Optimal Trajectory Planning of Drones for 3D Mobile Sensing

Transcription

1 Optimal Trajectory Planning of Drones for 3D Mobile Sensing Pengyu Zhao, Yuzhe Yang, Yuanxing Zhang, Kaigui Bian, Lingyang Song Pengpeng Qiao, Zhetao Li Abstract Mobile sensing is challenging in 3D space, as there are many inaccessible places where people rarely venture. Unmanned aerial vehicle (UAV), commonly known as drone, has greatly extended the scope of mobile sensing in 3D space, and pushed forward a variety of 3D mobile sensing applications, such as aerial photo- or video-graphy, 3D wireless signal survey, and air quality monitoring. However, the short battery life of drones has largely restricted the wide adoption of these applications. In this paper, we study the trajectory planning problem for optimizing the flight route in a given sensing space. We first divide the 3D space into an infinite three-dimensional network of observation locations (OLs), and model the sensing scope as a finite subgraph of 3D OL network. We formulate the problem as finding the optimal trajectory in the sensing scope. We propose an algorithm that finds trajectory in each divided 3D grid of the sensing scope by generating a nearly optimal dominating path, and finding the minimum dominating set in the dominating path. Then, we concatenate obtained trajectories in 3D grids to a nearly optimal trajectory in the sensing scope. Experimental results show that the proposed algorithm takes 24% less time to complete sensing the given space, and during the battery life it can cover 19% more sensing scope, than existing solutions. I. INTRODUCTION Unmanned aerial vehicle (UAV), commonly known as drone, is an aircraft without a human pilot aboard, which is commonly used in measurement and sampling in 3D space [8]. Compared to manned aircraft, drones are more suitable for data collections and mobile sensing applications that capture different dimensions of signals in the environment that are beyond the sensing capability and scope of human beings, such as aerial photography [4], 3D wireless signal survey [1], air quality index (AQI) measurement [5], [9]. However, civilian drones are still not widely used by customers in our daily life today, due to lack of killer applications, and the privacy/safety concerns (no-drone zones are defined to protect the residents privacy, or the safety of civil aviation system). Moreover, if one has actually tried flying a drone, he/she may feel that existing civilian drones are not ready for use in daily life for its limitation in hardware. For example, a drone s battery life is usually tens of minutes (less than half an hour) [7], which limits the scope of the drone when mobile sensing in a large scale space. In this study, we are not interested in the battery technology. Instead, in order to maximize the sensing scope under the limited battery life, we focus on minimizing the time consumed for sensing a given space. Since the drone can sense the signals in the cubic space around it, the sensing scope of a trajectory is the union of the cubic spaces that the drone Battery life (%) Flight Hovering Time consumed (s) Fig. 1: The flight mode consumes more battery power than the hovering mode when flying a DJI Phantom 3 quadcopter. has traversed along the trajectory. Therefore, the trajectory planning problem arises naturally: given a 3D sensing scope, then how to plan the drone s optimal trajectory with a number of stops for minimizing the flight time? The conventional way of addressing the trajectory planning problem for a mobile device (e.g., a robot) takes a two-step approach: (1) first select an arbitrary number of stops; and then (2) connect these stops to create a path. The first step is usually completed by a near-optimal solution to find a minimum vertex cover; and the second step can be done by finding the shortest path connecting those stops [6]. However, such a stop-selection-first strategy does not work well for planning a drone s trajectory. Even though an arbitrary number of stops is chosen first, due to the limited battery life, the drone may not be able to complete the expected path and traverse all stops before running out of battery. Hence, in a given space, it is necessary to first plan a path for the drone, and then select an appropriate number of stops within the path based on its remaining battery life. Moreover, the flight mode is usually dominant in a drone s trajectory, while the hovering mode (for collecting sufficient measurement data in mobile sensing) only takes a few seconds at each stop. It is known that the flight mode consumes more battery power than the hovering mode [7], and our experimental results shown in Figure 1 also confirm this when flying a DJI Phantom 3 quadcopter. These observations again motivate the design of a path-planning-first strategy for addressing the problem. In this paper, we study the optimal trajectory planning problem of a drone for 3D space mobile sensing applications. We divide the 3D space into an infinite observation location (OL) network, and formulate trajectory planning problem as finding the optimal trajectory in a connected subgraph of OL network. We propose a strategy that first determines a nearly optimal dominating path, and then selects an appropriate number of

