Optimization for the sensor placement problem in 3D environments

Proceedings of 2015 IEEE 12th International Conference on Networking, Sensing and Control Howard Civil Service International House, Taipei, Taiwan, April 9-11, 2015 Optimization for the sensor placement problem in 3D environments Nguyen Thi Tam Le Hoang Son Hai Dang Thanh Vinh Trong Le VNU University of Science, VNU University of Science, University of Dalat, VNU University of Science, Vietnam National University, Vietnam National University, 01 Phu Dong Thien Vuong Vietnam National University, 334 Nguyen Trai, Thanh Xuan, 334 Nguyen Trai, Thanh Xuan, , Da Lat, Viet Nam 334 Nguyen Trai, Thanh Xuan, Ha Noi, Viet Nam Ha Noi, Viet Nam haidt@dlu.edu.vn Ha Noi, Viet Nam vinhlt@vnu.edu.vn sonlh@vnu.edu.vn tamnt@vnu.edu.vn Abstract—In this paper, we propose a novel 3D sensing model for the sensor placement optimization problem given a three dimensional environment The model take into account the angles of a sensor, the distance between the sensor and a given point in the terrain, the Line-of-Sight (visibility) capability, the constraints of the terrain and the number of sensors needed to maximize the coverage over the terrain In order to generate optimal solutions to the model, we firstly present a novel Line-of-Sight (LoS) method aiming to determine the number of obstacles between a given sensor and a point in the region of interest using the ideas of adaptive lengths and linear regression Secondly, we propose a modification of PSO algorithm, where particles (sensors) update their velocity by using only local information coming from their neighbors The comparison and analyses of experimental results reveal that optimal solutions achieved from the 3D sensing model are better than those of the related work I I NTRODUCTION Wireless Sensor Network (WSN) consists of a number of sensor nodes, which can sense, measure and gather information from the environment and they can transmit the sensed data to the user [16] Each sensor node can be attached to some special types of sensors such as thermal, biological, chemical, optical, and magnetic sensors for the measurement of properties of the environment These measured data are transferred to a base station by a mean of radio wave in wireless communication WSN could be applied to various applications involving environmental based public safety hazards such as brush fires, biochemical accidents or attacks to obtain real time and accurate information about the hazards for immediate prevention A Previous works The Sensor Placement Optimization (SPO) problem contributes a great impact to WSN involving a large number of researches in recent years It can be represented as the maximization of the global coverage of WSN to a region of interest The basic principle is that each location in the region of interest should be within the sensing range of at least one of the sensor nodes (or in short sensors) Thus the interaction between points in the region of interest is always covered by WSN The maximization of such the interaction can be regarded as the assurance of high speed connection between those points A 978-1-4799-8069-7/15/$31.00 ©2015 IEEE 327 number of methods were proposed in the literature to handle the SPO problem Deterministic methods [5], [8], [10], [13], [15], [17] aimed to select a minimum size connected K-cover, which is defined as a set of sensors such that each point in the sensor network is covered by at least K different sensors, and their locations to guarantee that an area is K-covered and the network is connected Approximation algorithms were used to deliver sub-optimation solutions for this task Nonetheless, deterministic methods often rely on oversimplified sensing models and environmental factors Therefore the theoretical coverage shown in the deterministic methods may not hold true in practice Extensions of deterministic methods in 3D environments were presented to overcome the limitations above Dhillon et al [3] combined terrain modeling and a probabilistic sensing model for SPO Huang et al [6] formulated the SPO problem in 3D space and proposed a polynomial-time method for this problem Ma et al [9] utilized a virtual force mechanism and simulated annealing Topcuoglu et al [11] proposed a new formulation for the deployment of sensors in 3D environments Unaldi et al [12] proposed an algorithm based on a probabilistic sensing model, the Bresenham’s line of sight (LoS) algorithm and a guided wavelet