A GPS-less, outdoor, self-positioning method for wireless sensor networks

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A GPS-less, outdoor, self-positioning method for

wireless sensor networks

q

Hung-Chi Chu, Rong-Hong Jan

*

Department of Computer and Information Science, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 30050, Taiwan, ROC Received 3 May 2005; received in revised form 8 December 2005; accepted 9 March 2006

Available online 21 April 2006

Abstract

One challenging issue in sensor networks is to determine where a given sensor node is physically located. This problem is especially crucial for very small sensor nodes. This paper presents a GPS-less, outdoor, self-positioning method for wire-less sensor networks. In our method, a set of nodes, called reference points (RPs), are deployed in the sensor network with overlapping regions of coverage. The RP periodically broadcasts beacon frames which contain localization data. The sen-sor node collects the beacon frames from RPs and process the data in the frame; it can then easily localize itself. The anal-ysis of positioning accuracy is given to show how well a sensor node can correctly localize itself. In the optimal transmitting power, the worst-case accuracy for all data points is within 28.87% of the separation-distance between two adjacent RPs and the average accuracy is within 15.51%. The simulation results also show the robustness of the proposed method. Finally, we have implemented our positioning method on a sensor network test bed and the actual measurement show that the method can achieve average accuracy within 17.9% of the separation-distance between two adjacent RPs in an outdoor environment.

1. Introduction

The fast progress of micro-electro-mechanical systems (MEMS) technology and wireless commu-nications has enabled us to deploy a large number

of low-cost, low-power and networked sensors over wide areas. The sensor nodes can collect, store, and process the sensed data and communicate with neighboring nodes to provide observation of envi-ronmental systems. This makes monitoring and controlling the physical world more convenient and eﬃcient. In such sensor network systems, we need sensor nodes to be able to locate themselves in various environments. The location data of sensor nodes are useful for the centralized server or the managing node to analyze their sensing infor-mation. Not only sensor nodes but also other objects in the network need to be located. For

q

This research was supported in part by the National Science Council, Taiwan, under grant NSC 94-2219-E-009-005 and NSC94-2752-E-009-005-PAE, in part by the communication software technology of III, Taiwan, and in part by the Intel.

* _{Corresponding author. Tel.: +886 3 5731637; fax: +886 3}

5721490.

E-mail address:rhjan@cis.nctu.edu.tw(R.-H. Jan).

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example, the forest ﬁre detection system should detect exactly where the scene of a ﬁre is. In loca-tion-aware applications, localization enables the intelligent context selection includes tour guide

[1,2], points of interest, real-time traﬃc information and so on. In ad hoc networks, localization helps the transmitting node recognize where the commu-nicating node is and thus reduces the power con-sumption. Simply put, localization is important for many sensor network applications.

For localization systems, global positioning sys-tem (GPS)[3]is a good solution in outdoor environ-ments. However, it is not suitable to use GPS on all sensor nodes in sensor networks. This is because sensor nodes have size, cost, and power constraints. This paper focuses on the problem of GPS-less, out-door, low-cost localization for wireless sensor networks.

A survey of location systems can be found in[4]. Generally speaking, the localization can be divided into three major classes: self-positioning, remote positioning, and indirect positioning. The basic operations of these classes are summarized below:

A. Self-positioning system: The positioning recei-ver receives the appropriate signal measure-ments from geographically distributed transmitters and then uses these measurements to localize itself. Global positioning system (GPS)[3] is a typical self-positioning system. Recently, several self-positioning systems[5,6]

for sensor networks have been presented. In

[5], they measure the received signal strength and apply a triangulation method to localize moving sensors and handle dynamically chang-ing sensor topologies. In[6], some ﬁxed refer-ence nodes with overlapping regions of signal coverage are conﬁgured. These reference nodes transmit periodic beacon signals and then sen-sor nodes can localize themselves based on the received beacons. An ad hoc positioning system (APS)[7]is a distributed, hop by hop position-ing system. The sensor node uses the distance vector and the location information of land-marks to estimate its own location. In [8], point-in-triangulation test (PIT) is proposed to narrow down the possible region which a node resides in. In [9], a ring-overlapping approach is proposed. Based on received signal strength, a sensor node can determine an inter-section area where it resides and use the gravity of the intersection area as its position.

