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Definition 2.2 (Estimation rectangle): This method gets the intersection region of reference nodes with squares instead of circular areas
Pt
Pr. (1)
Note that the path loss exponent m will be set to 4 in the simulation later because the two-ray ground model is employed in our simulations for modeling radio propagation.
3.2.2 Position Computation
This component is responsible for estimating the location of the normal node based on the available information, e.g., positions of reference nodes and distances to them. In the lit-erature, bounding box [26], multilateration [17], trilateration [7], and triangulation [15,22]
are some techniques used to estimate the region of a normal node. Among these techniques, the bounding box method is the simplest one. As shown in Fig.2, the region estimation for normal node Nj is decided by the areas of reference nodes A1, A2, and A3, including the intersection region (IR) (see Fig.2a) which is the intersection region of the circular areas of reference nodes and the estimation rectangle (ER) (see Fig.2b). The ER can be formally defined as follows:
Definition 2.2 (Estimation rectangle): This method gets the intersection region of reference nodes with squares instead of circular areas.
3.2.3 Localization Algorithm
This component has an important role in manipulation of available information from the pre-vious two components to estimate positions of sensor nodes in the network. Although a lower estimation error and more settled nodes are the main objectives of localization, a good local-ization algorithm still needs to have the ability to reduce computation and communication costs.
Based on Definitions2.1and2.2and the three main components, the CDL-TAGS scheme is to be proposed in the next section.
4 The Proposed Localization Scheme
Let us first describe the tripodal anchor structure and basic concept of localization, then give the details of CDL-TAGS.
4.1 Tripodal Anchor Structure and Basic Concept of Localization
A tripodal anchor structure (see Fig.3a) is formed by three anchor nodes, say, A1, A2, and A3, connected to the centroid node C with the distance of and equal angle of 120◦. Each anchor node is equipped with the GPS device to obtain its location information. Note that
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(a)
(b)
Fig. 2 Region estimation for a sensor node. a Intersection region. b Estimation rectangle
the tripodal anchor structure should set as short as possible while being able to estimate positions of sensor nodes. To apply this structure, the precondition is that the system has randomly distributed normal nodes.
Our CDL-TAGS scheme has eight steps and is performed after all normal nodes and the tri-podal anchor structure are deployed. The basic concept of CDL-TAGS lies in the localization starting from the centroid node and then stretching outward as illustrated in Fig.3b.
4.2 Details of CDL-TAGS
The pseudocode of CDL-TAGS shown in Fig.4mainly consists of eight steps, i.e., calculat-ing the position of the centroid node and its virtual points, selectcalculat-ing reference nodes, gettcalculat-ing a virtual reference node, dividing the estimation rectangle into a grid matrix, grid scanning, estimating the position, quantifying the residual error, and broadcasting. Now, these steps are depicted as follows.
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(a)
(b)
Fig. 3 The tripodal anchor structure and basic concept of CDL-TAGS. a Tripodal anchor structure. b Basic concept
4.2.1 Calculating the Position of the Centroid Node and its Virtual Points
Each anchor node in the tripodal anchor structure first obtains its positionAi, i= 1, 2, 3, via the GPS device and disseminates a packet containing its position to the centroid node and its adjacent nodes. Then, the centroid node calculates the center of gravity (CoG)C to estimate its position after receiving packets from the three anchor nodes by
C =
3
i=1Ai
3 . (2)
After that, the centroid node generates three virtual points serving as virtual centroids (V Cs), say, V C1, V C2, and V C3, for normal nodes under the boundary error condition (to be explained in the second step of the localization scheme). V C positionsV Ci, i= 1, 2, 3, are derived via
V Ci = C + s(C − Ai), i = 1, 2, 3, (3) where s is a scaling factor and may be set to s= 2√33r . The centroid node finally broadcasts a packet containing its position and its virtual points to all sensor nodes. The aforementioned activities are shown in the pseudocode in Fig.4(see lines 9–13).
