We discuss the problem of maximizing the sensor field coverage for a specific number of sensors while minimizing the distance traveled by the sensor nodes. Thus, we define the movement task as an optimization problem that involves the adjustment of sensor node positions in a coverage optimization mission. We propose a coverage optimization algorithm based on sampling to enhance the coverage of 3D underwater sensor networks. The proposed coverage optimization algorithm is inspired by the simple random sampling in probability theory. The main objective of this study is to lessen computation complexity by dimension reduction, which is composed of two detailed steps. First, the coverage problem in 3D space is converted into a 2D plane for heterogeneous networks via sampling plane in the target 3D space. Second, the optimization in the 2D plane is converted into an optimization in a line segment by using the line sampling method in the sample plane. We establish a quadratic programming mathematical model to optimize the line segment coverage according to the intersection between sensing circles and line segments while minimizing the moving distance of the nodes. The intersection among sensors is decreased to increase the coverage rate, while the effective sensor positions are identified. Simulation results show the effectiveness of the proposed approach.
A wireless sensor network (WSN) consists of a large number of resource-limited (such as CPU, storage, battery power, and communication bandwidth) tiny devices, which are called sensors. These sensor nodes can sense task specific environmental phenomenon, perform in network processing on the on the sensing field and communicate wirelessly with other sensor nodes or with a sink (also called data gathering node), usually via multihop communications [
We analyze the coverage issues in 3D underwater sensor networks (USNs), where sensors are randomly deployed in a 3D field (Figure Sample random sampling is applied to the coverage problem in 3D space for dimension reduction. A quadratic programming mathematical model is established according to the intersection position of the sensing circles and sample line, position of the sensing circles, and radii of the sensing circles. We conclude that the sum of the node traveling distance for the unchanging node sequence is less than the sum of the changing node sequence. This conclusion is mathematically proven in this study. A nonlinear quadratic optimization problem is converted into a linear quadratic optimization problem by tightening constraints to obtain the suboptimum solution.
Underwater sensor networks.
The remainder of this paper is organized as follows: Section
The sensor coverage problem has been extensively studied in the field of multirobot systems and computational geometry the Art Gallery Problem (AGP) and the circumference coverage are related to the coverage of WSNs closely [
Coverage algorithm based on virtual force is one of research hotspots currently [
The application of linear programming in coverage problem focuses on point (or target) coverage, where the objective is to cover a set of points (targets) [
As described earlier, almost all of coverage strategies in 3D space are based on a premise that sensor nodes are deployed in a specified location and stay there so that the maximum coverage of 3D space is achieved. However, sensor nodes are usually deployed at random and are impossible suspended at specified location without any connection in water. Most of the researches on coverage and connectivity in WSNs are not suitable for underwater because of two-dimensional application only. We propose a 3D coverage optimization algorithm for underwater 3D sensor networks (USNs) in which sensor nodes are randomly deployed and are fixed by cables, in which quadratic programming is applied to coverage optimization problem in 2D plane and 3D space.
We assume that a large number of underwater sensors are distributed in an area of interest. A 3D USN model is described as follows [ A 3D USN contains two types of sensor nodes: one is used for communications and is deployed on the water surface; the other is used for sensing and is deployed underwater. All sensor nodes that are deployed underwater have homogeneous models; that is, all sensors have binary sensing coverage models. Thus, the sensing model is a sphere, and all sensors adopt the radius The underwater sensor node communicates with the surface node via a cable that connects the two sensors. Sensor node position and depth can be acquired via the sensor nodes deployed on the water surface and cables. The sensor nodes on the water surface are fixed in place with an anchor. However, the underwater nodes can move to a specified location vertically. Sensor nodes have enough residual energy to move the specified location because coverage algorithm is executed during network initialization.
We assume that sensor nodes are randomly deployed in a 3D cuboid space and denote the distance between two spots as a Euclidean distance.
Suppose that the distance between sensor node
Sensor node
The distance of underwater sensor node
An arbritrary plane in 3D space intersect the boundary of 3D monitoring space to form polygon MNOP, as shown in Figure
Coverage model in underwater 3D space.
Choose a line at random in sample plane and denote it as “
Sampling line.
Coverage rate
The proposed algorithm attempts to maximize the coverage of USNs while striving to minimize the movement distance of the nodes. This problem can be defined as follows: given that
The sample planes intersect with the sensing spheres. The distance of the center of the sensing spheres to the sample planes is less than
Sensing circle.
Sensing circles in a horizontal sample plane.
