Maneuvering target track prediction model

International Journal on Cybernetics & Informatics (IJCI) Vol. 4, No. 1, February 2015
DOI: 10.5121/ijci.2015.4104 41
MANEUVERING TARGET TRACK PREDICTION MODEL
Qingping Yu
1
, Xiaoming You
2
and Sheng Liu
3
1
College of Electronic and Electrical Engineering, Shanghai University of Engineering
Science, Shanghai 201620, China
2
School of Management, Shanghai University of Engineering Science
Shanghai, 201620, China
ABSTRACT
The issues about maneuvering target track prediction were discussed in this paper. Firstly, using Kalman
filter which based on current statistical model describes the state of maneuvering target motion, thereby
analyzing time range of the target maneuvering occurred. Then, predict the target trajectory in real time by
the improved gray prediction model. Finally, residual test and posterior variance test model accuracy,
model accuracy is accurate.
KEYWORDS
Current Statistical Model, Maneuvering Target, Kalman Filter, Gray Theory
1.THE STATE DESCRIPTION OF MANEUVERING TARGET
To build model of maneuvering target motion by current statistical model, to use Kalman filter
process tracking.
1.1 To build current statistical model for maneuvering target
Current statistical model is a nonzero mean time-dependent model. Current acceleration
probability density is described by the modified Rayleigh distribution. It is assumed that the target
acceleration (t)a satisfies the following relationship:
1(t) (t) (t)a a a= + （1）
'
1 1(t) (t) (t)a aα ω= − + （2）
In the above formula: a is the mean acceleration, and it is an estimate (a constant in each
sampling period) about previous time acceleration; 1(t)a is a first order Markov process of zero
mean, α is the reciprocal of the maneuvering time constant; (t)ω is zero mean Gaussian white
noise.
The discrete state equation of current statistical model is:

42
(k 1) F(k)X(k) U(k) (k 1) W(k)X a+ = + + + （3）
( )X k is state vector, ( )F k is state transition matrix, U( )k is Disturbance transfer matrix，
W( )k is zero mean white noise with variance ( )Q k ，radar scan for 1s, so the sampling period
T =1，takeα =1, the following expression:
' ''
(k) (k) (k) (k)X x x x =   （4）
2
1 ( 1 ) /
( ) 0 1 (1 ) /
0 0
T
T
T
T T e
F k e
e
α
α
α
α α
α
−
−
−
 − + +
 
= − 
 
  （5）
2
( / 2 1 (e ) / ) /
U( ) (1 e ) /
1 e
T
T
T
T T
k T
α
α
α
α α α
α
−
−
−
 − + + −
 
= − − 
 −  （6）
In the above formula, T is the sampling period.
Variance adaptive algorithm based on current statistical model, that is, the following equation
adaptive variance:
11 12 13
2
12 22 23
13 23 33
Q(k) E[W(k)W (k)] 2T
a
q q q
q q q
q q q
ασ
 
 = =  
   (7)
The variables are as follows:
2 3 3 2 2 5
11 [1 2 T 2 T / 3 2 T 4 ]/ 2T T
q e Teα α
α α α α α− −
= − + + − − (8)
2 2 2 4
12 [1 e 2 2 Te 2 T T ]/ 2T T T
q eα α α
α α α α− − −
= + − + − + (9)
2 3
13 [1 e 2 Te ]/ 2T T
q α α
α α− −
= − −
(10)
2 3
22 [4 3 e 2 T]/ 2T T
q e α α
α α− −
= − − + (11)
2 2
23 [ 1 2e ]/ 2T T
q e α α
α− −
= + − (12)
2
33 [1 e ]/ 2T
q α
α−
= −
(13)

