Ad Hoc & Sensor Wireless Networks, Vol. 28, pp. 97–114 Reprints available directly from the publisher Photocopying permitted by license only
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A Probabilistic Approach to Location Estimation in MANETs ˜ 1 , ROLANDO M ENCHACA -M ENDEZ1 , A NABEL P INEDA -B RISE NO E DGAR C HAVEZ2 , G IOVANNI G UZMAN1 , R ICARDO M ENCHACA -M ENDEZ3 , ROLANDO Q UINTERO1 , M IGUEL T ORRES1 , M ARCO M ORENO1 AND J. L. D IAZ - DE -L EON1 1 Instituto Polit´ecnico Nacional, M´exico E-mail:
[email protected],
[email protected],
[email protected],
[email protected],
[email protected],
[email protected],
[email protected] 2 Centro de Investigaci´on Cient´ıfica y de Educaci´on Superior de Ensenada, M´exico E-mail:
[email protected] 3 University of California, Santa Cruz, USA E-mail:
[email protected]
Received: October 6, 2013. Accepted: December 12, 2013.
Due to their simplicity, low energy cost and the fact that they do not require specialized hardware, range-free positioning algorithms are a convenient alternative to estimate the location of a set of nodes in a mobile ad hoc network (or MANET). This is particularly true when nodes are either not equipped with a GPS or the GPS system is not available. However, one of the main weakness of many of the rangefree positioning algorithms proposed up to date is that they employ trilateration which is well known to perform poorly if either the information regarding the location of the reference nodes or the distance estimates to them are noisy. Unfortunately, the latter is precisely the case in the context of range-free positioning systems for MANETs. In this paper we present the probabilistic multilateration method, a novel technique that can be used to estimate the location of a node based on the position of three or more references and noisy distance estimates to them. To asses the effectiveness of the proposed method, we present a detailed simulation-based analysis of four of the most representative range-free positioning algorithms, namely DV-Hop, Amorphous, Centroid and APIT, as well as modified versions of DV-Hop and Amorphous that employ probabilistic multilateration. We used 2D and 3D radio-signal propagation models to evaluate the performance of 2D and 3D versions of the aforementioned algorithms. Our experimental results show that the probabilistic multilateration method is superior to the traditional trilateration method.
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Keywords: Location estimation, MANET, range-free positioning.
1 INTRODUCTION A mobile ad hoc network (or MANET) is a collection of mobile nodes capable of establishing a temporal communication structure without the need of a predefined infrastructure. MANETs are self-configuring networks composed of tens to hundreds of autonomous nodes that can act as routers or endsystems [1, 2]. MANETs are flexible and resilient to individual node failures because they are specifically designed to tolerate continuous topological changes and most of their underlying algorithms (MAC and routing) are fully distributed. These properties make MANETs the ideal vehicle to support a wide class of distributed applications on the move [3, 4]. Among these applications, we can also find a growing class of ubiquitous applications [5,6] such as healthcare applications [7] or smart city applications [8] where positioning information is of prime importance. On the other hand, there are many scenarios where GPS is impractical due to its cost (e.g., sensor networks), but mainly to coverage (e.g., indoor) and energy consumption limitations [9]. In the literature, there is a large body of algorithms that have been proposed as a solution to the problem of determining the position of a MANET node. Most of these proposals can be classified as either range-based or range-free. Range-based algorithms usually measure physical attributes of the wireless signals such as the received signal strength indicator (RSSI) [10], the angle of arrival (AoA) [11, 12], the time of arrival (ToA) [11, 12] or the time difference of arrival (TDoA) [11, 12]. The main disadvantage of rangebased schemes is that they usually require every node to be equipped with additional specialized hardware which increments both deployment costs and the energy consumed by the mobile nodes. On the other hand, the range-free positioning algorithms usually employ the distance in hops to a reference (or beacon) as their main distance metric. Representatives of this type of algorithms are DV-Hop [13], Amorphous [14], Centroid [15] and APIT [16]. The main strengths of the range-free algorithms are their simplicity and low deployment costs. Their main limitations are that they usually suffer from higher localization errors than their range-based counterparts and that they usually assume dense and isotropic node distributions which is not always the case (see Figure 1(d)). Another source of error of range-free algorithms like DV-Hop and Amorphous is the fact that they employ trilateration at their last stage. It is well known that in situations where the distance estimations are not reliable, trilateration suffers from uncertainty (see Figure 1(a)) in which the three circumferences do not intersect in a single point, non-consistency (see Figure 1(b)) in which different groups of three beacons yield to different results and ambiguity (see Figure 1(c)) where the system of equations has
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FIGURE 1 Trilateration Deficiencies. (a) Uncertainty. (b) Ambiguity. (c) Non-consistency. (d) An example of an anisotropic network.
