3D Trajectory Design for UAV-Assisted Oblique Image Acquisition
Xiao-Wei Tang, Changsheng You, Shuowen Zhang, Xin-Lin Huang, Rui Zhang
11
3D Trajectory Design for UAV-Assisted ObliqueImage Acquisition
Xiao-Wei Tang,
Student Member, IEEE , Changsheng You,
Member, IEEE , Shuowen Zhang,
Member, IEEE , Xin-LinHuang,
Senior Member, IEEE , and Rui Zhang,
Fellow, IEEE
Abstract —In this correspondence, we consider a new un-manned aerial vehicle (UAV)-assisted oblique image acquisitionsystem where a UAV is dispatched to take images of multipleground targets (GTs). To study the three-dimensional (3D) UAVtrajectory design for image acquisition, we first propose a novelUAV-assisted oblique photography model, which characterizesthe image resolution with respect to the UAV’s 3D image-takinglocation. Then, we formulate a 3D UAV trajectory optimizationproblem to minimize the UAV’s traveling distance subject to theimage resolution constraints. The formulated problem is shownto be equivalent to a modified 3D traveling salesman problemwith neighbourhoods, which is NP-hard in general. To tackle thisdifficult problem, we propose an iterative algorithm to obtaina high-quality suboptimal solution efficiently, by alternatelyoptimizing the UAV’s 3D image-taking waypoints and its visitingorder for the GTs. Numerical results show that the proposedalgorithm significantly reduces the UAV’s traveling distance ascompared to various benchmark schemes, while meeting theimage resolution requirement.
Index Terms —Unmanned aerial vehicle, traveling salesmanproblem with neighbourhoods, oblique image acquisition.
I. I
NTRODUCTION
Unmanned aerial vehicles (UAVs) are expected to be widelydeployed in future networks to enable various new imageacquisition applications such as live broadcast, virtual reality,and so on, by leveraging their advantages of low cost, high mo-bility, as well as flexible deployment [1]. To maximize the ef-ficiency of UAV-assisted image acquisition, it is of paramountimportance to well design the UAV’s three-dimensional (3D)trajectory such that the images of ground targets (GTs) can becaptured with minimum traveling distance, while guaranteeingsatisfactory image quality. However, unlike the widely stud-ied UAV-enabled communications, this problem still remainsunaddressed in the literature, to the authors’ best knowledge.Particularly, there are two challenging issues that need to beresolved in designing the 3D trajectory for UAV-assisted imageacquisition. First, prior studies on UAV-assisted image acqui-sition mostly consider the vertical photography (VP) model,where the UAV-mounted camera is assumed to have a fixedshooting angle which is perpendicular to the ground. As such,the UAV needs to fly above each GT for image acquisition toensure that the GT is displayed at the center of the capturedimage [2]. However, this strategy may result in long travelingdistance and high energy consumption, especially for imageacquisition of multiple GTs that are far apart from each other[3]. Fortunately, this issue has been resolved in the latest UAV-mounted camera which is able to flexibly adjust its obliqueshooting angle according to the GT’s location [4]. However,
Xiao-Wei Tang and Xin-Lin Huang ( { xwtang, xlhuang } @tongji.edu.cn) arewith Tongji University, Shanghai, China.Shuowen Zhang ([email protected]) is with The Hong KongPolytechnic University, Hong Kong, China (corresponding author).Changsheng You and Rui Zhang ( { eleyouc, elezhang } @nus.edu.sg) arewith National University of Singapore, Singapore. to the best of our knowledge, there still lacks a tractablemodel to characterize the quality of the images captured bythe angle-rotatable camera. Moreover, the trajectory design forUAV-assisted image acquisition is substantially different fromthat in other missions such