WLAN Indoor Positioning Based on Euclidean Distances and Fuzzy Logic

Transcription

1 WLAN Inoor Positioning Base on Eucliean Distances an Fuzzy Logic Anreas TEUBER, Bern EISSFELLER Institute of Geoesy an Navigation, University FAF, Munich, Germany, (anreas.teuber, Abstract - This paper presents an inoor positioning approach base on pattern recognition of IEEE signal strength measurements. The approach is ivie into the calibration moe an an operational moe when a user emans to be positione. During the calibration moe signal strength measurements on sites with known coorinates are carrie out an store in a atabase. The current signal strength pattern of the measurements in the operational moe is then compare to the atabase by computing Eucliean istances. An effective processing of these Eucliean istances is unertaken by means of fuzzy logic methos in two stages. While in the first stage a number of caniates of proximate calibration sites is investigate the secon stage is responsible for an appropriate weighting of their coorinates an yiels the positioning result. First test results of two ifferent environments are presente. 1 Introuction In inoor environments WLAN accoring to the IEEE stanar can provie location information by using signal strength measurements to access points (AP) an hence offers a smart synthesis of communication an navigation functionality. Especially the approach of pattern recognition of signal strength fingerprints can obtain location accuracies of 1 to 5 m. Currently, the most acquainte inoor navigation system base on the IEEE which is commercially available is the Ekahau Positioning Engine (EPE). Accoring to the manufacturer s instructions the EPE combines signal strength pattern recognition with an attempt to recover the user s history (incluing bounary conitions like allowe paths an spee) in orer to etermine the current position [1]. Extensive tests have emonstrate the accuracy potential of the system. Uner optimal conitions, which especially means a sufficient ensity an an appropriate istribution of AP an an inoor environment which is highly subivie in ifferent rooms, the achieve accuracies have been even slightly better than the prospects mentione by the manufacturer [2]. However, poorer conitions lea to a worse accuracy. This coul be observe in regular structures like galleries, large rooms, an halls. In orer to correct the location results for occurring systematic errors a post-processing approach was propose [3]. Despite an increase of accuracy this approach is inaequate for real-time positioning. Since the proprietary EPE allows no unauthorize intervention with respect to the algorithms, the evelopment of an inepenent system was require. Instea of the absolute signal strength in this work the signal-to-noise ratio is use as input parameter. A software capable of recoring signal-to-noise ratio measurements of all available AP generates location fingerprints. Rather similar to the EPE uring the calibration moe signal strengths of a set of calibration points of known coorinates to be store in a atabase is establishe initially. During the operational moe the current signal-to-noise ratio pattern of the user is compare to the atabase. Processing algorithms to be evelope have the task to etermine a user position from the comparison of the signal-to-noise ratio patterns. The next chapters starting with the iea of Eucliean istances escribe how the processing algorithms work an how they are implemente. 159

2 2 Processing Eucliean Distances With respect to any etecte AP a signal-to-noise ratio ifference between the current recor an each calibration recor of the atabase can be compute. In the following the current signal-to-noise ratio recors are referre to as observation ata an the collecte signal-to-noise ratio recors in the atabase are referre to as calibration ata. The compute signal-to-noise ratio ifferences can be expresse in a signal-to-noise ratio ifference vector in which the number of elements represents the number of AP. The norm of this vector is well-known as the Eucliean istance: j,k AP AP1 AP 2 AP2 APi APi c s c s... c s (1) j k j k with: c calibration ata i total number of access points s observation ata j inex of calibration points Eucliean istance k inex of observation points Consierably, the compute minimum Eucliean istance j,k = min obtains a significant inication of the calibration point closest to the current observation point. Depening on the ensity of calibration points in the environment to be investigate, the coorinates of the calibration point connecte to min coul be aopte as the correct solution. However, in general, the calibration proceure is a timeconsuming work an therefore the ensity of calibration points will usually be as low as maintainable. Thus, it is recommenable to compute the user position from a weighte mean of a number of calibration points with lowest Eucliean istances: j k x k y k J j1 J j1 1 j, k 1 j, k 1 1 x1 x2 xj... 1,k 2,k J y 1 1,k y 2 2,k y...,k J J,k (2a) (2b) with: x x-coorinate J total number of calibration points to y y-coorinate be weighte Eucliean istance j inex of calibration point k inex of observation point The single calibration point coorinates serve as inputs which is weighte by the reciprocal of the Eucliean istance. The factor in roune brackets is necessary in orer to normalize the overall sum of the reciprocals of the Eucliean istances. The simple application of position etermination base on Eucliean istances leas to feasible results without the nee for extene computing capacities. For signal strength recors of only one epoch the application of the Eucliean istances is very simple. If recors of more than one epoch are available, it will be possible to etermine supplementary information like the mean, the meian, the maximum, the minimum or the stanar eviation of the signal strength measurements. In the following it is assume that these multi-epoch ata may contain aitional information exploitable for position etermination. This approach which will make use of the Fuzzy Set theory will be explaine in etail in chapter

