A New Algorithm for General Asynchronous Sensor Bias Estimation in

A New Algorithm for General Asynchronous Sensor Bias Estimation in Multisensor-Multitarget Systems Amir Rafati National Iranian South Oil Company Ahwaz, Iran [email protected]

Behzad Moshiri Control and intelligent processing centre of excellence University of Tehran Tehran, Iran (IEEE Senior member) [email protected]

Abstract - Errors due to sensor bias are often present in sensor data and can reduce the tracking accuracy and stability of multi-sensor systems. The other practical problem is that the target data reported by the sensors are usually not time-coincident or synchronous due to the different data. This paper deals with these problems and presents a new algorithm for estimation of both constant and dynamic biases in asynchronous multisensor systems. We use the measurements from asynchronous sensors into pseudomeasurements of the sensor biases with additive noises that are zero-mean, white and with easily calculated covariances. This algorithm is a Kalman filter based technique to estimate both the range and offset biases and is implemented recursively which is computationally efficient and provided real time estimation of asynchronous sensor bias. The Simulation results show the Cramer-Rao Lower Bound (CRLB) is achievable. This means the proposed estimation algorithm is statistically efficient. Keywords: Asynchronous Sensor, Bias Estimation, Pseudomeasurement Bias, Range bias, Scale Bias Dynamic Bias.

1

Introduction

Data fusion has been defined as a process of dealing with association, correlation and combination of data and information from single or multiple sources to achieve refined position and identity estimates [1]. Sensor registration is a part of the level one processing of data fusion which includes association, filtering and identifications. The effect of sensor registration errors is to introduce biases into fusion, generating ghost targets for multisensor signal processing [2]. It was found that, if the sensor registration errors are large, the integrated performance can be even worse than that of a single sensor application. So estimation of sensor biases is vital in multi-sensor systems to carry out data fusion. To estimate the bias vector, the classical approach is to augment the system state to include the bias vector as part of the state, and then implement an augmented state Kalman filter (ASKF) by stacking the state of all the

Javad Rezaei Control and intelligent processing centre of excellence University of Tehran Tehran, Iran (IEEE Student member) [email protected]

targets and the sensor biases into a single vector. The augmented Kalman filter may encounter difficulties in practical situations due to the prohibiting computational requirement. In [3], Blom et al. proposed an approach in which the augmented Kalman estimator is decoupled into separate filters: maintenance Kalman filters for the target states, and a Kalman-like filter for the sensor registration errors. By decoupling the augmented Kalman filter into two filters of smaller dimensions, the computation can be reduced significantly. Another commonly used approach [12] is the application of the two-stage Kalman estimator proposed by Friedland [13] and further developed by Ignagni [14]. The two-stage Kalman estimator is able to decouple the estimation of the target state and the bias parameters, leading to a substantial reduction in computation complexity. The two-stage Kalman filter is equivalent to but computationally more efficient than the augmented Kalman estimator because it involves state vectors of smaller dimensions. Lin, Kirubarajan and Bar-Shalom [7] presented the bias estimation based on the local unassociated track estimates at a single time, i.e., based on a single frame. In [8], the authors use a bias model which considers both offset biases and scale biases and then they extended the work of [7] to include the dynamic bias estimation based on the local track estimates at different times. All of these above papers are considering only synchronous sensors. In reality, sensors often operate asynchronously and deriving a common reference time for the measurement is often difficult. In [9], the bias estimation for asynchronous sensors with same sampling rate but with a phase offset is discussed. In [10], the authors extended the work of [9] to general asynchronous sensors and absolute registration was achieved under a relatively restrictive assumption that from n measurement in a defined time slot, the first n − 1 measurements must be from one sensor, and the last measurement must be from sensor 2. In addition, in first and last measurement, bias estimation can not be done. Rafati, et al [5] have assumed the biases are constant and presented a new algorithm for estimation of constant biases in asynchronous multisensor systems. But this

assumption is not true generally, the bias vector for sensor i can be modeled as a dynamic stochastic process. This paper extends this algorithm to estimate both constant and dynamic biases in asynchronous multisensor systems. We use the measurements from asynchronous sensors into state-independent pseudomeasurements of the sensor biases such that the pseudomeasurement noises of asynchronous sensor biases are zero-mean, white and with easily calculated covariances. This algorithm is a Kalman filter based technique to estimate sensor biases in a multiradar system. Both the range and offset biases are considered. At each sensor, a linear time-varying measurement model is obtained by using a first-order approximation and the near constant velocity model is used to describe the target motion dynamics. This paper is organized as follows: The bias model and the assumptions for bias estimation are discussed in section 2. The bias estimation of asynchronous sensors is proposed in section 3. Simulation results are presented in section 4. Conclusions are in section 5.