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Fig. 4 Pseudocode of CDL-TAGS
4.2.2 Selecting Reference Nodes
Each normal node needs at least two reference nodes and the information from the centroid node to estimate its position. As for how to select a pair of sensor nodes when more than two reference nodes exist, the two sensor nodes whose residual errors (see Fig.5a) are lower than the others will be selected because better estimation can be achieved. Here, the residual error is defined as a measurement of confidence for the reference nodes (quantified in the seventh step of the localization scheme). Because localization is started from the centroid node and then stretched outward, the two reference nodes selected should be in the proximity of the normal node and closer to the centroid node than the normal node. In Fig.5b, how to detect a pair of reference nodes in the proximity of the normal node and closer to the centroid node than the normal node is roughly illustrated. First, the normal node calculates distances from the two reference nodes, say, Rf (the farther one) and Rn (the nearer one), to the centroid node to make sure which one is closer to the centroid node. The normal node then finds the intersection points between the circle centered at reference node Rf with radius df, where df is the estimated distance to reference node Rf by the normal node, and the circle centered at the centroid node C with radius lf, where lf is the distance between reference node Rf
and the centroid node C. With the two intersection points, the normal node then calculates the distances from reference node Rn to them to get two estimated distances dmi nand dmax
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(a) (b)
Fig. 5 Selecting the best pair of reference nodes. a The two nodes with the lowest errors. b Checking that the two reference nodes are in the proximity of the normal node and closer to the centroid node than the normal node
(dmi n< dmax). The normal node now can check whether dmi n≤ dn ≤ dmax, where dnis the estimated distance to reference node Rnby the normal node. If yes, the two sensor nodes are accepted as the reference nodes; otherwise, the normal node excludes the two reference nodes selected and reselects two reference nodes based on the residual errors and rechecks for the previous condition until the best pair of reference nodes is got.
Note that a virtual point should be used as the substitute of the centroid node when the normal node is in the boundary error condition (see Fig.6a). This condition has two different scenarios. The first one is that the normal node is inside the circle centered at the centroid node with radius. For this scenario, no anchor nodes can be used as reference nodes because their distances to the centroid node are farther than that between the normal node and the cen-troid node. Hence, the normal node calculates its distance to each anchor node to determine the middle distance, say, the distance to anchor node Aj. The corresponding virtual point V Cj then serves as the centroid node so that the selected reference nodes can be closer to the new centroid node than the normal node. The second one is that the estimated distance to reference node Rf by the normal node is greater than two times distance between reference node Rfand the centroid node. In this scenario, dmi nand dmaxcan not be calculated because of no intersection points. Hence, the normal node calculates its distance to each anchor node and determines the smallest distance, say, the distance to anchor node Ak. The corresponding virtual point V Ckthen serves as the centroid node so that dmi nand dmax can be calculated and the selected reference nodes are closer to this centroid node than the normal node, too.
As shown in Fig.6a, N1and N2are two normal nodes in the two scenarios, respectively, of the boundary error condition; V C2serves as the new centroid node; A1, A3and A1, A2serve as reference nodes in the two scenarios, respectively.
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(a)
(b)
Fig. 6 Boundary error condition as well as virtual centroid nodes and a virtual reference node. a A normal node in the boundary error condition gets a virtual centroid node. b Determination of a virtual reference node
4.2.3 Getting a Virtual Reference Node
Using two reference nodes may result in a large estimated area for the normal node. To minimize it, an extra reference node is required. Based on the two reference nodes accepted previously, the normal node generates a virtual point to serve as the third reference node (see
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Fig.6b) as follows. First, the normal node creates a bounding box for each reference node and gets the intersection area of them by employing the estimation rectangle. Denoting P1 and P2as the two original reference nodes with estimated distances r1and r2, respectively, to the normal node. Without loss of generality, we assume that r1≥ r2. Let P3be the third reference node. It can be obtained by finding the farthest point from the centroid node lying on the side of the estimation rectangle and the line passing through the normal node and perpendicular to line P1P2at P0with coordinateP0expressed as
P0= P1+ γ (P2− P1)/dP1,P2, (4) whereγ = (r12−r22+dP21,P2)/(2dP1,P2), dP1,P2denotes the distance between P1and P2, and Pi, i = 1, 2, denote the coordinate of Pi. Finally, a bounding box of P3is created, making a smaller estimation rectangle.
4.2.4 Dividing the Estimation Rectangle into a Grid Matrix
Since the estimation rectangle does not fit the estimation region, dividing the estimation rectangle into a grid matrix with grid scale Sgis the easiest way to calculate the estimated location of the normal node. Assume that the four coordinates of the estimation rectan-gle are{xl, yt}, {xr, yt}, {xl, yb}, {xr, yb}, respectively, where subscriptsr,l,t,b denote right, left, top and bottom, respectively, xr > xl, and yt > yb. A matrixA = [Ai, j]mg×ng
represents the grids, where mg = (xr − xl)/Sg , ng = (yt − yb)/Sg , and Ai, j = (L, B) + ((i − 1/2)Sg, ( j − 1/2)Sg) with (L, B) being the coordinate of the left bottom corner of the estimation rectangle.