two mirror solutions [17]. Moreover, even with perfect distance estimates, trilateration can also fail if the node is colinear with any group of two beacons out of the three beacons used as references. As an alternative to the trilateration method, in this paper we present the probabilistic multilateration method, a novel technique that can be used to estimate the location of a node based on the position of three or more references and noisy distance estimates to them. The probabilistic multilateration method was specifically designed for situations where the distance estimations to the beacons are highly unreliable which is usually the case in the context of range-free algorithms where the precision of the estimations depends upon a number of factors such as the network topology itself and the quality of the topological information available to nodes. Unlike other probabilistic localization methods that either assume that the error probability distributions are known (e.g., [18]) or that take advantage of the physical properties of the radio-signals (e.g., [19]), the probabilistic multilateration method only employs the basic information available to range-free positioning algorithms to compute the nodes’ most likely location. On the other hand, despite the large body of work on distributed positioning algorithms for MANETs, most of the experimental results reported
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in the literature are based on centralized implementations that do not consider neither the effects of the physical layer nor the way in which the whole protocol stack of the nodes interact with each other. However, in realistic scenarios, packets are dropped due to radio-signal propagation effects, collisions and contention. The latter affects the quality of the topological information available to nodes and hence the precision attained by the positioning algorithms. In order to effectively assess the performance of such algorithms, in this paper we use 2D and 3D variants of fully distributed versions of DV-Hop, Amorphous, Centroid and APIT, as well as modified versions of DV-Hop and Amorphous that employ probabilistic multilateration. All these protocols were implemented in NS2 [20] and the source code can be downloaded from http://sourceforge.net/projects/posalgorithms/. The main contributions of this paper are as follows. (i) The introduction of the probabilistic multilateration method, a novel technique to estimate the position of a node which is specifically designed for situations where the distance estimations to the beacons are highly unreliable and where the error distribution is not known. (ii) Fully distributed 2D and 3D versions of DV-Hop, Amorphous, Centroid and APIT. (iii) A detailed simulation-based comparative analysis of range-free positioning algorithms using 2D and 3D radio-signal propagation models. The rest of this paper is organized as follows. Section 2 presents a succinct description of the original 2D versions of DV-Hop, Amorphous, Centroid and APIT. Section 3 introduces the Probabilistic Multilateration method. Section 4 describes the results of detailed simulation experiments used to study the performance of the 2D and 3D versions of DV-Hop, Amorphous, Centroid and APIT as well as versions of DV-Hop and Amorphous that use the proposed probabilistic multilateration method. Lastly, Section 5 presents our concluding remarks.