2 2.1

Unknown Constant Model for Sensor Biases

Then,  rit (t j )  ω ir (t j )   + Ci (t j ) βi (t j ) +  θ  zip (t j ) =  t θ i (t j )  ω i (t j ) 

(5)

where 1 0 ri t (t j ) 0  Ci (t j )    t θ i (t j )  0  0 1

(6)

is assumed known-the observed (or estimated) azimuth θˆ (t ) and range rˆ (t ) can be used without any loss of i

j

i

j

zi (t j ) = H (t j )x(t j ) + Bi (t j )Ci (t j ) β i (t j ) + wi (t j )

The problem is to estimate the bias vectors for all sensors such that when the bias estimation is completed, the bias estimate can be used to correct the state estimates of the targets. Then, the track-to-track fusion can be performed. Consider M sensors which measure the range and azimuth for N common targets in the surveillance region. The model [8] for the biased measurements in polar coordinates for sensor i at time t j is

In which x(t j ) = [ x(t j ) x (t j )

y(t j )

(1)

(7)

y (t j )]′ is the state

vector, H (t j ) is the measurement matrix and the matrix Bi (t j ) is a nonlinear function of the true range and

azimuth. If Bi (t j ) Ci (t j ) is constant, only the difference of the biases at different sensors is observable, which means "relative registration" or "incomplete observability". Using the observed (or estimated) azimuth θˆ (t ) and i

 ri (t j )  z (t j ) =   θ i (t j )  p i

j

range rˆi (t j ) from sensor i , one has the matrix Bi (t j ) in (7) as

where the range and azimuth at time k are ri (t j ) = 1 + ε (t j )  ri (t j ) + b (t j ) + ω (t j )

(2)

θ i (t j ) = 1 + ε iθ (t j )  θ it (t j ) + biθ (t j ) + ω iθ (t j )

(3)

t

(4)

performance. After transforming the measurements into Cartesian coordinates, the measurement equation for sensor i is

Bias Model

r i

bir (t j )   θ  bi (t j )  β i (t j )   r  ε i (t j )   θ  ε i (t j ) 

r i

r i

In the above ri t (t j ) and θ it (t j ) are the true range and azimuth; bir (t j ) and biθ (t j ) are the offset biases for the range and azimuth; ε ir (t j ) and ε iθ (t j ) are the scale biases of the range and azimuth, respectively; the measurement noises ω ir (t j ) and ω iθ (t j ) are zero-mean, white with corresponding variances σ r 2 and σ θ 2 and are assumed mutually independent of each other. Denote the bias vector for sensor i at time t j as

cos θˆi (t j ) rˆi (t j ) sin θˆi (t j )   Bi (t j ) =   sin θˆi (t j ) rˆi (t j ) cosθˆi (t j )

(8)

In (7) wi (t j ) is the measurement noise with the covariance in the Cartesian coordinates (omitting the time index in the measurements for simplicity):  r 2σ 2 sin 2 θ + σ r2 cos 2 θ i Ri =  i 2 r 2 2i  (σ r − ri σ θ ) sin θ i cos θ i

(σ r2 − ri 2σ θ2 ) sin θ i cos θ i   (9) ri 2σ θ2 cos 2θ i + σ r2 sin 2 θ i 

The polar to Cartesian conversion needed for (7) has been discussed extensively in the literature [4] together with the limit of validity of the standard transformation. If this limit is exceeded, then one can use the modified version [6] which eliminates the bias caused by the nonlinearities in the transformation and also provides the correct

covariance. Thus the linear measurement model (7) is exact (in the sense that the noises in it are zero mean, white and their actual covariance is available), even though it is obtained from a nonlinear coordinate transformation. The unbiased coordinate conversion of [6] takes care of all practical situations. In view of this, like [10], we call our sensor bias estimation technique “exact” because all the assumptions of the technique, namely, the noises in the linear pseudomeasurements of the sensor biases (to be derived in the next section) being zero-mean, white with known covariances, are satisfied exactly.

2.2

Dynamic Stochastic Process Model for Sensor Biases

In previous section, biases are assumed constant. But this assumption is not true generally and the bias vector for sensor i can be modeled as a dynamic stochastic process evolving according to bi (k + 1) = Fbi (k )bi (k ) + v bi (k )

(10)

where Fbi (k ) is the transition matrix of the bias vector bi (k ) , v bi (k ) is the process noise of the bias vector with zero-mean and variance Qbi (t j ) .