as UAV-assisted data collection.Specifically, the UAV should approach each GT such thatthe captured image can satisfy the resolution requirement, forwhich the feasible region is generally a complicated functionof the GT’s location. This makes the trajectory design fortaking images of multiple GTs fundamentally different fromthat for collecting data from multiple ground users (e.g., [5],[6]), where the feasible region for the UAV to meet thecommunication requirement is generally a cylindrical shapewith the ground plane centered at the user.Motivated by the above, we propose in this correspondencea novel oblique photography (OP) model to characterizethe resolution of the captured image. Based on this model,we study the 3D UAV trajectory optimization problem tominimize its traveling distance for taking images of multipleGTs, while guaranteeing a minimum resolution requirement ofeach captured image. The formulated problem is shown to be avariant of the traveling salesman problem with neighbourhoods(TSPN) [7], where the neighbourhood represents the feasibleregion for the UAV to capture the image with satisfactoryresolution. Note that although TSPN has been studied in[5], [6] for the two-dimensional (2D) case under disk-shapedneighbourhood, or [8], [9] for the 3D case under other regular-shaped neighbourhood, these algorithms cannot be applied toour problem with an irregular neighbourhood region. To tacklethis challenging problem, we simplify the UAV trajectory asline segments connected by multiple waypoints, each corre-sponding to the image-taking location of one GT. Then, wepropose an alternating optimization algorithm for finding asuboptimal solution to this simplified problem, by alternatelyoptimizing the waypoint locations and the GT visiting order.Numerical results show that the proposed scheme outperformsvarious benchmark schemes in terms of the traveling distance,while meeting the image resolution requirement.II. S YSTEM M ODEL AND P ROBLEM F ORMULATION
We consider a UAV-assisted image acquisition system withone UAV being dispatched to take images of K GTs, denotedby the set K = { , ..., K } . In the following, we first propose anovel UAV-assisted OP model which is tailored for the angle-rotatable camera, and then formulate the 3D UAV trajectoryoptimization problem. A. UAV-Assisted OP Model
In Fig. 1(a), we present a UAV-assisted OP model whererectangle A (cid:48) B (cid:48) C (cid:48) D (cid:48) is the camera’s image plane whose cov-erage region on the ground is an isosceles trapezoid (i.e., ABCD ). Points T (cid:48) and T represent the centers of the imageplane and GT k , respectively. Let [ w Tk , in meter (m) denotethe 3D coordinate of GT k where w k ∈ R × represents itshorizontal coordinate. For simplicity, we assume that GT k is adisk with a known radius r k in m. Let d u,k denote the distancefrom the UAV to GT k in m, i.e., | T (cid:48) T | , which is given by d u,k = (cid:113) (cid:107) q k − w k (cid:107) + z k , (1) a r X i v : . [ c s . MM ] D ec A B CD TA' B' C' T' O xyz
O' TA DB CH FED' E' TF EOT' T'' F' E'' xz O' F'' (a) The UAV-assisted OP model (b) The projection of the camera’s coverage region on the ground (c) The 2D profile of the OP model
Fig. 1 Illustration of UAV-assisted OP model. where q k ∈ R × and z k denote the UAV’s horizontal andvertical coordinate when taking image of GT k , respectively.In the conventional VP model, the image resolution ischaracterized by ground sample distance (GSD) that each pixelcan represent. However, with the angle-rotatable camera, theGSDs that the pixels can represent are different, thus renderingthe image resolution representation with GSD inapplicable tothe OP model. Therefore, we redefine the image resolutionas the ratio of GT k ’s area to the camera’s coverage area.Specifically, we denote by f the camera’s focal length, and w and l the width and length of the image plane, respectively.Let θ k denote the camera’s oblique angle (i.e., ∠ OO (cid:48) T inFig. 