Performance of UWB position estimation based on time ... - IEEE Xplore

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Performance of UWB Position Estimation Based on Time-of-Arrival Measurements Kegen Yu Centre for Wireless Communications University of Oulu, Finland e-mail: [email protected]

Absmcf-The paper reports on the development of a low-eort device for low data rate communications with tracking and positioning capabilities. We Investigate the performance of two different position estimation methods based on the estimate of the timcof-arrival (TOA) of the UWB signal at a set of receiverslsenson. The performance evaluation is performed io terms of the root mean-squared (RMS) error of the posltion coordinates estimation and the failure rate. We first study the dlrectcalculation method which gives exact solutions of a set of slmultaneous equations. We then study one of the classical nonlinear optimization techniques, the DavidonFletcher-Powell (DFP) quasi-Newton algorithm. Both the direct-calculation method and the nonlinear optimization algorithms do not require any knowledge of the TOA estimation error variance or distribution. Thls advantage would be attractive for practical applications. Index Terms-3-D position estimation, time of arrival, direct enleulatian, DFP formula, UWB.

1. INTRODUCTION

URFA (UWE RF ASIC) is a project jointly managed by CWC at University of Oulu and PJ Microwave. The intended application of the research is to develop a device which is capable of communicating and participate in a positioning network. The device must be very low cost so as to allow placement onto any moderate value assets such as parcels and clothing. The transceiver is envisaged to be sealed in an epoxy resin container case with its own power supply. The device is to be disposable, water-proof and bas an operational life of several months to a year. The physical dimensions of the device should be approximately the size of 1 EURO cent piece. This work focuses on the investigatioddevelopment of feasible position estimation algorithms. There are many position estimation techniques using radio signals for various pulposes under different scenarios. Signal strength, angle of arrival (AOA), time measurements (TOA, time of flight, and time difference of arrival (TDOA)) can all be exploited for the position estimation. The most straightfonvard way to estimate the position may be directly solving a set of simultaneous equations [I] based on the TDOA measurements. Therefore, exact solutions can be obtained for 2-D positioning

0-7803-8373-7/04/$20.0002004 IEEE

Ian Oppermann Centre for Wireless Communications University of Oulu, Finland e-mail: [email protected]

with three sensorsltwo TDOA measurements, and for 3D positioning with four sensors. For an over-determined system (with redundant sensors), Taylor series expansion may be used to iteratively produce a linearized leastsquare solution to the position estimate [2]. However, to maintain good convergence, the Taylor series method requires a quite accurate initial position estimate which is often difficult to obtain in some practical applications. To avoid the convergence problem, several different approaches have been proposed such as the spherical interpolation [3-51 and the double maximum likelihood (ML) method [6].The above mentioned hyperbolic positioning methods (except for the Taylor series method) share one common drawback of multiple solutions. Also they need the knowledge of the variance and/or the distribution of the TOA estimation error (except for the direct method). A different method for positioning comes from nonlinear optimization theory. The gradient-based algorithms may be employed for position estimation. One is the DFP quasi-Newton algorithm [7] which has been used in the UWB precision assets location system [SI developed by Multispectral Solution, Inc. In this paper, we are interested in the performance investigation of several estimation methods which employ the TOA measurements. We first discuss synchronization in Section 111, which is crucial due to its direct impact on the performance of the positioning system. We then present the expressions of the estimate of the position coordinates in the direct-calculation method in Section IV. Section V briefly examines the DFP quasiNewton algorithm, which may be used to improve the estimation accuracy especially in an over-determined system. Simulation results are finally conducted to show the performance of the methods in terms of the R M S of the estimation error and the failure rate, presented in Section VI. 11. BRIEFSYSTEM DESCRIPTION In the system there are a set of sensors which are perfectly synchronized by either sharing the same global clock or sharing the same local clock through a cable connection. The sensors are positioned at known coordinates in the area to be monitored. Also in the monitored

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area, there are certain number of active tags attached to assets which are to be located. The data transmission is packet based using a TDMA scheme. Due to the drift in the clock of the tag, synchronization between the sensors and the tag is performed once every second. This is achieved by broadcasting a beacon from one of the sensors to the tags. The TOA of the beacon is used as the reference clock for the tag to transmit data according to the pre-assigned time slot. Therefore, affordable oscillators can be used with this re-synchronization method'. Once the signal from the tag is received at the sensor, the TOA is estimated. The estimated TOA is passed to the position estimation algorithm to produce the position estimate. The impulse response of the channel model is given by L

h(t) =

Nf6(t - 71)

