By the advances of the Micro Electro Mechanical Systems (MEMS) and communication theory, wireless networking and embedded processing, ad-hoc networks of devices and sensors capabilities are becoming increasingly available for commercial applications such as environmental monitoring (e.g., traffic, habitat, security), industrial sensing (e.g., factory, appliances), and critical infrastructure protection applications (e.g., power grids, water distribution, waste disposal). For these purposes, each sensor node collaborates with others in sensing, monitoring, and tracking events of interests by exchanging acquired data, usually stamped with the time and position information. In many of these applications, knowledge about sensors’ geometrical positions are critical for many network protocols, e.g., topology control, geographical routing, and clustering [1]. It thus becomes one of the fundamental issues in Wireless Sensor Networks (WSNs) to acquire sensor position knowledge, called sensor localization problem.

In WSNs, positioning errors can often be masked by features such as fault tolerance, node redundancy, data aggregation and other means [2,3]. This makes coarse accuracy of sensor localization sufficient for many WSNs applications. Range-free localization is therefore pursued as a cost effective alternative for expensive range based scheme, by having no dependence on the availability or validity of hardware to provide range information. The key idea is to place a small fraction of anchors (i.e., special sensors with known positions) across the network. Positioning of sensors is obtained from the estimated distance to multiple anchors and their coordinates, according to trilateral algorithm.

Previous range-free localization work mainly focuses on regular sensor deployment areas [4], i.e., sensors are uniformly and densely distributed in a convex region. However, this assumption does not hold when a sensor network is deployed in irregular areas with obstacles, because the packet delivery path between two sensors can be detoured by obstacles and this shortest path distance is dramatically different from its geographical Euclidean distance. These detoured obstacles are inevitable in natural areas such as valleys where sensors are deployed for habitat monitoring, as well as in urban areas where sensors can be separated by buildings. Therefore, when applying range-free techniques to the concave areas, the position estimates may in fact contain large errors [5]. One response to this irregular area problem is to partially ignore the erroneous distance information by using an improved multi-hop algorithm [6]. Yet, distorted anchor information can mislead accurate position estimates. One way to improve the accuracy of localization would be to rule out distorted path information from some anchors, which however has two particular difficulties. First, because sensors do not have the global view of their network, they have no way of determining which path information is distorted and which is not. Second, anchors can rely on the information that they receive from other anchors that are in an unobstructed straight line path, because they are able to determine their mutual reliability based on the calculation of an expected hop length. However, anchors and sensors cannot rely on each other in this way because sensors do not know their own locations and so cannot make an expected hop-length calculation.

In this paper we introduce a novel method based on angular information of the detour transmission path to estimate the optimal path distance between any pair of sensor nodes in WSNs with arbitrary node density and low anchor to sensor ratio. The proposed algorithm leads to a good position determination for WSNs as compared with some existing positioning schemes such as pattern-driven [7], DV-Hop [8], Gang et al. [9], Chen et al. [10] and Hop-Count-based Neighbor Parition (HCNP) [11]. The main contributions of this paper are as follows:

The average localization error is approximately less than 0.3r (r is the radio range of sensors) and 0.35r in isotropic and anisotropic network respectively. This localization accuracy can satisfy the needs of many location-dependent protocols and applications, including geographical routing and tracking [2]. Compared with previous localization algorithms that declares to tolerate network anisotropy, our localization scheme excels in (1) higher accuracy stemming from its ability to tolerate multiple anisotropic factors, including the existence of obstacles, sparse sensor distribution, and anisotropic terrain condition; (2) localization accuracy guaranteed by theoretical analysis and simulation results; and (3) a distributed solution with comparative communication overhead but high accuracy and enhanced robustness to different network topologies and different degree of irregualrities.

The rest of the paper is organized as follows. We start by related work in Section 2. Section 3 describes the network model. Our proposed algorithm is presented in Section 4. In Section 5, simulation results are shown and localization performances are discussed. Finally, we present our conclusions in Section 6.

Many range-free approaches have been proposed to determine sensor locations in WSNs. For example, the Centroid method [12] is probably the earliest and simplest range-free approach, in which each node estimates its location by calculating the center of all the anchors it hears. APIT [2] lets each node estimate whether it resides inside or outside several triangular regions bounded by the anchors it hears, and refines the computed location by overlapping the regions a sensor could possibly reside in. In order to improve accuracy, APIT needs many anchors and assumes that the anchors have radio ranges that are 10 times larger than those of ordinary nodes. Another proposed space embedding approach [13] rely on Multidimensional Scaling (MDS) or Singular Value Decomposition (SVD) based techniques to project the node proximities into geographic distances.

