In the literature, there exist three main kinds of centralized localization algorithms [5]: multidimensional scaling (MDS), convex programming and stochastic optimization approaches. Shang et al. [7] proposed a centralized algorithm called MDS-MAP. By applying MDS technology to the matrix comprised by the distances or hop counts between all pairs of nodes, the relative positions of all nodes can be obtained. MDS-MAP is robust to measurement errors and only needs a small number of anchor nodes (three or more for 2D, four or more for 3D) to estimate the absolute coordinates of normal nodes. For a network that has n nodes, MDS-MAP needs to take O(n³) operations to compute all nodes’ coordinates. With the increase of network size, the operations of MDS-MAP increase dramatically. To make MDS-MAP more applicable to WSNs and have a better performance in irregularly-shaped networks, Shang et al. [11] improved MDS-MAP to a distributed fashion by using patches of relative maps, namely, MDS-MAP(P). The main idea of MDS-MAP(P) is to build a local map at each node of the immediate vicinity and then merge these maps together to form a global map. Since a large number of iterations are required for MDS-MAP(P) to converge, the communication and computation cost in map-merging process is high.

Doherty et al. [12] modeled the peer-to-peer communication of all nodes as a set of geometric constraints and yielded the global solutions of all unknown coordinates using convex optimization technology. They also gave a method for placing rectangular bounds around the possible positions for all normal nodes. Providing that the constraints are tight enough, the estimated values of this scheme are close to the actual positions of normal nodes. However, when the network density is small or the anchor nodes are not placed around the network boundary, the performance of this scheme would decrease significantly. Biswas et al. [8] formulated WSNs localization as a semi-definite programming (SDP) problem through relaxation. The optimization problem is set up so as to minimize the errors in sensor positions for fitting the distance measurements. Similar to MDS-MAP, SDP requires only a few anchor nodes to estimate the positions of all normal nodes in WSNs, but it still demands extensive storage and computation when the network size is large.

To solve the problem of flip ambiguity in WSNs localization, Kannan et al. [9] proposed a two-phase localization (SAL) algorithm based on simulated annealing, and it is still a centralized algorithm. Simulated annealing is a stochastic optimization technique that is robust against being trapped in local minima. In the first phase of SAL, simulated annealing is used to obtain the initial location estimation. Then, a second phase of optimization is performed only on those nodes that are likely to have flip ambiguity problems based on the neighborhood information of nodes. SAL gives better accuracy than SDP and does not propagate localization errors, but SAL may fail to identify the flipped node when the network density is low, and the computation and communication cost of SAL is higher.

Iterative localization schemes, such as the ad hoc localization system (AHLoS) [13], usually have a three-phase process. In the first phase, the normal nodes estimate the distances to their neighboring anchor nodes. In the second phase, the normal nodes compute their coordinates using the ranging information and the positions of their neighboring anchor nodes. In the third phase, any normal node that has estimated its position becomes an anchor node and assists other nodes in calculating their coordinates. This process iterates to estimate the positions of as many nodes as possible. Although iterative algorithms only need a small number of clustered anchors to localize the majority of normal nodes, they suffer from the propagation and accumulation of localization errors, especially in large-scale WSNs.

Most recent research works on iterative localization are focus on how to minimize the jeopardy of accumulated errors. Liu et al. [14] studied some questions such as where localization error comes from and how it propagates from a node to another one, and then developed an error control mechanism based on the characterization of node uncertainty and the active selection strategy of anchor nodes. The error control mechanism uses only local knowledge and can mitigate the effect of error propagation for both range and directional sensors to a certain extent. Yu et al. [15] proposed a two-stage localization scheme. First, localization starts from the nodes with the largest numbers of neighboring anchors and priority is always given to nodes with more neighboring anchors or localized nodes. Then, the locations of all neighboring nodes are exploited to improve localization accuracy. During the localization process, a number of measures are also taken to ensure the reliability of each location estimate to avoid abnormal errors and reduce error propagation. Vemula et al. [16] formulated the sensor localization from a probabilistic point of view and proposed four schemes that incorporate anchor position uncertainty to estimate the distribution (mean and covariance) of node coordinates, including iterative least squares (LS) and Bayesian (BS) methods, Monte Carlo importance sampling (IS) and cost-based (CS) methods. These schemes above have relatively good performance in inhibiting the accumulation of localization errors, but the high computational complexity and increased communication cost limit their application in practice.