B. Remote positioning system: A set of nodes with special radio frequency (RF) functions are deployed in some fixed place and measure the direction or the time delay of a signal which is originating from, or reflecting off, the transmitter nodes. After that, a centralized location server collects these measurements to determine the transmitter node’s location. Typical remote positioning systems are angle of arrival (AOA) [10,11], time of arrival (TOA)[11], time difference of arrival (TDOA)

[10,11], and received signal strength indicator (RSSI)[11]. The AOA measures the direction of the transmitter’s signals; the TOA measures the signal propagation time from transmitter to receiver; the TDOA measures the propaga-tion time difference from a signal traveling from transmitter to two different receivers; and the RSSI measures the received signal strength (RSS) and uses RSS to estimate the distance between transmitter and receiver. Such solutions do not require any modifica-tion to the objects but they have low posimodifica-tion accuracy and high network costs.

C. Indirect positioning system: The indirect posi-tion system combines self-posiposi-tioning and remote positioning systems. First, the node measures signal data and transfers it to the remote positioning system. Next, the remote positioning system collects these measure-ments, processes position bias, and then deter-mines the node’s position. Typical indirect positioning systems are assisted GPS (AGPS)

[12], diﬀerential GPS (DGPS) [13,14], and cell-based positioning [15] where AGPS and DGPS have the highest positioning accuracy. The cell-based positioning system [15] simply utilizes the characteristic of cell overlapping in geometry. However, it determines the location in a centralized server. When a sensor node needs to localize itself, it sends location requests to the location server. The location server determines the sensor’s location and then sends the location to the sensor node. Unfortunately, communications between the sensor and the location server require a lot of energy and thus are not suitable for wireless sensor networks. Based on the idea of cell overlap-ping, this paper presents a GPS-less, outdoor, self-positioning method for wireless sensor networks. In the proposed method, a set of nodes, called reference points (RPs), are deployed in the sensor

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network with overlapping regions of coverage. The RPs broadcast periodic beacon frames which con-tain localization data. The sensor node in the sensor network ﬁrst receives the beacon frames from RPs, then processes the information in the frame, and ﬁnally the localization can be determined by itself. The proposed method has the following charac-teristics:

1. It is a distributed GPS-less self-positioning sys-tem. That is, the location can be determined by the sensor node itself without GPS or centralized server.

2. Sensor nodes only use simple connectivity metric and localization data in the beacon frame to calculate their locations. That is, sensor nodes require little computation to localize by themselves.

The remainder of this paper is organized as fol-lows. In Section 2, we present the cell overlapping with an idealized radio model in detail. Section 3

gives the algorithm for the self-positioning system. The positioning accuracy analysis and simulation results are shown in Sections 4 and 5. A hardware implementation of the proposed method is given in Section 6. Finally, conclusions are given in Section7.

2. Cell overlapping model

Consider that a set of RPs are deployed in the sensor network with overlapping regions of cover-age. They are located at known positions and form a regular structure (e.g., hexagonal structure or meshed structure). As shown in Fig. 1(a), these RPs form a hexagonal structure. In our idealized radio model, we assume a perfect spherical radio propagation and identical transmission range for all reference points.1 The area covered by the RP is called a cell and each cell is circle-shaped. The sensor node (SN) can receive radio signals from the RP if it is within the signal coverage of that RP. For example, as shown in Fig. 1(a), an SN in region A1can listen to signals from RP P0; in region

B1, from RPs P0and P1; and in region C1, from RPs

P0, P1and P6. The localization region is deﬁned as

the region in which every SN can listen a unique set of RPs’ signals. As shown inFig. 1(a), the coverage of RP P0 has 13 localization regions, i.e., regions

A1, B1, . . ., B6, C1, . . ., C5and C6. (0,1) (0, -1) 2 1 2 3 ( , - ) 2 1 2 3 (- , - ) ( , )₂1 2 3 2 1 2 3 (- , ) A1 B₁ B2 B₃ B₄ B₅ B6 C₁ C₂ C₃ C₄ C₅ C6 (a) (b) A₁ B₁ B₂ B₃ B₄ B5 B₆ C₁ C₂ C₃ C₄ C₅ C₆ (x 1, y1) (x2, y2) (x₃, y₃) (x₄, y₄) (x₅, y₅) (x₆, y₆) (x₇, y₇) (x₈, y₈) (x₉, y₉) (0,0) P₁ P₂ P3 P₄ P5 P6 P₀

Fig. 1. The physical layout of reference points with a hexagonal structure.

