Doppler effect on target tracking in wireless sensor networks

Computer Communications 36 (2013) 834–848

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Computer Communications journal homepage: www.elsevier.com/locate/comcom

Doppler effect on target tracking in wireless sensor networks Youngwon Kim An a, Seong-Moo Yoo a,⇑, Changhyuk An b, B. Earl Wells a a b

Department of Electrical and Computer Engineering, The University of Alabama in Huntsville, Huntsville, AL 35899, United States PRA, Huntsville, AL 35801, United States

a r t i c l e

i n f o

Article history: Received 27 April 2012 Received in revised form 17 December 2012 Accepted 2 January 2013 Available online 10 January 2013 Keywords: Wireless sensor networks Doppler effect Kalman filter Tracking Detection

a b s t r a c t This paper presents a new detection algorithm and high speed/accuracy tracker for tracking ground targets in acoustic wireless sensor networks (WSNs). Our detection algorithm naturally accounts for the Doppler effect which is an important consideration for tracking higher-speed targets. This algorithm employs Kalman filtering (KF) with the weighted sensor position centroid being used as the target position measurement. The weighted centroid makes the tracker to be independent of the detection model and changes the tracker to be near optimal, at least within the detection parameters used in this study. Our approach contrasts with previous approaches that employ more sophisticated tracking algorithms with higher computational complexity and use a power law detection model. The power law detection model is valid only for low speed targets and is susceptible to mismatch with detection by the sensors in the field. Our tracking model also enables us to uniquely study various environmental effects on track accuracy, such as the Doppler effect, signal collision, signal delay, and different sampling time. Our WSN tracking model is shown to be highly accurate for a moving target in both linear and accelerated motions. The computing speed is estimated to be 50–100 times faster than the previous more sophisticated methods and track accuracy compares very favorably. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction Recent advancements of micro sensors technology have allowed a wide range of wireless sensor networks (WSNs) implementations to be realized. WSNs are used in monitoring and detecting specific events in many types of industrial and military environments. They are especially useful in such situations as tracking emergency rescue workers, tracking military targets in modern battlefield scenerios, and monitoring traffic in transportation systems [1]. WSNs, in general, are composed of a large number of micro sensors and a base station. Each micro sensor is battery powered and is constructed using analog sensor, microcontroller, memory, and communication component subunits. A WSN acoustic tracking system, the main subject of this paper, includes two parts. One consists of acoustic sensors which are distributed in a uniform grid (or some other known configuration) to detect target sound. The other is the fusion center where target detection information from the sensors is processed to track the target. In general, these sensors are not capable of performing complicated computation and communication because of the limited power and computing resources that are present at each node. Each sensor uses a single ⇑ Corresponding author. Tel.: +1 256 824 6858. E-mail addresses: [email protected] (Y.K. An), [email protected] (S.-M. Yoo), [email protected] (C. An), [email protected] (B. Earl Wells). 0140-3664/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.comcom.2013.01.002

bit to indicate that a target had been detected. When the sensor detects target sound power that is higher than a predetermined threshold value, it sends the detection information bit along with its ID and sound power information to the fusion center, otherwise it sends no information. The fusion center receives the signals from the sensors and records the receiving time along with the sensor ID and sound power information. Based upon the particular sensor layout, each sensor ID is translated into a corresponding target position that can be used along with the sound power information for tracking the target using a wide range of tracking methods. 1.1. Previous studies A number of tracking methods for a single target tracking have been proposed in the literature for WSN type applications. Mechitov et al. [1] and Kim et al. [2] used a path based approach that used the detection duration of each sensor as a weight and performed line fitting in a time window to estimate target position at each time step. The requirement of using detection duration information from each sensor may be a hindrance for real time application of the algorithm and reliable line fitting may require a high sampling rate by each sensor. Both methods also need time synchronization and information exchange between neighboring sensors and thus tend to further impose computing and communication loads to each sensor. Wang et al. [3,4] used a similar geometric method for target position estimation which requires each

Y.K. An et al. / Computer Communications 36 (2013) 834–848

sensor to store neighbor node identifier, intersection points of sensing circle of the node and its neighbor and an angle corresponding to the arc defined by these intersection points of the sensing circle. In noisy real-world environments accurate estimation of these parameters may be challenging. Also the applicability of the method to a target with a short sensing range covering only one or zero sensor or with a long sensing range covering many sensors is to be seen. Ribeiro et al. [5] developed Kalman filter (KF) based recursive algorithms for distributed state estimate based on the sign of innovations (SOI). Running the SOI-KF requires each sensor to detect the target sound, compute the state and observation predicted estimate, obtain SOI and send the information to all other sensors for computing the updated state. As each sensor has limited power and limited computing and communication resources, this algorithm may give too much load to each of the sensor nodes. Recently, a number of authors [6–8,11] chose the particle filters (PFs) over KF for target tracking in WSN environments. They claim that KF is not an optimal filter as the system and measurement process are not linear and Gaussian in real environments. PFs have become one of the most popular methods for stochastic dynamic estimation problems and the PFs can handle nonlinear/nonGaussian measurements and dynamics of target with high accuracy but at the expense of computational complexity [10]. For the application of PF to WSN, PFs with lower computational complexity were proposed [22,23] but the error propagation through the sensor network was regarded as the main drawback of the methods [9]. Similar results are reported by other authors [12–19]. As an improvement over PF, Teng et al. [9] applied a variational filter (VF) for target tracking. The filter uses a Monte Carlo method for a set of weighted samples to approximate the posterior distribution and adapts the variational approach. The authors claim that the algorithm is able to track targets in near random motion with high accuracy. All the previous studies mentioned above use the power law for simulating target sound power detection by the sensors and use the same power law for modeling the detection in their trackers. Their use of power law excludes the Doppler effect so that their detection model and track results are applicable only for very low speed targets. The power law model also makes the detection model in their tracker susceptible to mismatch with the detection by the sensors deployed in real environment [8]. Even though their choice of trackers yields high track accuracy in their simulation, their trackers do not guarantee the same accuracy when it is used for real WSN environments because of the mismatch. Furthermore, their computational complexity, especially for PF and VF, is too high that their trackers may be costly for real world WSN system. 1.2. Contribution of this paper The main contribution of this paper is to develop a realistic detection algorithm where the Doppler effect appears naturally depending on the target speed. For our tracker, we use the KF with the weighted centroid of detecting sensor positions as target position measurement, which makes KF near optimal. None of the previous studies used KF for acoustic WSN tracking systems, except the case of distributed state [5,24], by reasoning that KF is not optimal in a variety of real world situations. The weighted centroid also allows the tracker to be independent of sound detection models so that there is no detection mismatch between the tracker and the sensors in the field. With our realistic detection model incorporated into KF, we can track high as well as low speed targets accurately with very low computational complexity. The tracker also enables us to study various environmental effects on track accuracy, including Doppler effect, detection message delay and message collision at the fusion