3 3 Fuzzy Set Theory In science, it is common to work with accurate ata an moels. Since scientists are aware of the nature of imprecise information ue to the limitations an restrictions of the use sensors an the applie processing methos to obtain the necessary information they attempt to make an assessment of precision by using terms like variance an probability. However, in many situations of everyay life it is sufficient to characterize a conition by rather simple terms like the water is col, warm, or hot. In most of the cases this information is satisfying for further actions. Fuzzy logic theory assimilates this iea an every conition like col or warm is calle a membership function. A set of membership functions is connecte to a linguistic term, in this case the temperature. The main objective of processing fuzzy sets is to calculate the grae of accorance of the current ata set with the membership functions of the linguistic term. Initially, the user of fuzzy set theory has to efine the linguistic terms as well as their membership functions. In practice, the user has to escribe the conition warm quantitatively an qualitatively. Apart from the mean or centre temperature of warm the way of transition to the neighboure membership functions col an hot has to be efine. It is conceivable that a membership function yiels only a sharp output at a singular point like it is the case using triangular or Gaussian functions as membership functions (Fig. 1 left). Then the transition range embraces the whole area between the singular maximum points of the membership functions. The very most part of the temperature values is connecte to a fuzzy output value. If the membership function is a trapezoial function the fuzzy nature is restricte to those temperature values connecte to the eges of the trapezoi (Fig. 1 right). Fig. 1 Fictive linguistic term temperature an its membership functions col, warm an hot. Of course, the information on the temperature of the water is only one linguistic variable in orer to obtain for instance the efficiency of the run of a washing machine. Another linguistic term coul be the amount of the eploye washing power. In orer to achieve an assessment of the result (or the output) it is necessary to establish rules. The following trivial souning rule is conceivable: If the temperature of the water is hot, the cleaning result will be goo. The if-clause of the rule is always connecte to the relate membership function (e. g. hot) of the input, the main-clause of the rule is always connecte to the relate membership function (e.g. goo) of the output. By means of a set of these rules an the obligatory efinition of the output membership functions a confient statement on the cleaning result shoul be possible. All computations presente in the next chapters were conucte using the Fuzzy Logic Toolbox of MATLAB. Scientic papers on fuzzy set theory can be foun among others in [4],[5],[6]. 161

4 4 Two-Stage Fuzzy Set Design for WLAN Positioning (Airport Hangar Environment) 4.1 First Stage Since the propagation of the raio signals is massively affecte by multipath a esire property for communication purposes for WLAN positioning the principle of pattern recognition was preferre. Preparatory measurements ha le to the result that pattern recognition yiels accuracies twice better than the computation of free space loss. This is even more remarkable consiering that these tests ha been carrie out in a large hall (airport hangar) without obstacles or partition walls. It has to be emphasize that the fuzzy system escribe in this chapter 4, especially the secon stage, is esigne for this kin of environment. In chapter 2 the significance of the Eucliean istances has been pointe out. It has been foreshaowe that in case of multi-epoch ata aitional information of the signal-to-noise measurements can be use. For the approach of this paper these are: 1. the maximum signal-to-noise ratio (snrmax) 2. the stanar eviation of the signal-to-noise ratio (snrst) 3. the mean signal-to-noise ratio (snrmean) 4. the overall sum of all signal-to-noise ratios to all AP (snrsum) On the funament of these linguistic terms an their membership function a set of rules is establishe in orer to express human assessments like, e. g.: If the Eucliean istance of the mean of the signal-to-noise ratio is goo, the matching between the calibration an the observation point will be goo. The four linguistic terms mentione above were connecte to three membership functions each. Trapezoial functions were use as membership functions. For each of the four linguistic terms the quality criteria goo, meium an ba were assigne to the membership functions. In practice, most of the rules contain a combination of the linguistic terms. The rules were further connecte to five output membership functions with the criteria very goo, goo, satisfying, moerate an ba. The output membership functions are triangular functions of equal shape isochronously istribute on the output scale which also ranges from 0 to 1. Quality criteria goo meium ba Linguistic term snrmax snrst snrmean snrsum Tab. 1 Quality an quantity criteria of the membership functions in orer to achieve an equally istribute input. The next step was to assign the recore ata to the membership functions. Initially, there ha been no empirical ata on the magnitues of the Eucliean istances. For this reason the ata of a whole set of 162

