A Historical-Beacon-Aided Localization Algorithm for Mobile Sensor ...

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A Historical-Beacon-Aided Localization Algorithm for Mobile Sensor Networks Jen-Feng Huang, Guey-Yun Chang, and Gen-Huey Chen Abstract—Range-free localization approaches are cost-effective for mobile sensor networks (because no additional hardware support is required). However, existing range-free localization approaches for mobile sensor networks suffer from either sparse anchor node problem or high communication cost. Due to economic considerations, mobile sensor networks typically have sparse anchor nodes which makes most range-free localization algorithms inaccurate. On the other hand, due to the power limitation of mobile sensor nodes (i.e., they are battery-operated) and high power consumption by communication, high communication cost will significantly reduce the network life time. For solving these two problems, in this paper, we use historical beacons (i.e., anchor nodes’ announcements delivered in previous time slots) and received signal strength (RSS) to derive three constraints. By the aid of the three constraints, we introduce a low-communication-cost range-free localization algorithm (only one-hop beacon broadcasting is required). According to the theoretical analysis and simulation results, our three constraints can indeed improve the accuracy. Simulation results also show that our algorithm outperforms even in irregular-radio-signal environments. In addition, a hardware implementation running on sensor nodes, Octopus Xs, confirms theoretical analysis and simulation results. Index Terms—Ad-hoc network, localization, mobility, range-free, wireless sensor network

Ç 1

INTRODUCTION

L

OCALIZATION

is a critical issue in wireless sensor networks (WSNs). Although GPS has been widely used to assist location-based services [1], [2], [3], [4], [5], [6], [7], [8], it is impractical to equip each sensor node with a GPS device in large-scale WSNs. Therefore, localization algorithms for WSNs typically use a limited number of anchor nodes, which are aware of their locations, e.g., by the aid of GPS, while the other nodes (referred to as normal nodes) estimate their locations using the location information of anchor nodes. Such localization algorithms are anchornode-based, and they can be further divided into two categories [9]: range-based and range-free. A range-based localization algorithm calculates locations with absolute point-to-point distances, while a range-free localization algorithm calculates locations without these distances. However, distance estimation techniques usually require additional expensive hardware support (e.g., angle of arrival (AoA) [10] and time difference of arrival (TDoA) [11]), or have low accuracy (e.g., received signal strength (RSS)-based approaches). Due to the hardware limitations of WSNs, range-free solutions are being pursued as an alternative to range-based solutions. Most (e.g., [9], [12], [13], [14], [15]) of prior range-free localization algorithms were designed for static sensor networks and not applicable to mobile ones. Existing



J.-F. Huang and G.-H. Chen are with the Department of Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan. E-mail: [email protected], [email protected]. G.-Y. Chang is with the Department of Computer Science and Information Engineering, National Central University, Jhongli, Taiwan. E-mail: [email protected].

Manuscript received 20 Nov. 2013; revised 11 July 2014; accepted 26 July 2014. Date of publication 11 Aug. 2014; date of current version 1 May 2015. For information on obtaining reprints of this article, please send e-mail to: [email protected], and reference the Digital Object Identifier below. Digital Object Identifier no. 10.1109/TMC.2014.2346777

range-free localization approaches for mobile sensor networks usually suffer from sparse anchor node problem and high communication cost. Due to economic considerations, wireless sensor networks typically have sparse anchor nodes which makes most range-free localization algorithms inaccurate. On the other hand, in mobile sensor networks, sensor nodes are battery-operated and communication is the highest power consumption item. Prior localization algorithms achieve the required accuracy with high communication cost [12], [14], [15], [16], [17], [18], [19], [20], [21], [9], [22] and high communication cost will significantly reduce the network life time. Moreover, due to the rapid development of wireless technologies (e.g., Wi-Fi and Bluetooth) and quickly emerging applications, the ISM band, which is used by most WSNs, has become crowded and congested [23]. Hence, localization algorithms with high communication cost will be impractical in the near future. In this paper, we introduce a range-free localization algorithm for mobile sensor node networks. In order to address the sparse anchor node problem and high communication cost problem, our algorithm fully utilizes the advantages of the communication ranges (of nodes), historical beacons, and RSS (of beacons), which are free of communication cost. To the best of our knowledge, our algorithm is the first one to use the RSS of historical beacons in mobile sensor node localization. Our algorithm includes three new constrained regions (refer to Section 3.2). A constrained region is a region that can cover the location of the target normal node, e.g., the communication range of a one-hop neighboring anchor node (which is widely adopted in existing range-free algorithms [16], [17], [18], [19], [21]). According to the theoretical analysis and simulation results, the three constrained regions can indeed improve the localization accuracy. Besides, our algorithm has low communication cost (only one-hop beacon broadcasting is required). Simulation

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results also show that our algorithm outperforms even in irregular-radio-signal environments. In addition, a hardware implementation running on sensor nodes, Octopus Xs [24], confirms theoretical analysis and simulation results. The rest of this paper is organized as follows. Section 2 briefly surveys prior range-free localization algorithms. Section 3 presents the proposed algorithm and three constrained regions, and Section 4 illustrates the theoretical analysis of the constrained regions. Section 5 demonstrates the feasibility of our algorithm, through an hardware implementation using sensor nodes, Octopus Xs. Section 6 shows the simulations results and compares our algorithm with state-of-the-art algorithms. Finally, Section 7 concludes the whole paper.

2

PRIOR RANGE-FREE LOCALIZATION ALGORITHMS

In this section, prior range-free localization algorithms [16], [17], [18], [19], [21], [25], [26], [27] for mobile WSNs are reviewed. In subsequent discussion, each anchor node sends a beacon (which carries its location information) to its one-hop neighbors (called one-hop-beacon-broadcasting) [25] or its one-hop and two-hop neighbors (called two-hopbeacon-broadcasting) [16], [17], [19], [21] or all nodes[18]. Normal node a collects beacons from anchor nodes to determine its location. We use r to denote the communication radius of an anchor/normal node, and vmax to denote the maximal moving distance of a normal node during a time slot. Among prior range-free localization algorithms for mobile WSNs, two-hop-beacon broadcasting [16], [17], [18], [19], [21], normal node location exchange [17], [21], statistical models[21], [25], fuzzy logic [26], and RSS ordering [27] were used to improve the localization accuracy. In [16], the Monte Carlo Localization (MCL) algorithm was proposed, where a is located within the intersection (called the anchor-constrained region of a in [21]) of the communication ranges of a’s one-hop neighboring anchor nodes and ring areas. Each point in a ring area has distance to a a’s two-hop neighboring anchor node within ðr; 2r. The MCL algorithm randomly generates a set of points (called valid samples at time slot 0) in the deployment area, and then performs two phases: prediction and filtering, in each time slot. Consider time slot i with i > 0. In the prediction phase, a sample at time sloti (for a) is a point randomly picked from a sample circle. A sample circle has radius equal to vmax and is centered at a valid sample generated at previous time slot ði  1Þ. The MCL algorithm considers a sample (at time slot i) invalid (valid) for a, if it is outside (inside) the anchor-constrained region of a (at time slot i). In the filtering phase, invalid samples (generated at time slot i) for a are filtered out. The location of a (at time slot i) is then estimated as the centroid of all valid samples (at time slot i). For example, vmax ¼ 1, r ¼ 2, and points at (1, 1), (1, 0), and (0.5, 0.5) are chosen as valid samples at time slot 0 for a. Then there are three samples at time slot 1, each of which is randomly picked from a sample circle which has radius ¼ 1 and is centered at one valid sample at time slot 0. Suppose that points at (1.5, 1.5), (0.5, 0), and (0, 0.5) are chosen as samples at time slot 1. Assume that a has a one-hop

