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iet-spr.2013.0393

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Published in IET Signal Processing
Received on 26th September 2013
Revised on 22nd January 2014
Accepted on 12th May 2014
doi: 10.1049/iet-spr.2013.0393
ISSN 1751-9675
Track fusion in the presence of sensor biases
Hongyan Zhu, Shuo Chen
Automation Department, School of Electronic and Information Engineering, Xi’an Jiaotong University, Xi’an, 710049,
People’s Republic of China
E-mail: [email protected]
Abstract: A computationally effective approach is developed in this study to deal with the problem of track fusion in the presence
of sensor biases. Aiming at the case that sensor biases are implicitly included in the local estimates, a pseudo-measurement
equation is derived based on the Taylor series expansion firstly, which reveals the relationship explicitly between local
estimates and the sensor biases; and then, the bias estimates can be obtained in the rule of recursive least squares; finally,
based on the derived pseudo-measurement equation, the sensor biases can be removed from the original local estimates and
track fusion can be carried out directly and easily. Monte Carlo simulations demonstrate the efficiency and effectiveness of
the proposed approach compared with the competing algorithms.
1
Introduction
The potential advantages of fusing information from disparate
sensor systems to achieve better surveillance has been
recognised [1–3]. In the distributed fusion system, local
estimates are generated from individual sensors. At the
fusion centre, they are combined according to a certain
criterion to yield an improved estimate than local ones. An
important prerequisite for the successful fusion is the
transformation of sensor reports into a common spatial
reference frame [4]. The fusion process relies on the
accurate registration of sensors which is regarded as a
process to eliminate the effects caused by the sensor biases.
If uncorrected, sensor biases would lead to large tracking
errors and cause ghost tracks. Consequently, the fusion
performance cannot be guaranteed to be optimal.
The classical approach to deal with the problem is to
augment the system state to include the sensor biases as
part of the state vector, and then implement an augmented
state Kalman filter (ASKF). Friedland [5] proposed the idea
of implementing two parallel, reduced-order filters instead
of using an ASKF. Ignagni [6] generalised the two-stage
method of [5]. Van Doorn and Blom [7] gave an exact
solution for the augmented Kalman filter problem, but then
decoupled the equations using an approximation to make
the implementation feasible. In [8], under a rather restrictive
algebraic constraint, the optimal sate estimate can be
obtained by combining local bias-ignorant estimate and the
estimated sensor bias. Ignagni [9] derived the optimal form
of the separate-bias estimator for the general case in which
the bias vector is stochastic in nature. Okello and Ristic’s
work in [10] presented a batch maximum likelihood (ML)
registration algorithm for spatial alignment of multiple,
possibly dissimilar sensors. An exact solution was provided
for the multi-sensor bias estimation problem in [11] by
constructing the pseudo-measurements equation of the
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& The Institution of Engineering and Technology 2014
sensor biases with additive noises that are zero-mean, white,
and with easily calculated covariance. Okello and Challa
[12] proposed an equivalent measurement method for
registration at the track level by augmenting the state vector
with the sensor biases. Although it is easy to implement,
the accuracy is limited by the parallel extended Kalman
filter (EKF). In [13], track registration and fusion is
modelled as an ML joint estimation problem. In [11, 13], to
construct the pseudo-measurement equation of sensor
biases, lots of information besides local estimates, such as
the measurement matrix, Kalman gain, system matrix and
so on, is required which is not always available because of
the limitation of communication bandwidth in many
distributed fusion systems.
As we know, sensor registration is often performed at the
measurement level, since the sensor biases are directly
added to the sensor measurements in general. However, it is
not suitable to perform sensor registration at the
measurement level for distributed track fusion systems, in
which local sensors only send extracted tracks to the fusion
centre. In this respect, sensor registration at the track level
is needed more.
In this paper, we address the problem of track fusion in
which the sensor biases are added to the original sensor
measurements and implicitly included in the local estimates.
The main difficulties of the problem lie on how to model
and remove the sensor biases inherent in the local tracks.
For this purpose, we construct a pseudo-measurement
equation based on the Taylor series expansion, which
reveals the relationship explicitly between local estimates
and the sensor biases; after that, the bias estimates can be
obtained in the rule of recursive least squares (RLS);
Consequently, the biased track can be corrected by using
the estimated sensor biases according to the
pseudo-measurement equation and track fusion can be
implemented directly and efficiently.
IET Signal Process., 2014, Vol. 8, Iss. 9, pp. 958–967
doi: 10.1049/iet-spr.2013.0393
www.ietdl.org
The main merits of the proposed approach lie on the
following several aspects:
Firstly, the proposed approach requires far less information
than the competing algorithms when constructing the
pseudo-measurement equation. Secondly, the proposed
approach has the higher computational efficiency than the
competing algorithms. Finally, the proposed approach
enables the local biased track to be corrected easily by the
estimated sensor biases, as contributes to implementing
track fusion directly and efficiently. Selected simulation
results demonstrate the main advantages and features of the
proposed approach compared with the competing algorithms.