1

This idealized model has been checked by experimental measurements for its validity in [6]. They concluded that the idealized radio model may be considered valid for outdoor unconstrained environments.

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Consider a hexagonal structure as shown in

Fig. 1(a). The localization regions in the coverage of a RP can be divided into three types according to the number of receiving signals as follows:

• Type 1 region: The region is covered by only one RP’s signal, e.g., region A1.

• Type 2 region: The region is covered by two RPs’ signal coverage, e.g., regions B1, B2, B3, B4, B5,

and B6.

• Type 3 region: The region is covered by three RPs’ signal coverage, e.g., regions C1, C2, C3,

C4, C5, and C6.

Note that the radio coverage of RP is represented as a circle. By using simple geometry, we can ﬁnd all the intersections of the circles. For each localization region, we ﬁnd the centroid (xc, yc) of the region by

ðxc; ycÞ ¼ x1þ x2þ þ xn n ; y₁þ y2þ þ yn n ; where (x1, y1), (x2, y2), . . ., (xn, yn) are the vertices of

the region. If an SN can localize itself in the region, we use (xc, yc) to estimate the location of the SN.

For example, as shown inFig. 1(b), if an SN local-izes itself in region B1, the estimated location of SN

is x1þx2þx8þx9

4 ;

y1þy2þy8þy9

4

.

Given a set of RPs deployed in a hexagonal struc-ture in which the distance between two neighboring RPs is one unit and the transmission range of RP is r = 0.78, we can ﬁnd the centroids for all localiza-tion regions. The results are summarized inTable 1. 3. Self-positioning algorithm

As stated in the previous section, we can deploy RPs in a hexagonal structure and ﬁnd the

localiza-tion regions for each RP. The RP periodically broadcasts the beacon frame to notify all of the SNs staying in its signal coverage area. We assume that each RP knows all centroids of its localization regions. For example, RP P0knows the centroids of

13 localization regions. The centroids can be com-puted in the deployment stage. The beacon format contains the following data:

S ¼ ftn;ðtra;fðxc1; yc1Þ; . . . ; ðxca; ycaÞgÞ;

. . . ;ðtrk;fðxc1; yc1Þ; . . . ; ðxck; yckÞgÞg;

where tnrepresents the type of RP’s structure, (e.g.,

tn= 1 for hexagonal structure and tn= 2 for meshed

structure): tri represents the type of localization

region (e.g., tri 2 f1; 2; 3g for hexagonal structure);

and ðxci; yciÞ represents the centroid of the region.

Note that the type number of the region is equal to the number of signals that can be received in that region.

For example, as shown in Fig. 2, the beacon frames of RP 5 and RP 6 are

S5¼ f1; ð1; fMgÞ; ð2; fB; D; F ; H ; J ; LgÞ;

ð3; fA; C; E; G; I; KgÞg;

S6¼ f1; ð1; fW gÞ; ð2; fJ ; N ; P ; R; T ; V gÞ;

ð3; fK; I; O; Q; S; U gÞg;

where the symbols A, B, . . ., W represent the cent-roids of localization regions (e.g., M ¼ pﬃﬃ3

2 ; 1 2 , W = (0, 0)).

Then, the SN collects the beacon signals from the RPs and determines its location. The operations of SN are given as follows:

Table 1

The centroids of all regions in the hexagonal network structure

Region Centroid Region Centroid

A1 (0, 0) C1 ffiffi 3 p 6 ;12 B1 0;1₂ C2 ffiffi 3 p 6 ; 1 2 B2 ffiffi 3 p 4;14 C3 ffiffi 3 p 3 ;0 B3 ffiffi 3 p 4; 1 4 C4 ffiffi 3 p 6 ; 1 2 B4 0;1₂ C5 ffiffi 3 p 6 ; 1 2 B5 ffiffi 3 p 4 ;14 C6 ffiffi 3 p 3 ;0 B6 ffiffi 3 p 4 ; 1 4 – – 6 2 5 9 10 8 4 J W K I V T S R N P O Q 1 7 3 D A M C E B L F H G U (0,0) (0,1) (0,-1) ( , )₂3 ₂1

Fig. 2. An example of localization regions for hexagonal structure.

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1. Collect and store the beacon signal that it receives.

