Practical considerations of optimal threedimensional

Transcription

1 Practica considerations of optima threedimensiona indoor ocaization Jens Schroeder, Stefan Gaer, Kyandoghere Kyamakya, and Kaus Jobmann Abstract The probem of ocating objects and peope in indoor environments is atey gaining high interest in research and industry. In this context, the geometrica consteation of source and sensors pays an important roe and heaviy infuences the accuracy for the position estimate. This paper uses existing optima soutions of the probem of threedimensiona TDOA ocaization, and reates them to practica appication. The extension is necessary, since the optima soution is ony vaid for one singe position and not for a compete area, where a ocaization capabiity is desired. Therefore, it is shown by simuations that the optimum is very usefu for ocaization purposes in typica indoor scenarios. In addition, rea-word measurements using an Utra-Wideband (UWB) ocaization system have been carried out in two geometrica scenarios. It can be shown that the simuations, which are based on theoretica derivations, are a good approximation of the reaity. Finay, it can be demonstrated that the optima soution can be adapted to typica indoor environments and sti yied accurate three-dimensiona positioning. I. INTRODUCTION Appications, which use the position information of objects or peope, are becoming more and more attractive. This is mainy due to the increasing use of mobie devices that are equipped with communication capabiities. Different methods can be appied to sove the ocaization probem. One possibiity is to estimate the position of a radiating source via the anaysis of signas received at a certain number of sensors. This approach is normay based on measuring range differences between source and sensors, and subsequenty processing the range information into a position estimate via Time Difference Of Arriva (TDOA) agorithms. In this case, the ocaization accuracy mainy depends on two parameters: the error of the ranging differences and the geometrica arrangement of the sources and sensors. For exampe, to achieve decimeter accuracy of Manuscript received Apri. This work was supported in part by the DFG (German Research Foundation) under Grant KIOP, and the Minna- James-Heinemann Foundation. J. Schroeder, S. Gaer, and K. Jobmann are with the Institute of Communications Engineering (IKT), University of Hannover, Hannover, Germany (contact emai: jens.schroeder@ikt.uni-hannover.de). K. Kyamakya is with the Institute of Informatics Systems, University of Kagenfurt, Kagenfurt, Austria. position estimation it is necessary to use a radio technoogy, which can achieve decimeter accuracy in ranging. However, the impact of the geometry can be quite severe such that the goa is not reached. Therefore, in this paper the atter parameter is of interest, and previousy proposed optima geometries of sensors are further investigated. The probem of evauating the ocaization accuracy of a given consteation has been studied for a ong time. Especiay for passive source ocaization, but aso for positioning via sateites and atey for ceuar networks, anaysis has been carried out based on covariance study of ocaization errors and estimation theory [-]. In addition, research has been performed with the task of optimay pacing the sensors for different scenarios, i.e. Direction Of Arriva (DOA) [7-9] or two-dimensiona TDOA [9, ]. Most work is based on minimizing some kind of error variance of the estimation function. Ony atey the threedimensiona TDOA ocaization probem has been addressed [-] and an optima soution as been anayticay derived. Our paper uses the theoretica optimum and verifies its usefuness for practica appications. The extension is necessary, since the optima soution is ony vaid for one singe position and not for a compete area, where a ocaization capabiity is desired. Therefore, it is shown by simuations that the optimum is very usefu for ocaization purposes in typica indoor scenarios. In addition, rea-word measurements using an Utra-Wideband (UWB) ocaization system have been carried out in two geometrica scenarios. It can be shown that the simuations, which are based on theoretica derivations, are a good approximation of the reaity. Further, it can be demonstrated, that carefu adaptation of the optima geometrica soution to the instaation conditions of an indoor scenario, wi yied accurate position estimation. The paper is organized as foows: The signa setup and the optima geometry wi be introduced in section II. In section III, the soution is studied under practica considerations by means of simuation. Section IV describes the UWB measurement setup and anayses the different geometrica scenarios in which a source is ocated. Finay, chapter V derives the mentioned metric and discussed the comparison between the simuations and the measurements.

