^{†}

Based on “Grid-Scan-based Multi-hop Localization Algorithm for Wireless Sensor Networks”, by Xiaolei Guo, Ning Yu, Renjian Feng, Yinfeng Wu and Jiangwen Wan, which appeared in 2010 IEEE Sensors Conference Proceedings. © 2010 IEEE.

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

For large-scale wireless sensor networks (WSNs) with a minority of anchor nodes, multi-hop localization is a popular scheme for determining the geographical positions of the normal nodes. However, in practice existing multi-hop localization methods suffer from various kinds of problems, such as poor adaptability to irregular topology, high computational complexity, low positioning accuracy,

Recent advances in the fields of wireless communication, micro-electro-mechanical systems (MEMS) and embedded processing have enabled the emergence of wireless sensor networks (WSNs). WSNs consist of a large number of low-cost, low-consumption, small-size, and multi-functional sensor nodes. Usually, they are randomly deployed (e.g., nodes are scattered from the air) in complex environments to execute a wide variety of tasks, such as environmental monitoring, bush fire surveillance, wildlife behavior studies, target tracking, battlefield spying,

The task of WSN node localization is to determine the positions of sensor nodes without initial location information (normal or unknown nodes) based on the knowledge of sensor nodes with initial location information (anchor or beacon nodes) and inter-node distance or bearing measurements. Since anchor nodes usually obtain their coordinates from global positioning system (GPS) receivers or manual configuration in fixed places, raising the number of anchor nodes will significantly increase the cost of network deployment. They should therefore make up only a small proportion of nodes in large-scale WSNs. Thus, many normal nodes may fail to estimate their positions due to their short-range measurement. To solve this problem, three types of localization schemes are proposed, namely, centralized algorithms, recursive algorithms, and multi-hop algorithms.

In centralized algorithms, a powerful processing node collects all inter-node measurements to produce a global topology map of the WSN and then distributes all the nodes’ location information to the network. Typical centralized algorithms include MDS-MAP [

In this paper, we analyze the advantages and disadvantages of existing node localization schemes and propose a novel

To improve the topology adaptability and accuracy of multi-hop localization, we study the factors that influence the multi-hop distance estimation and give a quantitative rule for setting the weight of reference information, based on which a more realistic weighted constrained multi-hop localization model is constructed.

We come up with a novel approach to determine the scope of node coordinates. Due to the uncertainties in estimated distances, the normal nodes could not be localized in fixed points accurately. Usually, they could only be bounded in a certain region. In this paper, we define the feasible region as the intersection of bounding square rings. By computing the feasible region, we are able to restrict the candidates of node coordinates within a small scope.

We design a lightweight and local optimum-avoidable method for the estimation and refinement of node coordinates based on grid-scanning, which is very suitable to senor nodes of limited energy and computing power. Extensive simulations show that MLGS has higher localization accuracy and less computation cost than existing typical schemes, and can perform well, even in anisotropic networks.

The remainder of the article is organized as follows. Section 2 discusses some of the previous works on WSN node localization. Section 3 formulates the multi-hop localization problems and introduces the necessary definitions. Section 4 presents in detailed the MLGS algorithm procedure. Section 5 evaluates the performance of MLGS through experiments. Finally, Section 6 concludes this paper.

In the literature, there exist three main kinds of centralized localization algorithms [^{3}) 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

Doherty

To solve the problem of flip ambiguity in WSNs localization, Kannan

Iterative localization schemes, such as the

Most recent research works on iterative localization are focus on how to minimize the jeopardy of accumulated errors. Liu

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

Lim

Shang

_{m}_{n}_{m}_{n}_{m}_{n}_{n}_{n}_{p}_{p}_{p}_{p}^{T}_{p}_{q}_{pq}_{p}_{q}_{2}. The corresponding measurable distance is _{pq}_{pq}_{pq}_{pq}_{pq}_{pq}

In multi-hop scenarios, through hop by hop dissemination of the estimated distances to anchor nodes in a controlled flooding manner [_{a}_{ai}_{i}_{ai}

