A Location System Using Asynchronous Distributed Sensors

Transcription

1 A Location System Using Asynchronous Distributed Sensors Teng Li, Anthony Ekpenyong, Yih-Fang Huang Department of Electrical Engineering University of Notre Dame Notre Dame, IN 55, USA {tli, aekpenyo, Abstract This paper describes a location system using an asynchronous distributed sensor network The sensors clocks operate independently and are not explicitly synchronized The motion of the source, or the object, between two emitted pulses will cause the inter-arrival time of these two pulses at each sensor to vary This inter-arrival time is a function of the source location and motion and is independent of the clock offsets among sensors Thus, the independently calculated timedifference-of-arrival (TDOA) of successive pulses at each sensor provides a reliable source of information to estimate the source position as well as its motion The Cramer-Rao Lower Bound (CRLB) is derived to assess the performance of the proposed method and to show that its performance can approach that of a synchronous network given a sufficient number of sensors and moderate source movement Some sub-optimal but effective source trajectory tracking methods are proposed which further relax the assumption of source motion between consecutive pulses The simulation results show that the proposed estimation and tracking methods are quite good I INTRODUCTION Location systems are becoming increasingly important for many position-aware applications that include product tracking in supply chain management, locating equipment and/or people in hospitals, schools and offices Many systems use a lateration technique to determine the position of the desired object [1] A common approach is to compute the distance from the object (or signal source) to a sensor by measuring the time-of-flight of a signal emitted by the source A key requirement, therefore, is synchronization between the source and the sensor A different approach is to measure the time-differenceof-arrival (TDOA) of the signal between pairs of sensors [2], [3] In this case, synchronization is required between sensors Differences between sensor clocks will contribute significantly to TDOA estimation error In this paper, we consider a location system similar to Active Bat [] and Cricket [5] where an emitted acoustic signal is designed specifically for locating an object On the other hand, source localization schemes in [2], [3], [] locate a source emitting an arbitrary signal In the Active Bat and Cricket systems, an explicit notification of when the pulse is transmitted, is sent using a RF signal, to the sensors (or source) to achieve synchronization In localization systems implemented over a wireless sensor network, synchronization among distributed nodes is traditionally performed by a server or master station which broadcasts its time (solicited or unsolicited) to all dependent nodes This technique is prone to jitter due to delays in data processing at each sensor, or time-varying transmission times in contentionbased multiple access channels In [], an improved algorithm, termed Reference Broadcast Synchronization (RBS) algorithm [7] is used to mitigate jitter Synchronization is implemented by an exchange of physical layer beacon signals between receiver nodes An alternative precise synchronization scheme is to use the global positioning system (GPS) However, a clear view of the sky is required This would be impossible or at least quite challenging to implement for some systems such as an indoor location system Therefore, synchronization can be achieved by the methods mentioned above which add extra cost to the system, eg, increased signal processing, inter-node communication and synchronization protocols This cost is not desirable for a distributed sensor network in which each node is energy limited In addition, the synchronization errors will degrade the accuracy of the source location estimation An intriguing question is: can a location system be implemented with an asynchronous sensor network? In this paper, we propose a location system using an asynchronous network of wireless sensors 1 We consider a scenario where a moving source periodically transmits a specific signal pulse which is received by N sensors Each sensor independently counts the received pulses and estimates the time-of-arrival (TOA) in its local time It then transmits this estimate to the master station over a RF link There is no mandatory synchronization among source and sensors This system differs from Active Bat and Cricket in that the sensors