# Localization in 3D Surface Wireless Sensor Networks

**DOI:**https://doi.org/10.1007/978-3-319-32903-1_258-1

## Synonyms

## Problem Definition

Sensor network localization refers to the process of estimating the locations of sensor nodes with information between neighboring sensor nodes such as connectivity, local distance, and angle measurements. In real-world applications, many large-scale wireless sensor networks (WSNs) are deployed over complex terrains. Formally, a three-dimensional (3D) surface WSN is defined as follows.

### Definition 1 (3D Surface Wireless Sensor Network)

A *3D surface sensor network* consists of sensor nodes deployed on a 3D surface where wireless signals between nearby nodes propagate along the surface only.

### Definition 2 (Localization of 3D Surface Wireless Sensor Network)

Given a 3D surface sensor network with distance measurements between neighboring nodes within their communication range, the localization problem is to recover the 3D coordinates of each sensor node.

## Historical Background

Localization of a network deployed over a 3D surface generates a unique hardness compared with the well-studied localization of a network on 2D plane or in 3D volume. Specifically, due to limited radio range, the distance between two remote sensors deployed over a 3D surface can only be approximated by their surface distance, the length of the shortest path between them on the surface. Such surface distance is different from the 3D Euclidean distance of two nodes. A 3D surface is *localizable* if it exists a unique embedding up to a global rigid motion, under given constraints. Otherwise it is *non-localizable*. The following theorem claims that a network deployed over a 3D surface with surface distance information only is non-localizable, even if we assume accurate range distance measurement available (Zhao et al., 2012).

### Theorem 1

*A general 3D surface is not localizable, given surface distance constraints only.*

Given a wireless sensor network deployed on a 3D terrain surface with one-hop distance information available, a simple distributed algorithm introduced in Zhou et al. (2011) extracts a refined triangular mesh from the network connectivity graph. Vertices of the triangular mesh represent the set of sensor nodes. An edge between two neighboring vertices indicates the communication link between the two sensors. The state-of-the-art surface network localization methods are based on the extracted triangular mesh structure.

## Surface Network Localization with Height Information

Under a practical setting with estimated link distances (between nearby nodes) and nodal heights (i.e., Z-coordinates) obtained by measuring atmospheric pressure, a sensor network deployed on a single-value 3D surface is localizable. Briefly, a single-value surface is one on which any two points have different projections on the X-Y plane. The definition is in reference to X-Y plane since sensors’ heights are given as Z-coordinates. A planar projection of network deployed on a single-value surface converts a surface network localization problem to a planar one. The X-Y coordinates of nodes can be computed with well-known planar localization algorithms (Shang et al., 2003; Shang and Ruml, 2004; Vivekanandan and Wong, 2006; Lim and Hou, 2009; Jin et al., 2011). The localization result is then mapped back to 3D by adding the known Z-coordinates.

For sensor networks deployed on general 3D surfaces, a distributed localization algorithm, dubbed cut-and-sew, applies a divide-and-conquer approach to localize sensor nodes with height information available. The basic idea is to partition a general 3D surface network into single-value patches, localizing individual patch and then merging them into a unified coordinate system. Note that the number of single-value patches should be minimized to avoid unnecessary partitioning and merging, which are subject to linear transformation errors. The key is to identify non-single-value edges that guide the division of a 3D surface network into single-value patches.

Under practical sensor network settings, both surface distances and sensors heights are subject to measurement errors. The noisy distance and height measurements directly affect the identification of non-single-value edges, which may deviate from the ground truth and become isolated. One approach is to fuse nearby non-single-value edges to form a band and then cut the network along the medial axis of the band. This method effectively minimizes the impact of input errors on network partition and localization.