By approximating the length of the shortest path to the Euclidean distance, multi-hop localization schemes can infer the distances between any pairs of non-neighboring nodes. Based on the idea of Distance Vector (DV) routing and GPS positioning, Niculescu et al. [17] proposed DV-distance (range-based) and DV-hop (range-free) algorithms. They are the origination of multi-hop localization schemes for WSNs. In both algorithms, each anchor node first broadcasts a message that carries its location information to its immediate neighbors. Then, the message is propagated in WSNs in a controlled flood manner that is similar with the distance vector routing. At the same time, each normal node estimates the lengths of shortest paths or minimum hop counts to anchor nodes. If a normal node obtains the estimates to at least three (for 2D) or four (for 3D) anchor nodes, its position can be calculated by using multilateration. DV-distance and DV-hop are low-cost localization solutions, but their accuracy is built on the assumption that the shortest path between a pair of nodes is close to a straight line, which may not always be achievable in anisotropic or sparse networks.

Lim et al. [18] designed a proximity-distance map (PDM) to characterize the anisotropic features of WSNs. Actually, PDM is a semi-centralized algorithm. First, the anchor nodes derive an optimal linear transformation collaboratively to map the precise Euclidean distances and the proximities between pairwise anchors. Then, the map is sent to normal nodes to assist them in modifying their multi-hop distance estimations. The intuition of PDM is that the topology character of entire WSNs can be well represented by anchor nodes, but it is not the case in anchor clustered networks. Cheng et al. [19] investigated the effect of adverse placement and density of anchors on the accuracies of different algorithms, and developed an algorithm called hybrid localization (HyBloc) to provide reliable localization service with a limited number of clustered anchors. HyBloc combines two techniques, MDS-MAP and PDM. First, MDS-MAP is used to increase the number of anchor nodes in order to extend the anchor coverage of PDM. Then, the normal nodes are localized through PDM. HyBloc could give results as accurate as those of MDS-MAP and is less susceptible to the adverse effect of anchor placement, but it requires more communication and computation cost than PDM.

Shang et al. [20] studied the effect of anchor selection on multi-hop localization of WSNs. The experimental results show that using only the four nearest anchor nodes could get better localization performance in most cases. In the rest of this paper, we denote this algorithm as 4-Multihop. Wong et al. [21] proposed a density-aware hop-count localization (DHL) algorithm. In DHL, node density is considered and an empirical range ratio (the ratio of expected hop distance to node’s transmission range for a given local density) table is constructed to reduce the overestimation of multi-hop distances. Xiao et al. [22] proposed a novel scheme called reliable anchor-based localization (RAL) to eliminate the adverse impact of detoured paths from unreliable anchor nodes. Based on the theoretical analysis of the minimum hop length for uniformly distributed networks, a reliable minimal hop-length table that can help to judge whether a multi-hop path is severely detoured is constructed offline. At runtime, each node only utilizes the distance constraints obtained from reliable anchors to determine its position. Wang et al. [23] presented an improved multi-hop algorithm called i-Multihop to minimize the effect of erroneous multi-hop estimated distances on node localization. i-Multihop has higher computational complexity. First, the upper bound constraints are used to filter out the incorrect distance estimations and the estimated position is pinpointed to the intersection constrained by the correct distances. Second, the distance fitting is used to fit correct distance measurements, which makes the final estimated position is not affected by the layout of anchor nodes. Wan et al. [24] were concerned with the optimization problem for coordinate calculation in node localization and proposed three schemes based on least squares (LS) and multilateration, namely, Taylor-LS, weighted Taylor-based least squares (WLS) and constrained total least squares (CTLS). Moreover, a generalized Cramér-Rao lower bound (CRLB) is developed to theoretically analyze the performance of multi-hop localization approaches. Although these methods above can guarantee localization performance under certain conditions, most of them may expose certain problems in practice, which include: (1) lacking a local or global geometrical view of WSNs, they are vulnerable to irregular network topologies, (2) unreasonable to treat every reference information in a same priority, especially the one-hop information and the multi-hop one, their localization accuracy needs to be improved, (3) high computational complexity and easy to get stuck at local optimum. These problems inspire the work of this paper.