operate with asynchronous clocks and are autonomous distributed nodes Since the TOA estimates are computed in asynchronous local times, the traditional lateration techniques fail to provide an accurate source position estimate This paper proposes a novel method to estimate source location accurately using these estimates The basic idea is to exploit the inherent resources in a distributed location system, namely, the movement of the source, number of sensors and periodically emitted pulses When the source moves between two successive emitted pulses, its motion will cause a change in the inter-arrival time observed by each sensor Therefore, the inter-arrival time of 1 An asynchronous network is defined here as one with a level of synchronization several orders lower than that required for source location

2 two consecutively emitted pulses at a sensor is a function of the source location and its displacement We introduce the notion of a virtual sensor to explain this functional dependence Assume the source has displacement d between emitted pulses P1 and P2 spaced apart by an interval L Equivalently, it can be viewed that the source is fixed while the sensor is displaced by d relative to the source Thus P1 and P2 are received by the sensor and its displaced (virtual) version respectively, and, their TDOA can be calculated reliably since they are physically on the same node and, therefore, synchronized Source location is computed from TDOA estimates obtained from N pairs of sensors where one sensor is real and the other virtual As a result, unlike in traditional networks where there is networkwide synchronization, here each sensor operates independently to locate the source By designing the overall estimator using both position and displacement estimates, the assumption of source displacement between two consecutive pulses can be relaxed A similar idea of exploiting the source motion is used in [8], [9] to compute source location from Doppler-shifted frequency measurements In these papers, each sensor computes an estimate of the source location from sequential Doppler measurements and then a master station computes a more reliable estimate by combining the estimates from a group of sensors However, all sensors use the same clock and, furthermore, it is assumed that the source has a constant velocity We make no such assumptions in this paper and, in particular, we examine the tracking performance of our method for an arbitrary source trajectory The paper is organized as follows Section II describes an asynchronous distributed location system and identifies the synchronization parameters and their effects Section III establishes the functional dependence between the source location and the reliable pulse inter-arrival time observations The maximum likelihood estimate of both source location and displacement is also computed Section IV derives the Cramer-Rao Lower Bound (CRLB) for the proposed method and compares it to the synchronous case The proposed method is applied in Section V to track the source in some practical scenarios Several tracking methods are proposed based on the displacement estimate to obtain a better position estimate Section VI presents the simulation results and Section VII concludes the paper II PROBLEM FORMULATION The location system considered in this paper has N distributed and autonomous sensors at some fixed and known position x i R D where i =1, 2, N is the sensor index and D is the dimension of the position vector The objective of the system is to estimate the trajectory, denoted as x (t), of an object which emits some designed and known signal The N sensors process the received signal independently and send the observation to a master station to estimate the position of the object The system can track multiple objects at the same time if the sensors can separate the signals transmitted by each object Since the focus of the paper is to provide a solution for asynchronous sensors, we only consider a single source for simplicity The asynchronous nature of the system is due to the fact that the clocks in the sensors and the source are independent The clock at the j th node starts at some unknown time Ω j, where we have used j to index both the sensors (j =1, 2,,N) and the source (j = ) Its rate F j has some unknown fractional frequency drift, ɛ j, from the known nominal clock rate F s expressed as F j = F s (1 ɛ j ), (1) whereweassumeɛ j to be an independent Gaussian distributed random variable, ɛ j N(,σ 2 f ) The typical range of ɛ j is from 1 to 1 [1] Since the system makes no attempt