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Simplification of Unscented Kalman Filter for Orbit Object Tracking

Rongtai Cai , Qingxiang Wu, Jiangyong Cai, Jinqing Liu, Meigui Chen School of Physics, Optics, Electronic and Information Engineering,

Fujian Normal University Fuzhou, China

Abstract—Aim at the application of orbit object tracking; we adopt certain simplification technology to UKF (Unscented Kalman Filter), which reduce the computational complexity considerably. The state space equation in orbit object tracking is linear; the sigma sampling in unscented transform can be simplified as a composition-add process; the nonlinear transmit of sigma sampling, state vector, measurement vector and their correlation matrix are simplified by an MSUKF (UKF for Mixing system) algorithm. Experiment result shows that, compare with UKF, the proposed algorithm has the same calculation accuracy with considerable lower computational complexity. The amount of computation of proposed algorithm is only 43.33% of that of the UKF. Keywords— adaptive filter; Kalman filter; orbit trace; object tracking; Unscented Kalman Filter

I. INTRODUCTION Orbit object tracking is an important problem in space

science and aircraft engineering. The tracking of moving object, like missile and other aircraft is a basic task for trajectory measurement and control. There are many classical and novel algorithms in object tracking: tracking algorithm based on trace fitting [1], KF (Kalman filter) [2], PF (particle filter) [3] and so on. In the adaptive filter family, KF is a more reliable algorithm than other algorithms. But the original KF algorithm is powerless when it is used to solve nonlinear problems. EKF (Extended Kalman Filter) and UKF (Unscented Kalman Filter) were thus proposed. UKF is more effective to nonlinear filter than EKF, and it is consider as a good choice in object tracking. In object tracking (such as visual tracking and measurement), the measurement information is obtained in spherical coordinates and state equation is established in rectangular coordinate to reduce the coupling of measurement errors. In this way, the state equation is linear and measurement equation is nonlinear. In this situation, the UKF can be simplified for computational advantage. In this paper, we will develop a more feasible simplified UKF algorithm for orbit object tracking.

II. SPACE STATE MODEL FOR OBJECT TRACKING Suppose that the object is moving in a smooth trajectory,

and the trace can be fitted by a high order polynomial curve. We use low order polynomial going with noise to describe the orbit trajectory. In this experiment, we adopt the second order polynomial to describe the orbit trajectory. In rectangular coordinates, we have

20 1 2( ) ( )xx t a a t a t n t� � � � , (1)

2

0 1 2( ) ( )yy t b b t b t n t� � � � , (2) 2

0 1 2( ) ( )zz t c c t c t n t� � � � , (3) where x(t), y(t) and z(t) is the position of the object in x-axis, y-axis, and z-axis respectively; are noise respectively.

( ), ( ), ( )x y zn t n t n t

According to (1), (2) and (3), the space state model can be written as:

212

212

212

( 1) 1 0 0 0 0 0 0( 1) 0 1 0 0 0 0 0 0( 1) 0 0 1 0 0 0 0 0 0( 1) 0 0 0 1 0 0 0( 1) 0 0 0 0 1 0 0 0( 1) 0 0 0 0 0 1 0 0 0( 1) 0 0 0 0 0 0 1( 1) 0 0 0 0 0 0 0 1( 1) 0 0 0 0 0 0 0 0 1

x k T Tx k Tx ky k T Ty k Ty kz k T Tz k Tz k

� � �� �� �� �� � �� ��� ���� �

� �� ��� � ���� �

� �� ��� �� �� �

� �� ��� ��� �

�

��

�

��

�

��

( )( )( )( )( )( )( )( )( )

( )x

x kx kx ky ky ky kz kz kz k

k

� �� �� ��� ��� ����� ��� ��� ��� ��� ��� ��� ��

�n

�

��

�

��

�

��

� ,

(4) where T is sampling period, , , , , , , , ,x x x y y y z z z� �� � �� � �� are position, speed and acceleration components in x-axis, y-axis and z-axis respectively, k is time variable and nx(k) is noise.