2. Determine the number of RPs, denoted as m, that it can listen to. Then extract the centroid set with the type m from the beacon frames, denoted as Sm. Note that we can ﬁnd m diﬀerent centroid sets. For example, if an SN can receive beacons from RP 5 and RP 6, it extracts the cen-troid set with type 2 from the received beacon frames as follows:

S2₅¼ fB; D; F ; H ; J ; Lg; S2₆¼ fJ ; N ; P ; R; T ; V g.

3. The SN finds a centroid by intersecting the centroid sets as its location, i.e., find T_iSm_i. For example, S2₅\S2₆¼ fB; D; F ; H ; J ; Lg \ fJ ; N ; P ; R; T ; V g ¼ fJ g ¼ ffiffiffi 3 p 4 ; 1 4 ! ( ) .

4. Positioning accuracy analysis

Let the coordinate of the actual location of SN be (X, Y) where X and Y are random variables. In our proposed method, the SN localizes itself to the centroid of the localization region. Thus, the error distance D is D¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðX xcÞ 2 þ ðY ycÞ 2 q ;

where (xc, yc) is the centroid of the localization

re-gion (i.e., the estimated location of the sensor node). The precision e(r) can be deﬁned as the probability that the SN can localize itself within distance r. That is,

eðrÞ ¼ P fD < rg.

Assume that the SN falls equally likely to any point in the location region R. Then, the probability density function f(x, y) of (X, Y) can be written as follows: fðx; yÞ ¼ c ifðx; yÞ 2 R; 0 otherwise, where Z R Z fðx; yÞdx dy ¼ Z R Z cdx dy¼ 1. This gives c¼R 1 R R dx dy¼ 1 area of R. Therefore, the precision eðrÞ ¼ P fD < rg ¼ Z Cr Z fðx; yÞdx dy ¼ area Cr area of R; where Cr¼ ðx; yÞj ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx xcÞ 2 þ ðy ycÞ 2 q < r \ R.

4.1. The worst-case accuracy

Now, let us consider the shape of type 1 as shown inFig. 3. The precision e(r) is the area of Crover the

area of localization region R, if r is less then r1. If r

is greater than r1, the precision e(r) is 1. This means

that SN can localize itself within distance r1 with

probability 1. In other words, if SN localizes itself in the type 1 region and the tolerance of error dis-tance d is greater than r1, the position of SN can

be correctly determined. The radius r1is called the

critical radius. Furthermore, let r_{¼ maxfr}ð1Þ

1 ;

rð2Þ₁ ; rð3Þ₁ g where rðiÞ1 is the critical radius for type i

region. Thus, we can say that SN localizes itself correctly within distance r*_{. Note that r}* _{is the}

worst-case accuracy.

For example, consider that a set of RPs are deployed in a hexagonal structure in which the distance between two neighboring RPs is one unit and the transmission range of RP is 0.78. We can com-pute the precision ei(r) for each type i.Fig. 4shows the

precision ei(r) for type i = 1, 2, 3. Note that r¼

maxfrð1Þ1 ; r ð2Þ

1 ; r

ð3Þ

1 g ¼ maxf0:2685; 0:2993; 0:3088g ¼

0:3088. That is, for this hexagonal structure, SN localizes itself correctly within distance 0.3088.

r1

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Note that critical radius rðiÞ₁ is a function of RP’s transmission range d. Let fi(d) be the critical radius

for type i, i = 1, 2, 3. Then, the worst-case accuracy r*_{can be rewritten as r}*_{(d) = max{f}

1(d), f2(d), f3(d)}.

If the transmitting power of RP can be adjusted, then the transmission range of RP will vary. We assume that the radius d is bounded within

1_ﬃﬃ

3

p ;pﬃﬃ3

2

h i

.2Let us consider how to arrange the trans-mission range of RP such that the worst-case accu-racy is optimized. This problem is equivalent to ﬁnding a radius d such that r*_{(d) =}

max{f1(d), f2(d), f3(d)} is minimized. That is,

z¼ min 1ffiffi 3 p6d6pffiffi3 2 rðdÞ ¼ min 1ffiffi 3 p6d6 ffiffi3 p 2 maxff1ðdÞ; f2ðdÞ; f3ðdÞg. ð1Þ Fig. 5shows the functions f1(d), f2(d), and f3(d),

for 1_ﬃﬃ 3

p 6_{d 6} ﬃﬃ3

p

2. The function f1(d) is a decreasing

function and the function f3(d) is an increasing

func-tion where 1_ﬃﬃ 3

p 6_{d 6} ﬃﬃ3

p 2. Let d

* _{be the radius such}

that f1(d*) = f3(d*). Thus, maxff1ðdÞ; f2ðdÞ; f3ðdÞg ¼ f1ðdÞ if p1ffiffi₃6d 6 d; f3ðdÞ if d6d 6 ffiffi 3 p 2 (

and the minimum of max {f1(d), f2(d), f3(d)} occurs

at f1(d) = f3(d). By using the numerical method, we

ﬁnd d*_{= 0.7638 such that f}

1(d*) f3(d*) = 0.2887.