2 II. SETUP AND OPTIMAL GEOMETRY In a Cartesian coordinate frame the Eucidian distance d i between mobie source and sensor i is given by d i = x x i, i N () with source position x = [x,y,z] T th, i sensor position x i, number of sensors N, and denoting the L norm. The received signa s i (t) at sensor i can be modeed as an attenuated and deayed version of the radiated signa r(t) with additive noise: s () t = m ( d ) r( t t ) + n () t, i N, () i i i i i where m i (d i ) denotes a distance-dependent attenuation factor and t i the absoute time of arriva at sensor i. The radiated signa r(t) is assumed to be a stationary Gaussian process, the noises n i (t) are assumed to be identicay distributed zero-mean stationary Gaussian processes, and a are mutuay uncorreated. Time differences t ij are measured via e.g. crosscorreation of the received signas at sensor pair (i,j), and are converted to range differences dij by mutipying by the speed of ight c. d = c t = c( t t ) = d d, () ij ij i j i j where t i and t j are the absoute times of arriva at sensor i and j, respectivey. Substituting the distances d i of () into () yieds D = N(N )/ hyperbooid pseudo-range equations: c t ij = x xi x x j, i < j N. () From the non-inear system of D (usuay over-determined) equations, the genera probem is to sove for the mobie unit s position x, given the base stations positions x i and the TDOA measurements tij. The intersection of these hyperboas corresponds to the position of the mobie unit, if the TDOA measurements tij are exact. Therefore, one way to find a position estimate from TDOA measurements is the direct soution of the set of pseudo-range equations [- ]. The performance of such estimators is usuay evauated with the ower bound for the covariance matrix of unbiased estimators, the Cramer-Rao Lower Bound (CRLB) [7, pp.]: x ˆ Var { xˆ } J, where J ˆx is the Fisher information matrix. The CRLB has been derived for the case of position estimates via both, TDOA and Time-Of-Arriva (TOA) measurements [, 9,, 7, p., ]. Note, that no difference exists between the TOA and TDOA case [9]. In the navigation fied, a common expression for anayzing the geometrica consteation is the nondimensiona quantity set Position Diution Of Precision (PDOP) [, pp.,, pp.]. It describes how measurement errors transate into position errors: RMS position error PDOP =, () RMS measurement error and can easiy be reated to the CRLB [9]: PDOP = tr - { } J, () xˆ where tr stands for the trace operator. In this context, two other expressions are practicay reevant: The Horizonta Diution of Precision () describes the transation of measurement errors on the horizonta accuracy, i.e. the {x,y}-pane, and the Vertica Diution of Precision () describes the transation on the vertica accuracy, i.e. the z- direction. If the diagona eements of J x ˆ- are denoted by Jxx, Jyy, and Jzz, these expressions are defined as foows [, p. ]: xx yy = J + J and =. (7) Jzz Of course, the mathematica anaysis coud purey be based on the Fisher information matrix, for practica appications however, the use of the we known xdop expressions is quite convenient. The Fisher information matrix J ˆx for ˆx and the given mode is known as ([9, ]) xˆ σr T J = B M B, where () σ R can be interpreted as the mean-squared ranging error, which is a function of signa power and frequency, B =,,, b b b b bn b (9) is the matrix of source/sensor bearing vectors T x = [ xi, yi, zi ] = i x b i () di differenced by N N b= m b mi, () i i i= i= which is the weighted average of source/sensor bearing vectors, and M = diag(m,m,,m N ) is the diagona matrix of a attenuation factors. We now briefy review the optima array geometry under the condition that the PDOP is minimized, i.e. that for given measurement errors, the infuence of the geometrica consteation is minimized. In [] a nice soution has been