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neighboring anchor node b at (0, 0) and no two-hop neighbor, i.e., the anchor constrained region of a is b’s communication range (i.e., the circle which has radius ¼ 2 and is centered at (0, 0)) at time slot 1. Hence, there are two valid samples at time slot 1, i.e., points at (0.5, 0) and (0, 0.5). So, the location of a at time slot 1 is estimated as (0.25, 0.25). A defect of the MCL algorithm is that valid samples generated in the prediction phase may be not enough to estimate the location of a accurately [19]. Besides, in sparse anchor node environments, MCL has low localization accuracy. In [19], the Monte Carlo localization Boxed (MCB) algorithm was proposed. In MCB, one-hop communication ranges (of one-hop neighboring anchor nodes) and twohop communication ranges (of two-hop neighboring anchor nodes) are simplified as 2r  2r square areas and 4r  4r square areas, respectively. The intersection of these square areas is referred to as the anchor box. The samples that are generated in the prediction phase are from the intersection of the anchor box and the sample box (i.e., a box that covers the sample circle described in the MCL algorithm). Since the anchor box is a bare approximation of the anchor-constrained region, more valid samples may remain after the filtering phase. However, when the anchor density is high, the MCB algorithm may generate much more valid samples than necessary for estimating the location of a [21]. In the Mobile Static sensor network Localization (MSL ) algorithm [17], a is required to exchange location information with its one-hop and two-hop neighboring normal nodes in each time slot. The MSL algorithm considers a sample invalid (valid) for a, if it is outside (inside) the intersection of the anchor-constrained region of a, one-hop neighboring normal nodes’ communication ranges, and normal-ring areas. Here, a normal-ring area includes points whose distance to a two-hop neighboring normal node is within ðr; 2r. Although the MSL algorithm can estimate the location of a more accurate than the MCB and MCL algorithms, it incurs higher communication costs at the same time. Further, there is a refinement of the MSL algorithm, named the Improved Monte Carlo Localization (IMCL) algorithm, in [21]. The IMCL algorithm divides the communication region of a into eight sectors of equal size, and extends the radius of each sector to d þ r, where d is the maximal distance from a to the valid samples within the sector. The collection of the resulting sectors is referred to as the neighbor-constrained region of a. In the IMCL algorithm, a sample is considered valid (invalid) for a, if it is inside (outside) the intersection of the anchor-constrained region of a and all neighbor-constrained regions of the one-hop neighboring normal nodes of a. In [18], the Multi-hop-based Monte Carlo Localization (MMCL) algorithm considers connected mobile WSNs each of which has a large monitoring area and sparse anchor nodes. In the circumstances, the MMCL algorithm requires each anchor node to broadcast beacons to all normal nodes. In addition to the above MCL-based algorithms, a localization algorithm, named the Localization with Anchor History (LAH) algorithm, was proposed in [25]. The LAH algorithm calculates the location of a in each time slot, by the aid of the approximated mobility pattern of a and a regression model whose coefficients are determined according to

HUANG ET AL.: A HISTORICAL-BEACON-AIDED LOCALIZATION ALGORITHM FOR MOBILE SENSOR NETWORKS

the historical beacons of a. However, it is not easy in practice to predict the mobility pattern of a accurately. In [26], a fuzzy logic-based localization algorithm (FuzLoc) is proposed. To tolerate high degree of radio signal irregularity, anchor nodes build fuzzy rules by exchanging beacons, finding the distance between themselves and measuring the RSS. Each normal node a sends a query packet to two-hop anchor nodes to obtain the fuzzy distances to neighboring anchor nodes. Besides, to address sparse anchor node problem, each anchor node b also calculates the distances between a and some predefined locations (by the aid of the fuzzy distance from b to a, and the distances from b to these predefined locations), and provides these distances to a for improving localization accuracy. In [27], each normal node determines a 1-hop neighborhood ordering which is sorted by the decreasing RSS values. Using the differences of their 1-hop neighborhood orderings and node density, normal nodes can estimate their relative distances and locations among 2-hop neighborhood.

3

THE PROPOSED RANGE-FREE LOCALIZATION ALGORITHM

In this paper, we introduce a range-free localization algorithm, HitBall, for mobile sensor networks. Throughout this paper, mobile sensor networks are assumed to have mobile normal nodes and mobile anchor nodes. Each anchor node is assumed to broadcast a beacon that carries its location information to its one-hop neighboring normal nodes per slot (i.e., one-hop-beacon broadcasting). Each normal node a can determine the possible region (of its location) by the aid of collected beacons. A possible region of a’s location is a region which covers a’s location. Clearly, a smaller possible region implies higher localization accuracy. A beacon is called a current beacon if it is delivered in the current time slot, and a historical beacon otherwise (i.e., prior to the current time slot). Associated with each current beacon, there is a one-hop-anchor-constrained region, which is the communication range of the anchor node when it sent out the beacon. Besides, associated with each historical beacon, there is a historical-anchor-constrained region, which is a circle centered at the anchor node that sent out the beacon. If the historical beacon was delivered t time slots ago, then the circle has a radius of r þ vmax  t, where r is the communication radius of an anchor node and vmax is the maximum moving distance of a normal node during a time slot. The HitBall algorithm and MCL-based range-free localization algorithms (e.g., [16], [17], [18], [19], [21], [25]) determine the possible region of a’s location by finding the intersection of all one-hop-anchor-constrained regions of a with others constrained regions (e.g., historical-anchorconstrained regions and ring areas centered at two-hop neighboring anchor nodes of a [16], [19]). In these algorithms, more constrained regions can determine a smaller possible region of a’s location and hence improve the localization accuracy. In this paper, we introduce three RSS-constrained regions (explained later) for a. Our possible region of a’s location is the intersection of one-hop-anchor-constrained regions, historical-anchor-constrained regions, and the proposed three RSS-constrained regions.

1111

3.1 The HitBall Algorithm For ease of discussion, we have the following definitions. Let bt;i denote a beacon which is received by normal node a in time slot t from an anchor node bi . If a receives beacons b1;1 , b1;2 , b2;2 , and b2;3 in the first two time slots, then there are two historical beacons (i.e., b1;1 and b1;2 ) with respect to time slot 2, and four historical beacons with respect to time slot 3. For a node p, let Lt ðpÞ be the location of node p in time slot t. For a beacon bt;i , let Lt ðbt;i Þ be the location of the anchor node bi when it sent out bt;i , and let Lj ðbt;i Þ ¼ null if j 6¼ t. Our algorithm is a refinement of MCL and consists of three phases, sample generating phase, sample filtering phase, and location estimation phase. In the sample generating phase, a determines samples from the intersection of all one-hop-anchor-constrained regions and historical-anchorconstrained regions. However, the intersection mentioned above is difficult to calculate for resource-limited normal nodes [21]. Thus, in our algorithm, each one-hop-anchorconstrained region Xc (or historical-anchor-constrained region Xh ) is replaced with a minimum square Sc (or Sh ) which can cover Xc (or Xh ). In the sample filtering phase, normal node a filters out invalid samples by the aid of the proposed RSS constrained regions (see Section 3.2). In the location estimation phase, a estimates Lk ðaÞ to be the centroid of all valid samples (see Algorithm 1). Algorithm 1. The HitBall Algorithm /* Suppose that a wants to determine its location in slot k */ sample generating 1: For each current beacon bk;i (i.e., received from anchor bi in slot k), determine the minimum box, called Sc , which can cover the one-hop-anchor-constrained region associated with bk;i . 2: For each historical beacon bj;i0 (i.e., received from anchor b0i in slot j), determine the minimum box, called Sh , which can cover the historical-anchor-constrained region associated with bj;i0 . 3: Let rectangle I be the intersection of all Sc s and Sh s. 4: Rectangle I is divided into m squares, where m is the number of needed samples.1 The central point in each square is chosen as a sample. sample filtering 5: Construct an RSS-constrained region (see Sections 3.2.1, 3.2.2, and 3.2.3) for each beacon pair. 6: A sample is valid if it is inside all RSS-constrained regions constructed in step 5. location estimation 7: Estimate Lk ðaÞ to be the centroid of all valid samples.

3.2 RSS-Constrained Regions In this section, we use beacon pairs to derive three RSSconstrained regions. According to the types (i.e., current or historical) of beacons in a beacon pair, there are three types of RSS-constrained regions: current-current-RSS-constrained 1. The value of m depends on the size of rectangle I. Large m is adopted for large I [21].

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^ Lk ðaÞ, Li ðaÞ, and Li ðbi;1 Þ. Fig. 2. An example of Lk ðbÞ, Fig. 1. An example of CC-region of Lk ðaÞ. (a) The gray region is the CC-region of Lk ðaÞ corresponding to beacons bk;1 and bk;2 , where RSSðbk;1 Þ < RSSðbk;2 Þ. (b) The gray region is the CC-region of Lk ðaÞ corresponding to beacons bi;1 and bk;2 , when RSSðbi;1 Þ > RSSðbk;2 Þ. Clearly, it fails to cover Lk ðaÞ.

region (CC-region, for short), current-historical-RSSconstrained region (CH-region, for short), and historical-historical-RSS-constrained region (HH-region, for short). They are described in Sections 3.2.1, 3.2.2, and 3.2.3, respectively. Let RSSðbi;j Þ be the RSS value of the beacon bi;j . Let dx;y denote the distance from location x to location y. We have the following lemma: Lemma 1. Suppose that dLi ðaÞ;Li ðbi;1 Þ < dLj ðaÞ;Lj ðbj;2 Þ .