5 recors in an environment was analyze in post processing. All iniviual Eucliean istances were ivie by the maximum value of all compute Eucliean istances in orer to normalize them. In other wors, all compute Eucliean istances were now assigne to a scale from 0 to 1. Furthermore, the overall amount of compute Eucliean istances was ivie into three parts in a way that they were equally istribute to the three membership functions. With this information the membership functions coul also be efine quantitatively. Tab. 1 presents an overview of the quality an quantity criteria of the membership functions. The eges of the trapezoial functions were sprea over a range of 0.1 on the normalize scale by efinition. Besies, the sum of all membership functions was always 1. In the next step the rules were teste with respect to any collecte input vector. The input vector contains the normalize Eucliean istances of the four linguistic variables an can be seen in Fig. 2 in the lower left box (values ). The Rule Viewer of the MATLAB Fuzzy Logic Toolbox visualizes whether or not a rule can be applie. Usually, a number of 12 to 15 rules ha to be teste an the very majority of them were an-rules which means that every part of a rule ha to be fulfille cumulatively in orer to accept the result. The processing of the rules in orer to get an output is calle aggregation. The example in Fig. 2 shows that only rule #3 is met completely. This rule expresses that if only the linguistic terms snrst an snrmean are goo, the output will be very goo. In fact, the input for these two terms is 0.1 each an meets or falls below the require values of 0.12 an 0.1 (Tab. 1) for very gooness, respectively. In this rule #3 there is no requirement for the other two linguistic terms. Furthermore, it can be seen that some of the rules are met at least partially. This case occurs when the re line of the input hits the trapezoial function at its ege. Fig. 2 Rule Viewer of the MATLAB Fuzzy Logic Toolbox 163

6 The iniviual outputs have to be summarize to a single value. This process is calle efuzzification. The applie efuzzification metho is calculation of the centroi. For every observation point the proceure of testing all rules is one with every available calibration point. The result is a ranking of calibration points. It is again conceivable to aopt the coorinates of that calibration point of the smallest output value after efuzzification. Just as well it is possible to take a set of the best calibration points again an to weight them with the reciprocal of the belonging output values an to average them. The first trials have been unertaken in a rectangular part of an abanone airport hangar. The computation of an average out of a set of calibration point coorinates woul always lea to coorinates of a point surroune by the use calibration points. However, if a calibration point is locate at one of the eges of the rectangular structure one provokes a systematic error by this kin of averaging. For this reason it was preferre to a a secon stage of a fuzzy system which takes some basic topological issues into account. 4.2 Secon Stage The main objective of the secon stage is to fin an appropriate weighting of the output caniates of the first stage. All calibration points which have not been selecte as caniates in the first stage, are alreay exclue for the secon stage. The latter consists again of four linguistic variables, however, neither of them contains signal-to-noise information in a irect manner. They are: 1. the absolute value of the output of the first fuzzy stage (fuz1-absolute) 2. the quotient between the best an secon best output of the first fuzzy stage (fuz1-quotient) 3. the number of Delaunay neighbours (Delaunay-Neighbourhoo) 4. the periphery flag (periphery) The first two linguistic terms which eal with the output parameters of the first fuzzy stage play an important role. If the absolute value of the output of the first stage or the quotient between the first an the secon best output is extraorinary small it will be assume that the observation point is very close to the calibration point. As a consequence, the latter will get a very heavy weighting in a way that the resulting coorinates will only slightly iffer from that of the calibration point. Linguistic terms #1 an #2 are connecte to three membership functions calle high, meium an low. In contrast to the first fuzzy stage the parameters of the membership functions were set manually. In orer to avoi the shift of observation point coorinates locate at the periphery of an environment relatively to the centre by averaging the coorinates of the surrouning calibration points, some topological basis ata of the calibration points have to be acquire. For this reason a Delaunay triangulation of all calibration points of the environment was conucte. By means of this all Delaunay neighbours coul be etermine for each calibration point (Fig. 3). It is easy to unerstan that calibration points locate closer to the centre of the environment have more Delaunay neighbours than those locate more in the periphery. Therefore, the linguistic term 3 simply expresses the number of Delaunay neighbours. In orer to get a eeper insight of a peripheral location of a calibration point a further investigation was conucte. For each calibration point the azimuths to all Delaunay neighbours were etermine an sorte in ascening orer. Then the maximal angle between two neighboure azimuths was evaluate. If this value is small, e.g. below 120 eg, the point will efinitely not be locate at the periphery of a rectangular environment. However, if the angle is rather high, e.g. 180 eg, the calibration point will probably be locate in the periphery at one sie of the environment. If the angle is even higher, e.g. 270 eg, the calibration point will probably be locate in a corner of the shape of the environment. If a calibration point etecte to be at the ege of the environment is the best caniate of the first fuzzy stage, its coorinates will get a significantly higher weight in the secon 164