2 critical OLs within the path to perform measurement. According to the proposed strategy, we are able to draw the dominating path and select critical OLs in O(N) time where N is the total number of OLs in the sensing scope. Our experimental results reveal that the proposed algorithm outperforms existing algorithms in two aspects: (1) the proposed solution takes 24% less time than other solutions to complete sensing the same space; (2) during the battery life, the proposed solution can reach 19% more sensing scope than existing approaches. II. RELATED WORK A. Using drones in 3D mobile sensing Conventional mobile sensing systems rely on the mobile devices or ground-vehicles to perform the environmental sensing in a 2D plane, e.g., the mobile data analysis using robots [2]. Drones become more and more popular in 3D mobile sensing applications since they could capture different dimensions of signals in the 3D environment that are usually beyond our sensing capability. Aerial photo or video-graphy applications [4] facilitate the collection of images, and the analysis over the collected data. For example, it is feasible to locate anomalies in image data, and link particular image data to an address of the property where the anomaly is detected. DroneSense is a system for 3D wireless signal survey [1], i.e., measuring wireless signals in the 3D space, which provides us with an efficient method to quickly analyze wireless coverage and test their wireless propagation models. SensorFly is designed for indoor emergency response or inspections in inaccessible places where people cannot reach [5], which forms aerial sensor network platform that can adapt to node and network disruptions in harsh environments. However, these 3D mobile sensing applications of drones are constrained by the limited battery life, which motivates a more efficient measurement approach to better design the trajectory. B. Trajectory planning problem To address the trajectory planning problem, the genetic algorithm [11] and particle swarm optimization [3] are always proposed for real-time path planning, which could find an optimal or near-optimal path for robotics in both complicated static and dynamic environments. These approaches have been applied on the UAV platform which could ensure partial minimality between two nodes. Meanwhile, the trajectory planning for a large scale mobile sensing scenario is usually formulated as an Traveling Salesman Problem (TSP) [6]. Existing solutions take a twostep approach: (1) the first step is to find a vertex cover in the underlying network graph (i.e., a number of measurement locations that cover the given sensing area); and (2) the second step is to find the shortest path connecting these vertices (locations), along which the mobile device can traverse to complete the mobile sensing task in space. In this paper, we take a different approach by finding the optimal path first and then selecting the measurement locations, as the flight mode consumes more battery than the hovering mode for drone operations. III. SYSTEM MODEL AND PROBLEM FORMULATION In this section, we establish a multi-layer 3D network model for mobile sensing in the 3D space. Then, we analyze the time consumed for mobile sensing. Finally, we formulate the problem as a combination of the minimum dominating path problem and the constrained minimum dominating set problem. A. 3D network model Dividing the 3D space into cuboids: We divide the 3D space into basic cuboids that are a-meter long, b-meter wide and h-meter high. We define the center point of cuboid i as its observation location (OL) (as shown in Figure 2), which is denoted by the 3-tuple (longitude, latitude, and altitude), i.e., OL i = (x i, y i, z i ), where x i, y i, z i are 3D coordinates of OL i. Note that we call the OL at the center of cuboid i as OL i. 3D network of observation locations (OLs): OLs form an infinite 3D network graph in the 3D space. Specifically, the OL inside each cuboid i is considered as a vertex, and an edge (i, j) exists, if cuboid i is adjacent to cuboid j (i.e., they are the same in two coordinates and adjacent to each other in the third dimension coordinate). As a result, the OLs forms an infinite 3D grid. Moreover, the 3D grid could be divided into multiple layers of infinite 2D grids at different height. OL network embedding: We define