If _{a}_{i}_{a}_{i}_{ai}_{a}_{i}

If _{a}_{i}_{a}_{i}_{ai}_{a}_{i}

If _{a}_{i}_{ai}_{ai}

As shown in _{a}_{ai}_{a}

If _{ai}_{a}_{a}_{a}_{a}^{T}_{ai}_{a}_{1} between _{a}_{1} shown in

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:

_{a}_{a}_{a}_{a}_{a}

_{1}_{K}_{1} and _{K}_{1}, _{2}, ⋯, _{K}_{1}_{K}

_{1K} passes _{1K}’s multi-hop count is _{1K} =

_{ai}_{a}_{ai}

_{a}_{a}_{a}_{a}_{a}_{a}

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

(6)

In this section, we describe the proposed MLGS algorithm for WSN node localization. In general, MLGS can be divided into four phases: network initialization, construction of multi-hop localization model, estimation of node coordinates, and localization refinement (an optional phase). The details of each phase are given in the following.

Similar but not identical to the DHL algorithm proposed by Wong

_{p}_{q}

_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}

_{p}_{i}_{i}_{p}_{i}_{i}_{i}_{i}_{i}_{p}_{i}_{pi}_{i}

_{p}_{i}_{q}_{p}_{i}_{i}_{p}_{i}_{i}_{i}_{i}_{p}_{i}_{pq}_{i}

If _{i}_{pq}_{i}_{i}_{p}_{p}_{i}_{i}_{p}_{i}_{p}

If _{i}_{pq}_{i}_{p}_{i}

When normal node _{a}_{ai}_{a}_{a}_{a}_{a}_{ai}_{i}, _{ai}_{ai}_{ai}_{ai}_{ai}_{ai}

We mainly discuss the rules for setting _{ai}_{a}_{ai}_{ai}

When 1 < _{ai}

Firstly, multi-hop density is an important parameter that affects the multi-hop estimated distance. As can be seen from

The length of the shortest path _{ac}_{a}_{c}_{ac}_{ac}_{a}_{b}_{c}_{ac}_{d}_{e}_{ce}_{c}_{e}_{ae}_{a}_{e}_{ae}_{ai}_{ai}

Secondly, with the increase of _{ai}_{ai}_{ai}_{ai}_{ai}_{a}_{i}_{ai}_{ai}_{ai}

The base number in the third part of _{ai}_{ai}_{ai}

The change trend of average errors with varying

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.

Since the size of feasible regions can reflect the localization accuracy of normal nodes, it is an important task to determine the range of _{a}

_{a}

In _{a}_{1} are neighboring nodes. _{a}_{1}. The measurable distance is denoted as _{a1}. As the ranging error _{a}_{1} ∈ (−_{a1}, _{a1}), the Euclidean distance _{a}_{1} satisfies the following condition:

We can infer that _{a}_{a}_{1} of which the outer radius is _{a1} = _{a1} / (1 − _{a1} = _{a1} / (1 + _{a1} are respectively _{a1} = 2_{a}_{1} and
_{a}

In _{a}_{2} are non-neighboring nodes, but _{a}_{2}’s location information packet through multi-hop information transmission in the range of TTL. Without loss of generality, we take two hops for example. The shortest path _{a}_{2} between _{a}_{2} passes normal node _{b}_{a}_{2}, the Euclidean distance _{a}_{2} between them is bigger than _{a}_{b}_{ab}_{ab}_{b}_{2} are _{b}_{2} and _{b2}. Thus the estimated distance between _{a}_{2} is:

Based on −_{ab}_{ab}_{ab}_{b}_{2} ≤ _{b}_{2} ≤ _{b}_{2}, we can infer that:

Further:

According to _{a}_{2}>_{a}_{2}≤_{ab}_{b}_{2}, we have _{a2} ≤ _{a2} / (1 − _{a}_{a}_{2} of which the outer radius and inner radius are _{a2} = _{a2} / (1 − _{a}_{2} = _{a}_{a}_{2}.

_{a}_{ai}_{a}

Details of computing _{a}

After obtaining the feasible region _{a}

Suppose the grid granularity for coordinate estimation is _{a}_{a}_{a}_{a}_{a}_{a}^{2}.

