TPS: A Time-Based Positioning Scheme for Outdoor Wireless Sensor Networks Xiuzhen Cheng & Andrew Tbaeler Computer Science The George Washington University Washington, Dc 20052, USA {cheng,athaeler}@gwu.edu Guoliang Xue Dechang Chen Computer Science & Engineering Uniformed Services University Arizona State University of the Health Sciences Bethesda, MD 20817, USA Tempe, AZ 85287, USA xue@asu.edu Abstract-In this paper, we present a novel timebased positioning scheme (TPS) for efficient location discovery in outdoor sensor networks. TF'S relies on TDoA (TimeDifference-of-Arrival) of RF signals measured IocaUy at a sensor to detect range differences from the sensor to three base stations. These range differences are averaged over multiple beacon intervals before they are combined to estimate the sensor loeation through trilateration. A nice feature of this positioning scheme is that it is purely localized: sensors independently compute their positions. We present a statistical analysis of the performance of TF'S in noisy environments. We also identify possible sources of position errors with suggested measures to mitigate them. Our scheme requires no time synchronization in the network and minimal extra hardware in sensor construction. TPS induces no eommunication overhead for sensors, as they listen to three beacon signals passively during each beacon interval. The computation overhead is low, as the location detection algorithm involves only simple algebraic operations over scalar values. TF'Sis not adversely affected by increasing network size or density and thus offersscalability. We conduct extensive simulations to test the performance of TPS when TDoA measurement errors are normally distributed or uniformly distributed. The obtained results show that TPS is an effective scheme for outdoor sensor self-positioning. I. INTRODUCTION It is anticipated that wireless sensor networks will extend our sensory capability to every corner of the world. Distributed networks of thousands of collaborative sensors promise long-lived and unattended systems for many monitoring, surveillance and control applications. In this paper, we are going to examine the location discovery problem in outdoor wireless sensor networks. We will propose a novel time-based positioning scheme, henceforth referred to as TPS, that allows sensors to effectively determine their positions. Many applications of outdoor sensor networks require knowledge of physical sensor positions. For example, target detection and tracking is usually associated with location information [19]. Further, knowledge of sensor lo0-7803-8355-9/04/$20.00 02004 IEEE. dchen@usuhs.mil cation can be used to facilitate network functions such as packet routing [6], [161, [ZO], and collaborative signal processing [ll]. Sensor position can also serve as a unique node identifier, making it unnecessary for each sensor to have a unique ID assigned prior to its deployment [34]. However, location discovery in wireless sensor networks is very challenging. First the positioning algorithm must be distributed and localized in order to scale well for large sensor networks. Second. the localization protocol must minimize communication and computation overhead for each sensor since nodes have very limited resources (power, CPU, memory, etc.). Third, the positioning functionality should not increase the cost and complexity of the sensor since an application may require thousands of sensors. Fourth, a location detection scheme should be robust. It should work with accuracy and precision in various environments, and should not depend on sensor to sensor connectivity in the network. The TPS positioning scheme proposed in this research is designed to meet these challenges. The major contribution of this paper is twofold. First, we propose a time-based location detection scheme for outdoor sensor networks and demonstrate our algorithm by simulation. Second, we analyze the theoretical performance of our scheme in noisy environments and identify possible sources of error with measures to help mitigate them. We put very few restrictions on the network layout and propose a scheme suitable for general outdoor sensor networks. We rely on RF signal, which performs well compared to ultrasound, infrared, etc., in outdoor environments [29]. We measure the difference in arrival times (TDoA) of beacon signals. In previous research, Timeof-Arrival (ToA) has proven more useful than RSSI in location determination [32]. TPS does not need the specialized antennae generally required by an Angle of Arrival (AoA) positioning system. This time-based location detection scheme avoids the drawbacks of many existing systems for outdoor sensor location detection. Our sim- 2685 ulations show that TPS is potentially very effective and computationally efficient. Compared to existing schemes proposed in the context of outdoor sensor networks, our scheme bas the following characteristics and advantages: Time synchronization of all base stations and nodes is not required in TPS. Sensors measure the difference in signal arrival times using a local clock. Base stations schedule their transmissions based on receipt of other beacon transmissions and do not require synchronized clocks. Many existing location discovery systems for Sensor networks require time synchronization among base stations 1251, or between satellites and sensors 1151. Imperfect time synchronization can degrade the positioning accuracy. There are no requirements for an ultrasound receiver [8], [32], second radio [15] or specialized antennae [ 5 ] , [23], [251 at base stations or sensors. Our scheme does not incur the complexity, power consumption and cost associated with these components. (TPS sensors do require the ability to measure the difference in signal arrival times with precision.) Our algorithm is not iterative-and doesn’t require a complicated refinement step as does [281, [311,[331. We refine position estimates by averaging time difference measurements over several beacon intervals prior to calculating position. This is useful to mitigate the effects of momentary interference and fast fading. l h i s averaging requires less computation than repeatedly solving linear system matrices, least squares or multilateration algorithms. TPS has low computation cost. .Our location detection algorithm is based on simple algebraic operations on scalar values. On the other hand, multilateration based systems [151, [171, 1321, [331 require matrix operations to optimize the objective functions (minimum mean square estimation or maximum likelihood estimation), which induces higher computation overhead at each sensor. Sensors listen passively and are not required to make radio transmissions. Base stations transmit all the beacon signals. This conserves sensor energy and reduces RF channel use. Connectivity based systems often require global flooding [26] or global connectivity information E351 to estimate range. This paper is organized as follows. Section I1 presents current location discovery techniques for outdoor sensor networks. Section Iii presents the network model. Section IV proposes TPS, a time-based location detection scheme. Its theoretic performance analysis is given in Section V. Simulation results are reported in Section Vi. . . . C-7803-8355-9M20.00 OZW4 BEE. We conclude ow paper in Section VII. 11. AN OVERVIEW ON CURRENT LOCATION DETECTIONTECHNIQUES FOR SENSOR NETWORKS A. Sensor Location Detection Techniques The majority of current sensor location detection schemes contain two phases: (i) range or angle measurement between sensors and baw stations; and (ii) calculations which transform these. measurements in to a position estimate. Some schemes perform a refinement phase after generating an initial estimate. In this subsection, we are going to examine related location discovery techniques. Range Estimation and Angle Measurement Popular techniques for range estimation include Timeof-Arrival (ToA), Time-Difference-of-Aval (TDoA), and Received-Signal-Strength-Indicator (RSSI). Angleof-Arrival (AoA) involves measurement of the angle at which a signal arrives at a base station or a sensor. If there is no direct communication between base stations and sensors, network connectivity can be used for range estimation [261. ToA and TDoA measure the signal arrival time or the difference of arrival times and calculate distance based on transmission times and speeds. They can be applied to many different kinds of signals such as RF, acoustic, ultrasound, etc. ToA has a disadvantage compared to TDoA as processing delays and non-LOS propagation can introduce errors [5]. ToA also requires synchronization to accurately measure time-of-flight. RSSI computes distance based on transmitted and received power levels, and a radio propagation model. RSSI is mainly used with RF signals. Due to multipath fading in outdoor environments, range estimation with RSSI can be inaccurate [32]. AoA is an attractive method due to the simplicity of the subsequent calcuhtions (triangulation). But AoA can be difficult to measure accurately if a sensor is surrounded by scattering objects [5]. Further, measuring AoA requires sensors or base stations to be equipped with directive antennaes or antennae arrays, which may be prohibitive due to cost and form factors. Our time-based location detection scheme computes range based on TDoA with no requirement for time synchronization. We actually detect the range differences from a sensor to three base stations (one is termed the master base station). If a sensor can not receive signals from enough base stations ( 2 2 for AoA, 2 3 for ToA, TDoA, and RSSI), none of the previous techniques will work. In this case, network connectivity can be exploited for range estimation [26], [31]. DV-hop, DV-distance, and Euclidean are three range detection methods in this category. In DV-lrop [26], 2686 [31] base stations flood their positions to all nodes in the network Sensors compute the minimum distance in hops to several base stations. Rase stations compute an average disfunceper hop to other base stations. The base stations then flood this information to the whole network allowing nodes to calculate their positions. DV-distance [26] replaces hop counts with cumulative range estimates in meters estimated from RSSI. Both techniques provide coarse range estimation to base stations and both require the expensive global flooding to compute the shortest path. Euclidean[26] estimates a sensor's distance to a base station based on the distance to two of its neighbors, the distance between the neighbors, and the distance from the neighbors to the base station. The Euclidean algorithm uses basic trigonometry to calculate distance to the base station. Each sensor needs to execute the Euclidean algorithm twice for two pairs of neighboring sensors to unambiguously determine its range to any base station. Fig. 1. Range or angle combining techniques: (i) triangulation, (ii) ailateration, (iii) multilateration. to use the centroid of multiple base stations to approxiLocation Computation from Range or Angle Measure- mate the sensor location. In this subsection, we are going to overview in detail several works designed for outdoor ments Triangulation, trilateration, and multilateration are the sensor networks. For a taxonomy of location systems for three techniques for combining ranges and angles. Tri- ubiquitous computing we refer the readers to [ 121. Ref. [32] proposes a TDoA based scheme (AHLQS) that ungulation is the simplest. As in Fig. l(i), if the angles requires base stations to transmit both ultrasound and RF ((U and 0) to base stations A and B are known, the location of S is where lines from A and B intersect. Thus signals simultaneously. The RF signal is used for synfor AoA, at least two base stations are required. Tri- chronization purposes. A sensor will measure the differluterution computes the intersection of three circles, as ence of the arrival times between the two signals and deshown in Fig. I(ii). If the range to each base station is termine the range to the base station. Multilateration is not accurate, the three circles may not have a common applied to combine range estimates to generate location intersection point leading to ambiguous solutions. Mul- data. Testhed experiments demonstrate that AHLoS protiluteration uses an objective function to minimize the vides fine-grained localization capability. However, ultradifference between the estimated position and real po- sound transceivers can only cover a short range (several sition of a sensor. -For example in Fig. l(iii), we can meters) and large numbers of base stations may be reuse m i n C , ( D ~%DsZ)*to compute (z,y) for S, where quired to cover large areas. Other contributions by [32] Ds, = J ( x - z,)~ (y - Y ~ ) ~D, s ~is the estimated include the introduction of iterative multilateration and range from S to i, z = A, B , C, D, E. This technique can collaborative multilateration. In iterative multilateration, improve accuracy but involves higher computation over- a sensor becomes a base station after its position is deterhead. For details on multilateration, we refer the readers mined. Whenever a sensor has range estimates to at least to [32]. Both trilateration and multilateration require at three base stations, multilateration is used to compute its least 3 base stations. TPS uses trilateration with range position; otherwise, it continues to listen to beacon sigdifference information. We compute a sensor's position nals from base stations. If it is impossible for a sensor to find 3 base stations, collaborative multilateration can be and its range to the master base station at the same time. used. In collaborative multilateration two or more sensors (which can be multiple hops apart) can form an overdetermined system of equations with a unique solution B. Existing Sensor Location Detection Schemes set. Feasible conditions for collaborative multilateration GPS is the most popular localization system but may are further explored in FIX], [33]. Ref. [I71 compares the not be desirable in a sensor network due to cost, form fac- performance of different multilateration methods by simtor, energy consumption, and the requirement for a sec- ulation and proposes a new and fast iterative improvement ond radio. GPS-less localization techniques have been algorithm to optimize location discovery. Ref. [XI designs researched extensively. For example, Ref. [3] proposes and analyzes an acoustic ranging system for robotics ap- + plications and embedded sensor technology. This paper examine methods to detect and eliminate various types of interference. As mentioned earlier, AoA techniques require special antennae and may not perform well due to omnidirectional multipath reflections. To avoid requirements for directional antennae, Ref. 1251 first transforms TDoA measurements in to AoA information and then applies triangulation to compute location. This scheme requires at least 3 base stations with synchronized rotating directional antennae. Non-zero antennae beam width and imperfect synchronization contribute to decrease system accuracy. A prototype navigation system based on AoA measurements for autonomous vehicles is presented in [23]. It estimates AoA by means of a set of optical sources and a rotating optical sensor. This system is not suitable for out'door sensor networks due to its cost and complexity. Our scheme is similar to the one in Ref. [25] in that TPS measures TDoA at each sensor and has no additional special requirements for sensors. However, we do not use directional antenna in base stations and we do not require any kind of synchronization in the whole network. The works mentioned above are all based on siraightline range estimation to base stations. Ad Hoc Positioning system (AI'S) [261 first estimates ranges based o n DV-hop, DV-distance, or Euclidean, and then applies hilateration to compute the location of each sensor. If enough base stations are available, location errors for A P S with DVhop can he about 30% of radio range in a dense and regular topology. For sparse and irregular network topologies, the accuracy degrades to roughly the radio range. For DVdistance and Euclidcan, the performance of APS also dcpen& on the accuracy of the distance measured between neighboring sensors. Ref. [31] goes one step further: it refines location estimates computed by APS with DV-hop by using neighboring sensor position and distance estimates to help convergence to a better solution. To mitigate error propagation, a confidence weight from 0 to 1 is associated with each estimated position. With measured distance errors of 5% , [311 produces an error of 33% of radio range on average for random gaphs. Another work is [33], which uses DV-distance to compute range and Min-Max to compute position. To refine position estimates, [331 uses a computation tree.Ref. [181 compares [261, [31], and [331 in simulation. Locating a subscriber in cellular networks or PCS systems has been well studied in literature [l], 151, [211. The techniques involved produce coarse location granularity (tens or hundreds of meters), thus may not he suitable for outdoor sensor networks. Research on indoor or in-building localization is on-going and many interesting &7803-8355-9/04/%20.00 02004 IEEE. systems have been designed. Examples include Active Badge [36], Active Bat [lo], RADAR [2], Cricket [27], SpotON [13], to name a few. Some of these systems require location surveys which are not possible with an airdeployed outdoor sensor network. For a brief overview on these systems, we refer the readers to [17]. Other interesting works in sensor networks include [7], [35], [30]. Ref. [7] proposes apositioning algorithm based on convex optimization. Ref. [351 describes a localization scheme based on multidimensional scaling. Both are centralized and both rely on connectivity information. Ref. [30] applies error detection and correction coding theory to location detection in emergency indoor sensor networks. 111. NETWORKMODEL We assume that the sensors are deployed randomly over a 2-dimensional monitored area (on the ground). (However, Our proposed sensor positioning scheme can be easily extended to 3-dimensional space.) Each $ensor has limited resources (battery, CPU, etc), and is equipped with an omni-directional antenna. Three base stations A, B, C, with known coordinates (za, ya), (zb, y b ) , and (zc,y,.) respectively, are placed beyond the boundary of the monitored area, as shown in Fig. 2. Let us assume A he the master base station. Assume the monitored area is enclosed within the angle LBAC. Let the unknown coordinates of a sensor be (z, y), which will he determined by TPS. Each base station can reach all sensors in the monitored area. One restriction on the placement of these base stations is that they must be non-collinear, as otherwise, the sensor locations will be indistinguishable. 0. Monitored Arc. 0 Fig. 2. An example sensor network. Note that these base stations will transmit RF heacon signals periodically to assist each sensor with location discovery. They have long-term power supplies and can receive RF signals from each other. Note that there is no time synchronization among these three base stations. However, we require base stations to detect signal arrival 2688 times with precision and to accurately calculate total turnLet A be the master base station, which will initiate a around delay. This calculated turn-around delay consists beacon signal every T seconds. Each beacon interval beof arandom delay combined with known system transmis- gins when A transmits a beacon signal. Consider any heasion and reception delays. con interval i, at times t!, t:, t:, sensor S, base stations B and C will all receive A's beacon signal respectively. At Remark: If the monitored area is so large that 3 base sta- time $2. which is ti, B will reply to A with a beacon tions can not cover the whole area completely, we can al- signal conveying information t t - t i = At:. This sigways divide the area into smaller subareas and place more nal will reach S at time ti. After receiving beacon signals base stations. from both A and B, at time t:, C will reply to A with a beacon signal conveying information 6," - t; = At:. This IV. TPS: A TIME-BASED POSITIONING SCHEME signal will reach S at time ti. Based on triangle inequality, ti < ti < ti. Let Atf = ti - tf, At; = t i - tf, we In this section, we propose TPS, OUT time-based po- obtain sitioning scheme for outdoor wireless sensor networks. dab + dab - d8, f U At: = v . At! (1) This scheme consists of two steps. The first step detects the time difference of signal arrival times from three base d, ds, - ds, + U At: = v . At;, (2) stations. We transform these time differences in to range differences from the sensor to the base stations. In the sec- which gives ond step, we perform trilateration to transform these range dsb = d,, 2) . At'; - dab - 21 . At; = d,, k'; ( 3 ) estimates into coordinates. d,, = d,, + U At; - d, - U . Atf = d,, +IC:, (4) > + + where d,,, dab and d,, are positive real numbers and A. A Time-Eased Location Detection Scheme Given the locations (z,, y,), (zb,yb), and (zc,yc) of base stations A, B, and C, respectively, we are going to determine the location (z, y) of sensor S, as shown in Fig. 3. Let U be the speed of RF beacon signals from A, B, and C. Let dab be the distance between base stations A and B and d, be the distance between base stations A and Thus d d = d(z, - $&)2 (&I, - yb)2 and d, = d(za - %I2(ya - yc)2. Let Q,, dabr and d,, be the unknown distances from S to A, B, and C respectively. Our time-based location detection scheme TPS consists of two steps. c. kf = U . At: k: = U . At; B \ ' (% Yd d At; - U . At: -U - dab, - dac. (5) (6) Averaging kf and k: over I intervals gives (7) + + + We are going to apply trilateration with IC1 and IC2 to compute coordinates (z, v) for sensor S in the next step. Remarks: (i) All arrival times are measured locally. In other words, tl, t 2 , t~ are measured based on sensor S's local timer; t b and tb are based on B's local timer and known system delays; while t, and t: are based on C's local timer and known system delays. There is no global synchronization. (ii) We require A to periodically initiate the beacon signal transmission for two reasons. First, avc (%. YJ eraging ICE and over multiple beacon intervals helps to Fig. 3. Sensor S will measu~eIhe TDoA of bexon signals from base decrease the measurement error. ' h e number of beacon stations A. B. and C locally. S also will receive the turn-around delay intervals I can be a trade-off between potential accuracy information from B and C . B's transmission will stan after it receives A s beacon signal, while C's transmission will start after it receives improvement and power consumption. Second, sensors both A and B's beacon signals. This procedure will be repeated once may sleep to save energy; or they may be deployed at difevery T seconds. ferent times; or they may move during their lifetime. The periodic beacon signals from A and the reply signals from Step 1: Range Detection. B and C can facilitate location discovery at any time. \ \ I 078034355-9/04/520.M) 02004 IEEE. 2689 Step 2: Location Computation. From Eqs. (3), (4), (7) and (8), we have dsb = d m Suhstituting Eqs. (20) and (21) into (14), we obtain -1k i , Q, = d,, kz. (9) (10) + od;, + (Y - ya)’ ( x -.b12 + (Y - ~ (x -ZJ2 + (Y - Yc)’ = d2 sa + kl12 (11) (12) (&a + k2)* (13) = ) ~(dm 6 = In the next Subsection, we will show how to compute x , y and d,, efficiently. We will also give the conditions under which the solution set is unique. B. An EBcient Solution for Location Detection by Trilat- eration Without loss of generality, we assume the three base stations are located at (0,0), (x1,0), and (22,y2), respectively, where x~ > 0, yz > 0. In other words, X 2 , and Ye = y2. yb = 0, xb = X i , 2, X n = ya Let sensor S he located at ( x ,y). Note that we can always transform real positions to this coordinate system through rotation and translation. We want to compute the location for s. From Eqs.(ll),(lZ),and (13). we have Z= X’ x2 + y2 = d:,, - 22x1 X: y2 = d:, 2d&1 x2 - 2222 xf f y= - 2yy2 f y; = dz, 2d,k2 kg. + + + + + IC:, + + (14) (15) (16) Subtracting Eq. (14) fromEq. (15), we obtain - k: 2x12 = -2kld,, + x:. (17) Subtracting Eq. (14) from Eq. (16), we obtain 2x23: + 2y2y = -2kzd,, - kz + 22 + yz. (18) Multiplying Eq. (18) with z l and submcting the product of Eq. (17) with 2 2 , we obtain 2x1~= 2 ~(28122 - 21cz~i)d,,+ k:x2 +X$Xl f Since 2 1 > 0, yz > 0, E& y;x1 - x1x2. 2 - kzxi (19) (17) and (19) can be rewritten as 0-7803-8355-9/04/$20.W Q2oDt BEE. (22) where Based on trilateration, we obtain three equations with three unknowns x,y, and ds,, where d,, > 0. (Z- 2,)’ + Pdsn + y = 0, CY = p = + ( k i ~ 2- I c z ~ i -) ~Z;Y$], il[ki(kY - XY)Y/,’+ ( k t X 2 - k 2 Z l ) ( k :~ 82x1 + +&xi - 4[kYyz 2 7 = (23) % : ~ 2 ) ] ,(24) ~z.1 2 2 2 (k? - X I ) Y 2 + ( k y z z - k$x1 +x;xi+ y;x1 -~ 7 ~ 2 ) ~ . ( 2 5 ) w. Theorem NI: (22) has a unique positive root for d, if and only if one of the following three conditions holds. 1) CY = 0 , < 0 , a n d y > 0; 2) CY^ < 0; 3) cup < n,y = PROOF.We prove the theorem by case study. First, we consider the case where both a and p are zero. In this case, (22) is either satisfied by all values of d (when y = 0) or violated by every value of d (when y # 0). Next we consider the case where cy = 0 and p # 0. In this case, (22) has a unique root d,, = Since y 2 0, is positive if and only if p < 0 and y > 0. This corresponds to the first condition in the theorem. In the rest of the proof, we will consider the cases where CY # 0. Consider the case where CY^ < 0. This implies that y > 0 and CY < 0. It also implies that p2 - 40.7 > p2. Therefore (22) has a unique positive rootd,, = -’-. 2a This corresponds to the second condition in the theorem. In the case where 4127 > p2, the equation does not have any root in the real field. In the case where 0 <: 4ay < p2, the equation has and &) --P-two roots d,,(1) - -o+@=G E. -3. -3 la la which have the same sign. In the case where $ a y = p2, the unique root of the equation is d,, = -$ which is positive if and only if p < 0. This corresponds to the third condition in the theorem. Next consider the case where y = 0. Note that y = 0 implies that = xf,which in turn implies that k:xz +%,XI 2 + y 22 z l -x1x2 2 = 0. Therefore7 = 0 implies that p = 0. In this case, the equation does not have a positive root. This completes the proof of the theorem. 0 Substituting the value of d,, into Eqs. (20) and (21), we will have the coordinates x and y for S. In the above solution, we have used the square root function. Note that computing the square root X of a positive number N only requires a few iterations of Newton’s 2690 method [24] in the form of X := 0.5 * ( X + N / X ) . Our simulation results show that four iterations are sufficient to produce accurate solutions. and kz are the averaged results over I bcacon intervals, and based on the Central Limit Theorem, kl and k2 are approximately normally distributed when I is large. Therefore, without loss of generality, we may assume k l and k2 are distributed according toN(jb1,U : ) andN(p2, U ; ) , respectively. In this Section, we first give a statistical error analysis of sensor coordinate estimation. We then identify the major sources of errors affecting TPS’s location detection accuracy based on the network model described in Section 111. Remarks: (i) Newton’s method converges quadratically, thus solving trilateration functions can be done in a fast fashion. (ii) Compared to the other location detection methods in literature [17], [321, [331, [l51, our scheme has an important advantage: we improve performance by refining in the first step - averaging time differences over multiple beacon intervals, which involves only simple algebraic operations. Refining through popular strategies A. Theoretical Error Analysis like maximum likelihood or minimum mean square require To simplify the elaboration, we consider the case when more computation. base stations A, B, and C are located at (0, 0), (R,0), and We note that data collected may have errors. When (0, R ) , respectively. This base station placement corresolving a system of linear equations such as those defined sponds to condition number 1, which results in the most by (20) and (21), solutions are more accurate when the stable system. To further simplify the analysis, we concondition number (the condition number of a system of sider the case when S is equidistant to any base station. linear equations is the ratio of the largest eigenvalue over The general case can be analyzed similarly. the smallest eigenvalue) is small [91. We note that the conIn our case, it is reasonable to assume jbl = pz = 0, dition number of the system of linear equations (20) and and thus k l / R N 0, k z / R N 0. To facilitate our analysis, (21) is max{$, When designing the system, it is we further assume thaf kl and kz are independent. (In better to choose the locations of base stations so that rhe general, one can introduce correlation between kl and kz.) ratio is as close to 1aspossible. It is interesting to note Plugging x1 = R, x2 = 0, and y2 = R into Eqs. (23), that the value of xz does not affect the condition number (24), and (25), and simplifying the solution to Eq. (22) by of the system. In practice, we may choose the locations of approximating k:/Rz and k:/R2 with 0, we end up with the base stations so that they are sitting at the vertices of an equilateral triangle. In this case, the condition number 2R2 2kik2 - (kl k z ) will be 1.155, which is veIy close to 1,resulting in a very (28) dsa = 2 stable system. To ensure the unique positive solution ford,,, it suffices Substituting the above into Eq. (20) yields to have a y < 0. From Eq. (25), y > 0. Thus the sufficient condition is reduced to CY < 0. That is g}. 2 ’+ k:d + ( h X 2 - k 2 X d 2 < x:y;, + (26) which gives In our simulation, this condition is satisfied in all cases where sensors are not in close Droximitv to or behind a base station. Near the base stations (interior to triangle), the solutions for d,, are both positive. If the position that corresponds to our measurements is interior to the tri&gle, d,,(’) - e THEORETICAL PERFORMANCE C-7803-8355-9x)4/s20.00 02004 IEEE. = 3 - 4.Similarly, from Eq. (21) we have R kikz ya-+-2 2R 6. is the correct calculation. ANALYSIS The trilateration equations (1 11, (12). and (13) determine coordinates (z, U) for sensor S based on the measured values kl and kz. The inaccuracies of kl and k2 cause sensor position errors. From Eqs. (7) and (8). k-1 V. where k,‘ (31) where k ; = Since (x,y) is used to estimate the location of S, the error in the estimation must be addressed. There are several ways to do this. The following is a common practice, where the variance of each variable is computed and the size of the variance or standard deviation is used as a measure of estimation error. 2691 As kl has a Gaussian distribution with mean p1 and variance U:, and kz has a Gaussian distribution with mean p2 and variance ug,the linear combination k ; has a Gaussian distribution with mean % - @and variance and k$ has a Gaussian distribution with mean &k - A, &. and variance Denote by E ( X ) and V ( X ) the mean and variance of a random variable X . We have, from Eq. (30). V(x) V(k1k;) E(k&)2 - [E(k&)]Z = E ( ! - C ; ( ~ ;-) ~ [E(k1k;)I2. ) K5 = V ( y ) x &2, showing that the variance of x is dependent on that of kl while the variance of y is dependent on that of k2. Fourth, if U: = U;, the variances of x and y can be treated the same in practice. Wc note that the above discussion is based only on the first two moments of the random variables kl and k2. We have not taken advantage of the normality assumption of these two variables. In fact, with additional normality assumption on k~ and kp, we can obtain approximations to the distrihutions of z and y. For example, since kl and kg are independent, the CDF P ( z 5 a ) of x can he approximated by (for any real number a). (32) By the independence between kl and k g , we have E(kik;) = E ( h ) E ( k ; ) E(k?(k;)') = E(k;)E(k2*2) = [ V ( k i )+ ( E ( h ) ) 2 1 [ V ( k ; )+ (E(G))'I. (33) where (34) Therefore substitution gives 2rl+Jz = (&)"R2( = ($)"2R2 2 ) (1 + a v ) . 02 2 Since p1 = pp = 0,the U2 V ( x ) x -2. 2 above reduces to U:.; +-4R2 2 (35) u2 (37) The above results will help us to explain simulation results. In ow simulation study (Section VI), we consider the cases when the errors of TDoA measurements at the sensor are normally distributed or uniformly distributed. The variance of TDoA measurements determines the variances of kl and k 2 . Simulation results show that position error strongly depends on the variance of TDoA measurements. B. Sources of Errors There are three major sources of errors for ow timebased location detection scheme: the receiver system delay, the wireless multipath fading channel, and the nonline-of-sight (NLOS) transmission. The receiver system delay is the time duration from which the signal hits the receiver antenna until the signal is decoded accurately by the receiver. This time delay is determined by the receiver N -+of U;.; - U; (1 . (36) electronics. Usually it is constant or varies in very small 2 4R2 2 scale when the receiver and the channel is free from inFrom the above analysis, we have the following obser- terference. This system delay can he predetermined and vations. First, the variance of both x and y depend on the be used to calibrate the measurements. For example, base variances of kl and k2. Second, the variance of kl con- stations B and C can always eliminate the system delay tributes more to that of z than the variance of k ~ And from A$ and At: before these values are conveyed to ; the sensors in their reply messages to A's beacon signal. the variance of kg contributes more to that of y than the variance of k l . Thud, when R is large, V ( x ) N 4 / 2 , Meanwhile, as At; and At; arc measured by one sensor, + &) the effect of receiver system delay . may . cancel out. Thus in our model, if base stations B and C can provide precise a priori information on receiver system time delay, their effect will be negligible. The wireless multipath fading channel will greatly influence the location accuracy of any location detection system. Major factors influencing multipath fading [291 include multipath pmpagation, speed of the receiver, speed of the surrounding objects, and the transmission signal bandwidth. Multipath propagation refers to the fact that a signal transmitted from the sender can fullow a multiple number of propagation paths to the receiving antenna. In our system, the performance is not affected by the speed of the receivers since all sensors and base stations are stationary. However, a moving tank in the surrounding area can cause interference. There are two important characteristics of multipath signals. First, the multiple non-direct path signals will always arrive at the receiver antennae. latter than the direct path signal, as they must travel a longer distance. Second, in LOS transmission model, non-direct multipath signals will normally be weaker than the direct path signal, as some signal power will be lost from scattering. If NLOS exists, the non-direct multipath signal may be stronger, as the direct path is hindered in some way. Based on these characteristics, scientists can always design more sensitive receivers to lock and track the direct path signal. For example, multipath signals using a pseudo-random code arriving at the receiver later than the direct path signal will have negligible effects on a high-resolution DS-BPSK receiver [4]. Our location detection scheme mitigates the effect of multipath fading by measuring TDoA over multiple beacon intervals. ‘I‘DoA measurements have been very effective in fading channels, as many detrimental effects caused by multipath fading and processing delay can be cancelled [5]. Another factor related to wireless channels that causes location detection errors is NLOS transmission. To mitigate NLOS effects, base stations can be placed well above the surrounding objects such that there are line-of-sight transmission paths among all base stations and from base stations to sensors. In the next section, we are going to study the performance of our TPS positioning scheme over fading channels. We will consider the inaccuracy of TDoA information measured at sensors only. The sources of errors under consideration include multipath fading and NLOS. Thus we are going to assume the TDoA measurements are either normally distributed or uniformly distributed. These assumptions are popular in literature for TDoA measurements [SI,[17] in fading channels. 0-7803-8355-YD4/$20.CHlOZW4 IEEE. VI. SIMULATION Eqs. ( 1 I), (12), and (13) compute coordinates x and y for sensor S based on kl and k2, which are determined by the time-related values at the sensor (At; and At:) and base stations B (At;)and C (At;)over beacon interval z (see Eqs. (7) and (8)). Thus the errors of x and y result from the measuring errors of At:, At:, At;, and At:. In this simulation, we assume the measuring errors of At: and At: are negligible. This is reasonable, as we can always take possible measures (see Subsection V-B). to decrease the measuring errors of base stations when the number of base stations is small (only 3 in our case). For example, base stations can be placed well above the surrounding objects to avoid multipath fading and NLOS transmission, and the system delay can be predetermined to calibrate the ToA measurements. (In this case, the sensor network resides in a 3-dimensional space. TPS needs to be modified accordingly.) On the other hand, Eqs. (7) and (8) tell us that the measuring error of At: (At;) plays the same role as that of At; (At;) in the computation of kl (k2). Thus in our simulation study, we only consider the measuring errors of At; and At;, which are termed TDoA measuring errors in the following description. We will study the influence of I and u2 upon position error. where I is the number of beacon intervals used to compute kl and k2, uz is the variance of the TDoA measuring error. We use Matlab to code TPS. This tool provides procedures to generate normally distributed and uniformly distributed random numbers. Note that we do not use the sqrt function in Matlab. Instead, we use Newton’s method described in Subsection IV-B. We found that 4 iterations generally yielded good results. We first check the correctness of our scheme. In this simulation, no measuring errors are introduced. Base station A is the master base station. We randomly place sensors within the open area formed by the acute angle LBAC, as shown in Fig. 2. This area is termed feasible area. We found that sensors close to the base stations may have two computed locations: one within the feasible area, and one outside. This is because Eq. (22) generates two positive roots for d,. But if we throw away the solution that is outside the feasible area, we can always compute the location for each sensor correctly (uniquely). Thus in the following simulation, we only consider the solutions that are within the open area formed by LBAC. This is reasonable, as the base station locations and the master base station are known to each sensor. Now we study the distribution of position errors over a 2D planar monitored area. Fig. 4 is drawn in 3D space which demonstrates position errors vs. the real positions 269‘3 Comparison of computed location emxs by position. The sensors are placed at 19 x 19 grid points and u2 = 0.05. The base srationsacelocatedat(0,0),(20,0)and(0,20). Fig. 4. of sensors. In this simulation scenario, the three base stations are located at (0,0), (20,O) and (0,20). Sensors are placed at grid points (i 0.5,j 0.5). where i ,j = 0,1, . . . ,19. We average the sensor location results computed from our scheme over 10000 trials. For each trial, I = 4. The measuring errors of At; and At: are normally distributed according to N(0,0.05). This corresponds lo a TDoA deviation of f 0 . 2 2 unit. Note that we conduct extensive simulation over different U' and achieve very similar results. Clearly this simulation shows that as the distance from a sensor to all three base stations become larger, the position error will become larger correspondingly. Also when the sensor is closer to any of the three base stations, the error becomes larger. We also observe that the sensor at location (6.5,6.5), which is close to the intersection of the three angle bisectors of AABC, has the smallest position error and the sensors at its neighboring area also demonstrate quite low position errors. Interestingly, Refs. [25] and [3] provide similar results in their simulation study. Intuitively this is because the geometry of the intersection of the range circles is poor when the sensors are far away from any base station or when the sensors are close to any base station. From this analysis, we conclude that the position error is related to the placement of base stations. Careful studies will be conducted in the future as the results can be applied to guide the deployment of base stations for better performance. For these reasons, in the following simulation, we intentionally enforce an allowable shortest distance (1.0 unit) from any randomly generated sensor to any base station. This means the three base stations are placed some distance away from the boundary of the monitored area. Next we consider the scenario when sensors are randomly deployed in a square region with lower-left comer (1,l) and upper-right comer (20,20). The three base sta- + 0-7803-8355-9/04620.~02004IEEE. + tions are still located at.(O, O), (20,0),and (0,20), respectively. We consider two error models: normal distribution according to N(0,a2)and uniform distribution over the range [-b, b], which gives the variance u2 = b2/3. For each variance value, we try 10000 random sensor positions. The averaged results are reported in Figs. 5 and 6. Note that we average over such a large number of sensor positions for each variance in order to take the whole monitored area into consideration. Also note that we report the simulation results when the number of beacon intervals I used for position computation is chosen from the set {4,8,16,32,64,128}. We obtain three observations from Figs. 5 and 6. First, as I increases, position error decreases. This is because averaging over larger number of beacon intervals to compute k l and k2 can better smooth out the effects of measuring errors in TDoA measurements At: and At:, thus produce improved result. A detailed theoretical explanation comes from Subsection V-A. As I increases, U: and U; will decrease, and thus V(z) and V(y) will decrease. Then the errors from estimating the coordinates of sensors by x and y will decrease, implying that the position error will become smaller. Second, position error increases as variance u2 increases. This is reasonable as variance corresponds to the measuring error. Again, this can be well explained by Subsection V-A. In fact, if U' increases, U: and U; will increase. Then V(z) and V(y) will increase so that the errors from estimating the coordinates of sensors by 3: and y will increase. Thus the larger the TDoA measuring error, the larger the position error. Thud, for the same variance, position error is smaller when the TDoA measuring error is normally distributed. This is particularly hue for small values of I. An explanation is given as follows. When I is small, kl and k2 are still normally distributed if TDoA measuring errors are normally distributed. However, for small I,kl and k2 are not normal variables if the measuring errors are uniformly distributed. Following Subsection V-A, we can show that with fixed confidence, the predictive intervals of x and y for normally distributed measuring errors are narrower than those for uniformly distributed measuring errors. This shows that for normally distributed TDoA measuring errors, the position error is smaller. Note that shifting the square monitored area within LBAC, we obtain very similar results. In the following, we report the simulation results when the base stations form a triangle such that LBAC 5 90". In this simulation, three base stations are located at (0, O), (z1,0, and (zz,y2), where ZI, 2 2 , and y2 are randomly drawn from [5,20]. 100 sensors are randomly placed within the overlapping area formed by LBAC and the 2694 Fig. 5. Position m r v s . variance ua.The TDoA measurements are normally distributed. 08 Fig. 7. Position erroI vs. Mliance U ' . The TDoA measurements axe normally distributed. Base stations are placed randomly %thin a fixed ma. I Fig, 6 . Position ermr "S. variance The TDoA are Fig. 8. Position e m r vs. variance U'. The TDoA measuremcnts are uniformly distributed. Base stations are placed randomly witbin a fixed area. square with corners (0,O) and (20,20). we also throw away sensors whose distance to any base station is < 1.0 unit, which means we only count sensors that are not too close to any base station. Fig. 7 reports the result when the TDoA measuring errors are normally distributed while Fig. 8 reports the result when the TDoA measuring errors are uniformly distributed. It is obvious that Figs. 7 and 8 are very similar to Figs. 5 and 6, respectively. We obtain the same observations. However, for the same TDoA measuring error (same u2),we always achieve better performance in Figs. 7 and 8. Thus LBAC = 90° is not the optimal base station.placement. This induces a network optimization problem, the base stalion placement problem, for us to explore in thc future. putation overhead and scalability. To evaluate the performance ofTpS, we conduct& bo& theoretical and simulations, ourscheme is simple and effective, uniformly distributed. ACKNOWLEDGMENT The research of G . Xue was supporled in part by ARO grant DAAD19-00-1-0377 and NSF ITR grant ANI0312635. ' Transocrions on Wireless Communications, In this paper. we presented TPS, a time+=* positioning scheme for outdoor sensor networks. 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