3

Bias Estimation for Asynchronous Sensors

In reality, sensors often operate asynchronously and deriving a common reference time for the measurement is often difficult. So, we can not use synchronicity-based algorithms for bias estimation in multi-sensor systems. In [10], the authors extended the work of [9] to general asynchronous sensors and absolute registration was achieved under a relatively restrictive assumption that from n measurement in a defined time slot, the first n − 1 measurements must be from one sensor, and the last measurement must be from sensor 2. In addition, in first and last measurement, bias estimation can not be done. This paper presents a new algorithm for multi-sensor bias estimation in asynchronous sensors based on [8],[10] that it seems to be more realistic and practicable. Assume the dynamic equation of targets is x(t j ) = F (t j , ti )x(ti ) + v(t j , ti ) zi (t j ) = H (t j ) x(t j ) + Bi (t j )Ci (t j ) βi (t j ) + wi (t j )

transition

the

zero-mean white process noise with covariance Q (t j , ti ) . We assume the locations of sensors are known. Assume the measurement time-invariant matrix H (t j ) and the time-varying transition matrix F (t j , ti ) are defined as 1 0 0 0 H (t j ) =  H 0 0 1 0

1 t j − ti  1 0 ( , ) = F t j ti 0 0  0 0

(13)

0   0 0  1 t j − ti   0 1  0

(14)

A state-independent pseudomeasurement of the biases requires at least three measurement sets. In each measurement we consider the last previous measurement from both sensors and use the pseudomeasurements of bias with fusion of these three measurements. The problem is to estimate the bias vector for the last measurement at t n . Assume without loss of generality, this measurement is from sensor 1, z b1 (t n ) , that must be fused with previous measurements of sensor 1 and sensor 2 those are z1b (t n-1 ) and zb2 (t m ) , respectively and are  x (t )  z1b (tn ) =  n  + B1 (tn )C1 (t n ) β1 (tn ) + w1 (tn )  y (tn ) 

(15)

 x(t ) − (t n − t n-1 ) x(tn )  zb1 (t n -1 ) =  n  + B1 (t n-1 )C1 (tn -1 ) β1 (tn -1 )  y (tn ) − (t n − t n-1 ) y (tn ) 

+ A(tn , t n -1 )v (tn , tn -1 ) + w1 (tn -1 )

(16)

 x (t ) − (t n − t m ) x (t n )  z b2 (t m ) =  n  + B 2 (t m )C 2 (t m ) β 2 (t m )  y (t n ) − (t n − t m ) y (t n ) 

+A (t n , t m )v (t m +1 , t m ) + w 2 (t m )

(17)

in which A (t j , t k ) is defined as

(11) (12)

That is a linear time-varying measurement model [obtained by using a first-order approximation in which the state vector at time is tj x(t j )  [x (t j ) x (t j )

v(t j , ti )  vx (t j , ti ) vx (t j , ti ) v y (t j , ti ) v y (t j , ti )′ is

y (t j ) y (t j )]′ , F (t j , t i ) is the

matrix from time ti to time t j , and

 −1 t j − t k A (t j , t k )   0 0

 −1 t j − t k  0

0

(18)

Notice that in this algorithm only in first measurement of each sensor, Bias estimation can not be done. In other measurements, to obtain a state-independent pseudomeasurements of the biases, we use the combination of these three measurements: z1b (t n ) ,

z1b (t n-1 ) and zb2 (t m ) such that the true states of target can be canceled. Define the pseudomeasurement z b (t n ) as zb (tn ) = zb2 (tm ) − α1 (tn ) zb1 (tn ) + α 2 (tn ) zb1 (tn −1 ) 

(19)

Then, by appropriately selecting the parameters α1 (tn ) and α 2 (tn ) the true states of target can be canceled.

