Connectivity Based Location Estimation Scheme for Wireless Ad Hoc Networks

Transcription

1 Connectivity Based Location Estimation Scheme for Wireless Ad Hoc Networks Niveditha Sundaram Parameswaran Ramanathan Department of Electrical Computer Engineering University of Wisconsin-Madison Madison, WI ( Abstract In this paper, we propose a method to estimate the location of a mobile user We assume that mobile users carry portable devices such as laptops or personal digital assistants equipped with off-the-shelf stard wireless interfaces The infrastructure has a few fixed nodes whose locations are assumed to be known The location estimates rely on neighborhood relationships gathered by each user through message exchanges over a wireless ad hoc network We show that by incorporating non-neighbor constraints, one can substantially improve the accuracy of location estimation as compared to only utilizing neighbor relationships We also compare our approach without the non-neighbor constraints with the results from a paper in literature based on the same model Empirical evaluation of our algorithm for several numerical examples are also included I INTRODUCTION There is a growing dem for supporting location-based services to mobile users These services depend on each user knowing his/her current geographic location For example, a mobile user at an airport may want a route to the nearest food court from his/her current location One possible way for a user to keep track of his/her current location is through Global Positioning System (GPS However, GPS typically works only in outdoor environments Also due to issues such as costs, form factors, energy consumption, etc, providing GPS support may not be feasible even in outdoor environments The other approach, especially in indoor environments, is to rely on ultrasound beacons to accurately localize a user [1], [2], [3], [4] These approaches rely on specialized listening hardware at the mobile user to measure the one-way propagation delay of an ultrasound pulse thereby measure the distance to the fixed beacon It has been argued that the beacons the listening devices are not very expensive [2] Despite these advantages, we contend that they are an overkill for many applications For example, mobile users in an airport may not be interested in carrying these listening devices just to accurately determine their location Therefore, in this paper, we focus on The work reported here was supported in part by US Army Research grant DAAD under a subrecipient agreement S01-24 from The Pennsylvania State University Any opinions, findings, conclusions or recommendations expressed in this publication are those of the author(s do not necessarily reflect the views of the US Army Research Office location estimation using off-the-shelf wireless hardware such as Bluetooth In particular, we assume that the mobile users carry a portable device such as a personal digital assistant (PDA or a laptop Furthermore, we assume that there are fixed devices in the infrastructure to offer a variety of services to mobile users over a wireless ad hoc network For example, an airport may deploy Bluetooth access points to provide connectivity information services to its users Our goal is to estimate each user s location using a software application running in the portable devices in the infrastructure without any specialized hardware support other than the stard wireless connectivity The closest related research is the one by Doherty et al in [] As in this paper, the scheme in [] relies exclusively on connectivity-induced constraints It uses a convex optimization based approach to estimate a node s location on a twodimensional surface The constraints in [] are solely based on neighborhood relationships In contrast, in this paper, we also consider non-neighbor relationships to reduce the estimation error The primary problem in considering non-neighbor relationships is that sometimes, physical neighbors may appear to be non-neighbors due to reasons such as poor channel quality obstruction between the users If this happens, the proposed algorithm may produce an incorrect location estimate for a user Arguably, there are applications which can tolerate occasional incorrect location estimates Also, one can use historic information about a user to detect reduce the likelihood of an incorrect estimate For applications that cannot tolerate an incorrect estimate, we also adapt our scheme to ignore nonneighbor relationships We can show that our approach, after ignoring non-neighbor relationships, is provably superior to the one in [] In fact, our approach is an iterative algorithm the solution in [] is equivalent to the result from the first iteration albeit the solution method is quite different The rest of this paper is organized as follows In Section II we formulate the problem discuss our solution Proof of correctness is given in Section III Implementation issues are discussed in Section IV Results of an empirical evaluation of our scheme are presented in Section V The paper concludes in Section VI Bluetooth is a trademark owned by TelefonaktiebolagetL M Ericsson, Sweden