2 RANGE-FREE POSITIONING ALGORITHMS In this section we present the description of four of the most representative range-free positioning algorithms for MANETs. All these algorithms assume that only a fraction of the nodes, known as beacons or anchors, know their location. The locations of these nodes are used as reference to estimate the position of the remaining nodes. DV-Hop [13] and its variants (e.g., [21, 22]) work similar to distancevector routing protocols [23] like DSDV [24] in the sense that some nodes (the beacons in the case of DV-Hop) periodically flood the network with control packets that establish an ordering on the nodes based on their distances in hops to the beacons and inform every node in the connected component about
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the current position of each beacon. This way, every node becomes aware of the location of the beacons and of its shortest distance in hops to them. Then, every beacon uses the information received from the other beacons to compute their estimated average length of one hop (average size of one hop in DV-Hop’s terminology). In a second phase, beacons disseminate their estimations of the average length of one hop so that nodes can use these estimates to compute their own estimations of their Euclidean distances to the beacons. With the position of at least three beacons and estimates of Euclidean distances to them, nodes can use trilateration to estimate their locations. In Amorphous [14], the average length of one hop is computed off-line based on the assumption that the node density is homogeneous and constant through time and hence, that it can be computed a priori. As in the case of DV-Hop, beacons flood the network in order to publish their locations and to establish an ordering over the nodes. Then, nodes use trilateration to estimate their own locations. With Centroid [15], beacons broadcast location messages to their one-hop neighbors and nodes simply estimate their location as the centroid of the polygon with vertices at the beacons. In APIT (Approximate Point-In Triangulation) beacons also broadcast location messages to their one-hop neighbors. Then, nodes generate all the 3-combinations of the known beacons and apply an approximate PIT (Point-In Triangulation) [16] test to determine if they are located inside of each triangle. Lastly, nodes employ a grid-scan algorithm [25] to determine the area defined by the maximum number of intersecting triangles. The estimated location is computed as the centroid of this area. The Appendix A of the Supplementary Material presents fully distributed formulations of the 3D versions of DV-Hop, Amorphous, Centroid and APIT.
3 PROBABILISTIC MULTILATERATION The probabilistic multilateration method is a technique aimed to estimate the location of a point, based on the position of three or more references and noisy distance estimates to them. The objective of the proposed method is to alleviate the deficiencies of the trilateration method such as uncertainty, non-consistency and ambiguity [17] that arise when the information regarding either the location of the references or the distance estimates to them is not accurate. For each reference β, the probabilistic multilateration method defines a probability density function Pβ (x, y) that assigns a probability of finding a node to every point in the plane (or space). Since in the case of range-free positioning algorithms the error probability distribution is not known, the probabilistic method employs a generic density function based
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on a Gaussian function with parameters that depend on the location of the reference, a distance estimate to the reference, and the hop distance to that reference. Now, since the actual positions of the references and nodes are independent, we can assume that the probability of finding a node in a given (x, y) point, defined by function Pi (x, y), is also independent of the probability defined by another function P j (x, y) for the same point. Therefore, as shown in (1), the probability of finding a node in a point (x, y), given the information provided by a set B of n references can be computed as the product of the probabilities defined by the n probability density functions.
PB (x, y) =
n
Pi (x, y)
(1)
i=1
Lastly, the position estimated by the probabilistic multilateration method will be the (x, y) point where PB (x, y) has its maximum.
3.1 Position Estimation in 2D For the 2D case, we propose the probability density function defined in (2) which is based on Gaussian functions with parameters μβ and σβ . In (2), xβ and yβ are the coordinates of the position of reference β, μβ is the estimated distance to reference β and σβ is proportional to the estimation error of the distance to reference β. For the case of range-free positioning algorithms, σβ is a function of the hop distance to reference β and of the radio range. Nβ is a normalization constant. As it can be seen in (2), the probability of finding a node in a given point in the plane is simply a function of its distance to the references. Figure 2(a) shows a plot of this probability density function where it can be observed that is has an infinite number of maximums located over the circumference with center in (xβ , yβ ) and radio μβ . 1 − 12 Pβ (x, y) = e Nβ
√(x−xβ )2 +(y−yβ )2 −μβ 2 σβ
(2)
With information from a set B of three or more references, we can substitute (2) into (1) to get the probability density function (pdf) defined in (3). Figure 2(b) shows a plot of three overlapping pdfs defined by the information of three references whereas Figure 2(c) shows a plot of the product of three pdfs which has a single maximum that will be the position estimated by the
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FIGURE 2 Probabilistic multilateration using Gaussian functions. (a) Probability density function defined by a single reference β. (b) Three overlapping probability functions. Each one defined by a different reference. (c) Probability density function of the localization of a node given three references. (d) Four overlapping probability density functions.
probabilistic multilateration method.