e= 1 where L is the number of the total channel paths, 01 is the eth path gain of lognormally distributed, and 71 is the time delay of the Pth path. This model may be considered as a simplified version of the IEEE 802.1S.SG3a channel model final report [9], since we only consider one cluster of rays and lognormal shadowing is not considered. 111. TOA ESTIMATION To achieve accurate position estimation, we first need to acquire accurate TOA measurements. There exists numerous TOA estimation algorithms in the literature. A quite comprehensive literature review on code acquisition and delay estimation can he found in [IO, 111. Recently, several researchers have proposed some techniques to obtain rapid synchronization for the UWB system. For the correlator-type synchronization, serial search techniques such as the look-and-jump search and the bit reversal search have been proposed in [12]. Special code design has been employed in [13]. Chip-level post-detection integration (CLPDI) has been proposed in [14] and applied to UWB in [IS]. On the other hand, frequency-domain treatment of UWB synchronization using the spectral estimationhas been considered in [16]. In this work we employ the correlator for synchronization. Since the system operates in the single-user channel. A binary m-sequence is employed as the spreading code2. All tags share the same code but have distinct tag ID numbers. When a signal is detected at the sensor, the

incoming signal is correlated with the restored template waveform of different code offsets. Assume that a group of correlators are available at each sensor. The possible range of code offsets is divided into sectors. Each sector of code offsets is approached with one specific correlator. We choose the maximum selection criterion (MSC) for assessing correlation values, involves testing all code offsets and then picking the offset with the maximum correlation value [17-191. This MSC hypothesis test rule is suitahle for the situation where there are not too many offsets to be tested'. We then employ the general two-stage principle, i.e. search stage and verification stage [20, 211. When an offset with the maximum correlation value is found, it is temporarily chosen as the desired one and then the verification stage starts. In the verification, if the chosen offset has the maximum correlation value at least I E ~times out of the K tests, then the hypothesis is accepted, otherwise the search recommences. The initial acquisition may be locked to any path in the multipath channel. Further search is required to obtain synchronization with the first path. This may he achieved by backward search for a number of offsets not larger than the number of paths. Once the initial code acquisition of the first path is accomplished, fine acquisition may be pursued by testing several sub-chip offsets to obtain a more accurate TOA estimate. IV. DIRECTCALCULATION In the Cartesian system, the range (distance) between sensor i and the tag is given by

J ( z - z;)Z

i = 1, 2, 3, 4 (1) where c is the speed of light, t; is the TOA at sensor i, and t o is the transmit time at the tag. In the development of the expressions, we ignore the difference between the hue and the measured TOAs for simplicity. Squaring both sides of (1) gives (z - z;)2

+ (y - Yi)2 + ( 2 - 2 i ) Z = c y t i - t o y , i = 1, 2, 3, 4 (2)

Subtracting (2) for i = 1 from (2) for i = 2, 3, 4 produces

'Alternatively, whm the packet duration is entrcmcly s h m compared to the packet period at cach tag and the system load is not too heavy, the random BEECSS technique may be employed due to the small probability of collision betwem packets. Thiri random BCE~SSscheme has bcm exploited in [E]. Of COUISC, retransmission is required once a collision OECUIS. To decrease the chance of collision, the channel sensing facility may be equipped at caeh tag. 'In a multiple access channel, some particular codes such as the Gold codes and tbs Kascode would be employed to obtain the desirable cross-comlation of the codes.

+ (y - y;)2 + (2 - Z i ) 2 = c(t; - t o ) ,

do = ;(tl+ti)+

1 2C@l

~

ti)

(al-2zilz-2yily-22il.),

i = 2, 3, 4 (3) 'Whcn the number of ofEets is considerably large. B hybrid search may be applied. The Sectors are processed in parallcl, and insih each sector B serial search is employed. la the serial search, threshold erassing d e in used far hypothesis testing and the look-and-jump search fcchniquc [12,13] may be Ulicd to speed up the synchronization.

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Substituting (6), (7) and (8) back into (1) for i = 1 followed by squaring yields

G~~+ H ~ I+= o

(9)

where

+1- E ~ G = +c2 H = 2 [ A ( B- 51) C(D - vi) - 21 - E F ] I = ( B - XI)* (D - yl)' z? - F 2

+

+

+

The two solutions to (9) are z=

-25 2G *

JW

(10)

The two estimated z values (if both are reasonable) are then substituted back into (6) and (7) to produce the estimate of the coordinates x and y, respectively. However, there is only one desirable solution. We get rid of the one either with no physical meaning or beyond the monitored area. If both solutions are reasonable and they are very close, we may choose the average as the position estimate. Otherwise, an ambiguity occurs. Other cases of no acceptable results include two complex solutions, both solutions of beyond the monitored area. To increase the probability of the existence of one reasonable position estimate, we may add a fifth sensor. Then we have five different combinations, producing five different results. The benefits from an extra sensor will be shown in Section VI, but at the cost of increased system complexity and computation complexity In practice, there may exist more than five sensors. In this case, we may choose the five sensors with the highest received signal powers. This simple method is particularly suitable when the noise level is similar in the received signals. When a sequence of measurements are available at each sensor, accuracy may be further improved by first processing the sequence of measurements such as averaging. V. NONLINEAR OPTIMIZATION METHOD The objective function is defined as