RSSðbi;1 Þ > RSSðbj;2 Þ.

Then

Note that Lemma 1 holds if the difference between RSSðbi;1 Þ and RSSðbj;2 Þ (i.e., jRSSðbi;1 Þ  RSSðbj;2 Þj) is large enough. The threshold of jRSSðbi;1 Þ  RSSðbj;2 Þj for guaranteeing that Lemma 1 holds is discussed in Section 6.6. For simplicity, in the rest of this paper, RSSðbi;1 Þ > RSSðbj;2 Þ also means that the difference between RSSðbi;1 Þ and RSSðbj;2 Þ is large enough to guarantee that Lemma 1 holds.

3.2.1 CC-Region Consider the case that normal node a receives two current beacons bk;1 and bk;2 from anchor nodes b1 and b2 , respectively, in current time slot k. By Lemma 1, Lk ðaÞ is closer to Lk ðbk;1 Þ if RSSðbk;1 Þ > RSSðbk;2 Þ, and Lk ðaÞ is closer to Lk ðbk;2 Þ otherwise. For ease of the discussion, we have the following definition. Given three points x, y, and z. Clearly, the perpendicular bisector of line segment xy determines two half-planes (one including points closer to x and the other including points closer to y). Define Hðx; y; zÞ to be one of these two half-planes which covers z. So, Lk ðaÞ 2 HðLk ðbk;1 Þ; Lk ðbk;2 Þ; Lk ðaÞÞ:

In the next section, we aim to use historical beacons to derive CH-region and HH-region.

3.2.2 CH-Region Suppose that normal node a received a current beacon bk;2 from anchor node b2 in current time slot k and a historical beacon bi;1 from anchor node b1 in previous time slot i ( dLk ðaÞ;Lk ðbk;2 Þ if RSSðbi;1 Þ < RSSðbk;2 Þ: dLk ðaÞ;Lk ðbÞ ^ ¼ dLi ðaÞ;Li ðbi;1 Þ < dLk ðaÞ;Lk ðbk;2 Þ

(1)

if RSSðbi;1 Þ > RSSðbk;2 Þ:

Below, HðLk ðbk;1 Þ; Lk ðbk;2 Þ; Lk ðaÞÞ is called the CC-region (Hcc ) of Lk ðaÞ corresponding to two current beacons bk;1 and bk;2 (see Rule 1).

^ if RSSðbi;1 Þ > RSSðbk;2 Þ, So, Lk ðaÞ should be closer to Lk ðbÞ and Lk ðaÞ should be closer to Lk ðbk;2 Þ otherwise. Hence, we have the following lemma.

Rule 1. CC-region if RSSðbk;1 Þ < RSSðbk;2 Þ then CC-region Hcc ¼ fp j dp;Lk ðbk;1 Þ > dp;Lk ðbk;2 Þ g else if RSSðbk;1 Þ > RSSðbk;2 Þ then CC-region Hcc ¼ fp j dp;Lk ðbk;1 Þ < dp;Lk ðbk;2 Þ g end if

Lemma 2. Suppose that normal node a has received beacons bi;1 and bk;2 from anchor nodes b1 and b2 in time slots i and k, respectively. Then

CC-regions are wildly used for localizing static normal nodes (e.g., [15]). Keep in mind that a CC-region is determined by two current beacons. However, due to sparse anchor nodes in mobile sensor networks, a mobile normal node usually receives rather few beacons during a time slot.




dLk ðaÞ;Lk ðbÞ ^ > dLk ðaÞ;Lk ðbk;2 Þ dLk ðaÞ;Lk ðbÞ ^ < dLk ðaÞ;Lk ðbk;2 Þ

if RSSðbi;1 Þ < RSSðbk;2 Þ; if RSSðbi;1 Þ > RSSðbk;2 Þ:

Recall that Hðx; y; zÞ is one of the two half-planes (one including points closer to x and the other including points closer to y) which covers point z. So, ^ Lk ðaÞÞ: Lk ðaÞ 2 HðLk ðbk;2 Þ; Lk ðbÞ;

(4)

HUANG ET AL.: A HISTORICAL-BEACON-AIDED LOCALIZATION ALGORITHM FOR MOBILE SENSOR NETWORKS

Fig. 5. An example RSSðbi;1 Þ > RSSðbk;2 Þ.

^ Fig. 3. The gray region denotes possible locations of Lk ðbÞ.

^ ¼ Li ðbi;1 Þ þ ðLk ðaÞ  Li ðaÞÞ (obtained by However, Lk ðbÞ rearranging Formula (2) is unknown since Lk ðaÞ  Li ðaÞ is unknown. Note that dLi ðaÞ;Lk ðaÞ  vmax  ðk  iÞ, where vmax is the maximum moving distance of a normal node during a ^ should be inside the circle C which is time slot. So, Lk ðbÞ centered at Li ðbi;1 Þ and has radius vmax  ðk  iÞ, referred to Fig. 3. That is,

of

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dLk ðaÞ;Li ðbi;1 Þ

and

dLk ðaÞ;Lk ðbk;2 Þ

when

Proof. Below we divide the proof into the following two cases: Case 1. RSSðbi;1 Þ > RSSðbk;2 Þ. By Lemma 1, dLk ðaÞ; Lk ðbk;2 Þ  dLi ðaÞ;Li ðbi;1 Þ > 0. By Formula (3), we have dLi ðaÞ;Li ðbi;1 Þ ¼ dLk ðaÞ;Lk ðbÞ ^ : Hence, we have dLk ðaÞ;Lk ðbk;2 Þ  dLk ðaÞ;Lk ðbÞ ^ > 0:

(6)

Refer to Fig. 5, we have dLk ðbÞ;L ^ i ðbi;1 Þ  vmax  ðk  iÞ

(5)

^ Accordand each point p 2 C is a possible location of Lk ðbÞ. ing to Formula (4), it is not difficult to see that Lk ðaÞ should be inside the union ofSall such HðLk ðbk;2 Þ; p; Lk ðaÞÞs, denoted by Hch , i.e., Hch ¼ 8p2C HðLk ðbk;2 Þ; p; Lk ðaÞÞ. Below, we call Hch as the CH-region of Lk ðaÞ corresponding to beacons bi;1 and bk;2 . Theorem 1 shows that when dLi ðbi;1 Þ; Lk ðbk;2 Þ > vmax  ðk  iÞ, Hch should be the region determined by an arm of hyperbola X, where hyperbola X has Li ðbi;1 Þ and Lk ðbk;2 Þ as its two focal points, and transverse axis length ¼ vmax  ðk  iÞ (referred to Fig. 4). More precisely, when RSSðbi;1 Þ > RSSðbk;2 Þ, Hch is the union of part A and part B in Fig. 4 (i.e., Hch ¼ fp j dp;Li ðbi;1 Þ  dp;Lk ðbk;2 Þ < vmax  ðk  iÞg). When RSSðbi;1 Þ < RSSðbk;2 Þ, Hch is the union of part B and part C in Fig. 4 (i.e., Hch ¼ fp j dp;Lk ðbk;2 Þ  dp;Li ðbi;1 Þ < vmax  ðk  iÞg). Theorem 1. Suppose that normal node a has received beacons bi;1 and bk;2 from anchor nodes b1 and b2 in previous time slot i and current time slot k, respectively. If dLi ðbi;1 Þ;Lk ðbk;2 Þ > vmax  ðk  iÞ, then
Hch ¼

h1 ; if RSSðbi;1 Þ > RSSðbk;2 Þ ; h2 ; if RSSðbi;1 Þ < RSSðbk;2 Þ

where h1 ¼ fp j dp;Li ðbi;1 Þ  dp;Lk ðbk;2 Þ < vmax  ðk  iÞg ði.e., the union of part A and part B in Fig. 4Þ and h2 ¼ fp j dp;Lk ðbk;2 Þ  dp;Li ðbi;1 Þ < vmax  ðk  iÞg ði.e., the union of parts B and C in Fig. 4Þ.