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center, and different sampling time steps. Our simulation study of these environmental effects demonstrates that the track accuracy is sensitive to these effects. Using our KF, our track accuracy is comparable to the accuracy of PF and VF and the computing speed is estimated to be nearly 100 times faster than the speed of those filters. The high computing speed and high track accuracy are critical factors for implementing the system in real world. This study is meant to lay the groundwork for more detailed systems study for implementing a WSN tracking system in real world scenarios. This paper is organized as follows: Section 2 describes the detection framework. It gives a detailed description of Doppler effect in the sensor detection simulation. In Section 3, we describe the tracking framework demonstrating that KF becomes near optimal. An algorithm for tracking a target in acceleration is also presented. In Section 4, we show the simulation results for track accuracy of target in linear and accelerated motions, and different environmental effects. Section 5 compares our results with those of previous works. Section 6 concludes the paper. 2. Detection framework We assume that the wireless sensors are simple passive sensors without range measurement capability. Each sensor has its own unique integer ID that translates into its position. Each sensor has own internal noise and also encounters ambient noise from the sensing region. A threshold value for the noises is set for each sensor before the deployment. Target sound over the threshold value will be detected by the sensor as a signal from a target and the detection information will be sent to the fusion center. 2.1. Model for target sound propagation with Doppler effect Most of previous studies modeled the received target sound power using the following simple power law equation. a

Ps;t ¼ P0;t =ds;j þ ms;t

ð1Þ

where P0,t is the target sound power at time t, Ps,t is the target sound power detected by the sensor, s, at time t, and ms,t is the detection noise at the sensor s which is assumed to be a zero mean Gaussian distribution. The distance from the sensor, s, to a target position at a time index j is expressed as ds,j in Eq. (2) and a is an attenuation parameter that depends on the environment of the sensing region.

 2  2 2 ds;j ¼ xs  xtj þ ys  ytj

ð2Þ

Here, (xs, ys) is the sth sensor position and ðxtj ; ytj Þ is the target position at a time index j. If the detected target sound power, Ps,t is one of measurement vector, the relationship between the measurement, ~ zs;t and target state vector, ~ xt is a non-linear function as

~ zs;t ¼ hð~ xt Þ þ ms;t

ð3Þ

Eq. (3) causes the Kalman filter (KF) to be suboptimal and then the PFs are appropriate for target tracking in WSN. We note that Eq. (1) is not an accurate detection model for high speed targets as the equation does not take into account the Doppler effect. As the sound speed is 340 m/s, target speed can be a nonnegligible fraction of the sound speed and the Doppler effect is important for high speed targets. Below, we are going to derive an equation of target sound propagation in which the Doppler effect is included and estimate the maximum target speed that allows us to use Eq. (1) as an accurate target sound detection model. Fig. 1(a) shows the target position moving from left to right with a constant speed at times t0, t1, t2, and t3 and the sound wave propagation from respective target positions in the time step, Dt.

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Y.K. An et al. / Computer Communications 36 (2013) 834–848 2

R2 ¼ d þ r 2  2rd cos h

ð5Þ

From Eqs. (4) and (5), we have

pffiffiffiffi r2 X  2rb cos h X þ b2  1 ¼ 0

ð6Þ

where X = Ps,t/P0,t and b = vt/Vc. By computing the root of Eq. (6), we have the equation for target sound detection which includes the Doppler effect as

Ps;t ¼

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P0;t 2 2 2 1 þ b ð2 cos h  1Þ þ 2bj cos hj 1  b2 sin h r2

ð7Þ

For this derivation we assume that the attenuation parameter, a of Eq. (1), equals 2 for simplicity. The terms with b of Eq. (7) come from the Doppler effect. As b < 1 for most ground vehicles, we can neglect b2 terms and Eq. (7) becomes Fig. 1. (a) Sound wave propagation from a moving target. Target position and the center of each sound propagation at each time step are shown with solid and clear diamond respectively. Each arrow shows sound propagation from each propagation center. (b) Sound detection regions at each time step are shown with shades. Different shades are from different propagation centers.

At t = t0, the target at x = x0 starts to send sound and at t = t1 the target is at x = x1 and sound propagates by d0 = VcDt from x0. Here Vc is the speed of sound. At t = t2, the target moves to x = x2 and sound from x0 propagates by d0 = Vc2Dt and sound from x1 propagates by d1 = VcDt. At subsequent time steps, target and sound propagate in similar fashion as at previous time steps. Because the target speed is set to be more than half of the sound speed in the figure, the Doppler effect is clearly seen where sound wavelength in the forward direction is shorter (so the sound frequency is higher) than that in the rear direction and the target position is shifted to forward from each center of the propagation. Next, we are going to derive an equation for target sound power distribution from the target position at the current time t by taking into account the Doppler effect. Fig. 2 shows target positions x0 and x at a previous time t0 and the current time t respectively and the sound propagation from x0 at t. The distance between x0 and s is R and between x and s is r, and the distance between x0 and x is d. From the following three equations we can derive an equation of target sound propagation.

d ¼ v t  ðt  t0 Þ R ¼ V c  ðt  t 0 Þ

Ps;t 

P0;t ð1 þ 2bj cos hjÞ r2

ð8Þ

If b = 0 (or Vc = 1), Eq. (8) becomes identical to Eq. (1) with a = 2 except the noise term. For target speed vt = 30 m/s and the sound speed of 340 m/s, the Doppler effect is too big to neglect as 2b = 0.18. On the other hand, for vt = 10 m/s, the Doppler effect is insignificant as 2b = 0.06 implying that Eq. (1) is a good approximation for the speed. As target speed increases to higher than 10 m/s, the mismatch between the detection model of Eq. (1) and the detection by the sensors deployed in real environment will increase and causes increasing track error. The Doppler effect is not the only cause of detection model mismatch with real sensor detection. The attenuation parameter may be variable depending on the environmental condition of the sensing region. The mismatch of the attenuation parameter between the tracker and real environment will also cause high track error when the tracker is deployed in the field as studied by Ahmid et al. [8]. This simple analysis shows that the track results of the previous studies mentioned in the introduction are valid only for target speed less than 10 m/s. We note that Eq. (7) merely shows the target sound power distribution from the current target position when the Doppler effect is taken into account. Target sound detected by a sensor is more complicated than Eq. (7) because a sensor can detect target sound from previous multiple target positions at a given time as can be seen from Fig. 1(b). An algorithm for target sound detection by the sensors is shown in the next section.

ð4Þ 2.2. Algorithm for target sound detection with Doppler effect

Ps;t ¼ P 0;t =R2 From the trigonal relation, we have

Fig. 1(b) shows target sound detection regions with shaded areas at each time. If the sensors are distributed evenly and the sensing range, rs, is set to be large to cover multiple sensors in the range, the sensors in the shaded areas of Fig. 1(b) detect target sound as long as the sound signal is higher than the threshold values of the sensors. We note that some sensors in the detection region receive the sound signal from more than one previous target positions. We see some gaps at t = t2 and t3 in which the sensors do not detect the sound. The gaps are due to the finite time step. The time step, Dt, is set to be 0.005 s in our simulation. For the simulation, target motion is described by the following discrete equations.

s R

r θ

x0

x

~ xtk þ ~ v tk  Dt þ 0:5  ~atk  Dt2 xtkþ1 ¼ ~ ~ v tkþ1 ¼ ~ v tk þ ~atk  Dt d t0

ð9Þ

~ atk atkþ1 ¼ ~ t

Fig. 2. Target position x0 and x at t = t0 and t and the sound propagation from x0 at t.

Here, ~ xtk ; ~ v tk ; ~atk are target position, velocity, and acceleration at a time index k. The sound propagation from the target position at each previous time step can be expressed by the following equation.

