Extended Filtering for Self-Localization over RFID Tag Grid Excess Channels II

Transcription

1 Extended Filtering for Self-Localization over RFID Tag Grid Excess Channels II Moises Granados-Cruz, Yuriy S. Shmaliy, and Sanowar H. Khan Abstract In the first part of this paper, we have modified the extended Kalman filter () algorithm and developed a new extended unbiased finite impulse response () filtering algorithm for mobile robot self-localization over radio frequency identification (RFID) tag grid excess channels. In the second part, we provide simulations and show that redundant information captured from the tags allows increasing both the localization accuracy and system stability. The common factor here is that the number of tags required to increase accuracy is limited in the target nonlinear medium, by about six in our case. It is also shown that target state observation over the RFID tag excess channels allows mitigating effect of the imprecisely defined noise statistics on the performance and preventing divergence in. Keywords RFID tag information grid, extended unbiased FIR filter, extended Kalman filter, self-localization. I. INTRODUCTION MOBILE robot self-localization over the radio frequency identification (RFID) tag information grids require highest accuracy. In such grids, information delivery is typically combined with sensing and wealth of information captured critically depends on accuracy of the information receptor localization. To increase the localization accuracy, various modifications are exploited of the Kalman filter (KF) [] [], particle filter (PF) [6] [8], and unscented KF (UKF) [9]. Algorithms utilizing the extended Kalman filter () require white noise approximation as well as known noise statistics, initial conditions, and initial error statistics in order for the to be suboptimal. Otherwise, accuracy provided by may be low [] and unacceptable for information grids. Another flaw is that can be unstable and demonstrate divergence under the uncertainties [] and large nonlinearities with intensive noise []. The problem of not exactly known noise statistics also arises in UKF, although this filter demonstrates better performance than for highly nonlinear systems. The PF is free of many disadvantages peculiar to. However, PF based on the Monte Carlo approach often requires large data and time and cannot always be used in real-time localization. It is also known that the Gauss s least squares (LS) often give accuracy that is superior to the best available []. This investigation was supported by the Royal Academy of Engineering under the Newton Research Collaboration Programme NRCP//. Moises Granados-Cruz and Yuriy S. Shmaliy are with the Department of Electronics Engineering, Universidad de Guanajuato, Mexico ( shmaliy@ugto.mx). S. H. Khan is with the Department of Electronics Engineering, City University London, London, UK, S.H.Khan@city.ac.uk. Thus, methods of averaging implemented in LS and finite impulse response (FIR) filters may be more preferable. So, there is still room for discussion of the best estimator for RFID tag information grids. The FIR filter has been under the development for decades [], [] []. It has been shown that this filter is more robust than KF under the unbounded disturbances [8]. The FIR filter is also lesser sensitive to noise [9] and produces smaller round-off errors [] owing to averaging. Of practical importance is that complex optimal FIR (OFIR) structures [], [] do not demonstrate essential advantages against simple unbiased FIR (UFIR) ones [9] which ignore the noise statistics [], []. The effect is due to averaging leveling the difference between OFIR and UFIR on large averaging horizons. The latter has made the UFIR filter a strong rival to the Kalman filter. Recently, the UFIR algorithm was developed in [] to the extended UFIR () algorithm following the same strategy as for the Kalman filter. First applications of the filter to localization problems [], [] have already shown some promising results. It was revealed [] that the filter initiated by can be much more successful in accuracy and stability in the triangulationbased localization. It was also noticed [] that the filter has much stronger protection against divergency and instability than in the RFID tag grid-based localization. In the first part of this paper [], we have discussed the extended filtering algorithms for