4.2. The average-case accuracy

Given that the location (x, y) of SN falls in the type i area, the expected accuracy Diis

E½Di ¼ Z ðx;yÞ2Ri ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx xciÞ 2 þ ðy yciÞ 2 q fðx; yÞdx dy;

where Ri is the localization region of type i and

ðxci; yciÞ is the centroid of Ri. Thus, the expected

accuracy of D for the network with hexagonal struc-ture can be found by

E½D ¼X

3

i¼1

p_iE½Di;

where piis the probability that SN falls in the type i

area. By this way, we can evaluate the average accu-racy of the proposed method.

Note that the average accuracy E[D] is also a function of RP’s transmission range d. Let g(d) be the average accuracy E[D] for the RPs with hexago-nal structure having transmission range d. Let us consider how to arrange the transmission range of RP such that the average accuracy is minimized. The problem is to find a radius d such that z¼ min1ffiffi 3 p6d6 ffiffi3 p 2 gðdÞ.

We can evaluate the average accuracy E[D] by simulation. In our simulation, 10,000 sensor nodes were generated in the working area of 100· 100 square units. The SNs are placed in the working

2

This is because (1) if d < 1_ﬃﬃ 3

p_{, then there are some areas not} covered by RP’s signal; (2) if d >pﬃﬃ3

2, then the type 2 area will be separated into two sub-areas.

0.6 0.65 0.7 0.75 0.8 0.85 0 0.1 0.2 0.3 0.4 0.5 Transmission range Accuracy f 1 (d) f 2 (d) f 3 (d) d*

Fig. 5. The worst-case accuracy for hexagonal structure.

0 0.15 0 0.6 1 Accuracy Precision Type 1 Type 2 Type 3 0.2685 0.3088 0.2993 r₁(1) r(2) 1 r₁(3)

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area with a uniform distribution. We assume that all RPs are deployed in a hexagonal structure with transmission range ^d and their locations are known in advance. By the proposed self-positioning method, each SN can localize itself at position (xc, yc). Thus, the positioning error can be found.

By this way, we can evaluate the average accuracy gð^dÞ. Furthermore, we find g(d), for 1_ffiffi

3

p 6d 6pﬃﬃ3

2,

as shown inFig. 6. Note that function g(d) is a con-vex function. We ﬁnd the minimum of g(d) is 0.1551 where d = 0.744.

5. Positioning accuracy for imperfect RPs

In order to show the robustness of the proposed method, we assume that RPs are not perfect. Con-sider the example given in Section 3. Assume that an SN is in the region J (seeFig. 2) and RP 6 fails. The SN only receives the beacon frame S5=

{1, (1, {M}), (2, {B, D, F, H, J, L}), (3, {A, C, E, G, I, K})} from RP 5. As a result, the SN localized itself at M ¼ pﬃﬃ3

2 ;

1 2

. That is, the accuracy error becomes large.

We evaluate the average accuracy for imperfect RP by simulation. In our simulation, 10,000 sensor nodes were generated in the working area of 100· 100 square units. Then, SNs are placed in the working area with a uniform distribution. We assume that all RPs are deployed in a hexagonal structure with transmission range 0.744 and their locations are known in advance. We consider three cases of imperfect RPs. That is, case 1 has a 1% of failure rate of RPs; case 2 has 5%; and case 3 has 10%.Fig. 7shows the average accuracy of the pro-posed method with imperfect RPs. From Fig. 7,

note that the proposed method with imperfect RPs having 1%, 5%, and 10% failure rates can locate SN to within 0.3088 unit distance for 98.68%, 92.74% and 85.6% of measurements, respectively. Because of the failure of RPs, some sensor nodes in the working area may not localize themselves. When the RPs failure rates are 1%, 5%, and 10%, the probabilities that the sensor nodes cannot localize themselves are 0.21%, 1.21%, and 2.36%, respectively. That is, the probability that the sensor node cannot localize itself is very small and the decease in positioning accuracy is very lim-ited for the network with imperfect RPs having a 10% failure rate.