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Y.K. An et al. / Computer Communications 36 (2013) 834–848

 2  2 2 dj;k ¼ xj;k  xtj þ yj;k  ytj

1200

ð10Þ

dj;k ¼ ðk  jÞ  V c

1100

where Vc is sound speed, j(=1, 2, . . ., k  1) is the index of previous time steps, dj,k is the sound propagation distance at the current time of k from the target position at j, ~ xj;k is the sound wave front position at the time of k propagating from the target position at j, and ~ xtj is the target position at j. The distance between a sensor of s and the target position at j can be expressed by Eq. (2). Any sensor that satisfies the following two conditions simultaneously at the current time step of k can detect the target sound.

dj;k1 6 ds;j 6 dj;k Ps;t ¼

X

1000 900 800 700

ð11Þ 600

a

P0;t =ds;j P cs

ð12Þ

j

where cs is the sensor threshold value. Algorithm 2.2 shows the pseudocode of the detection model described above.

Algorithm 2.2 Target detection Require: Sensor positions ~ xs and target sound power, Ps,t = 0 (s = 1, . . ., Nt) 1. for k = 1:kmax % Index of current time step 2. Compute target motion using (9) 3. for j = 1:k  1%Index of previous time step 4. Compute dj,k using (10) 5. Compute dj,k1 using (10) 6. for s = 1:Nt % Nt = total number of deployed sensors 7. Compute ds,j using (2) 8. Check if dj,k1 = cs (s = 1, . . ., N), save s into array ~ xd . 14. Send detection_info (~ xd , Ps,t) to the fusion center 15. end

Multiple target detection can be simulated by adding a for-loop between line 1 and 2 to compute each target motion and modifying the single target sound detection shown in the algorithm for multiple targets. A robust multiple targets tracking algorithm requires not only a new detection model but also effective data association [29,30]. Fig. 3 shows a simulation example of the detecting sensors at three different times for a target of sensing range of 75 m and its speed of 50 m/s. The sensor separation (the distance between neighboring sensors) is 25 m. The detecting sensor is marked with a symbol ‘‘⁄’’ and the target is shown as a diamond among the detecting sensors. The figure shows that the target is a little shifted to the front due to the Doppler effect at the high speed. The Doppler effect will be more clearly seen for various target speeds in the next section. 3. Tracking framework In Sections 2.1 and 2.2, we show that Eq. (1) is not accurate as a target sound detection model because the equation does not take into account the Doppler effect. Eq. (7) is not accurate either for target detection model as a sensor detects target sound from more than one of previous target positions. None of the two equations

500 400 200

300

400

500

600

700

800

900

1000

1100

Fig. 3. Detecting sensors marked by ‘‘⁄‘‘ surround the target of sensing range 75 m and speed of 50 m/s at three different times. The target is marked by a solid diamond.

can avoid mismatch with the target sound detection by the sensors deployed in the field because the attenuation parameter, a, can be variable depending on the environmental condition. To avoid these adverse effects of the detection models, we use the centroid of the detecting sensor positions as target position measurement and feed the measurements to our tracker. For the centroid to be a good detection model for tracker, the detection error should be a mean zero Gaussian distribution. We show in this section that the target position measurement through the weighted centroid has near zero mean Gaussian error even with the Doppler effect for reasonable target speeds. We will also show an algorithm for tracking a target in acceleration with KF. 3.1. Target position measurement with centroid of detecting sensors For passive acoustic sensors with no range measurement capability, the target position can be measured by computing the centroid of the detecting sensors. We are going to consider two different centroids, one is the unweighted (or geometric) and the other is the weighted centroid. The unweighted centroid is an arithmetic average of the detecting sensor positions as shown in Eq. (13). Here, N is the number of detecting sensors at a given time step.

~ xc ¼

N X ~ xs

, N

ð13Þ

s¼1

The geometric centroid computes the target position at the geometrical center of the detecting sensors which is equivalent to the detection model of Eq. (1). Geometric centroid is a good target position measurement for a low target speed as implied by Eq. (8). For a high target speed, as the target position is shifted forward from the geometric center, the unweighted centroid (geometric centroid) will give a target position measurement bias. The centroid weighted with the detecting target sound power is a better target position measurement because the centroid is closer to the true target position and the measurement bias will be much reduced. For the weighted centroid, each detecting sensor scales the detecting power (Ps) with its threshold value (cs), turns it into an integer as

ws ¼ floorðPs =cs Þ

ð14Þ

and sends the integer value to the fusion center. Here, floor(x) changes the real value x into a nearest integer smaller than x and

Y.K. An et al. / Computer Communications 36 (2013) 834–848

9

9

6

6

3

3

dc (m)

dc (m)

838

0

0

-3

-3

-6

-6 -9

-9 0

20

40

60

80

100

0

120

20

40

60

80

Time (sec)

Time (sec)

(a)

(b)

100

120

Fig. 4. The difference between the target truth position and centroid, ~ dc , for a target of sensing range 75 m traveling with speed of 10 m/s. (a) Weighted centroid, (b) Geometric centroid.

subscript s stands for each detecting sensor. The integer weight is intended to lower the number of bits when transmitting the information to the fusion center. The fusion center computes a weighted centroid as

~ xc ¼

N X

, N X ws~ ws xs

s¼1

ð15Þ

s¼1

Similar ideas for weighted centroid are used in binary sensor network [2] and radio frequency identification systems [27,28,31]. For reliable target position measurements with the centroids, we may need the sensing range larger than the distance between neighboring sensors so that more than two sensors are included in the centroid computation. 3.2. Doppler effect on the target position measurements The Doppler effect on the target position measurement through the centroids is to be shown in this section. We compute the difference between the target true position (~ xt ) and the centroid as

~ dc ¼ ~ xt  ~ xc

ð16Þ

3.3. Near optimality of Kalman Filter with the weighted centroid KF becomes optimal filter if the following three conditions are satisfied [10,20];

12

12

9

9

6

6

dc (m)

dc (m)

The difference is considered as measurement error. Figs. 4 and 5 show the variation of ~ dc as a target moves with velocity of 10 and 50 m/s, respectively, and (a) is for with and (b) is for without the weighting factor. The sensing range of the target is 75 m. The target speed of 50 m/s is very high for any ground vehicles but the speed is chosen for demonstration purpose. When the target speed is 10 m/s, Fig. 4(a) shows that ~ dc fluctuates with time around zero but Fig. 4(b) shows that the fluctuation

center is a little bit shifted to positive. In other words, the target position measurement has an unbiased zero mean with weighted centroid but is a little bit biased for geometric centroid. For target speed of 50 m/s, Fig. 5 shows that the measurement bias is significant for geometric centroid and the bias is noticeable even for weighted centroid. It is an expected result as the truth target position is far ahead from the geometric center for the high speed due to Doppler effect. However, the implication of the result needs to be addressed. The power law detection model of Eq. (1), which is equivalent to target position measurement with geometric centroid, will have measurement bias if their trackers are used in real world for target speed >10 m/s. The results also show that the measurement bias is unavoidable even for weighted centroid for target speed >50 m/s. But as most of ground vehicles have speed less than 50 m/s we may say that the measurement is almost unbiased with the weighted centroid as shown in Fig. 6. Fig. 6 shows the time average of detection error using weighted centroid vs. target speed for sensing range 50 and 30 m. The mean of dc varies from 0.09 to 0.4 for sensing range of 50 m and from 0.049 to 0.045 for sensing range of 30 m as target speed varies from 10 to 30 m/s. Within the speed range, the maximum mean value is less than 0.8% and 0.17% of the sensing range of 50 and 30 m respectively. The study shows that within a reasonable target speed, the mean of dc is near zero for various target sensing ranges.