to synchronize these independent clocks, each node only knows its local time l j which is in clock ticks The local time l j can be linearly related to its corresponding global time t j, which is in seconds, by t j = Ω j + l j T j = Ω j + l j T s 1 (1 ɛ j ) Ω j + l j T s (1 + ɛ j ) (2) where T j =1/F j is the clock interval and the last approximation follows from the Taylor expansion, discarding the second and higher order terms since ɛ j is small {ɛ j } N j= and {Ω j} N j= are the unknown synchronization parameters The object periodically emits a known signal pulse s(t) with a known and constant propagation speed, c The pulses are emitted at source local time l = {pl} p= where p is the pulse index and L is the interval, in source local time, between consecutive pulses L is a design parameter and is known at the master station Thus, the master station knows the expected interval between successive pulses, LT s, but the instantaneous value, LT s (1+ɛ ), is unknown The position of the source is only observable when a pulse is transmitted This implies that the continuous source motion, x (t), is sampled at rate 1 T s(1+ɛ ) which gives a discrete sequence of the source motion trajectory From (2), this sequence can be expressed in global time as {x (plt s (1 + ɛ )+Ω )} p= (3) which is denoted by {x(p)} p= Each sensor samples the received signal and estimates the TOA of the signal Since the signal pulse s(t) is known to all sensors, one way to estimate the TOA is to correlate the received signal with a locally generated waveform and look for the peak position of the output Again, since our emphasis is on the performance of an asynchronous network, it is sufficient to estimate the TOA in clock ticks although it is possible to estimate it in fractions of a clock tick At the i th sensor, the TOA estimate of the p th pulse, ˆl i (p), is given by ˆli (p) =l i (p)+n i (p) ()

3 Ω Fig 1 p th pulse l i (p) l (p) l (p +1) L (p +1) th pulse sensor local time l i l i (p +1) source local time l global time t Ω i t (p) t i (p) t (p +1) t i (p +1) Time associated with two consecutive pulses where l i (p) is the true TOA in local time and n i (p) is the estimation error which is assumed to be an iid Gaussian random variable, n i N(,σ 2 n) Each sensor sends a short message composed of its index i, the TOA estimate ˆl i (p) and the pulse index p to the master station The problem is to estimate the source locations {x(p)} p= given a set of observations {ˆl i (p)} p=, i = 1,, N In order to associate a correct pulse index with its TOA estimation, the sensors must count the pulses synchronously This could be easily achieved by inserting a special pulse at the start of source transmission The sensors detect the special pulse and can then count the subsequent pulses synchronously In Fig 1, the time relationship for two consecutive pulses, p and p+1, is illustrated The p th pulse is emitted at source local time l (p) =pl which corresponds to a global time of t (p) =plt s (1 + ɛ )+Ω Its time of propagation to the i th sensor is x(p) xi c and it arrives at global time t i (p) = x (p) x i + plt s (1 + ɛ )+Ω c This time can be converted to the i th node s local time using (2) and is estimated as ˆl i (p) from () Thus, the local TOA estimate is related to the source position as ˆli (p) = x (p) x i (1+ɛ i ) +pl1+ɛ + Ω Ω i 1+ɛ i T s (1+ɛ i ) +n i(p) a) b) x (p) x i (1 ɛ i ) +pl(1 + ɛ ɛ i )+ Ω Ω i (1 ɛ i )+n i (p) x (p) x i +pl(1 + ɛ ɛ i )+ Ω Ω i + n i (p) (5) T s In a), a Taylor expansion is performed similar to (2) while in b), ɛ i is ignored in the first and third terms since it only causes a negligible change However, ɛ i is kept in the second term since p can increase without bound The TDOA between nodes i and j can be calculated using T s (5) as ˆli (p) ˆl j (p)= x (p) x i x (p) x j + pl(ɛ j ɛ i )+ Ω j Ω i +n i (p) n j (p) () T s From () we can see that the observation is the true TDOA with some additive error terms The second term is the accumulated error caused by different frequency offsets while the third term is the error caused by different time offsets The second term is not negligible since p can be unbounded and the third term can also be significant since there could be a large difference between Ω i and Ω j Therefore, the TDOA estimate between two asynchronous sensors is not reliable III PROPOSED METHOD In this section, a location algorithm is developed which does not estimate the TDOA between sensors Instead, we compute the TDOA estimation between consecutively received pulses at a single node which is ˆli (p +1) ˆl i (p)= x (p +1) x