α1 (tn ) + α 2 (tn ) = 1

(20)

−(tn − tm ) − α1 (tn ) × 0 − α 2 (tn ) ( tn − tn −1 )  = 0

(21)

We have

α1 (tn ) =

(tm − tn −1 ) (t n − t n −1 )

(22)

Equation (27) indicates that asynchronous pseudomeasurement noise of the biases w b (tn ) is zero mean, white noise with covariance Rb (t n , t n )  var [w b (t n ) ] = R 2 (t m ) + α1 (t n ) 2 R1 (t n ) + α 2 (t n ) 2 R1 (t n -1 )

+A (t n , t m )Q (t n , t m )A (t n , t m )′ +α 2 (t n ) 2 A (t n , t n -1 )Q (t n , t n -1 ) A (t n , t n -1 )′

(29)

Equation (27) also shows that asynchronous pseudomeasurement noise is correlated with previous asynchronous pseudomeasurement noises of both sensors, w b (tn -1 ) and w b (tm ) . If t m < t n-1 covariance matrix between these pseudomeasurement noises will be Rb (t n , t n -1 )  cov [w b (t n ),w b (t n -1 )]

(t − t ) α 2 (t n ) = n m (t n − t n −1 )

(23)

Then

= R 2 (t m ) − α 2 (t n )α1 (t n -1 ) R1 (t n -1 )

(30)

Rb (t n , t m )  cov [w b (t n ),w b (t m ) ]

z b (t n ) = B 2 (t m ) C 2 (t m ) β 2 (t m ) + w 2 (t m )

= −α1 (t m ) R 2 (t m )

+A (t n , t m )v (t m+1 , t m ) − α1B1 (t n )C 1 (t n )β1 (t n )

And if tm ≥ tn -1

−α 2 B1 (t n-1 )C 1 (t n-1 ) β1 (t n-1 ) − α1 w 1 (t n ) − α 2 w 1 (t n-1 ) −α 2 A (t n , t n-1 )v (t n , t n-1 )

(24)

If the biases are time-invariant constants,  b (tn ) zb (tn ) = H(tn )b + w

(25)

Rb (t n , t n -1 )  cov [w b (t n ),w b (t n -1 )]

= −α 2 (t n )R1 (t n -1 )

H(tn ) = [ −α1 (t n ) B1 (tn ) C1 (tn ) − α 2 (tn ) B1 (tn −1 ) C1 (tn −1 ),

(26)

w b (t n ) = w 2 (t m ) + A (t n , t m )v (t m +1 , t m ) − α1 (t n )w 1 (t n ) −α 2 (t n )w 1 (t n -1 ) − α 2 (t n )A (t n , t n -1 )v (t n , t n -1 ) (27) and b is bias vector : β  b   1 β2 

(28)

(32)

Rb (t n , t m )  cov [w b (t n ),w b (t m ) ]

= −α1 (t m ) R 2 (t m ) − α 2 (t n )R1 (t n -1 )

Where

B2 (tm ) C2 (tm )]

(31)

(33)

In [10] the bias estimation can not be done in fist and last measurements of time slots and this can made a bad estimation specially when there are considerable difference in sampling rates of asynchronous sensors. But in proposed algorithm, only in first measurement of each sensor, Bias estimation can not be done and in other measurements, this algorithm estimates a stateindependent pseudo-measurement of the biases. All noises in deriving pseudomeasurement of asynchronous bias estimation are assumed zero-mean, white noises with known covariances, so, this method is "exact". Also, according to (11-18), for bias estimation in this algorithm any approximation is not used, and bias estimation and state estimation are separate processes and this algorithm can be implemented recursively which is computationally efficient.

4 4.1

Simulation Estimation of Unknown Constant Sensor Biases

Consider a scenario with two asynchronous sensors with different sampling rates. Sensor 1 reports measurements at 1 s interval. Sensor 2 reports measurements at 3s interval, and there is 2.5s time offset between reporting times of the two sensors. But it is assumed that the first measurement of both sensors is at the same time. So there is 1.5s between the first and second measurement of sensor 2 and 2.5s between other measurements of this sensor, as illustrated in Figure 1.

and the power spectral densities qx = q y = 6 m 2 s 3 . The measurement matrix and the transition matrix are defined in (112) and (113). The initial bias estimate of sensor i is zero with the initial bias covariance 2 2 2 ∑ i ( 0 | 0 ) = diag (100m) , (200mrad) , (0.01) , (0.1)2  . The initial state estimates are generated randomly with 2 2 N (xt , P (0 | 0)) where P (0 | 0) = ( 200m ) ( 20 m s )  2 2 ( 200 m s ) ( 20 m s )  , xt , is the true state and both of process and measurement noises of sensors are assumed mutually independent.