2 > II PROPOSED SOLUTION The problem addressed in this paper can be stated as follows We are given a set of mobile users fixed nodes that can communicate over a wireless ad hoc network The locations of the fixed nodes are known The mobile users observe other users identify neighborhood relationships This information is used to determine the set of all possible locations for each user Our solution approach is based on the following software architecture Each mobile user runs a software client, called LocEst Client, at the application layer This client is responsible for participating in the location estimation algorithm providing the node s location estimate to other applications that need it One of the fixed nodes implements a software server called LocEst Server, that implements the proposed centralized location estimation algorithm The server interacts with the clients provides each client a location estimate of its user LocEst Client: The LocEst Client, at each node, periodically broadcasts a message of the form I am at node This message is only heard by nodes in the radio range of node The LocEst Client at node locally maintains a neighbor set which contains the nodes that are within its radio range Specifically, when node hears a message of the form I am at node, it adds node to the set Likewise, if node does not hear the I am at node message for a preset timeout period, it removes node from the set The LocEst Client periodically sends its neighbor list to the LocEst Server It also periodically queries LocEst Server to receive the current location estimate for its node LocEst Server: The LocEst Server receives the neighbor set from all the nodes Neighbor sets from different nodes can be inconsistent due to incomplete or stale information in each node s neighbor set The inconsistencies can be removed by incorporating certain implementation details in gathering the neighbor sets We discuss these issues in Section IV For now, assume that LocEst Server implements a scheme to make the neighbor sets accurate consistent That is, node belongs to node s neighbor set, if only if, it is within the radio range of node The LocEst Server then executes the algorithm shown in Figure 1 The algorithm produces a location estimation for each node Upon receiving a request from LocEst Client, the LocEst Server responds with the location estimate The basic idea of the algorithm is as follows Let denote the set of nodes whose location is exactly known The algorithm uses multiple iterations to converge on a location estimate for each node The location estimate is not a single number but, given the neighborhood relationships, it is the set of all possible locations where the node can be present If the neighbor set is accurate, Theorem 1 shows that the correct location of each node is always contained in its location estimate Let denote the location estimate of node in the iteration In line 1, the algorithm initializes for each node as follows If, then is set to the known location Otherwise, equals the entire region over which all the nodes are expected to be located In each iteration, the algorithm computes two sets,, for each node Given the current estimate of node s locations, ie,, the set contains the set Algorithm LocEst With NonNeigh 1 2 3! #%$'& if ( *,+-0/1/ + *,+-0/1/ + otherwise 4 done = FALSE 9: 6 while (done == FALSE do 7 done = TRUE 8 foreach node < 9,+-=?> 10 $ /2/4367& if ( $ /2/4867& if otherwise otherwise /1/ + >C/2/D36 /1/ + >C/2/D86 & EGFIHKJML,N*O RQEHTSH JVU L,NWO P Z[ if ( 13 done = FALSE 14 endfor endwhile then such that such that Fig 1 Proposed algorithm for location estimation exploiting non-neighbor information of all possible locations where a neighbor of can be located C6B Likewise, given, the set contains the set of all possible locations where a non-neighbor of can be located \?B After computing, is refined to obtain a new estimate The algorithm continues to iterate until the location estimates cannot be refined any further To better underst the idea in the refinement step, let us look at lines 9, 10, 11 of the algorithm in more detail Line 9 states that a neighbor of node can possibly be at location +, XP RY ]^ thus + belongs to, if there is a location > which is within the radio range of + This is because, node may be at location >, in that case, a neighbor of can be at location + Similarly, line 10 states that a non-neighbor of node can possibly be at location + only if there is a location [_B which is beyond the radio range Again, this is because node may be at location > in that case, a nonneighbor of can be at location + Z9_B Line 11 refines as follows Node can be at location +, thus +, if only if all its neighboring non-neighboring relationships allow to be at location + That is, is the intersection of all locations permitted by the neighbors non-neighbors of node Example: For simplicity, consider a one-dimensional scenario where all nodes locations are in the range %`B Suppose 9_abc%`#dec, P 96Bfgheci`\jkc that, the radio range is 1 Then, the neighbors of cannot be located in (0, 3 (90, 100 because they would be out of range from every possible location for Furthermore, the neighbors could lie in (3, 90 because there is at least one location for in (0, 7 that is within the radio range Consequently, Klhci`Mme Through similar arguments, one can show E:n%`#ce that P Now consider the non-neighbors of A non-neighbor of

