2007 IEEE International Conference on Granular Computing
Nonlinear Target Identification and Tracking Using UKF
D. G. Khairnar, S. Nandakumar, S. N. Merchant and U. B. Desai
SPANN Laboratory
Department of Electrical Engineering
Indian Institute of Technology, Bombay,
Mumbai-400 076, India
phone: +(9122) 25720651, email:[email protected]
Abstract
In [7] Julier and Uhlmann presented Unscented transformation method to propagate mean and covariance information through nonlinear transformations. EKF is generally
used to track nonlinear targets, but linearization introduces
a bias and there is no guarantee that even the second order
terms can compensate for such errors. UKF is a straight
forward extension of the Unscented transformation which
overcomes the limitations of EKF; has been the motivation
behind the research and development work that is elaborated in this paper.
In this paper, we implemented target identification algorithm using Dempster Shafer Theory (DST) and tracking
algorithm using Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF). A tracking filter is developed
and simulated based on UKF to track nonlinear, ballistic
and reentry targets. Comparison of EKF and UKF for nonlinear targets tracking are presented based on the simulation results. Simulated results gives more supporting points
to use UKF for nonlinear target tracking rather using EKF.
2 Unscented Kalman Filter
The Unscented Kalman Filter (UKF) addresses the approximation issues of the EKF. The state distribution is
again represented by a Gaussian Random Variable (GRV),
but is now specified using a minimal set of carefully chosen sample points. The UKF is a straight forward extension
of the Unscented transformation to the recursive estimation
in Kalman filter equations [9]. After Unscented transformation of the state variables, UKF equations are formed as
following: The predicted state
and its covariance
are estimated as
1 Introduction
Radar target identification and tracking is primary function of any radar which makes it as an active research area.
Radar signal processing consists of digital pulse compression, droppler estimation, SNR estimation. Radar data processing consists of tracker, classifier/identifier, schedular
etc. Smitch and Goggans [1] discussed the signal level
theory for radar target identification in the aspect of High
Resolution Range (HRR) radar. The radar target identification based on complex image analysis for identifying aircrafts, ground vehicles in Synthetic Aperature Radar (SAR)
application with an indication of how these methods can be
applied to missiles, rockets and satellites in [2]. Different
methods of radar recognition is presented in [3]. Denoeux
[4] have attempted the problem of classiflying an unseen
pattern on the basis of its nearest neighbours in recorded
data set from the point of view of Dempster Shafer Theory. Engin Avci, Ibrahim Turkulugu and Mustafa [5] is
presented an intelligence target recognition system for target recognition and a wavelet packet neural network model
is used for RTID. Simon Julier and Jeffrey Uhlmann suggested better method for nonlinear target tracking in [6].
0-7695-3032-X/07 $25.00 © 2007 IEEE
DOI 10.1109/GrC.2007.97
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(b) Non-aircraft Target
(c) High Altitude Target
(d) Unknown Target
(6)
(7)
For linear functions, the UKF is equivalent to the Kalman
filter, The computational complexity of the UKF is the same
as the EKF, but it is more accurate and does not require the
derivation of any Jacobians [9].
Figure 1. Probabilities of target types for Aircraft, Non-aircraft, High altitude and Unknow
Targets
3 Implementation and Testing of Identification and Tracking Algorithm
Identifying the target type for tracking radar is important
that according to the type of target necessary action has to
be taken at that moment. If it is identified wrongly it will
lead to serious disaster in case of defence applications [8].
RTID algorithm is implemented and real, simulated targets
are used to test the algorithm. Aircraft, Non-aircraft, High
altitude target and unknown target trajectories are given as
input to the algorithm. basic belief masses (BBMs) for each
elements of the target possibilities are calculated with corresponding bound values of parameters. This bound values are determined by the type of target like for aircraft
maximum velocity considered is 660m/s. The parameters
and
vary for aircraft, Non-aircraft and
High altitude targets used in the simulation. The probabilities of the four target types vary according to the type
of target, which can be seen in the Figures 1-(a), (b), (c)
and 1-(d). When the probabilities of any of the target types
don’t cross the threshold, then target type is identified as
unknown.
Tw\ \
u uyx uz
(a) Aircraft Target
reentry targets. The reference trajectory is generated using general dynamics model. Because of drag, gravitational
acceleration and Coriolis force nonlinearity is involved in
tracking this type of target. Sampling frequency is
and the target dynamics is modeled using RK method 4th
order. The simulation parameters are shown in Table 1.
Measurement noise covariance and process noise covariance are updated for each measurement update. States estimation of position for high altitude ballistic and Reentry
targets are shown in Figures 2-(a), (b), (c) and (d).