2
Problem formulation
Fig. 1 depicts the configuration geometry with two sensors and
one target. The location of each sensor {xom, yom)}(m = 1, 2)
is assumed to be known exactly. The local Cartesian
coordinate systems (LCCS) have the rotation angles f1 and
f2, respectively, with respect to the global Cartesian
coordinate system (GCCS) denoted by O. It is assumed that
the target T is detected by both of the two sensors (S1 and
S2), respectively.
The measurement process is implemented in the local polar
coordinate system (LPCS). Since there exist systematic
biases
and random errors, the original measurement rkm , um
from
k
sensor m at time k can be modelled as
m
m
rkm = rm
k + Drk + vk,r
umk
(1)
m
m
= uk + Dum
k + vk,u
(2)
m
where rm
k and uk denote the real range and angle about the
observed target, Drkm,i and Dum,i
are the systematic biases,
k
m
vm
k,r and vk,u are the random errors.
m
Furthermore, it is assumed that the biases Drm
k and Duk are
invariant across the overall tracking time, that is
Drkm
= Dr ,
m
Dum
k
= Du
In this way, the original measurement
rkm ,
umk
m
m
rkm = rm
k + Dr + vk,r
(4)
umk = uk + Dum + vmk,u
(5)
m
m
m
m T
is
The random measurement noise Ṽ k = vk,r vk,u
modelled as independent white Gaussian noises in the range
2 m 2
and angle with zero-mean and variances of sm
, su .
r
The bias vector of the two sensors is denoted by
T
T
T
T
h = h1
, where hm = Drm Dum
is the
h2
bias for sensor m.
Each sensor m produces
itsown local tracks based on the
biased measurements rkm , um
k , and each local track at time
m
m
k is represented by a two-tuples {x̂m
k , P k } (m = 1, 2). x̂k
m
and P k mean the state estimate and error covariance,
respectively. The goal of track fusion is to combine local
estimates to yield a better estimate than the local ones. Note
that, local trackers cannot produce the bias estimates based
on their own local measurements. Therefore, local estimates
are biased estimates. The direct fusion of biased local
estimates will not produce a satisfactory result.
3 Review of existing main approaches to
sensor registration
3.1 Exact multi-sensor dynamic bias estimation
with local tracks
In what follows, the ‘Exact method’ given in [11] for bias
estimation is reviewed. Suppose that the dynamic model of
the target is
xk+1 = F k xk + G k vk
(3)
from sensor m
(6)
After transforming the measurements from LPCS into LCCS,
the measurement equation for sensor m is
m
m m
m
zm
k = H k xk + Bk h + vk
m
can be rewritten as
(7)
where m is the sensor index, Fk is the transition matrix, ωk and
vm
k are zero-mean, white Gaussian noises with covariance Qk
m
m
and Rm
k , respectively. H k and Bk are the measurement matrix
and the weighted matrix for the bias vector ηm.
Because the local tracker has no the information about the
sensor biases, the measurement model assumed by local
trackers is of the following bias-ignorant formulation
m
m
zm
k = H k xk + vk
(8)
From the local estimate x̂m
k+1 from sensor m, one has
m
m
m
m
x̂m
k+1 = F k x̂k + K k+1 zk+1 − ẑk+1|k
m
m
m
= F k x̂m
k + K k+1 H k+1 F k xk + G k vk + Bk+1 h
+vm
− H m F k x̂m
k+1 m k+1
k m
= I − K k+1 H m
k+1 F k x̂k
m
m
m
m
+ Km
k+1 [H k+1 F k xk + G k vk + Bk+1 h + vk+1 ]
(9)
Fig. 1 Sensor configuration geometry
IET Signal Process., 2014, Vol. 8, Iss. 9, pp. 958–967
doi: 10.1049/iet-spr.2013.0393
Consequently, the following pseudo-measurement for sensor
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www.ietdl.org
P h,k+1|k = F h,k P h,k F Th,k + Qh,k
m is defined by
+ m
m
m
m
zm
x̂m
h,k+1 = K k+1
k+1 − I − K k+1 H k+1 F k x̂k
(10)
m
m
m
= Hm
k+1 F k xk + G k vk + Bk+1 h + vk+1
+
is the pseudo-inverse of the Kalman gain
where K m
k+1
Km
k+1 .
Then the pseudo-measurement equation of the sensor
biases can be given by
zh,k+1 = Hk+1 h + ṽk+1
where
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩ Rh,k+1
3.2.2 State estimate and covariance update: The
predictive state, covariance matrix and the predictive
measurements are
(11)
+
zh,k+1 = z1h,k+1 − H 1k+1 H 2k+1 z2h,k+1
+
Hk+1 = [ B1k+1 −H 1k+1 H 2k+1 B2k+1 ]
T
h = h1 T h2 T
+
ṽk+1 = v1k+1 − H 1k+1 H 2k+1 v2k+1
+
+ T
= R1k+1 + H 1k+1 H 2k+1 R2k+1 H 1k+1 H 2k+1
After having the above pseudo-measurement equation, the
optimal bias estimates can be obtained in the rule of RLS
when given an initial estimate ĥk , P h,k . (see (13))
(15)
zk = H k xk + Bk h + vk
z1
H 1k
Bk = diag B1k , B2k ,
where zk = k2 , H k =
2 ,
zk
Hk
1
1 2
vk
vk = 2 , Rk = diag Rk , Rk .
vk
The covariance matrices for process noise ωk and ωη,k are
Qk and Qh,k , respectively. The decoupled Kalman filter
consists of the following steps.