Figures 1 and 2 show two different types of WSNs. The solid dots ‘ ’ and hollow dots ‘○’ represent anchor nodes and normal nodes, respectively. In Figure 1, all nodes are randomly scattered in a 200 × 200 square area to form an isotropic network, while all nodes in Figure 2 are deployed in an H-shaped area to form an anisotropic network. In practice, the anisotropic characteristic results from certain unavoidable reasons, such as non-convex deployment region, node failure or movement, different node densities, obstacle interfering, etc. A typical example of anisotropic network is that WSNs are deployed in streets of urban areas where nodes may be separated from each other by buildings, which results in H-shape topology. Without loss of generality, we consider a network consisting of m anchor nodes and n normal nodes. The identities (IDs) of anchor nodes are from 1 to m and those of normal nodes are from m + 1 to m + n. Each node’s communication and ranging radius is R. The network graph can be defined as G = (V_m ∪ V_n, E), where V_m and V_n are respectively anchor node set and normal node set. E is measurable distance set of all pairs of neighboring vertexes (i, j), i, j ∈ V_m ∪ V_n. The objective of WSNs localization is to recover the coordinates of the vertexes in normal node set V_n under the constraints of edge set E and anchor node set V_n. The coordinates of node N_p can be described by X_p = [x_p, y_p]^T. The Euclidean distance from N_p to its neighbor N_q is d_pq = ‖X_p–X_q‖₂. The corresponding measurable distance is d′_pq = d_pq + ɛ_pq, where the ranging error ɛ_pq ∈ (−αd_pq, αd_pq). The ranging error factor α reflects the ranging capability of sensor nodes.

In multi-hop scenarios, through hop by hop dissemination of the estimated distances to anchor nodes in a controlled flooding manner [17], the normal node N_a can estimate the distance d′_ai to the anchor node N_i. d′_ai includes three cases:

If N_a and N_i are neighboring nodes, N_a can measure the distance to N_i. Thus d′_ai is the measurable distance between N_a and N_i.

As shown in Figure 1, when N_a gets enough estimated distances d′_ai (i = 1, 2, ⋯, K to anchor nodes, a system of Euclidean equations can be set up: (1) { ( x a − x 1 ) 2 + ( y a − y 1 ) 2 = d a 1 ′ ( x a − x 2 ) 2 + ( y a − y 2 ) 2 = d a 2 ′ ⋮ ( x a − x K ) 2 + ( y a − y K ) 2 = d a K ′where K represents the maximum number of estimated distances N_a gets (in Figure 1, K = 6). In 2D scenarios, the node localization problem can be seen as solving an intersection point among several circles. In this case, at least three estimated distances are required to determine a normal node’s position. Similarly, in 3D scenarios, it requires at least four spheres to determine an intersection point, i.e., K should be no less than four. In our paper, we mainly consider the 2D localization problem.

If d′_ai is accurate, solving (1) can obtain the true value of N_a’s coordinates X_a = [x_a, y_a]^T. However, due to ranging errors and approximations of multi-hop distances, d′_ai suffers from certain uncertainty which directly leads to the localization errors of sensor nodes. Especially in anisotropic networks, the shortest paths between pairs of non-neighboring nodes may be distorted by concave area and deviate far away from the Euclidean distances (e.g., the shortest path P_a1 between N_a and N₁ shown in Figure 2). In this case, approximating the lengths of shortest paths to Euclidean distances would give rise to erroneous localization results. How to mitigate the influence of irregular network topology on node localization and improve the localization accuracy is one topic of our study.