3

3

0

0

-3

-3 -6

-6 0

10

20

30

40

0

10

20

Time (sec)

Time (sec)

(a)

(b)

30

40

Fig. 5. The difference between the target truth position and centroid, ~ dc , for target speed of 50 m/s. (a) Weighted centroid, (b) Geometric centroid.

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Y.K. An et al. / Computer Communications 36 (2013) 834–848

0.45 SR=50

Mean of dc (m)

0.35

SR=30

0.25

0.15

0.05

-0.05 10

15

20

25

30

Speed (m/s) Fig. 6. Variation of the mean value of dc with target speed for sensing range 30 and 50 m. Weighted centroid is used for this computation.

 The measurement and state noises are zero mean,  They have Gaussian distribution, and  The motion and measurement equations are linear. In this section, we will show that the KF satisfies approximately the three optimality conditions with the weighted centroid as target position measurement. We note, however, that this simulation based empirical study of near optimality is limited to the detection model and the detection parameters that were chosen for ground vehicles in this paper. These parameters are considered to be very reasonable, though, in terms of speeds, sound characteristics, and typical tracks for a number of valid scenarios as discussed later. In the previous section, we showed that the mean error of target position measurement with weighted centroid is less than 0.8% of the sensing range for speeds equal to or less than 30 m/s (more than 65 mph). The confidence interval is from 0.14909 to 0.15439 where sensing range is 30 m, speed is 20 m/s and mean of dc is 0.00265. We thus can say that the measurement error is near mean zero for reasonable speed of ground vehicles. Fig. 7(a) and (b) shows the histograms of the measurement error, dc, with weighted centroid overlaid with Gaussian curves for targets traveling with speed of 20 m/s for sensing range 30 and 50 m, respectively. The histograms were made with 1468 data points. The averaged values of Fig. 7(a) and (b) are 0.0027 and 0.25 m respectively. Considering that the sensing ranges are 30 and 50 m, the averaged values are considered to be nearly zero. The histograms are curve fitted by adjusting the width and amplitude of the Gaussian function. The histograms of Fig 7(a) and (b) are fitted well with the Gaussian curves. The v2 test, which is the most important test for the goodness of fit [21], on Fig. 7(b) shows

that the confidence level is 94.86% where the degree of freedom is 238. It is noted that the data distribution fits to narrower Gaussian curves as the sensing range increases from 30 to 50 m. We also computed the distribution for sensing range of 75 m showing further narrower Gaussian distribution. The above studies demonstrate that the measurement error (dc) has nearly Gaussian distribution with zero mean for reasonable sensing ranges and target speeds. However, environmental effects such as directional winds, obstacles between sensors and environmental noises may degrade the Gaussianity of the distribution. Gaussianity may not also be obtained for target with sensing range shorter than the sensor separation distance as only one or two sensors are involved for centroid calculation. We find, however, in Section 4.1 that the KF gives high track accuracy for the short sensing range as long as the mean measurement error is low. The effect of environmental noise on track accuracy is our next research project. Another condition for the KF to be optimal is that the motion and measurement equations are linear. Below, we are going to show the linearity of the equations. For our tracking problem in WSN, target motion is expressed in a discrete form as follows.

^xðk þ 1=kÞ ¼ FðkÞ  ^xðk=kÞ þ lðkÞ where F(k) is the transition matrix,

2

Dt 60 1 0 6 FðkÞ ¼ 6 40 0 1 0 0 0

3

Dt 7 7 7 0 5 1

^xðk=kÞ ¼ ðxðkÞ yðkÞ

ð18Þ

v x ðkÞ v y ðkÞÞ0

ð19Þ

where A0 is the transpose of A. If we use the centroid of detecting sensors as target position measurement, the measurement vector is ~ zðk þ 1Þ ¼ ~ xc ðk þ 1Þ and the relation between the measurements and state vector can be expressed as follows:

^zðk þ 1=kÞ ¼ Hðk þ 1Þ  ^xðk þ 1=kÞ where

Hðk þ 1=kÞ ¼



1 0 0

0 0

ð20Þ

 ð21Þ

1 0 0

Eqs. (17)–(21) show that the motion and measurement equations are linear functions. The results shown above demonstrate that the KF is near optimal for tracking in WSN contrary to the

30

Number of distribution

Number of distribution

0

1 0

and ^xðk=kÞ; ^xðk þ 1=kÞ are updated and predicted state vectors, respectively. The term, l(k), in (17) is a process noise term and acceleration is considered as a process noise later. The updated state vector is

20

15

10

5

0 -6

ð17Þ

-4

-2

0

2

4

6

25 20 15 10 5 0 -6

-4

-2

0

dc (m)

dc (m)

(a)

(b)

2

4

6

Fig. 7. The histogram of dc for a target of speed 20 m/s for weighted centroid. (a) Sensing range is 30 m. (b) Sensing range is 50 m. The smooth curve is the Gaussian curve.

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Y.K. An et al. / Computer Communications 36 (2013) 834–848

previous authors’ assertion, if weighted centroid is used for position measurement. For the completeness of this paper, we present the remaining KF equations below.

Pðk þ 1=kÞ ¼ FðkÞPðk=kÞFðkÞ0 þ QðkÞ

600 550

ð23Þ 0

ð24Þ

Sðk þ 1Þ ¼ Hðk þ 1ÞPðk þ 1=kÞHðk þ 1Þ0 þ Rðk þ 1Þ

ð25Þ

Wðk þ 1Þ ¼ Pðk þ 1=kÞHðk þ 1Þ0 Sðk þ 1Þ1

ð26Þ

^ þ 1=kÞ mðk þ 1Þ ¼ zðk þ 1Þ  Zðk

ð27Þ

ytrajectory

ð22Þ

^xðk þ 1=k þ 1Þ ¼ ^xðk þ 1=kÞ þ Wðk þ 1Þmðk þ 1Þ Pðk þ 1=k þ 1Þ ¼ Pðk þ 1=kÞ  Wðk þ 1ÞSðk þ 1ÞWðk þ 1Þ

650

500 450 400 350 300 250

Eq. (22) is the predicted state covariance with Q(k) being the covariance matrix of the process noise, l(k) of Eq. (17). Eqs. (23) and (24) are the updated state vector and updated state covariance matrix, S(k + 1) is the sum of measurement error and state error covariance, W(k + 1) is the Kalman gain, (k + 1) is the innovation or measurement residual, and R(k + 1) is the measurement error covariance. The initializations of the state vector, state covariance, and measurement error covariance are determined empirically after running the tracker with numerous realistic parameters. The empirical determination is a calibration process of the tracker. The initial x and y positions are selected from the x and y centroid computed at the start of track and initial x and y component of velocity are set to be 10 m/s as the velocity is reasonable for most of ground vehicles. The initializations of state (Eq. (28)) and measurement error covariance matrix (Eq. (29)) are based on the simulation results of various target speed and sensing ranges. The initial values of the process covariance Q(0) are determined knowing that Eq.17 describes the linear target motion very accurately.

xð0Þ ¼ ðxc

yc

10 10Þ

Pð0Þ ¼ diagð52

52

Rð0Þ ¼ diagð52

52 Þ

Q ð0Þ ¼ diagð104

52

104

ð28Þ 52 Þ

ð29Þ ð30Þ

105

105 Þ

ð31Þ

Here, diag stands for diagonal elements of the matrix. 3.4. Tracking target in acceleration In the previous section, we find that KF becomes near optimal if we use the weighted centroid as target position measurement. In this section, we present an algorithm for tracking a target in acceleration with KF. The ability of KF for tracking targets in acceleration accurately is an important factor to confirm that our KF is a good tracker for WSN tracking. For the study, numerous target trajectories for accelerated motion are generated and one of them is shown in Fig. 8. Target acceleration is considered as a perturbation (or noise) in Eq. (17) and is treated by adjusting the process noise covariance Q (see (22) and (31)). When the target motion changes from a linear to an accelerated motion, the value of each element of Q should be increased so that the tracker gives higher weight to the measurements than to the predicted state vector. When the target returns to a linear motion, the Q values should return to the initial low value. The tracker, then, controls the divergence of the track state vector and follows the target closely.