7 fuzzy stage in orer to avoi the effect of centralization. The maximal angle of each calibration point is also visualize in Fig. 3. The linguistic terms #3 an #4 are connecte to four membership functions each which were calle high, above average, moerate an low. Like terms #1 an #2 they were set manually, too. <60 60 to < to <225 >225 Fig. 3 Delaunay triangulation of calibration points an classification of maximal angles The secon fuzzy system runs in a sequential manner with respect to the calibration point hierarchy of the first fuzzy stage. Finally, the normalize reciprocals of the outputs of the secon fuzzy stage are irectly use as weighting factors for the esire coorinates of the observation point. 4.3 Results The first test run of the system was conucte in one part of approximately 20 m by 20 m of the abanone airport hangar which ha a total imension of 40 m by 60 m. In the 400 m² part of the hangar five access points of the IEEE b stanar were installe, one in each corner an one in the centre. Three of the access points were Netgear ME 102 an the two remaining SMC 2671W. 16 calibration points were uniformly istribute in the test area. This ajustment was not one exactly but accoring to visual jugement. The average istance between two calibration points was approximately 5 m. Another 16 observation points were locate in a manner that they alternate with the calibration points. Again, the fixing of the sites was one by visual jugement. Consequently, the approximate istance between two observation points was also 5 m. Each calibration an observation point was reference to millimetre accuracy to a local coorinate system by an electro-optical tachymeter. Fig. 4 provies a schematic overview of the arrangement of access points, calibration points an observation points. 165

8 Signal-to-noise ratio measurements were then conucte on the calibration points in orer to establish a atabase. In orer to provie the require four parameters for the first fuzzy stage a sample of about ten epochs was collecte on each point. Subsequently, the signal-to-noise measurements on the observation points were conucte in exactly the same manner. Fig. 4 Arrangement of access points, calibration points an observation points in the hangar environment The results presente in the following were obtaine with the settings of the fuzzy systems escribe above whereas the best four outputs of the first fuzzy stage were selecte for the secon stage. Of course, it is possible to vary this number. For simplicity, as parameter for error consieration an comparison of the results the average 2D position error of all iniviual results is given here. After the first stage the average 2D position error was 3.59 m. This is significantly better than a simple weighte computation of Eucliean istances without application of the fuzzy approach with an average 2D position error of 4.47 m. After the secon stage the error ecrease to 3.27 m. The best iniviual result is a 2D error of 0.35 m, the worst case is 6.58 m. There is no significant ifference between observation points at the periphery an closer to the centre. The main results are presente in Tab. 2. Number of calibr. points Average 2D position error [m] Fuzzy Stage 1 Fuzzy Stage 2 EPE Eucliean Distance No ata Tab. 2 Accuracy of the ifferent inoor positioning approaches in the hangar environment The results of the fuzzy approach have also been compare to the commercial Ekahau Positioning Engine (EPE). It is well-known that the EPE yiels better results in heterogeneous environments where partition walls support the iea of the efine allowe paths an the user position history. In fact, with the EPE it is not possible to achieve the results of the fuzzy logic approach, the average 2D position error is 4.40 m whereas the iniviual errors vary from 0.96 m to 9.07 m. This is even more surprising since the results are not irectly comparable. This is ue to the fact that with the EPE on every observation point a calibration recor was conucte an every 36 points were calculate. If this metho was applie to the fuzzy logic approach, the error woul ecrease to 2.48 m after the first fuzzy stage an even 1.61 m after the secon stage. There has been only slight moification to the stanar settings of the fuzzy system so far. A thorough review of the applie rules coul be a key to a further improvement of the result. However, the extension to a higher number of membership functions was evaluate an le to a eterioration of the position accuracy since the user is in anger to lose the unerstaning of the whole system. As a consequence, the user is no longer able to express the necessary number of evient rules. 166