i x (p) x i +L(ɛ ɛ i +1)+n i (p+1) n i (p) (7) By subtracting L from both sides of (7) and defining y i (p) ˆli (p+1) ˆl i (p) L, we get y i (p)= x (p +1) x i x (p) x i +L(ɛ ɛ i )+n i (p+1) n i (p) = x (p)+d(p) x i x (p) x i +e i (p) = f i (x (p), d(p)) + e i (p) (8) where d(p)=x (p+1) x (p) is the displacement of the source between the p th and (p+1) th pulses and f i (x, d)=( x + d x i x x i )/ is a deterministic function of the source position and displacement vectors e i (p) = L(ɛ ɛ i )+n i (p+1) n i (p) is the effective noise comprised of estimation errors and ( frequency offset) and is Gaussian distributed, e i (p) N, 2(L 2 σf 2 + σ2 n) It may be observed that the time offset error term in (), (Ω j Ω i )/T s, is eliminated while the frequency offset error, pl(ɛ j ɛ i ), is reduced to L(ɛ ɛ i ) and is now bounded Therefore, we can treat the small frequency offset in (8) as an additive error All N sensors compute the TDOA estimate between consecutive pulses and these estimates can be stacked to form an N-dimensional vector equation y = f (x, d)+e (9) where e is a zero mean Gaussian random vector with covariance matrix Q = E[ee T ]=σqi 2 N σ 2 Q is the effective noise variance and is equal to 2(L 2 σf 2 + σ2 n) We have dropped the pulse index p for simplicity

4 x (p) d(p) source x (p +1) x 2 d(p) x 3 d(p) x 2 x 3 x (p +1) d(p) x 1 d(p) x (p) source x N d(p) x i (p) d(p) virtual sensor i x i(p) sensor i x 1 virtual sensor pair Fig 3 Network composed of N virtual pairs x N Fig 2 Description of a virtual sensor pair We give a geometrical interpretation of the above estimation method by defining x i (p) x i(p) d(p) and re-writing f i (x, d) as f i (x, d) = x x i x x i (1) This is of the same form as the general hyperbolic location estimation [3] with two sensors at x i, x i In light of (1), another sensor, the virtual sensor at position x i, is paired with the i th sensor at position x i This virtual sensor is merely the i th sensor shifted by d(p) due to the motion of the object between the p th and (p+1) th pulses as shown in Fig 2 It has exactly the same clock as the i th sensor The combination of a sensor and its virtual sensor is termed a virtual pair The TDOA can be calculated reliably for the virtual pair Therefore the TDOA of two consecutive pulses received at the i th sensor is interpreted as the TDOA of a single pulse received at the i th sensor and its virtual sensor, plus the constant interval L We can now view the asynchronous sensor network as a set of N fully synchronized sensor pairs with the same but unknown displacement vector d(p) as shown in Fig 3 The source location problem can therefore be re-cast as a joint estimation of the source position and displacement vector which can be combined into one vector parameter θ = [ x d ], θ R 2D The likelihood function for the observation vector is 1 p(y; θ)= (2π) N/2 Q 1/2 exp { (1/2)[y f (θ)] T Q 1 [y f (θ)] } where denotes the determinant of a matrix The maximum likelihood estimate (MLE) of θ is [ ˆθ MLE = arg min y f (θ)] T Q 1 [y f (θ) ] (11) θ However, since f( ) is a nonlinear function of θ, one solution is to linearize it by a Taylor series expansion around an initial guess, ˆθ, for the true parameter vector [11] Hence, we have f(θ) f(ˆθ )+G(ˆθ )(θ ˆθ ) (12) where only the first two terms of the expansion have been retained and G(ˆθ )= θ f(θ) ˆθ, is the gradient of f( ) θ= evaluated at θ = ˆθ Solving (11) using (12) we then obtain ( ) 1 ˆθ=ˆθ + G T (ˆθ )Q 1 G(ˆθ ( ) ) G T (ˆθ )Q 1 y f(ˆθ ) (13) An initial guess close to the true solution is not normally available, thus (13) is solved iteratively as ( ) 1 ˆθ k+1 =ˆθ k + G T (ˆθ k )Q 1 G(ˆθ k ( ) ) G T (ˆθ k )Q 1 y f(ˆθ k ) (1) where ˆθ k is the solution at the k th iteration This is a gradient descent technique to find ˆθ MLE This iterative descent method suffers from the problem of local minima due to the nonlinearity of f (θ) A more general grid search method can be applied as in [9] However, for the case of source tracking, where a previous position estimate can serve as a good initial guess, the iterative descent method can perform well There are 2D unknowns in N equations as can be observed from (9) Hence it is necessary that the number of sensors, N, is no less than 2D for G T Q 1 G to be full rank At every pulse instant p, the parameter vector ˆθ(p) =[ˆx (p) T, ˆd(p) T ] T is found by using the iterative procedure in (1) which uses the previous estimate ˆθ(p 1) as an initial guess IV CRAMER-RAO LOWER BOUND It is well known that the variance of an unbiased estimator is bounded below by the CRLB The CRLB is used in this section to study the performance of the proposed location method The CRLB is defined in terms of the Fisher information matrix which is given as I(θ) =E [ ( )( ) ] T ln p(y; θ) ln p(y; θ) (15) θ θ