1(a)) and we have cos θ k = z k d u,k and tan θ k = (cid:107) q k − w k (cid:107) z k . As such, the camera’s coverage area, denoted by S ck , can beexpressed as below with the detailed derivation presented inAppendix A: S ck = S vk × φ ( θ k ) = 4 z k b b × − b tan θ k ) cos θ k , (2)where b = f w and b = f l are constants determined by thecamera’s parameter setting. Note that S vk is the camera’scoverage area when taking the image right above GT k , whichis proportional to z k , and φ ( θ k ) is defined as the coveragescaling factor which is monotonically increasing with respectto (w.r.t.) θ k . It is worth mentioning that the proposed OPmodel reduces to the conventional VP model when θ k = 0 .Based on the above, the image resolution, denoted by I k , canbe characterized by the UAV’s 3D image-taking location as I k = S GT k S ck = a k ( z k − b (cid:107) q k − w k (cid:107) ) ( (cid:107) q k − w k (cid:107) + z k ) z k , (3)where a k = b b πr k . In this correspondence, we consider aminimum resolution requirement for each GT k denoted by I k , thus the UAV’s image-taking location [ q Tk , z k ] shouldsatisfy I k ≥I k . Moreover, to let each GT k be completelyprojected in the camera’s coverage region, [ q Tk , z k ] should alsosatisfy r k ≤ min( d ,k , d ,k ) , where d ,k and d ,k represent thedistances from point T to AD and BC (see Fig. 1(b)), whichare defined in Appendix B. It is also worth noting that [ q Tk , z k ] should satisfy b z k −(cid:107) q k − w k (cid:107)≥ to meet the focal lengthrequirement of the camera, as explained in Appendix A. B. Problem Formulation
We aim to optimize the 3D UAV trajectory for capturingthe images of the K GTs, to minimize the UAV’s travelingdistance from given initial to final points denoted by [ w TI , z I ] The UAV-mounted camera can rotate its oblique shooting angle towardsthe GT with the angle θ k as shown in Fig. 1(a). (a) The 3D view of the neighbourhood (b) The vertical profile of the neighbourhood Fig. 2 Demonstration of the neighbourhood. and [ w TF , z F ] , respectively, while ensuring a sufficiently highimage resolution for all GTs. Note that since the UAVtrajectory is continuous, this problem involves an infinitenumber of variables, thus making the problem difficult tosolve. To simplify the trajectory design, we assume that theUAV trajectory consists of K +1 consecutive line segments ina similar manner as [10], with K waypoints each denoting theUAV’s image-taking location for one GT.Let ψ ( k ) ∈ K denote the index of the k -th visited GT and [ q Tψ ( k ) , z ψ ( k ) ] denote the location of the waypoint at which theUAV takes the image of GT ψ ( k ) . For consistence, we define [ q Tψ (0) , z ψ (0) ] = [ w TI , z I ] and [ q Tψ ( K +1) , z ψ ( K +1) ] = [ w TF , z F ] .For notational convenience, we define Ψ ∆ = [ ψ (1) , ..., ψ ( K )] , Q ∆ = [ q Tψ (1) , ..., q Tψ ( K ) ] , and Z ∆ = [ z ψ (1) , ..., z ψ ( K ) ] . The UAV’straveling distance is thus given by D ( Q , Z , Ψ )= K (cid:88) k =0 (cid:113) (cid:107) q ψ ( k +1) − q ψ ( k ) (cid:107) +( z ψ ( k +1) − z ψ ( k ) ) . (4)Let I ψ ( k ) denote the resolution requirement of GT ψ ( k ) . Then,under the given resolution constraints, the 3D UAV trajectoryoptimization problem can be formulated as (P1) min Q , Z , Ψ D ( Q , Z , Ψ )s . t . ψ ( k ) ∈ K , ∀ k ∈ K , (5a) K ∪ k =1 ψ ( k ) = K , (5b) I ψ ( k ) ≥ I ψ ( k ) , ∀ k ∈ K , (5c) r ψ ( k ) ≤ min( d ,ψ ( k ) , d ,ψ ( k ) ) , ∀ k ∈ K , (5d) b z ψ ( k ) −(cid:107) q ψ ( k ) − w ψ ( k ) (cid:107)≥ , ∀ k ∈ K , (5e)where (5a)-(5b) specify the feasible set of the GT visitingorder Ψ , and (5c)-(5e) specify the the feasible region