Unscented Extended Kalman Filter for Target Trakcing

Journal of Systems Engineering and Electronics Vol.22,No.2,April2011,pp.188–192
Available online at
Unscented extended Kalmanfilter for target tracking
Changyun Liu1,2,*,Penglang Shui1,and Song Li2
1.National Laboratory of Radar Signal Processing,Xidian University,Xi’an710071,P.R.China;
2.Missile College of Air Force Engineering University,Sanyuan713800,P.R.China
Abstract:A new method of unscented extended Kalmanfilter (UEKF)for nonlinear system is presented.This new method is a combination of the unscented transformation and the extended Kalmanfilter(EKF).The extended Kalmanfilter is similar to that in a conventional EKF.However,in every running step of the EKF the unscented transformation is running,the deterministic sample is caught by unscented transformation,then posterior mean of non-linearity is caught by propagating,but the posterior covariance of nonlinearity is caught by linearizing.The accuracy of new method is a little better than that of the unscented Kalmanfilter(UKF), however,the computational time of the UEKF is much less than that of the UKF.
Keywords:unscented transformation(UT),extended Kalmanfil-ter(EKF),unscented extended Kalmanfilter(UEKF),unscented Kalmanfilter(UKF),nonliearity.
DOI:10.3969/j.issn.1004-4132.2011.02.002
1.Introduction
The Kalmanfilter is an optimalfilter for estimating a lin-ear system[1,2].The simplicity,recursive structure,and mathematical rigour of the derivation of the Kalmanfilter make it well-suited and attractive for use in many practical applications.However,many real-world systems are non-linear in nature.The extended Kalmanfilter(EKF)is de-veloped to help account for these nonlinearities.The EKF accounts for nonlinearities by linearizing the system about its last-known best estimation with the assumption that the error incurred by neglecting the higher-order terms is small in comparison to thefirst-order terms.Although the EKF has been used successfully in many applications,it has sev-eral shortcomings.The EKF is a suboptimal nonlinearfil-ter due to the truncation of the higher-order terms when linearizing the system[3–6].The computational time of the EKF is similar to that of the Kalmanfilter.
The UKF attempts to remove some of the short comings of the EKF in the estimation of nonlinear systems.The UKF is an extension of the traditional Kalmanfilter for the estimation of nonlinear systems that implements the Manuscript received August31,2009.
*Corresponding author.unscented transformation.This approach gives the UKF better convergence characteristics and greater accuracy than the EKF for nonlinear systems[3].However,the com-putational time of the UKF is more than that of the EKF. For these above reasons,a new method for tracking a maneuvering target,is proposed.The basic idea is to combine the unscented transformation(UT)with the EKF, namely UEKF.The sigma points and weights are calcu-lated from the a priori mean and covariance of the state through UT,then the sigma points are propagated through the nonlinear system,and the posterior mean of the state is determined from the statistics of the propagated sigma points.The posterior covariance is determined from the sigma points through linearity of the nonlinear system.The filter update is performed using the standard Kalmanfilter equations.The simulation shows that the accuracy of the UEKF is a little better than that of the UKF,and both are more accurate than the EKF.However,the computational time of the UEKF is much less than that of the UKF.
2.System model
The basic framework of the target tracking involves esti-mation of the state of a discrete-time nonlinear dynamic system.
X k+1=F k(X k,v k)(1)
Z k+1=H k+1(X k+1,w k+1)(2) where X k represents the state of the system and Z k is the only observed signal.The process noise v k drives the dynamic system,and the observation noise is given by w k.The system dynamic models F k and H k are assumed known.F k and H k may be linear or nonlinear function.
3.UEKF algorithms
3.1Unscented transformation
The unscented transformation is a method for calculating the statistics of a random variable which undergoes a non-linear transformation[4,5].Given the mean and covari-
Changyun Liu et al.:Unscented extended Kalman filter for target tracking 189
ance of the state vector at step k −1are known.
ˆX
k −1=E {X k −1}(3)P ˆX k −1=E {(X k −1−ˆX k −1)(X k −1−ˆX k −1)T }
(4)
The sigma vector
χi k −1
and the corresponding weights W i
(i =0,1,2,...,2L ,and L denotes the dimension of X k )are calculated from the priori mean and covariance of the state,according to the following [7–9]
χ0
k −1=ˆX k −1(5)χi k −1=ˆX k −1+(
(L +λ)P ˆX k −1)i ,
i =1,2,...,L χi k −1=ˆX k −1−(
(L +λ)P ˆX k −1)i ,
i =L +1,...,2L W m
0=λ/(L +λ)
W c
=λ/(L +λ)+(1−α2
+β)
W m i =W c i =1/(2(L +λ))
where λ=α2(L +κ)−L is a scaling parameter,αdeter-mines the spread of the sigma points around ˆX
k −1and is usually set to a small positive value,κis a secondary scal-ing parameter which is usually set to 0,and βis used to
incorporate prior knowledge of the distribution of ˆX k −1,(
(L +λ)P ˆX k −1)i is the i th row of the matrix square root.