3.2.1 Bias prediction: Given the initial bias estimation ĥk
and its covariance Pη,k, the predictive bias and the
corresponding covariance can be computed by
ĥk+1|k = F h,k ĥk
(16)
P k+1|k = F k P k|k F Tk + G k Qk G Tk
(19)
ẑk+1|k = H k+1 x̂k+1|k + Bk+1 ĥk+1|k
(20)
+ Rk+1
(22)
The Kalman gain for the system state xk + 1
−1
K x,k+1 = P̃ xz,k+1|k P̃ zz,k+1
(23)
The updated state estimates and the corresponding covariance
are computed by
x̂k+1 = x̂k+1|k + K x,k+1 [zk+1 − ẑk+1|k ]
P k+1 = I − K x,k+1 H k+1 P k+1|k
(24)
(25)
3.2.3 Bias estimates and covariance update: The bias
vector can be updated by
ĥk+1 = ĥk+1|k + K h,k+1 [zk+1 − ẑk+1|k ]
P h,k+1 = I − K h,k+1 Bk+1 P h,k+1|k
T
= P h,k+1|k − K h,k+1 P̃ hz,k+1
(26)
(27)
where the Kalman gain for the bias vector ηk can be computed
by
−1
K h,k+1 = P̃ hz,k+1 P̃ zz,k+1
(28)
P̃ hz.k+1 ≃ P h,k+1|k BTk+1
(29)
where
4 Proposed approach to sensor registration
and track fusion
Since the sensor biases {Δrm, Δθm} are implicitly included in
m
local estimates {x̂m
k , P k }, the key problem is to construct a
pseudo-measurement equation which can reveal the
⎧
−1
⎪
⎨ ĥk+1 = ĥk + P h,k HTk+1 Hk+1 P h,k HTk+1 + Rh,k+1 [zh,k+1 − Hk+1 ĥk ]
−1
⎪
T
T
⎩ P
Hk+1 P h,k
h,k+1 = P h,k − P h,k Hk+1 Hk+1 P h,k Hk+1 + Rh,k+1
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(21)
P̃ zz,k+1 ≃ H k+1 P k+1|k H Tk+1 + Bk+1 P h,k+1|k BTk+1
The accumulated measurement process for two sensors is
formulated by
(18)
P̃ xz,k+1|k ≃ P k+1|k H Tk+1
Decoupled Kalman filtering for bias estimation
The decoupled Kalman filtering for sensor registration is
earliest presented in [7], and is extended in [14] (called
‘KVDB’). The dynamic equations of the system state xk
and bias vector ηk are formulated by
xk+1 = F k xk + G k vk
(14)
hk+1 = F h,k hk + vh,k
x̂k+1|k = F k x̂k
where the covariance between the state and the
measurements, and the measurement prediction covariance
are
(12)
3.2
(17)
(13)
IET Signal Process., 2014, Vol. 8, Iss. 9, pp. 958–967
doi: 10.1049/iet-spr.2013.0393
www.ietdl.org
relationship explicitly between local estimates and the bias
vector to estimate.
4.1
measurement
xm
k
ym
k
in LCCS (see (34))
where
Notations
ym
xm
rm
rm
(35)
k =
k cos (uk ), k =
k sin (uk )
^m
2 ^m
2
2 m 2
m
rk =
x k − xom + y k − yom = xm
+ yk
(36)
k
m
Firstly, we introduce some notations for the sake of clarity.
m m
T
vm
xk vx,k ym
xm
: real state in LCCS.
k
y,k
k = ^m ^m ^m ^m T
^m
x k = x k v x,k y k v y,k : real state in GCCS.
m m
T
vx,k ym
vm
xm
: equivalent measurement in
k
y,k
k = xk
LCCS.
m
^m
^
^m
^m
^m T
x k = x k v x,k y k v y,k : equivalent measurement
in GCCS.
m m
T
v̂x,k ŷm
v̂m
: state estimate in GCCS.
x̂m
k
y,k
k = x̂k
In all the notations given above, the first and third
components mean the position, and the second and fourth
components mean the velocity. The state estimate x̂m
k
is affected by both sensor biases and random error.