of theUAV’s image-taking locations for each GT, which is termedas the “neighbourhood” for simplicity. In Fig. 2, we illustratethe neighbourhood of the GT located at the original point witha minimum resolution requirement of I = 0 . , where the 3Dview of the neighbourhood is depicted in Fig. 2(a), whichappears to be a spherical sector , and the vertical profile of theneighbourhood is shown in Fig. 2(b), which has the shape of a crescent moon . Note that unlike UAV-assisted data collectionwhere the communication quality generally increases as theUAV approaches the ground user, the UAV needs to maintaina certain distance from the GT for ensuring the image qualitydue to the non-convexity of the neighbourhood region asshown in Fig. 2. It is also observed from Fig. 2(b) that theimage resolution gradually increases from the outside to theinside of the neighbourhood, which intuitively indicates that the UAV may prefer to take the image at the surface of theneighbourhood for reducing its traveling distance.Notice that (P1) can be shown to be a modified 3D TSPNproblem [7], where the neighbourhood of each GT is acomplicated function of the UAV’s 3D location as well as theimage resolution requirement. It is worth noting that such a3D TSPN problem is generally NP-hard, and is more involvedas compared to the 2D TSPN problem studied in e.g., [5]. Inthe following, we propose an efficient iterative algorithm forfinding a high-quality suboptimal solution to (P1).III. P ROPOSED S OLUTION TO (P1)In this section, we propose an iterative optimization algo-rithm for solving (P1) by alternately optimizing one betweenthe 3D UAV waypoints and the GT visiting order with theother set of variables being fixed at each time. Specifically,we first introduce the two subproblems, and then present theoverall algorithm and analyze its computation complexity.
A. 3D Waypoint Location Optimization
First, we present the subproblem for optimizing the 3Dwaypoint locations with given Ψ , which is formulated as (P2 .
1) min Q , Z D ( Q , Z , Ψ )s . t . (5 c ) − (5 e ) . Since Q and Z are coupled with each other and theconstraints (5c)-(5d) are non-convex, (P2.1) is a non-convexoptimization problem, which is difficult to solve in general.To make the problem more tractable, we make some trans-formations to (5c)-(5d). For simplicity, we define (cid:96) ψ ( k ) ∆ = q ψ ( k ) − w ψ ( k ) , ∀ k ∈ K , and L ∆ = [ (cid:96) Tψ (1) , ..., (cid:96) Tψ ( K ) ] . By takinglogarithm of both sides of (5c), we obtain −
32 ln( (cid:107) (cid:96) ψ ( k ) (cid:107) + z ψ ( k ) ) − z ψ ( k ) ≥ f ( z ψ ( k ) , (cid:96) ψ ( k ) ,r ψ ( k ) ) , (6)where f ( z ψ ( k ) , (cid:96) ψ ( k ) , r ψ ( k ) ) (cid:44) ln I ψ ( k ) a ψ ( k ) − z ψ ( k ) − b (cid:107) (cid:96) ψ ( k ) (cid:107) ) is a convex function w.r.t. z ψ ( k ) and (cid:107) (cid:96) ψ ( k ) (cid:107) , respectively.Next, by taking the square of both sides of (5d), (5d) canbe equivalently transformed into z ψ ( k ) +2 z ψ ( k ) (cid:107) (cid:96) ψ ( k ) (cid:107) + (cid:107) (cid:96) ψ ( k ) (cid:107) ≥ f ( z ψ ( k ) , (cid:96) ψ ( k ) , r ψ ( k ) ) , (7)where f ( z ψ ( k ) , (cid:96) ψ ( k ) , r ψ ( k ) ) (cid:44) r ψ ( k ) max(( b z ψ ( k ) + (cid:107) (cid:96) ψ ( k ) (cid:107) ) , b z ψ ( k ) +(1+ b ) (cid:107) (cid:96) ψ ( k ) (cid:107) ) is a convex function w.r.t. z ψ ( k ) and (cid:107) (cid:96) ψ ( k ) (cid:107) , respectively. Inthe following, we apply the block coordinate descent (BCD)technique to decouple the joint optimization for Q and Z into two subproblems for Q and Z , separately, each of whichis sub-optimally solved by using the convex approximationtechnique as in [11].
1) Optimizing Z with given Q : With given Q and hence L , (P2.1) reduces to the following optimization problem overthe altitudes of the K waypoints in Z :. (P2 . .