Various optimization approaches have been proposed to solve the multilateration problems, among which nonlinear least squares solver (e.g., Levenberg-Marquardt method) and Taylor-series estimator are the most commonly used. However, most of these optimization methods are complex and resource-intensive and therefore usually not applicable to resource-limited sensor nodes. In addition, these methods contain an iterative operation procedure which usually converges to a local minimum close to the initial point. To get a better solution, they need an ideally initial point that is approaching to node’s actual position, but it is not an easy task to obtain such a point. Therefore, reducing the computational complexity and preventing the local optimum from emergence is another main topic of this paper.

Before describing our MLGS algorithm, we introduce some necessary definitions:

Local density (LD) [21]: the number of neighboring nodes per node’s communication area. If N_a has T_a neighboring nodes, we denote the local density of N_a as LD_a = T_a. Given a network consisting n nodes, its network connectivity is defined as the average value of n nodes’ local densities.

The details of bounding square ring and feasible region will be discussed in Section 4.3.1.

(6) Grid granularity (g): a normal node’s feasible region can be divided into some sub-grids of equal size, the size of a sub-grid is called grid granularity. It can be represented by the ratio of side length of a sub-grid to node’s communication radius. The details of grid granularity will be described in Section 4.3.2.

Similar but not identical to the DHL algorithm proposed by Wong et al. [21], in MLGS, the network is initialized in a controlled flood manner that is aware of path-length (in distance) and multi-hop density. We also set a TTL field for propagation packets to reduce the communication cost of sensor nodes. The steps of network initialization are shown as follows (see Figure 3):

Step 1. Each node first broadcasts a challenge packet ‘I’m N_p. Who is my neighbor?’. Any node that receives the challenge packet then sends a response packet ‘I’m N_q. I’m your neighbor.’ to the corresponding node. All nodes count the number of respond packets they receives to get their local densities. At the same time, all nodes measure the distances to their neighboring nodes.

Step 2. Each anchor node N_i broadcasts a location information packet F_i = {i, X_i, MHC_i, MHD_i, d′_i} that contains its ID i and coordinates X_i. Here, MHD_i is multi-hop density of the shortest path to N_i. Its initial value is N_i’s local density LD_i. MHC_i and d′_i are multi-hop count and length of the shortest path to N_i, respectively. Both initial values are set to 0.

Step 3. When node N_p receives F_i that is directly transmitted by N_i (N_p is N_i’s neighboring node), it upgrades F_i to {i, X_i, MHC_i + 1, MHD_i + LD_p, d′_i + d′_pi}, then stores and forwards the new F_i.

Step 4. When node N_p receives F_i that is forwarded by its neighboring node N_q (N_p is not N_i’s neighboring node), it first examines whether it received F_i before. If not, same to step 3, N_p updates F_i to {i, X_i, MHC_i + 1, MHD_i + LD_p, d′_i + d′_pq}, stores and forwards the new F_i. Otherwise, there are two cases:

If d′_i + d′_pq < d′_i(old), where d′_i(old) is the multi-hop estimated distance N_p has stored, N_p updates F_i and stores it. When MHC_i + 1 < TTL, N_p forwards the new F_i. Otherwise, N_p doesn’t forward it.

Step 5. Repeat steps 3 and 4 until there is no message exchange in the network. Finally, the normal nodes can get the multi-hop counts, the multi-hop densities and the lengths of shortest paths to the anchor nodes that do not exceed the range of TTL.

When normal node N_a gets enough estimated distances d′_ai(i = 1, 2, ⋯, K to anchor nodes, a multi-hop localization model can be constructed based on the principle of weighted constrained least squares estimator: (3) X ^ a ( 0 ) = arg min X a ∑ i = 1 K w ai ( ‖ X a − X i ‖ 2 − d ai ′ ) 2 subject to   X a ∈ FR awhere X ^ a ( 0 ) is the estimative value of N_a’s coordinates X_a. FR_a is the feasible region of N_a. The weight w_ai of reference information {X_i, d′_ai} is in inverse proportion to distance estimation error Δ_ai. The bigger Δ_ai is, the smaller w_ai is, i.e., w_ai∝(1/Δ_ai).