We define
^m
m
the equivalent measurements xk in LCCS (or x k in GCCS) as
one part of the state estimate, which is affected by sensor
biases only. Several transformations about the position
components between different coordinate systems are
m
introduced as follow.
xk
Conversion from equivalent measurement
in LCCS
^m ym
k
x
to the equivalent measurement ^km in GCCS
yk
^m m xk
xk
xom
+
(30)
=
R
m
wm y
^m
yom
k
y
k
where the rotation matrix is
cos (wm ) −sin (wm )
=
sin (wm ) cos (wm )
R wm
Conversion from real state
in LCCS
^m x k
^m
y k
(31)
in GCCS to real state
^m
x k − xom
−1 xm
k
= Rwm ^m
ym
k)
y k − yom
xm
k
ym
k
(32)
where the rotation matrix is
R−1
wm
cos (wm )
=
−sin (wm )
Conversion from real state
xm
k
ym
k
xm
k
ym
k
sin (wm )
cos (wm )
m
4.2 Construction of the pseudo-measurement
equation
m
To establish the link between the local estimate x̂m
k , P k and
the sensor bias, we firstly decompose the local estimate into
different parts (from the real state, the sensor bias and
random error, respectively) with a simple and easily
calculated formulation. After that, a pseudo-measurement
equation of the bias vector is derived, which serves to
further bias estimation.
the meaning of the biased local estimate
mBy analysing
x̂k , P m
,
it
is
seen
that it is affected by two kinds of
k
errors: sensor biases and random errors. Moreover, the
covariance P m
k only reflects the effect from random errors.
Therefore, local estimate x̂m
k can be represented by the sum
^m
of the equivalent measurement x k (affected by sensor
biases only) in GCCS and the random error em
k
^m
^m
m
k , hm + em
x̂m
k = x k + ek = fm x
k
(37)
^m
where fm x k , hm is the part influenced by the sensor biases
ηm, and em
k is the part influenced by random noise that can
be regarded as a random vector with zero-mean and
covariance P m
k . The following theory provides a formulation
for local estimate decomposition.
Theorem 1: Any local estimate x̂m
k in GCCS can be
approximated by the following linear form
x̂m
k
^m
= x k + Jhm
Drm
+ em
k
Dum
where Jhm is the Jacobian matrix defined by
∂f T
Jhm = mm m ∂h (x̂k , 0 0 )
(38)
(39)
(33)
in LCCS to equivalent
The Jacobian matrix can be expressed by components of local
estimate x̂m
k as follows (see (40) at the bottom of the next page).
Proof: See Appendix
m
m
m
m
m
rk + Drm cos (uk ) cos (Dum ) − sin (uk ) sin (Dum )
rm
cos uk + Dum
k + Dr
m
= m
= m
m
m
rk + Drm sin (uk ) cos (Dum ) + cos (uk ) sin (Dum )
rk + Drm sin uk + Dum
⎡
⎤
xm
ym
m
m
m
m
m
m
m
k
k
xm
cos
(D
u
)
−
y
sin
(D
u
)
+
cos
(D
u
)Dr
−
sin
(D
u
)Dr
k
⎢ k
⎥
rm
rm
k
k
⎥
=⎢
m
m
⎣
⎦
yk
xk
m
m
m
m
m
m
m
m
yk cos (Du ) + xk sin (Du ) + m cos (Du )Dr + m sin (Du )Dr
rk
rk
IET Signal Process., 2014, Vol. 8, Iss. 9, pp. 958–967
doi: 10.1049/iet-spr.2013.0393
(34)
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& The Institution of Engineering and Technology 2014
www.ietdl.org
If two local estimates x̂1k and x̂2k are assumed from the same
^1
^2
ĥk , P h,k , the update process is given by
⎧
−1
⎪
T
T
⎪
ĥ
=
ĥ
+
P
A
A
P
A
+
P
⎪
k+1
k
h
,k
k+1
h
,k
k+1
k+1
k+1
⎪
⎪
⎪
⎨
[Y k+1 − Ak+1 ĥk ]
−1
⎪
T
T
⎪
P
=
P
−
P
A
A
P
A
+
P
⎪
h
,k+1
h
,k
h
,k
k+1
h
,k
k+1
k+1
k+1
⎪
⎪
⎪
⎩A P
k+1 h,k
target, that means x k = x k . Thus, we have
x̂1k
−
x̂2k
= Jh1
2
Dr1
Dr
− Jh2
+ e1k − e2k
Du1
Du2
(41)
Rearranging the terms in (41) and recalling the definition
h = [ Dr1 Du1 Dr2 Du2 ]T leads to
x̂1k − x̂2k = Jh1
−Jh2 h + e1k − e2k
(48)
(42)
4.4
In this way, we obtain the following Corollary, which
provides a pseudo-measurement equation revealing the
relationship between local estimates and the bias vector. □
Corollary 1: Given two
local
estimates and the corresponding
m
covariance matrices x̂m
,
P
k
k (m = 1, 2) from the same target,
the following pseudo-measurement equation of the bias
vector can be obtained
Correction and fusion of local estimates
having the bias estimates ĥk , by
Drm
m
ŷm
based on (38), we have
k = x̂k − Jhm
Dum
After
^m
x k + em
ŷm
k = k
letting
(49)
Y k = Ak h + ek
(43)
Equation (49) indicates that ŷm
k is affected by the random error
m
em
k only. As a result, ŷk can be regarded as the unbiased local
estimate. If we ignore the effect resulting from the
cross-correlation between the bias estimation errors and the
state estimation errors, the covariance of ŷm
k can be
approximated by
Y k = x̂1k − x̂2k
(44)
T
m
m
P
k = P k + Jhm Phm Jhm
where
Ak = Jh1
−Jh2
(50)
(45)
ek = e1k − e2k
(46)
The covariance of the noise vector ek is
12 T
P k = P 1k + P 2k − P 12
k − Pk
(47)
where P hm is the sub-matrix of Pη, k with respect to the bias
vector ηm.
