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IEEE Int. Conf. Neural Networks & Signal Processing

Nanjing, China, December

14-17, 2003

A NOVEL MOTION MODEL AND TRACKING ALGORITHM

Dang, Jianwu Huang, Jianguo

College

of

Marine Engineering,Northwestem

Polytechnical

University

Xi'an, Shaanxi, 710072, China

ABSTRACT

A novel motion model and adaptive algorithm for tracking

maneuvering target are proposed, in which the

acceleration of maneuvering targets is considered

as

a

time-correlation random process with non-zero mean

values and the probability density of the acceleration is

assumed Gaussian diseibution. The mean value of the

distribution function is the optimal estimation of the target

acceleration at present and its variance is directly

proportional to the square of the differential c oefficient of

the optimal estimations of the target acceleration at present.

The Monte Carlo simulation results show that the model

and adaptive algorithm proposed in this paper can estimate

the position, velocity and acceleration of a target well and

requires less computation than the others, no matter what

the target is maneuvering at any form .

1. INTRODUCTION

During the last three decades, the model of maneuvering

targets

has been studied and som e useful results have been

achieved.

In

1970, R. A . Singer [ I ] put forward a

time-correlation model of maneuvering targets with zero

mean values, i.e. Singer model, in which the acceleration

of m aneuvering targets was considered as results force d by

random noise. Based on the above, the statistic model of

maneuvering targets w as given. In

1979,

R.

L.

Moose, etc.

[2]

brought forward the Semi-Markov statistic model of

maneuvering targets with random switch-mean, which was

more advanced than Singer model.

In 1983,

R. F. Berg

[3]

suggested revised Singer model, which introduced the

differential coefficient of the mean values of targets

acceleration to Singer model. In

1984,

Dr Zhou H.

R. [4]

given the Current statistic model of maneuvering target

based on the Singer model,

in

which the non-zero mean

values of acceleration was inducted and the acceleration

of

maneuvering target was thought the revised Raly-Markov

process. In

1984,

H. A.

P.

Blom

[SI

proposed the

interacting multiple model (IM M) of maneuv ering targets

that is based on the single model of maneuvering motion.

Thereafter Blom, Barshlom, etc

[ 6 ] , [7], [IO],

and [ I l l

developed and completed the IMM tracking algorithm in

theory. In

1989,

B a n g B oya n [ 8 ] developed the Truncated

Normal model of maneuvering targets, which differed

from the Current model in that probability density of

targets acceleration was assumed truncated normal

distribution.

In this paper, only single model of .maneuvering

target is discussed and a novel model, i.e. Adaptive

Gaussian model (AGM) was presented, in which the

acceleration of a unknown target was considered

as

a

Gauss-Markov random process with non-zero mean values

and its probability density was assumed normal

distribution. The mean values of the normal distribution

was considered

as

the optimal estimation

of

target

acceleration at present and its variance is proportional to

the square of the differential coefficient of the optimal

estimations of the acc eleration at present.

2.

ADAPTIVE GAUSSIANMODEL

2.1. First order t imecorre la t ion

model

with non-zero

mean

values

When a target maneuve rs, its acceleration is not always a

stationary random process. Assuming the acceleration

following a fm t order time-correlation random process

with non-zero mean values, i.e.:

1)

wherex(t) is the position of the target at t,

2( t )

s the

acceleration,

i

s

the mean value of acceleration and

~ ( r )s the colored acceleration noise with zero-mean

values. Letting the time-correlation function of a(t) be

the form with exponential attenuation [I]

R ~ ( . ) = E [ a ( f ) a ( f + r ) ] = u ~ e - " " ' a>O)

2)

where is the variance of acceleration and

a

s the

frequency of targets maneuver, i.e. the reciprocal of the

maneuvering time constant.