The radii of the sensing circles increase as the underwater sensor node moves closer to plane
Suppose that the center of the
For
In this section, we derive the optimization coverage in a vertical sample plane (Figures
Vertical sample plane in underwater 3D space.
Vertical sample plane.
Coverage circle
Upper and lower intersection points.
The sensing circles and line segments intersect at
The sensing line segments move with the corresponding underwater sensor nodes moving in
It is difficult to find the optimal solution because the previous problem is nonlinear. Therefore we seek the feasible solution by tightening constraints. First, we analyze the affection of the sensing circles’ position on optimization algorithm.
The sum of the traveled distance of the nodes is small when the order of the nodes remains the same compared with other nodes.
We assume that only two sensing circles
Both
Thus,
Position of two sensing circles.
On the basis of the previous analysis, we assume that the relative order of the sensing circles after optimization is the same before optimization. We propose different constraint conditions depending on the length of the sensing line segments. The length of the sensing line segment is
The new positions of the sensing circles are
Before optimization
After optimization
Before optimization
After optimization
We find a feasible solution after limiting the constraints of the optimization algorithm in vertical planes. Moreover, the 2D plane coverage optimization in the
The pseudocode of the COS algorithm and the single sample plane coverage function are presented in Algorithms (
COS algorithm %Input: %Output: Main procedure: (1) (2) While (3) For (4) If (the sensing sphere and sample plane intersect) (5) calculate the radius of the sensing circle according to ( to (6) End (7) end (8) % the single sample plane coverage optimization function and its procedure are shown in Figure (9) Update the position set (10) (11) Empty the set (12) End (13) (14) While (15) For (16) If (the sensing sphere and sample plane intersect) (17) calculate the radius of the sensing circle according to ( circle to (18) End (19) end (20) (21) Update the position set (22) (23) Empty the set (24) End
Plane coverage function
(1) (2) while (3) % the sample line coverage optimization function and its flowchart is shown in Figure (4) (5) (6) end
Straight line coverage flowchart.
The COS algorithm is performed by the sink node on land. First, a plane that is parallel to the
Given the sampling plane function, the position of the initial sampling line, boundary of the plane, step length of the sampling, and positions and radii of the sensing circles, the subprogress can be used to optimize the coverage of the sampling plane. Line 1 is the line function of the initial sampling line. Line 3 performs the line coverage optimization function and calculates the new positions of the sensing circles. Additional optimization sequences are conducted until the sampling line is beyond the boundary of the sampling plane.
We describe the single-line coverage optimization function in a flowchart (Figure
In this section, we describe the simulation setup, performance metrics, and performance results of the study. We assume that all messages can be transmitted/received without any errors and that the sensor nodes are uniformly distributed in a 100 m
Figure
Initial position of the sensing circles after random distribution.
Position of the sensing circles after execution of the COS algorithm.
Figure
Coverage rate of sample planes for different distributions.
We have conducted experiments with different node quantities. The results shown in Figure
Coverage rate for the number of nodes.
Figure
Coverage rate for iterations.
Similarly, when sensing range is increased while the number of nodes is fixed as 60 and 80, respectively, coverage increases as seen in Figures
Coverage versus the sensing radius for 60 nodes.
Coverage versus sensing radius for 80 nodes.
This study proposes a 3D COS to enhance the coverage of USNs. We convert a 3D space coverage into a 2D plane coverage for heterogeneous networks by a sampling plane in the target 3D space. The positions of the sensing circles in the sample plane are calculated by the COS algorithm to enhance coverage. Thereafter, the sensor nodes that correspond to the sensing circles are redeployed. The simulation results show that the COS algorithm can improve the coverage provided by an initial random placement. To a certain degree, the coverage in the 3D space improves when the iterations increase. However, the coverage will remain unchanged or decrease when the number of iterations surpasses a certain value for the adjacent sample planes. The performance of the COS algorithm is affected by two aspects: for the limitation of the network model, the coverage cannot be significantly improved when the wireless sensor nodes are nonuniformly distributed, the COS algorithm is performed on each sample plane alone because we do not consider the relativity between the adjacent sample planes.
In the future, we will consider the relativity of the sample planes to ensure the flexibility of the COS algorithm as well as adjust the positions of the wireless sensor nodes on the water surface to distribute uniformly the sensor nodes and enhance their coverage in 3D space.
The subject is sponsored by the National Natural Science Foundation of China under Grant nos. 61373139, 61300239 and 61171053; the Doctoral Fund of Ministry of Education of China under Grant no. 20113223110002; Science & Technology Innovation Fund for higher education institutions of Jiangsu Province (CXZZ12_0481).