0

200

400

600

800

1000

xtrajectory Fig. 8. A trajectory of a target in acceleration with speed 20 m/s.

Here, an important question is how the tracker detects the onset and termination of acceleration and automatically switches Q back and forth to a high and low value. It is found that the normalized innovation squared (NIS) value, em(k),

em ðkÞ ¼ mðkÞ0 SðkÞ1 mðkÞ

ð32Þ

is sensitive to velocity change as can be seen in Fig. 9(a). Here, m(k) is a measurement residual (or innovation) shown in (27) and S(k) is shown in (25). Fig. 9(a) shows steep increase of NIS when the target velocity changes at t = 23 s. If the tracker does not control the process noise covariance, Q, NIS keeps increasing to reach nearly 300 and the state vector runs away from the truth value and the track position error reaches nearly 50 m. Fig. 9(b) shows that the track position error, dx, overlaid with the state error covariance marked by ‘‘⁄’’. Note that we cut off the line of NIS larger than five and the line of dx larger than eight to see the sudden increases of the two track quantities at the onset of acceleration. To control the run-away increase of track error and improve the track accuracy significantly when the target is in acceleration, the tracker monitors the change of NIS and adjusts the process noise covariance, Q. If the value of NIS exceeds a certain threshold, the tracker switches the initial Q value, Q(0) of (31), to a predetermined higher value, Q2. Next, a method for determining the threshold value and adjusting Q is explained in detail. In our simulation, unlike the methods described in [20], the mean (em ) and standard deviation (r1) of em(k) are used to determine the threshold of em(k). From our numerous simulations for various acceleration trajectories we define the threshold value, emth, as

emth ¼ em þ 3  r1

ð33Þ

The mean (em ) and standard deviation (r1) values are computed and the threshold value is updated for every tracking time step until NIS is detected larger than the updated threshold value. The same updated threshold value is then used throughout the remaining track time for monitoring the change of NIS and adjusting Q value. From our numerous tests with various trajectories, we find that the threshold value of (33) detects the target velocity change promptly and reliably. Adjusting Q value based on NIS change is a little subtle. In our simulation, four diagonal elements of Q are initialized as shown in (31). The four diagonal elements are for x and y components of position and velocity process noises. After running the tracker with various realistic initial values of the state and covariance

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5

8

4

6

dx (m)

NIS

3 2

4

2

1

0

0 10

20

30

40

50

60

70

80

0

20

40

Time (sec)

Time (sec)

(a)

(b)

60

80

Fig. 9. Change of NIS, and track position error, dx, upon acceleration without Q adjustment. (a) NIS (ev(k)). (b) Track position error, dx, overlapped with the position error covariance marked with ⁄.

matrices and for various trajectories as a calibration process, we determine the diagonal elements of Q2 empirically as shown in (34). 3

Q 2 ¼ diag½10

3

10

1

10

1

ð34Þ

10 

The determination of the diagonal terms is based on that the perturbation of the linear motion is mainly caused by the perturbation of the velocity from their originally set values. Because acceleration and velocity are vector quantities, the direction of their changes should be taken into account when adjusting Q value. When acceleration occurs only in the x-direction, the x-components of Q value should be adjusted to the x-components of Q2 with all the y-components of Q unchanged. To determine which components of Q value should be adjusted for an unknown acceleration, the change of the x and y component of the innovation of (27) is monitored. The x and y component of the innovation, emx and emy, are defined as

emx ¼ mðk; 1Þ

ð35Þ

emy ¼ mðk; 2Þ

ð36Þ

It should be noted that emx and emy are not normalized values as they are more sensitive to velocity change than with normalization. If the tracker detects em(k) > emth, and also detects emx > emxth and emy < emyth, the tracker switches the x-components of Q(0) to the xcomponent of Q2 and keeps the y-components of Q(0). If emx and emy are all higher than their threshold values, the tracker switches all the elements of Q(0) to those of Q2. The threshold values, emxth and emyth, are chosen with the means (emx , emy ) and the standard deviation (rx, ry) of emx and emy as shown in (37) and (38).

ev yth ðkÞ ¼ emy þ 2  ry

ð37Þ

ev xth ðkÞ ¼ emx þ 2  rx

ð38Þ

We choose the factor 2, instead of 3, as a multiplication factor to the standard deviation because the factor 2 picks up the change of the innovation components better than 3. Our numerous tests with various trajectories confirm that these threshold values of (37) and (38) work well. This adaptive process for Q is applicable to any smoothly accelerated motions that most ground vehicles are expected to be. For random accelerating motions, however, PF and VF will work better than the KF of adaptive Q. 4. Simulation results In this section, we will study the effects of different target velocities, sensing range, and different sensor layouts on track accuracy for linear and accelerated motions. We will also study various environmental effects, such as different sampling time steps, collision of detection messages at the fusion center, and delay of message by hopping on the track accuracy. These are important environmental effects in real world WSN tracking systems which strongly affect track accuracy. We assume that the wireless sensors are distributed in a uniform grid of 50 by 50 with the sensor separation (the distance between neighboring sensors) being 25 m in the sensing region. This distribution is reasonable for the detection and tracking of ground vehicles with speeds of 10–30 m/s and sensing range of 15–75 m. Here, the sensing range stands for the maximum distance from the target at which the sensor can detect the target sound above the threshold. Most of our target tracking simulations are performed in this uniform sensor layout with sensor separation of

Table 1 Simulation parameters. Parameter

Unit

Description

Section

dx,hdxi V SR dist rand NIS Dt DT Delay N ts Dts,p Dtmin DTr

Meter m/s Meter Meter Meter

Track position error, averaged value Target speed Target sensing range Sensor separation distance in uniform grid Displacement of sensor in random direction from a uniform grid Normalized innovation squared Simulation time step Sensor sampling time step Message delay percentage by hopping Number of sensors sending target message to fusion center Target sound arrival time to sensor s Detection message arrival time difference from sensor s and p The minimum message arrival time difference Fusion center time resolution

All All All 4.1 4.1 3.4, 4.2 2.3, 4.3.1 4.3.1 4.3.3 3.1, 4.3.1, 4.3.4 4.3.4 4.3.4 4.3.4 4.3.4

Sec Sec % Sec Sec Sec Sec

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Y.K. An et al. / Computer Communications 36 (2013) 834–848

10

10

dx (m)

14

dx (m)

14

6

2

2 0

6

5

10

15

20

25

30

0

35

5

10

15

20

Time (Sec)

Time (Sec)

(a)

(b)

25

30

35

Fig. 10. Track position error with speed V = 50 m/s. (a) Geometric centroid. (b) Weighted centroid. Sensing range is 50 m for both cases. The state position error of thin line is plotted with the position state covariance of ⁄.