Showing colored noise on Eq. (1) and

2) as

the

following result

driven

by w hite noise:

i t )= E

+

a(t)

where

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4)

( 5 )

1

s + a

H ( s )

=

o s ) = L[R, r )J = ao:

here

&(q

means the correlation function of white noise

W t ) '

Based on the above, fust order time-correlation

model of maneuvering acceleration with zero mean values

is

i ( t )

=

-ua(t) w ( t )

where the variance of input white noise is

According to Eq. (1) and 6), fmt order time-correlation

model of mane uvering acceleration with non-zero mean is

8)

6)

a: = 2 a a : 7)

X ( t )

=?it )

a ( t ) = - a a ( f )+

w t )

2.2. Adaptive Gaussian model

If prior knowledge about the statistical characters

of

maneuvering targets acceleration can't be obtained, the

probability density of first order time-correlation model of

maneuvering acceleration with non-zero mean described

in Eq.

8)

may be regarded as the Gaussian distribution

with mean value ind variance , i.e.

X ( f ) -

N(z,a:) a( t )

- N ( O , ( T ) and

~ ( t )

N ( 0 , 2 a u : ) .

In terms

of

the characteristic

of

Gaussian distribution,

the

PDF of

maneuvering acceleration

X ( t )

is

In view of the estimating theory, letting the mean

value

of

maneuvering target acceleration equal to the

condition expectation of the maneuvering acceleration as

the observation values of a system is

z ( t ) ,

i.e. the

optimal estimation of state variable x ( t ), nd setting

i t )

/a, = b i.e.

B t)

b =

I O )

where

i t )

is the differential coefficient of the optimal

estimation of maneuvering acceleration : I ) ,

b

is the

variance adaptive coefficient to be a constant.

Adaptive Gaussian model of maneuvering target

acceleration is expressed completely in Eq S),

9)

and

10).

3. ADAPTIVE

TRACKING

AL GOR IT HM

To

simplify the illumination in this paper, only one

dimensional tracking in a rec tangular coordinate is studied.

The other two dimensional cases are similar.

In view of Eq.

S),

the state equation of a target

tracking system is

k t)= X ( t ) + B ; i + C w ( t ) 11)

where

x ( r )

,

q r , q r ) and

a

is respectively the position,

velocity and acceleration of the maneuvering target at t

Letting T be sampling period, the discretized-time

state equation of the continuous-time system in E q . (11)

can be obtained as following:

( 1 2 )

where

( 1 3 )

and W ( t )-

N (O ,Za

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(24)

11

421 411

Q ( k )

=

E [ w ( k ) w r ( k ) l= 412 4 3 2 .2ao :

[ , 23

ql = I

-e?

W

dT3/3-2aiT2

-@e?)/(&)

(25)

q ?

=(e-*

1 - 2 8

+ W e - - 2 a ~ + a ' ~ ~ ) / ( ~ a ' )

26)

28)

ql , = I

e-*=

-2aTe- T)/(2a')

(27)

q2*

= ( 4 ~ ' -2a' +2 d 7 / ( 2 2 )

q21=(e-* + I

- 2e- )/(2a2)

(29)

q33= I -e-' ')/(2a2) 30)

Letting n= k ) and B

=

( g ( k / k )

(k

-

/ k

- ) ) / T

According to Eq. (12),

(14), (15)

and (20), the following

equation can be achieve d

where

k ( k + I / k )= I D , ( k +

I , k ) 2 ( k , k ) U , ( k ) t

3 1 )

T 71/2 (-~+ar'/2+(1-8)/a)/a

, q k + L k ) = O

o

I T

1

T-;W )/a 1 3 2 )

When using

Eq.

(22) to calculate

P ( k + l / k ) ,

let

:

= ( ( a ( k / k ) - a ( k -

l / k - l ) ) / T ) 2 / b 2

33)

The tracking algorithm of the system shown in Eq.

(12) and 17) based on AGM can be obtained from Eq. (19)

to Eq.

33).

4.