3 can be located in (0, 3 (90, 100 because it is out of range from every possible location for Furthermore, the interval (3, 90 cannot be ruled out as a possible location for a nonneighbor because it is out of range for at least one location for For instance, a non-neighbor can be located at 40 because node may be located at 7 location 40 is out of range Ei`Be from 7 Thus, Finally, consider the non-neighbors of In this case, line i`mh]`c%`b 10 will determine [P &, ie, a nonneighbor of cannot be located in (30, 0 This is because every location in (30, 0 is within radio range of every possible location for Therefore, no matter where is, a nonneighbor of cannot be in (30, 0 III PROOF OF CORRECTNESS The following lemmas theorems can be proved about the algorithm in Figure 1 The proofs are shown in the Appendix Theorem 1: For all nodes, for all %`e`#n%` :, where is the true location of node The above theorem implies that correct location of each node is always contained in the location estimate determined by the proposed algorithm ' Lemma 1: If C6, then 6 Lemma 2: If, then Lemma 3: If, for some 9_, for all nodes 6B, then for all nodes B for all Theorem 2: For all nodes, %`e`#n%` The above theorem implies that the error in the location estimation is non-increasing with iteration Therefore, the proposed algorithm is guaranteed to terminate The number of iterations required to terminate can be fairly large in general However, for the numerical examples we have considered, the algorithm terminates in fewer than 1 iterations Also when it terminates, the error in location estimation is not always zero These are shown for several numerical examples in Section V IV IMPLEMENTATION ISSUES A Runtime Complexity The pseudo-code in Figure 1 is a logical description of the algorithm Direct implementation of the pseudo-code can be computationally expensive However, the runtime complexity can be substantially reduced for specific instances of the problem For example, if we assume that the nodes are located in one-dimension, then lines 9 10 can be implemented very efficiently One-dimension may be adequate for location estimation in an airport, since the corridors in the airport are often long but not wide In this case, one can represent the set as a,` ]`` V]` union of intervals, say `BX`D & Then, the following lemma can be proved bb_ Lemma 4: In the one-dimensional case, if `#V`B,` V]` `BX`i &, then R! ` ` $# `! Likewise simplifications are possible for the two-dimensional case B Incomplete Inconsistent Neighbor Sets In proving the correctness convergence of the proposed algorithm we assumed that the neighbor sets of all nodes are complete accurate In this section, we discuss some of the difficulties in satisfying this assumption how one can cope with them Inaccurate non-neighbor set: Note that, node considers node to be a neighbor only if it hears a I am at node message Otherwise, the algorithm as described in Figure 1 will treat as a non-neighbor of thus assume that are not within radio range of each other However, in practice, it is possible that are within radio range but still is unable to hear the message I am at node This can be due to factors such as poor channel quality physical obstruction between Incorrectly assuming that node is a nonneighbor of will result in wrong location estimation of the nodes, ie, the location estimation computed by the algorithm may not contain the true location of the node If the application cannot tolerate occasional location estimation errors, one can modify our algorithm to ignore nonneighbor relationships Specifically, lines 3 10 in Figure 1 can be deleted line 11 can be modified as follows 11 H JML,NWO P In the rest of this paper, we refer to this modified algorithm as Algorithm LocEst Without NonNeigh Note that, this algorithm does not rely on convex optimization techniques Furthermore, we can prove that the solution in [] is equivalent to the output from the first iteration of our algorithm Therefore from Theorem 2, it follows that Algorithm LocEst Without NonNeigh is better than the one in [] Incomplete neighbor set: The neighbor sets can be incomplete due to various reasons For example, node may belong to whereas node may not be in P This may be due to the fact that the LocEst Client at nodes send I am at node % message asynchronously without coordinating with the other nodes the LocEst Server Therefore, it is possible that node just moved into the range of node its LocEst Client sent a I am at node message heard by node However, node may not have seen the I am at node message because it may have been out of range of node when the message was last sent by If the LocEst Server runs the proposed algorithm using such inconsistent neighbor sets, the location estimates may be incorrect The LocEst Server can detect these problems rectify the situation by deleting node from putting on a new list denote &, so that is neither a neighbor nor a nonneighbor of At the same time, must also be put on & P so that is also not considered a neighbor or a non-neighbor of This may result in larger location estimation errors but the estimate will be correct Inconsistent neighbor sets: The LocEst Server does not run the location estimation algorithm continuously Also, the nodes collect the neighbor set information asynchronously only periodically As a result, there can be inconsistencies between the neighbor sets For example, at some point in time, node may have been in the radio range of node, therefore,! P Node may then have moved away