The sigma points are propagated through the nonlinear system function
χi k |k −1=F k (χi
k −1),
i =0,1,2,...,2L (6)
The posterior mean ¯X
k |k −1,and covariance ¯P k |k −1are determined from the statistics of the propagated sigma points as follows
¯X
k |k −1=2L i =0
W m i χi k |k −1
(7)
¯P
k |k −1=2L i =0
W c i (χi k |k −1−¯X k |k −1)(χi k |k −1−¯X k |k −1)
T (8)
3.2Extended Kalman filter
Unlike the UT,the EKF is to linearize the nonlinear func-tions in the state dynamic and observation models.The nonlinear functions in (1)and (2)are approximated by the first term in their Taylor series expansion.The EKF is based on the assumption that local linearization of the above equations may be a suf ficient description of non-linearity.The prediction equation of the EKF is done as follows
¯X
k +1|k =F k (ˆX k |k )(9)¯P k +1|k =ˆF k P k |k ˆF T k
+Q (10)
The update equation is performed using the standard
Kalman filter equation as follows
K k +1=¯P k +1|k ˆH T k [ˆH k ¯P k +1|k ˆH T k
+R ]−1(11)
ˆX
k +1|k +1=¯X k +1|k +K k +1[z k +1−ˆH k ¯X k +1|k ](12)ˆP
k +1|k +1=(I −K k +1ˆH k )¯P k +1|k (13)
where ˆF
k is the Jacobian of F k (X k )evaluated at the filter estimate ˆX
k |k ,ˆH k is the Jacobian of H k +1(X k +1)evalu-ated at the prediction ¯X
k +1|k ,as follows ˆF k +1=[∇X k F T k (X k )]|X k =ˆX k |k (14)ˆH k +1=[∇X k
H T k +1(X k )]|X k =¯X k +1|k (15)
3.3Unscented EKF (UEKF)
The UEKF is to combine the UT with the EKF.The sigma points and weights are calculated using (5),then the sigma points are propagated using (6),and the posterior mean of the state is determined from the statistics of the propagated sigma points using (7),however we do not calculate the posterior covariance using (8).We utilize the i th sigma point χi
k −1to linearize F k .ˆF i k +1=[∇X k F T k (X k )]|X k =χi k −1
,i =0,1,2, (2)
(16)
The i th prediction covariance is performed according to the following
¯P i k +1|k =ˆF i k P k |k ˆF i T k
(17)
So,the posterior covariance of the state are determined
as follows
¯P
k +1|k =2L i =0
W c i ¯P i k +1|k +Q
(18)
The filter update is performed using the standard
Kalman filter equations (11)–(13).
The detailed algorithms of the UEKF is utilized as fol-lows:
(i)Initalize with
ˆX
0=E {X 0}P 0=E {(X 0−ˆX
0)(X 0−ˆX 0)T }(ii)Prediction and update of the system state
k =1
190Journal of Systems Engineering and Electronics V ol.22,No.2,April2011 (a)Calculate sigma points and weights
χ0
k−1
=ˆX k−1
χi
k−1=ˆX k−1±(
(L+λ)PˆX
k−1
)i W m0=λ/(L+λ)
W m i=1/(2(L+λ))
(b)Propagating the sigma points
χi k|k−1=F k(χi k−1),i=0,1,2, (2)
(c)Time update
ˆF i k+1=[∇X
k
F T k(X k)]|X
k=χi k−1¯P i
k+1|k
=ˆF i k P k|kˆF i T k
¯X
k|k−1=
2L
i=0
W m iχi k|k−1
¯P
k+1|k =
2L
i=0
W c i¯P i k+1|k+Q
(d)Measurement update equation
K k+1=¯P k+1|kˆH T k[ˆH k¯P k+1|kˆH T k+R]−1
ˆX
k+1|k+1
=¯X k+1|k+K k+1[z k+1−ˆH k¯X k+1|k]
ˆP
k+1|k+1
=(I−K k+1ˆH k)¯P k+1|k
k+1→k
go to(a)
4.Analysis of computational cost
Let the X k is N dimensional vector and the Z k is M di-mensional vector.In general,M is much less than N. The computational cost of EKF mainly comes from (10)–(13).In(10),the computational cost is2N3times of real multiplications and additions.In(11),the compu-tational cost is mainly N2M times of real multiplications and additions.In(12),the computational cost is mainly N2 times of real multiplications and additions,and in(13),the computational cost is mainly N3times of real multiplica-tion and additions.So,the real multiplications and addi-tions about EKF is mainly3N3times.