We mainly discuss the rules for setting w_ai in this sub-section. How to determine the feasible region FR_a will be discussed in Section 4.3.1. Since the multi-hop distance estimation errors mainly arise from the approximations between the lengths of the shortest paths and the Euclidean distances, they are usually larger than the direct ranging errors. Consequently, the multi-hop reference information should be assigned to a smaller weight in multi-hop localization. According to multi-hop count MHC_ai, we set w_ai as follows: (4) w ai = { 1 if     MHC ai = 1 0 < p ≤ 1 if     1 < MHC ai ≤ TTL 0 if     MHC ai > TTL

When 1 < MHC_ai ≤ TTL, the value of p is complex. Each of the shortest paths should be assigned different weights according to the bending degrees of broken lines, i.e., a winding path should have smaller weight than a straight one. In the following, we analyze it in detail.

Firstly, multi-hop density is an important parameter that affects the multi-hop estimated distance. As can be seen from Figure 4(a), in a dense network, an approximately straight multi-hop path is likely to exist between pairwise nodes.

The length of the shortest path P_ac between N_a and N_c is close to their Euclidean distance. Smaller distance estimation error makes P_ac a higher confidence level. In contrast, if nodes are sparsely deployed [see Figure 4(b)], it is difficult to find a direct multi-hop path between a pair of non-neighboring nodes. The shortest path P_ac is generally more winding than that in Figure 4(a). Figure 4(c) is a combination of the two cases above. Nodes N_a, N_b and N_c have lower local densities, the shortest path P_ac is a winding broken line. However, nodes N_d and N_e have higher local densities, the shortest path P_ce between N_c and N_e is close to a straight line. When we evaluate the bending degree of the shortest path P_ae between N_a and N_e, all nodes’ local densities in P_ae should be considered together. Generally, the larger the multi-hop density is, the more accurate the multi-hop estimated distance is. Therefore, the weight w_ai should be proportional to MHD_ai.

Secondly, with the increase of MHC_ai, the number of line segments in P_ai rises, which reduces the probability that P_ai is close to a straight line. In this case, approximating the length of P_ai to the Euclidean distance d_ai between N_a and N_i would bring larger localization errors. In order to mitigate the influence of the multi-hop distance estimation errors on node localization, we should lower the weight of multi-hop reference information. Therefore, w_ai should be in inverse proportion to MHC_ai. In addition, with the increase of ranging error factor α (i.e., nodes’ ranging capability declines), we should appropriately raise the confidence level of multi-hop reference information to weaken the impact of direct ranging results on node localization, since the weight of one-hop reference information remains a constant value of 1 in (4). Based on the analysis above and the results of numerous simulations, the weight w_ai of multi-hop reference information can be set as follows: (5) w ai = exp ( α ) * ( 1 MHC ai ) r * ( MHD ai ( MHC ai + 1 ) ∗ LD a ) t      ( r , t = 1 , 2 , 3 , ⋯ )

The base number in the third part of Equation (5) represents the ratio of all nodes’ average density to the target node’s local density. In most cases, Equation (5) could satisfy the requirement that the weight w_ai of multi-hop reference information is no more than 1. If extreme case of w_ai > 1 appears, we set w_ai = 1. To determine the optimal values of indexes r and t, we have done numerous simulations in various network environments. The results show that we could usually get more ideal localization accuracy when both r and t are set to 1. Taking the scenarios shown in Figures 1 and 2 as examples, we vary r and t from 1 to 10 and use classical weighted least squares estimator to compute the coordinates of normal nodes. The average localization errors in both scenarios are shown in Figures 5 and 6, respectively.

The change trend of average errors with varying r and t in isotropic network is nearly consistent to that in anisotropic network. And the minimum points in both figures usually appear in the lower-left corners of the curved surfaces where both r and t are equal to 1.

To simplify the computational complexity of node localization and prevent getting stuck at local optimum, we propose a novel method to estimate the coordinates of normal nodes. First, the feasible regions of normal nodes are determined by calculating the intersection of bounding square rings. Then, the coordinates of normal nodes are estimated through a lightweight grid-scanning procedure. The details of this method are shown as follows.

After node N_a gets its initial coordinates X ^ a ( 0 ), it can optionally step into the phase of localization refinement through collaborating with its neighbors. The following are the refinement procedure.