5 For the Gaussian case, it is expressed as [12] I(θ)= f(θ) T 1 f(θ) Q θ θ =G T (θ)q 1 G(θ) (1) where I(θ) is evaluated at the true value of θ For simplicity, from now on, we will only consider the results in a two dimensional space (the extension to three dimensions is straightforward) The position vector and displacement vector can be written as x j =[x j,y j ] T and d =[d x,d y ] T G(θ) can then be expressed as G= f 1 f 1 f 1 f 1 x y d x d y f N x = 1 f N y f N d x f N d y R x 1 R x1 R y 1 R y1 R x 1 R y 1 R x N R xn R y N R yn R x N R y N (17) where R x i = x+dx xi x, R +d x i x i = x xi x, x i R y i = y+dy yi x, +d x i R yi = y yi x x i for i =1,,N The variance of any element of ˆθ is then bounded below as E[(ˆθ k θ k ) 2 ] [I 1 (θ)] kk k =1,, 2D (18) where [I 1 (θ)] kk is the k th diagonal element of I 1 (θ) We may also derive the bounds for the range and bearing in the two-dimensional case Define g(θ) =[r, α] T where r, α are the range and bearing respectively We have that r =(x 2 + y) 2 1/2, α = tan 1 y x The CRLB for ĝ(θ) is given as [12] Cov(ĝ(θ)) g(θ) θ I 1 (θ) g(θ) T [ θ x y x = x ] I 1 (θ) y x x 2 x 2 [ x y x x y x x 2 x 2 ] T (19) where the matrix notation A B means A B is positive semi-definite Various plots of the CRLB are discussed in detail in Section VI Since the estimation scheme given in Section III depends on the assumption that the source moves between consecutive pulses, a natural question is: what is the performance of the estimator if the source moves slowly and/or stops? We will derive the asymptotic bound on the variance of the position and displacement estimate as the displacement goes to zero Let the source be at the origin of the coordinate system as shown in Fig The angles of the vectors pointing from the source to the i th sensor and its virtual sensor are denoted as ψ i and ψ i respectively It is straightforward to show that G in (17) can be written as Fig source y α d ν i ψ i x + d x i ψ i d sin(α + ψ i ) x x i i th virtual sensor d i th sensor Source-sensor illustration to derive an asymptotic bound on variance cos ψ 1 cos ψ 1 cos ψ G = 1 N cos ψ T N sin ψ 1 sin ψ 1 sin ψ N sin ψ N cos ψ 1 cos ψ N sin ψ 1 sin ψ N As shown in Fig, let ν i = ψ i ψ i for i =1,,N d sin(α+ψi) d and since sin ν i = x +d x i x +d x i we have ν i as d By using the trigonometric identities for cos Ψ i cos Ψ i and sin Ψ i sin Ψ i and the limits of cos ν i 1, sin ν i ν i as ν i, the diagonal entries of the Fisher information matrix are obtained as diag(i(θ)) = 1 σ 2 Q x i ν2 i sin2 ψ i i ν2 i cos2 ψ i i cos2 ψ i (2) i sin2 ψ i The CRLB can be further lower bounded by the inverse of the diagonal entries of the Fisher information matrix [12, p5] to give var(ˆθ k ) [ I 1 (θ) ] kk 1 k =1,, 2D (21) [I(θ)] kk From (2) and (21), we have that and lim var(ˆx σ 2 Q ) lim = (22) ν i νi i ν2 i sin2 ψ i var( d ˆ σ 2 Q x ) (23) i cos2 ψ i The same condition holds for the corresponding y coordinate components From (22), the variance of the position estimate goes unbounded as d (or equivalently ν i ) goes to zero This could be explained by the virtual sensor interpretation Since the spacing of the virtual sensor pair is d, when d is too small, the sensor pair is unable to estimate the position well However, from (23), the variance of the displacement estimate

6 Fig 5 ˆx (m) ˆd(m) ˆd(m +1) ˆx (m +2) ˆx (m +1) ˆD(m; p) ˆd(p) ˆx (p) ˆx (p +1) Illustration of source movement used for the sequential estimator is unaffected by ν i as ν i, which implies that our method still has a good estimate of d even when d The MLE proposed in Section III generally does not achieve the CRLB due to the nonlinearity of f (θ) However, it may achieve the CRLB asymptotically, when the number of observations ie sensors, is large enough [12] It may also achieve the CRLB if the linearization in (12) is quite accurate V TRACKING From Section IV, it is possible to design a good estimation algorithm which can estimate source position well even when the source displacement is small In this case, the current position estimate would be more reliably determined from the information provided by previous displacement and position estimates We define the estimates obtained in Section III as instantaneous estimates while the position estimate obtained by utilizing past instantaneous estimates is called a sequential estimate We will