3 derived with two necessary and sufficient conditions under the assumption that m =, i : C: b = and C: B B = ( N /) I, i T i.e. matrix B has orthogona row vectors with equa row norm. From these conditions foows, that the optima threedimensiona array geometry must consist of vectors b i, which form an uniform anguar array, meaning that they are equay distributed on a unit spherica surface with the source ocated in the center. As [] points out, the five soutions are the Patonic soids tetrahedron, octahedron, cube, icosahedron, and dodecahedron; depending on the number of sensors. Note that in these consteations the + vectors b i coud be scaed to any ength α b i, α and the optimum wi sti be achieved. Since the soution is based on the mode in (), the effect of mutipath is not incuded. Aso, since we assume that the R ranging accuracy σ is equa for a sensors, signa power and frequency is assumed to be equa at a sensors. However, the authors do not see these as imitations to the soutions, since in this paper ony the infuence of the geometry shoud be anayzed. When necessary, specia care is taken in the foowing. within a cube, which is spanned by the sensors in a tetrahedron arrangement. Some resuts are shown in fig. and fig., where the and are potted parae to the {x,y}-pane at different heights. As it can be seen, measurement errors are amost equay transated to position errors at a ocations of the area. A compete simuation with a tight, three-dimensiona grid of positions confirmed that the does not exceed. and the does not exceed. at any position (except of singuarities at the ocations of the sensors, which is regarded as a practicay irreevant). This shows that the tetrahedron consteation is a good choice for instaing sensors for accurate, threedimensiona ocaization Fig.. Contour pots of and in the midde of a cube spanned by a tetrahedron consteation of sensors (trianges). III. PRACTICAL CONSIDERATIONS The goa is now to verify and adapt the theoretica resuts under practica considerations. The regarded scenarios are ocaization appications for typica indoor environments, e.g. offices, inventory stores, production pants. To insta a three-dimensiona ocaization system in these areas, the pacement of the sensors must be carefuy seected. Especiay, the choice of the sensors heights is crucia to have good resoution possibiity in the z-direction. How important this is can be easiy shown for the case, that four sensors woud be intuitivey paced in the four top corners at the ceiing of a room. In this panar configuration the goes to infinity, which needs to be interpreted as the impossibiity to resove the z-coordinate, even in the case of very sma measurement errors. Therefore, we use the theoretica optimum as a basis for the best pacement of such sensors. In the foowing, the number of sensors is restricted to N =. This corresponds to the minima number of sensors in a TDOA setup with the capabiity to resove three-dimensions. In section II, the optima soution has been derived for the origin of a vectors b i, i.e. the center of the tetrahedron. However, no statement has been made about other positions within the tetrahedron. Yet for practica appications, every position in a desired area shoud have a good ocaization possibiity as it is not appicabe to rearrange the sensors when the source it moving. Therefore, it is investigated how accurate such an optimay arranged ocaization system can perform. Formuas (7) and () were used to cacuate and at different positions Fig.. Contour pots of and at top and bottom of a cube spanned by a tetrahedron consteation of sensors (trianges). IV. MEASUREMENT SETUP Measurements using an Utra-Wideband (UWB) ocaization system have been performed to experimentay verify the good performance of the tetrahedron geometry as we as the appicabiity of the simuation environment regarding reaity. UWB has been chosen as technoogy, since it is capabe of offering decimeter accuracy in ranging. The measurement setup is shown in fig.. A Matab PC, which serves as a centra controer and processing unit, triggers the mobie unit, which emits a bi-phase, pseudorandom-noise UWB puse train. Four antennas are connected to a digita samping oscioscope, which operates as receiver. A detaied description of the system can be found in []. The received signa waveforms s i (t) are crosscorreated