transform (WT) in which the sensor movements are carried out within the mutation phase of the genetic algorithms Zhao et al [18] proposed a new coverage model called surface coverage in which the targeted Field of Interest is a complex surface in 3D space and sensors can be deployed only on the surface Akbarzadeh et al [2] developed a probabilistic sensing model for sensors with LoS-based coverage consisting of membership functions for sensing range and sensing angle, which takes into consideration sensing capacity probability as well as critical environmental factors such as terrain topography to tackle the SPO problem Other solutions could be referenced in the overviews [1], [4], [6] B Limitations of the previous works The existing 3D models designed for the SPO problem have some limitations such as, • They not take obstacles into consideration; • The binary sensing coverage was used; • Constraints related to the 3D terrain were not taken into account ◦ nrows and ncols: the number of rows and columns of DEM respectively C Contributions of the article Firstly, a novel 3D sensing model for the SPO problem that remedies the disadvantages, as pointed out above, are proposed in Section Those models are the generalization of that of Akbarzadeh et al [2] where constraints are provided in the problem, and the formulas to calculate membership values are adjusted to express more accurately the placement problem in a terrain and to be of parameter-free; Secondly, a novel Line-of-Sight (LoS) method aiming to determine the number of obstacles between a given sensor and a point in the region of interest using the ideas of adaptive lengths and linear regression is presented in section III-B; Thirdly, a modification of PSO algorithm using the ideas of Particle Swarm Optimization (PSO) [7] and local information coming from their neighbors is introduced in section III-C to generate optimized solutions for the proposed models PSO works in the same way as genetic algorithms and other evolutionary algorithms Similar to evolution algorithm, PSO algorithm adopts a strategy based on particle swarm and parallel global random search PSO differs from these algorithms by simulating the social behavior and moment dynamics of a swarm Each swarm always moves to the own local optimum solution and the global optimum solution Finally, swarm finds the good optimum solution However, it has better performance than early intelligent algorithms on calculation speed and memory occupation, and has less parameter Fig Represent a point in terrain • W SN = {s1 , s2 , , sN } is a sensor network where: sj = {(xsj , yjs ), hsj (xsj , yjs ), αj , θj , ξj , βj } ∀j ∈ [1, 2, , N ] (1) ◦ (xsj , yjs ) is the coordinate of sj in Oxy; ◦ hj (xsj , yjs ) is the heigh of sj in position (xsj , yjs ) ◦ rjs is the sensing radius of sj ; ◦ θj is the pan angle of sj around the vertical axis (X direction); ◦ αj is the angle to define the orientation of the directional sensor sj around X direction, ≤ αj ≤ 2π; ◦ ξj is the tilt angle sj around the horizonal axis (Z direction); ◦ βj is the angle to define the orientation of the directional sensor sj around Z direction, ≤ βj ≤ 2π The angles of a sensor is represented as shown in Fig D Organization of the article The rest of the paper is organized as follows In section II & III, we present the main contributions Section V presents the experimental results and Section VI makes conclusions and future works II A NOVEL 3D SENSING MODEL Giving a target area and sensor network include N sensors, the problem need be solved is that how to cover this area If all the points in the target area are covered by sensor network, the target area is covered On the other hand, there are infinite number of points to deal with within the single target area To overcome this, sampling method was used, where only a fix number of points are used to evaluate the coverage One the commonly used sampling methods is grid In grid method, the target area is divided into uniform size grid If all grid points are covered then the entire target area is covered The proposed 3D sensing model is stated as follows Suppose that we have: • T is a Digital Elevation Model (DEM) terrain, which is a matrix whose values representing for the elevations of grid points as shown in Fig Some parameters are: ◦ cellsize: the size of grid cell; 328 Fig Represent angles of the sensor • R = {r1 , r2 , , rH } is a set of physical holes where can not put sensors where, ri = {(xsi , yis ), (xfi , yif )} ∀i ∈ [1, 2, , H] (2) ◦ Each hole is represented by a rectangle as shown in Fig 3; ◦ (xsi , yis ) is coordinate of top vertex of rectangle at left; ◦ (xfi , yif ) is coordinate of bottom vertex of rectangle at right • E = {e1 , e2 , , eM } is the sampling set, ei = {(xei , yie ), hei (xei , yie ), wi }, ∀i ∈ [1, 2, , M ] (3) ◦ M is the number of sampling points which is not in physical holes; ◦ (xsi , yis ) is coordinate of point ei in Oxy; ◦ wi is the weight of ei Fig Represent holes in terrain • A point ei is said to be convered by sensor si if and only if the following conditions are satisfied: ◦ The Euclidean distance between the location of sensor sj and point ei less than or equal sensing radius of sj ; ◦ The angle between the sensor sj and point ei along the X direction less than or equal the pan angle of sj ; ◦ The angle between the sensor sj and point ei along the Z direction less than or equal the tilt angle of sj ; ◦ Visibility from the sensor sj to point ei Therefore, the sensing model mainly depends on distance, orientation, and visibility ◦ µd is the binary function to measure the distance between sj and ei : 1, d(sj , ei ) ≤ rsj (4) µd = 0, otherwise d(sj , ei ) = (xei , yie , hei ) − (xsj , yjs , hsj ) ◦ µp is the binary function to measure the coverage capabilities of sensor sj to the point ei by angle of the sensors along vertical axis; µp = y e −y s arctan( xie −xjs ) i j 1, 0, otherwise ∈ [αj , αj + θj ] (5) y e −y s where arctan( xie −xjs ) is the angle between the sensor sj i j and the point ei along the X direction ◦ µt is the binary function to measure the coverage capabilities of sensor sj to the point ei by angle of the sensors along horizontal axis; he −hs µt = 1, arctan( d(si j ,eij) ) ∈ [ξj , βj + ξj ] 0, otherwise (6) he −hs where arctan( d(si j ,eij) ) is the angle between the sensor sj and the point ei along the Z direction ◦ vij represent visibility between sj and ei ; vij = 0, µd = or µt = 1+num Obstacles(sj ,ei ) , or µd = otherwise (7) where num Obstacles(sj , ei ) is the number of obstacles between sensor sj and point ei , it is determined by LoS method, • The coverage C(sj , ei ) of sj at point ei can be defined as functions of distance µd , pan angle µp , tilt angle µt and visibility vij from sensor; C(sj , ei ) = àd ì àt ì àp × vij (8) 329 Given a set of sensors and a point, the probability that the sensor will detect an event at a given point can be calculated A point can be covered by one or more sensors C(sj , ei ) represents the probability of coverage, and hence, − C(sj , ei ) gives the probability of non-coverage When two or more sensors will happen at the same time and sensors are independent then the special rule of multiplication law is used to find the joint probability To calculate the probability that sensors would not cover at target point, the multiplication law of probability is used to define the miss probability (1 − C(si , q)) In case, we are looking i=1,N for probability of coverage Then, the probability of the environment that covers point ei is Cei (W SN, ei ) = − (1 − C(si , q)) (9) i=1,N • The global coverage wi Cei (W SN, ei ) Cg (W SN, E) = ei ∈E M (10) • Global coverage is maximum Cg (W SN, E) → max (11) (xsj , yjs ) ∈ / R, ∀j ∈ [1, 2, , N ] (12) • Constraints: III PROPOSED METHODS TO FIND OPTIMIZED SOLUTIONS A Interpolation high of a point We will use bilinear interpolation to determine height of point p having coordinate (x, y) To determine height of any point in DEM terrain, we will perform following steps: Step 1: We find coordinate of four grid points which are in grid cell contain p To determine coordinate of four grid points, we will perform following: x1 = f loor(x/cellsize) × cellsize y1 = f loor(y/cellsize) × cellsize x2 = x1 + cellsize y2 = y1 x3 = x1 y3 = y1 + cellsize x4 = x1 + cellsize y4 = y1 + cellsize where, f loor(a) returns the largest integer value less than or equal to a Step 2: Assume h1 , h2 , h3 , h4 are height of four grid points which is given by DEM matrix For example, to determine the height hi at (x, y), the elevations at y on the vertical boundaries of the grid cell can be linearly interpolated between h1 and h3 at , and h2 and h4 at hb : ≈ y × h1 + (1 − y) × h3 hb ≈ y × h2 + (1 − y) × h4 Step 3: The required elevation at (x, y) can be linearly interpolated between and hb h = × x + hb × (1 − x) The bilinear function is akin to fitting a hyperbolic paraboloid to the four vertices of the grid cell It is usually