Fig. 4. The gray region denotes the CH-region of Lk ðaÞ, Hch , corresponding to beacons bi;1 and bk;2 when RSSðbi;1 Þ > RSSðbk;2 Þ. The hyperbola has Li ðbi;1 Þ and Lk ðbk;2 Þ as its two focal points, and transverse axis length ¼ vmax  ðk  iÞ.

dLk ðaÞ;Li ðbi;1 Þ  dLk ðaÞ;Lk ðbk;2 Þ < ðdLk ðbÞ;L ^ i ðbi;1 Þ þ dL ðaÞ;L ðbÞ ^ Þ  dLk ðaÞ;Lk ðbk;2 Þ k k ðby triangle inequalityÞ ¼ dLk ðbÞ;L ^ i ðbi;1 Þ  ðdLk ðaÞ;Lk ðbk;2 Þ  dLk ðaÞ;Lk ðbÞ ^Þ < dLk ðbÞ;L ^ i ðbi;1 Þ ðby Formulað6ÞÞ By Formula (5), we have dLk ðbÞ;L ^ i ðbi;1 Þ  vmax  ðk  iÞ. So, In other dLk ðaÞ;Li ðbi;1 Þ  dLk ðaÞ;Lk ðbk;2 Þ < vmax  ðk  iÞ. words, Lk ðaÞ should be inside Hch ¼ h1 . Case 2. RSSðbi;1 Þ < RSSðbk;2 Þ. Again, by Lemma 1, dLi ðaÞ;Li ðbi;1 Þ  dLk ðaÞ;Lk ðbk;2 Þ > 0. By Formula (3), we have dLi ðaÞ;Li ðbi;1 Þ ¼ dLk ðaÞ;Lk ðbÞ ^ : So, dLk ðaÞ;Lk ðbÞ ^  dLk ðaÞ;Lk ðbk;2 Þ > 0:

(7)

Similarly, dLk ðaÞ;Lk ðbk;2 Þ  dLk ðaÞ;Li ðbi;1 Þ < dLk ðaÞ;Lk ðbk;2 Þ  ðdLk ðaÞ;Lk ðbÞ ^  dLk ðbÞ;L ^ i ðbi;1 Þ Þ ðby triangle inequalityÞ ¼ dLk ðbÞ;L ^ i ðbi;1 Þ  ðdL ðaÞ;L ðbÞ ^  dLk ðaÞ;Lk ðbk;2 Þ Þ k k < dLk ðbÞ;L ^ i ðbi;1 Þ ðby Formulað7ÞÞ: Again, by Formula (5), we have dLk ðbÞ;L ^ i ðbi;1 Þ  vmax  ðk  iÞ. So, dLk ðaÞ;Lk ðbk;2 Þ  dLk ðaÞ;Li ðbi;1 Þ < vmax  ðk  iÞ: In other words, Lk ðaÞ should be inside Hch ¼ h2 . Since dLi ðbi;1 Þ;Lk ðbk;2 Þ > vmax  ðk  iÞ (the if-condition in Theorem 1), it is not difficult to see that Hch can be determined by an arm of hyperbola X, where hyperbola X has Li ðbi;1 Þ and Lk ðbk;2 Þ as its two focal points, and transverse axis length ¼ vmax  ðk  iÞ (referred to Fig. 4). More precisely, Hch ¼ h1 is the union of part A and part u t B, and Hch ¼ h2 is the union of parts B and C. Keep in mind that Theorem 1 is applicable when the criterion dLi ðbi;1 Þ;Lk ðbk;2 Þ > vmax  ðk  iÞ is satisfied. In fact, when the criterion above is not satisfied, Hch is useless for reducing possible region of Lk ðaÞ (i.e., useless for improving

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Fig. 6. The gray region denotes the HH-region (i.e., Hhh ) of Lk ðaÞ corresponding to beacons bi;1 and bj;2 when RSSðbi;1 Þ > RSSðbj;2 Þ.

localization accuracy). This is because Hch ¼ V when dLi ðbi;1 Þ;Lk ðbk;2 Þ  vmax  ðk  iÞ, where V is the 2D area which the WSN deployed.

3.2.3 HH-Region Consider the scenario that normal node a has received two historical beacons bi;1 and bj;2 from anchor nodes b1 and b2 at previous time slots i and j, respectively. Let X be a hyperbola with Li ðbi;1 Þ and Lj ðbj;2 Þ as its two focal points, and transverse axis length ¼ vmax  ð2k  i  jÞ (referred to Fig. 6). Below Theorem 2 show that Lk ðaÞ should be inside a region determined by an arm of hyperbola X. For ease of the discussion, the region mentioned above is called HHregion (i.e., Hhh ) of Lk ðaÞ corresponding to beacons bi;1 and bj;2 . When RSSðbi;1 Þ > RSSðbj;2 Þ, Hhh is the union of part A and part B in Fig. 6 (i.e., the set of all point ps which satisfy dp;Li ðbi;1 Þ  dp;Lj ðbj;2 Þ < vmax  ð2k  i  jÞ. When RSSðbi;1 Þ < RSSðbj;2 Þ, Hhh is the union of part B and part C in Fig. 6 (i.e., the set of all point ps which satisfy dp;Lj ðbj;2 Þ  dp;Li ðbi;1 Þ < vmax  ð2k  i  jÞ.

Fig. 7. An example RSSðbi;1 Þ > RSSðbj;2 Þ.

of

dLk ðaÞ;Li ðbi;1 Þ

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dLk ðaÞ;Lj ðbj;2 Þ

when

Case 1. RSSðbi;1 Þ > RSSðbj;2 Þ. By Theorem 1, Lj ðaÞ should be inside Hch ¼ fp j dp;Li ðbi;1 Þ  dp;Lj ðbj;2 Þ < vmax  ðj  iÞg, i.e., dLj ðaÞ;Li ðbi;1 Þ  dLj ðaÞ;Lj ðbj;2 Þ < vmax  ðj  iÞ:

(9)

Refer to Fig. 7, we have dLk ðaÞ;Li ðbi;1 Þ  dLk ðaÞ;Lj ðbj;2 Þ  ðdLj ðaÞ;Li ðbi;1 Þ þ dLk ðaÞ;Lj ðaÞ Þ  ðdLj ðaÞ;Lj ðbj;2 Þ  dLk ðaÞ;Lj ðaÞ Þ ðby triangle inequalityÞ ¼ ðdLj ðaÞ;Li ðbi;1 Þ  dLj ðaÞ;Lj ðbj;2 Þ Þ þ 2dLk ðaÞ;Lj ðaÞ < vmax  ðj  iÞ þ 2dLk ðaÞ;Lj ðaÞ ðby Formulað9ÞÞ  vmax  ð2k  i  jÞðby Formulað8ÞÞ: In other words, Lk ðaÞ should be inside Hhh ¼ h1 . Case 2. RSSðbi;1 Þ < RSSðbj;2 Þ. Again, by Theorem 1, Lj ðaÞ should be inside Hch ¼ fp j dp;Lj ðbj;2 Þ  dp;Li ðbi;1 Þ < vmax  ðj  iÞg, i.e., dLj ðaÞ;Lj ðbj;2 Þ  dLj ðaÞ;Li ðbi;1 Þ < vmax  ðj  iÞ:

(10)

So,

Rule 2. CH-region

dLk ðaÞ;Lj ðbj;2 Þ  dLk ðaÞ;Li ðbi;1 Þ

if dLi ðbi;1 Þ;Lk ðbk;2 Þ > vmax  ðk  iÞ then if RSSðbi;1 Þ > RSSðbk;2 Þ then Hch ¼ fp j dp;Li ðbi;1 Þ  dp;Lk ðbk;2 Þ < vmax  ðk  iÞg else if RSSðbi;1 Þ < RSSðbk;2 Þ then Hch ¼ fp j dp;Lk ðbk;2 Þ  dp;Li ðbi;1 Þ < vmax  ðk  iÞg end if end if

 ðdLj ðaÞ;Lj ðbj;2 Þ þ dLk ðaÞ;Lj ðaÞ Þ  ðdLj ðaÞ;Li ðbi;1 Þ  dLk ðaÞ;Lj ðaÞ Þ ðby triangle inequalityÞ ¼ ðdLj ðaÞ;Lj ðbj;2 Þ  dLj ðaÞ;Li ðbi;1 Þ Þ þ 2dLk ðaÞ;Lj ðaÞ < vmax  ðj  iÞ þ 2dLk ðaÞ;Lj ðaÞ ðby Formulað10ÞÞ < vmax  ð2k  i  jÞðby Formulað8ÞÞ:

Theorem 2. Suppose that normal node a has received two historical beacons bi;1 and bj;2 from anchor nodes b1 and b2 at time slots i and j, respectively. If dLi ðbi;1 Þ;Lj ðbj;2 Þ > vmax  ð2k  i  jÞ, then
Hhh ¼

h1 ; if RSSðbi;1 Þ > RSSðbj;2 Þ ; h2 ; if RSSðbi;1 Þ < RSSðbj;2 Þ

where h1 ¼ fp j dp;Li ðbi;1 Þ  dp;Lj ðbj;2 Þ < vmax  ð2k  i  jÞg (i.e., the union of part A and part B in Fig. 6Þ and ði.e., h2 ¼ fp j dp;Lj ðbj;2 Þ  dp;Li ðbi;1 Þ < vmax  ð2k  i  jÞg the union of parts B and C in Fig. 6Þ. Proof. Due to the speed constraint of normal nodes (i.e., each normal node has speed vmax ), we have dLk ðaÞ;Lj ðaÞ  vmax  ðk  jÞ:

(8)

Below we divide the proof into the following two cases.

In other words, Lk ðaÞ should be inside Hhh ¼ h2 . Consider the case that dLi ðbi;1 Þ;Lk ðbk;2 Þ > vmax  ð2k  i  jÞ. Since Hhh is either h1 or h2 , it is not difficult to see that Hhh can be determined by an arm of hyperbola X, where hyperbola X has Li ðbi;1 Þ and Lj ðbj;2 Þ as its two focal points, and transverse axis length vmax  ð2k  i  jÞ (referred to Fig. 6). More precisely, Hhh ¼ h1 is the union of part A and part B, and Hhh ¼ h2 is the union of parts B and C. u t

Rule 3. HH-region if dLi ðbi;1 Þ;Lj ðbj;2 Þ  vmax  ð2k  i  jÞ then if RSSðbi;1 Þ > RSSðbj;2 Þ then Hhh ¼ fp j dp;Li ðbi;1 Þ  dp;Lj ðbj;2 Þ < vmax  ð2k  i  jÞg else if RSSðbi;1 Þ < RSSðbj;2 Þ then Hhh ¼ fp j dp;Lj ðbj;2 Þ  dp;Li ðbi;1 Þ < vmax  ð2k  i  jÞg end if end if

HUANG ET AL.: A HISTORICAL-BEACON-AIDED LOCALIZATION ALGORITHM FOR MOBILE SENSOR NETWORKS

Fig. 8. Improvement by the aid of CC-region. (a) The gray region denotes Jcc . (b) The gray region is the intersection of Jcc and the CCregion of Lk ðaÞ corresponding to beacons bk;1 and bk;2 when RSSðbk;1 Þ > RSSðbk;2 Þ.

Keep in mind that Theorem 2 is applicable when the criterion dLi ðbi;1 Þ;Lj ðbj;2 Þ > vmax  ð2k  i  jÞ is satisfied. In fact, when the criterion above is not satisfied, Hhh is useless for reducing the possible region of Lk ðaÞ. This is because Hhh ¼ V when dLi ðbi;1 Þ;Lj ðbj;2 Þ  vmax  ð2k  i  jÞ, where V is the 2D area which the WSN deployed.

4

PERFORMANCE ANALYSIS

4.1 Performance of CC-Region Recall that CC-region of Lk ðaÞ is determined by two current beacons (e.g., bk;1 and bk;2 ). According to the state of the art, the possible region of Lk ðaÞ determined by the two beacons mentioned above, called Jcc , is the intersection of the onehop-anchor-constrained regions associated with beacons bk;1 and bk;2 (i.e., the intersection of the communication ranges of anchor nodes b1 and b2 in time slot k, see Fig. 8a). Refer to Fig. 8b, CC-region can improve localization accuracy, i.e., the intersection of Jcc and the CC-region has size equal to half of Jcc .

Fig. 9. Improvement by the aid of CH-region. (a) The gray region denotes Jch . (b) The gray region is the intersection of Jch and the CHregion of Lk ðaÞ corresponding to beacons bi;1 and bk;2 when RSSðbi;1 Þ > RSSðbk;2 Þ.

Section 3.2.2) which is closer to Lk ðbk;2 Þ, called armx . To prove that the intersection of Jch and CH-region corresponding to Li ðbi;1 Þ and Lk ðbk;2 Þ is smaller than that of Jch , it is sufficient to show that an vertex on armx , called c, is inside Jch (see Fig. 9b). Without loss of generality, assume that c is the intersection of armx and the line segment Li ðbi;1 ÞLk ðbk;2 Þ. Note that hyperbola X has transverse axis length ¼ vmax  ðk  iÞ. Clearly, point c is at a distance of vmax  ðk  iÞ=2 from the midpoint of line segment Li ðbi;1 ÞLk ðbk;2 Þ (¼ the midpoint of transverse axis of hyperbola X), i.e., c ¼ Li ðbi;1 Þ þ

ðdLi ðbi;1 Þ;Lk ðbk;2 Þ þ vmax  ðk  iÞÞ=2 dLi ðbi;1 Þ;Lk ðbk;2 Þ

*

ðDÞ; (11)

*

where D¼ Lk ðbk;2 Þ  Li ðbi;1 Þ. Let ai and ak be the two intersection points of the boundary of Jch and the line passing through Li ðbi;1 Þ and Lk ðbk;2 Þ (see Fig. 9b). It is not difficult

4.2 Performance of CH-Region Recall that CH-region of Lk ðaÞ is determined by one current beacon (e.g., bk;2 ) and one historical beacon (e.g., bi;1 ). According to the state of the art, the possible region of Lk ðaÞ determined by the two beacons mentioned above, called Jch , is the intersection of the one-hop-anchor-constrained region associated with beacon bk;2 (i.e., communication range of anchor node b2 in time slot k) and historicalanchor-constrained region associated with beacon bi;1 (i.e., the circle which is centered at Li ðbi;1 Þ and has radius r þ vmax  ðk  iÞ, see Fig. 9a). Below we show that when RSSðbi;1 Þ > RSSðbk;2 Þ, CH-region of Lk ðaÞ associated with beacons bi;1 and bk;2 can improve localization accuracy, i.e., the intersection of Jch and the CH-region mentioned above has size smaller than that of Jch . The correctness is proved in Theorem 3.

1115

to

see

that

ai ¼ Li ðbi;1 Þ þ

dLi ðbi;1 Þ;L ðb Þ r k k;2 dLi ðbi;1 Þ;L ðb Þ k



k;2

ðLk ðbk;2 Þ  Li ðbi;1 ÞÞ and ak ¼ Li ðbi; 1Þ þ rdþ vmax  ðkiÞ  Li ðbi;1 Þ;Lk ðbk;2 Þ

ðLk ðbk;2 Þ  Li ðbi;1 ÞÞ. According to Formula (11), c is the midpoint of line segment ai ak . Hence, the theorem follows. u t

Theorem 3. Suppose that normal node a has received two beacons bi;1 and bk;2 at time slots i and k, respectively. If RSSðbi;1 Þ > RSSðbk;2 Þ, then the intersection of Jch and CHregion of Lk ðaÞ corresponding to bi;1 and bk;2 is smaller than that of Jch .

4.3 Performance of HH-Region Recall that HH-region of Lk ðaÞ is determined by two historical beacons (e.g., bi;1 and bj;2 ). According to the state of the art, the possible region of Lk ðaÞ determined by the two historical beacons mentioned above, called Jhh , is the intersection of the historical-anchor-constrained region associated with bj;2 (i.e., the circle which is centered at Lj ðbj;2 Þ and has radius r þ vmax  ðk  jÞ) and the historical-anchorconstrained-region associated with bi;1 (i.e., the circle which is centered at Li ðbi;1 Þ and has radius r þ vmax  ðk  iÞ), see Fig. 10a. Below we show that when i  j and RSSðbi;1 Þ > RSSðbj;2 Þ, HH-region has obvious improvement in localization accuracy, i.e., the intersection of Jhh and the HH-region has size smaller than that of Jhh . The correctness is proved in Theorem 4.