-c(t* - t o ) } 2 Then, substitution of (6) and (7) back into (3) for i = 2 produces c(t1 - t o ) = E z F (8)

+

where

(11)

Where

P = [x, Y,2, tolT is the vector of the unknown position coordinates (x, y, z) to be estimated and the unknown transmit time to. Also ti is the known receive time of the ith sensor. This objective function has been considered in [SI for UWB precision asset position location with the DFP quasi-Newton algorithm. Clearly, the objective function is the summation of the squared range errors

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of all sensors. The optimization purpose is to minimize this objective function to produce the optimal position estimate. Let us briefly study the DFP formulae. The solution to the position estimate is solved iteratively by

'

2.5

1

1

P ~ += I ~k - OBkgr

where p h is the vector of the estimated position coordinates and the estimated transmit time at the kth iteration, and IY is the step size. Also gk is the gradient of the objective function given by

0.5

-

-m-

~

B

-

DlRECTran OFPrsn OIRECTsel DFPsel

.I=.a 0' sts ndad Oeviatlon of TOA Ermr

And also Bk is the so-called inverse Hessian which is updated according to

Fig. 1. Root mean squared emor of position ertimstion. D F P m and DFPsel denote the average R M S enor of the x, y and z coordinate estimation using the DFP algorithm with 5 sensors with randomly chosen, and ~clectivslychosen sensor positions respsotivcly. DlRECTran and DlRECTsel are the correspondingly results with the direct calculation using 5 sensors. The re~ultsof the direct cslculation method using 4 semors are nearly the same Bs the case of 5 SenSOTs.

where hk = P ~ + I ~b 9 k = gk+l gk

To stalt the iteration, the initial position coordinates and the initial transmit time are required. The initial estimated values of the position coordinates may he chosen to be the mean position of the all active sensors or the area under being monitored. The initial estimated transmit time may be chosen to be some time point earlier than the earliest receive time. How much earlier will depend on the dimension of the monitored area. At each iteration both the step size CY and the first partial derivatives g (the gradient) are updated.

the failure rate decreases dramatically with five sensors compared to the case of four sensors, although the accuracy is nearly the same. The performance gain is achieved at the cost of increased computation complexity and system complexity. Figs. 3 and 4 show the results for the position estimation combined with delay estimation. We choose the direct-sequence spreading with spreading gain 31 (i.e. 31 chips per data bit). The pulse width is.O.4 nanosecond and the duty cycle is 1/14. The sampling rate is 6 GHz which could he feasible at the base stations. A three-path channel model is employed.

VI. SIMULATION RESULTS The monitored area has a dimension of 40(1) x 30(w) x 5(h) m. The position of the sensors and the tag of interest is randomly generated to obtain the average performance. The performance evaluation is first performed by assuming that the TOA measurement error is the i.i.d. random variable of Gaussian disrributed with zero mean and variance oz. Note that the two position estimation algorithms do not rely on this particular distribution of the TOA estimation errors. At each test point (c),1000 runs are conducted with new random position of the sensors and the tag at each run. The performance is then averaged. The case of one particular location of the base stations is also examined. This location is randomly chosen &om 50 locations of the base stations which produce the hest results. Figs. I and 2 show the corresponding root mean square (RMS) ermr of the coordinates estimation and failure rate. Clearly, for the direction-calculationmethod,

Standard DeviaUon of TOA Ermr

Fig. 2. Fdurs rate of position estimation.Either four (DIRECT(4)) or five (DIRECT(5)) sensors arc examined for the direct calculation method

- 4n3 -

-

DFPran DIRECT%

-1.! E

I

i 5

10 Average SNR (dB)

15

R M S errors of position ertimation combined with TOA

Fig. 3.

sstilnetion.

-0-

-

DFPran DlRECTran

+ DFPsel

0.14

DiRECTrel

0 12

0

-..M

.

L

0.0 s-...

OO

5

10

Average SNR (dB)

1

~ . . 15

FBilure rate of position estimation combined with TOA Fig, 4. estimation.

VII. CONCLUSIONS

In this paper we investigated the performance of a couple of practical position estimation methods. The aim of this work is to investigateldevelop position estimation techniques for an UWB s y s t e m envisaged for communication and 3-D positioning. Ongoing work includes investigation of more practical and efficient TOA estimation techniques. REFERENCES [l] B. T.Fang, ”Simple rolutions for hypsrbolic and related position fixcs:’ IEEE Trans. Aemsp. Elecchon. Syrl., vol. 26, pp. 748753, Sept. 1990.

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