PB (x, y) =
√ 2 n 2 i ) −μi 1 − 12 (x−xi )2 +(y−y σi e Ni i=1
(3)
In order to find a maximum of (3), we can compute the solutions to the partial derivatives ((4) and (5)) of (3). However, since (4) and (5) equal zero only if the last summation equals zero, we just have to solve (6) and (7) simultaneously. The latter is the same as solving (8) where (6) and (7) are squared and then added. Please note that the values of the normalization constants are not needed to find the maximum of (3). This is important because it greatly reduces the complexity of the method. √(x−x )2 +(y−y )2 −μ 2 n n 1 i i i ∂ 1 i=1 − 2 σi PB (x, y) = e ∂x N i i=1 n ( (x − xi )2 + (y − yi )2 − μi )(x − xi ) − σi (x − xi )2 + (y − yi )2 i=1
(4)
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√(x−x )2 +(y−y )2 −μ 2 n n 1 i i i ∂ 1 i=1 − 2 σi PB (x, y) = e ∂y Ni i=1 n ( (x − xi )2 + (y − yi )2 − μi )(y − yi ) − σi (x − xi )2 + (y − yi )2 i=1
(5)
n ( (x − xi )2 + (y − yi )2 − μi )(x − xi ) 0= σi (x − xi )2 + (y − yi )2 i=1
(6)
n ( (x − xi )2 + (y − yi )2 − μi )(y − yi ) 0= σi (x − xi )2 + (y − yi )2 i=1
(7)
n ( (x − xi )2 + (y − yi )2 − μi )(x − xi ) 2 0= + σi (x − xi )2 + (y − yi )2 i=1 n ( (x − xi )2 + (y − yi )2 − μi )(x − xi ) 2 σi (x − xi )2 + (y − yi )2 i=1
(8)
Lastly, we can employ a root-finding algorithm to compute a root of (8). For the sake of simplicity, we used the Newton method [26] with the centroid of the positions of the references as starting point. It is important to point out that if (8) has more than one root, the Newton method can find any of them. However, our experimental results showed that even when arbitrarily selecting any local maximum of (3), the probabilistic multilateration method provides a better position estimation than the one provided by the trilateration method. It is also worth noticing that if the derivative of (8) is continuous and nonzero in the neighborhood of the root, the Newton method converges quadratically. 3.2 Position Estimation in 3D The probabilistic multilateration method can be easily extended to estimate the position of a node in a three dimensional space based on the information provided by a set B of four or more beacons. In three dimensions, the probability density function based on Gaussian functions, defined by a set of n beacons is given by (9). √ n 1 − 12 (x−xi )2 +(y−yσi )2 +(z−zi )2 −μi 2 i PB (x, y) = e Ni i=1
(9)
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From (9), we can proceed analogously to the two dimensional case and compute the partial derivatives with respect to x, y and z. Then, the three equations have to be solved simultaneously to find the points that maximize (9). As in the two-dimensional case, this problem is equivalent to that of finding the roots of (10) that can also be done using the Newton method [26]. n ( (x − xi )2 + (y − yi )2 + (z − z i )2 − μi )(x − xi ) 2 0= + σi2 (x − xi )2 + (y − yi )2 + (x − xi )2 i=1 n ( (x − xi )2 + (y − yi )2 + (z − z i )2 − μi )(z − z i ) 2 + σi2 (x − xi )2 + (y − yi )2 + (y − yi )2 i=1 n ( (x − xi )2 + (y − yi )2 + (z − z i )2 − μi )(z − z i ) 2 σi2 (x − xi )2 + (y − yi )2 + (y − yi )2 i=1
(10)
3.3 Sensitivity to Collinearity/Coplanarity In this section we present a simple sensitivity analysis of the trilateraton and probabilistic multilateration methods to the level of collinearity and coplanarity among references. For this analysis we employed the most common instantiation of the trilateration method which basically consist of solving the system of Equations (11) by means of (12). Both methods were implemented using standard double-precision floating-point arithmetic operations. The results presented in Figure 3 are based on exact distances to the references. (x − xβ1 )2 + (y − yβ1 )2 = dβ21 (x − xβ2 )2 + (y − yβ2 )2 = dβ22 (x − xβ3 ) + (y − yβ3 ) = 2
2
(11)
dβ23
2
(dβ − dβ2 ) − (xβ2 − xβ2 ) − (yβ2 − yβ2 ) 2(yβ2 − yβ1 )
2 1 2 1 2
1
(d 2 − d 2 ) − (x 2 − x 2 ) − (y 2 − y 2 ) (yβ − yβ )
3 1 β1 β3 β1 β3 β1 β3
x=
2(xβ2 − xβ1 ) 2(yβ2 − yβ1 )
2(xβ − xβ ) 2(yβ − yβ )
3 1 3 1