DV-Hop employs a constant number of anchors and relies on the heuristic of proportionality between the distance and hop count in isotropic networks. The system estimates the average distance per hop from anchor locations and the hop count among anchors. Each node measures the hop count to at least three anchors and translates these into distances. By triangulation, the location is then calculated. However, the DV-Hop method yields high localization errors in anisotropic networks, where the existence of holes breaks the proportionality between the distance and hop count and thus leads to inaccurate location estimates.

To modify the disadvantage of existing DV-Hop localization algorithm, the relevant literature proposed many improved algorithms [9,10,14,15]. In [9], the location accuracy is improved bymodifying the network average hop distance based on minimum mean square error criteria as , where d_ij is the straight line distance between the anchor node i and j, h_j is the hop segment number between the anchor nodes i and j. Another algorithm in [10] calculates the error e^ij as , where is the estimated distance between anchor nodes i and j, is is the Euclidean distance between anchor i and j. The average hop distance is finally adjusted by where m is the closest anchor node to anchor node i and HopSize_i is calculated as where (x_i, y_i) (x_j, y_j) are the coordinates of anchor node iand j and h_ij is the number of hops between anchor iand j. The Algorithms [9,10] made improvements on distance estimation and localization of the DV-Hop algorithm. There are still some disadvantages in the improved algorithms, such as no obvious improvement on localization accuracy, especially when the transmission route is not straight but detoured.

Another pattern driven localization scheme is proposed in [7] to tolerate network anisotropy. The paper proposes three different methods of anchor-sensor distance calculation based on three patterns, namely Concentric Ring (CR): isotropic pattern, Centrifugal Gradient (CG): anisotropic but slightly detoured, and Distorted Gradient (DG): anisotropic and strongly detoured. For the CR pattern, it utilizes the last hop distance for overall distance calculation. The method is based on the neighbor node degree of a sensor node. It requires high node density to work properly. For the CG pattern, the author proposes DiffTriangle to revise the anchor-sensor distance estimates with the assistance from the nearest anchor to the sensor (namely Reference Station), which exhibits the CR pattern. This implies that these reference stations should appear in normal sensors’ CR category and thus the distance from sensors to their dominating reference stations should be no more than three or four hops. This assumption implicitly places a high demand for anchor distribution density. In DG pattern, the anchors which falls in DG category are dropped and no longer be used for location estimation. However it may be impossible in practice to accurately recognize the slightly detoured anchors from the strongly detoured anchors or even moderately detoured anchors, without the global knowledge on network topology, i.e., network boundary and obstacles shapes.

In the REndered Path (REP) algorithm [16], the authors assumed that the boundaries of holes in the network have been detected and every node knows if it is a boundary node or not. The authors proposed an approach for computing the straight line distance between two nodes by arguing corresponding shortest path with virtual holes. In HCNP [11], the author proposed source to destination distance estimation algorithm based on the observation that the neighbors of a destination can have different hop counts with respect to the same source and such information is used to improve the localization accuracy only in isotropic network.

When sensor nodes are randomly deployed in WSNs, we cannot assume any regularity in spacing or pattern of the sensors. This is due to the fact that most of the cases sensors are deployed from the low flying airplanes or unmanned ground vehicles. However anchors can be placed randomly or in the form of regular tile across the network so as to help in estimating the sensors positions [17]. In this paper we place anchor and sensor nodes randomly, which is more practical.

Consider a WSNs in a 2D plane with N sensors, denoted by a set where s_i is the i^th sensor node. All sensor nodes are uniformly and independently deployed in a square area A = L× L. Such a random deployment results in a 2D Poisson distribution of sensors with sensor density . All sensors are assumed to be homogeneous and stationary and omnidirectional. Therefore the network can be seen as static or regarded as a snapshot of mobile ad hoc sensor networks. In order to simplify the discussion, we are not concerned with the issues of energy consumption and robustness of sensor nodes. We believe that these missing issues do not invalidate the correctness of the proposed method.

Definition 1: Let a (s_i,r) define the transmission range or radio coverage area of a sensor s_i ,where the center is sensor s_i and the transmission radius is r. The radio coverage area of any sensor is assumed to be circular and symmetrical. So a (s_i,r) = πr². Any sensor nodes that are located within this area can directly communicate with each other and is defined as neighbors of each other. It is because we assume that all the sensors have the same transmission capability.