Consequently,
trackfusion based on corrected local tracks
1 1
2k can be done according to the BC
ŷk , P k and ŷ2k , P
(Bar-Shalom and Campo) fusion formula [16]
ŷ =
ŷ1k
!
T "−1
1
12
1
2
12
12
Pk + Pk − Pk − P
+ Pk − Pk
(ŷ2k − ŷ1k )
k
1
A recursive formula of the cross-covariance P 12
k between ek
2
and ek can be found in [15].
(51)
The corresponding covariance is
4.3 Bias estimation based on recursive least
squares
!
T "−1
1
2
12
1k − P
12
12
=P
1k − P
P
+
P
−
P
−
P
P
k
k
k
k
k
In this paper, the sensor biases are modelled as the unknown
constants, hence, we can obtain the RLS estimator based on
the pseudo-measurement (43). Given an initial bias estimate
!
x̂m
k − xom
⎢
m
2 m
2
⎢
−
x
+
ŷ
−
y
x̂
om
om
k
k
⎢
⎢ m
m
m
⎢ ŷ − y 2 v̂m − x̂m − x
om
om ŷk − yom v̂y,k
x,k
k
⎢ k
⎢
2 m
2 3/2
⎢
+ ŷk − yom
x̂m
⎢
k − xom
⎢
m
=⎢
ŷk − yom
⎢
⎢
m
2 m
2
⎢
⎢
−
x
+
ŷ
−
y
x̂
om
om
k
k
⎢
m
m
m
⎢ m
⎢ x̂k − xom 2 v̂m
y,k − x̂k − xom ŷk − yom v̂x,k
⎢
⎢
3
⎣
2 m
2 m
x̂k − xom + ŷk − yom 2
⎡
Jhm
T "
1
12
Pk − P
k
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(52)
⎤
− yom ⎥
⎥
⎥
⎥
⎥
⎥
m
−v̂y,k
⎥
⎥
⎥
⎥
m
⎥
⎥
x̂k − xom ⎥
⎥
⎥
⎥
⎥
⎥
m
⎥
v̂x,k
⎥
⎦
−
ŷm
k
(40)
IET Signal Process., 2014, Vol. 8, Iss. 9, pp. 958–967
doi: 10.1049/iet-spr.2013.0393
www.ietdl.org
k between two estimates ŷ1k and
where the cross-covariance P
2
ŷk can also be computed as in (47).
12
4.5
Discussions
4.5.1 Extension to the multi-target case or the case of
detection probability less than one: Here, we only
address the problem of bias estimation at the track level in
the single-target scenario. It is of great importance and
challenging to extend the proposed approach to the
multi-target case or the case of the detection probability less
than one. It is well known that one needs sensor reports
from the common target to perform bias estimation.
However, in the multi-target case, it is expected to
implement data association to determine the correspondence
between local sensor reports. Therefore, sensor registration
is highly conditioned on the results of track-to-track
association. Moreover, note that if sensor biases are
estimated and removed from sensor reports, one can
determine the correspondence more accurately and easily.
In this sense, we can say that track-to-track association and
sensor registration are highly coupled with each other. An
integrated way may be promising to implement data
association and bias estimation jointly. Related work about
this topic can be found in [17–19], where the sensor biases
are assumed to be a direct additive term on local estimates,
and only relative biases can be estimated.
In addition, in the case of detection probability less than
one, the target sets detected by different sensors do not
coincide. Here, we refer to a local track from one sensor as
‘outlier’ when the track has no the corresponding track
Fig. 2 Real and estimated track
from another sensor. Such ‘outliers’ further complicate an
already difficult track-to-track association problem. In this
case, dummy tracks can be introduced which can be used to
associate with the ‘outliers’. Aiming at the problem of
performing sensor registration at the track level and
track-to-track association jointly, the detailed construction
and solution of the joint optimisation model are beyond the
scope of the work, and will be covered in the future work.