Step 1. N_a first broadcasts its estimated coordinates and the total number U of samples it get in the previous phase. Then, a weighted refinement model for N_a can be constructed based on the broadcast coordinates of N_a’s neighbors and the measurable distances between N_a and its neighbors: (13) X ^ a ( t + 1 ) = arg ⁡   min ⁡ X a ∑ b = 1 J v ab ( ‖ X a − X ^ b ( t ) ‖ 2 − d a b ′ ( t ) ) 2where X ^ b ( t ) ( b = 1 , 2 , ⋯ , J ) is the broadcast coordinates of N_a’s neighboring node N_b. J is the number of N_a’s neighbors. d ab ′ ( t ) is the measurable distance between N_a and N_b. v_ab is the weight of reference information { X ^ b ( t ) , d ab ′ ( t ) }. And t is the iteration number of refinement, with an initial value of 0.

Here, we briefly discuss how to set the value of weight v_ab. When N_b is an anchor node, we should set v_ab a bigger value. In contrast, if N_b is a normal node, v_ab should be set a smaller value according to the estimated accuracy of N_b’s coordinates. Since the localization accuracy of normal nodes can be evaluated by the size of feasible regions, v_ab should be in inverse proportion to the total number U of samples in 4.3.2. In collaborative refinement, this weighting mechanism not only contributes to the improvement of localization accuracy, but also helps to prevent the localization error from propagation.

Step 2. The grid-scanning scheme can also be used to seek the close to optimal value of model (13). As can be seen from Figure 9, a square of which the center coordinates and the size length are respectively X ^ a ( t ) = [ x ^ a ( t ) , y ^ a ( t ) ] T and L (generally no more than R) is regarded as N_a’s feasible region for localization refinement, and it is denote as: (14) FR a ( t + 1 ) = [ x ^ a ( t ) − 1 2 L , x ^ a ( t ) + 1 2 L ] × [ y ^ a ( t ) − 1 2 L , y ^ a ( t ) + 1 2 L ]

Suppose the grid granularity for localization refinement is r (r ≤ g). According to FR a ( t + 1 ), a sample set Θ_a can be obtained by dividing FR a ( t + 1 ) into a number of sub-grids: (15) Θ a = { S a ( 1 ) , S a ( 2 ) , S a ( 3 ) , ⋯ , S a ( V ) }where the number of samples is V = L²/(rR)². Through scanning Θ_a, the close to optimal value of model (13) can be obtained, denoted as X ^ a ( t + 1 ). N_a upgrades its estimated coordinates to X ^ a ( t + 1 ).

Step 3. N_a broadcasts its new coordinates. Set t = t + 1. Repeat steps 1 and 2 until the accuracy ( ‖ X ^ a ( t + 1 ) − X ^ a ( t ) ‖ 2 ≤ r) is satisfied or the maximum iteration number is reached, whichever comes earlier. The final X ^ a ( t + 1 ) is the refinement coordinates of N_a.

First, we analyze the distribution of node localization errors in the default environments. The localization errors are represented by the ratio of the Euclidean distances between estimated coordinates and actual coordinates to node’s communication radius. Figures 10 and 11 present the distribution boxplots for isotropic and anisotropic networks, respectively. The y-axis of both figures is drawn in log-scale. In isotropic network, 4-Multihop gives the worst performance. Its average and median errors are respectively 23.24% and 13.36%, and its maximum outlier is even close to 800%. The average and median errors of i-Multihop are almost the same as those of MLGS. But the errors of i-Multihop are more scattered. The maximum outlier of i-Multihop reaches 200.50% (compared with 103.29% of MLGS). MLGS(R) has an average error of 7.17% and a median error of 4.08%, which is a significant improvement in localization accuracy. In anisotropic network, 4-Multihop is less affected by irregular shape because it only uses the four nearest anchor nodes in calculating the coordinates of normal nodes. Compared with Figure 10, we can see that the accuracy of i-Multihop declines vastly. The average error of i-Multihop reaches 21.67%, but it is still lower than that of 4-Multihop (26.53%). MLGS and MLGS(R) are robust to irregular topologies. Their average errors, median errors and maximum outliers in anisotropic network are almost the same as those in isotropic network, and much smaller than those of 4-Multihop and i-Multihop.