We rewrite (4) as: ( 1) ( ) ( )xk k k� � �x Ax n , (5)

where is state variable, A is state transition matrix. And suppose that is Gaussian white noise with zero-mean. The correlation matrix of the noise is defined by

( )kx( )kxn

1 0 00 1

( )0

0 0

nx xxk

1

� � � �� � � �� �

R

�

� �

� � �

�

, (6)

where xx is correlation coefficient of state variable ( )x k . According to the translation between spherical coordinates

and rectangular coordinates, we can derive the measurement equation as below:

2010 Second International Conference on Information Technology and Computer Science

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DOI 10.1109/ITCS.2010.26

82

2 2 2

2 2

( ) ( ) ( )( ) ( )( )( ) arctan ( )( )

( ) ( )( )arctan

( ) ( )

R

V

H

x k y k z kR k ny kV k n kx k

k

H kz k

x k y k

� �� �

� �� �� � �� �� � �� �� �� � �� �� � �� �� �

� �� ���

n k

����

��

, (7)

where x(k), y(k)and z(k) are position components in x-axis, y-axis and z-axis respectively, R(k),V(k) and H(k) are radial distance, azimuth angle and elevation angle respectively,

are noise in radial distance, azimuth angle and elevation angle respectively.

( ), ( ), ( )R V Hn k n k n k

Equation (7) can be written as below for conciseness:

y( ) [ ( ), ] ( )k h k k k� �y x n , (8) where ny(k) is noise matrix, and suppose the correlation matrix of the noise is

2

2

2

R

ny V

H

R�

��

� ��

� �� ��

, (9)

where 2R� , 2

V� and 2H� are variance of noise in the 3 channels

respectively.

III. UNSCENTED KALMAN FILTER

A. Key Idea of KF The original KF including 2 steps: 1) The time update of

state vector , measurement vector and their correlation matrix and , where

means prediction for time k from time (k-1). 2) The measurement update of the above variables and matrix. Noted as , , and respectively, where (k| k) means update in time k with measurement information in time k.

( | 1)k k�x

( | )y k k P

( | 1)y k k�

( | -1)k k( | 1)xx k k �P

( | )k kxx xyP

xyP

)

( | 1)k k�

( |k kx ) ( |k k

B. Key Idea of EKF and UKF The original KF can be used only with linear state equation

and measurement equation. To be used in nonlinear application, the original KF must be improved and adjusted with certain technology. The EKF algorithm uses a linearization technology by means of Taylor expansion of space transition matrix A and measurement matrix h at ˆ( -1| -1)x k k and respectively. ˆ ( | 1)k k�x

The UKF employs an UT (Unscented Transform) technology to deal with the nonlinear problem. In the UKF algorithm, UT technology is used to nonlinear transition of state matrix x , measurement signal y and their correlation matrix xxP and xyP . But the state matrix and correlation matrix is expanded by the state noise and measurement noise. Thus, we have

[a T T Tx y�x x n n ]T

0

(10) And

0 000 0

xxaxx nx

ny

� � �� � �� �

PP R

R (11)

And the nonlinear transition of state matrix x and measurement signal y by UT is realized by � sampling, then the other matrix xxP and xyP are obtained by linear composition of the � sampling value.

So it is obvious that the UKF algorithm composed with 3 steps: unscented sampling, time update and the measure update.

C. Simplification of UKF for Orbit Object Tracking

1) Simplification of � sampling for UKF:

Considering the state equation and measurement equation descript by (4), (5), (7) and (8), the � sampling algorithm in UKF can be descript by a composition-add-decomposition process, see Table 1.

TABLE I COMPOSITION-ADD-DECOMPOSITION PROCESS OF � SAMPLING

1 input , ,xx nx ny,x P R R

2 composition xx

axx nx

ny

� � �� � �� �

P 0 0P 0 R 0

0 0 R

3 Cholesky decomposition

( )

xxchol

axx nx

ny

�

� � �

� � � �� �

P 0 0

P 0 R 0

0 0 R

4 composition

4 2 1n m

a

� �� � � � �� � � � �� �

x x0 0

x

0 0

�����

�

� � �

�

1xx xx

axx nx nx

ny ny

� �� � �� � �

� �� �

0 P 0 0 P 0 0

P 0 0 R 0 0 R 0

0 0 0 R 0 0 R

5 add a a axxf� � � �x P1

6 decomposition

xTT T Tn n n

a x nx ny nx

ny

�� � � � �

�

�� � � �� � � � �� � �

In Table 1, n is the dimension of state vector, and m is the dimension of measurement vector.

Note that nx� and ny� are fixed in the composition-add-decomposition process in each time of the iteration, so we can

83

compute nx� and ny� before the iteration, which can reduce many computation. The computing formula of nx� and ny� can be derived from (6), (9), (11) and Table I.