TL}q~|
u|{
Table 1. Simulation parameters of EKF/UKF
ballistic and Reentry target tracking
Coordinate system
Target type
Dynamics of target
Filter
States
States Measurement
Measurement error
Initial State
Initial covariance
4 High altitude Ballistic and Reentry targets
Comparison of filters implemented for tracking nonlinear targets is presented to analyze the performance of UKF
for tracking application. The need for moving from KF to
EKF and from EKF to UKF is presented when the target to
be tracked is nonlinear. Targets considered are ballistic and
ECEF
High altitude ballistic/Reentry
Nonlinear
EKF/UKF
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(a) EKF (Ballistic Target)
(b) EKF (Reentry Target)
(c) UKF (Ballistic Target)
(d) UKF (Reentry Target)
(a) Position Error X
(c) Histogram Position Error (d) Histogram Velocity Error
Figure 2. Filtered and True positions by EKF
and UKF for Ballistic and Reentry Targets
Figure 3. Filtered Position, Velocity and Histogram Errors by EKF and UKF for Ballistic
Targets
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Estimated states by EKF and UKF for ballistic target, are
compared and the results are shown in Figures 3-(a),(b),(c)
and (d). In Figure 3-(a), the position error of state looks
same for EKF and UKF, but the histogram in Figure 3-(c)
and (d), shows that the error given by EKF has got the bias
and the mean value of error given by EKF is quite greater
than the mean value of error given by UKF. In case of velocity error, steady state error given by UKF is less than by
EKF and when the position and velocity error statistics, in
Tables 2 and 3 respectively, are compared, it is clear that
the UKF is a better option than EKF for tracking this type
of high altitude ballistic target.
Estimated states by EKF and UKF for reentry target are
compared and the results are shown in Figures 4-(a),(b),(c)
and (d). In Figure 4-(a), the position error of state given
by EKF have got bias, but the errors given by UKF don’t
have bias. This bias is due to linearization of nonlinear system in EKF, because linearization leads to approximation
of distribution. But in UKF this problem is avoided, this
can be seen in the histogram as shown in Figure 4-(c). In
case of velocity error, steady state error given by UKF is
almost same by EKF, but histogram in Figure 4-(d), show
that the Gaussian distribution of error is maintained closely
in UKF but not in EKF. The position error statistics, in Table
4, show that the position error given by UKF is less than by
EKF, and this comparison says that UKF is a better option
than EKF for tracking this type of nonlinear targets.
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Table 2. Statistics of estimation error for position of ballistic by EKF and UKF
State error
X by EKF
X by UKF
Y by EKF
Y by UKF
Z by EKF
Z by UKF
Mean (m)
-94.809249
8.152945
156.791740
-26.674749
-156.031930
20.272383
(b) Velocity Error X
STD (m)
173.063523
172.096793
442.839395
271.295848
167.652937
102.907127
5 Conclusion
The identification and tracking algorithms were evaluated on a number of simulated targets like aircraft, missile
and satellite. Tracking of Ballistic and Reentry targets by
UKF is better than by EKF when observing position errors.
By seeing the above performance analysis it gives more
supporting points to use UKF for nonlinear target tracking
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Table 3. Statistics of estimation error for velocity of ballistic by EKF and UKF
State error
Vx by EKF
Vx by UKF
Vy by EKF
Vy by UKF
Vz by EKF
Vz by UKF
Mean (m/s)
0.836867
-0.414676
-4.672585
2.236967
-2.706268
1.108465
STD (m/s)
9.405423
4.207772
4.585036
7.110487
11.115798
1.648443
(a) Position Error X
(b) Velocity Error X
Table 4. Statistics of estimation error for position of reentry by EKF and UKF
State error
X by EKF
X by UKF
Y by EKF
Y by UKF
Z by EKF
Z by UKF
Mean (m)
9.041741
0.106275
-0.213563
0.104656
18.666802
-0.035020
STD (m)
0.009733
0.260220
0.046350
0.189262
0.047140
0.137230
(c) Histogram Position Error (d) Histogram Velocity Error
Figure 4. Filtered Position, Velocity and Histogram Errors by EKF and UKF for Reentry
Targets
rather using EKF. Even though UKF has clear advantages
over EKF, the computational complexity of both algorithms
are to be compared. Results show that the error given by
UKF is higher in the beginning of tracking, this may be due
to sigma point sampling of the distribution.
[6] S.J. Julier, and J.K. Uhlmann,; A new extension of
the Kalman filter to nonlinear systems, Proceedings
of AeroSense: The 11th International Symposium on
Aerospace/Defence Sensing, Simulation and Controls,
(1997).
References
[7] S.J. Julier, and J.K Uhlmann,; Unscented filtering and
nonlinear estimation, Proceedings of IEEE, Vol. 92, No.
3,(Mar. 2004) 401-422.
[1] C.R. Smith and P.M. Goggans,; Radar target identification, IEEE Antenna and Propagation Magazine, Vol.35,
No.2,(April 1993) 27-38.
[8] D. Seshagiri, S. Ravind, and G. M. Cleetus,; Target
classification using data fusion based on DempsterShafer theory of evidences, International Radar Symposium India, (Dec. 2003).
[2] A.W. Rihaczek, S.J. Hershkowitz,; Theory and Practice
of Radar Target Identification, Artech House Publishers, Boston., (2000).
[9] S. Haykin,; Kalman Filtering and Neural Networks,
John Wiley and Sons, New York, (2001).
[3] V.G. Nebabin,; Methods and Techniques of Radar
Recognition, Artech House Publishers, Boston.,
(1995).
[4] T. Denoeux,; A k-nearest neighbor classification rule
based on Dempster-Shafer theory, IEEE transactions on
Systems, Man and Cybernetics, Vol. 25, No. 5,(1995)
804-813.
[5] E. Avci, I. Turkolugu, and M. Poyraz,; Intelligent target recognition based on wavelet packet neural network, Elsevier Expert Systems with Applications, Vol.
29,(2005) 175-182.
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