4 with each other for three-dimensiona positioning. Time differences are obtained by taking the argument of the peak of the crosscorreation functions between two respective base stations and are regarded as the TDOA estimate dˆij. At every measured position, ine of sight paths between source and sensors were present. Then, Bancroft s agorithm is used as a direct soution soving the hyperboic ocaization probem []. It has been shown, that this method sits sighty above the CRLB []. that most of the times an instaation ike this wi ead to ess non-ine-of-sight situations, which woud aso be preferabe for accuratey estimating ranges. Aso, in a typica indoor environment an instaation of the sensors can be difficut at ower heights due to machinery, sheves, etc. The respective heights of the antennas are [z z z z ] = [....]m. The simuations in fig. show that the vertica diution of precision increases quite bady, even though the remains sti acceptabe in most of the areas. Activation si() t i < j N Fig.. UWB measurement setup using a samping oscioscope and a Matab PC as a digita receiver. Measurements were conducted in an industria environment. The source was moved at height.m in cm steps aong a ine with ength.9m, resuting in L = recorded positions. The ine is indicated as an arrow in fig. and fig.. Measurements were carried out in two geometrica scenarios. In the first one, representing an as good as possibe configuration, four antennas were instaed at the heights of [z z z z ] = [....]m, which corresponds to a squeezed tetrahedron geometry. The heights and the mounting positions in the {x,y}-pane were mainy determined by the area geometry. Athough the tetrahedron is strongy deformed, simuation resuts in fig. show that both, the vertica and horizonta diution of precision at a height of.m are sti quite acceptabe. Hence, with a decimeter ranging accuracy, the horizonta position accuracy woud be about two, the vertica about five decimeters. Fig.. Contour pots of and at height of.m with an amost panar consteation of sensors (represented by the numbered squares) V. RESULTS AND DISCUSSION To compare the simuated resuts with the measured ones, an estimate for the xdops at position x must be cacuated from the range measurements using () and (7). Therefore, the RMS range measurement error at position x is estimated as foows: r ˆ ex = ( dij dij ) /, () i= j= with dˆij being the measured and dij the rea range difference at position x. The horizonta RMS position error is estimated as foows: Fig.. Contour pot at height of.m with a squeezed tetrahedron consteation of sensors (represented by the numbered squares) The second geometry foows the more intuitive approach to insta a sensors at the ceiing of the room in an amost panar configuration. This is supported by the fact h ˆ ex = ( xˆ x ) + ( y y ), () the vertica RMS position error ex v simiary. Both errors of a recorded measurements are depicted in fig.. The degradation in vertica direction of the panar configuration can easiy be observed. To gain a statistica basis, an arithmetic mean of a L = xdops is used as an approximation for the infuence of the geometrica consteation over the compete measurement ine: e = L x h r L = ex L ex v and = r L. () e = x

5 Horizonta pos. error e h x [m]... Tetrahedron Panar Index of measurement Vertica pos. error e v x [m]... Tetrahedron Panar Index of measurement Fig.. Position estimation errors in horizonta (eft) and vertica (right) direction of each recorded position. Finay, to compare the simuated with the estimated resuts, the power and frequency dependence of σr is removed, by using a ratio expression for the xdops: ratio panar =. () tetrahedron The estimate of the ratio is cacuated simiary. Note, that such a comparison is ony vaid if the summarized RMS ranging error of both, the tetrahedron and the panar consteation, are comparabe. In fact, this is the case with panar RMS tetrahedron RMS e =.m and e =.m. A xdop ratios are summarized in Tabe I, and two concusions can be drawn. On the one hand, the measurements show that the simuations, which are based on theoretica derivations, are a good approximation of the reaity. On the other hand, the strong deterioration of the vertica position estimation is based on the amost panar arrangement of the sensors. TABLE I COMPARISON OF SIMULATED AND MEASURED XDOP-RATIOS ratio ratio SIMULATED,,7 MEASURED,, VI. CONCLUSION The practicabiity of previousy derived optima geometries for three-dimensiona TDOA ocaization has been investigated. With simuations and rea-word measurements in typica indoor scenarios it has been shown, that the optima soutions can be bent and sti are quite appicabe. Athough the resuts can be appied for instaing three-dimensiona indoor ocaization systems, the tradeoff between ranging accuracy and geometry has aways to be taken care of: striving to arrange the sensors as good as possibe in a tetrahedron can ead to non ine of sight situations, which degrade the ranging accuracy and therefore inherenty deteriorate the position accuracy. 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