( 1

( 1

n n

n n

� �

� �

0 )

) ( )

nxx n m n n nx

n m n nx n

� � �

� � ��

= R 0 0 -

0 - R

n

nx0 R

n m

m

�� �� �� �� �

R 0

0, (12)

(2 1m n� � ) 2ny m n ny�0 R 0 -ny� � � �� �R . (13)

And computing of x� can be descript by a Composition-Add process, see Table II.

TABLE II COMPOSITION-ADD PROCESS FOR

x� SAMPLING

1 input xx,x P

2 Cholesky decomposition ( )choleskyxx xx

������P P

3 expand ,

4 2

x x��� 1n m� �� � �� �� �

x�

�a

1 ( ) (� �� �

)axx n n m xx n n m� � �� �� �P P 0 P 0n0 xx

4 Add x a axx� � � fx P

2) Simplification of UKF for object tracking: The simplification of UKF for orbit object tracking is

according the linear state space model (4). Because the system is a mixing of linear equation (4) and nonlinear system (7), we call this simplification algorithm: MSUKF (UKF for Mixing System). The simplification is taken by:

(A) The update of state is done by linear operation directly ˆ ˆ1) ( 1| 1)k k� � � �x Ax( |k k . (14)

(B) The � transmit is done by ( | 1)

( 1 |k k

( 1 | 1)

1) ( 1 |

x x

a a nxxx

k k k k

f k k

� �

1)

nx�

�

� � � �

� � � � �

A

A P

)

� �ai

�

+

Ax. (15)

(C) Note that , and from (14), (15) we have

ˆ ˆ( 1 | 1) ( 1 | 1k k k k� � � � �x x

� �

� ��

1

�

ˆ( | ) 1)

( 1

ˆ1) ( 1|

( 1

ˆ1) ( 1 )

| 1)

xi

ai

a nxxx i

iai

a nxxx i i

i

a nxxx i

k kk k

f k k k

k k

f k k k

k

�

�

�

�

� � �

� � �

� � � � � � � �

� � �

� � � � � � � �

� � � � �

xA x

A P A x

A x

A P A x

A P

( |

| 1)

( 1|

| 1)

( 1|

( 1

k k

f k

�

�

�

�

�

1)

| 1a

k

k

�

�

i

From (16), we can derive the time update formula of correlation matrix of state vector x

. (16)

� �� �� � � �

0

0

0

ˆ[ ( | 1) ( | 1)( | 1)

ˆ( | 1) ( | 1)]

[ ( 1 | 1)

( 1 | 1) ]

( | 1) ( | 1)

c xLi i

xx x Ti i

c a nxL i xx i

i

a nx Tixx i

iL T

c a ai xx xx

i ii

w k k k kk k

k k k k

w f k k

f k k

w k k k k

�

�

�

�

�

�

�

� �� �

� �

� � ��

� � �

� � �

�

�

�

- x ]P

[ - x

A P ]

[ A P

P P

, (17)

where ( | 1) ( 1 | 1)a axx xxk k f k k nx�� � � � �P A P .

In the same way, we can derive the time update formula of correlation matrix between state vector x and measurement vector y, and that between measurement vector y and itself:

0

( | 1)

ˆ( | 1) ( | 1) ( | 1)]

xy

Lc a y Ti xx i i

i

k k

w k k k k k k��

� �

� � ��

P

P [ - y , (18)

0

ˆ( | 1) [ ( | 1) ( | 1)]

ˆ [ ( | 1) ( | 1)]

LP y

yy i ii

y Ti

k k w k k k k

k k k k

�

��

� � � �

� � �

�P y

y

�. (19)

Compare with the standard UKF, the proposed MSUKF reduce the computation of “add-subtract” of state vector x and measurement vector y, and thus reduce the computation complexity. According to the above work, there is the summary of MSUKF, see Table III.

TABLE III MSUKF ALGORITHM

1 Initial and Compute nx� and ny� by (12), (13) 2 loop

2.1 Cholesky decomposition of ( 1 | 1)xx k k� �P2.2 Expand, see Table II 2.3 � sampling

ˆ( 1 | 1) ( 1| 1) ( 1| 1)x a axxk k k k f k k� � � � � � � � �x P

2.4 Time update Compute (14), (16), (17) (18) and (19), and:

� �ˆ( | 1) ( | 1) ( | 1)x ai x

ik k k k k k� � � � � �x P x

ny

( | 1) [ ( | 1), ( ), ( )]y xi ik k k k k k� � �� � �h u

0

ˆ ( | 1) ( | 1)L

m yi i

ik k w k k�

�

� � ��y

2.5 Measurement update 1( ) ( | 1) ( | 1)xy yyk k k k k�� � �G P P

ˆ ˆ ˆ( | ) ( | 1) ( )[ ( ) ( | 1)]k k k k k k k k� � � � �x x G y y

( | ) ( | 1) ( ) ( | 1)Txx xx xyk k k k k k k� � � �P P G P

End loop

IV. EXPERIMENTS AND ANALYSIS We use (20), (21) and (22) to simulate the trace of an orbit

object, and apply UKF and MSUKF to the tracking of the orbit object.