9 5 Moifications for a Heterogenous Environment (Office Builing) Both the fuzzy logic an the EPE system have also been teste in a combine office an laboratory builing. This builing has a similar imension of approximately 20 m by 20 m. However, it is separate into ten offices an laboratories of ifferent sizes, all of them separate to each other by massive, but relatively thin partition walls. A former investigation with the EPE le to excellent 2D position results of 1 m an better [2]. This was ue to the two basic techniques of the EPE: first, the path esign which requests the user to ascertain the allowe user paths in a builing in avance on a igital map in orer to avoi trajectories crossing walls (thick, continuous lines in Fig. 5 right); secon, the consieration of the user history an a preefine maximum user velocity which rules out suen user movements from one epoch to the next. The goo results in [2] ha been achieve since all observation points were locate more or less exactly on the preefine paths. For the comparison with the fuzzy approach the observation points were locate on all accessible points, regarless of their istance to the paths. Similar to the hangar, the office environment was equippe with five access points. The main features of the test esign can be gathere from Fig. 5. Fig. 5 Unfavourable Delaunay trinangulation in the office builing (left) an the remey (right): usage of the EPE paths combine with aitional lines Soon it became obvious that for the fuzzy system a Delaunay triangulation is fruitless since most of the triangle sies will cross the walls of the builing (ashe lines in Fig. 5 left). For this reason the topological information ha to be aapte to the specific environment. In the case of the office builing the paths of the EPE calibration were use an supplemente by some more lines epicte in Fig. 5 right as otte lines. Despite of the aaptation of the topological information in the fuzzy system the EPE remaine superior with respect to position accuracy. Again the total number of points was 32, 16 calibration points an 16 observation points. A total number of eleven points coul be irectly compare to the EPE. The average 2D position error of the fuzzy system is 3.14 m incluing one outlier of more than 10 m. After elimination of this outlier the average is 2.37 m, but still worse than the EPE result of 1.94 m. 167

10 It has to be pointe out that the presente results consier only position computations where the observation points o not coincie with the calibration points since this is the bigger challenge for systems base on pattern recognition algorithms. 6 Conclusion an Further Work An approach of using fuzzy inference systems in orer to etermine positions on the basis of WLAN signal-to-noise ratio measurements an of pattern recognition was presente. The first results of the two-stage system are encouraging an show less variation with respect to ifferent environments than the commercial EPE. If the position computation is limite to observation points which o not coincie with calibration points at all, the accuracy will be about 3 m for both environments. If observation points coincie with the calibration points, the accuracy will be significantly better. However, the processing of the topological information has to be aapte to the environment. It is conceivable that this aaptation coul be realize automatically, possibly by another fuzzy logic system. The investigation in the two presente environments are ongoing an will be extene to further environments. In the short term the investigations also comprise a better unerstaning of the fuzzy inference systems which coul provie a (limite) improvement of accuracy. Furthermore, there are challenges with respect to the number of epochs necessary for an appropriate positioning. Of course, a moving user woul prefer to etermine his position by one single signal-to-noise ratio measurement. This is still contraictory to the principle of the first fuzzy stage where statistical information of the signal-to-noise ratio ata are use as inputs. 7 References [1] Ekahau, Ekahau Positioning Engine 2.0: base Wireless LAN positioning system. An Ekahau Technology Document, November [2] Eissfeller, B., D. Gänsch, S. Müller an A. Teuber, Inoor Positioning Using Wireless LAN Raio Signals in Proceeings of ION-GNSS 2004, Long Beach, California, September 2004, pp A. N. Eitor an A. Nother (Es.), An Eite Book, Wiley, Chichester, U.K, [3] Teuber, A., G.W. Hein, Analysis an Reuction of Systematic Errors in Wireless LAN Positioning in Workshop Proceeings of WPNC 05, Hannover, Germany, 17 March 2005, pp [4] Zaeh, L. A., Fuzzy Sets, Information an Control, [5] Zaeh; L. A., Fuzzy Logic, Neural Networks, an Soft Computing, Communications of the ACM, Issue 37, No. 3, pp , [6] Haberler, M., Einsatz von Fuzzy Methoen zur Detektion konsistenter Punktbewegungen, Geowissenschaftliche Mitteilungen Nr. 73, Schriftenreihe er Stuienrichtung Vermessung un Geoinformation, Technische Universität Wien, Institut für Geoäsie un Geophysik,