the OL network embedding of sensing scope as the OLs inside the sensing scope, which forms a finite connected subgraph G = (V ; E ) of the infinite 3D grid. Since the sensing scope could be covered by its inner OLs, and each OL could cover a cuboid area in 3D space, sensing scope could be covered by a union of cuboids. For the convenience of proof, we assume that the sensing scope is a union of cuboids, otherwise we could fill the sensing scope into the ideal shape. The coverage set of OL i: We define the coverage set of OL i as the set of OLs including those located in the neighbor cuboids of the cuboid i, plus OL i itself, as shown in Figure 2. We denote the coverage set of vertex v as Cov(v), and the coverage set of a vertex set S as Cov(S). Correlation between OLs: We assume that the sensed data at two OLs in the same 3D space may have certain correlation. Intuitively, the more distant two OLs are located, the less correlation they may have. That is, two adjacent OLs have the strongest correlation in their sensed data; the sensed data at two adjacent OLs are almost the same if the distance between them is small. Hence, if the drone has sensed at one OL, it can skip sensing at its adjacent OLs. Critical OL (COL): Given a specific sensing scope and its OL network embedding G = (V ; E ), it is inefficient for 2

3 Unmeasured Location covers all OLs in V, and then select a set V C of COLs from V P as the dominating set. 3-Dimension Spatial Grid Observation Location Coverage Set Fig. 2: The divided cuboids of a 3D space; a center OL in the cuboid in the green color; and the OLs and their cuboids in the coverage set of the center OL. a drone to traverse all OLs in V. Based on the correlation between OLs, we could select a number of critical observation locations (COLs) in V where the drone can perform the measurement. B. Time consumed The time consumed by the 3D mobile sensing of drones consists of two parts: (1) the flight time, and (2) the measurement time. Let V C V denote the set of COLs in the sensing scope with embedding G = (V ; E ). Let V P V denote the set of vertices in the drone s trajectory which forms a path of OLs in the sensing scope. We have V C V P since the trajectory should contain the COLs. Flight time: the flight time T F is time consumed by the drone s flight. In the formed 3D grid, we use Hamiltonian distance to characterize the distance between OLs. So the flight time is proportion to the length of the trajectory, which is written as T F = t F V P, where t F is the flight time for a unit length in the coordinate system. Measurement time: the measurement time T M is time consumed by the drone for hovering and measurement. For the same sensing task (e.g., WiFi signal survey), we could assume that measurement time is the same for all OLs. So the total measurement time is proportion to the number of COLs, which can be written as T M = t M V C, where t M is the measurement time at each OL. Therefore, the total time T consumed by the drone is C. Problem formulation T = T F + T M. We formulate this problem as a combination of a minimum dominating path problem and a constrained minimum dominating set problem in a connected finite subgraph of infinite 3D grid. Problem 1 (Trajectory planning by both flight time and measurement time). Given a specific sensing scope in which OLs forms a connected subgraph G = (V ; E ) of infinite 3D grid graph, assume the time required for completing a drone s trajectory is dependent on the drone s flight time and measurement time, we seek to find the minimum dominating path V P V which minimize V P, V C subject to Cov(v) = V, v V P Cov(v) = V, v V C V P is a path, V C V P. IV. TRAJECTORY PLANNING MECHANISM In this section, we address Problem 1 by proposing an algorithm which divides OL network embedding of the sensing scope into multiple 3D grids first. For each 3D grid, we construct a nearly optimal trajectory by generating a dominating path, and then selecting critical OLs in it. Finally, we concatenate trajectories in 3D grids into the trajectory of sensing scope. A. Sensing scope division and trajectory planning For a given sensing scope in 3D space, its OL network embedding G is a finite subgraph of infinite 3D OL grid. As the shape of sensing scope is arbitrary, specifically, a union of cuboids, it is difficult to plan a optimal trajectory for the sensing scope. We will show the whole process for the sensing scope in Figure 3. Thus, we propose the following proposition: Proposition 1. Sensing scope could be divided into multiple disjoint 3D cuboids. Hence, its OL network embedding G could be divided into multiple disjoint 3D grids. Proof. We divide sensing scope in the same way as the division of 3D space in Section III, and merge adjacent cuboids until no two adjacent cuboids could be