Figures 12 and 13 show the comparison results of average localization errors and localization coverage rates with various TTL. With the increase of TTL, the normal nodes could get more and more reference information for their localization, so the localization coverage rates of four algorithms grow gradually. When TTL reaches 5, the coverage rates in both networks are approaching to 100%. Since raising TTL could increase the communication cost in localization, we try to keep TTL a smaller value in the premise of localizing most nodes. That is the reason why we set the default value of TTL to 5 in simulations.

In isotropic network, 4-Multihop performs the worst. Its average error varies significantly when TTL ≤ 5 and remains generally stable (about 23%) after TTL > 5. The localization accuracies of MLGS, MLGS(R) and i-Multihop are less affected by TTL and are always better than that of 4-Multihop. Among them, MLGS(R) gives the smallest localization error of about 10%. Through the accuracy of i-Multihop slightly exceeds that of MLGS when TTL ≥ 6, but it is always lower than that of MLGS(R). In anisotropic network, 4-Multihop still has the lowest accuracy. i-Multihop is greatly affected by irregular network, and its average error is about 5% higher than that of MLGS when TTL ≤ 5. With the increase of TTL, the number of distance constraints in i-Multihop rises, and the accuracy of i-Multihop is gradually near to that of MLGS. Through refinement, MLGS(R) can increase the localization accuracy by more than 5%. And in most cases, its average error is less than 10% of the communication radius of sensor nodes.

In this part, we vary the communication radius of sensor nodes and get the accuracy comparisons of four algorithms under different network connectivity, ranging from 6 to 15 (see Figures 14 and 15). In general, the probability that a shortest path between pairwise nodes is close to a straight line grows as network connectivity increases, which directly results in the improvement of multi-hop localization accuracy. In isotropic network, the performance of MLGS(R) is better than those of three other algorithms. It gives an average error of less than 3.5% for high network connectivity (no less than 12). When network connectivity is smaller than 9, the accuracy of i-Multihop is higher than that of MLGS. However, the average localization error of i-Multihop gradually converges to about 11% and exceeds those of MLGS and 4-Multihop. In anisotropic network, the variation trend of average localization errors of four algorithms is similar to that in isotropic network, but the gap among the four algorithms becomes more evident. The localization accuracies of MLGS and MLGS(R) are always better than those of i-Multihop and 4-Multihop. When network connectivity is 6, the average error of MLGS is less than 40%, above which the localization error will significantly affect the application performance of WSNs [25]. When network connectivity reaches 10, the average error of MLGS is below 10% and it can be further reduced to less than 5% through refinement, while the two other algorithms give larger average errors of more than 20%.

Figures 16 and 17 show the statistics for performance of four algorithms with different ranging error factors. With the increase of ranging errors, the accuracies of four multi-hop algorithms drop gradually. Among them, i-Multihop is the most sensitive to ranging errors. In isotropic network, the average error of i-Multihop is near to 10% when α < 0.1, while that of MLGS is about 12%. However, when α increases to 0.1, the two algorithms produce similar results. And after that, the average error of i-Multihop increases substantially and even exceeds that of 4-Multihop when α = 0.35.

As can be seen from Figure 16, we can infer that MLGS and MLGS(R) are robust with respect to high ranging errors. When α increases to 0.5, the average localization error of MLGS is still smaller than 40% of node’s communication radius. Through refinement, the average error can be further reduced to 30%. In anisotropic network, MLGS and MLGS(R) perform consistently better than 4-Multihop and i-Multihop under all ranging error factors considered. Compared with the latter two algorithms, MLGS can improve localization accuracy by 10%∼20%, and further increase by 5% after refinement. When α = 0.25, the average errors of 4-Multihop and i-Multihop are more than 30%, while those of MLGS and MLGS(R) are only less than 18%.