1) min Z D ( Q , Z , Ψ )s . t . (5 e ) , (6) , (7) . Problem (P2.1.1) is still hard to solve due to the non-convexconstraints (6)-(7). However, note that with given L , the firstterm of the left-hand side (LHS) of (6) is convex w.r.t. z ψ ( k ) ,and so is the second term w.r.t. z ψ ( k ) . As such, we can applythe convex approximation technique to approximate the twoterms by their lower bounds as follows by using the first-order Taylor expansion at the given local point z ( i ) ψ ( k ) of the i -th iteration: −
32 ln( (cid:107) (cid:96) ψ ( k ) (cid:107) + z ψ ( k ) ) ≥ ϕ ( z ψ ( k ) ) (cid:44) − (cid:107) (cid:96) ψ ( k ) (cid:107) +( z ( i ) ψ ( k ) ) ) − (cid:107) (cid:96) ψ ( k ) (cid:107) +( z ( i ) ψ ( k ) ) ) ( z ψ ( k ) − ( z ( i ) ψ ( k ) ) ) , (8) − z ψ ( k ) ≥ ϕ ( z ψ ( k ) ) (cid:44) − z ( i ) ψ ( k ) − z ( i ) ψ ( k ) ( z ψ ( k ) − z ( i ) ψ ( k ) ) , (9)where the equality holds at the point z ψ ( k ) = z ( i ) ψ ( k ) . With (8)and (9), we approximate (6) with the following constraint: ϕ ( z ψ ( k ) ) + ϕ ( z ψ ( k ) ) ≥ f ( z ψ ( k ) , (cid:96) ψ ( k ) , r ψ ( k ) ) . (10)For the constraint (7), it can be shown that its first and secondterms on the LHS are both convex w.r.t. z ψ ( k ) . This allows usto lower-bound the two terms as follows: z ψ ( k ) ≥ ϕ ( z ψ ( k ) ) (cid:44) ( z ( i ) ψ ( k ) ) +4( z ( i ) ψ ( k ) ) ( z ψ ( k ) − z ( i ) ψ ( k ) ) , (11) z ψ ( k ) (cid:107) (cid:96) ψ ( k ) (cid:107) ≥ ϕ ( z ψ ( k ) ) (cid:44) z ( i ) ψ ( k ) ) (cid:107) (cid:96) ψ ( k ) (cid:107) +4 z ( i ) ψ ( k ) (cid:107) (cid:96) ψ ( k ) (cid:107) ( z ψ ( k ) − z ( i ) ψ ( k ) ) , (12)where the equality holds at the point z ψ ( k ) = z ( i ) ψ ( k ) . Therefore,we approximate (7) by replacing its LHS with its lower boundas the following constraint: ϕ ( z ψ ( k ) )+ ϕ ( z ψ ( k ) )+ (cid:107) (cid:96) ψ ( k ) (cid:107) ≥ f ( z ψ ( k ) , (cid:96) ψ ( k ) , r ψ ( k ) ) . (13)As such, (P2.1.1) can be reformulated into an approximateform given below, with the LHSs of (6) and (7) replaced bytheir respective lower bounds: (P2 . .
2) min Z D ( Q , Z , Ψ )s . t . (5 e ) , (10) , (13) . (P2.1.2) is a convex optimization problem, which can beefficiently solved via existing software, e.g., CVX. Moreover,it can be shown that the optimal solution to (P2.1.2) isguaranteed to be a feasible solution for (P2.1.1).
2) Optimizing Q with given Z : With given Z , (P2.1)reduces to the following problem for optimizing the horizontalwaypoint locations Q (or equivalently L ): (P2 . .