combined. Therefore, sensing scope could be divided into multiple disjoint 3D cuboids. Obviously, the OLs in each divided 3D cuboid forms a 3D grid, so G could be divided into multiple disjoint 3D grids. Following the proposition, the OL network embedding G could be divided into multiple disjoint 3D grids. And from the definition of the division, each 3D grid could be further divided into 2D grids in different layers. Then, for a given sensing scope, we could divide OLs in it into several multilayer 2D grids. Let L m,n denote a 2D grid with m rows and n columns. Since 2D grids of a 3D grid are the same in different layers, our proposed algorithm works by following steps. First, we divide the OL network embedding of sensing scope into several 3D grids. For each 3D grid, we generate a nearly optimal dominating path by connecting minimum dominating sets of different layer 2D grids. Then, we select COLs from the dominating path in the 3D grid. Finally, we merge trajectories in 3D grids into the trajectory of sensing scope. 3

4 Fig. 3: The top figure is an irregular sensing scope of a real 3D space. The bottom figure shows the sensing scope division of the above space. We will show how to construct trajectory in detail in the following sections. B. Generating nearly optimal dominating path in a 3D grid For each 3D grid, since 2D grids in different layers are the same, the minimum dominating set in each 2D grid is also the same. We will show that those dominating sets could be connected into a nearly optimal dominating path of the 3D grid. Proposition 2. Given a specific 3D grid G = (V; E) = L m,n,p, denote G i = L m,n as the layer i 2D grid of G, and denote VS i = {(x 1, y 1, i), (x 2, y 2, i),..., (x V i S, y VS i, i)} Gi as the minimum dominating set of G i, where (x j, y j, i) is the intersection of row x j and column y j in layer i. Then, we have (1) p 1 V S i forms a dominating set in G. (2) p 1 V S i could be concatenated into a dominating path (V P ) of G. (3) V P is optimal when m, n and p tend to infinity. Proof. We denote V S as the size of all VS i because the minimum dominating sets are the same for 2D grids in different layers. Therefore, (1) since VS i is a dominating set in Gi, and G = p 1 Gi, p 1 V S i is a dominating set of G. (2) As we have proved the domination property in (1), we could generate a dominating path as follows: First, for j = 1, 2,..., V S, we connect vertices in the same position of different layers and get a dominating path P j = {(x j, y j, 1), (x j, y j, 2),..., (x j, y j, p)}. Then, we concatenate P 1, P 2,..., P V S on the top layer and the bottom layer to a dominating path (V P ) of G, as shown in Figure 4(c). (3) Because every vertex has at most 7 neighbors (including itself) in 3D space, vertices in every dominating path could cover only 5 vertices (including itself) other than its predecessor and successor. Therefore, the size of the minimum dominating path in G should be greater than m n p 5. From [1], when m and n tend to infinity, we have lim V (m + 2) (n + 2) S = lim 4. (1) m,n m,n 5 Since those vertices which are used to connect P i in both the top layer and the bottom layer could not exceed a single 2D grid L m,n, we have V S V P P i + m n i=1 p = V i S + m n (2) i=1 = p V S + m n. Thus, when m, n and p tend to infinity, we have lim V P lim p V S + m n m,n,p m,n,p (m + 2) (n + 2) = lim p 4 + m n m,n,p 5 pmn = lim m,n, p n + 2 5m + 1 p 16 mn = lim m,n, p = lim m,n, p pmn 5 1 pmn 5. (3) Therefore, V P converges to the size of the minimum dominating path in G, which means the proposed dominating path is optimal. Following the above method, we could obtain a nearly optimal dominating path in a 3D grid. Figure 4 (a)-(d) shows the dominating path generation in a divided 3D cuboid. C. Selecting COLs from obatined dominating paths In the previous section, we find a nearly optimal dominating path in the 3D grid. We will give an approach to show how to select COLs from the obtained dominating path in a specific 3D grid G = (V; E) = L m,n,p. We use the same symbols as Proposition 2. First, we select all vertices in VS i as COLs, for i = 3, 4,..., p 2. Then, for v = (x j, y j, 1) VS 1, if there exists a vertex u = (x j, y j, 1) V P such that u is adjacent to v, while (x j, y j, 2) is the only one vertex which could only be covered by w = (x j, y j, 2), we select u as COL. Otherwise, we select v and w as COLs. Similarly, for v = (x j, y j, p) V p S, if there exists a vertex u = (x j, y j, p) V P such that u is adjacent to v, while (x j, y j, p 1) is the only one vertex which could only be covered by w = (x j, y j, p 1), we select u as COL. Otherwise, we select v and w as COLs. Since the vertices in selected COLs could cover the coverage set of all vertices in p 1 V S i, while from Proposition 2 p 1 V S i is a dominating set of G, the selected COLs forms a 4