4.5.2 Extension to asynchronous sensor registration
and track fusion: In this work, it is assumed that local
Fig. 3 Estimated sensor biases
a Estimated range bias for sensor 1
b Estimated range bias for sensor 2
c Estimated angle bias for sensor 1
d Estimated angle bias for sensor 2
IET Signal Process., 2014, Vol. 8, Iss. 9, pp. 958–967
doi: 10.1049/iet-spr.2013.0393
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www.ietdl.org
Fig. 4 RMSE for track fusion
a RMSE in position
b RMSE in velocity
c RMSE in position
d RMSE in velocity
tracks from different sensors are synchronous. In many
practical applications, sensor reports are usually not timecoincident because of different data rates of local sensors.
Hence, it is meaningful to implement bias estimation and
track fusion to the case of asynchronous sensors [20, 21].
The key point is, based on asynchronous local tracks, to
establish the pseudo-measurement equation of the bias
vector (including the asynchronous pseudo-measurement,
the asynchronous pseudo-measurement matrix, the
asynchronous bias vector and pseudo-measurement noise),
and evaluate the statistical property of the pseudomeasurement noise. Future work will include the extension
of the proposed work to asynchronous sensor registration
and track fusion.
5
Simulation results
In this section, we provide some selected simulation results to
illustrate the performance of the proposed approach.
5.1
Simulation scenario
We consider a two-dimensional tracking scenario with two
sensors and one target. The initial state of the target is at
(35,135) km with a velocity of (0.2128,0.1786) km/s. The
motion of the target follows the following dynamic equation
xk+1 = F k xk + G k vk
964
& The Institution of Engineering and Technology 2014
(53)
where the state xk = [xk , ẋk , yk , ẏk ]T means the position and
velocity of the target at x–y plane,
Fk = diag{Fk,1
, 2Fk,1},
1 T
T /2
, G k,1 =
Gk = diag{Gk,1, Gk,1}, F k,1 =
.
0 1
T
The sample interval T is set to be 10 s, and the surveillance
time period is 2000 s. The covariance matrix of the
zero-mean process noise ωk is designed as Q = (0.0002 km/
s2)I, where I is an 2 × 2 identity matrix. Each sensor
measures the range rkm and angle um
k to the target by the
measurement equation
rkm
^m
2 ^m
2
x k − xom + y k − yom + Drkm + vm
=
k,r
#
$^m
$ y − y
m
m
om
uk = arctg %^km
− wm + Dum
k + vk,u
x k − xom
(54)
(55)
^m ^m where x k , y k means the target’s real state in GCCS,
(xom, yom) is the sensor’s position. In this scenario, sensor 1
is located at (xo1, yo1) = (20, 50) km in GCCS and the
corresponding LCCS has a rotation angle of f1 = 0.242 rad;
sensor 2 is located at (xo2, yo2) = (400, 100) km in GCCS
and the corresponding LCCS has a rotation angle of f2 =
m
0.375 rad. The random measurement errors (vm
k,r and vk,u )
for both of sensors are modelled as white Gaussian with
IET Signal Process., 2014, Vol. 8, Iss. 9, pp. 958–967
doi: 10.1049/iet-spr.2013.0393
www.ietdl.org
variances (0.01 km)2 and (0.001 rad)2, respectively. The
range and angle biases for sensor 1 and sensor 2 are (−1
km, −0.0042 rad) and (1.2 km, 0.0035 rad), respectively.
Both of the sensors employ EKF to obtain the local biased
tracks based on their own measurement information. The
fusion centre receives local tracks from different sensor, and
implements sensor registration and track fusion. The
measurements are supported by 200 Monte-Carlo runs
performed on the same target trajectory but with
independently generated measurements for each trial. The
initial bias vector is given by ĥ0 = [ 0 0 0 0 ]T , Pη,0 =
diag ([106, 104, 106, 104]). The simulation results are given
by implementing the proposed method and the competing
algorithms (exact method in [11] and ‘KVDB’ in [11, 14]).
Fig. 2 shows the real trajectory of the target and local
biased tracks by different sensors at a time interval,
respectively.
Fig. 3 shows the estimation results for the range and angle
biases. It is shown that the proposed method and ‘Exact
method’ perform similarly, and yield better estimates than
‘KVDB’.
The average root mean-square error (RMSE) in the position
and velocity component over the whole simulation time are
shown in Fig. 4. Since the ‘Exact method’ cannot be used
to implement track fusion directly, we illustrate the fusion
result based on the proposed approach, the ‘KVDB’ and
direct fusion of biased local tracks (named by ‘bias-ignorant
fusion’). It is seen that the proposed method outperforms
the competing ones. Among them, Figs. 4c and d eliminate
the result given by ‘bias-ignorant fusion’ so as to make the
performance comparison between the proposed approach
and the ‘KVDB’ more clearly.
5.2
Computation cost
The executing time of these algorithms is summarised in
Table 1. It is evaluated based on the following computer
configuration: CPU: Intel (R) Core(TM) i5-2430M
[email protected] GHz 2.40 GHz; RAM:2 GB; Operating system:
Windows 7.
From the above table, it is seen that that the proposed
approach has a significant improvement in computational
efficiency. The reason why the proposed approach is so
computationally effective will be summarised in the
following subsection.