43
2
max
2
2
max
4
[a (k)] (k) 0
4
[a (k)] (k) 0
a
a a
a a
π
πσ
π
π
−
−
− >
= 
− − <
 (14)
maxa
is the maximum positive acceleration， maxa− is the maximum negative acceleration.
Measurement equation：
(k) H(k)X(k) V(k)Z = + (15)
In the above formula，H(k) is measurement matrix, V(k) is white noise that it's zero mean and
variance is (k)R 。
1.2. Kalman filter based on current statistical model
Kalman filter based on current statistical model smooth the state of the target on the past and
present time, while predicting the target movement in the future time, including the location of
the target, velocity and acceleration parameters.
State variable method is a valuable method to describe a dynamic system, using this methods, the
system input-output relationship is described in the time domain by the state transition model and
output observation model. Input can be determined by dynamic model consisted of a function of
time and unpredictable variables or random noise process to describe. Output is a function of the
state, which often disturbed by random observation errors, can be described by measurement
equation.
Dynamic equations of discrete-time systems (state equations) can be expressed as：
(k) (K)X(k 1) (k) (k) W(k)X F a U= − + + (16)
The measurement equation of discrete-time systems：
Z(k) (k)X(k) (k)H V= + (17)
(k)H is the measurement matrix， (k)V is zero mean Gaussian white noise sequence. Different
time measurement noise is independent
.
The formulas of Kalman filter algorithm based on the current statistical model are：
(k | k 1) (k)X(k 1| k 1) (k) (k)X F a B− = − − + (18)
P(k | k 1) (k)P(k 1| k 1)F (k)T
F Q− = − − + (19)

44
X(k | k) (k | k 1) (k)[Z(k) (k)X(k | k 1)]gX K H= − + − −
(20)
1
(k) P(k | k 1)H (k)[(H(k)P(k | k 1)H (k) (k)]T T
gK R −
= − − +
(21)
P(k | k) [I K (k)H(k)]P(k | k 1)g= − −
(22)
Wherein, (k) ~ N(0, )W Q , V(k) ~ N(0,R) .
Kalman filter does not require saving measurement data in the past, when new data measured,
according to new data and various valuation in previous time, by means of state transition
equation of the system, according to the above recurrence formula, thereby calculating the
amount of all the new valuation.
2. TRACK PREDICTION BASED ON THE GRAY THEORY
To solve track forecast problem of aerial target, an aerial targets track prediction method based on
gray theory is proposed in this paper. Grey system theory is based on new understanding about
objective system. Although insufficient information on some systems, but as a system must have
specific functions and order, but its inherent laws are not fully exposed. Some random amount, no
rules interference component and chaotic data columns, from gray system point of view, is not
considered to be elusive. Conversely, in gray system theory, a random amount is regard as a gray
amount within a certain range, according to the proper way to deal with the raw data, grey
number is converted to generation number, and then get strong regularity generation functions
from generation number.
To achieve a real-time online track prediction and improve the prediction accuracy, an improved
GM forecasting model is established. In the case of limited data, the model is used to predict the
track effectively. The basic steps of aerial target track grey prediction: 1) the raw data
accumulated generating 1; 2) the establishment of GM (1, 1) prediction model; 3) to test the
model; 4) establishment of a residual model; 5) target track forecast.
2.1. Target track prediction
The current track graphics, coordinate information is known.
Figure 1. The current track data

45
Provided the original data sequence：
(0) (0) (0) (0) (0) (0) (0) (0)
(x (1),x (2),...,x (n)), (y (1), y (2),..., y (n))X Y= = ,
(0) (0) (0) (0)
(z (1),z (2),...,z (n))Z = ,
(0) (0) (0)
(x (i), y (i),z (i)) is the coordinates of the target
(Figure1 data) in rectangular coordinate system with the center of earth.
(0)
x (i) 0,i 1,2,...m> =
， (0)
(i) 0,i 1,2,...my > = ， (0)
(i) 0,i 1,2,...mz > = , using the data sequence, general
procedures to establish GM (1,1) model is：
The first step: to do the first order accumulated generating of the original data series
(0)
X (i.e., 1-
AGO), get accumulated generating sequence:
(1) (1) (1) (1)
(x (1),x (2),...,x (n))X = (23)
Wherein,
(1) (0) (1) (0)
1
x (1) x (1),x (k) x (i),(k 2,3,...,n)
k
i=
= = =∑
，
The second step: the first order accumulated generating sequence
(1)
X established GM (1, l)
model, getting related albino differential equations:
(1)
(1)(t)
(t) b
dx
ax
dt
+ =
（24）
Where, a is the development factor, reflecting the development trend between
$(1)
x and
$(0)
x . B is
gray action，the gray action of GM (1,1) model is mined from the background data, it reflects the
relationship to data changes, so the exact meaning is gray.
The related gray differential equation form as:
(0) (1)
(k) (k) b,k 2,3,...x ad+ = = （25）
The third step: Solving the parameters a, b. Parameter sequence
(1)
(1)
(1)
(2) 1
(3) 1
(n) 1
d
d
B
d
 −
 