Proof. Recall that when RSSðbi;1 Þ > RSSðbk;2 Þ, the CHregion of Lk ðaÞ corresponding to bi;1 and bk;2 is determined by the arm of hyperbola X (defined in

Theorem 4. Suppose that normal node a received two beacons bi;1 and bj;2 at time slots i and j, respectively. If i  j and RSSðbi;1 Þ > RSSðbj;2 Þ and 2r þ vmax  ðj  iÞ > dLi ðbi;1 Þ;

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Fig. 10. Improvement by the aid of HH-region. (a) The gray region denotes Jhh . (b) The gray region is the intersection of Jhh and the HHregion of Lk ðaÞ corresponding to beacons bi;1 and bj;2 when RSSðbi;1 Þ > RSSðbj;2 Þ.

Lj ðbj;2 Þ, then the intersection of Jhh and HH-region of Lk ðaÞ corresponding to bi;1 and bj;2 is smaller than that of Jhh . Proof. Recall that when RSSðbi;1 Þ > RSSðbj;2 Þ, the HHregion is determined by the arm of hyperbola X (defined in Section 3.2.3) which is closer to Lj ðbj;2 Þ, called armx , see Fig. 10b. To prove that the intersection of Jhh and HH-region corresponding to Li ðbi;1 Þ and Lj ðbj;2 Þ is smaller than that of Jhh , it is sufficient to show that an vertex on armx , called c, is inside Jhh , see Fig. 10b. Without loss of generality, assume that c is the intersection of armx and the line segment Li ðbi;1 ÞLj ðbj;2 Þ. Note that hyperbola X has transverse axis length ¼ vmax  ð2k  i  jÞ. Clearly, point c is at a distance of vmax  ð2k  i  jÞ=2 from the midpoint of line segment Li ðbi;1 ÞLj ðbj;2 Þ (¼ the midpoint of transverse axis of hyperbola X), i.e., c ¼ Li ðbi;1 Þ þ

ðdLi ðbi;1 Þ;Lj ðbj;2 Þ þ vmax  ð2k  i  jÞÞ=2 ! ðD Þ; dLi ðbi;1 Þ;Lj ðbj;2 Þ

(12) ! where D ¼ Lj ðbj;2 Þ  Li ðbi;1 Þ. Let ai and aj be the two intersection points of the boundary of Jhh and the line passing through Li ðbi;1 Þ and Lj ðbj;2 Þ (see Fig. 10b). It is not difficult to see that ai ¼ Li ðbi;1 Þ þ dLi ðbi;1 Þ;Lj ðbj;2 Þ ðrþvmax ðkjÞÞ dLi ðbi;1 Þ;Lj ðbj;2 Þ

 ðLj ðbj;2 Þ  Li ðbi;1 ÞÞ and aj ¼

Li ðbi;1 Þ þ r dþ vmax  ðk  iÞ  ðLj ðbj;2 Þ  Li ðbi;1 ÞÞ. Since 2r þ Li ðbi;1 Þ;Lj ðbj;2 Þ

vmax  ðj  iÞ > dLi ðbi;1 Þ;Lj ðbj;2 Þ , we have that there exists a positive real number k such that ðaj  cÞ ¼ kðc  ai Þ, which implies that c is on the line segment Li ðbi;1 ÞLj ðbj;2 Þ. Hence, the theorem follows: u t

5

EXPERIMENT

In this experiment, we implemented three algorithms HitBall, MCL, and IMCL in a mobile sensor network. The mobile sensor network consists of sensor nodes (Octopus Xs [24]), and all normal nodes have mobility.

5.1 Static Anchor Nodes The mobile sensor network consists of five sensor nodes, four static anchor nodes and one mobile normal node, and

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Fig. 11. Two experiment results. The gray objects are obstacles. The black solid lines denote the moving paths of the normal node, and dashed lines denote three protocols’ results. (a) Straight-line moving path. (b) Curve-line moving path.

is deployed in a 15 m  15 m square region of a building (referred to Fig. 11). Each anchor/normal node has communication radius 5 m. The normal node has moving distance 0:25 m per timeslot. The experiment is carried out for 100 time slots. Note that at the final time slot (i.e., time slot 100), the normal node does not receive any beacon from anchor nodes. The normal node has two kinds of moving paths: straight line (refer to Fig. 11a) and curve line (refer to Fig. 11b). In Table 1, our HitBall algorithm has high accuracy because three RSS-constrained regions (CC-region, CH-region, and HH-region) are useful in narrowing down the possible region. Note that in the straight-line scenario (Fig. 11a), the localization errors of MCL, IMCL, and HitBall at the final time slot are 0:96r, 0:4r, and 0:84r, respectively, while in the curve-line scenario (Fig. 11b), the localization errors of MCL, IMCL, and HitBall at the final time slot are 1:612r, 0:806r, and 0:538r, respectively, where r is the communication radius. In the straight-line scenario, IMCL outperforms than others at the final time slot because it adopts the assumption that normal nodes’ moving direction do not change too much in two consecutive time slots. In this curve-line scenario, IMCL does not outperform than others at the final time slot because the aforementioned assumption causes more errors.

5.2 Mobile Anchor Nodes In this experiment (referred to Fig. 12), the mobile sensor network consists of 18 mobile anchor nodes and four mobile normal nodes, and is deployed in a 16 m  33 m region. Each anchor/normal node has ideal communication radius ¼ 12 m (i.e., real communication radius 12 m) and distinct moving path. Each normal node has moving distance 2:4 m per timeslot. The experiment is carried out for 15 time slots. RSS of received beacons vary from 0 TABLE 1 Average Localization Error Algorithms HitBall MCL IMCL

Scenarios Straight-line Curve-line 0:477r 0:704r 0:488r

0:618r 1:265r 0:726r

HUANG ET AL.: A HISTORICAL-BEACON-AIDED LOCALIZATION ALGORITHM FOR MOBILE SENSOR NETWORKS

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TABLE 2 Simulation Parameters Simulation parameters

Value

Communication radius Deployment area Number of Node Mobility Model

50 m 500  500 m2 350 Random-way-point model [28]

the average location of all current beacons (received by a) as the location estimate (of a).

6

Fig. 12. Experiment scenario. The gray objects are obstacles. (a) The red/black/blue/dark blue lines denote the moving paths of the four normal nodes. (b) Black objects denote anchor nodes. The places which deploy anchor nodes are their initial locations and red lines are their moving paths.

to 60 dbm. The RSS threshold of our HitBall is set to be 40 dbm. Fig. 13 shows the average localization error of these four normal nodes. Obviously, our HitBall algorithm has high accuracy (although there do exist beacon pairs which satisfy RSS threshold, but do not satisfy Lemma 1). Note that the localization errors is high in time slots 1-3 and time slots 10-15. This is because there are more obstacles in the front part and the end part of normal nodes’ moving paths. The baseline approach uses the average location of top-k-strongest-RSS current beacons as the location estimate. More precisely, the location estimate of normal node a at slot t is the average location of Lt ðbt;1 Þ, Lt ðbt;2 Þ, . . . , and Lt ðbt;k Þ, where bt;1 , bt;2 , . . . , and bt;k are the top-k-strongest-RSS current beacons received by a at slot t. In this experiment, the value of k is set to be 7 (which is the average number of current beacons received by a normal node in a time slot, i.e., k is the average anchor node density, since each anchor node sends a beacon in a time slot). When there are fewer than k current beacons (received by a) in a time slot, the baseline approach uses

Fig. 13. Average localization error for four mobile normal nodes, when average anchor node density is approximately equal to 7.

SIMULATION

To evaluate the performance of the proposed algorithm, the algorithm is implemented in NS version 2.34. For the sake of fairness, we divide the localization algorithms into two categories: one-hop-beacon-broadcasting approaches and two-hop-beacon-broadcasting approaches. In the former, beacons are broadcasted to one-hop neighbors (of anchor nodes) only, i.e., normal nodes do not relay beacons, while in the latter, beacons are broadcasted to two-hop neighbors (of anchor nodes). The former category includes HitBall and MCL-1-hop, while the latter includes MCL, IMCL, MCL+ HitBall, and IMCL+HitBall. MCL-1-hop is the same as MCL except the beacon broadcasting. MCL+HitBall and IMCL+ HitBall are hybrid algorithms that integrate our three RSSconstrained regions into the sample filtering phase. There are 350 mobile nodes (including 322 normal nodes and 28 anchor nodes unless specified otherwise) deployed in an 500  500 m2 region. Each sensor node has communication radius 50 m, and follows the random-way point mobility model. The simulation parameters are shown in Table 2.