5 (a) (b) (c) (d) (e) Fig. 4: A nearly optimal trajectory planning in a 3D cuboid. Figure (a) shows the 3D grid (blue vertices) in the cuboid. For each layer of 3D grid, we select the minimum dominating set (black vertices) in it as shown in (b), where we hide the blue vertices for better visualization. We connect vertices in the same position of different layer dominating sets (grey lines), as shown in (c). We then concatenate obtained vertical paths on the top layer and the bottom layer (with dark blue vertices as connecting vertices), and get a nearly optimal dominating path in the cuboid as shown in (d). Finally, we select COLs (orange vertices) from the dominating path with the proposed approach as shown in (e). dominating set of G. Then, we will show that the obtained COLs is optimal. dominating set. Thus, we assume there is only one vertex u = (x j, y j, 2) in v s independent coverage set. Then, we choose w = (x j, y j, 1) to cover u if w VP. If we just replace v by w, the size the obtained dominating set is the same as Ð p of i. However, since l = (x, y, 1) s independent V j j 1 S coverage set is one layer below v s independent coverage set, l s independent coverage set contains w only. Therefore, we could use w to replace both v and l which belong to 1 VSi if the above condition is fulfilled. 3) Since the replacement in both layer p 1 and p is the same as layer 1 and 2, we would not repeat here. Proposition 3. The selected COLs is a constrained minimum dominating set of G, which is restricted in VP. Proof. Again, we use the same symbols as Proposition 2. Moreover, we define connecting vertices as the relative complement of 1 VSi with respect to VP. And we define independent coverage set of a vertex v in VP as the coverage set of v which could not be covered by other vertices in VP except connecting vertices. Since 1 VSi is a dominating set in VP, we would like to consider the replacement of vertices in 1 VSi to a constrained minimum dominating set in VP. That is, if 1 VSi is not the smallest constrained dominating set, some vertices in it could be replaced by a smaller subset (or empty set) in VP while keeping the coverage set unchanged. Specifically, v VSi might be replaced by vertices between layer i 1 and i + 1. We will show the replacement of vertices in different layers below. 1) When 2 < i < p 1, for v VSi, v s independent coverage set contains at least one vertex, because from [1], v covers at least one vertex (other than v) which is beyond the coverage set of other vertices in layer i of VP (which is indeed VSi ), not to mention vertices in layer i 1 and layer i + 1. Therefore, we could not replace v by any other vertices in the dominating path. This implies that the constrained minimum dominating set in VP contains all vertices in VSi for 2 < i < p 1. 2) We consider dominating set in both layer 1 and 2. If we want to replace a vertex v in layer 1, we would need one more connecting vertex in the same layer since v s independent coverage set contains at least one vertex. Therefore, we could not just replace vertices in layer 1. If we want to replace v = (x j, y j, 2) VS2, since v s independent coverage set contains at least one vertex, we could only choose connecting vertices in layer 1 to cover v s independent coverage set. If v s independent coverage set needs more than two vertices to cover, we would not acquire a smaller Note that the result of replacement is the same as the selected COLs. Therefore, the selected COLs is a constrained minimum dominating set of G. Following Propositions 2 and 3, we could get a nearly optimal trajectory in a 3D grid as shown in Figure 4. D. Concatenating trajectories in 3D grids to the trajectory of sensing scope For each 3D grid divided from sensing scope, we could construct a nearly optimal trajectory by finding a nearly optimal dominating path, and selecting a constrained minimum dominating set in the dominating path with above method. Then, we will show how to generate the trajectory of sensing scope. First, we could simply construct the set of COLs in the sensing scope by joining COLs of all divided 3D grids. Based on the domination property of the dominating path and the connectivity of sensing scope, every pair of adjacent 3D grids must intersect in their minimum dominating paths or in the neighborhood of their minimum dominating paths. Therefore, we propose a heuristic algorithm that traverses obtained dominating paths in 3D grids in the spatial order. Specifically, if any unvisited 3D grid is in the neighborhood of current position, we traverse it recursively, mark it visited, and move back to the origin position after traversal. Since the OL network embedding of sensing scope is connected, 5