Figure 1. Asynchronous sensors with different sampling rates

The time-invariant biases for the two sensors are offset and scale biases for range and azimuth,

β1 = β 2 =  20m 2mrad 3 ×10 -5 2 × 10 −4 ′

Figure 2. The Geometry of 32 Targets and Two Asynchronous Sensors

(34)

The geometry of the targets and sensors is shown in Figure 2 and the targets are moving at nearly constant velocity with x = y = 20 m s . The standard deviation of the measurement noise variances are σ r = 10m and σ θ = 1mrad for the range and the azimuth measurements, respectively. Assume the locations of the sensors are known. The dynamics of the target are modeled using Discretized Continuous White Noise Acceleration (DCWNA) models [11] where the process noise covariance between time ti and t j is 0  Qx (t j , ti ) Q (t j , ti ) =  Qy (t j , ti )  0

1 3  3 (t j − t i ) Qx (t j , ti ) =   1 (t − t )2  2 j i 1 3  3 (t j − ti ) Q y (t j , t i ) =   1 (t − t ) 2 j i  2

1  (t j − ti )2  2  q x (t j − ti )   1  (t j − ti )2  2  q y − t t ( j i)  

(35)

(36)

The RMS errors and the Cramer-Rao Lower bound (CRLB)[2] of the range offset biases are shown in Figure 3-7. In a considered 30s interval, sensor 1 and sensor 2 have 30 and 10 measurements. The bias estimation is done in each measurement but the figures show the estimation errors in 3s intervals. The RMS estimation error of range biases are shown in Figure 3 and the significant improvement in estimation errors can be seen. These RMS estimation errors are about 4m and 4.5m at step k = 30s for sensor 1 and sensor 2, respectively. Similar significant improvement in performance can also be seen in the azimuth bias RMS estimation errors shown in Figure 5, that are 0.16 mrad and 0.44mrad for sensor 1 and sensor 2, respectively. In Figure 6 the RMS estimation error of range scale biases are 1.45 ×10 −5 and 1.1× 10 −4 . Finally in Figure 6, the azimuth scale bias estimation errors for sensor 1 and sensor 2 are 7.13 ×10 −5 and 2.18 × 10 −4 , respectively. The figures show the estimation of biases for sensor 1 is better than sensor 2 and this is predictable because sensor 2 has less measurement in this interval and its convergence is slower.

4.2 (37)

Estimation of Dynamic Sensor Biases

In this scenario, the sensor biases are modeled as stochastic processes (10).

The parameters of this dynamic equation are Fbi = 0.99 , Qbi = diag (1.5m)2

(0.3mrad) 2

2

(10−5 ) 2  .

(5 × 10 −5 ) 2

−3 2

σ CRLB: εθ1 RMS: εθ2 σ CRLB: εθ2

1.4

1.2

1

0.8

0.6

0.4

0.2

Range Offset RMS Error

0

100

RMS: εθ1

1.6

RMS

(30mrad) , (5 × 10 ) , (2 × 10 )  . The steady state RMS values for the Range and azimuth offset biases are 20 m and 2 mrad and Range and azimuth offset biases are 4 × 10 −4 and 3 × 10 −4 , respectively. The other parameters and noises are the same as previous scenario. −3 2

Azimuth Scale RMS Error

-3

1.8

The initial bias estimate of sensor i is zero mean with the initial bias covariance ∑ i ( 0 | 0 ) = diag (100m) 2 , 2

x 10

3

6

9

12

15

RMS:br

1

90

18

21

24

27

30

step k

σ CRLB:br1 RMS:br2

Figure 6. Azimuth scale RMS estimation error

σ CRLB:br2 80

70

Range Offset RMS Error 150

RMS:br1

RMS(m)

60

σCRLB:br1 RMS:br

2

σCRLB:br2

50

40 100

RMS(m)

30

20

10

3

6

9

12

15

18

21

24

27

50

30

step k

Figure 3. Range offset RMS estimation error 0

Azimuth Offset RMS Error

-3

x 10

7

3

6

9

12

15

18

21

24

27

30

step k

RMS:bθ

1

σ CRLB:bθ1 RMS:b2θ σ CRLB:bθ2

6

Figure 7. Range Offset RMS Errors (Dynamic Sensor Biases) -3

5

6

Azimuth Offset RMS Error

x 10

RMS:bθ1

σCRLB:bθ1 RMS:bθ2 σCRLB:bθ1

RMS(rad)

4 5

3 4

RMS(rad)

2

3

1 2

0

3

6

9

12

15

18

21

24

27

30

step k

1

Figure 4. Azimuth offset RMS estimation error 0

3

6

9

12

15

18

21

24

27

30

step k -3

1.2

Range Scale RMS Error

x 10

RMS: εr1

Figure 8. Azimuth Offset RMS Errors (Dynamic Sensor Biases)