Definition 2: Let N_E be the average number of sensors located in the radio coverage area. The average connectivity denoted by C_E is defined as the average number of neighbor sensors located in the sensors transmission range. Following definition 1, the transmission coverage area is a (s_i,r) = πr². The sensor density of the network is . Therefore N_E = λπr² and C_E = N_E − 1.

In real life scenarios, the sensor nodes are deployed randomly from the airplanes in areas where obstacles or holes may exist. Because of obstacles or holes, the packet transmission paths among anchor and sensor nodes are not always straight but detoured. In range-free localization, the exact hop distance calculation is the key problem for estimating location. The average hop distance of the network heavily depends on the data transmission path, i.e., if the transmission path is almost straight, then the average hop distance is almost accurate, otherwise deviated from its actual value if there is an obstacle in between. We observe that, if we can associate the angle of the detoured transmission path with the calculation of the average hop distance, then we can accurately calculate the average hop distance and hence the localization. Utilizing this concept of angular information of the detoured transmission path, we propose the DPAI algorithm for precise localization.

A series of simulations are conducted to evaluate the performance of our proposed scheme in isotropic and anisotropic WSNs, where anchor nodes are deployed randomly. For anisotropic network, we consider O-shape and C-shape network topology. For each of the different types of network, we run the simulation for 100 rounds and take the average over 100 runs, during which the quantity of the deployed sensor nodes is kept unchanged. However, the topology of the network varies because we establish connectivity between pair of sensor nodes randomly. To measure the accuracy of localization, the average localization error is used.

The average localization error is defined as the ratio of the differences between the estimated location and the real location (x_i, y_i) to the communication range of sensor nodes. In this paper, the average localization error is expressed relative to the radio range r. The average localization error △ of the sensor network, which is composed of |V_S| sensor nodes, is expressed as follows: (8) The initial network parameters for simulations are shown in Table 2. We vary the number of anchors and sensor density (the average number of sensors per sensor radio area) when necessary to observe their impact on localization errors.

For simulation, we compared our proposed algorithm with DV-Hop [8], Gang et al. [9], Chen et al. [10], HCNP [11] and pattern-driven [7]. The results are shown in the Figure 4 and Figure 5 for isotropic network, Figure 6 and Figure 7 for O-shape, and Figure 8 and Figure 9 for C-Shape network. Also we investigate the impact of radio irregularity on the localization performance. The result is shown in Figure 10. From the results, we can say that, our scheme has higher localization accuracy than HCNP [11] (which cannot tolerate anisotropic factor) and pattern-driven [7] (which can tolerate multiple anisotropic factor), DV-Hop [8], Gang et al. [9], and Chen et al. [10]. Our scheme is robust in sparse isotropic network (Figure 11), anisotropic (O-Shape (Figure 12) and C-Shape (Figure 13)) network, and under realistic system configurations where the radio transmission pattern of a sensor node varies per unit degree change in the direction of radio propagation as shown in Figure 14 and Figure 15. The other three methods degrades severely in anisotropic network and under different degree of radio irregularity. HCNP [11], which is only designed for isotropic network, degrades severely in different anisotropic network conditions.

Locating sensors is necessary for many location dependent applications in WSNs. The existing range-free schemes suffer from poor localization accuracy when sensor density and the numbers of anchor nodes are low. Without the help of a large number of uniformly deployed anchor nodes and higher sensor density, those schemes fail in isotropic as well as in anisotropic WSNs. We have proposed a novel localization algorithm based on angular information of the detour path that improves the basic DV-Hop algorithm as well as three other existing algorithms, particularly when the network is anisotropic as well as isotropic. In particular, a DPAI scheme, proposed in this paper, performs best when irregular radio patterns, low sensor density and random node placement are considered. The localization error of our proposed algorithm is decreased significantly by using the angular information of the curved transmission path between two anchor nodes in such a way that we can calculate the average hop distance more accurately. Thus DPAI achieves better performance even with few number of anchor nodes, which implies the cost effectiveness of our algorithm in both isotropic and anisotropic network condition. The extensive simulation results in isotropic, O-shape and C-shape network prove the effectiveness of our proposed algorithm. We believe that our design of DPAI algorithm for both anchors and sensors will make the range-free localization schemes more practical for ad-hoc networks and WSNs.