4.2.5 Grid Scanning
Grid scanning is a process to examine whether grids in the estimation rectangle are used to represent the location of the normal node or not. Let grid value Gi, j indicate the number of reference nodes with communication regions able to cover gridAi, j (see Fig.7). Initially, each grid is associated with grid value 0. The grid scanning is examined by the two reference nodes selected previously and a virtual reference node generated previously. Furthermore, we can consider two trianglesP1P0P3andP2P0P3as shown in Fig.6b to decide where
Fig. 7 Grid scanning
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the normal node is inside any of them. As indicated in [25], the position of the normal node is near the circular boundary of the reference node with smaller estimated distance to the normal node. As shown in Fig.6b, r2 < r1implies that the position of normal node is near the circular boundary of P2. Hence, triangleP1P0P3can be selected for the estimation.
The triangle scanning in [24] can then be used to examine whether gridAi, j is inside the selected triangle or not. When two reference nodes have the same estimated distance to the normal node i.e., r2= r1, the triangle scanning can be ignored.
4.2.6 Estimating the Position
The coordinate of the normal node can be derived from gridAi, j in the valid grid area where the normal node may reside. Assume that there are m valid grids with the greatest grid value, the position of the normal node is then estimated bym1
valid i, jAi, j. 4.2.7 Quantifying the Residual Error
Once the normal node estimates its position for being a settled node, it also checks the residual error e via
e=
l∈ref set
(xl− ˆx)2+ (yl− ˆy)2− dl, (5)
where( ˆx, ˆy) is the estimated position of the normal node, (xl, yl) is the position of the lth ref-erence node, r e f set is the set containing the refref-erence nodes, and dlis the distance acquired by the RSSI measurement for the corresponding reference node. Note that the residual error of a normal node is indeed influenced by the estimated distance, estimated position of that normal node, and positions of the reference nodes possibly with additional biases. Because of this fact, the policy to select reference nodes for our proposed scheme is to choose the best pair of reference nodes with minimum residual errors as described in Sect.4.2.2.
4.2.8 Broadcasting
Since the normal node only uses reliable settled nodes with low residual errors to serve as the reference nodes, the other settled nodes with high residual errors do not need to broadcast packets to their adjacent nodes. With the setting of an error threshold,2the communication overhead among sensor nodes can then be reduced.
5 Complexity Analysis
This section provides the complexity analysis of CDL-TAGS and the related localization schemes in terms of communication, time, and space complexities.
5.1 Communication Complexity
In both CDL-TAGS and RPE, anchor nodes and the normal nodes able to estimate their positions and broadcast packets to the normal nodes with unknown positions for serving as
2The error threshold should be properly selected, in particular, in a sparse WSN (note that such a situation seldom occurs in most of applications of WSNs).
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Table 1 Time and space complexities for different schemes
Localization scheme Time complexity Space complexity
CDL-TAGS O(b2+ g) O(b + g)
EWCL O(b2) O(b)
RPE O(b3) O(b2)
Grid Scan O(b(1 + g)) O(b + g)
reference nodes. In RPE, each sensor node only broadcasts one packet during the execution of the whole algorithm. Therefore, the communication complexity of RPE is O(nn+na), where naand nn are the number of anchor nodes and the number of normal nodes, respectively, in the network. However, the normal nodes in CDL-TAGS only use reliable settled nodes with low residual errors to serve as the reference nodes. With the setting of an error threshold, the settled nodes with high residual errors have no need to broadcast packets to their adjacent nodes. This can reduce the communication overhead among sensor nodes in CDL-TAGS and the communication complexity of CDL-TAGS can be lower than that of RPE.
5.2 Time and Space Complexities
First, note that both Grid Scan and CDL-TAGS need to calculate the estimation rectangle and scan the grid matrix. Let us assume that the average number of neighboring reference nodes of a normal node is b. Since Grid Scan needs to derive the estimation rectangle and scan the grid matrix for b neighboring reference nodes on average, but CDL-TAGS only needs to do these for two reference nodes and a virtual point, the time and space complexities for Grid Scan are O(b(1 + g)) and O(b + g), respectively, while these two complexities for CDL-TAGS are all O(g) in this step, where g is the number of grids. Moreover, each normal node in CDL-TAGS needs to select a best pair of reference nodes. Hence, the time and space complexities of CDL-TAGS in this step are O(b2) and O(b), respectively. For EWCL, the same time and space complexities as those of CDL-TAGS can be derived, i.e., O(b2) and O(b). Consequently, the total time and space complexities for CDL-TAGS are O(b2+ g) and O(b + g), respectively. Table1summarizes the time and space complexities for the related schemes, where RPE has the highest time complexity (O(b3)) and space complexity (O(b2)) because of the QR factorization to solve the localization.
6 Simulation Results and Discussions
In this section, we evaluate CDL-TAGS and some related localization schemes using the mannasim [14] framework based ns–2 [21] simulator. In the following, let us elaborate on the simulation environment and parameters first.