6 Start and then select COLs from path. The two algorithms are introduced below. 1) COL prior algorithm: We select the minimum dominating set in the sensing scope as COLs, and find a optimal dominating path to connect those vertices following the layer-first strategy which chooses the next COL as the closest one in the same layer. 2) Path greedy algorithm: We select an OL as start point. Then, we use a greedy algorithm to construct the dominating path in the sensing scope which chooses the next vertex v as the closest one that can lead to the maximum coverage, such as: min max v, where End d(v,v ) N [v]\n [P] v V is the last vertex of path of V P V. Next, we select the minimum dominating set constrained in V P as COLs. Performance metrics: We use the following two metrics for evaluating the algorithm performance. Time consumed: The time consumed is defined as total time of flight and measurement of the drone to cover the space of the trajectory under a specific algorithm. Coverage space: The coverage space is defined as the maximum coverage area in the 3D space during entire battery life, by following the trajectory under a specific algorithm. Fig. 5: The nearly optimal trajectory in the whole sensing scope generated by the proposed algorithm. Orange vertices are COLs, blue vertices are connecting vertices. Fig. 6: DJI Phantom 3 quadcopter. we could finally obtain a nearly optimal trajectory of sensing scope by traversing dominating paths of all 3D grids. Thus, given a specific 3D sensing scope, we could generate a nearly optimal trajectory for the drone. Figure 5 shows the obtained trajectory in sensing scope shown in Figure 3. B. Experimental Results We divide the 3D space into cuboids, and choose an irregular open space in a university campus as sensing scope. Using sensing scope division algorithm in Section IV-A, sensing scope could be divided into 5 disjoint 3D cuboids, which are 7 7 7, 9 5 8, 4 6 5, 1 5 5, 3 6 7, respectively as shown in Figure 3. Since the time consumed is the sum of flight time and measurement time, we consider the impact of different measurement times in experiments, which are set as, 1, 2, 5 seconds at each COL. The trajectory generated by proposed algorithm is shown in Figure 5. Time consumed: Figure 7 shows the time consumed under compared algorithms with different granularity of the measurement time. We observe that our proposed algorithm consumes less time for all measurement granularities. Furthermore, the gap between our algorithm and compared algorithms increases as the measurement time increases. Coverage space: Figure 8 shows the coverage space in the sensing scope under compared algorithms with different granularity of the measurement time during the battery life. We limit battery life under which trajectories generated by the proposed algorithm and compared algorithms would not exceed sensing scope. We observe that our proposed algorithm could cover more space than compared algorithms during the whole battery life, when the unit measurement time are, 1, 2 seconds. While in the case that the unit measurement time is 5 seconds, path greedy algorithm outperforms other algorithms, since it utilizes a short-sighted strategy, which means it could cover more space in the early stages, but also face a performance drop in the final stages. As the V. E VALUATION A. Experiment Setup Prototype: The prototype consists of two parts: (1) a drone with an API that can configure its trajectory; and (2) a measurement sensor board or a smartphone is installed into a plastic box with vent-holes made by a laser engraving machine, which is attached at the bottom of the drone. The drone we use is a DJI Phantom 3 quadcopter, as shown in Figure 6. There is an intelligent flight mode that allows the customization of the trajectory by configuring a number of stop locations such that the drone will autonomously follow this trajectory to hover over each stop and complete the flight. The GPS sensor, ultrasonic sensor, and the motion sensors collectively determine the current observation location of the drone and guide the drone to the next stop in the trajectory. At each stop, the drone hovers for a few seconds, t M, to collect sufficient measurement data, before traversing to the next stop. Compared algorithms: We compare the proposed algorithm with two other conventional algorithms. In proposed algorithm, we choose dominating paths in divided 3D grids, select COLs from paths, and merge obtained trajectories into the trajectory of sensing scope. In contrast, the first conventional algorithm we considered will select COLs first, and then choose a path to connect the selected COLs, while the second conventional algorithm we considered will choose a dominating path in the whole sensing scope greedily first, 6