3) min L D ( Q , Z , Ψ )s . t . (5 e ) , (6) , (7) . Since the first term on the LHS of (6) is convex w.r.t. (cid:107) (cid:96) ψ ( k ) (cid:107) , it is lower-bounded by the first-order Taylor expan-sion at the given local point (cid:96) ( i ) ψ ( k ) of the i -th iteration as −
32 ln( (cid:107) (cid:96) ψ ( k ) (cid:107) + z ψ ( k ) ) ≥ ϑ ( (cid:96) ψ ( k ) ) (cid:44) −
32 ln( (cid:107) (cid:96) ( i ) ψ ( k ) (cid:107) + z ψ ( k ) ) − (cid:107) (cid:96) ( i ) ψ ( k ) (cid:107) + z ψ ( k ) ) ( (cid:107) (cid:96) ψ ( k ) (cid:107) − (cid:107) (cid:96) ( i ) ψ ( k ) (cid:107) ) , (14)where the equality holds at the point (cid:96) ψ ( k ) = (cid:96) ( i ) ψ ( k ) . Then, (6)can be approximated as ϑ ( (cid:96) ψ ( k ) ) − z ψ ( k ) ≥ f ( z ψ ( k ) , (cid:96) ψ ( k ) , r ψ ( k ) ) . (15) Similarly, we can derive the lower bounds of the secondand third terms on the LHS of (7) as follows by using thefirst-order Taylor expansion: z ψ ( k ) (cid:107) (cid:96) ψ ( k ) (cid:107) ≥ ϑ ( (cid:96) ψ ( k ) ) (cid:44) z ψ ( k ) (cid:107) (cid:96) ( i ) ψ ( k ) (cid:107) +4 z ψ ( k ) ( (cid:96) ( i ) ψ ( k ) ) T ( (cid:96) ψ ( k ) − (cid:96) ( i ) ψ ( k ) ) , (16) (cid:107) (cid:96) ψ ( k ) (cid:107) ≥ ϑ ( (cid:96) ψ ( k ) ) (cid:44) (cid:107) (cid:96) ( i ) ψ ( k ) (cid:107) +4 (cid:107) (cid:96) ( i ) ψ ( k ) (cid:107) ( (cid:96) ( i ) ψ ( k ) ) T ( (cid:96) ψ ( k ) − (cid:96) ( i ) ψ ( k ) ) . (17)With (16) and (17), the constraint in (7) is approximated asfollows by replacing its LHS with its lower bound: z ψ ( k ) + ϑ ( (cid:96) ψ ( k ) )+ ϑ ( (cid:96) ψ ( k ) ) ≥ f ( z ψ ( k ) , (cid:96) ψ ( k ) , r ψ ( k ) ) . (18)As a result, (P2.1.3) can be reformulated into the followingapproximate form: (P2 . .
4) min L D ( Q , Z , Ψ )s . t . (5 e ) , (15) , (18) . Since the constraints in (15) and (18) are convex, (P2.1.4)is a convex optimization problem, which can be efficientlysolved via existing software, e.g., CVX.With the convex approximation technique, the objectivevalue of (P2.1) can be shown to be non-increasing over theiterations similarly as in [11], which is also lower-bounded bya finite value. Therefore, the proposed algorithm for optimizingthe 3D waypoint locations is guaranteed to converge.
B. Visiting Order Optimization
With given waypoint locations ( Q , Z ) , the subproblem foroptimizing the GT visiting order Ψ is recast as (P2 .