4 from node into the range of node thus P Now suppose that the LocEst Server runs the location estimation algorithm before nodes realize that they have moved out of range of each other Further suppose that the distances between the nodes are such that a node cannot simultaneously be a neighbor of both That is, it is inconsistent to have As a result of this inconsistency, the location estimation algorithm will not be able to find any possible location for (since no location can satisfy both constraints Furthermore, there is no easy way to detect that the problem is between,, because the estimates of all neighbors of,, will also be affected One possible remedy is for the LocEst Server to direct all nodes to verify the neighbor set synchronously when the algorithm fails to find a possible location for a node This can be accomplished by broadcasting a message to all LocEst Clients which in turn can trigger I am at node % from all LocEst Clients within a short time window Note that, this problem will not occur if the nodes are stationary V EVALUATION In this section, we present results from an empirical evaluation of the proposed algorithm The results are obtained from a simulation carried out as follows The parameters of the simulation correspond to an airport application The users (fixed mobile are located in a 120 meter corridor The number of fixed nodes is varied (3, 6, 12 The radio range is assumed to be 10 meters The location of the fixed nodes are such that when there are six fixed nodes each mobile user is guaranteed to be in the range of exactly one of them When there are three fixed nodes some users may not be in the range of any fixed node The location estimation errors presented here are an average obtained from 10,000 distinct runs Successive runs model a scenario where approximately 10% of the mobile users depart approximately the same number of new mobile users arrive In the first run, the location of each mobile user is chosen romly from an uniform distribution in the range (0, 120 In the subsequent runs, the locations for new mobile users are chosen as above the locations of the other mobile users are preserved Figure 2 shows a plot of the average location estimation error as a function of number of mobile users for six different cases Curves with solid lines correspond to Algorithm LocEst With NonNeigh the other three curves correspond to Algorithm LocEst Without NonNeigh For each of these algorithms, the three curves differ in the number of fixed nodes, namely 3, 6, 12 The following observations can be made from these curves As the number of mobile users increases, the average location estimation error decreases in all cases This is because our algorithms exploit the additional neighborhood relationships to reduce the location estimation errors As expected, both algorithms perform better (ie, have smaller location estimation error when the number of fixed nodes increase Algorithm LocEst With NonNeigh has considerably smaller location estimation error than Algorithm Lo- Average Location Estimation Error fixed 3 fixed 6 fixed 6 fixed 12 fixed 12 fixed Number of mobile users Fig 2 Comparison of LocEst With NonNeigh LocEst Without NonNeigh cest Without NonNeigh This shows that it is clearly beneficial to utilize non-neighborhood constraints for location estimation VI CONCLUSIONS In this paper, we presented an algorithm for estimating the location of mobile users The algorithm does not rely on special hardware such as beacons to localize a user It uses nonneighbor information to improve the accuracy of location estimation The algorithm is particularly well-suited for indoor environments where GPS capabilities are not easily available The strength of the algorithm is that the accuracy of location estimation improves when the number of mobile users increases REFERENCES [1] N Bulusu, J Heidemann, D Estrin, GPS-less low cost outdoor localization for very small devices, IEEE Personal Communications Magazine, vol 7, no, pp 28 24, October 2000 [2] N Priyanatha, A Chakraborty, H Balakrishnan, The Cricket location support system, in Proceedings of MOBICOM, Aug 2000, pp [3] P Bahl V N Padmanabhan, RADAR: An in-building RF-based user location tracking system, in Proceedings of INFOCOM, Mar 2000, pp [4] A Ward, A Jones, A Hopper, A new location technique for the active office, IEEE Personal Communications Magazine, vol 4, no, pp 42 47, October 1997 [] L Doherty, K S J Pister, L El Ghaoui, Convex position estimation in wireless sensor networks, in Proceedings of INFOCOM, Apr 2001, pp Appendix Proof of Theorem 1: From the definition of, it follows that e for all nodes (see line 1 Assume, for all nodes, [ B for some Consider any node For each, /1/ P /1/43 (see line 9 Therefore, from the definition of P [6 P P P for each

5 , /1/ P the definition of P [ B P ZP XP Likewise, for each (see line 10, /1/486 Therefore, from for each From the above two observations, The theorem follows by induction for any since for (see line 11 The theorem also holds for : for all `Mni` E C6 Proof of Lemma 1: The lemma holds trivially for any Therefore, consider Consider any + B From line 9, + B implies that there exists a > such that /1/ + >0/1/e36 [AB Since, this implies that there exists a > 4 B such that /2/ + >0/1/e36 Therefore, from line 9, + Proof of Lemma 2: The lemma holds trivially for any Therefore, consider < Consider any + From line 10, + implies that there exists a > such that /2/ + >C/2/486 ' Since, this implies that there exists a > such that /1/ + >C/1/a8 Therefore, from line 10, + Proof of Lemma 3: From Lemmas 3 2 the hypothesis, Z $: for all nodes $ Therefore, from line 11, for any $:Bf Furthermore, for any G Hence, the lemma,, Proof of Theorem 2: Consider For any \6 \ e For all, Z (see lines 9 10 Thus, for all \ \ (see line 11 Consequently, for all nodes The theorem follows from Lemma 3