithm 1 is to solve problem (P3). Note that solving (P3) includes determining and the waypoints . (P3) is essentially equivalent to the TSP with neighborhoods (TSPN), which is known to be NPhard [13]. To solve (P3), we propose an efficient method for waypoints design based on TSP method and convex optimization. Specifically, the visiting order for the SNs in is first determined by simply applying the TSP algorithm over the SNs in while ignoring the coverage (disk) region of each SN. Since the initial and final points of the UAV are fixed, can be obtained by using a variation of the TSP method (NoReturnGivenOriginAndEnd TSP) [10]. Various algorithms have been proposed to find highquality solutions to TSP efficiently, e.g., with time complexity [14]. With the visiting order determined, the optimal waypoints can be obtained by solving the following problem,
s.t.  (11) 
Note that the objective function of (P4) is a convex function with respect to , and the coverage area is a convex set. Thus, (P4) is a convex optimization problem, which can be solved by standard convex optimization techniques or existing software such as CVX [15], with polynomial complexity. Let . The trajectory design algorithm for (P3) with given is summarized in Algorithm 2.
Iv Numerical Results
We consider a WSN with SNs, which are randomly located within an area of size km km. The following results are based on one random realization of the SN locations as shown in Fig. 2. The UAV’s initial and final locations are respectively set as and , and is set as m/s. We assume that the communication range of different SNs is identical, i.e., . If not stated otherwise, we set m. For performance comparison, we also consider two benchmark schemes, namely stripbased and zigzag line trajectories for the UAV, as described in the following.
For the stripbased trajectory, the area of interest is partitioned into rectangular strips that are perpendicular to the line connecting and . Furthermore, each strip has width so that all SNs within the strip will be visited as the UAV travels along the strip, as shown in Fig. 2. If the rectangular strips exceed the boundary of the area of interest, then the UAV just travels along the intersection between the borderlines of the area and these rectangular strips, which can be uniquely determined. With such stripbased trajectory, the number of visited SNs increases with the height of the strips. Therefore, a bisection search method can be used to determine the maximum height of the strips so that the total UAV flying distance is no greater than . On the other hand, the zigzag line trajectory is similar to the stripbased trajectory, but with the difference in that rectangular strips are replaced with zigzag lines. The two benchmarks lead to rather intuitive UAV trajectories for different values of . For example, when is small, say , the two benchmarks yield the same path that directly connects and . When increases, the heights of the strips and zigzag lines increase since the width of the strip and parallel zigzag lines are fixed as . Therefore, when is sufficiently large, both benchmark schemes result in UAV trajectories covering the entire area of interest, so that all SNs will be visited by the UAV.
The optimized trajectories with different schemes with s are shown in Fig. 2. It is observed that with our proposed solution, the UAV can visit more SNs than the two benchmark schemes. In Fig. 3(a), we compare the number of visited SNs by our optimized trajectory with the two benchmark trajectories for different . As expected, our proposed design significantly outperforms both benchmarks. It is observed that the performance gain is more substantial with small . As becomes sufficiently large, all the three trajectories can visit all SNs, but our proposed scheme requires much less time to visit all SNs. It is also observed that the stripbased trajectory gives better performance than the zigzag line trajectory. This is because the zigzag line trajectory in general has smaller coverage areas than the stripbased trajectory with the same traveling distance or (see Fig. 2).
Furthermore, we study the effect of the SNs’ communication range on the system performance. Fig. 3(b) plots the number of visited SNs versus when s. It is observed that for all the three schemes, the number of visited SNs increases with , as expected; and the proposed trajectory outperforms the two benchmarks significantly, especially for small .
V Conclusion
This paper studies the trajectory design for distributed estimation in a UAVenabled WSN to minimize the MSE for the estimation, which is shown equivalent to maximize the number of SNs with successful data collection by the UAV. Although the formulated problem is NPhard, we reveal that the optimal UAV trajectory consists of connected line segments only. We then propose a greedy algorithm with low complexity based on TSP method and convex optimization to obtain a suboptimal trajectory solution. Numerical results demonstrate that the proposed design significantly improves the number of visited SNs and hence the estimation performance, as compared to the benchmark schemes.
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