6.1 Simulation Environment and Parameters
In the following simulations, the IEEE 802.11 medium access control (MAC) protocol is employed in the MAC layer, while mica2 radio device with the two-ray ground propaga-tion model is utilized for the physical layer. Here, we assume that the MAC layer protocol can solve the collision problem so that each sensor node can correctly receive information from adjacent nodes. For EWCL and Grid Scan, non-homogeneous sensor nodes are set (namely, the communication range of anchor nodes is longer than that of normal nodes),
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while homogeneous sensor nodes are set for RPE and CDL-TAGS with the same communi-cation range of 15 m. For all schemes, immobile normal nodes and anchor nodes are randomly distributed in a region of 100× 100 m2. Each sensor node is forced to stay in the active state during the localization process. After the sensor node obtains its location, the sensor node could enter the low-power mode for power efficiency. As for the other parameters, they are described as follows. The RSSI error3varies among 0.01–0.1%. The number of anchor nodes for RPE and that for Grid Scan are set to 5 and 40%, respectively, of the normal nodes (the number of normal nodes varies among 250–1000), while EWCL has 40 anchor nodes. For CDL-TAGS, the number of anchor nodes varies among 3–60, the number of centroid nodes varies among 1–20, and the arm length varies among 2–9 m. As for the error threshold of CDL-TAGS, it is set to 1.5m. Finally, the time that a normal node has to wait for more packets from its neighbors is set to 1 s. The reason is stated as follows. We assume that a normal node should receive the necessary broadcast information required by our proposed scheme (see lines 14–18 of the pseudocode) from its neighbors being reference nodes within this specified time period.
6.2 Performance Metrics
Two performance metrics are utilized to evaluate the performance of localization schemes as follows.
6.2.1 Average Location Error¯el
It is the average location error between the estimated position( ˆxi, ˆyi) and the actual position (xi, yi) of all sensor nodes and is defined by
¯el =
nn
i=1
( ˆxi− xi)2+ ( ˆyi− yi)2
nn . (6)
6.2.2 Ratio of Locatable Nodes Rl
It is the ratio of the number of locatable nodes nland the total number of normal nodes nn, i.e.,
Rl = nl
nn. (7)
6.3 Determination of Some Parameters for CDL-TAGS
Let us now investigate the performance of CDL-TAGS to get the best setting for the position and arm length of the tripodal anchor structure.
6.3.1 Position of the Tripodal Anchor Structure
Considering four positions denoted by I, II, III, and IV as shown in Fig.8, we examine the impact caused by the placement of the tripodal anchor structure in Fig.9. Shown in Fig.9a
3Actually, the variation of the received signal strength (RSS) can be captured by its variance etc. Similarly, the variation of RSSI can be revealed by its variance, too. However, the RSSI error here denotes the maximum variation in percentage caused by the variation of RSS (or the inaccuracy of the RSS measurement) associated with the estimated distance according to the RSSI approach governed by (1).
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Fig. 8 Positions of the centroid node considered in the simulation
is the average location error. Obviously, the average location error is the highest when the tripodal anchor structure is near the corner of the field (position I), while it is the lowest when the tripodal anchor structure is close to the center of the field (position IV). The reason why such a phenomenon is exhibited is that a more symmetric position of the tripodal anchor structure has more uniformly distributed settled nodes to achieve better assistance to localize sensor nodes. Furthermore, more sensor nodes in the field cause smaller average location errors as shown in Fig.9a because more reference nodes exist for normal nodes. In Fig.9b, ratios of locatable nodes are shown. With a similar reason to that of the phenomena exhibited in Fig.9a, the ratio of locatable nodes is the highest when the tripodal anchor structure is put to the center and it is the lowest when the tripodal anchor structure is near the corner. Based on the aforementioned observations, we shall put the tripodal anchor structure to the center of the field for the remaining simulations.
is the average location error. Obviously, the average location error is the highest when the tripodal anchor structure is near the corner of the field (position I), while it is the lowest when the tripodal anchor structure is close to the center of the field (position IV). The reason why such a phenomenon is exhibited is that a more symmetric position of the tripodal anchor structure has more uniformly distributed settled nodes to achieve better assistance to localize sensor nodes. Furthermore, more sensor nodes in the field cause smaller average location errors as shown in Fig.9a because more reference nodes exist for normal nodes. In Fig.9b, ratios of locatable nodes are shown. With a similar reason to that of the phenomena exhibited in Fig.9a, the ratio of locatable nodes is the highest when the tripodal anchor structure is put to the center and it is the lowest when the tripodal anchor structure is near the corner. Based on the aforementioned observations, we shall put the tripodal anchor structure to the center of the field for the remaining simulations.