2) min Ψ D ( Q , Z , Ψ )s . t . (5 a ) , (5 b ) . (P2.2) is equivalent to a classic TSP, for which a high-quality suboptimal solution can be found with low compu-tational complexity via binary integer programming [12]. C. Overall Algorithm and Computational Complexity
The proposed algorithm for (P1) is summarized as fol-lows. First, we initialize the visiting order Ψ by solvingthe TSP in (P2.2) based on q k = w k , z k = 0 , ∀ k ∈ K ,i.e., considering waypoint locations at the GTs. Then, weiteratively optimize the 3D waypoint locations and GT visitingorder based on Sections III-A and III-B, respectively. Next,with given Q and Z , Ψ is optimized by solving (P2.2).The proposed algorithm stops until we cannot find a bettersolution within a prescribed precision requirement or a maxi-mum number of iterations is reached. The overall complexityof the proposed algorithm is analyzed as follows. For thesubproblem of waypoint locations optimization, Q and Z areiteratively optimized by using the convex software based onthe interior-point method, and their individual complexity canbe represented as O ( K . ) and O ( K . ) , respectively. Then,let I denote the number of iterations for the BCD method,the total computation complexity for optimizing the waypointlocations is O ( K . I ) . For the subproblem of visiting orderoptimization, the complexity for solving the classic TSP with Note that the proposed iterative algorithm can be generally extended tosolve the TSPN problems in e.g., [6], [8], [9] by modifying the neighbourhood-related constraints in (P1). (a) 3D trajectories (b) Horizontal trajectories(c) Vertical trajectories (d) Traveling distance versus number of iterations
Number of iterations T r av e li n g d i s t a n ce ( m )
2D UAV trajectory under the convention VP model2D UAV trajectory under the proposed OP model3D UAV trajectory under the proposed OP model
The visiting order optimizationThe visiting order optimization
Fig. 3 Comparison of the optimized UAV trajectories and travelingdistances by different schemes. the algorithm in [12] is O (2 K K ) . As such, the overallcomplexity is O (cid:0) (2 K K + K . I ) I (cid:1) , where I denotes thenumber of inter-subproblem iterations.IV. N UMERICAL R ESULTS
In this section, we provide numerical results to showthe effectiveness of the proposed OP model as well asthe corresponding 3D trajectory design. The parameters areset as f =0 . m, w =0 . m, l =0 . m [13], and [ w TI , z I ]=[ w TF , z F ]=[0 , , m. We consider GTs withthe same radius of r k =20 m, k ∈ K , which are randomlydistributed in a square area of × m . The resolutionrequirement of each GT, i.e., I k , k ∈ K , is independentlyand randomly set within [0 . , . . Two benchmark schemesare considered: 1) 2D UAV trajectory under the conventionalVP model, and 2) 2D UAV trajectory under the proposed OPmodel. For the two benchmark schemes, the UAV altitude isfixed (i.e., z k = 100 m, ∀ k ∈ K ) to ensure that at least afeasible waypoint can be found for each GT to satisfy theimage resolution requirement. Specifically, in the benchmarkscheme 1, the image-taking waypoint for each GT is rightabove the GT and thus the UAV trajectory can be obtained bysolving the TSP problem based on these waypoints. Whilein the benchmark scheme 2, the UAV trajectory can beobtained via the iterative algorithm proposed in Section IIIby alternately optimizing the horizontal waypoint locations in Q and the GT visiting order in Ψ .Figs. 3(a)-(c) show the UAV trajectories obtained by differ-ent schemes. It is observed from Fig. 3(b) that compared to theUAV trajectory under the conventional VP model, the UAV’shorizontal flight range can be greatly reduced by adoptingthe proposed OP model, even with the fixed altitude (as inbenchmark scheme 2). Moreover, it is observed from Fig. 3(c)that under the proposed OP model, the 3D UAV trajectoryin general has a lower altitude than the 2D UAV trajectory.This is because the UAV can also satisfy the resolutionrequirement when taking the image of the GT at a loweraltitude by exploiting a larger horizontal distance away from the GT, thus resulting in a shorter traveling distance, whichcan be inferred from Fig. 2(b). Fig. 3(d) shows the travelingdistances versus the number of iterations for all schemes.Specifically, the circled dots denote the traveling distanceobtained by optimizing the GT visiting order, while other dotscorrespond to the traveling distances obtained by optimizingthe horizontal/vertical waypoint locations in Q or Z . It isobserved that the proposed OP model with 2D trajectory yieldsmuch shorter distance than the conventional VP model with2D trajectory, which is further reduced by the proposed OPmodel with 3D trajectory due to additional degrees-of-freedomin the vertical trajectory optimization (see Fig. 3(c)).V. C ONCLUSIONS