In the previous simulation, we assume the com-munication range is an ideal circle. In reality, the coverage of RP is irregular due to multipath propa-gation eﬀects. Thus, we construct a simulation using the shadowing model [16] as its radio model. The shadowing model3can be represented by

PrðdÞ Prðd0Þ dB ¼ 10b log d d0 þ XdB;

where Pr(d) (Pr(d0)) is the received signal power at

distance d (d0), b is the path loss exponent, and

XdB is a Gaussian random variable with l = 0

and standard deviation rdB. Note that the

shadow-ing model extends the ideal circle model to a statistic model. For outdoor environments, we set

0.6 0.65 0.7 0.744 0.8 0.85 0 0.05 0.1 0.1551 0.2 0.25 Transmission range Accuracy 1 3 3₂

Fig. 6. The average accuracy for hexagonal structure.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 Accuracy Precision Perfect 1% failure 5% failure 10% failure

Fig. 7. The average accuracy for imperfect RPs in a hexagonal structure.

3

This model does not include the effects of multipath fading. These effects can be significant when working with narrowband signals.

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rdB= 11 and b = 2 (free space) or b = 3

(shad-owed urban area) in our simulation [17]. The SN can receive the beacon frame if the received signal power is greater than the value of Pr(d) where d is

0.744 unit distance. A unit distance is equal to 20 m in the simulation. The working area was 100· 100 square units and RPs were deployed with hexagonal structure. For randomly generating 100,000 SNs to be located in a working area,

Fig. 8 shows the average accuracy of the proposed method. For outdoor, free space environment (i.e., (rdB, b) = (11, 2)), the accuracy curve is almost

the same as the accuracy curve of perfect model. For outdoor, shadowed urban area (i.e., (rdB, b) = (11, 3)), the SN can localize itself to

within 0.3 unit distance for 87% of measure-ments. Thus, the proposed method still worked well in the outdoor, shadowed urban area.

6. Hardware implementation

The proposed self-positioning method was imple-mented over a collection of MICA2 sensor nodes

[18]to verify its feasibility and estimate its accuracy in a real-world environment. The resource con-straints of MICA2 are listed inTable 2. We placed MICA2 sensor nodes as RPs on an outdoor skating rink in our campus. The topology is shown in

Fig. 9(b) in which seven black dots represent seven RPs. The distance between two adjacent RPs is

about 10 m. The transmission power of each RP was tuned such that its transmission range is about 8 m. Each RP broadcasts a beacon frame every 200 ms. The contents of beacon frames are listed inTable 3. A white dot with coordinate (x, y), where x and y are integers, in Fig. 9(b) represents a test point. Each time we placed a MICA2 sensor node on a test point (white dot) and then the sensor node collected beacon frames for 9600 ms. Let Nabe the

total number of beacon frames collected at test point a and Na(i) be the number of beacon frames

collected at test point a that were issued from RP i. The sensor node at test point a discards the bea-con frames from RP i ifNaðiÞ

Na is less than 0.1. Based

on the beacon frames it collected, the sensor node localized itself by the proposed positioning method. In our experiment, we measured 276 test points as shown inFig. 9.

Fig. 10 shows the average accuracy for the experimental and simulation results. We use

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 Accuracy Precision Perfect model Shadowing modelβ=2 Shadowing modelβ=3

Fig. 8. The average accuracy for the shadowing propagation model.

Table 2

The parameters and hardware information about MICA2 Mote

Component Description

Processor Atmel ATMega 128L

Program ﬂash memory 128 KB

Conﬁguration EEPROM (Data) 4 KB

Radio frequency 868–870 MHz

Radio transceiver Chipcon CC1000

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RP 1 (15,25) RP 2 (24,20) RP 3 (24,10) RP 0 (15,15) RP 4 RP 1 RP 6 RP 5 RP 2 RP 3 RP 0 (a) (b) RP 6 (6,20) Reference point Test point X Y (0, 0) RP 5 (6,10) RP 4 (15,5)

Fig. 9. The topology of RPs.