MONTE CARLO SIMULATION

In order to evaluate the AGM of the maneuvering target

presented in this paper, some Monte Carlo simulation are

conducted on the condition of typical underwater targets

maneuver. The values of the parameters and the initial

values

of

maneuvering target in Monte Carlo simulation

are as follows:

( I ) . When a target maneuvers at a constant speed

~ ( 0 ) 500 m, y (0 )

=

0 , k 0)=

0,

v r = 20mis, y y = 0,

Y r = ~ , w = ~ 0 m / s 2 ,a=0.1, ~ = t .

(2). When a target maneuvers at a constant acceleration

~ ( 0 )

500

m, y 0) = 0 , k 0)= 0, v

=

IOmis, vy

=

0,

V , = O , ~ , = I O ~ / S ~ , = o . I , = I , a ,

= 5m/s2.

(3). When

a target maneuvers in snaking

a = 0.1,

T =

1.

The amplitude values

of

target

maneuvering is 5001x1 and i ts radian 6equ en cy is 2 n 1200.

In Monte Carlo simulation, when a target maneuvers

in the difference forms, the Singer model in [ I ] , the

Current Statistic model in [4], the Truncated Normal

Probability Density model

in

[9] and AGM in this paper

were respectively simulated by sixty runs. Because of the

limitation of paper pages and the similarity of the

simulation results, only typical figures of the simulation

results compared with the Current Statistic model

~ 0 ) = 5 0 0 m, y(0) = O , h ~ ) = ,

a,

= 10mis2,

presented in

[4]

are show n in this paper.

mean square error and ME mea ns the mean error.

In the Fig.1, 2, and 3,

RMSE means the mo t

Fig.1.A target maneuvers in snaking

-

0

f ,

a

Fig.2.

A

target maneuvers at a constant acceleration

nrbnugapndpallm

Fig.3.

Atarget maneuvers at

a

constant velocity

The conclusions can be found from the Fi g. l,2 and

3

that no matter what a target maneuvers

in

any of three

forms, the AGM has better performance of adaptive

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hacking and

its

estimate errors on the position, velocity,

and acceleration is less th m that

of

the other models.

5.

CONCOLUSION

In thi s paper, a no vel target motion model and its adaptive

hacking algorithm was given, and a large number of

Monte Carlo simulation experiments were made. The

theory analyzing and simulation results show that the

AGM presented in this paper characterizes exactly the

maneuver of target and the physical meaning

of

the mean

and

the variance of the Gaussian distribution. When any

prior knowledge about the characteristic of maneuvering

target can't be obtained, The

AGM

has better performance

of adaptive hack and less tracking error than that

of

the

other models.

REFERENCES

[ I ]

R. A

Singer, Estimating Optimal Tracking Filter

Performance for Manned Maneuvering Targets,

IEEE

Transactions on Aerospnce and Elecmnic Systems, 1970,

6 4): 473--483.

R.

L. Moose, etc., Modeling and Estimation for Tracking

Maneuvering Targets,

IEEE Transactions

on

Aerospace

Elec mn Systems, 1979, 15 3), 448--456.

R.

F. Berg, Estimation and Prediction for Maneuvering

T w e t Trajectories,

IEEE Transactions

on

Automatic

[Z]

[3]

r41

[71

C o i m l , 1983,28 3)

Zhou Honmen. Kumar

K

S P. A current statistical model

I

and adaptive algorithm

for

estimating maneuvering

targets.

AIAA

J d n

Gidawz, C 1 and

@f b,

984,7 5):

5 w .

H.

A. P.

Blom, An effrcient filter for abruptly changing

systems,

Proc IEEE CDC, 984,

pp656.

H.

A. P.

Blom and

Y.

Bar-Shalom, The IMM algorithm for

system with Markovian switch ing coefficien ts, IEEE

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Y. Bar-Shalom, K. C. Chang and

H.

A. P. Blom, Tracking

a

maneuvering target using input estimation . versus

interacting multiple model algorithms,

IEEE Transactions

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T.

L.

Song, Suboptimal Filter Design with

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Zhang boyan, Cai qingyu

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Bar-shalom, Precision

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B.Chen and J.K.Tugnait, Interacting Multiple Model

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