6.1 Communication Cost In this simulation, each beacon packet has size 32 bytes. The simulation result is shown in Table 3. Since HitBall and MCL-1-hop require one-hop beacon broadcasting only, their communication cost is OðmÞ per time slot, where m is the number of anchors. The MCL and MCL+HitBall additionally require normal nodes (which are neighboring to anchor nodes) to relay beacons, and thus the communication cost is Oðm þ m  n Þ, where n is normal node density. The IMCL and IMCL+HitBall not only require normal nodes (which are neighboring to anchor nodes) to relay beacons (from anchors), but also require normal nodes to exchange their own localization results (i.e., neighbor-constrained region). So, the total number of packets send in each of these two algorithms is Oðm þ m  n þ nÞ, where n is the number of normal nodes. Note that in our simulation, a packet carrying the localization result of a normal node has size ¼ 64 bytes (to describe the size and the shape of a possible region). 6.2 Anchor Node Density Here we consider the impact of anchor node density on localization accuracy. The anchor node density varies from 1 to 15 percent. In one-hop-beacon-broadcasting approaches, our algorithm has localization error lower than that of MCL-1-hop, see Fig. 14. In two-hop-beacon-broadcasting approaches, by the aid of our algorithm, MCL+

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TABLE 3 Communication Cost Types one-hop-beacon broadcasting two-hop-beacon broadcasting

Algorithms (per time slot) HitBall MCL-1-hop MCL MCL+HitBall IMCL IMCL+HitBall

Number of packet (per time slot) 28 28 441 441 763 763

HitBall and IMCL+HitBall have at least 28% ¼ ð2:5r  1:8rÞ=ð2:5rÞ and 38% ¼ ð2:1r  1:3rÞ=ð2:1rÞ reduction, respectively, in localization errors. Refer to Figs. 14 and 15, localization error decreases as anchor node density increases. Note that the MCL+HitBall has localization error lower than that of IMCL when anchor density ¼ 1. This is because IMCL implements the neighbor-constrained region (i.e., each normal node a should be inside the communication regions of it’s one-hop neighboring normal nodes) by the aid of the estimated locations of one-hop neighboring normal nodes (which has large localization error in extremely low anchor density environments).

Number of bytes

Communication cost

896 896 14,112 14,112 34,720 34,720

OðmÞ OðmÞ Oðm þ m  dn Þ Oðm þ m  n Þ Oðm þ m  n þ nÞ Oðm þ m  n þ nÞ

Fig. 14. The impact of anchor node density in one-hop-beaconbroadcasting approaches.

6.3 Moving Speed Let the moving speed ratio (MSR) be the ratio of the moving speed of anchor nodes to the moving speed of normal nodes. Here we have two simulations: one studies the effect of the moving speed of normal nodes when MSR is fixed and the other studies the effect of MSR when the moving speed of normal nodes is fixed. In the first simulation, MSR ¼ 2 and the moving speed of normal nodes varies from 0:5 to 3 m=s, see Figs. 16 and 17. Fig. 16 shows the results of one-hop-beacon-broadcasting approaches. Our algorithm has performance better than that of MCL-1-hop. Besides, our localization error increases in high-normal-node-speed environments. This is because the number of beacon pairs which satisfies the if-conditions of Theorems 1 and 2 decreases as the number of fast normal nodes increases. Fig. 17 shows that in two-hop-beacon-broadcasting approaches, algorithms with the aid of our three constraints (i.e., MCL+HitBall and IMCL+HitBall) have better performances (no matter whether normal nodes move fast or not). In the second simulation, the maximum moving speed of normal nodes is 0:5 m=s, and MSR varies from 1 to 5. Fig. 18 shows that in one-hop-beacon-broadcasting approaches, our algorithm has better performance in high-anchor-nodespeed environments. This is because the number of beacon pairs which satisfies the if-conditions of Theorems 1 and 2

Fig. 15. The impact of anchor node density in two-hop-beaconbroadcasting approaches.

Fig. 17. The impact of normal node speed in two-hop-beaconbroadcasting approaches.

Fig. 16. The impact of normal node speed in one-hop-beaconbroadcasting approaches.

Fig. 18. The impact of MSR in one-hop-beacon-broadcasting approaches.

HUANG ET AL.: A HISTORICAL-BEACON-AIDED LOCALIZATION ALGORITHM FOR MOBILE SENSOR NETWORKS

Fig. 19. The impact of MSR in two-hop-beacon-broadcasting approaches.

Fig. 21. The approaches.

impact

of

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DOI

in

one-hop-beacon-broadcasting

increases when the moving speed of anchor nodes is significantly faster than that of normal nodes. Fig. 19 also shows that in two-hop-beacon-broadcasting approaches, the MCL +HitBall and IMCL+HitBall have performances better than those of MCL and IMCL.

6.4 Number of Historical Beacons Fig. 20 shows the effect of the number of historical beacons on localization accuracy. Clearly, more historical beacons can make more RSS constrained regions. More RSS constrained regions result in higher localization accuracy. On the other hand, the RSS-constrained region of a historical beacon received long time ago has less improvement on localization precision. In Fig. 20, the localization accuracy of the HitBall with 10 latest historical beacons is nearly the same as that of HitBall with all received historical beacons. In other words, a normal node can achieve acceptable accuracy (e.g., nearly the same as the localization accuracy which HitBall with all received historical beacons has) by keeping a small number of historical beacons. 6.5 Irregular Radio Signal In this experiment, the communication region of an anchor/normal node is irregular and follows an irregular radio model. The degree of irregularity is denoted by DOI. Assume that the communication range under the different direction is randomly chosen from ½ð1DOIÞ  r; r [21]. In Fig. 21, the localization errors of one-hop-beacon-broadcasting approaches increase significantly as DOI increases. This is because when DOI increases, the number of beacons received by normal nodes decreases (since normal nodes within the ideal communication radius r of an anchor node b may be unable to receive beacons from b). In Fig. 22, the localization errors of two-hop-beacon-

Fig. 22. The impact of DOI in two-hop-beacon-broadcasting approaches.

broadcasting approaches also increase significantly as DOI increases. Note that under the DOI model, an anchor node which is farther from normal node a may locate within the communication region of a, while an anchor node closer to a may not. Therefore, the ring areas (normal node a should be within [r, 2r) distance from its two-hop anchor nodes) which are adopted in the sample selection phases of MCL and IMCL may result in fewer legal sample points.

6.6 Toleration of Irregular Radio Signal Due to the environmental factors (e.g., path loss, multi-path, and shadowing), the transmission distance corresponding to the given RSS value (and the given transmission power) is a range [29], [30]. Consider that every normal node has sufficient knowledge about the transmission distance ranges for various RSS values, e.g., Fig. 23. The RSS knowledge mentioned above can be obtained by the aid of anchor nodes, e.g., by exchanging beacons between anchor nodes, and determining RSSs and the transmission distances of received beacons (since anchor nodes know their locations). Based on the knowledge, a normal node can determine whether Lemma 1 holds for a given beacon pair, e.g., bi;3 and bj;5 . Without loss of generality, assume that the

Fig. 20. The impact of the number of historical beacons. The deployment area has size 300  300 m2 . (a) There are 40 anchor nodes. Normal nodes and anchor nodes have moving speed not greater than 10 and 50 m=s, respectively. (b) There are 20 anchor nodes. Normal nodes and anchor nodes have moving speed not greater than 10 and 50 m=s, respectively. (c) There are 20 anchor nodes. Normal nodes and anchor nodes have moving speed not greater than 20 and 100 m=s, respectively.