25 m. We also simulate target tracking in different sensor layouts; uniform grids with different sensor separations (20, 25, and 30 m) and random distribution of sensors. The simulation parameters used in this section is shown in Table 1. 4.1. Track accuracy for linear motions

6

6

5

5

4

4

dx (m)

dx (m)

In this section, track accuracy is studied for a target in linear motion with speed of 50 and 10 m/s. The speed of 50 m/s is rather high for ground vehicles but we take this number for comparison purpose. The centroids of detecting sensor positions computed with and without weight are fed to the tracker as target position measurements. As discussed in Section 3.3, weighted centroid causes the KF near optimal by being closer to the true target position and leads to the relatively accurate target tracking. Fig. 10 shows clearly the Doppler effect on the track accuracy for a high speed target (50 m/s) with the sensing range 50 m. Note that the sensing range is two times the distance between sensors. Fig. 10(a) is for the centroid without weight and (b) is for with weight. The position state error covariance (marked by ⁄) overlays the position track error. The figure shows that the track error shown in Fig. 10(a) of geometric centroid is three times higher than that in Fig. 10(b) of weighted centroid and that the position state error covariance fits to the position error well for Fig. 10(b) but not for Fig. 10(a). The high track error for Fig. 10(a) is caused by the position measurement bias of the geometric centroid due to the Doppler effect. When the speed is low, the Doppler effect is reduced and the difference between the weighted and geometric centroid decreases. For the low speed (10 m/s), Fig 11(a) and (b) show that

the tracking error for geometric centroid is about 1.5 times higher than for weighted centroid and the position error covariances fit to the track error well for both cases. The Doppler effects are clearly reflected for high and low speed targets in the tracking results above. Next, we are going to study the effects of different velocities and sensing ranges on the track accuracy. For the study of different sensing ranges’ effect on track accuracy, we choose two sensing ranges, 50 and 30 m in Fig 12(a) and compute the mean track position error, hdxi, for the speeds of 10–50 m/s. The figure shows that as the speed becomes faster, the mean track error becomes larger for both of the sensing ranges but the error increase rate is higher for larger sensing range. Fig 12(b) shows position track error for target speed of 20 m/s and various sensing ranges of 15–75 m. Note that the distance between neighboring sensors is 25 m and the sensing range of 15 m is much shorter than the sensor distance. In the previous section, we mentioned that the measurement error distribution is not near Gaussian for sensing range shorter than the distance between sensors. However, Fig. 12(b) shows that the track accuracy is very good for the sensing ranges shorter than the sensor distance. The results may be understood with Fig. 12(c) which shows the mean measurement error vs. sensing range. For sensing ranges equal or less than 30 m, the mean measurement error is much lower than 0.1 m but the error increases to 0.38 and 0.6 m as the sensing range increase to 50 and 75 m. Fig. 12(b) and (c) imply that as long as the mean measurement error is near zero, the track accuracy is still good even if the error distribution is off the Gaussian. The results shown above are for a sensor layout in a uniform grid with 25 m of sensor separation. Track accuracy may also be

3

3

2

2

1

1

0 20

30

40

50

0

20

30

40

Time (Sec)

Time (Sec)

(a)

(b)

50

Fig. 11. Track position error with speed V = 10 m/s. (a) Geometric centroid. (b) Weighted centroid. Sensing range is 50 m for both cases. The state position error with multiple peak is plotted with the position state covariance.

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Y.K. An et al. / Computer Communications 36 (2013) 834–848

SR=50 SR=30

1

(m)

(m)

1.2 0.8 0.6 0.4 0.2

0.7

0.7

0.6

0.6

0.5

0.5

(m)

1.4

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0 0

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0 0

20

40

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80

0 0

20

40

Velocity (m/s)

SR (m)

(a)

SR (m)

(b)

(c)

60

80

Fig. 12. (a) Averaged track position error, hdxi, in meter vs. target speed for sensing ranges 50 and 30 m. (b) Averaged track position error, hdxi, vs. sensing range (SR) for target speed 20 m/s. (c) Measurement error vs. sensing range (SR).

(a)

(b)

(c)

Fig. 13. Three different sensor layouts. (a) Sensors are distributed in a uniform grid with 25 m of sensor separation. (b) Sensors are shifted from the uniform grid by 5 m in random directions. (c) Sensors are shifted from the uniform grid by 10 m in random directions. The sensors designated with ⁄ in red are the sensors that detect target sound above the threshold. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

affected by different sensor separations in uniform grids and by random layouts of the sensors. We have studied the track accuracy for sensors in a uniform grid with sensor separations of 20, 25, and 30 m respectively and for three different sensor layouts as shown in Fig. 13. Fig. 13(a) is a uniform grid with sensor separation of 25 m and Fig. 13(b) and (c) are random distributions of sensors generated by shifting each sensor by 5 and 10 m respectively from the uniform grid in random directions. Fig. 14(a) shows the mean position track error vs. target speed for sensors in uniform grids with sensor separation 20, 25, and 30 m. The target sensing range is set to be 40 m. For target velocity lower than 30 m/s, higher sensor separation gives higher track error as the measurement error increases with higher sensor distance. For target speed 30 m/s or higher, the track error for sensor separation of 30 m is slightly lower than the track error for sensor separation of 25 m showing intricate coupling between Doppler effect and higher sensor distance. Fig. 14(b) shows track error vs. target speed for the three different sensor layouts shown in Fig. 13. The figure shows that track error increases as the random shift increases from zero to 5 and10 m for target velocity less than 30 m/s. The Doppler effect on track error appears for target velocity P30 m/s for the random layouts. This study shows that the track accuracy is within a comparable range for various sensor density and layouts.

4.2. Track accuracy for accelerated motions Fig. 15 shows the result of controlling Q when the target switches back and forth between linear and accelerated motions as shown by the trajectory of Fig. 8. Fig. 15(a) shows NIS change and Fig. 15(b) shows drastic improvement of track accuracy when Q value is adjusted based on the change of NIS as explained above.

Fig. 16 shows the track results for target sensing ranges of 30 and 50 m with speed of 20 m/s. Here, we fix the sensor sampling time to be 0.1 s and change the track time step and compute the averaged position error for the target trajectory described in Fig. 8. The averaged errors generally increase with increased tracking time step and are higher than for the linear trajectory. The figure shows that the sensing range of 50 m gives a little higher track error than the sensing range of 30 m. The result shows that the accuracy of our KF compares favorably with other trackers that have higher computational complexity for targets in acceleration. 4.3. Environmental effects The uniqueness of our realistic detection model enables us to study the following environmental effects on the target tracking. We will show that the track accuracy is sensitive to these environmental effects. 4.3.1. Different sampling time steps As the sensors in WSN have limited power and communication resources, it is desirable for each sensor to have longer sampling time step for sending detection message to the fusion center as long as track accuracy is not degraded. However, we have found that the increase of sampling time step gives an adverse effect on the track accuracy and the effect is more serious than previous studies indicated. Before going into details, we note that we use two different time steps; one is the simulation time step, Dt, with which target motion and sound propagation are simulated and the other is the sampling time step, DT. In our simulation, the simulation time step is fixed to be 0.005 s and the sampling time step varies from 0.1 to 1.0 s. The effect of different sampling time steps on the target position measurement through the centroid computation can be explained in a simple way as follows.