### Prototyping and Experiments

Every sensor periodically broadcasts a beacon message that contains its node ID to its neighbors. Based on received beacon messages, a node builds a neighbor list with the RSSI of corresponding links. RSSI is used to estimate the length of links by looking up a RSSI-distance table established by experimental training data. The preliminary test shows that, under low transmission power, such estimation has an error rate about 20%. At the same time, the ground truth of surface distances and sensor coordinates is manually measured. Figure 3b illustrates the triangulation based on ground truth inputs. Figure 3c shows the localization result. The combined patches largely restore the original 3D surface network, with an average location error of about 14*%*.

## Surface Network Localization with Digital Terrain Model

Integrating height measurement into every sensor of a network is not always practical and affordable, especially for a large-scale sensor network. However, a 3D representation of a terrain’s surface, called digital terrain model (DTM), is available to public with a variable resolution up to one meter. DTMs are commonly built using remote sensing technology or from land surveying. A DTM is represented as a grid of squares, where the longitude, latitude, and altitude (i.e., 3D coordinates) of all grid points are known. It is straightforward to convert a grid into a triangulation, e.g., by simply connecting a diagonal of each square. Therefore a triangular mesh of a terrain surface can be available before we deploy a sensor network on it. On the other hand, a refined triangular mesh can be extracted from the connectivity graph of network deployed on the terrain surface with local distance information. The constraint that the sensors must be on the known 3D terrain surface ensures that the triangular meshes of terrain surface and network approximate the same geometric shape. Theoretically, the two triangular meshes share the same conformal structure. We can construct a well-aligned conformal mapping between them. Based on this mapping, each sensor node of the network can easily locate reference grid points of the DTM to calculate its own location.

*f*

_{1}and

*f*

_{2}respectively, to map the two triangular meshes to plane as shown in Fig. 4b, d respectively. However, the two mapped triangular meshes on plane are not aligned. Three anchor nodes, sensor nodes equipped with GPS, marked with red as shown in Fig. 4c are deployed with the network to provide a reference for alignment. Based on the positions of the three anchor nodes, we construct a M\(\ddot {o}\)bius transformation, denoted as

*f*

_{3}, to align the mapped triangular meshes of the network and the terrain surface on plane. Combining the three mappings, \(f_1^{-1} \circ f_3 \circ f_2\), induces a well-aligned conformal mapping between the two triangular meshes shown in Fig. 4a, c, respectively. Based on the well-aligned mapping, each sensor node of the network simply locates its nearest grid points, vertices of the triangular mesh of the terrain surface, to calculate its own geographic location.

### Deployment of Anchor Nodes

*μ*), median (\(\tilde {x}\)), and standard deviation (

*σ*) of localization errors under different deployments of anchor nodes of each network. The positions of three anchor nodes affect the performance of the localization algorithm slightly. In general, the more scattered the three anchor nodes are deployed in a network, the lower the localization error is.

The distribution of localization errors under different sets of anchor nodes (Yang et al., 2014)

Error | DTM I | DTM II | DTM III | DTM IV |
---|---|---|---|---|

| 0.2579 | 0.1356 | 0.0951 | 0.2098 |

\(\tilde {x}\) | 0.2306 | 0.1343 | 0.0956 | 0.1512 |

| 0.1089 | 0.1717 | 0.0158 | 0.0352 |

### Size of Anchor Nodes

## Applications

Geographic location information is imperative to a variety of applications in WSNs, ranging from position-aware sensing to geographic routing. While global navigation satellite systems (such as GPS) have been widely employed for localization, integrating a GPS receiver in every sensor of a large-scale sensor network is unrealistic due to high cost. Moreover, some application scenarios prohibit the reception of satellite signals by part or all of the sensors, rendering it impossible to solely rely on global navigation systems. In real-world applications, many large-scale WSNs are deployed over complex terrains, such as the volcano monitoring project (Werner-Allen et al., 2006). The introduced state-of-the-art surface network localization algorithms recover the coordinates of sensor nodes with assumption of the availability of nodal height measurements (Zhao et al., 2012, 2013) or digital terrain model, a 3D representation of terrain’s surface (Yang et al., 2014). They are all distributed and scalable to large-scale sensor networks deployed over general 3D terrain surfaces.

## Cross-References

## References

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