written as: h = a00 + a10 x + a01 y + a11 xy, where, a00 = h1 a10 = h2 − h1 a01 = h3 − h1 a11 = h1 − h2 − h3 + h4 distance between sensor sj and point ei , k1 , k2 are the userdefined B The LoS method The aim of this section is to answer the question “how many obstacles are there between a sensor and a point in the region of interest?” This requires the understanding of the characteristics or more specifically the morphology of the DEM terrain If an answer is determined, it can be used to calculate vij in equation (7) In fact, the main factor that affects the visibility between a point and a sensor is the elevations of all points in the straight line connecting them This information is provided by DEM, which is basically a two dimensional matrix, where each cell stores the elevation of the corresponding location in the real environment 1) The old LoS method: In order to calculate the visibility between a point and a sensor, a list of cells in the line-ofsight matrix between them should be specified Some main steps are drawn as follows Firstly, we divide the line segment connecting the point and the sensor into several split points according to the size of grid cell Secondly, each point in the list is checked versus the cell containing it If its elevation is smaller than the elevation of the cell then the number of obstacles is increased In Fig 4, the line of sight is shown as a line which connect between sj and ei 2) The new LoS method: However, the drawback of this method is its computational complexity In order to achieve high accuracy, a large number of split points must be checked; thus increasing the computational time of the algorithm If less number of split points is used then some obstacles could be omitted and the calculation of vij is inaccurate For this reason, our idea is to use the adaptive lengths between split points to reduce the number of those points but still keeping the accuracy In the old approach, two consecutive split points are equally apart by the size of grid cell Our observation is that if there is no obstacle measured in a split point then it is likely no obstacle in the next split point taking into account the differences between the altitudes of the split points and the cell in two consecutive split points Thus, a linear regression could be used to predict when the line connecting the point and the sensor could intersect the obstacle If so, record the distance at that point and the beginning point as the new length, set the new point as the beginning one and continue to find another adaptive lengths until the new point is over the sensor point The pseudo-code in Table I describes the idea in details In this algorithm, h(cellP oint) denotesthe height of grid cell contain the split point, h(splitP oint) denotes the height of split point, numberOf P oints denotes the number of split points, dij is 330 Fig (a) LoS query returning visible, (b) LoS query returning not visible TABLE I P ROCEDURE OF NEW L O S METHOD d ij numberOf P oints ← cellsize while numberOf P oints > X1 ← h(splitP oint) − h(cellP oint) numberOf P oints ← numberOf P oints − X2 ← h(splitP oint) − h(cellP oint) if X1 < AND X2 < X3 ← k1 × X1 + k2 × X2 While X3 > AND numberOf P oints > X1 ← X2 10 X2 ← X3 11 X3 ← k1 × X1 + k2 × X2 12 numberOf P oints ← numberOf P oints − 13 Else if X1 > 14 count ← count + 15 Else if X2 > 16 count ← count + C Finding optimized solutions by PSO This section presents the optimization method to determine the optimized solutions of the 3D sensing model (1-12) by Particle Swarm Optimization (PSO) [7] PSO is a population based stochastic optimization technique developed by Dr Eberhart and Dr Kennedy in 1995, inspired by social behavior of bird flocking or fish schooling Generally, it is based on the principle: “The best strategy to find the food is to follow the bird which is nearest to it” Indeed, in PSO, each single solution is a “bird” or “particle” in the search space All particles have fitness values which are evaluated by the fitness function to be optimized, and have velocities which direct the flying of the particles The particles fly through the problem space by following the current optimum particles Motivated by the virtual forces algorithm (VFA) in [14], in this article, we present the VFA based PSO algorithm to find the optimized solutions Some notations is used in PSO algorithm: • Xi = (X1i , X2i , , XN i ) and Vi = (V1i , V2i , , VN i ) represent for the position and the velocity