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1.6

1.2

1.4

1

(m)

(m)

1.2 0.8 0.6 0.4

dist=20

0.6

0

10

20

30

40

rand=5

0.2

dist=30

0

rand=0

0.4

dist=25

0.2

1 0.8

0 50

rand=10 0

10

20

30

Velocity (m/s)

Velocity (m/s)

(a)

(b)

40

50

Fig. 14. Track position error vs. target velocity for different sensor layouts. (a) Track errors for sensors in uniform grids with distance between sensors (dist) 20, 25, and 30 m. (b) Track errors for three different sensor layouts shown in Fig. 13. The data label ‘‘rand’’ stands for sensor shift in meter in random direction from the uniform grid.

2.5

8

2

(m)

6

NIS

1.5 1

2

0.5 0

4

0

20

40

60

80

0

0

20

40

Time (sec)

Time (sec)

(a)

(b)

60

80

Fig. 15. Change of NIS and track position error upon acceleration with Q adjustment. (a) NIS. (b) Track position error overlapped with the position error covariance marked with ⁄.

3.5

SR=50

3

SR=30

(m)

2.5 2 1.5 1 0.5 0 0

0.2

0.4

0.6

0.8

1

Time step (sec) Fig. 16. Averaged position error vs. different track time steps for targets of sensing range 30 and 50 m in accelerated motion.

For simplicity, we assume that the sampling time is the same as the detection message sending time, which means that a sensor sends a detection message to the fusion center as soon as it samples a target sound higher than the threshold. For a given simulation time, t = t0, N0 number of sensors are assumed to be in the detection region (the shaded area of Fig. 1(b)). If N1 out of the N0 number of sensors sent the detection messages to the fusion center in t0  DT < t < t0, those sensors are excluded for sampling at t = t0 and only N = N0–N1 number of sensors send the detection message to the fusion center. When the sampling time step, DT, increases, N1 also increases so the number of sensors, N, that send the detec-

tion message at t = t0 decreases in average. When the fusion center computes the centroid based on the received message at t = t0 the centroid does not represent an accurate target position measurement because of the missing messages. Fig. 17 shows how different sampling times affect the number of sensors which send detection message at a given simulation time. Each of Fig 17(a)–(c) shows the number sensors for the sampling time step 0.1, 0.6, and 1.0 s, respectively, and each frame shows the sensors at five different times. The target is in the middle of the sensors designated with a diamond. The target is moving diagonally from the left lower corner to the right upper corner of the sensing area. As the time step, DT, is larger, the simulation shows that the number of message sending sensors decreases as mentioned above. Fig. 18(a) and (b) show the number of sensors sending the detection message to the fusion center every 0.1, 0.6, and 1.0 s during the target motion described above for target sensing range 75 and 30 m respectively. For the sensing range 30 m and DT = 0.6 and 1.0 s, sometimes there is no sensors which send detection message to the fusion center. In this case, the tracking is skipped until at least one sensor sends the detection message to the fusion center. Thus, for short sensing range targets, the KF tracking time step is variable. For sensing range of 15 m which is shorter than the distance between neighboring sensors, the number of sensors that send detection message can often be zero for several seconds of time interval. Fig. 19 shows the dependence of the mean position track error, hdxi, on DT, for the sensing range of 15, 50, and 75 m. The figure shows that increasing DT gives higher track error because of the loss of the detection messages for centroid calculation. This study

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Y.K. An et al. / Computer Communications 36 (2013) 834–848

1200

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200 0

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1200

0

200

400

600

(a)

800

1000

1200

200

0

200

400

600

(b)

800

1000

1200

(c)

Fig. 17. Each figure shows the sensors which send detection signal to the fusion center every DT s for five different simulation times. The target position is designated with a diamond in the middle of the sensors. (a) DT = 0.1 s, (b) DT = 0.6 s, (c) DT = 1.0 s. The target sensing range is 75 m and target velocity is 30 m/s.

40

7

Δτ = 0.1

30 25 20

Δτ = 0.6

15 10

Δτ = 1.0

5

Δτ = 0.1

6

0

Number of sensors

Number of sensors

35

5 4

Δτ = 0.6

3 2

Δτ = 1.0

1 0

0

20

40

60

0

20

40

Track time (sec)

Track time (sec)

(a)

(b)

60

Fig. 18. The number of sensors that send detection message to fusion center every 0.1, 0.6, and 1.0 s for two different target sensing ranges. (a) Sensing range = 75 m, (b) Sensing range = 30 m.

shows that for accurate tracking, DT should be as short as possible, in other words, saving sensor energy requires significant sacrifice of track accuracy especially for sensing range 15 m. For sensing range 15 m, the track error suddenly increases from 2.5 m to nearly 8 m when DT increases from 0.8 to 1.0 s. Ahmed et al. [8] studied the track accuracy for different values of DT and found that best tracking results are obtained with 0.5 and 1.0 s of sampling time. The discrepancy of their result from ours may be attributed to their

10

sr=75 sr=50

8

(m)

sr=15 6

4

2

0 0

0.2

0.4

0.6

0.8

1

Sending time step (sec) Fig. 19. Dependence of track accuracy on the sampling time steps (detection message sending time steps), DT, for various target sensing ranges (SR), 75, 50, and 15 m. hdxi is the mean position track error in meter.

simple power law detection model and very low target speed of 0.35 m/s.

4.3.2. Delay of detection message at the fusion center due to hopping As each of the wireless sensor nodes has short transmission range, the detection message needs to be relayed to a remote fusion center by nearby sensors, in other words, the message signal may require several hopping to reach the fusion center. The hopping delays the signal to reach the fusion center and the delay will affect the tracking accuracy. In this section, we will study how the message delay affects the track accuracy. For the study, we select the delay time at the fusion center randomly in 0–0.1 s and choose the delayed messages randomly with the ratios of 0%, 60%, and 100% of the detecting sensors at each simulation time. The ratio of 100% means all the detecting sensors’ messages are delayed and the ratio of 0% means none of the detecting sensors’ messages are delayed. We also assume that nearly equal numbers of the delayed messages at each time step are filled with the delayed messages in previous time steps. The delay time that we choose for this study does not mean to reflect any real sensor delay time but for simulation purpose only. The accurate delay time might be WSN systems dependent. Because of the mix of the detection messages from current and previous simulation time steps, the target position measurement estimate with the messages will have a bias depending on the number of delayed messages. Fig. 20 shows the averaged position track error vs. target speed for three different ratios of delayed messages. The figure shows

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Y.K. An et al. / Computer Communications 36 (2013) 834–848

compute the time difference between the messages that arrived at the fusion center and then find the minimum difference at each simulation time step as follows. For all the sensors that satisfy the target sound detection conditions shown by (11) and (12), the sound arrival time to each sensor is calculated as

8 delay=100%

7

delay=60%

6

(m)

delay=0%

5

ts ¼ ds;j =V c

ð39Þ

We, then, compute the arrival time differences between the detecting sensors and find the minimum value from the differences at each simulation time step.