2(xβ2 − xβ1 ) (dβ2 − dβ2 ) − (xβ2 − xβ2 ) − (yβ2 − yβ2 )
1 2 1 2 1 2
2(xβ − xβ ) (d 2 − d 2 ) − (x 2 − x 2 ) − (y 2 − y 2 )
3 1 β1 β3 β1 β3 β1 β3
y=
2(xβ2 − xβ1 ) 2(yβ2 − yβ1 )
2(xβ − xβ ) 2(yβ − yβ )
3 1 3 1
(12)
For the 2D case, we fixed the positions of two references and of the node that is trying to estimate its location. Then, as shown in Figure 3(a), the position of the third reference is changed until it is collinear with the other two
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FIGURE 3 Sensitivity to Collinearity/Coplanarity. (a) Positions of three references and a node in two dimensions. (b) Location error when moving the third reference. (c) Positions of four references and a node in three dimensions. (d) Location error when moving the fourth reference.
references. The location error incurred by the two methods is shown in Figure 3(b). From the figure we can observe that the probabilistic multilateration method is far less sensitive to collinearity than trilateration whose location error increases rapidly as the third reference becomes collinear with the other two and the denominators in (12) tend to zero. On the other hand, even when the three references are collinear (see Figure 4(a)), a maximum of the probability density function defined by the probabilistic multilateration method is located close by the real node position. In this situation, however, this probability density function is not unimodal (see Figure 4(b)) and hence the probabilistic multilateration method could estimate that the node is located in the other local maximum. Please note that in these cases, all the local maximums are located inside of the polygon of smallest area that contains all the circumferences defined by each probability density function of each individual reference. Therefore, the location estimated by the probabilistic method can not be arbitrarily bad as it is the case with the multilateration method. For the 3D case, we fixed the positions of three references and of the node that is trying to estimate its location. Then, as shown in Figure 3(c),
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FIGURE 4 Probability density functions of the location of a node when the references are collinear. (a) Probability density functions defined by each individual reference. (b) Probability density function given the information of the three references.
the position of the fourth reference is changed until it is coplanar with the other three references. From Figures 3(c-d) we can observe that the results are analogous to the those of the 2D case.
4 EXPERIMENTAL RESULTS We present simulation results comparing the distributed versions of DV-Hop, Amorphous, Centroid and APIT, as well as modified versions of DV-Hop and Amorphous that use the probabilistic multilateration (PM) method. In our experiments, all the protocols run on top of IEEE 802.11 DCF [27] with 802.11b as physical layer. We use location error, coverage and network overhead as our performance metrics. The location error is computed as the distance between the real node location and the location estimated by the algorithm. All the graphs report this distance in terms of the radio range. The coverage is computed as the proportion of the nodes that are able to estimate their position and the network overhead is the average total number of packets transmitted by nodes and beacons. We used the discrete event simulator NS2 [20] version 2.34, that provides realistic simulations of the physical layer, and a well-tuned version of IEEE 802.11 DCF. Each simulation was run for ten different seed values. All the protocols use the same period of three seconds to refresh their location information. For all the experiments, the value of σ used in the implementation of the probabilistic multilateration method was set to 0.25×estimated average hop length. We employ random waypoint as our mobility model. The node speeds used in the simulation experiments vary from 1 to 20 m/s (or 3.6 to 72 Km/h) to cover pedestrian and vehicular speeds.
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Section 4.1 presents results when nodes are located in a two-dimensional plane and the radio-signals propagate according to the two-ray ground propagation model and Section 4.2 when nodes are located in a three-dimensional space and radio-signals propagate according to the Durkin’s [28] propagation model that considers the effects of irregular terrains. 4.1 Positioning in Two Dimensions In this section we present the results of a series of experiments where we evaluate the effect of node mobility and number of beacons (nodes that know their position a priori) over the performance of the 2D versions of the different protocols when radio-signals propagate according to the two-ray ground propagation model and mobile nodes follow the random waypoint mobility model in a plane of 1000m × 1000m. Both, regular nodes and beacons have a transmission range of 250m. Table 1 lists the details of the simulation environment. In the implementations of DV-Hop and Amorphous used in these experiments, nodes select the three nearest beacons in hops which are not collineal. Figures 5(a-i) show the results obtained in three different scenarios where the number of mobile nodes is increased. In Scenario 1-2D, nodes and beacons are static; in Scenario 2-2D, beacons are static and nodes are mobile and; in Scenario 3-2D, beacons and nodes are mobile. In all the scenarios the number of beacons is increased from 3 to 31. Figures 5(a-c) show the location error attained by the protocols as the number of mobile nodes is increased. From the figures, we can observe that across the three scenarios the location error attained by Centroid and APIT is less than half of those of DV-Hop and Amorphous. At the same time Amorphous consistently outperformed DV-Hop by incurring in location errors of close to half the magnitude of that of DV-Hop. A detailed analysis of the results showed that the on-line estimation of the average hop length performed by DV-Hop is much more sensitive to irregular topologies and node mobility than the offline approach taken by Amorphous. It is also important to highlight that the reduced error attained by Centroid and APIT is due to
Simulation area Total nodes Physical layer model Transmission range Transmission rate Min.-Max. Vel.
1000m × 1000m 100 IEEE 802.11b 250m 11000000bps 1-20m/s
TABLE 1 Simulation Environment (2D).
Simulation time Number of beacons Propagation model Transmission. power Mobility model Pause time
300 seconds 3,7,15,31 Two-ray ground 0.28 W Random waypoint 10 seconds
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FIGURE 5 Effect of mobility and number of beacons when using a 2D propagation model.
their conservative approach of only using beacons located one-hop away from the nodes. This approach, however, comes with the prize of a reduced coverage as shown by Figures 5 (d-f). Figures 5 (d-f) also show that the PIT-test performed by APIT is quite sensitive to node mobility. When nodes move, almost all the PIT-test failed and hence nodes were not able to estimate their positions. The latter is reflected by the fact that the coverage attained by APIT on Scenarios 2-2D and 3-2D is almost zero. From Figures 5(a-c) it can also be observed that as the number of beacons increases, the estimation error attained by DV-Hop and Amorphous decreased. These results were expected because one of the main components of the overall estimation error for both protocols is the error incurred when computing the average hop length, which is very sensitive to the distance in hops from nodes to beacons. As the number of beacons increases, the probability of finding close-by beacons also increases and therefore the estimation error decreases. Another intuitive result shown by Figures 5(a-c) is the fact that the estimation error attained by Centroid and APIT is less sensitive to the number of beacons. On the other hand, as the number of mobile nodes is increased, the estimation error of all the protocols also increases. This is more apparent in protocols such as DV-Hop and Amorphous that rely more heavily
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on topological information because mobility tends to reduce the accuracy of the topological and location information available to nodes. Figures 5(g-i) show the overhead induced by the different protocols. These experimental results are consistent with the analysis of the network complexity presented in the Appendix A of the supplementary material. As expected, the overhead increased as the number of beacons increased because beacons initiate the dissemination of control information. In the case of Centroid and APIT this information is transmitted to one-hop neighbors only, whereas DV-Hop and Amorphous flood the whole network making them much more costly in terms of energy consumption, contention and congestion. The most resource consuming protocol is DV-Hop because it has to flood the network twice every refreshing period, the first time to inform every node in the network about its distance in hops to the beacon as well as the current location of the beacon, and a second time to disseminate the average hop length. It is important to point out that the overhead also has a negative impact over the location error attained by DV-Hop and Amorphous because more packets are lost due to collisions and queue congestion. The latter induces errors when computing the distances in hops towards the beacons. From Figures 5(a-c) we can also notice the remarkable performance gains attained by simply replacing the traditional trilateration method with the proposed probabilistic multilateration method. In the figures, the graphs labeled “DV-Hop (PM)” and “Amorphous (PM)” correspond to versions of DV-Hop and Amorphous that use probabilistic multilateration. DV-Hop (PM) attained an estimation error from half to one fourth of that of DV-Hop whereas Amorphous (PM) consistently attained smaller estimation errors than Amorphous. In fact, the precision delivered by DV-Hop (PM) and Amorphous (PM) is comparable to that of APIT and Centroid but with a much better coverage. Lastly, since DV-Hop (PM) and Amorphous (PM) use exactly the same distributed algorithms as DV-Hop and Amorphous, respectively, they incur in exactly the same control overhead. These results confirm our hypothesis that the probabilistic method effectively reduces the impact of the uncertainty, non-consistency, and ambiguity, which negatively affect the performance of the traditional trilateration method. 4.2 Positioning in Three Dimensions In this section we present the results of a series of experiments where we evaluate the effect of node mobility and number of beacons over the performance of the 3D versions of the different protocols when radio-signals propagate according to the Durkin’s [28] propagation model and mobile nodes follow the random waypoint mobility model in a terrain of 1000m × 1000m × 200m. The terrain is of ravine type as provided by [29]. For these experiments we used the 2.31 version of NS2 because Durkins model was
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FIGURE 6 Effect of mobility and number of beacons when using a 3D propagation model.
implemented and evaluated for that version. Similar to the 2D case, the 3D versions of DV-Hop and Amorphous used in these experiments select the four nearest beacons in hops which are not coplanar. Unless otherwise stated, the remaining simulation parameters are the same as the those presented in Section 4.1. Figures 6(a-i) show the results obtained in three different scenarios where the number of mobile nodes is increased. In Scenario 1-3D, nodes and beacons are static; in Scenario 2-3D, beacons are static and nodes are mobile and; in Scenario 3-3D, beacons and nodes are mobile. In all the scenarios the number of beacon is increased from 3 to 31. Figures 6(a-c) show the location error incurred by the different protocols. As can be noticed from the figures, the trends are similar to those observed in the 2D case, namely, the performance of Centroid is much better than those of DV-Hop and Amorphous, being DV-Hop the worst performing algorithm. APIT performs well only on static conditions and when the number of beacons is larger than seven. The figures also show that the performance gains attained by DV-Hop (PM) and Amorphous (PM) over the basic versions of DV-Hop and Amorphous are significant. Similar to the 2D case,
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DV-Hop (PM) attained an estimation error from half to one fourth of that of DV-Hop whereas Amorphous (PM) consistently attained smaller estimation errors than that of Amorphous. The latter is particularly apparent in Scenario 3-3D where the localization error attained by Amorphous (PM) is from half to one third of that of Amorphous. As shown in Figures 6(d-f), the coverage attained by the algorithms was lower than that of the 2D experiments. The reasons are as follows. Since the network is composed of the same 100 nodes placed in a 1000m × 1000m× 200m volume, the node density and the average one-hop neighborhood size of these scenarios are lower than those of the 2D scenarios. The latter also implies a reduction in the average number of beacons in the one-hop neighborhood of nodes which particularly impacts the performance of the algorithms, such as APIT, that require position information from at least four beacons located one-hop away. For this reason, nodes running APIT were able to estimate positions only on the static scenario (Figure 6(d)) with 15 or more beacons. In general, the reduced node density implied an increase in the average distance from nodes to beacons which reduces the quality of the topological information available to nodes. From Figures 6(d-e) we can also notice that in Scenarios 1-3D and 2-3D, Amorphous (PM) attained consistently more coverage than Amorphous. A detailed analysis revealed that in situations where Amorphous nodes did not find enough non-coplanar beacons, Amorphous (PM) nodes were still able to estimate position with a reasonably good precision. Lastly, as shown in Figures 6(g-i), the overhead induced by the 3D versions of the algorithms is completely analogous to that of their 2D counterparts.