derive some simple but effective causal sequential estimators which perform well in some common motion scenarios The sampled source trajectory in (3) is re-written in the form of a discrete state-space representation as x (p) =x (p 1) + d(p 1) (2) This suggests a simple linear sequential estimator where x (p) =ˆx (m)+ ˆD(m; p), m p (25) ˆD(m; p)= { p 1 ˆd(n) n=m if m<p otherwise (2) is the accumulated displacement estimate as illustrated in Fig 5 m is a design parameter in the range from to p and could change for different p When m = p for all steps, the sequential estimate is simply the instantaneous position estimate x (p) =ˆx (p) (27) When m =for all steps, the sequential estimator uses only the first instantaneous position estimate ˆx () and adds to it all subsequent displacement estimates ˆd(n) for n =,,p 1 The m =sequential estimator is thus x (p) =ˆx () + p 1 n= ˆd(n) (28) A Hybrid Estimator The asymptotic CRLB of position and displacement estimates in (22) and (23), has an important implication that when source displacement ˆd is too small, the position estimate is unreliable but the displacement estimate is still reliable This observation suggests a simpler method to choose m, ie switch m between p and p 1 according to the value of ˆd(p) This leads to a sequential estimator, called the hybrid estimator expressed as { ˆx (p) if ˆd(p) d th x (p) = x (p 1) + ˆd(p (29) 1) otherwise where d th is a threshold to be designed The hybrid estimator avoids using ˆx (p) when it has a large variance It also avoids the accumulation of the error in ˆd(n) by switching to the position estimate when ˆd(p) exceeds the threshold B Cumulative Estimator When the source always moves slowly, it is possible that d(p) never exceeds the threshold The hybrid estimator then becomes (28) All subsequent position estimates are dependent on the initial position estimate ˆx () which has a large variance Thus, the error is never corrected One solution is to combine multiple small movements into a larger one which can give an improved estimate This is called a cumulative estimator The operation of the cumulative estimator is described as follows: 1) Given a sequence of instantaneous estimates {ˆx (n), ˆd(n)} starting from n = q, and the initial sequential estimate x (q) from previous steps, we find the first instant p where the cumulative displacement ˆD(q; p) exceeds a threshold D th 2) For n = q +1,,p 1, the sequential estimate is x (n) = x (n 1) + ˆd(n 1) 3) At n = p, a reliable position estimate could be obtained by defining a cumulative TDOA y i (q; p) ˆl i (p) ˆl i (q) (p q)l Substituting this value for y in (9), we get i =1,,N y (q; p) =f(x (q), D(q; p)) + e (3) where e i =(p q)l(ɛ ɛ i )+n i (p) n i (q) This equation gives a refined estimate of ˆx (q) and ˆD(q; p) Solving (11) now using (3), the solution of (11), denoted by ˆx (q) and ˆD (q; p), provides a better position estimate at q as shown in Fig The sequential estimate at n = p can then be given as x (p) =ˆx (q)+ ˆD (q; p) ) The process continues by setting q = p The cumulative estimator starts at q =and its initial value is given by x () = ˆx ()

7 ˆd(q) ˆd(q +1) ˆx (q +2) ˆx (q +1) ˆd(p) x (p) 1 position displacement ˆx (q) ˆD(q; p) D th Re-estimate x (p +1) 1 1 ˆx (q) Fig ˆD (q; p) Movement combining in the cumulative estimator rms CRLB (m) d = 3m d = 1m d = 3m synchronous case Magnitude of displacement d (m) rms CRLB for position 1 1 Fig 8 CRLB as a function of d (m) x coordinate of position range rms CRLB (m) Number of sensors, N Fig 7 CRLB for x-coordinate (m) vs number of sensors, N VI RESULTS For all simulations, we assume the source emits acoustic pulses with c = 33 m/s We assume an audio sampling rate of F s = 8 KHz and a pulse separation interval of one second giving L =8 1 3 clock ticks From [1], we choose a standard deviation for the clock error in the range [1, 1 ] to obtain an effective frequency drift deviation of 2L 2 σf 2 =8 clock ticks per second The effective TOA measurement noise variance is also chosen as 2σn 2 =1clock tick A CRLB simulation In the first set of simulations, we calculate the CRLB as a function of different parameters The sensor positions are randomly generated in a rectangular region defined by coordinates (, ) and (1, 1) in meters The initial position of the source is at the center (5, 5) of the array of sensors and its displacement d is set to be d [cos ψ sin ψ] T where ψ is drawn from a uniform distribution, [, 2π] The CRLB is calculated and averaged over 5 independent realizations of the sensor positions The CRLB of the position