7 1 2 5 Time consumed (s) Measurement time (s) Fig. 7: Time consumed of three algorithms with different granularity of measurement time. Coverage space (m 3 ) Measurement time (s) Fig. 8: Coverage space of three algorithms with different granularity of measurement time. Time consumed (s) No Loads With Cell Phones With AQI Sensors Fig. 9: Time consumed of three algorithms with different loads. Coverage space (m 3 ) No Loads With Cell Phones With AQI Sensors Fig. 1: Coverage space of three algorithms with different loads. measurement time increases, the coverage area by any of three algorithms decreases, since the measurement time consumes part of the battery. Time consumed and the coverage space with different loads: In a real-world application, drones might be used to delivery packages or other goods. Hence, we also evaluate the time consumed and the coverage space with different loads when we only consider the flight time. We fix the unit measurement time, which is 2 seconds at each COL. Results in Figure 9 and 1 show that as the load increases, the time consumed of three algorithms to complete the sensing task of a given grid increases; meanwhile, the coverage space of three algorithms during the battery life decreases. Note that the proposed algorithm performs the best in all scenarios under different loads on the drone. Results in different-size spaces: In above experiments, we only test the performance of compared algorithm in a given sensing scope with the fixed size. Next, we evaluate the algorithms performance in different-size spaces. Specifically, we fix the height of sensing scope, and test the performance in different-size 3D grids. Figure 11 shows the flight time in spaces with different width and length. We observe that our algorithm consumes less time to complete the sensing task in each grid. When fixing the width (or length) of the grid, the gap among algorithms increases as the length (or width) increases. VI. CONCLUSIONS In this paper, we study the optimal trajectory planning problem for mobile sensing in 3D space, where a drone can cover the sensing scope of the space by following the minimum-cost trajectory. We divide the 3D space into a grid of observation locations, and consider the sensing scope as a finite subgraph of the infinite 3D grid. We formulate the problem as finding the minimum dominating path to cover the sensing scope in 3D space with the minimum number of stops. We propose a nearly optimal algorithm that first finds the minimum dominating path and critical OLs in the dominating path in each 3D grid divided from the sensing scope embedding, and then concatenates obtained trajectories in 3D grids to the trajectory of the sensing scope. Experimental results show that the proposed algorithm can save 24% less time than existing approaches to complete sensing a given space; during the battery life, it can cover 19% more sensing scope than existing solutions. Fig. 11: Comparison of the proposed, COL prior, and path greedy algorithms on time consumed in different-size grids. REFERENCES [1] S. Alanko, S. Crevals, A. Isopoussu, P. Östergård, and V. Pettersson. Computing the domination number of grid graphs. the electronic journal of combinatorics, 18(1):P141, 211. [2] R. Brooks. A robust layered control system for a mobile robot. IEEE journal on robotics and automation, 2(1):14 23, [3] Y. Fu, M. Ding, and C. Zhou. Phase angle-encoded and quantumbehaved particle swarm optimization applied to three-dimensional route planning for uav. IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans, 42(2): , 212. [4] D. L. Newman. Drone for collecting images and system for categorizing image data, 214. [5] A. Purohit, Z. Sun, F. Mokaya, and P. Zhang. Sensorfly: Controlledmobile sensing platform for indoor emergency response applications. In International Conference on Information Processing in Sensor Networks, pages , 211. [6] K. Savla, F. Bullo, and E. Frazzoli. On traveling salesperson problems for dubins vehicle: stochastic and dynamic environments. In Decision and Control, 25 and 25 European Control Conference. Cdc-Ecc 5. IEEE Conference on, pages , 25. [7] D.-J. I. Science and L. D. Technology Co. Phantom 3 professional. Website, [8] K. P. Valavanis and G. J. Vachtsevanos. Handbook of unmanned aerial vehicles. Springer Publishing Company, Incorporated, 214. [9] Y. Yang, Z. Zheng, K. Bian, Y. Jiang, L. Song, and Z. Han. Arms: a fine-grained 3d aqi realtime monitoring system by uav. In IEEE GLOBECOM, 217, pages 1 6. IEEE, 217. [1] E. Yu, X. Xiong, and X. Zhou. Automating 3d wireless measurements with drones. In Tenth ACM International Workshop on Wireless Network Testbeds, Experimental Evaluation, and Characterization, pages 65 72, 216. [11] S. C. Yun, V. Ganapathy, and L. O. Chong. Improved genetic algorithms based optimum path planning for mobile robot. In International Conference on Control Automation Robotics & Vision, pages ,