5 CONCLUSIONS We introduced the probabilistic multilateration method for estimating the position of a node based on the position of three or more references and noisy estimates of distances to them. Unlike previous probabilistic approaches, the probabilistic multilateration is well suited for range-free positioning algorithms for MANETs because it does not make strong assumptions about the properties of the distance estimates used to compute the nodes locations. Our method has four main advantages over the traditional trilateration method. (i) It eliminates the non-consistency problem because an arbitrary number of references can be used at the same time to estimate the position of a node. (ii) It is more resilient to collinearity (coplanarity) because the probability density function defined in (3) does not have local maximums outside of the polygon (polyhedron) of minimum area (volume) that contains the
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circumferences (spheres) defined by the probability density functions of the individual references and hence, the probabilistic multilateration can not provide arbitrarily bad estimations as it is the case of the trilateration method. (iii) The probabilistic mutilateration method performs well even if the circumferences (spheres) defined by the references do not intersect or intersect at many points. (iv) With perfect distance estimates, the probabilistic multilateration method performs just as well as trilateration with no extra computational cost. These properties were confirmed by our experimental results that showed that the protocols that use probabilistic multilateration clearly outperform those that use trilateration under a wide variety of conditions. We also presented a detailed simulation-based analysis of four of the most representative positioning algorithms for MANETs, namely, DV-Hop, Amorphous, Centroid and APIT. In our experimental analysis we considered scenarios with different radio-signal propagation models, as well as increased number of beacons and node mobility. Our results reveled that due to its simplicity, Centroid is very robust to the different network conditions. However, since Centroid only uses references located within the radio range of nodes; it attains significantly lower coverage that those of DV-Hop and Amorphous which can estimate positions based on references located many hops away from nodes. Our results also showed that the PIT test is quite sensitive to node mobility and that nodes running APIT are not able to estimate their positions when either nodes or beacons are mobile. It is also apparent that DV-Hop and Amorphous suffer from the deficiencies of the trilateration method which was not designed to cope with noisy distance estimates to the references. In general, protocols that flood the whole network such as DV-Hop and Amorphous attain higher coverage than protocols like Centroid and APIT that transmit to one hop neighbors only, but they are also more costly in terms of bandwidth utilization. ACKNOWLEDGMENTS This work was sponsored in part by a grant from the University of California Institute for M´exico and United States (UC MEXUS) and the Consejo Nacional de Ciencia y Tecnolog´ıa de M´exico (CONACyT), by the Instituto de Ciencia y Tecnolog´ıa del Distrito Federal (ICyT-DF) and by Instituto Tecnol´ogico de Matamoros. REFERENCES [1] Chlamtac I., Conti M., Liu J.J.N. Ad Hoc Networks 1 (2003) 13. [2] Hoebeke J., Moerman I., Dhoedt B., Demeester P. Journal of the Communications Network 3 (2004) 60.
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