estimate in the x coordinate for both asynchronous and synchronous sensors, as a function of the number of sensors, is shown in Fig 7 Different curves are shown for several values of d for the asynchronous case For d = 3 m, the root mean square (rms) error is smaller than 1 m when N 8 When d =3m, the rms CRLB for bearing (deg) x Standard deviation of the frequency offset, σ f x 1 Fig 9 CRLB for x-coordinate (m), range (m) and bearing (degree) performance of the asynchronous network with 1 sensors is close to that of the synchronous network with 5 sensors This demonstrates the good location performance of the proposed method using asynchronous sensors The performance loss compared to using synchronous sensors is tolerable given a large number of sensors and sufficient displacement From the curve, we can also determine the minimal number of asynchronous sensors required by an efficient estimator to meet a performance target in the error variance In Fig 8, the CRLB as a function of d is shown Notice from the curve that the CRLB for the x-coordinate is unbounded as d while the bound for the d estimate is still small This validates our theoretical analysis in Section IV When d is over m, the position estimate has a rms bound below 5 m Thus d = m can be used as our threshold in the hybrid estimate It is also interesting to note that the variance of the displacement estimate is smaller than that of the position estimate until d exceeds m This fact can be used as a guideline to design the sequential estimators

8 12 1 sensor postions m=p sequential estimator m= sequential estimator 12 1 sensor postions hybrid estimator, d th =1m cumulated estimator, D th =1m B A 8 8 y ( meter ) y ( meter ) C 2 E D x ( meter ) x ( meter ) Fig 1 Tracking a constant speed 1 m/s, random direction source Fig 12 Tracking a source with linearly varying speed using the hybrid and cumulative estimator 12 sensor postions m=p sequential estimator m= sequential estimator A 5 m=p sequential estimator m= sequential estimator 1 B 8 55 y ( m ) C 5 2 E D x ( m ) Fig 11 Tracking a source with varying speed using the m =and m = p sequential estimator Fig 13 Tracking a slowly moving source using the m =and m = p sequential estimator The effect of random frequency offset errors on our scheme is shown in Fig 9 for d =1m We plot the CRLB against the standard deviation of the frequency offset At d =1m, the error variance bound for the range and bearing is low even for high frequency offset B Tracking simulation The second set of simulations examines the performance of a practical scheme using the iterative gradient descent method as an instantaneous estimator and the various sequential estimators discussed in Section V Eight sensors are placed at randomly generated positions in a rectangular region defined by coordinates (, ) and (1, 1) in meters The source moves inside this region Three realistic source motion scenarios are considered: 1) constant speed with random direction, 2) acceleration and deceleration, 3) very slow speed The TOA estimation errors are randomly generated from a Gaussian distribution for each measurement The frequency offsets are Gaussian random variables The source tracking is a single run simulation without any averaging Case 1 (constant speed, random direction): The source maintains a speed of 1 m/s The performance of the sequential estimator is shown in Fig 1 It may be seen that the sequential estimator has good performance for both m= and m=p since the displacement between consecutive pulses is large enough Case 2 (acceleration and deceleration): In Figs 11 and 12, the source accelerates from 1 m/s at point B and reaches a maximum speed of 1 m/s at C, then decelerates to 1 m/s at D and then accelerates to 1 m/s again at E Fig 11 verifies that

9 y ( meter ) hybrid estimator, d th =m cumulative estimator, D th =3m cumulative estimator, D th =5m cumulative estimator, D th =9m x ( meter) B A Fig 1 Tracking a slowly moving source using the hybrid and cumulative estimators the m = p sequential estimator performs well when the speed is large enough at C and E, but exhibits large error near regions where the source moves slowly (at B and D) On the other hand, the m =sequential estimator has a large bias which does not vanish due to the error in the first position estimate ˆx () at A The performance degradation due to slow motion can be reduced by using the hybrid or cumulative estimator as shown in Fig 12 Initially around A, both estimators have large deviations Once the source accelerates over the threshold d th = m, the hybrid estimator