The computational cost of UKF mainly comes from(4) and(8).In(4),the computational cost is mainly N4times of real multiplications and additions,and in(8),the com-putational cost is mainly2N4times of real multiplications and additions,there are several N3times of real multipli-cations and additions.So the computational cost of UKF is mainly3N4times of real multiplications and additions.
The computational cost of UEKF mainly comes from (11)–(13).In the above analysis,in(11)–(13),the com-putational cost is N3mainly times of real multiplications and additions,in(17)and(18),the computational cost is mainly2N3times of real multiplications and4N4times of real additions.So the computational cost of UEKF is3N3 mainly times of real multiplications,but the computational cost of real additions is4N4times.
From the above analysis,among EKF,UEKF and UKF, the EKF runs fastest,the UEKF takes second place,the UKF runs slowly.
5.Simulation
5.1Simulation model
We adopt dynamic model of a ballistic object on reentry in[10],for the theoretical Cramer-Rao bounds of the dy-namic model is presented.Utilizing the CRLB,the per-formances of three nonlinear trackingfilter are compared against each other and against the CRLB.The considered filters include the standard EKF,the UKF,and the UEKF. The state of a discrete-time nonlinear dynamic system is described as following.
The state equation is
X k+1=
⎡
⎣
1−τ0
010
001
⎤
⎦X k−
⎡
⎣
τ
⎤
⎦[D(X k)−g]+v k and
D(X k)=
g×1.754e−1.49×10−4×X k(1)X2k(2)
2X k(3)
The measurement equation is below(considering mea-surement intervals is T)
Z k+1=
100
X k+1+w k+1
where X k=
hυβ
,h is altitude,υis velocity,βis the ballistic coefficient.X k(i)denotes the i th element of state vector X k,w k+1and v k are zero-mean white Gaus-sian noise and mutually independent.The detailed param-eter is analyzed in[10,11].
5.2Simulation result
The initial mean and covariance matrix can be written re-spectively as
X0=
61304836500
T
P0|0=
⎡
⎣
σ2rσ2r/T0
σ2r/T2σ2r/T20
00σ2β
⎤
⎦
whereσr is200m,σβis14814kg/ms2.
Changyun Liu et al.:Unscented extended Kalmanfilter for target tracking191
A typical target trajectory is shown in Fig.1.All error curve corresponding to threefilters(EKF,UKF,UEKF) are obtained by Monte Carlo simulations with100inde-pendent runs,and the mean and the standard deviation of the estimation error are shown in Fig.2and Fig.3,respec-tively.The absolute error mean(AEM)of the height,veloc-ity and the computational time of threefilters are shown in Table
1.
Fig.1Ideal target trajectory of a ballistic object versus time
Analyzed from the simulation results,such as Fig.2and Fig.3,the performance of the UKF and the UEKF is bet-ter than the EKF,and the performance of the UEKF is a little better than that of the UKF.Seen from Table1,the computational time of the EKF is lest,and the compu-tational time of the UEKF is much less than that of the
UKF.
Fig.2Mean error in
estimating
192Journal of Systems Engineering and Electronics V ol.22,No.2,April
2011
Fig.3
Standard deviation of estimation error
Table 1
The computational time,RMS of the height and the veloc-
ity from three various filtering methods(simulation condition:CPU 2.5GHz,DDR memory:2G)
EKF UKF UEKF AEM(height)/m 57.971311.44259.9499AEM(velocity)/(m/s)35.31989.97139.9633Computational time/s
1.178771
8.252716
3.227560
6.Conclusion
Combining the UT with the EKF,a novel method,namely UEKF,is presented.The simulation results show the per-formance of the UEKF is a little better than that of the UKF,and the computational time of the UEKF is much less than that of the UKF.
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Biographies
Changyun Liu was born in 1973and is an as-sociate professor.He received his B.S in 1996and M.S in 2002respectively from Air Force Engineering University.Now he is a Ph.D.can-didate of National Laboratory of Radar Signal Processing in Xidian University.His researches include signal processing,target tracking and time-frequency analysis.
E-mail:icemissile@
Penglang Shui was born in 1967,he is a pro-fessor and doctor supervisor in National Lab-oratory of Radar Signal Processing of Xidian University.He has been working on modern radar signal processing,wavelet theory,time-frequency analysis and image processing.
E-mail:plshui@
Song Li was born in 1977.He is a Ph.D.can-didate in Air Force Engineering University.His researches include ISAR signal processing and imaging.
E-mail:lisong@。