From previous investigations, we draw a conclusion that MLGS produces better results in most cases. Here, we discuss the impact of grid granularity g on localization accuracy of MLGS under various ranging error factors (see Figures 18 and 19). In both figures, the decimals in the legends represent the values of grid granularity g for coordinate estimation. Generally, the localization accuracy of MLGS improves with grid granularity g declining. However, when g reduces to a certain extent, any further decrease of g does not yield any significant improvement in accuracy. In isotropic network, the average localization error with α = 0.1 can be reduced by 4.97% as g decreases from 0.4 to 0.2, while only 3.23% as g decreases from 0.2 to 0.1. In anisotropic network, the corresponding descents of average errors with α = 0.1 are 5.64% and 2.06%, respectively. In addition, with the increase of α, the impact of grid granularity g on localization accuracy drops gradually. For example, MLGS(0.1) and MLGS(0.2) nearly have the equivalent performance when α = 0.4. In the phase of coordinate estimation, there is U ∝ (1/g²), where U represents the number of samples. Thus, reducing g would make U grow significantly, which further leads to the increase of computation cost required in localization. The quantitative analysis of computation cost will be present in Section 5.6. In practice, we should determine the optimal grid granularity g based on the trade-off of localization accuracy and computation cost.

In this part, we discuss the computation cost of 4-Multihop, i-Multihop and MLGS with the metric of total computation time for calculating the coordinates of all normal nodes under different degrees of network connectivity (see Figures 20 and 21). As i-Multihop employs the complex constrained nonlinear programming solver to estimate the coordinates of normal nodes, its computation cost is more than 30 times of those of 4-Multihop and MLGS. When network connectivity is low, the computation cost of MLGS(0.1) (no more than 0.35 s in both scenarios) is slightly smaller than that of 4-Multihop (0.4 s and 0.45 s in isotropic and anisotropic networks, respectively).

However, with the increase of network connectivity, MLGS(0.1) performs faster and faster, while 4-Multihop keeps a constant computation time. That is because higher network connectivity would enhance the constraints of sensor nodes and diminish the feasible regions of normal nodes in MLGS. For MLGS, reducing grid granularity g could lower the computation cost evidently in sparse networks. For example, in isotropic network with connectivity of 6, the computation time of MLGS(0.1) is 0.33 s, that of MLGS(0.2) is 0.14 s and that of MLGS(0.4) is only 0.06 s. But in networks with high connectivity, with the increase of g, the variety of computation cost is not so obvious. When network connectivity reaches 10, the computation cost of MLGS in different g drops to below 0.2 s. It is worth noting that MLGS(0.2) and MLGS(0.4) nearly have a constant computation time in various network connectivity.

Finally, we evaluate the performance of the MLGS by comparing it with MDS-MAP [7] and the improved iterative algorithm with error control mechanism similar to [14]. The comparison results are shown in Figures 22–24, in which the circles represent true positions of nodes (solid circles for anchor nodes and empty circles for normal nodes), the triangles represent estimation positions of normal nodes, and the lines represent localization errors. For MLGS and Iterative algorithms, if a normal node can’t get enough reference information to computing its coordinates, a square will be drawn around it. Table 2 gives the average localization errors of three algorithms in isotropic and anisotropic networks.

In isotropic network, MLGS has the best localization performance. Its average error is below 15% and the error distribution is uniform. One unlocalized node and a few normal nodes with bigger localization errors are mainly concentrated in the upper-left corner, where fewer anchor nodes exist. MDS-MAP has an average error of 23.1% and a localization coverage rate of 100%. The localization accuracies of edge nodes are worse than those of middle nodes. The iterative algorithm with average error of 25.8% and localization coverage rate of 91.7% performs the worst. The iterative process stops at the lower-right corner where sensor nodes are sparsely deployed. Furthermore, the impact of error accumulation is not totally eliminated in the improved iterative algorithm. As can be seen from the Figures 22 and 24, the localization accuracy of MLGS and iterative algorithm is not obviously affected by network topology. In anisotropic network, the average errors of both algorithms are 12.7% and 26.1%, respectively, which are close to those in isotropic network. However, the average error of MDS-MAP (62.1%) is much larger than that in isotropic network. That is because MDS-MAP needs to approximate the lengths of shortest paths to Euclidean distances between all pairs of non-neighboring nodes. Irregular network topology would make the approximation large errors, especially between pairwise nodes that are far apart.