5.3
Algorithm analysis
As shown in Fig. 3, it seems that the ‘Exact method’ and the
proposed approach behaves similarly in estimating the sensor
biases. However, the proposed approach has its own
important merits. Compared with the ‘Exact method’, the
advantages of the proposed approach lie on the following
several aspects.
Firstly, the proposed approach requires far less information
than the ‘Exact method’ when constructing the
pseudo-measurement equation. The ‘Exact method’ requires
lots of information from local trackers, such as the Kalman
gain, system matrix, measurement matrix, covariance of
Table 1 Executing time of algorithms
Proposed approach
0.0073 s
Exact method
KVDB
0.0268 s
0.0245 s
IET Signal Process., 2014, Vol. 8, Iss. 9, pp. 958–967
doi: 10.1049/iet-spr.2013.0393
measurement noise, local estimates, and so on. However,
the proposed method requires local estimates and its
covariance only. Secondly, the proposed approach has the
higher computational efficiency than the ‘Exact method’
(shown in Table. 1). Although the process of deriving the
pseudo-measurement equation seems a little bit complex,
the final (43) is so easy. However, the ‘Exact method’
requires computing the pseudo-inverse of the matrix when
constructing the pseudo-measurement equation, which is
not necessary in the proposed approach. Thirdly, the
proposed approach has a wider use scope. The ‘Exact
method’ requires that local tracker is a linearised model, the
proposed approach does not have this restriction. Finally,
the proposed approach enables the local track to be
corrected easily by the estimated sensor biases, as
contributes to implementing the track fusion easily, but the
‘Exact method’ cannot.
Compared with the ‘KVDB’, the advantages of the
proposed approach lie on the following several aspects.
Firstly, the proposed approach can deal with sensor
registration at the track level, which is needed in the
distributed information fusion system. Secondly, the
proposed approach has a higher computational efficiency
than ‘KVDB’(shown in Table. 1), since the latter requires a
two-stage filter process. Thirdly, about the bias estimation,
the proposed approach behaves more stable than ‘KVDB’.
Finally, about the fusion performance, the proposed
approach outperforms the ‘KVDB’ in RMSE for track
fusion (shown in Fig. 4c–d ).
6
Conclusion
In the distributed multi-sensor tracking system, sensor
registration is expected to be performed at the track level
rather than at the measurement level. However, it becomes
quite difficult to remove the biases implicitly included in
local tracks. In this paper, a novel approach to bias
estimation and track fusion is proposed based on the
carefully devised pseudo-measurement equation. Simulation
results demonstrate that the proposed approach is
computationally efficient than competing algorithms. Unlike
‘Exact method’, the proposed approach enables it easy to
implement the track fusion. Moreover, the proposed
approach outperforms the ‘KVDB’ in bias estimation and
track fusion. Further work includes the extensions of the
proposed method to some challenging cases, such as the
multi-target case, asynchronous sensor case, the case of
detection probability less than one or time-varying sensor
biases and so on.
7
Acknowledgments
The work is supported by the State Key Program for Basic
Research of China (2013CB329405) and the National
Natural Science Foundation of China (No. 61203220).
8
References
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Canada, July 2007, pp. 1–8
9
Appendix
9.1
computed by
^m m xk
xom
xk
+
=
R
wm
^m
yom
ym
yk
k
⎡ m
m
xk cos (Dum ) − ym