In this correspondence, we proposed a novel OP model tocharacterize the resolution of images captured by an angle-rotatable camera mounted on a UAV. Under the proposedOP model, we formulated a 3D UAV trajectory optimizationproblem to minimize the UAV’s traveling distance whilemaintaining a given resolution requirement for the imagestaken from multiple GTs. The formulated problem was shownto be a modified 3D TSPN problem, for which we proposedan iterative algorithm for finding an efficient solution, byalternately optimizing the image-taking waypoints and thevisiting order of the GTs. Numerical results were presentedto show the effectiveness of the proposed scheme comparedto other benchmark schemes.A
PPENDIX
AFor ease of explanation, we illustrate in Fig. 1(c) the 2D pro-file of Fig. 1(a). Specifically, we have | E (cid:48) F (cid:48) | = w , | T (cid:48) O (cid:48) | = f ,and ∠ OO (cid:48) T = θ k . According to the triangle similarity theorem (TST), we have | O (cid:48) E (cid:48)(cid:48) || O (cid:48) O | = | E (cid:48) E (cid:48)(cid:48) || OE | where | O (cid:48) O | =( d u,k − f ) cos θ k , | O (cid:48) E (cid:48)(cid:48) | = f cos θ k − w sin θ k , and | E (cid:48) E (cid:48)(cid:48) | = f sin θ k + w cos θ k .To ensure | O (cid:48) E (cid:48)(cid:48) |≥ , θ k should satisfy that ≤ θ k ≤ arctan f w ,which is equivalent to b z k − (cid:107) q k − w k (cid:107) ≥ , (19)where b = f w . Then, | OE | can be obtained as | OE | = ( d u,k − f ) cos θ k × f sin θ k + w cos θ k f cos θ k − w sin θ k . (20)Similar to | OE | , | OF | is given by | OF | = ( d u,k − f ) cos θ k × f sin θ k − w cos θ k f cos θ k + w sin θ k . (21)Since | EF | = | OE |−| OF | , | EF | can be derived as | EF | = f w ( d u,k − f ) cos θ k f cos θ k − w sin θ k . (22)Similarly, | AD | and | BC | can be obtained as follows with thederivation omitted for brevity. | BC | = l ( d u,k − f ) cos θ k f cos θ k − w sin θ k , | AD | = l ( d u,k − f ) cos θ k f cos θ k + w sin θ k . (23)According to the trapezoid area formula, i.e., S ck = · | EF | · ( | AD | + | BC | ) , the camera’s coverage area is given by S ck = 4( d u,k − f ) b b × − b tan θ k ) cos θ k , (24)where b = f l . Since f (cid:28) d u,k , we have ( d u,k − f ) ≈ d u,k = z k cos θ k . Thus, (24) can be rewritten as S ck ≈ z k b b × − b tan θ k ) cos θ k . (25) A PPENDIX
BAs shown in Fig. 1(b), it is required that both | F T | (denotedby d ,k ) and | HT | (denoted by d ,k ) derived in the followingshould be no smaller than r k . According to the TST, we have | AD || BC | = | FT || ET | where | ET | + | F T | = | EF | . Therefore, we obtain d ,k = | F T | = | AD | × | EF || AD | + | BC | . (26)With | AD | , | BC | , and | EF | given in Appendix A, d ,k can beapproximated as d ,k ≈ z k + (cid:107) q k − w k (cid:107) b z k + (cid:107) q k − w k (cid:107) . (27)The area of ABCD is given by S ABCD = 12 × ( | AD | + | BC | ) × | EF | = 12 | AD | × | F T | + 12 | BC | × | ET | +2 × | AB | × | T H | , (28)where | AB | = (cid:112) ( | BC | − | AD | ) / | EF | . (29)Therefore, according to (28)-(29), d ,k can be obtained as d ,k = | HT | = | AD | × | BC | × | EF | ( | AD | + | BC | ) × | AB | . (30)With | AD | , | BC | , and | EF | given in Appendix A, d ,k can beapproximated as d ,k ≈ z k + (cid:107) q k − w k (cid:107) ( b z k + (1 + b ) (cid:107) q k − w k (cid:107) ) . (31)Please note that we approximate d u,k − f as d u,k in (27) and(31) as that in Appendix A for simplicity.R EFERENCES[1] Y. Zeng, Q. Wu, and R. Zhang, “Accessing from the sky: A tutorial onUAV communications for 5G and beyond,”
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