Modeling Sensor Position Uncertainty for Robust Target Localization in Wireless Sensor Networks Zhenxing Luo and Thomas C. Jannett Department of Electrical and Computer Engineering, The University of Alabama at Birmingham, Birmingham, AL, 35294, USA Abstract — In wireless sensor networks, sensor position uncertainty degrades the accuracy of the energy-based target localization achieved using maximum likelihood estimation (MLE) methods. In this paper, we developed a new MLE approach that incorporates a model of sensor position uncertainty. Simulations demonstrated that our new MLE approach outperformed an MLE approach that does not account for sensor position uncertainty. Root-mean-square (RMS) estimation errors for target localization were close to the Cramer-Rao lower bound (CRLB). Index Terms — Target localization, wireless sensor network, sensor position uncertainty, maximum likelihood estimation, Cramer-Rao lower bound.
I. INTRODUCTION Target localization in wireless sensor networks (WSNs) has become a popular research topic [1-3]. Many target localization algorithms exist, such as time delay of arrival (TDOA), direction of arrival (DOA), and energy-based target localization [1]. Energy-based target localization has advantages, since it does not require direction information from sensors that is required for DOA methods or sensor synchronization that is required for TDOA methods [1]. We consider an energy-based target localization approach in WSNs. Each sensor compares strength of the signal it receives against a pre-determined threshold to make a decision. A fusion center employs a maximum likelihood estimation (MLE) method to estimate the target position based on decisions received from all sensors and knowledge of the sensor positions. In practice, the fusion center may have inaccurate information about sensor positions due to errors in the measurement of sensor positions or due to sensor movement. Since the sensor position influences the received signal strength and the decision made by each sensor, inaccurate information about sensor positions will degrade the target localization performance achieved at the fusion center. In this paper, we develop a MLE approach that incorporates a model of random sensor position errors in order to reduce the performance degradation caused by sensor position uncertainty. In Section II, we represent sensor position errors using a line error model (LEM) in which actual sensor positions are located along the line passing through the assumed sensor position and the target loca-
tion. We extend the MLE target localization approach described in [1] to account for sensor position errors represented using the LEM. In Section III, we apply the new LEM MLE approach as an approximate MLE approach for other sensor position error models in which the actual sensor position is located within a circle that has the assumed sensor position as its center. In Section IV, we present the simulation setup, followed by simulation results in Section V. Discussion and conclusions are provided in Section VI. II. ENERGY-BASED TARGET LOCALIZATION ALGORITHM FOR THE LEM An exemplary field of N = 441 uniformly distributed sensors is shown in Fig. 1. As presented in [1], the target emits a signal that decays according to the model
ai2 =
Gi P0'
di
d0
n
(1)
where ai is the signal amplitude at the ith sensor, Gi is the gain of the ith sensor, and P0' is the power of the target measured at a reference distance d0 . Here, n is the power decay exponent, which is set to 2 in this paper. The distance between the target and the assumed position of the ith sensor is di = (xi - xt ) 2 + (yi - yt ) 2
(2)
where (xt , yt ) is the target position and (xi , yi ) is the assumed position of the ith sensor. In this paper, we assume that the target is at least d0 meters away from any sensor. For Gi 1 and d0 1 , the model (1) can be simplified as ai2 =
P0 . d i2
(3)
However, errors in the assumed sensor position affect di in (2) to alter the received signal strength (3) and degrade the performance of current MLE target localization approaches, which assume exact knowledge of the sensor positions [1]. Our approach is to model the effects of uncertainty in the assumed sensor position and incorporate
the model into the MLE framework for target localization presented in [1].
f (si ) =
Sensor Field 100 Sensors True target location
80
f ai ( ) f wi ( si )
di b P0
diP a
60 Sensor Field Y-coordinate (m)
P0 di a P0
0
di b
P0
e
2 (b a )
( si )2 2 2
2
40 20
0 -20
P0
P0
2 (b a )
di a P0
e
di b
d
(7)
( si ) 2 2 2
2
d .
The sensor quantizes the received signal si according to a set of pre-determined thresholds ηi = [ηi0 ,ηi1 ,...,ηiL ] (8)
-40 -60 -80 -100 -100
-80
-60
-40 -20 0 20 40 Sensor Field X-coordinate (m)
60
80
100
Fig. 1. Sensor field layout.
Fig. 2 shows the LEM in which actual sensor positions are located along the line passing through the assumed sensor position and the target location. The distance between the target and the actual position of sensor i is modeled as the sum of di and a uniformly distributed random distance error, mi , such that ai2 =
P0
di mi
n
,
mi u[a b] .
(4)
to produce a decision Di . Here, ηi0 and ηiL . The quantization process is denoted by 0 1 Di L 2 L 1
si i1
i1 si i 2
(9)
i ( L 2) si i ( L 1) i ( L 1) si .
Given [ P0 xt yt ]T , the probability that Di assumes value l is i ( l 1) pil (i , ) R(il ) R(i (l 1) ) f (si )dsi il
The received signal at sensor i is (5)
si = ai + wi
where R( x) is defined as
where wi is a Gaussian noise with distribution N 0,σ 2 .
R ( x ) f(si )dsi x
mi
di
Fig. 2. Representation of sensor position uncertainty using the LEM.