k sin (Du )
m
m
⎢ xk
⎢ + cos (Dum )Drm − yk sin (Dum )Drm
⎢ rm
rm
⎢
k
= R wm ⎢ m k
m
m
⎢ yk cos (Du ) + xk sin (Dum )
⎢
⎣ ym
xm
+ mk cos (Dum )Drm + mk sin (Dum )Drm
rk
rk
xom
+
yom
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(57)
and since
^m
−
x
x
xm
−1
om
k
, R−1
= Rwm ^km
wm
ym
)
k
y k − yom
cos (wm ) sin (wm )
=
− sin (wm ) cos (wm )
(58)
Combining (36), (57) and (58) leads to
⎡ ^m
^m
⎤
x k − xom cos Dum − y k − yom sin Dum
⎢⎛
⎥
⎞
⎢
⎥
⎢
⎥
m
⎢⎜
⎥
⎟
Dr
⎢ ⎜1 + ⎟ + x
⎥
⎢⎝
⎥
om
⎠
2
2
m
m
⎢
⎥
^
^
^m ⎢
⎥
−
x
+
y
−
y
x
k
k
om
om
⎢
⎥
xk
⎢
⎥
=
^m
m
^m
m ⎥
⎢ ^m
yk
⎢ y k − yom cos Du + x k − xom sin Du
⎥
⎢
⎥
⎞
⎢⎛
⎥
⎢
⎥
⎢
⎥
m
⎟
⎢⎜
⎥
Dr
⎢ ⎜1 + ⎟ + yom
⎥
⎣⎝
⎦
^m
2 ^m
2 ⎠
x k − xom + y k − yom
(59)
The velocity component can be computed by
Appendix 1: proof of Theorem 1
^m
xk
Note that the equivalent measurement
written by
⎡
^m
xk
in GCCS can be
^m
=
∂x k
^m
v x,k
^m ∂x k
^m
+
∂x k
^m
∂y k
^m
v y,k
^m
= cos(Dum ) + [(y k − yom ) cos(Dum )
⎤
⎢
⎥
⎢ ^m ⎥
⎢
^m
⎢ v x,k ⎥
⎥
^m
m
⎥
x k = fm x k , h = ⎢
⎢ ^m ⎥
⎢ y ⎥
⎢ k ⎥
⎣
⎦
^m
v x,k
^m
+ (x k − xom ) sin(Dum )]
^m
(56)
^m
v y,k
×
^m
[(x k
Drm (y k − Dum )
− xom )2 +
,
^m
(y k − yom )2 ]3/2
^m
^m
v x,k
(60)
+ − sin(Dum ) − [(y k − yom ) cos(Dum )
^m
+ (x k − xom ) sin(Dum )]
Next, we aim to express each component of the equivalent
^m
xk
^m
^m
Drm (x k − xom )
,
^m
v y,k
in GCCS by that of x k in GCCS. According
measurement
^m
to (30) and (34), the position components of x k can be
×
966
& The Institution of Engineering and Technology 2014
IET Signal Process., 2014, Vol. 8, Iss. 9, pp. 958–967
doi: 10.1049/iet-spr.2013.0393
^m
[(x k
− xom )2 +
^m
(y k
− yom )2 ]3/2
www.ietdl.org
^m
v x,k
^m
^m
v y,k
where
and
are the velocity components of the x k in
x and y coordinate, respectively.
Similarly, we have
^m
v y,k
^m
=
∂y k
^m
v x,k
^m ∂x k
^m
+
∂y k
^m
v y,k
^m
∂y k
^m
= sin(Dum ) − [(y k − yom ) cos(Dum )
^m
− (x k − xom ) sin(Dum )]
,
^m
×
Drm (x k − xom )
^m
^m
v x,k
^m
[(x k − xom )2 + (y k − yom )2 ]3/2
^m
+ cos(Dum ) + [(y k − yom ) cos(Dum )
(61)
^m
− (x k − xom ) sin(Dum )]
,
^m
×
^m
[(x k
Drm (y k − yom )
− xom )2 +
^m
(y k
− yom )2 ]3/2
^m
v y,k
xk , h ) =
fm (
m
fm (x̂m
k,
[0
^m
∂fm m
· [x k − x̂k ] + m ∂h (x̂m ,[0
k
∂f
T
m
0] ) + ^m ∂x k (x̂m ,[0
k
0]T )
·[
Drm
Dum
0]T )
∂fm T
Jhm = m m ∂h (x̂k , 0 0 )
m
⎡
x̂k − xom
⎢ m
2 2
⎢
x̂k − xom + ŷm
⎢
k − yom
⎢ ⎢ ŷm − y 2 v̂m
⎢
om
k
x,km
m
⎢
⎢ − x̂m
−
x
k
om ŷk − yom v̂y,k
⎢
3/2
⎢ ⎢ x̂m − x 2 +ŷm − y 2
⎢ k
om
om
k
m
=⎢
⎢
ŷk − yom
⎢
⎢ m
2 2
⎢
x̂ − xom + ŷm
⎢
k − yom
⎢ k
2
⎢
⎢ x̂m
− xom v̂m
k
y,k
⎢
⎢ −x̂m − x ŷm − y v̂m
⎢
om
om x,k
k
k
⎣ 2 m
2 3/2
m
x̂k − xom + ŷk − yom
(62)
]
^m
T
m
m · x
k − x̂m
x̂m
+ Jhm
k = fm (x̂k , 0 0 ) + J^
k
xk
m
Dr
+ em
·
k
Dum
∂fm ^m | m ∂x k x̂k , 0
0
T = I
IET Signal Process., 2014, Vol. 8, Iss. 9, pp. 958–967
doi: 10.1049/iet-spr.2013.0393
⎤
(65)
T Since fm x̂m
,
0
0
= x̂m
k
k , we have
x̂m
x k + Jhm
k = The Jacobian matrix is computed by
x k
− yom ⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
m
⎥
−v̂y,k
⎥
⎥
⎥
⎥
m
⎥
⎥
x̂k − xom ⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
m
v̂x,k
⎦
ŷm
k
In this way, the local estimate x̂m
k can be represented by
^m
J ^m =
−
(64)
In general, the sensor biases are assumed to be small
quantities. By making the first-order
^m
Taylor series
m
expansion about the function fm x k , h
at the point
T
(x̂m
k , 0 0 ), we have the following approximation
^m
where I is the identity matrix, and
Drm
+ em
k
Dum
(66)
(63)
Thus, Theorem 1 is proven.
□
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& The Institution of Engineering and Technology 2014
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