Table 3

The beacon content of RPs

RP Beacon content RP 0 {1, (1, {(15,15)}), (2,{(15,20),(19,18),(19,13),(15,10),(11,13),(11,18)}), (3,{(12,20),(18,20),(21,15),(18,10),(12,10),(9,15)})} RP 1 {1, (1, {(15,25)}), (2,{(15,30),(19,28),(19,23),(15,20),(11,23),(11,28)}), (3,{(12,30),(18,30),(21,25),(18,20),(12,20),(9,25)})} RP 2 {1, (1, {(24,20)}), (2,{(24,25),(28,23),(28,18),(24,15),(19,18),(19,23)}), (3,{(21,25),(27,25),(30,20),(27,15),(21,15),(18,20)})} RP 3 {1, (1, {(24,10)}), (2,{(24,15),(28,13),(28,8),(24,5),(19,8),(19,13)}), (3,{(21,15),(27,15),(30,10),(27,5),(21,5),(18,10)})} RP 4 {1, (1, {(15,5)}), (2,{(15,10),(19,8),(19,3),(15.0),(11,3),(11,8)}), (3,{(12,10),(18,10),(21,5),(18,0),(12,0),(9,5)})} RP 5 {1, (1, {(6,10)}), (2,{(6,15),(11,13),(11,8),(6,5),(2,8),(2,13)}), (3,{(3,15),(9,15),(12,10),(9,5),(3,5),(0,10)})} RP 6 {1, (1, {(6,20)}), (2,{(6,25),(11,23),(11,18),(6,15),(2,18),(2,23)}), (3,{(3,25),(9,25),(12,20),(9,15),(3,15),(0,20)})} 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 Accuracy Precision Perfect model Shadowing model Implementation

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10 m as a unit distance in our experiment. From

Fig. 10 we note that the SN can localize itself to within 0.3 unit distance (i.e. 3 m) for 91.67% of measurements in our outdoor experiments. The experimental results also agree with the simu-lation results using the shadowing model (b = 3, rdB= 11). In Fig. 11, the positioning error

obtained from experiments is plotted as a function of the test points. The positioning error is lowest for the test points at the centroid of the regions and increases towards the edges of the regions. The average positioning error was 1.79 m and the standard deviation was 0.86 m. The minimum error was 0 m and the maximum error was 4.12 m across 276 test points.

7. Conclusions

In this paper, we proposed a GPS-less, outdoor, self-positioning method for wireless sensor net-works. In our method, a set of RPs with overlap-ping regions of coverage are arranged in a hexagonal structure or meshed structure in the sensor network and broadcast the beacon frames. Sensor nodes only collect the beacon frames from RPs and use the localization data in the beacon frame to calculate their locations. Note that sensor nodes require little computation to localize by them-selves. This kind of localization system, with its low cost and easy computation, is very suitable for sensor networks.

In the optimal transmitting power, the worst-case accuracy for all data points is within 28.87% of the separation-distance between two adjacent RPs and the average accuracy is within 15.51%. The simula-tion results also show the reliability and robustness of our proposed method. Regarding system robust-ness, the proposed method with imperfect RPs can locate SN to within 30.88% of the separation-dis-tance between two adjacent RPs for 85.6% of mea-surements even though 10% of RPs failed. Finally, we have also implemented our positioning method on a sensor network test bed to verify its feasibility. The actual measurements show that it can achieve average accuracy within 17.9% of the separation-distance between two adjacent RPs in a outdoor environment.

Although the proposed positioning method is based on a regular structure, it might be extended to solve the positioning problem based on irregular structures, under the condition that each RP’s posi-tion and coverage can be precisely determined and no two localization regions receive the same set of RP beacon frames. Furthermore, the analysis of the positioning accuracy and optimization of RP’s coverage for irregular structures might be interest-ing for possible future work.

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Hung-Chi Chu received the B.S. and M.S. degrees in Computer Science and Engineering from Tatung University, Taiwan, in 1995, 1997. Since 2001, he has been working toward the Ph.D. degree in Computer and Information Science at National Chiao Tung University, Tai-wan. His research interests include

wireless networks and artiﬁcial

intelligence.

Rong-Hong Jan received the B.S. and M.S. degrees in Industrial Engineering, and the Ph.D. degree in Computer Sci-ence from National Tsing Hua Univer-sity, Taiwan, in 1979, 1983, and 1987, respectively. He joined the Department of Computer and Information Science, National Chiao Tung University, in 1987, where he is currently a Professor. During 1991–1992, he was a Visiting Associate Professor in the Department of Computer Science, University of Maryland, College Park, MD. His research interests include wireless networks, mobile computing, distributed systems, network reliability, and opera-tions research.