3 2 1 0 0

20

40

ð40Þ

Dtmin ¼ minðDt s;p Þ

ð41Þ

If the time resolution of the fusion center, DTr, is lower than the minimum difference, Dtmin, the fusion center can process all the messages without missing any of them. But if DTr is larger than Dtmin, some of the messages will be lost. The minimum difference, Dtmin, fluctuates with simulation time in the range of 0–103 s as the distance between the moving target and each of the detecting sensors varies with time. The distribution of Dtmin throughout the target track time varies depending on the size of sensing range as shown in Fig. 21. Fig. 21(a)–(c) show the percentage distribution of Dtmin for target speed 30 m/s and sensing range 75, 50, and 30 m respectively. The horizontal axis is for Dtmin and the vertical axis is for the percentage distribution of Dtmin. We compute the percentage with total about 360 samples. The average number of detecting sensors at each simulation time is 28, 13, and 5 for sensing ranges 75, 50, and 30 m respectively. The figure shows that 40% of the samples have Dtmin 6 5  105 s for sensing range 75 m. For sensing range of 50 and 30 m, 28% and 78% of the samples have Dtmin P 103 s respectively. The results imply that if the time resolution of the fusion center, DTr, is 5  105 s, it can handle most of the messages without collisions for all the sensing ranges shown above. The effect of the collision on the track accuracy is studied with the following process. From Eqs. (40) and (41), we compute the time difference, Dts,p for a given detection message s and p. If Dts,p < DTr, then the message s is discarded, otherwise, the message is kept to send it to the fusion center. This process continues for all s = 1, 2, . . ., N  1 and p = s + 1, . . ., N, and send all the messages that survive from this selection process to fusion center. Fig. 22 shows the effect of fusion center time resolution, DTr, on the track accuracy for the three different sensing ranges. The horizontal axis is for different time resolutions and the vertical axis is for mean track position error in meter. The figure shows that longer sensing range gives higher track error for all time resolutions. The track result matches well with the prediction from Fig. 21 as

Fig. 20. The effect of detection message delay at the fusion center due to hopping on averaged position track error for the delay ratio of 100%, 60% and 0%. The target sensing range is 75 m and delay time is 0–0.1 s.

that the track errors increase with target speed and the rate of increase is higher for higher delay ratio. The track errors of 100% and 60% of delay ratio are nearly 6 and 4 times higher than the track error of no delay. If the delay time is shorter than 0.1 s, the track errors will decrease below the levels shown in the figure. This study suggests that message delay by hopping can be fatal for track accuracy unless the delay time is reduced to much less than 0.1 s by having local fusion centers. Some authors [9,25,26] suggested various ways of handling multiple hoppings or local fusion centers but we do not consider them in detail in this paper.

4.3.3. Collision of detection messages at the fusion center For a target with sensing range of 75 m, there are 25–30 sensors at each simulation time step that detect the target sound. When these sensors send the detection message to the fusion center on detection, there will be collisions between the messages depending on the time resolution of the fusion center. In this section, we are going to study how different sensing ranges affect the collision and how collisions affect the track accuracy. For a given simulation time step, we have, for example, N number of sensors that detect target sound. Because the distance between the target and each of the sensors varies depending on sensor position, the detection time of each sensor is different by sensor position. We assume that each sensor sends a detection message as soon as it detects the target sound and neglect message delay by hopping for simplicity. The detection time, then, can be considered as the message arrival time to the fusion center. We

30

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10

0

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min-time (sec)

(a)

0.8

1 -3

x 10

SR=30m

distribution (%)

SR=50m

distribution (%)

SR=75m

0

Dts;p ¼ jt s  t p j ; ðs ¼ 1; 2; . . . ; N  1Þ and ðp ¼ s þ 1; . . . ; NÞ

60

Target Speed (m/s)

distribution (%)

ðs ¼ 1; 2; . . . ; NÞ

4

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0

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(b)

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60 40 20 0

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min-time (sec)

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1 x 10-3

(c)

Fig. 21. The distribution of the minimum time difference, Dtmin, through the course of the target travel with the speed of 30 m/s. Sensing range (SR) is (a) 75 m, (b) 50 m, and (c) 30 m.

Y.K. An et al. / Computer Communications 36 (2013) 834–848

2

(m)

1.6 1.2 SR=75 0.8

SR=50 SR=30

0.4 0 0.E+00

2.E-04

4.E-04

6.E-04

8.E-04

1.E-03

Time resolution (sec) Fig. 22. The effect of fusion center time resolution, DTr, on track position error for sensing range 75, 50, and 30 m. Target speed is 30 m/s.

mentioned above showing that the time resolution of 5  105 s can handle all the collisions without degrading track accuracy much. Ahmed et al. [8] studied packet losses occurred in a real multihop network deployment due to channel contention, collisions, and queues overflows in the buffers. At this moment, it is unclear how their experimental result of collision is compared with our simulation result as we lack detailed knowledge of their experiments. Comprehensive simulation studies for the effects of these packet losses on track accuracy with realistic parameters are important work to perform in the future.

5. Comparison with previous work The main difference between our work and the previous studies mentioned in the introduction is that we have developed a more realistic model for target sound propagation and detection that accounts for the Doppler effect. Previous studies used a power law for sound power propagation and detection model in their trackers neglecting the Doppler effect. As the power law detection models in their trackers are susceptible to the mismatch with the sound power detection by the sensors deployed in the field, we expect that the track error by PF and VF of the previous studies may increase by factor of two or three higher than reported in their papers when used in real world WSN. Ahmed et al. [8] studied the track accuracy for different sensor sampling times and found that best tracking results are obtained with 0.5 and 1.0 s of sampling time. Our result in Section 4.3.2 shows, however, that the maximum time step that gives reasonably good tracking results is 0.2 s, above which the track accuracy deteriorates rapidly with increasing time steps. The discrepancy between our result and the result of Ahmed et al. may be attributed to their simple detection model and their low target speed. This comparison demonstrates the importance of using realistic detection model and realistic target speed for the WSN tracking simulation. We have also studied the effects of message delay due to hopping, and message collisions on the track accuracy but none of the previous work studied these environmental effects. Any direct comparison of our track accuracy and computational complexity with other trackers is beyond the scope of this paper as our KF tracker is integrated with our detection model while the previous studies used a simple power law detection model. However, Teng et al. [9] compared their VF results with the PF and SOI-KF results and found that the VF gives higher accuracy than the PF and SOI-KF results. The computation time of the VF is reported to be similar to that of the PF but is nearly 50 times higher than the computing time of the SOI-KF. Ristic et al. [10] also com-

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pared the track accuracy and computing time for tracking a reentry object by the PF and extended KF (EKF) methods. They showed the track accuracy improvement by the PF method over EKF but the PF computing time was 100 times longer than EKF time. Considering that our KF is less complex than EKF and SOI-KF, these studies suggest that our KF is about 100 times faster than the PF and VF trackers. For track accuracy, our position error of a target in acceleration is 1.2–3 m for sensing range 30 and 50 m (see Fig. 14) which is compared well with the errors of 1.85 m for VF, 5.64 m for PF, and 11.01 m for SOI-KF [9]. We note that the comparison is tentative as the target speed and sensing range are not explicitly specified in [9] and the target trajectories used for both studies are not the same.

6. Conclusion The implementation requirements of WSN trackers in the real world are low computing complexity with high track accuracy and robust tracking in real environments. The tracker is also required to be verified and validated using simulations in which realistic detection model and environmental effects are implemented. To satisfy the requirements of WSN, we have developed a realistic WSN detection algorithm from which the Doppler effect naturally appears depending on target velocity. For a tracker of low computing complexity, we use the KF with weighted centroid of detecting sensors as target position measurement. The weighted centroid makes our tracker independent of detection models and becomes impervious to the mismatch of detection between the tracker and the sensors deployed in the field. We also find that the KF becomes near optimal filter with the weighted centroid for the detection model and detection parameters used in this study. The high computing speed and high track accuracy will make the WSN tracking system inexpensive to implement in the real world, which is one of crucial factors for WSN. With our detection model and KF tracker, we plan to expand this study to multiple targets tracking in noisy environments to verify the robustness of the tracker in real environments.

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