switches to the current position estimate which is accurate for d m When the cumulative displacement is over the threshold D th =1m, the cumulative estimator begins to lump multiple displacements as one in order to obtain a better estimate of source position Thus its estimate converges to the true value from point B Note that both the hybrid and cumulative estimators perform well by using the displacement estimates at D when the source slows down Case 3 (slow moving source): Figs 13 and 1 depict the performance for a slow moving source (note the scale of the figure) It initially moves at 5 m/s, then maintains a constant speed of 1 m/s Both m =and m = p sequential estimators behave as expected in Fig 13 The m = p sequential estimator has large variance while the m =sequential estimator has a non-vanishing bias term As shown in Fig 1, the hybrid estimator performs the same as the m =sequential estimator since the source s motion between consecutive pulses never exceeds the threshold The cumulative estimator can still approach the true position around B since the cumulative displacement can exceed the threshold and give a reliable position estimate using an equivalently larger displacement The cumulative estimator uses thresholds of 3 m, 5 m and 9 m It may be observed that, for a small threshold, the estimator eliminates the initial bias fast, however it also results in larger errors On the contrary, the large threshold gives a better estimation after more steps However if the threshold is too large, the error due to the frequency offset will begin to build up These simulations show that it is possible to use our method to design a practical location systems using asynchronous sensors where the source has some motion VII CONCLUSION This paper proposes a location system using asynchronous sensors and examines its performance We have shown that for such a system, synchronization among all sensors is not indispensable Instead, good source trajectory estimation is achievable if the location system can utilize the source motion and there is a sufficient number of sensors in the network The communication and energy cost of the system is small because all sensors just passively listen to the source and only transmit a short TOA package to the master station Thus, this method is especially suitable for energy limited sensor networks or any system where the synchronization cost is substantial ACKNOWLEDGMENT This work has been supported, in part, by U S Department of Army under Contract DAAD 1-2-C-57-P1, and, in part, by the Indiana 21st Century Fund for Research and Technology REFERENCES [1] J Hightower and G Borriello, Location systems for ubiquitous computing, IEEE Computer, vol 3, no 8, pp 57, August 21 [2] J O Smith and J S Abel, Closed form least-squares source location estimation from range-difference measurements, IEEE Transactions on Acoust, Speech, Signal Processing, vol ASSP-35, no 12, pp 11 19, December 1987 [3] Y T Chan and K C Ho, A Simple and Efficient Estimator for Hyperbolic Location, IEEE Transactions on Signal Processing, vol 2, no 8, pp , August 199 [] A Ward, A Jones, and A Hopper, A new Location Technique for the Active Office, IEEE Pers Commun, pp 2 7, October 1997 [5] N B Priyantha, A Chakraborty, and H Balakrishnan, The Cricket Location-Support System, in Proc th Annual ACM/IEEE Int Conf Mobile Computing MobiCom2, Boston, MA, August 2, pp 32 3 [] J C Chen et al, Coherent Acoustic Array Processing and Localization on Wireless Sensor Networks, Proc of the IEEE, vol 91, no 8, pp , August 23 [7] J Elson, L Girod, and D Estrin, Fine-Grained Network Time Synchronization using Reference Broadcasts, in Proceedings of the Fifth Symposium on Operating Systems Design and Implementation (OSDI 22), Boston, MA, December 22 [8] Y T Chan and F L Jardine, Target localization and tracking from Doppler-shift measurements, IEEE J Oceanic Eng, vol 15, no 3, pp , July 199 [9] Y T Chan and J J Towers, Sequential Localization of a Radiating Source by Doppler-Shifted Frequency Measurements, IEEE Trans Aerosp Electron Syst, vol 28, no, pp , October 1992 [1] J R Vig, Introduction to Quartz Frequency Standards, SLCET- TR-92-1 (rev 1), Army Research Laboratory, Electronic and Power Sources Directorate, Fort Monmouth, NJ, at //wwwieeeuffcorg/freqcontrol/quartz/vig/vigtochtm, October 1992 [11] D J Torrieri, Statistical theory of passive location systems, IEEE Trans Aerosp Electron Syst, vol AES-2, pp , March 1989 [12] S M Kay, Fundamentals of Statistical Signal Processing - Estimation Theory, Prentice Hall, Englewood Cliffs, NJ, 1993