In _{a}_{a}_{a}_{a}

After getting _{a}_{a}_{a}

This grid-scanning approach only needs simple arithmetical and comparison operations. It not only has low computational complexity, but also can prevent getting stuck at local optimum. In addition, when the number of reference information increases, it only requires a modest increase in memory consumption and arithmetical operations over those of the classical optimization methods. Therefore, it is a lightweight and efficient method. And it is very suitable to sensor nodes with limited computing and storage capability.

After node _{a}

_{a}_{a}_{a}_{a}_{a}_{b}_{a}_{a}_{b}_{ab}

Here, we briefly discuss how to set the value of weight _{ab}_{b}_{ab}_{b}_{ab}_{b}_{ab}

_{a}

Suppose the grid granularity for localization refinement is _{a}^{2}^{2}. Through scanning _{a}_{a}

_{a}_{a}

In this section, we conduct extensive simulations to study the performance of MLGS algorithm in the isotropic network shown in

The proposed algorithm without refinement phase, denoted as MLGS, in which the grid granularity

The proposed algorithm with refinement phase, denoted as MLGS(R), in which the grid granularity

The 4-Multihop algorithm proposed by Shang

The i-Multihop algorithm proposed by Wang

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.

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

As can be seen from

From previous investigations, we draw a conclusion that MLGS produces better results in most cases. Here, we discuss the impact of grid granularity ^{2}), where

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

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

Finally, we evaluate the performance of the MLGS by comparing it with MDS-MAP [

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

In this paper, we present a novel multi-hop localization algorithm called MLGS, which is shown to be able to enhance the adaptability to irregular network topology, improve the positioning accuracy, as well as reduce computational cost for multi-hop localization in large-scale WSNs. We first analyze the factors that influence the multi-hop distance estimation and give a quantitative rule for setting the weight of reference information. Then, the close to optimal values of node coordinates are efficiently searched and obtained in the feasible regions of normal nodes through a lightweight grid-scanning scheme, which avoids solving the complex constrained nonlinear programming and prevents getting stuck at local optimum. MLGS is very suitable for sensor nodes of limited energy and computing power. Through extensive simulations in isotropic and anisotropy networks, we demonstrate that MLGS outperforms the typical multi-hop localization schemes in many aspects. Compared with MDS-MAP and iterative algorithm, MLGS can also do better in localization accuracy and topology adaptability. In most cases, MLGS could achieve better performance, even without refinement phase. Therefore, the phase of node collaboration refinement is optional. Reducing the grid granularity

The authors would like to thank the anonymous reviewers for their comments. This work is supported by the National Natural Science Foundation of China under Grant No. 60873240, No. 60974121 and No. 61001138.

As shown in _{ai}_{a}

Division of bounding square ring.

Pseudo-codes of determining _{a}

Isotropic network.

Anisotropic network (H shape).

Network initialization procedure.

Impact of multi-hop density on multi-hop distance estimation.

Average localization error as a function of

Average localization error as a function of

_{a}

Intersection of bounding square rings.

Samples for localization refinement.

Distribution boxplots of node localization errors (isotropic network).

Distribution boxplots of node localization errors (anisotropic network).

Average localization error

Average localization error

Average Localization error

Average localization error

Average localization error

Average localization error

Average localization error

Average localization error

Computation cost

Computation cost

Localization results of MLGS.

Localization results of MDS-MAP.

Localization results of iterative algorithm.

Default parameters of WSNs.

Network deployment area (m) | 200 × 200 | 200 × 200 |

Network holes (m) | No apparent hole | 66.7 × 66.7 (×2) |

Number of nodes | 200 | 200 |

TTL | 5 | 5 |

Percentage of anchor nodes | 10% | 10% |

Node’s communication radius (m) | 25.6 | 24.2 |

Network connectivity | 9 | 9 |

Ranging error factor | 0.1 | 0.1 |

Average errors of MLGS, MDS-MAP and iterative algorithms.

MLGS | 13.4% | 12.7% |

MDS-MAP | 23.1% | 62.1% |

Iterative algorithm | 25.8% | 26.1% |