Resource-Aware Localization in Sensor Networks
6.3 Positioning Systems and Localization Algorithms
6.3.5 Proximity-Based Techniques .1 Nearest Beacon
Simply choosing the nearest beacon position, for example, by determining the highest received signal strength indication (RSSI), is a possibility to solve the localization problem. Although, this can be achieved with minimal computation effort, the localization error may be very high. Nevertheless, the estimated position can be used as start value for iterative methods that increase the precision [BRBT].
6.3.5.2 Coarse-Grained Localization
Centroid localization (CL) was first published by Bulusu as “GPS-less low cost outdoor localization for very small devices” in [Bul]. The algorithm completely avoids explicit distance measure-ments, but assumes a grid-based beacon placement with constant distances q between each other as demonstrated in Figure .. To localize nodes, b beacons B(x; y)⋯Bb(xb; yb)are aligned in a grid.
Thetransmission range of beacons is assumed to be circular.
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Relative error r = q fAv = 0.20 q fmax = 0.48 q
2q
q q
Ai
2q
r
0.4q
0.3q
0.2q
0.1q
0
0 q 2q 0 0 q 2q 0
FIGURE . Scenario with a × grid of beacons: (a) resulting overlaps and (b) localization error. (From Blumen-thal, J. and Timmermann, D., th IEEE/ACM International Conference on Distributed Computing in Sensor Systems (DCOSS ), pp. –, Santorini, Griechenland, June . With permission.)
CL starts on every beacon by sending a packet including its own position and network address.
Sensor nodes within the beacon’s transmission range receive corresponding packets and save its con-tent. At the end of this phase, every sensor node received n packets and determines the position with the centroid formula:
˜S(˜x; ˜y) = n
∑n
j=Bj(xj; yj) (.)
If no beacon position is received, a sensor node’s position cannot be estimated. Analysis in an outdoor testbed with a quadratic sensor field of side length a = m, four beacons at the corners and test points resulted in an averaged localization error of fAv=. m with standard deviation σfAv =. m [Bul]. Additional analysis showed that the error strongly depends on the beacons transmission range and placement [BRHT,Rei]. For that, [RBT] presents a simple equation to determine the optimal transmission range in terms of a specific grid width. Moreover, Bulusu et al. and Salomon suggest strategies for a better beacon placement, which also reduces the localization error significantly [BHE,SB]. In finite sensor networks with high transmission ranges, the CL is characterized by a very high localization error near the borderlines of the network. This error can be reduced significantly as proposed by Blumenthal [Blu].
6.3.5.3 Weighted Centroid Localization
Weighted CL (WCL) is an extension of CL by including distances in form of weights in the centroid formula. WCL was first published in [BRT] and features randomly distributed beacons opposite to the grid alignment CL assumes.
Figure . shows a sensor node, which was localized by CL and also WCL. Four beacons are placed at the corners of the sensor field. Whereas CL leads to a high error, WCL reduces the error, because nearer beacons pull the sensor node’s position stronger to their own position than to farer beacons.
This also works in case of noisy distance measurements. Beacons have constant and ideal concentric transmission ranges. In a first phase, beacons send their position and network address to all sensor nodes in range. These nodes save this information. Contrary to CL, the distances to all beacon must be measured by one of the techniques described in Section .. After a defined timeout, phase starts.
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Centroid
Weighted centroid S (x; y) B4(x4; y4)
B1(x1; y1) B2(x2; y2)
B3(x3; y3)
S (x; y) d4
d1 d2
d3
~ ~ ~
S (x; y)~ ~ ~
FIGURE . WCL compared to CL. (From Reichenbach, F., Resource aware algorithms for exact localization in wireless sensor networks, PhD thesis, University of Rostock, Rostock, Germany, December . With permission.)
Thisimplies calculating the position at every sensor node by a weighted centroid determination:
˜Si(˜xi; ˜yi) =
∑n
j=(wi j⋅Bj)
∑n j=wi j
(.)
In Equation ., the parameter wi jrepresents the weight between sensor node i and beacon j.
To achieve the above-mentioned characteristics, the weight can be calculated by the inverse of the distance.
wi j(di j) =
di jg (.)
In Equation ., a new parameter was introduced—the degree g, which amplifies shorter distances to beacons.
Simulation results illustrated in Figure . show the localization error with several minima depending on the ratio between transmission range and degree. In terms of minimal energy con-sumption, all sensor nodes can estimate a position if r is greater than rmin = .q using a weight function w = /d. If r < rmin=.q (critical area) not all sensor nodes are able to estimate a posi-tion, because not enough beacons are in range. However, if high precision instead of minimal energy consumption is demanded, then the optimum exists at ropt = .q, because there, the smallest localization error is achieved. Although a degree of g = yields in the best results, it is highly fluctuating and thus critical to apply. Higher degrees are more stable, which may be better in practice.
In conclusion, WCL is very resource aware, because every beacon sends only one short packet.
Further on, a sensor node must receive n packets, measures the distance and executes a calculation with a time complexity of O(n − ).
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rmin= 0.71q
Comparison finite sensor networks (3q ´ 3q) with different degree of weights
FIGURE . Localization error using different degrees. (From Blumenthal, J., Grosmann, R., Golatowski, F., and Timmermann, D., st European ZigBee Developer’s Conference (EuZDC), Munich, Germany, June . With permis-sion; Blumenthal, J., IEEE International Symposium on Intelligent Signal Processing, WISP , Madrid, Spain, October
. With permission.)
6.3.5.4 Range-Free Localization
Therange-free localization algorithm is also known as “approximate point in triangulation” (APIT) and bases on triangular surfaces. The algorithm was first published in [HHB+]. He et al. reduce the influence of absolute distance measurements, due to the high error that can be expected.
Likewise in CL, every beacon transmits its position in the sensor field. By permuting all beacon positions b!/! ⋅ (b − !), every sensor node determines all resulting triangles. Then, each trian-gle is checked by the “point in triangulation” (PIT) test. This check allows to make a decision if the sensor node is placed on the triangle surface or not. After this test is finished, sensor nodes know all triangles on which they are placed. All beacons constructing these triangles are used to estimate the sensor nodes position by a centroid calculation with their positions. This is shown in Figure .a.
Thecenterpiece of APIT is the PIT test. Theoretically, a sensor node is outside a triangle of three beacon positions if the following assumption is true: A mobile sensor node is slightly shifted in any direction ∆d, then all three distances to these three beacons must increase or decrease simultaneously.
Otherwise, the node must be inside a triangular (Figure .b).
He et al. take advantage of a large node density in sensor networks. Figure .c exemplary illus-trates that a sensor node may have connection to four very close neighbors. These neighbors measure the RSSI to all three required beacons and send this information to the sensor node under test. This
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Triangle surface
(b)
Theoretical PIT
Practical PIT
Outside Inside
Outside Inside
(c) (a)
FIGURE . (a) Overlapping of triangles are basis of APIT, (b) theoretical PIT test with distances and a moveable sensor node, and (c) practical PIT test with neighbors and RSSI values. (From Reichenbach, F., Born, A., Bill, R., and Timmermann, D., Lokalisierung in Ad Hoc Geosensornetzwerken mittels geodatischer Ausgleichungstechnik, GIS:
Zeitschrift für Geo Informationssysteme, ABC Verlag GmbH, Heidelberg, Germany, , pp. –. With permission;
Reichenbach, F., Resource aware algorithms for exact localization in wireless sensor networks, PhD thesis, University of Rostock, Rostock, Germany, December . With permission.)
node checks if all four neighboring RSSI are higher or lower simultaneously. This is similar to the theoretical PIT test and avoids absolute distances as well as moving capabilities.
6.3.5.5 Bounding Box Algorithm
Thebounding box algorithm was published in [SS] and is also based upon beacons. In this al-gorithm, beacons transmit their position with a specific transmission power, respectively within a well-defined transmission range r. Next, sensor nodes receive beacon packets and determine dis-tances to beacons by one of the measurement techniques described in Section .. Then, each sensor node calculates a quadratic surface Ai = (di j)including every Bi(xi; yi), where Bi(xi; yi)is the middle of the square (Figure .a). By comparing the minimal and maximal coordinates (Equation
.) of all resulting squares Ai, sensor nodes narrow down the surface on which they are placed.
Thisremaining square ASis called bounding box (Figure .b). These rough position estimates can later be used for iterative approaches.
xmin=max(xi−di j) ≤xj≤min(xi−di j) =xmax
ymin=max(yi−di j) ≤yj≤min(yi−di j) =ymax (.) A disadvantage of the bounding box algorithm is that a small localization error can only be achieved if Sjis placed within a convex, by beacons limited, hull AS. Otherwise, the bounding box, and therefore the localization error, increases as shown in Figure .c. Moreover, the algorithm behaves instable if distance measurements are very noisy, because probably overlapping surfaces are missed.
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A1 and (c) bounding box AS (S ∉ AS,hul l). (From Blumenthal, J., Resource aware and decentral localization of au-tonomous sensor nodes in sensor networks, PhD thesis, University of Rostock, Rostock, Germany, , submitted.
With permission.)
6.3.6 Optimization Methods