The probability density function (PDF) of the sum of two independent random variables is equal to the convolution of their PDFs [4]. Therefore, the PDFs f (ai ) and f ( si ) are f (ai )
P0 ai
2
f (mi )
P0 ai
2
P0 P0 1 ai , (6) ba di b d i a
(10)
(11)
and l is any integer from 0 to L 1 . After receiving the decision vector D [ D1 , D2 , , D N 1 DN ] ,
(12)
the fusion center estimates [ P0 xt yt ]T by maximizing N L-1 lnp(D θ ) = δ (Di - l )ln pil (ηi ,θ )
(13)
i=1 l=0
where 1, x = 0 . δ(x) = 0, x = 0
(14)
The maximum likelihood estimator is θˆ max lnp D θ . θ
(15)
For an unbiased estimate of , the Cramer-Rao lower bound (CRLB) is given by
E{[θˆ (D) - θ ][θˆ(D) - θ ]T } J 1
(16)
J E T ln p D θ .
(17)
For the log-likelihood function (13), the (1, 1) element of matrix J can be derived using an approach similar to that employed in [1] 2 2 ln p D θ ( Di l ) pil (i , ) 2 P02 pil (i , ) P0 i l 2 (18) ( Di l ) pil (i , ) . 2 P0 pil (i , ) Because E ( Di l ) pil (i , ) , the expected value of
(18) is
pil (i , )
i ( l 1)
P0
2 (b a ) The derivative of pil (i , ) is pil (i , ) P0 il
P0 di a P0 di b
1
2
e
( si )2 2 2
(19)
R
d dsi . (20)
mi
di
Fig. 3. Representation of sensor position uncertainty using the TVCEM and CEM.
(21) dsi .
2 ln p D θ 2 . J (1,1) E P02 pil i l
We applied the LEM MLE approach as an approximate target localization method for sensor position errors described using the TVCEM and the CEM. Our idea is to approximate the distribution of mi in the TVCEM or the CEM by a uniformly distributed random distance error mi u[b b] where the value of b is set such that var(mi ) var(mi )
(23)
var(mi ) b 2 3 .
(24)
and The variance equation (23) holds for di R . We
Finally, we have
Other elements of J can be derived similarly.
Target
Actual Sensor Position
P0 1 ( si )2 d i a 2 e 2 2 d d P0b i i ( l 1) 2 2 (b a ) P 0 il P0 2 P ( si ) ( si 0 )2 di a di b 2 2 2 ( d i b ) e 2 ( d i a )e 2 2 (b a ) Po
di
Assumed Sensor Position
We used the LEM MLE approach as the basis for an approximate MLE approach that is applicable for models of sensor position error that are more general than the LEM. In these models, the actual sensor position is located within a circle of radius R that has the assumed sensor position as its center (Fig. 3). The distance between the target and the actual position of sensor i is modeled as the sum of di and a random sensor position error mi . We consider a time-varying circle error model (TVCEM) in which actual sensor positions drift randomly over time within the circle, such that values of mi vary over time. We also consider a circle error model (CEM) in which knowledge of sensor positions is uncertain due to errors in the initial layout of the sensor field. In the CEM, sensors are located within the circle, but the values of mi do not vary over time.
2 2 p D θ 1 pil E 2 . 2 pil P0 i l P0 The expression for pil (i , ) is
III. APPROXIMATE MLE APPROACH FOR CIRCLE ERROR MODELS
(22)
compute the constant value approached by var(mi ) for di >> R, and then solve (24) for the value of b needed to determine pil (ηi ,θ ) in the likelihood function (13). If the distance from the target to any sensor is smaller than b , the distance is set to b in order to avoid problems with carrying out the integration in (7).
4000 Po
We used Monte Carlo simulation (1000 runs) to compare our new LEM MLE target localization approach, based on the model (4), to the original MLE approach. We considered sensor position uncertainty described by the LEM, the TVCEM, and the CEM. The original MLE approach, based on the model (3), did not account for the uncertainty of the sensor positions. We used binary quantization with 4 . The target position ( xt , yt ) was
model sensor position uncertainty. Our new MLE approach that incorporates a model of sensor position uncertainty can alleviate this problem.
Po
x, m y, m
15
20
25
25
30
0
5
10
15
20
25
30
0
5
10 15 20 Radius of the circle, R
25
30
4 2
y, m
4 2
Original MLE
3000
LEM MLE
2000 1000 0
5
10
15
20
25
30
0
5
10
15
20
25
30
0
5
10 15 20 Radius of the circle, R
25
30
x, m
6 4 2
6 4 2
30
REFERENCES 0
5
0
5
10
15
20
25
30
25
30
5
0
20
Fig. 6. Estimation errors for sensor position uncertainty described by the CEM.
5
0
15
Fig. 5. Estimation errors for sensor position uncertainty described by the TVCEM.
Po
Original MLE LEM CRLB LEM MLE
10
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6
y, m
For sensor position uncertainty described by the LEM, RMS localization errors achieved using our LEM MLE approach were lower than those achieved using the original MLE algorithm (Fig. 4). Localization performance for the LEM MLE approach was close to the CRLB (16). Our approximate MLE approach also outperformed the original MLE algorithm for sensor position uncertainty described by the TVCEM (Fig. 5) and the CEM (Fig. 6).
5
5
4000
V. RESULTS
0
0 6
was computed using (24) such that var(mi ) var(mi ) .
0
2000 1000
(12,13) , P0 10000 , n 2 , and σ 2 1 . For the LEM, the drift of sensor position followed a uniform random distribution over the interval from a to b, mi u[a b] , along the line from the target to the assumed position of each sensor. In the simulation, a 0 and b was changed in steps from 0 to 30. For sensor position uncertainty described by the TVCEM and by the CEM, R was changed in steps from 0 to 30 and we set mi u[b b] , where b
5000
Original MLE LEM MLE
3000
x, m
IV. SIMULATION SETUP
10 15 20 Range of distance error, b
Fig. 4. Estimation errors and the CRLB for sensor position uncertainty described by the LEM.
VI. DISCUSSION AND CONCLUSION Sensor position uncertainty degrades target localization achieved using the current MLE approaches that do not
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