self-localization over RFID tag grid access channels. To learn effect of redundant information captured from excess tags on the localization accuracy, below we consider a vehicle travelling on an indoor floorspace in different RFID tag environments. Each tag has a circular detection area with the detection range r. Both the short range tags (r < cm) [] and long range tags (r < m) [] can be used. In our case, we suppose that a vehicle has a reader which is able to measure distances to k n tags at once employing the maximum RSSI given by the Friis equation η P r = P t [], [], in which P Di r is the received power, P t is the transmitted power, D i is a reader-to-(ith)tag distance, and η is a coefficient dependent on the transmitter and received antenna gains, system loss factor, and wavelength. A FOG installed on a vehicle directly measures a pose angle Φ n. All noise sources are supposed to be additive, stationary, zero mean, white Gaussian, and uncorrelated. Accordingly, we introduce the estimation error variances σx, σy, and σφ and specify the noise covariance matrix with the main diagonal diag Q = [ σx σy σφ ] and all other components zeros. For the noise variances σl and σ R in the inputs d Ln and d Rn, we specify the relevant noise covariance matrix with the main diagonal diag L = [ σl σ R ] and all other ISBN:

2 components zeros. Finally, for the measurement noise variances σvi, i [, k n], we specify the covariance R n as diag R n = [ σv σv... σvk n σφ ] with all other components zeros. II. SELF-LOCALIZATION USING RFID TAGS NESTED ON INDOOR BOUNDARIES We first consider a scheme in which a vehicle is localized on an indoor floorspace with tags mounted with equal intervals of m on the indoor space boundaries (Fig. ). We assume that a reader has range of r = 6 m and is thus able to detect simultaneously several tags at the signal-to-noise ratio (SNR) level of db. Two tags, Tag (, ) and Tag (, 6 m), are used to form linear measurements y n as will be shown below. The state and input noise standard deviations are set as σ x = σ y = σ L = σ R = mm and σ Φ =.. Note that, in [], σ x, σ y, and σ Φ are supposed to be zeroth. Following [8], we allow noise in the measured distances to have σ v = σ v = cm and let σ φ =. We also suppose that test measurements are available and thus the test trajectory x n is known. Simulations are conducted at points. In the nonlinear problem considered in [], linear measurements of x n and y n are unavailable. Because a vector y n combining linear measurements is required by the algorithm, we exploit distances D and D measured to tags T and T which have only one nonzero coordinate µ = 6 m and solve the inverse problem to define linear measurements x n and ỹ n of x n and y n as x n = ỹ n = A n µ (µ A n + A n ), () (µ A n + A n ), () µ where A n = Dn c, A n = Dn c, and c = c = m. Here, c and c correspond to tags T and T respectively. We then unite x n, ỹ n, and Φ n in a vector y n = [ x n ỹ n Φn ] T. Provided accurate values of x n via test measurements, a straightforward way to find N opt is to minimize the MSE by N as [] N opt = arg min [ tr P (N) ], () N where tr P (N) means the trace of P defined by () in [] with the third state removed, because the third state is defined in angular units while the first and second states are in meters. Using (), we find N opt = 89. An example of measured vehicle location for a spiral trajectory is given in Fig.. Here, we also show the and estimates of location for exactly known noise covariances R n, Q, and L, initial state ˆx = y, initial error P =, and N opt = 89. Because such conditions are ideal, the estimates sketched in Fig. are the best available by the filter and. As can be seen, measurements are too rough here for straightforward localization. In turn, the estimates are quite accurate and consistent with each other. A more precise look at the localization errors is thus required. Below we consider several special cases. y, m T T T6 T7 T T T T T Measurement x, m Fig.. and filtering estimates of location of a vehicle travelling spirally on an indoor floorspace. The indoor space is equipped with nested RFID tags. Measurement is obtained from Tag and Tag. Localization is provided for exactly known noise statistics and N opt = 89. TABLE I. SETS (k n) OF OBSERVED SAW TAGS (t) CORRESPONDING TO FIG. k n t A. Fully Known Noise Statistics To investigate effect of excess tags interacting