σ CRLB: εr1 RMS: εr 2 σ CRLB: εr2

1

RMS

0.8

0.6

0.4

0.2

0

3

6

9

12

15

18

21

24

27

step k

Figure 5. Range scale RMS estimation error

30

The range and azimuth offset bias RMS errors are shown in Figures 7,8 and the range and azimuth scale bias RMS errors are shown in Figures 9,10. The significant improvement in RMS errors can be seen in these figures. The RMS errors and the Cramer-Rao Lower bound (CRLB)[2] of the range offset biases are shown in Figure 7-10. In a considered 30s interval, sensor 1 and sensor 2 have 30 and 10 measurements. The bias estimation is done in each measurement but the figures show the estimation errors in 3s intervals. The RMS estimation error of range biases are shown in Figure 8 and the significant improvement in estimation errors can be seen. These RMS estimation errors are about

4m and 4.5m at step k = 30 s for sensor 1 and sensor 2, respectively. Similar significant improvement in performance can also be seen in the azimuth bias RMS estimation errors shown in Figure 9, that are 0.16 mrad and 0.44mrad for sensor 1 and sensor 2, respectively. In Figure 9 the RMS estimation error of range scale biases

This algorithm is computationally and statistically efficient and is implemented recursively. So, it can be used for real time bias estimation of asynchronous multisensor systems.

are 1.45 × 10−5 and 1.1× 10−4 . Finally in Figure 10, the azimuth scale bias estimation errors for sensor 1 and

[1] D. L. Hall, Mathematical Techniques in MultiSensor Data Fusion, Norwood, MA: Artech House, 1992.

sensor 2 are 7.13 × 10−5 and 2.18 × 10−4 , respectively. The figures show the estimation of biases for sensor 1 is better than sensor 2 and this was predictable because sensor 2 has less measurement in this interval and its convergence is slower. -3

1.4

Range Scale RMS Error

x 10

RMS: εr1

σCRLB :εr1 RMS: εr 2 σCRLB :εr2

1.2

1

References

[2] Y. Bar-Shalom, X. R. Li and T. Kirubarajan, Estimation with Applications to Tracking and Navigation: Theory Algorithms and Software, Wiley, 2001. [3] H. A. P. Blom, R. A. Hogendoorn and B. A. van Doorn, Design of a multi-sensor tracking system for advanced air traffic control, in Multitarget-Multi-sensor Tracking: Applications and Advances, vol.2, Y. BarShalom, Ed. Norwood, MA: Artech House, 1990 (reprinted by YBS Publishing, 1998), pp. 31-64,1992.

0.8 RMS

[4] Y. Bar-Shalom and X. R. Li, MultitargetMultisensor Tracking: Principles and Techniques, Storrs, CT: YBS Publishing, 1995.

0.6

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30

step k

Figure 9. Range Scale RMS Errors (Dynamic Sensor Biases) -3

1.8

Azimuth Scale RMS Error

x 10

RMS: εθ1

σCRLB :εθ1 RMS: εθ2 σCRLB :εθ1

1.6

Rafati, B.Moshiri, K.Salahshoor and [5] A. M.Tabatabaei pour, Asynchronous Sensor Bias Estimation in Multisensor-Multitarget Systems, 2006 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems- MFI2006, Heidelberg, Germany, September 3-6, 2006, pp 402-407. [6] L. Mo, X. Song, Y. Zhou, Z. Sun, and Y. BarShalom, Unbiased converted measurements in tracking, IEEE Transaction on Aerospace and Electronic Systems, 34(3), pp. 1023–1027, July 1998.

1.4

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[7] X. Lin, T. Kirubarajan, and Y. Bar-Shalom, Multisensor Bias Estimation with Local Tracks Without a Priori Association, Proceeding of SPIE Conference on Signal and Data Processing of Small Targets, vol. 5204, San Diego, CA, August 2003.

Figure 10. Azimuth Scale RMS Errors (Dynamic Sensor Biases)

5

Conclusions

This paper presents a new algorithm for multi-sensor state-independent estimation constant and dynamic biases in asynchronous sensors. In this algorithm, both offset and scale biases are considered. We use the measurements from two asynchronous sensors to obtain the pseudomeasurements of the sensor biases such that the pseudomeasurement noises of asynchronous sensor biases are zero-mean, white and with easily calculated covariances. So, this method is "exact". Also, for bias estimation in this algorithm any approximation-except in linear dynamic modeling of targets- is not used, and bias estimation and state estimation are separate processes.

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