pi where, Xji = (xji , yji ) and Vji = (vji , vji ) represent for the position and the velocity of sensor sj in particle pi ∀j = 1, N , • pbesti = {pbest1i , pbest2i , , pbestN i } denotes the best particle of particle ith , where pbestji the best position of sensor sj in particle pi , • gbest = {gbest1 , gbest2 , , , gbestN } the best particle in the swarm, where gbestj the best position of sensors sj in history of the swarm, • f (gbest) is the best fitness value of the swarm as in equation (13) pushes them apart  dave −dij  if dave > dij  2×dij (xi − xj , yi − yj ) d −d ave ij Fij = − 2×dij (xi − xj , yi − yj ) if dave < dij   if dave = dij Details of this algorithm are shown below Step 1: Initialization The beginning population is initiated with Npop particles, where Npop is a designated parameter Each particle is randomly initiated their position and velocity Step 2: Calculate fitness values of all particles The procedure is shown as in Table II TABLE II C ALCULATE FITNESS TABLE IV Cg (W SN, E) ← Cg (W SN, E) + Cei (W SN, ei ) Cg (W SN,E) M Fij (14) U PDATE THE VELOCITIES AND POSITIONS OF PARTICLES IV Step 3: Update pbest and gbest The procedure is shown as in Table III TABLE III Σ si ∈adj(sj ) Step 5: Repeat the whole process from Step to Step until the maximal interation step (PSO MaxIter) is reached j=1,N 10 Fj = For i = to Npop For j = to N Vij = w × Vij + r1 × c1 × (pbestij − Xij) + r2 × c2 × (gbestj − Xij ) r3 × c3 × Fj Xij + Vij if Xij + Vij ∈ /R Xij = Xij otherwise For i = to M For j = to N Calculate µd according to (4) Calculate µp according to (5) Calculate µt according to (6) Calculate vij according to (7) C(sj , ei ) àd ì àt ì àt ì vij Cei (W SN, ei ) ← − (1 − C(sj , ei )) 11 return (13) Finally, the total virtual force action on a sensor is, U PDATE pbest AND gbest PROCESS A NALYSIS OF COMPLEXITY 1) The LoS method The largest number of split points num is calculated as following: (cellsize × nrows)2 + (cellsize × ncols)2 cellsize (15) where nrows and ncols are the number of rows and columns of DEM matrix, respectively It is clear that in this case two points are located on main diagonal of the DEM matrix The computational time complexity of the LoS algorithm in the worst case is O(num) because each node is checked only once 2) The PSO algorithm • Step 1: Generation Npop particle, each particle includes N sensors, each sensor is not in holes Then, complexity of this step is T1 = O(Npop × N × L), where L is the number of holes • Step 2: The computational time complexity of this step is T2 = O(M × N × num), where M is the number of sampling points • Step 3: The computational time complexity of this step is T3 = Npop × N × M × num • Step 4: The computational time complexity is T4 = O(Npop × N × L) num = For i = to Npop If f (pi ) > f (pbesti ) f (pbesti ) ← f (pi ) For j = to N pbestji ← Xji For i = to Npop If f (gbest) < f (pbesti ) f (gbest) ← f (pbesti ) For j = to N gbestj ← pbestji Step 4: Update the velocities and positions of particles by virtual forces The procedure is shown as in Table IV Some terms is used • d(si , sj ) is the Euclidean distance between sensors, • adj(si ) is the adjacency set of sensor si , sensor sj is called adjacency of si sensor if and only if d(si , sj ) ≤ rc , where rc is communication radius, rc = × rs , • Fij is the virtual force exterted by the neighborhood sj on si , • Fi is the total virtual force action on sensor si , • dave is the average distance between two sensors when they are evenly distributed in the area, In modification of PSO algorithm, sensors update their velocity and position by using information coming from their neighbors Virtual Force approach has ability to ”position” sensors with no overlap, by using attractive and repulsive forces based on the distance between sensors The basic idea of the virtual force based PSO algorithm is based on attributes of sensors which are electromagnetic particles: when two electromagnetic particles are too close to each other, a repulsive force calculated 331 The worst-case computational time complexity of the PSO algorithm in each iteration is: T = T1 +T2 +T3 +T4 = O(Npop ×N ×num×(M +L)) (16) V EXPERIMENT A LoS method In the following section, we implemented the proposed LOS method using the C programming language We performed with 100, 200, 500, 1000, 5000, 10000, 20000 couple of points These methods were run 30 times and the giving average results in Table V TABLE V R ESULT OF NEW The number of couple of points 