with a target, we increase the reader sensitivity and add more detected tags as shown in Table I. For each of these cases, we repeat measurements and estimates times and compute the localization errors by the trace of P (N opt ). The results are sketched in Fig. a and Fig. b for and filter respectively along with the average errors. As can be seen, the localization errors reach peak-values in both filters by k n =, undergo reduction with k n 7, and then become constant in average by further increase in k n. What else flows from this simulation and previously published outcomes is the following: Extra tags interacting with a target create an environment for error reduction. In linear systems with white Gaussian noise, the output noise variance is reduced by the factor of k n. In nonlinear systems such as that formalized in Fig. in [], the error reduction function is rather complex and the effect can be lesser pronounced. It implies a finite value for k n which is about six in Fig.. T8 T9 T ISBN:

3 ..9 Extended KF Extended KF kn.. Extended FIR p = kn Extended KF k n Fig.. Instantaneous (circles) and averaged (bold) localization errors tr P (N opt ) as functions of the number k n of the observed tags in Fig.. Localization is provided for exactly known noise statistics and N opt = 89: and filter. Because noise in the extended model is not actually white, extended filtering techniques may not be very successful in accuracy due to the bias error. In white Gaussian medium generated for PF, hybrid structures such as PF/ and PF/ may be more accurate owing to excess tags. An evidence for this statement can be found in [8] where the authors pointed out that the PF/ exploited in the short-range (r = cm) tag environment has reduced the localization error from. cm to. cm by increasing k n from to. B. Not Fully Known Noise Statistics The noise statistics are typically not well-known to the engineer []. To evaluate a maximum effect of imprecisely defined Q, R n, and L on the output accuracy, we introduce a correction coefficient p as p R n, Q/p, and L/p [] p = k n Fig.. Localization errors tr P (N opt ) in the as functions of k n and p corresponding to Fig.. estimates provided with N opt = 89 are dashed. Localization is obtained for ˆx = y : p =,,,, and p =.,.,.,.. and compute the trace of P (N opt ) as function of p. The localization errors are shown in Fig.. As expected, the is more accurate here than filter when p = (ideal case). But the error difference between two filters is very small: about several mm. Some other useful observations can also be made: It is only when. < p < that errors in are smaller than in filter. Moreover, estimates do not seem to be acceptable with p >. Similarly to Fig., error reduction is provided here by increasing k n for any reasonable p. Error reduction is more noticeable when p > (Fig. a) and lesser appreciable if p < (Fig. b). For example, with p = (Fig. a) the localization error of cm by k n = reduces to.6 cm by k n = 6. ISBN:

4 9 m m m Fig.. Schematic diagram of a vehicle platform travelling on an indoor passway in the RFID tag environment with 8 tags mounted on a ceiling and 8 tags mounted on a floor. III. ROBOT TRAVELLING ON A PASSWAY A most common organization of the RFID tag-based environment is a square grid with tags nested on a floor or/and ceiling []. This environment is simple, since the tag coordinates can easily be predicted based on its geometrical location. We construct such an environment in an indoor passway with the tags nested as in Fig.. We suppose that a reader is able to detect 8 tags at once at each floorspace point. It is also implied that not each tag is available at each floorspace point by some reasons (out of service, isolated by furniture, low power, etc.). However, at least two tags are always available. The following noise statistics are allowed: σ x = σ y = mm, σ L = σ R =. mm, and σ Φ =.. Because the tags mounted on a ceiling are most far distanced from a vehicle platform, we set different measurement noise statistics as σ v = = σ v8 = mm and σ v9 = = σ v6 = mm. The noise standard deviation in measured Φ n is set to be σ φ =. Using (), we then find N opt =. In such an environment, simple inverse solution to the measurement equation is unavailable to form a linear measurement vector y n. We therefore run the (Table I in []) with roughly set covariances (p = ) and initial values (allowing errors of %) and use its output ˆx n as y n in the filter (Table II in []) on a horizon of first N opt points. Thereby, we exploit the /Kalman algorithm. Taking into account that not all 8 tags can be detected at each floorspace point, we voluntary change the observed tag-sets each m along a passway referring to the nearest tags as in Table II. The instantaneous localization errors are sketched in Fig. for the case of exactly known noise statistics. Even a quick look at this figure suggests that the and estimates are almost identical so that the /Kalman filter ignoring the noise statistics is virtually as successful in accuracy as. Minimal localization errors in Fig. a correspond to detected tags (second interval) and maximal errors to detected tags (third interval). Although this deduction is supported by Fig. and Fig., we admit that the effect may vary in other nonlinear models that requires further investigations. TABLE II. RFID TAGS DETECTED IN 6 INTERVALS (IN M) ALONG THE PASSWAY SHOWN IN FIG.. TAGS, 6 WERE NOT AVAILABLE. Estimation error, - rad m Tag x x x x x x x x x 6 x x x 6 8 x x x x 8 x x x x x Tags Tags Tags Tags Tags Tags 6 8 Coordinate y Initial state for 6 8 Heading 6 8 (c) Fig.. Typical localization errors in 6 intervals specified in Table?? for exactly known noise covariances and N opt = : coordinate x, coordinate y, and (c) heading Φ. ISBN:

5 divergence.. p Fig. 6. Typical localization errors by and filter as functions of p. The filter is p-invariant. The diverges when p <.. TABLE III. AVERAGE LOCALIZATION ERROR STATISTICS (EXCLUDED N opt = INITIAL POINTS OF TRANSIENTS) CORRESPONDING TO FIG. FOR FILTER AND. x, cm y, cm Φ, rad bias σ bias σ bias σ (p = ) (p = ) (p =.) Localization error, m Localization error, m 6 8 (c) Fig. 7. Failures of due to imprecisely defined noise covariances with p <.: local instabilities, single divergence, and (c) multiple divergence. T/T T/T T6/T T8/T6 A situation changes dramatically if to define the noise covariances imprecisely owing to typically insufficient knowledge about noise. As follows from Fig. 6, the retains here a slight advantage in accuracy of several mm, but only within a narrow region of.7 < p <.. Beyond this region, errors in the grow rapidly and the filter becomes more advantageous. As an evidence, Table III gives typical average errors for the filter (N opt = ) and assuming p =, p =, and p =. with % of inaccuracy in the appointment of the initial state. What was unexpected that the will diverge with p <. (Fig. 6). The known causes of divergence in are larger nonlinearities and intensive noise []. Figure 6 definitely points out at another sourse of divergence errors in the noise covariances. To learn it in more detail, we repeat the estimates for p <. with different noise realizations and select three specific appearances shown in Fig. 7. One can recognize here local instabilities (Fig. 7a) which may not be treated as divergence. Figure 7b demonstrates a brightly pronounced single divergence between m and 6 m. Note that, within this interval, the reader detects tags (Table II) and the localization error is maximal (Fig. ). Multiple divergence is illustrated in Fig. 7c and we notice that diverges here every time when a target interacts with a small number of the tags. A conclusion one may arrive at is the following: an increase in the number of tags interacting with a target T/T9 T/T T/T T7/T Fig. 8. Typical appearance of the divergence between 8 m and m with tags (T, T6, and T) in a view. The diverges due to imprecisely defined noise covariances with p <.. prevents the divergence caused by imprecisely defined noise statistics. Just to illustrate the divergence in the x y plane, in Fig. 8 we give an example of a single effect along the passway. Note that the /Kalman algorithm (Table II in []) is p-invariant and thus protected against such kind of failures. However, it may fail during the first N opt points if the initiating diverges in this interval as shown in []. IV. CONCLUDING REMARKS In general, the,, and /Kalman algorithms become more successful in accuracy by increasing the number of detected tags. However, this number (six in our case) is limited in the target nonlinear model. Besides, if the noise statistics are specified imprecisely with p >, then an increase in accuracy in is most noticeable. It was also revealed that the becomes addicted to divergence if p <. Hereby, ISBN:

6 we state that deviations from the actual noise covariances in nonlinear state-space models may cause divergence in. Note that the filter is p-invariant and thus protected against such sort of failures. But, the filter requires about N times more computation time to complete iterations. We finally notice that the /Kalman algorithm developed in this paper should be tested by uncertainties and non- Gaussian noise often peculiar to applications and compared to the PF and UKF. Divergence in caused by errors in the noise covariances also needs further investigations. REFERENCES [] S. S. Saab and Z. S. Nakad, A standalone RFID indoor positioning system using passive tags, IEEE Trans. Ind. Electron., vol. 8, no., pp , May. [] E. DiGiampaolo and F. Martinelli, A passive UHF-RFID system for the localization of an indoor autonomous vehicle, IEEE Trans. Ind. Electron., vol. 9, no., pp, 96 97, Oct.. [] V. Savic, A. Athalye, M. Bolic, and P. M. Djuric, Particle filtering for indoor RFID tag tracking, in Proc. IEEE Statist. Signal Process. Workshop (SSP),, pp [] M. Boccadoro, F. Martinelli, and S. Pagnotelli, Constrained and quantized Kalman filtering for an RFID robot localization problem, Auton. Robots, vol. 9, no. -, pp., Nov.. [] J. Pomarico-Franquiz, M. Granados-Cruz, and Y. S. Shmaliy, Selflocalization over RFID tag grids excess channels using extended filtering techniques, IEEE J. of Selected Topics in Signal Process., vol. 9, no., pp. 9 8, Mar. [6] S. Park and H. Lee, Self-recognition of vehicle position using UHF passive RFID tags, IEEE Trans. Ind. Electron., vol. 6, no., pp. 6, Jan.. [7] A. Howard, Multi-robot simultaneous localization and mapping using particle filters, Int. J. of Robotics Research, vol., no., pp. 6, Dec. 6. [8] E. DiGiampaolo and F. Martinelli, Mobile robot localization using the phase of passive UHF-RFID signals, IEEE Trans. Ind. Electron., vol. 6, no., pp. 6 76, Jan.. [9] F. Martinelli, Robot localization: comparable performance of and UKF in some interesting indoor settings, in Proc. 6th Mediterranean Conf. on Contr. Autom., Ajaccio, France, June -7, 8, pp. 99. [] B. Gibbs, Advanced Kalman Filtering, Least-Squares and Modeling, New York: Wiley,. [] Y. S. Shmaliy, An iterative Kalman-like algorithm ignoring noise and initial conditions, IEEE Trans. Signal Process., vol. 9, no. 6, pp. 6 7, Jun.. [] R. J. Fitzgerald, Divergence of the Kalman filter, IEEE Trans. Autom. Control, vol. AC-6, no. 6, pp , Dec. 97. [] F. Daum, Nonlinear filters: beyond the Kalman filter, IEEE Aerosp. Electron. Syst. Mag., vol., no. 8, pp. 7 69, Aug.. [] W. H. Kwon and S. Han, Receding Horizon Control: Model Predictive Control for State Models. London: Springer,. [] Y. S. Shmaliy, Linear optimal FIR estimation of discrete timeinvariant state-space models, IEEE Trans. Signal Process., vol. 8, pp , Jun.. [6] A. M. Bruckstein and T. Kailath, Recursive limited memory filtering and scattering theory, IEEE Trans. Inf. Theory, vol. IT-, no., pp., May 98. [7] W. H. Kwon, Y. S. Suh, Y. I. Lee, and O. K. Kwon, Equivalence of finite memory filters, IEEE Trans. Aerospace Electron. Syst., vol., no. 8., pp , Jul. 99. [8] A. H. Jazwinski, Stochastic Processes and Filtering Theory, New York: Academic, 97. [9] Y. S. Shmaliy, Unbiased FIR filtering of discrete time polynomial state space models, IEEE Trans. Signal Process., vol. 7, no., pp. 9, Apr. 9. [] Y. S. Shmaliy, Suboptimal FIR filtering of nonlinear models in additive white Gaussian noise, IEEE Trans. Signal Process., vol. 6, no., pp. 9 7, Oct.. [] J. Pomarico-Franquiz, S. Khan, and Y.S. Shmaliy, Combined extended FIR/Kalman filtering for indoor robot localization via triangulation, Measurement, vol., pp. 6, Apr.. [] J. Pomarico-Franquiz and Y.S. Shmaliy, Accurate self-localization in RFID tag information grids using FIR filtering, IEEE Trans. Ind. Inform., vol., no., pp. 7 6, May. [] M. Granados-Cruz and Y. S. Shmaliy, Extended filtering for selflocalization over RFID tag grids excess channels II, in Proc. The Int. Conf. on Systems, Control, Signal Process. Informatics, Barcelona, Spain, April 7-9,. [] J. Zhou and J. Shi, RFID localization algorithms and applications a review, J. Intell. Manuf., vol., no. 6, pp , Dec. 9. [] F. Ramirez-Echeverria, A. Sarr, and Y. S. Shmaliy, Optimal memory for discrete-time FIR filters in state space, IEEE Trans. Signal Process., vol. 6, no., pp. 7 6, Feb.. ISBN: