The Effect of Stale Ranging Data on Indoor 2-D Passive Localization

Transcription

1 The Effect of Stale Ranging Data on Indoor 2-D Passive Localization Chen Xia and Lance C. Pérez Department of Electrical Engineering University of Nebraska-Lincoln, USA Steve Goddard and Robert S. Sprick Department of Computer Science and Engineering University of Nebraska-Lincoln, USA April 2, 2007 Abstract This report provides a theoretical analysis on the staleness effect in a passive localization system. The relationship is found between the position error and the velocity and the inter-arrival time of range signals. Results from the analysis make it possible to parameterize the localization system, such as the beacon deployment, required sampling rate and benchmark performance. Simulations are performed based on some trajectories generated by constant and non-constant velocity movements. The outcome

2 of the simulation is consistent with the analysis results. The simulations show that the Linear Least Square (LSQ) has a tradeoff effect between the staleness effect and its ability to combat the range measurement noise. From the results, a scheme using LSQ and linear Kalman filter shows better performance than the extended Kalman filter scheme. Experiments in the Cricket system has verified the observations from the analysis and the simulations. 1 Introduction Localizing the position of an object is one of the core functions of indoor assisted living environment. In a 2-D localization system, at least three range measurements are required between the object and some fixed reference points to compute the object position. For most of the applications in indoor environment, sensor nodes are mounted on the ceiling of the rooms as reference points and communicate with a sensor node attached to the object. The range measurements can be obtained by transmitting range signals, such as RF signals, ultra-sonic signals [11] or infra-red signals [16], between a reference point and the object. In general, the transmitting node is called beacon and the receiving node is called listener. The listener node uses time-of-arrival (TOA) [19], time-difference-of-arrival [14], received signal strength [10] or angle of arrival [9] of the range signals to compute the distance between the listener node and the beacon node. To guarantee the accuracy of the 2-D localization, it is generally required that the three or more ranges can be measured simultaneously. The most direct solution to the simultaneous range reception is to use the active architecture [5, 16, 7, 2], in which a beacon node is attached to the object and multiple listener nodes are mounted on the reference points. The active beacon node on the object transmits range signals periodically so that different listener nodes receive the signal at approximately the same time. In practical applications, the active architecture has some shortcomings, such as scalability, privacy and process latency. Thus, the passive architecture is introduced to overcome these shortcomings [14]. In such architecture, the listener is attached to the object and multiple beacons are mounted on

3 the reference points. The beacons transmit range signals to the listener according to some specific MAC protocol. Thus, it is generally difficult for the listener node to receive multiple range signals simultaneously if only one transmission channel is available. This might not be a problem for the accuracy of localization for a stationary object. However, if the object is moving, the stale ranging data due to the non-simultaneous signal reception may introduce serious localization error. Thus, this effect is called staleness in this report and efforts are made to analyze the effect of stale ranging data on the position error of 2-D localization. Since the stale ranging data generally appears in passive architecture, the staleness problem is mostly addressed as non-simultaneity in the localization systems [4, 14] or virtual environment systems using the passive architecture. In the field of virtual environment applications [18, 17, 1, 12, 15], non-simultaneity of range measurements may appear when a video camera is mounted on the moving object and range measurements are computed by taking pictures of multiple reference points sequentially. In these studies, the localization algorithms are based on the simultaneity assumption that the range measurements are taken at the same time. Due to the obvious performance degradation, methods are taken to avoid using multiple non-simultaneious range measurements in localization to satisfy the simultaneity assumption. The most common approach is to use a single constraint at a time (SCAAT) and apply extended Kalman filter (EKF) to obtain the current coordinates of the object [14, 18, 17, 1, 12]. However, the effect of the non-simultaneous range measurements has not been fully understood yet. The earliest analysis of this effect is found in Burton s Ph.D. dissertation dated back to 197 [4]. A similar analysis is found in [17] for the -D object tracking in virtual environment applications. In these studies, non-simultaneity is only discussed with simple examples and no deep theoretical analysis has been given yet. A rigorous analysis on the staleness effect considering all kinds of object movements and inter-arrival times of the range signals may provide significant direction for designing an accurate 2-D or -D localization system using sensor nodes. Also, the analysis could provide certain implications on the beacon node deployment. Thus, it is the target of this report to provide such a theoretical analysis on the relationship between the staleness, movement

4 pattern and inter-arrival time. The analysis is based on the assumption of three beacon nodes and Linear Least Square (LSQ) algorithm for localization. Further solution of the staleness problem is proposed by using a linear Kalman filter [] after LSQ to process the position estimates. Simulation results show that this approach is effective in improving the accuracy of 2-D localization that surpasses the performance of the SCAAT approach using EKF when the object is moving with higher speed. This report is structured as follows. In Section 2, related work of the localization system and the staleness effect is discussed. In Section, description of the 2-D localization system and assumptions about the analysis are provided. The analysis on the staleness effect for 2-D localization using LSQ is given. In Section 4, simulation over different movement trajectories are shown and simulation results are compared with the analytical results. In Section 5, a scheme using LSQ and standard Kalman filter is proposed that shows significant improvement under the staleness effect. The performance is compared with an EKF scheme proposed in [14]. In Section 6, experiments with the Cricket system is implemented to verify the theoretical analysis. In Section 7, conclusions from the analysis and simulations are drawn, and future research works are proposed. 2 Related Work For the localization and tracking systems, lots of work has been done in the area of assisted living environment, the virtual environment and the GPS system. The GPS system [19] is one of the most mature tracking systems. In such system a large number of localization and filtering techniques have be applied effectively, such as LSQ, iterative LSQ (ILSQ), linear Kalman filter, EKF and iterative EKF (IEKF) []. However, since the signal provided by the GPS satellite uses coded division multiple access (CDMA) scheme, the GPS system is a passive localization and tracking system with multiple channels. Thus, the staleness effect is not a concern in such system. In this report, filtering techniques popular in all kinds of GPS receivers, such as linear Kalman filter and EKF, are considered

5 in the indoor localization system to mitigate the staleness effect. In the field of Virtual Environment, the staleness effects is considered as the non-simultaneity assumption that the required ranging data are assumed to be received simultaneously. General approach in this area to solve this problem is to avoid breaking this assumption by using the SCAAT scheme [18, 17, 1, 12]. Lots of work has been done to improve the performance of the non-linear Kalman filter or EKF. Some research has addressed the non-simultaneity problem by using some simple examples [4, 17]. However, a thorough analysis is still missing. In this report, a rigorous analysis of the staleness effect is done based on some simple assumptions. The staleness effect is measured by the root mean square (rms) error of the estimated position as a function of the object s movement pattern including velocity and acceleration, and inter-arrival time of the range signals. According to such relationship, the localization system can be parameterized given a required level of rms error. Rules for the beacon deployment and transmission scheduling are also implied by the analysis results. Simulations are performed for LSQ over a large number of specific movement trajectories, inter-arrival times and range noise levels. The results show high consistency with the analysis results. In the area of indoor applications about the assisted living environment, the passive localization systems has been studied in [14, 8]. In [14], a tracking scheme using Cricket motes and EKF is proposed for the passive localization. In this scheme, the beacon mote transmits both RF and ultrasonic signal and the listener uses the times difference of arrival of the two signals to compute the ranges. The beacon coordinates are carried with the RF signal to the listener. For the EKF, a single range is taken at a time and the estimate of the current object position is output. The EKF is based on a mixture of position model (P-model) that assumes constant velocity and a position-velocity model (PV-model) that treats velocity as a state of the filter. Due to its non-linearity, the EKF is not stable and its states may diverge during the tracking procedure. Thus, an outlier rejector is implemented to detect the bad state and reset the EKF state using LSQ output. For this reason, though the EKF does not introduce stale ranging data, staleness will be introduced by LSQ, which

6 deteriorates the performance when EKF resets. In this report, experiments are performed in the Cricket system to verify the effectiveness of the theoretical analysis of the passive localization. The linear Kalman filter is used to reduce the localization error. Results show that this scheme is effective and performs better than the EKF scheme in [14]. The transmission of the Cricket motes uses a scheme similar to the CSMA/CA [1] protocol. Thus, the inter-arrival time between each two consecutive range signals varies. This is different from scheme used in [8], where the Cricket motes use TDMA to schedule the transmissions in the passive architecture. The comparison between the analysis results and the experiment results shows that scheduling of the transmissions in a TDMA manner is necessary. Analysis In the design of an indoor passive localization system, three important questions need to be answered. First, how does the object move in a given indoor space? Second, for a specific velocity and indoor space, what is the required sampling rate to have the estimation error lower than a given level? Third, how should the beacons be deployed so that the performance is optimized. The first question can be answered by the requirement of a specific indoor assisted living environment that generally deals with human velocities. To answer the other two questions, a relationship needs to be found between the localization error, the target velocity and the sampling rate. It is the object of this analysis to answer these questions. The localization error is measured by the distance between the estimated position and the actual position. This analysis computes the error of LSQ as a function of the object s velocity and the sampling rate. The analysis considers a 2-D space of 2w 2h cm 2. The x-coordinates range from w to w and the y-coordinates range from h to h. To make the analysis reasonably simple, it is assumed that the position of the target is uniformly distributed in this space. A simple scheduling scheme of transmissions is assumed that the beacons take turns to

7 C 2 C 1 B2 B 1 r 2 C r 1 B r Figure 1: The localization system with triangulation. transmit range signals according to a constant inter-arrival time T s = 1 f s and f s is called sampling rate. Thus, there is no collision of signals if a single transmission channel is available. For the three consecutive sampling times, t 1, t 2 and t, the coordinates of the sampled positions are P 1 (t 1 ) = P 1, P 2 (t 2 ) = P 2 and P (t ) = P. The localization algorithm computes the estimate of P, denoted by ˆP (t ) = ˆP, using the three ranges R 1 (t 1 ) = r 1, R 2 (t 2 ) = r 2 and R (t ) = r measured at time t 1, t 2 and t, respectively. Assume the ranges, r 1, r 2 and r, are the distances between the target and the three beacons B 1, B 2 and B, respectively. The coordinates of the beacons are known to the localization algorithm. The range measurements are assumed to be noisy, and the distribution of the measurement noise is assumed to be zero-main Gaussian noise with variance σ 2. The localization algorithm with three ranges is described in Figure 1, where the target moves from the time t 1 to t with an initial velocity at t 1 and a constant acceleration. Given the three ranges, the possible locations of P 1, P 2 and P are on the circles C 1, C 2 and C that are centered at B 1, B 2 and B, respectively. The localization algorithm finds an intersection of the circles C 1, C 2 and C, which is the estimated position ˆP. The three circles can be expressed in math by C 1 : (ˆx a 1 ) 2 + (ŷ b 1 ) 2 = r1, 2 C 2 : (ˆx a 2 ) 2 + (ŷ b 2 ) 2 = r2, 2

8 C : (ˆx a ) 2 + (ŷ b ) 2 = r 2, (1) where (ˆx, ŷ) are the estimated coordinates of P and (a i, b i ) are the coordinates of the beacon B i. This localization algorithm is called triangulation when the target is static and the range measurements are accurate and there is a unique solution of the equations in (1). In the presence of stale ranging data or measurement noise, triangulation fails to find a solution of the equations in (1) since the circles may not intersect at a single point. On the other hand, LSQ will provide the coordinates of ˆP by subtracting C 2 and C from C 1 and computing an intersection of the two resulting lines. This result is given by ˆx = v 1 r v 2 r v r 2 + v 4, ŷ = v 5 r v 6 r v 7 r 2 + v 8, (2) where the constants v 1 -v 8 are computed from (a 1, b 1 ), (a 2, b 2 ) and (a, b ). Note that in the absence of measurement noise or staleness, LSQ provides the same result as given by triangulation. For both triangulation and LSQ, if the three beacons are located on a line, no solution of (1) can be found, which is denoted as colinear case. In the analysis, it is assumed that the colinear case never happens and LSQ is considered for the localization algorithm to guarantee that the estimated position ˆP can always be found. In deriving the localization error as a function of velocity and sampling rate. The velocity and sampling rate are assumed to be known. An example in analyzing the problem is described in Figure 2. At the time instants t 1, t 2 and t, the possible coordinates of P 1, P 2 and P are on the circles C 1, C 2 and C. Due to the velocity, all points on the circle C 1 will shift to C 1 along a specific trajectory in the first sampling period. A constant velocity is assumed in this example, so the trajectory is linear. Thus, the position P 2 is also on the circle C 1 and it should be one of the intersections of C 1 and C 2. In the second sampling period, all points on C 1 and C 2 will shift along the linear trajectory to the circles C 1 and C 2. So, the accurate position P is at the intersection of C 1, C 2 and C. In Figure 2, the estimated position ˆP from LSQ is also shown. In the presence of measurement noise or stale ranging data, P and ˆP can be different. The squared distance between P and ˆP is

9 C 2 d = S / f s B2 C 2! S B 1 P 1 C C 1 B1 P 2 B 2 B C 1 B1 P C 1 C 1! C 2 C 1! C P Figure 2: Localization based on the stale ranging data. computed by Err(ˆP, P ) = (ˆx x ) 2 + (ŷ y ) 2. () Given the known beacon coordinates, there are two approaches to express the error in () as a function of velocity and sampling rate. - Given a set of ranges r 1, r 2 and r, for each velocity and sampling rate, the estimated position ˆP is computed from (2) and the true position P is equal to the intersection of C 1, C 2 and C. Thus, ˆP and P are functions of velocity, sampling rate and ranges. As a result, err( ˆP, P ) in () is a function of velocity, sampling rate and the ranges. - Given an ending position P, for each velocity and sampling rate, the first two positions P 1 and P 2 can be computed. Next, compute the ranges r 1, r 2 and r from P 1, P 2, P and the beacon coordinates. Based on the assumption that the range measurements are noised by the Gaussian noise, r 1, r 2 and r are also functions of n = {n 1, n 2, n }, where n 1, n 2, n are the zero-main Gaussian noises with variance σ 2. The estimate ˆP

10 can be computed from (2) using the ranges. Thus, the error is a function of velocity, sampling rate, P and n. As a result, the error err( ˆP, P ) is a function of velocity, sampling rate, P and n. For the first approach, given an arbitrary set of r 1, r 2 and r, the values of velocity and sampling rate may not guarantee that the circles C 1, C 2 and C have a single intersection, since the given ranges may be inconsistent with the values of velocity and sampling rate. So, the true position P may not be found. Hence, for each velocity and sampling rate values, cares should be taken such that the ranges are given to be reasonable. For the second approach, there is no consistency problem, and it is in fact equivalent to the first approach. For simplicity, this analysis uses the second approach. Assume the velocity is a function of time, v(t) = s(t) θ(t), with constant acceleration a = s θ. In math, the velocity can be computed by v(t) = s(t) θ(t) = (S 0 + st) (θ 0 + θt), (4) where S 0 and θ 0 are the initial speed and initial direction at time t 1. Given the position P = (x, y ), the positions P 1 = (x 1, y 1 ) and P 2 = (x 2, y 2 ) can be computed by x 1 = x α 1 cos(θ 0 ) β 1 sin(θ 0 ), y 1 = y + β 1 cos(θ 0 ) α 1 sin(θ 0 ), x 2 = x α 2 cos(θ 0 ) β 2 sin(θ 0 ), y 2 = y + β 2 cos(θ 0 ) α 2 sin(θ 0 ), (5) where α 1, α 2, β 1 and β 2 are functions of S 0, f s, S and θ. Details in computing α 1, α 2, β 1 and β 2 are given in Appendix I. Thus, the noisy range measurements are ˆr 1 = ˆr 2 = ˆr = (x 1 a 1 ) 2 + (y 1 b 1 ) 2 + n 1, (x 2 a 2 ) 2 + (y 2 b 2 ) 2 + n 2, (x a ) 2 + (y b ) 2 + n, (6)

11 The estimated position ˆP = (ˆx, ŷ) can be computed by (2) using ˆr 1, ˆr 2 and ˆr in (6) as ranges. So, ˆP is a function of P = (x, y ), v, f s and n. Given a constant acceleration a, the localization error from LSQ can be computed by () and then expressed in the function Err( ˆP, P ) = f( v, f s, P, n). (7) Note that, the error function f( v, f s, P, n) depends on the position P and the Gaussian range noises. In order to study the localization error as a function of velocity and sampling rate, it is necessary to overlook the effects of P and n over the localization error. Based on the assumptions about the distributions of P and n, the error function is averaged over P and n. Further, velocity v is a function of the initial speed S 0 and angle θ 0. Note that, in investigating the effect of stale ranging data on the localization error, it is more important to decide the relation between the error and the speed. For these reasons, the error function is averaged over the Gaussian noises, the position P and the movement angle θ 0. It is assumed that the angle θ 0 follows a uniform distribution between 0 and 2π. Finally, the localization error is measured by the rms error over P, θ 0 and n, which is a function of S 0 and f s as Err rms (S 0, f s ) = { w 1 2π w 1 2w h h 2π 0 1 2h dθ 0dx dy E n [f( v, f s, P, n)] } 1 2, (8) where E n ( ) is the expectation over the Gaussian noises. A closed form of the function Err rms (S 0, f s ) is derived in Appendix II. To show the dependancy of the localization error on velocity and sampling rate, the 2-D space with 2w 2h cm 2 is considered. Three beacons are located on a circle with radius r = 200 cm and centered at (0, 0), as shown in Figure 4(a). The beacons form an equilateral triangle with B 1 on the x-axis. The acceleration of the target is fixed at S = 0 cm/s 2 and θ = π 2 rad/s2. In Figure, Err rms is computed as a function of f s by fixing S 0 to 20 cm/s. Both w and h are set to 400 cm. As a comparison, the high velocity results are shown in Figure (b), where the Err rms is computed as a function of f s by fixing S 0 to 100 cm/s

12 (a) S 0 = 20 cm/s (b) S 0 = 100 cm/s ! 2 =0cm 2! 2 =1000cm 2! 2 =2000cm 2! 2 =000cm err rms cm ! 2 =0cm 2! 2 =1000cm 2! 2 =2000cm 2! 2 =000cm 2 err rms cm f s Hz f s Hz Figure : err rms with S = 0cm/s 2, θ = π 2 rad/s2 and (a)s 0 = 20 cm/s, w = h = 500 cm, (b) S 0 = 100 cm/s, w = h = 2000 cm. and w = h = 2000 cm. For both the low speed and high speed cases, the variance of the measurement noise changes from 0 to 000 cm 2. The results in Figures (a) and (b) show that the localization error decreases while the sampling rate is increasing. When the noise level is high, there is a high error floor. The Err rms converges to the error floor faster when the range noise is highe. For example, in Figure (a) the Err rms converges to the error floor for f s > 4Hz when σ cm 2, while it converges to the error floor after 10 Hz when σ = 0. The reason is that there is a tradeoff between the effect of ranges noise and the effect of staleness. For small range noise, the stale ranging data affects the performance significantly. Thus, increasing the sampling rate mitigates staleness and lowers the localization error. When the noise level increases, the range noise affects the performance more seriously and increasing the sampling rate does not help in combating the range noise. By comparing the results for low speed and high speed in Figures (a) and (b), it shows that the high speed movement results in higher localization error. This is because that the staleness problem is more serious for high speed case. The tradeoff between range noise and

13 S!=" / 4 B2 o r B1 2h m B 2w m Figure 4: A 2-D square space and the deployment of 4 beacons. staleness can also be observed in this comparison. For the low speed case, the convergence of Err rms by increasing f s is faster than the high speed case. This implies that for high speed the staleness effect becomes dominant and increasing sampling rate helps reduce the localization error. The deployment shown in Figure 4 provides a way to investigate the relationship between the localization error and the beacon deployment. For this purpose, the size of the 2-D space is set to 800 cm 2. The acceleration a is set to 0. The Err rms is computed as a function of S 0 and f s when the radius of the circle is set to r = 100 cm and r = 400 cm. Results are compared in Figure 5, which shows that the localization error is much lower when r = 400 cm. This implies that it is desirable to have the beacons spread out over the given space such that the target always moves in the center of the area circumscribed by the three beacons. On the other hand, if the target is far from the beacons and out of the area circumscribed by the beacons, the error is large on average. Thus, in order to have a small localization error, it is preferred that the distance between each two beacons is as large as possible in the given area.

14 r=100cm r=400cm rmse cm rmse cm S cm/s f s Hz S cm/s f s Hz 20 Figure 5: Err rms for constant velocity, varying the distance between beacons. 4 Simulations In computing the Err rms as a function of velocity and sampling rate, it is assumed that the target can appear in any position in the given space with a uniform distribution. In reality, the object is generally moving on a specific trajectory and appears in specific positions in the space. Thus, it is necessary to test the consistency between theoretical analysis and practical scenarios. For this purpose, some movement trajectories are simulated and the Err rms are computed over these trajectories based on the following system configuration. 4.1 System Configuration To deploy the beacons in this space, the circle described in Figure 4 is applied again with r = 200 cm. K beacons are deployed on the circle and the coordinates of the i th beacon is (a i, b i ) where a i = r cos ( ) 2πi, K

15 ( ) 2πi b i = r sin. (9) K As assumed in the analytical model, transmissions of the beacons are scheduled in a pipeline style that the beacons take turns transmitting around the circle with T s = 1 f s seconds between each two consecutive transmissions. Each sampling point is defined as the time the listener receives a range signal. For example, the transmissions of the 4 beacons at sampling points {t k,..., t k+7 } are scheduled as {..., B, B 4, B 1, B 2, B, B 4, B 1, B 2,...}. At each sampling point, the target takes the three most recent ranges from different beacons as the input to LSQ. In this example, the range sets input to the localization algorithm at sampling points {t k,..., t k+7 } are t k : {r 1 (t k 2 ), r 2 (t k 1 ), r (t k )}, t k+1 : {r 2 (t k 1 ), r (t k ), r 4 (t k+1 )}, , t k+7 : {r 4 (t k+5 ), r 1 (t k+6 ), r 2 (t k+7 )}, where r j (t k+i ) denotes the range received from the beacon B j at the sampling point t k+i. Three categories of trajectories are simulated: linear trajectories, circular trajectories and random walk. For the linear trajectories, the target starts from the position ( 500cm, 500cm) or (-20cm, -20cm) and moves for 1 minute with the constant speed S 0 = 20 or 50 cm/s and angle θ 0 = π. For the circle trajectories, the target starts from the position (0, 100cm) or 4 (0, 500cm) and moves along the trajectory with initial speed S 0 = 20 or 50 cm/s, initial angle θ 0 = 0, speed acceleration S = 0 and angle acceleration θ = π 16 rad/s2. For the random walk, the target starts from position (0, 0) and moves with random velocities for 1 minute with an average speed of 27 cm/s. For a fair comparison between the simulation results and analytical results, in computing the analytical rmse by (19), the size of the 2-D space is set to the area that covers the trajectory exactly. As an example, the low speed linear trajectory is shown in Figure 6 as

16 / o 9:71,279: /059 ;o:171o <1 <2 < (+,)!100!200!00!400!500!600!600!500!400!00!200! ( (+,) Figure 6: Trajectory of a movement with constant velocity. S = 0.2 cm/s, θ = π 4. the solid line. The deployment of three beacons is also shown here. The trajectory estimated by LSQ is described with the dashed curve in Figure 6. Note that the smallest space area to cover the trajectory is around cm 2. Thus, for the analysis, w and h are set to 500 cm. For the same reason, w and h are set to 2000 cm for the high speed linear trajectory, 100 cm for the low speed circle trajectory, 550 cm for the high speed circle trajectory and 200 cm for the random walk. 4.2 Results As the simplest case, three beacons are considered and the comparison is presented in Tables 1 and 2 for σ 2 = 0 and 1000 cm 2. As shown in these result, the analytical results are consistent with the simulation results for all the trajectories and for both noiseless and noisy ranges. The difference between the analytical results and simulation results is due to the randomness introduced in the simulations. Though this difference is large for high speed trajectories, it is generally within 0% of the Err rms from the analysis. This consistency verifies that the

17 Err rms obtained in the analysis is effective even though it assumes that the target position is uniformly distributed in the space. Table 1: Comparison of Err rms from analysis and the simulation when σ 2 = 0. Err rms (cm) from analysis f s = f s = f s = 1Hz 5Hz 20Hz Err rms (cm) from simulation f s = f s = f s = 1Hz 5Hz 20Hz Low Speed Linear high Speed Linear Low Speed circle high Speed circle random walk Table 2: Comparison of Err rms from analysis and the simulation when σ 2 = 1000 cm 2. Err rms (cm) from analysis f s = f s = f s = 1Hz 5Hz 20Hz Err rms (cm) from simulation f s = f s = f s = 1Hz 5Hz 20Hz Low Speed Linear high Speed Linear Low Speed circle high Speed circle random walk , In practical applications, more than three beacons are deployed so that a large space area can be covered by the beacons signals. Thus, the simulation is extended to use more than three beacons. Due to the consistency between the simulation and analysis seen from

18 the previous discussion, simulation is performed to show the performance of localization algorithm using LSQ. Further, LSQ can consider more than three ranges for the localization at each sampling point. In the simulation,, 5 and 7 beacons are deployed in the 2-D space according to (9) for r = 200cm. Based on such deployment, when there are more beacons deployed, the distance between each two neighboring beacons are smaller. According to the transmission scheduling discussed in Section 4.1, the beacons considered by each sampling point circumscribe a smaller area if the radius of the circle is fixed when more beacons are deployed on this circle. Thus, increase of Err rms is expected due to the decrease of distance between beacons. If more than three ranges are input to LSQ, more stale data are introduced. In this sense, it is expected that LSQ will perform worse compared with the case when three ranges are used. On the other hand, if more than three ranges are considered, LSQ has the property to draw the estimates closer to the exact positions when there is measurement noise. Thus, in the noisy range situation, a tradeoff effect is expected between the staleness introduced by LSQ and the effect of LSQ to combat range noise. In Figures 9 and 10, simulation results are shown for LSQ when the random walk trajectory is considered and the sampling rate is 1Hz. The X-axis denotes the number of ranges considered by LSQ. The red curves are provided by LSQ with three ranges. It can be observed that when there is no measurement error (σ 2 = 0cm 2 ), LSQ performs worse when more ranges are considered. When noise is presented in the range measurements, the tradeoff effect is obvious for LSQ, which implies that 4 is the best number of ranges to approach the benchmark performance in this case. In Figures 11 and 12, simulation results are shown when the sampling rate is increased to 5Hz. When the measurement noise is presented, LSQ can reduce the estimation error. Since the sampling rate in this case is higher, the staleness problem is not serious. The tradeoff effect is not obvious even when 7 ranges are used. However, it can be expected that when there are more than 7 ranges considered in LSQ, the Err rms will start to increase.

19 r j E"# Prediction previous state Correction predicted NO predicted..., r j!2, r j!1, r r j r j j Outlier Rejection Badstate? YES LSQ Reset State estimated (x, y) Figure 7: EKF scheme using a single range at a time. 5 Solutions of Staleness Effect: Kalman Filters From previous analysis and simulation results, it is obvious that staleness seriously degrades the performance of the localization system when high sampling rate cannot be achieved. Thus, methods must be taken to reduce the position error introduced by staleness. A usual approach in existing localization systems is to use single range at a time and apply EKF to process the input range as shown in Figure 7. In this scheme, the use of stale ranging data in localization is avoided. However, since a single constraint is used in 2-D localization and tracking, which generally requires at least constrains, accuracy of the EKF scheme cannot be high. This problem becomes more serious when the object is moving at high speed and the sampling rate is low. Another approach to solve this problem is to use a linear Kalman filter after the localization algorithm, such as LSQ. This scheme, denoted by LSQ+SKF, is shown in Figure 8, where the SKF stands for linear Kalman filter or standard Kalman filter. Simulations are performed to compare the performance of LSQ, LSQ+SKF and EKF.

20 Kalman Filter r, r,..., r 1 2 j Localization pre!estimated (x,y) Correction estimated (x,y) predicted (x,y) Prediction Figure 8: Localization plus standard Kalman filter scheme. EKF is implemented using the scheme in [14]. Since the outlier rejector is used for the EKF scheme, the EKF uses LSQ results to reset its state when the outlier rejector detects a bad state. As a result, the number of ranges used in LSQ affects the performance of EKF slightly. For the LSQ+SKF scheme, since the SKF takes the LSQ output as input, the performance of this scheme is significantly affected by the number of ranges used in LSQ. In Figures 9-12, results of LSQ, LSQ+SKF and EKF are compared for the random walk trajectory. From the comparison, it shows that the LSQ+SKF scheme performs consistently better than the LSQ and EKF. Most of the time, the EKF scheme even performs worse than the LSQ. It could be explained by two reasons. First, the EKF uses a single range as input that make the localization results worse since ranges are required for an accurate 2-D localization. Second, the EKF is not stable and needs to reset its states using LSQ s result from time to time. Since the LSQ introduces staleness when the object is moving, the EKF is reset with an inaccurate localization value. Moreover, it is much more difficult for the EKF to converge to the correct state when it resets while the object is moving. Thus, the performance of EKF could become worse and worse after many resets. As a conclusion, the results presented in Figures 9-12 suggest that the LSQ+SKF scheme outperforms the EKF scheme that uses a single range at a time.

21 Err rms (cm) LSQ,beacons LSQ+SKF,beacons EKF+LSQ,beacons LSQ,5beacons LSQ+SKF,5beacons EKF+LSQ,5beacons LSQ,7beacons LSQ+SKF,7beacons EKF+LSQ,7beacons Number of ranges used in localization Figure 9: Err rms over random walk trajectory when f s = 1Hz, σ = 0 cm LSQ,beacons LSQ+SKF,beacons EKF+LSQ,beacons LSQ,5beacons LSQ+SKF,5beacons EKF+LSQ,5beacons LSQ,7beacons LSQ+SKF,7beacons EKF+LSQ,7beacons Err rms (cm) Number of ranges used in localization Figure 10: Err rms over random walk trajectory when f s = 1Hz, σ = 1000cm 2.

22 Err rms (cm) LSQ,beacons LSQ+SKF,beacons EKF+LSQ,beacons LSQ,5beacons LSQ+SKF,5beacons EKF+LSQ,5beacons LSQ,7beacons LSQ+SKF,7beacons EKF+LSQ,7beacons Number of ranges used in localization Figure 11: Err rms over random walk trajectory when f s = 5Hz, σ = 0cm Err rms (cm) LSQ,beacons LSQ+SKF,beacons EKF+LSQ,beacons LSQ,5beacons LSQ+SKF,5beacons EKF+LSQ,5beacons LSQ,7beacons LSQ+SKF,7beacons EKF+LSQ,7beacons Number of ranges used in localization Figure 12: Err rms over random walk trajectory when f s = 5Hz, σ = 1000cm 2.

23 6 Experiments in the Cricket System The theoretical analysis and simulations results provide certain rules for parameterizing a practical localization system. To verify these rules, experiments using the Cricket system are performed. To be specific, the rules are listed below. - Beacons should spread out over the given space such that the distance between each two beacons is as large as possible. - Required sampling rate for a required level of position error can be found by the theoretical analysis. - An optimal number of ranges exists for LSQ to approach the best performance. - LSQ+SKF performs better when sampling rate and object speed is high. The experiments are performed in a room of cm 2. A cricket mote is attached to a small train running on a square oval track as listener. Information about the track is provided in Figure 1. The train is moving on the track with two speeds: 16cm/s and 50cm/s. -8 cricket motes are deployed in the room as beacons. As shown in Figure 1, the following five beacon deployments are used. - D1: beacons on a right triangle with small edge length. - D2: 4 beacons on the corners of a rectangle with small edge length. - D: beacons on a right triangle with large edge length. - D4: 4 beacons on the corners of a rectangle with large edge length. - D5: 8 beacons deployed as a composition of D2 and D4. Since the cricket motes use a random access protocol similar to CSMA/CA [1] and the transmissions of the beacons are not scheduled, the sampling period and sampling rate varies for different sampling points. This is different from the transmission scheme assumed by the theoretical analysis and simulations. However, since the sampling rate in the experiments varies within a given region, the performance obtained from the experiments is a compromise between a high sampling rate performance and a low sampling rate performance. The timestamp and the corresponding range from the most recent received range signal are extracted from the listener as a sampling point. This range is also the input of EKF. A

24 D4 D D5 (!126, 85) (0, 108) y (55, 85) D D4 D5 D2 D1 D5 (!149, 0) D1 D2 D5 (!41, 27) (!4,,1) (!67,!9) (!2,!7) D1 D2 D5 D5 D2 o (76, 0) x D D4 D5 (!126,!97) (0,!117) (56,!97) D4 D5 Figure 1: Track information and beacon deployments. history of the received range signal is kept for each beacon and three or more most recently received ranges from different beacons are input to LSQ for localization. The number of ranges used by LSQ is different for the deployments. For D1 and D, only three ranges can be used since there are only three beacons. For D2 and D4, or 4 ranges can be used since there are 4 beacons. For D5, up to 8 ranges can be used since there are 8 beacons. The output of LSQ for each sampling point is input to the SKF. Timestamps of the sampling points are also input to SKF and EKF since they need this information to compute the time difference between sampling points and predict the next state according to the velocity model. To test the LSQ performance, the newest ranges are used for D1 and D. The or 4 newest ranges are used for D2 and D4. The -8 newest ranges are used for D5. To compute the Err rms, the localization results and the results given by SKF and EKF are compared to the accurate positions of each sampling point, which is provided by a synchronized video analysis of the train movement. Note that this video analysis can achieve an accuracy of less than 5cm within the exact position.

25 60 Average sampling rate:2.628 Hz 50 Number of sampling points Inter arrival time (ms) Figure 14: Inter-arrival time between sampling points for D1 Statistics of the inter-arrival time between each pair of consecutive sampling points are shown for D1 in Figure 14. The average sampling rates achieved for D1 is 2.628Hz. For the other deployments, the histograms have the similar shape. The average sampling rate for these deployments are summarized in Table. Table : Average sampling rate achieved by D1-D5. D1 D2 D D4 D5 f s 2.628Hz.618Hz 2.444Hz.1747Hz 4.66Hz Due to the random access scheme in the cricket system, more beacons lead to lower inter-arrival time and thus higher sampling rate. Thus, D1 and D, or D2 and D4 have the similar sampling rate. Since three beacons are deployed, the sampling rate of D1 and D is the lowest. D5 provides the highest sampling rate since 8 beacons are deployed. Paralleled with the experiments, the theoretical analysis is performed using LSQ for D1, D2, D and D4. The space size is set to the size of the square oval track. Based on previous experiments on the distribution of range measurements in the Cricket system [6], it

26 is assumed that the measurement error has the variance σ = 10cm 2. The average sampling rates shown in Table are used for the four deployments such that the results can be compared to the experiment results. For D2 and D4, since there are 4 beacons, the analysis choose any three of them as the beacons, which makes sense since any subset of beacons in D2 and D4 will form a triangle with same edge lengths and thus the distances between the beacons are the same. In the analysis, assume that the movement is with constant velocity since most of the time the train runs on the straight edges of the square oval track with constant speed. Results from the analysis is shown in Table 4. Table 4: Err rms (cm) from analysis. Deployment speed: 16cm/s speed: 50cm/s D D D D In Tables 5 and 6, the simulation results are provided, where the set size is the number of ranges input to LSQ. The results with set size are close with the analysis results shown in Table 4. However, the Err rms found by the analysis are generally lower then the experiment results. The reason is that the analysis assumes uniform sampling rate and thus uniform inter-arrival time for the samples. In the Cricket system, the inter-arrival time varies and may be much higher than the average level, which deteriorates the performance seriously. This implies that the transmission of the beacons need to be scheduled. A TDMA based MAC scheme will improve the performance significantly. Moreover, the optimal MAC scheme needs to schedule the beacons such that the newest ranges come from the beacons with large distances between them. The best performance from the experiment for low speed is given by D4. This observation is consistent the the analysis result that D4 provides the best performance. For the high speed, staleness effect will affect the performance seriously. In this case, the higher

27 sampling rate will mitigate the staleness effect and improve the performance largely. Since the sampling rate of D4 is lower than D2 and D5, its Err rms is higher than D2 and D5 for high speed. Also, performance of D is better than the performance of D1 since the distance between beacons in D is larger. Tradeoff effect of LSQ is obvious in the experiment results. For D2 and D4, the Err rms is higher for all the schemes when 4 ranges are used since the sampling rate is low in this case and the staleness effect affects the performance seriously. For D5, the best tradeoff is achieved when 4 ranges are used. In this case, the sampling rate is higher and staleness is not as serious as in that D2 and D4. Thus, LSQ reduces estimation error and improves the performance. When more ranges are used, the staleness effect starts to affect the performance more. Thus, Err rms increases when more than 4 ranges are used. The experiment results show that the LSQ+SKF scheme improves over LSQ and EKF. Though EKF provides a decent Err rms for D5 and low speed, that is because the sampling rate is high in this case. In most of other cases, even LSQ is better than EKF. 7 Conclusion and Future Work In this report, the theoretical analysis of the staleness effect of ranging data in an indoor 2-D localization system with passive-architecture is discussed. This analysis has considered both the constant velocity movements and the non-constant velocity movements with constant acceleration. Though the analysis only considers LSQ and it is based on many impractical assumptions, such as the three beacons constraint, uniform sampling rate and uniform object position in a given space, it provides certain rules concerning the design of a localization system. First, the analysis provides a relationship between the Err rms of the estimates, the movement speed and the sampling rate. This is the first time such relationship is found for localization systems and it provides an effective approach to decide the sampling rate given the required level of Err rms and the object velocity. Second, the analysis implies that the best beacon deployment should make the distance between beacons as large as possible such

28 Table 5: Err rms (cm) for experiment with speed 16cm/s. Deployment Set Size TRI LSQ LSQ+SKF EKF D D D D D D D D D D D D

29 that the object is always moving in the area circumscribed by the beacons. Besides the theoretical analysis, simulations based on all kinds of trajectories and movement patterns have been performed to show that the analysis is effective even when the object is moving along specific trajectory in specific spots of the space. The simulation results presented in this paper are consistent with the theoretical analysis. These results provide some insight about LSQ and filtering techniques such as the linear Kalman filter and the extended Kalman filter. One of the most important observations from these results is that LSQ has the tradeoff effect between its ability to combat the range noise and the staleness it introduces. Generally, an optimal number of ranges needs to be decided so that the Err rms from LSQ is the lowest. The simulation results also suggest that the LSQ+SKF scheme can mitigate the staleness effect and improve the system performance. In fact, the performance achieved by LSQ+SKF scheme has surpassed the EKF scheme that uses a single range at a time. Table 6: Err rms (cm) for experiment with speed 50cm/s. Deployment Set Size TRI LSQ LSQ+SKF EKF D D D D D D D D D D D D

30 The experiments in the Cricket system also shows that the theoretical analysis provides a reliable approach to parameterize the system. The results verified that LSQ s tradeoff effect happens in the practical system and the LSQ+SKF scheme can achieve better performance than EKF. Since the theoretical analysis assumes uniform inter-arrival time of the samples, the computed Err rms are better than the experiment results which are featured by the varying inter-arrival times. Thus, a better design of a localization system requires to schedule the transmissions in a TDMA manner. Also, the scheduling needs to guarantee that the newest ranges come from the beacons with large distances between each other. For the future research, solutions should be taken to attack the staleness effect directly. A possible approach is to use a non-linear Kalman filter with multiple-range inputs. The traditional EKF scheme uses a single range each time in order to avoid using stale ranging data. However, the accuracy of the scheme is reduced because less constraints are used in the filtering. Given a non-linear Kalman filter with multiple-range inputs, there are more constraints used for localization and the filter can use certain velocity models to adjust the input ranges so that the staleness can be reduced. A Computation of P 1 and P 2 From the definition of velocity and acceleration in (4), if the target starts from P 1 = (x 1, y 1 ), at the time instant t, the position of the target, (x(t), y(t)), can computed by x(t) = x 1 + t 0 t s(t) cos(θ(t))dt = x 1 + (s 0 + st) cos(θ 0 + θt)dt 0 [ 1 = x 1 + θ (s 0 + st) sin( θt) + s ] s 2 cos( θt) θ θ 2 cos(θ 0 ) +[ 1 θ (s 0 + st) cos( θt) s 0 θ s θ 2 sin( θt) ] sin(θ 0 ), (10)

31 t y(t) = y t s(t) sin(θ(t))dt = y 1 + (s 0 + st) sin(θ 0 + θt)dt 0 [ = y θ (s 0 + st) cos( θt) + s 0 θ + s θ 2 sin( θt) ] cos(θ 0 ) 1 +[ θ (s 0 + st) sin( θt) + s ] s 2 cos( θt) θ θ 2 sin(θ 0 ). (11) Let α(t) = 1 θ (s 0 + st) sin( θt) + s s 2 cos( θt) θ θ 2, β(t) = 1 θ (s 0 + st) cos( θt) s 0 θ s θ 2 sin( θt), (12) then (10) and (11) become x(t) = x 1 + α(t) cos(θ 0 ) + β(t) sin(θ 0 ), y(t) = y 1 β(t) cos(θ 0 ) + α(t) sin(θ 0 ). (1) Note that the time between t 1 and t 2 is 1 f s, and the time between t 1 and t is 1 f s. Thus, given P = (x, y ), P 1 and P 2 can be computed from (1) as x 1 = x α(t t 1 ) cos(θ 0 ) β(t t 1 ) sin(θ 0 ) = ( ) ( ) 2 2 x α cos(θ 0 ) β sin(θ 0 ), fs fs y 1 = y + β(t t 1 ) cos(θ 0 ) α(t t 1 ) sin(θ 0 ) = ( ) ( ) 2 2 y + β cos(θ 0 ) α sin(θ 0 ), fs fs x 2 = x 1 + α(t 2 t 1 ) cos(θ 0 ) + β(t 2 t 1 ) sin(θ 0 ) = ( ) ( ) 2 2 x α cos(θ 0 ) β sin(θ 0 ) fs fs ( ) ( ) 1 1 +α cos(θ 0 ) + β sin(θ 0 ) fs fs

32 ( ( ) ( )) 2 1 = x α α cos(θ 0 ) fs fs ( ( ) ( )) 2 1 β β sin(θ 0 ), fs fs y 2 = y 1 β(t 2 t 1 ) cos(θ 0 ) + α(t 2 t 1 ) sin(θ 0 ) = ( ) ( ) 2 2 y + β cos(θ 0 ) α sin(θ 0 ) fs fs ( ) ( ) 1 1 β cos(θ 0 ) + α sin(θ 0 ) fs fs = y ( ( ) ( )) 1 2 β β cos(θ 0 ) fs fs ( ( ) ( )) 2 1 α α. (14) fs fs Let ( ) 2 α 1 = α, fs ( ) 2 β 1 = β, fs ( ) ( ) 2 1 α 2 = α α, fs fs ( ) ( ) 2 1 β 2 = β β, (15) fs fs then, (14) becomes x 1 = x α 1 cos(θ 0 ) β 1 sin(θ 0 ), y 1 = y + β 1 cos(θ 0 ) α 1 sin(θ 0 ), x 2 = x α 2 cos(θ 0 ) β 2 sin(θ 0 ), y 2 = y + β 2 cos(θ 0 ) α 2 sin(θ 0 ). (16) To simplify (x 1, y 1 ) and (x 2, y 2 ) further, introduce the following notations u 1 = α1 2 + β1, 2 u 2 = α2 2 + β2, 2 ( ) ɛ α1 = arc cos, u 1 ɛ = θ 0 ɛ,