100 200 500 1000 5000 10000 20000 METHOD LOS Accuracy 87.5240 % 86.6202 % 86.5912 % 85.0920 % 86.1447 % 84.4549 % 84.7576 % • The proposed models use 71 and 232 sensors, each sensor have sensing radius of 10m achieving the coverage percentages of 93.339% and 92.194% respectively The optimized solutions achieved from 3D sensing models are better than those of the relevant works • Sensing radius of sensors is used 10m in in our model while one model [2] is 30m When we increase sensing radius of sensors to 30m, we only use 105 sensors and achieving the coverage percentange of 97.845% with DEM size of 225m× 225m It is obvious that our proposal uses a smaller number of sensors than the probabilistic model [2] • The proposed models are more generalized than that of the probabilistic model [2] They could achieve a higher coverage percentage if a suitable number of sensors and the sensing range are provided VI Fig Comparison time of new method LoS with old method LoS Fig compares time of new method LoS with old method LoS It is observed that as the number of couple of points increases, time of old method LoS tend to climb very fast B PSO Algorithm We have implemented the proposed algorithms using the C programming language and executed them on a Linux Cluster 1350 with eight computing nodes of 51.2GFlops Each node contains two Intel Xeon dual core 3.2GHz, 2GB Ram These algorithms were run against the DEM terrains of BolzanoBolzen province, Italy in 2005 Parameters of the algorithm are set as: • rjs = 10m, αjs = 0, βjs = 0, θjs = 180o , ξjs = 90o , ∀j = 1, , N , • Npop = 100, P SO M AXIT ER = 100 We made several tests with different numbers of sensors (N ) and numbers of points in the restricted region (L) as in Table VI In each test, each model is run 10 times and the coverage percentages in Table VI are the average result Some remarks extracted from the experiments are shown as follows • The probabilistic model given in [2] ◦ Size of target area is 200m × 200m; ◦ Using 131 senros, each sensor have sensing radius of 30m; ◦ Coverage percentages is up to 95.98% 332 CONCLUSION This paper has made several contributions to the sensor placement optimization problem, such as i) designing some novel 3D sensing models that are the generalization of the existing models; ii) proposing a novel Line-of-Sight (LoS) method aiming to determine the number of obstacles between a given sensor and a point in the region of interest; iii) designing a modification PSO algorithm using the ideas of Particle Swarm Optimization and local information coming from neighbors of sensors The experimental results show that the optimized solutions achieved from the 3D sensing models are better than those of the relevant works For future works, we may consider the following approaches to further extend our proposed models • Considering the region of interest and the restricted regions in the model as different polygon shapes; • Examining various types of DEM terrains and morphologies; • Taking the hybrid sensors/BS into the wireless networks; • Designing variants of the LoS/ PSO methods R EFERENCES [1] [2] [3] [4] [5] [6] [7] Amac Guvensan, M Gokhan Yavuz, On coverage issues in directional sensor networks: A survey, Ad Hoc Networks, 9(7), 1238-1255 V Akbarzadeh, C Gagne, M Parizeau, M Argany, M.A Mostafavi, (2013), Probabilistic sensing model for sensor placement optimization based on line-of-sight coverage, IEEE Transactions on Instrumentation and Measurement, 62(2), 293-303 S.S Dhillon, K Chakrabarty, 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Computer Communications and Networks (ICCCN 2004) (pp 373-378) M.C Zhao, J Lei, M.Y Wu, Y Liu, M Shu, W (2009), Surface coverage in wireless sensor networks, INFOCOM 2009 (pp 109-117) 333 c2 = 1, c1 = c3 = 89.0625 % 90.6205 % 92.1875 % 100.00 % 92.968 % c2 = 1, c1 = c3 = 85.5556 % 94.2222 % 96.7467 % 95.5556 % 93.828 % ... that point and the beginning point as the new length, set the new point as the beginning one and continue to find another adaptive lengths until the new point is over the sensor point The pseudo-code... in equation (7) In fact, the main factor that affects the visibility between a point and a sensor is the elevations of all points in the straight line connecting them This information is provided... area If all the points in the target area are covered by sensor network, the target area is covered On the other hand, there are infinite number of points to deal with within the single target