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Fig. 25. The big black circle denotes the ideal disk communication range of the black node. The red and blue curves denote the boundaries of real communication ranges of the black node in different environments. Fig. 23. The relationship of RSS and transmission distance for sensor nodes, Octopus X [24], when communication radius r ¼ 5 m. Thirteen solid lines denote thirteen distinct measurement results. Dotted lines show the upper bound and the lower bound on the measured RSS values, respectively. (a) An indoor environment with humidity 68 percent, temperature 25 C, and wind speed is 2.8 m/s. (b) An outdoor environment with humidity 80.6 percent, temperature 17.8. C, and wind speed 2.5 m/s.

transmission distances corresponding to RSSðbi;3 Þ and RSSðbj;5 Þ are in the range ½di;1 ; di;2  and ½dj;1 ; dj;2 , respectively. That is, the distance between beacon bi;3 and the target normal node (i.e., dLi ðaÞ;Li ðbi;3 Þ ) is in the range ½di;1 ; di;2  and the distance between beacon bj;5 and the target normal node (i.e., dLj ðaÞ;Lj ðbj;5 Þ ) is in the range ½dj;1 ; dj;2 , respectively. Clearly, Lemma 1 is satisfied by beacon pair ðbi;3 ; bj;5 Þ if ½di;1 ; di;2  \ ½dj;1 ; dj;2  is an empty set. For example, RSSðbi;3 Þ ¼ 20 dbm, RSSðbj;5 Þ ¼ 38 dbm, and the relationship of RSS and corresponding transmission distance is as shown in Fig. 23a. According to Fig. 23a, the transmission distances corresponding to RSSðbi;3 Þ and RSSðbj;5 Þ should be in the ranges [0.2, 2.4] and [3, 5], respectively. Since ½0:2; 2:4 \ ½3; 5 ¼ ? , beacon pair ðbi;3 ; bj;5 Þ satisfies Lemma 1 (because the closer beacon, i.e., bi;3 , has larger RSS value). Formally, we define a threshold as follows: A beacon pair ðbi;3 ; bj;5 Þ is adopted to determine the RSS constrained region if ½di;1 ; di;2  \ ½dj;1 ; dj;2  is an empty set, where ½di;1 ; di;2  and ½dj;1 ; dj;2  are ranges of the transmission distances corresponding to RSSðbi;3 Þ and RSSðbj;5 Þ, respectively. By the aid of the RSS threshold defined above, our HitBall can achieve the near-ideal result. Refer to Fig. 24. There are two curves for HitBall with DOI ¼ 0 (i.e., without irregularity of RSS), each of which denotes the ideal result when the relationship of RSS and transmission distance is a blue dashed line in Fig. 24. Obviously, with the aid of the RSS threshold defined above, the localization

Fig. 24. Localization error with the aid of our RSS threshold, when the number of anchor nodes ¼ 20, communication radius r ¼ 5 m, and the size of the deployment area ¼ 30  30 m2 . (a) The experiment result when the relationship of RSS and transmission distance is as shown in Fig. 23a. (b) The experiment result when the relationship of RSS and transmission distance is as shown in Fig. 23b.

accuracy of our HitBall (see the curve for HitBall with DOI > 0 and RSS threshold) is approximately equal to that of ideal one. Note that when sufficient RSS knowledge is unavailable, beacons which satisfy the RSS threshold mentioned above may not satisfy Lemma 1 (called the failure of the RSS threshold). To tolerate the failure of the RSS threshold, we determine the reliabilities of RSS constrained regions. Intuitively, the reliability of a RSS constrained region can be defined to be the probability that the corresponding beacon pair satisfies Lemma 1. Based on the RSS knowledge (mentioned in the first paragraph of this section), each normal node a can determine the probability that a beacon pair ðbi;3 ; bj;5 Þ satisfies Lemma 1 by itself. Without loss of generality, assume that RSSðbi;3 Þ > RSSðbj;5 Þ. Then, the probability that a beacon pair ðbi;3 ; bj;5 Þ satisfies Lemma 1 is the probability that dLi ðaÞ;Li ðbi;3 Þ < dLj ðaÞ;Lj ðbj;5 Þ . For example, RSSðbi;3 Þ ¼ 20 dbm, RSSðbj;5 Þ ¼ 30 dbm, and the relationship of RSS and corresponding transmission distance is as shown in Fig. 23a. It is not difficult to see that distance dLi ðaÞ;Li ðbi;3 Þ is in the range [0.2, 2.4] and distance dLj ðaÞ;Lj ðbj;5 Þ is in the range [1.4, 5]. Note that the intersection of [0.2, 2.4] and [1.4, 5] is [1.4, 2.4]. Clearly, dLi ðaÞ;Li ðbi;3 Þ < dLj ðaÞ;Lj ðbj;5 Þ , if one of dLi ðaÞ; Li ðbi;3 Þ and dLj ðaÞ;Lj ðbj;5 Þ is not in the intersection range [1.4, 2.4], i.e., dLi ðaÞ;Li ðbi;3 Þ < dLj ðaÞ;Lj ðbj;5 Þ , if dLi ðaÞ;Li ðbi;3 Þ is in the range [0.2, 1.4), or dLi ðaÞ;Li ðbi;3 Þ is in the range [1.4, 2.4] and dLj ðaÞ;Lj ðbj;5 Þ is in the range (2.4, 5]. When both dLi ðaÞ;Li ðbi;3 Þ and dLj ðaÞ;Lj ðbj;5 Þ are in the range [1.4, 2.4], the probability of dLi ðaÞ;Li ðbi;3 Þ < dLj ðaÞ;Lj ðbj;5 Þ can be roughly estimated as 50 percent. So, the probability of dLi ðaÞ;Li ðbi;3 Þ < dLj ðaÞ;Lj ðbj;5 Þ 2:41:4 can be estimated as follows: ð1:40:2 2:40:2  1  100%Þ þ ð2:40:2  52:4 2:41:4 2:41:4 51:4  100%Þ þ ð2:40:2  51:4  50%Þ 0:9368. That is, the RSS constrained region of beacons bi;3 and bj;5 has reliability approximately 0.9368. According to the discussion above, normal nodes can adopt the RSS constrained regions with high reliability to determine their possible regions. Besides, they can determine the reliability of their possible regions. In this paper, to tolerate irregular radio signals, a bigger disk communication range D which can cover various communication ranges (in diverse environments) is chosen, see the black circle in Fig. 25. Although such a disk may also cover the points that the anchor node (at the center of the disk) can’t communicate with, it can guarantee that the normal node (which receives its beacon) is inside the disk. In our algorithm, we use the intersection of disks (one-hop-anchor constrained regions) and the enlargement of disks (historical-anchor

HUANG ET AL.: A HISTORICAL-BEACON-AIDED LOCALIZATION ALGORITHM FOR MOBILE SENSOR NETWORKS

constrained regions), and RSS constrained regions to determine the possible region. Clearly, disks and enlargement of disks can cover the location of the normal node, and thus their intersection is useful in improving localization precision. On the other hand, if a disk communication range D0 which cannot cover various communication ranges in diverse environments (i.e., D0 is smaller than D ) is chosen, the reliability of D0 should be provided to indicate the percentage of the real communication range inside D0 . Since the communication range varies in diverse environments, the reliability of D0 could be estimated as ðr0 =r Þ2  100%, where r0 and r are the radiuses of D0 and D , respectively.

7

CONCLUSION

Range-free localization algorithms for mobile sensor networks usually suffer from sparse anchor node problem and high communication cost. To overcome the problems mentioned above, in this paper, we use RSS values of beacons to derive three constrained regions, CC-region, CH-region and HH-region. Developing the three constrained regions requires extremely low communication cost (only one-hop-beacon broadcasting). According to the theoretical analysis and simulation results, the proposed three constrained regions do indeed improve the localization accuracy.

ACKNOWLEDGMENTS G.-Y. Chang is the corresponding author.

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Guye-Yun Chang received the BS and MS degrees in computer science and information engineering from National Taiwan Normal University, Taiwan, in 1995 and 1998, respectively, and the PhD degree in computer science and information engineering from National Taiwan University, Taiwan, in 2005. She is currently an associate professor in the Department of Computer Science and Information Engineering, National Central University, Taiwan. Her research interests include cognitive radio networks, wireless sensor networks, combinatorial optimization, and system-level diagnosis.

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Gen-Huey Chen received the PhD degree in computer science from National Tsing Hua University, Taiwan, in January 1987. He joined the faculty of the Department of Computer Science and Information Engineering, National Taiwan University, in February 1987, and has been a professor since August 1992. His current research interests include data mining, wireless communication and mobile computing, graph theory and combinatorial optimization, and design and analysis of algorithms. " For more information on this or any other computing topic, please visit our Digital Library at www.computer.org/publications/dlib.