84

TABLE IV MSE OF UKF AND MSUKF ( ) (1 sin ) ( )4 VV t t n t�� � � , (20)

( ) (1 sin 2 ) ( )4 HH t t�� � � n t , (21)

( ) (1 cos ) ( )6 RR t t n t�� � � . (22)

where, t is time variable, and 3[0, ]14t � , sampling rate is

set by 800dt �� ; ,( )V t ( )H t

~ (0,N

and are radial distance,

azimuth angle and elevation angle respectively; , and are noise in radial distance, azimuth angle and

elevation angle respectively, suppose that , and .

( )R t

)

( )Vn t

0,0.01)

( )Hn t

( ) ~Vn t

( )Rn t

(0,0.01N ) ( )Hn t 0.01 ( ) ~ (Rn t N

Items Methods

radial distance R

azimuth angle V

elevation angle H

UKF 0.0149 0.0049 0.0049 MSUKF 0.0149 0.0049 0.0049 Suppose that the dimension of the state vector is n, and the

dimension of measurement vector is m, and n>m. we can derive the computational complexity of UKF and that of the MSUKF [4, 5], see Table V.

TABLE V COMPUTATIONAL COMPLEXITY OF UKF AND MSUKF

Methods Computational ComplexityUKF 3(30 )O n MSUKF 3(13 )O n

There are learning curves of UKF filter and MSUKF filter after 100 times of Monto Carlo simulation, see Fig 1.

0 20 40 60 80 100 120 140 160

10-5

100

Iteration

MS

E

UKFMSUKF

The amount of calculation of MSUKF is only 43.33% of that of UKF. It is obvious that the proposed MSUKF can reach the same calculation accuracy with considerable lower computational complexity.

V. CONCLUSIONS In this paper, we develop a simplified UKF algorithm:

MSUKF, in which the � sampling in unscented transform can be simplified as a composition-add process; the nonlinear transmit of � sampling, state vector, measurement vector and their correlation matrix are simplified. The proposed MSUKF algorithms can reach the same calculation accuracy as the UKF with considerable lower computational complexity, which means that the MSUKF can find wide range application in real time engineering.

(a) learning curve in radial distance

0 20 40 60 80 100 120 140 160

10-5

100

Iteration

MS

E

UKFMSUKF

ACKNOWLEDGMENT

This work was supported by the Science-Technology Project of Education Bureau of Fujian Province, China (Grant No. JA09040) and the Natural Science Foundation of Fujian Province, China (Grant No. 2009J05141).

(b) learning curve in azimuth angle

0 20 40 60 80 100 120 140 160

10-5

100

Iteration

MS

E

UKFMSUKF

REFERENCES [1] Cai Rongtai, Wang Yanjie. Predictive visual tracking based on least

absolute deviation estimation [J]. Chinese Optics Letters, 2008, 6(1):35-37

[2] C. K. Chui, Guanrong Chen. Kalman Filtering: With Real-Time Applications [M].Springer, 2008.

(c) learning curve in elevation angle [3] S. Jean Marc Valin, Francois Michaud, and Jean Rouat. Robust localization and tracking of simultaneous moving sound sources using beamforming and particle filtering [J].Robotics and Autonomous Systems, 2007, 55(3):216-228.

Fig. 1 Learning curve of UKF and MSUKF filter

From Fig 1, it is obvious that the proposed MSUKF filter reaches the convergence point at about the same time of the UKF filter; however, the standard UKF filter curve oscillates larger than the proposed MSUKF. The reason is that the proposed filter has less computation than that of the standard UKF filter, which reduces errors in the computational process.

[4] Cai Rongtai. Application of nonlinear adaptive filter in TV tracking [D]. Doctoral dissertation of Chinese Academy of Science, 2008, 6. (in Chinese)

[5] Simon Haykin. Adaptive filter theory, fourth edition [M].Prentice Hall, 2002

The average MSE of these two filters is listed in Table IV.

From Table IV, we can see that the Calculation accuracy of MSUKF is equal to that of the standard UKF.

85