− =
 
 
− 
M M
can be
determined by the least squares method： 1
[B ,B]T T
B A−
Φ = ,

46
(1)
(1)
(1)
(2) 1
(3) 1
(n) 1
d
d
B
d
 −
 
− =
 
 
− 
M M
，
(1) (1) (1)1
(k) [x (k) x (k 1)]
2
d = + −
,
(0) (0) (0)
(x (2),x (3),...,x (n))T
A = 。
The fourth step: In the initial conditions
$(1)
(1) (0)
x (1) x (1) x (1)= = ，Generate data series models
are available:
$
$
$
$
(1)
(0) (k 1)
x (k) (x (1) ) (k 2,3,...,n)ab b
e
a a
− −
= − + =
$ $
（26）
The fifth step: In the initial conditions
$(1)
(1) (0)
x (1) x (1) x (1)= = ，Generate data series models
are available:
$ $ $
$
$(0) (0)
(0) (0) (k 1)
x (1) x (1),x (k) (1 e )(x (1) )e ,k 2,3,...,na ab
a
− −
= = − − =
$
（27）
Namely,
$(0)
(0)
x (1) x (1)= ，
$ $
$
$(0)
(0) (k 1)
x (k) (1 e )(x (1) )ea ab
a
− −
= − −
$
， 2,3,...,nk =
So, 2,3,...,nk = is substituted into the formula, getting the fitted values of the initial data;
when k n> , to get the gray model predictive value for the future.
The sixth step:
(0) (0) (0) (0)
(y (1), y (2),..., y (n))Y = ， (0) (0) (0) (0)
(z (1),z (2),...,z (n))Z = repeat
steps one to step five。
2.2. Model accuracy test
Gray model prediction test are generally residual test and posterior variance test.
First, the relative size of the error test method, it is a straightforward arithmetic test methods
comparing point by point. In this method, prediction data is compared with the actual data,
observing the relative error whether satisfies the practical requirements.
The actual data set is used for modeling:
(0) (0) (0) (0) (0)
(x (1),x (2),x (3),...,x (n))X = ，according to GM (1,1) modeling method to obtain
$ $ $ $(1) (1) (1) (1) (1)
(x (1),x (2),x (3),...,x (n))X = ,
(1)
X do a regressive generate transformed into
(0)
X
，Model value of actual data： $ $ $ $(0) (0) (0) (0) (0)
(x (1),x (2),x (3),...,x (n))X = 。
Computing residuals, residual sequence is:

47
(0)
(0)
( (1),e(2),...,e(n)) X XE e= = − （28）
where， $(0)
(0)
(i) x (i) x (i)e = − ， 1,2,...,ni =
Calculate the relative error, the relative error is：
$
(0) (0)
(i) (i) (i)
(i) 100% 100%
(i) (i)
e x x
x x
ε
−
= × = ×
（29）
(0)
(i)
(i) 100%
(i)
e
x
ε = ×
is the origin of the error， 1
1
(i) (i)
m
im
ε ε
=
= ∑
is the average relative error
of GM（1,1）model; ( (k) ) 100%o
p ε ε= − × is the model accuracy of GM（1,1）model，i.e.,
minimum error probability, and it generally requires 80%o
p > , best 90%o
p > .
Second, posterior deviation test belongs to the statistical concept, it is according to the probability
distribution of residual test.
If the actual model data used was:
(0) (0) (0) (0)
(x (1),x (2),...,x (n))X = ，getting model value of
actual data with the GM (l, l) modeling method:
$ $ $ $(1) (1) (1) (1) (1)
(x (1),x (2),x (3),...,x (n))X = ,
Assuming variances of the actual data sequence
(0)
X and residuals sequence E separately
are
2
1S and
2
2S ,
(0)2 (0)
1
1
1
(x (i) x )
n
i
S
n =
= −∑
，and
(0) (0)
1
1
(i)
n
i
x x
n =
= ∑
2
2
2
1
1
(e(i) )
n
i
S e
n =
= −∑
，and 1
1
(i)
n
i
e e
n =
= ∑
calculating Posterior variance ratio：
2 1C S S=
(30)
Determine the model level, indicators such as Table 1:
Table 1. Model accuracy class
Model accuracy class Small Error probability P Posterior inferior C
I >0.95 <0.35
II >0.8 <0.5
III >0.7 ,0.65
IV ≤ 0.7 ≥0.65

48
Level Description: C values as small as possible, i.e., 1S
is much smaller than the 0S
.raw date
discrete is large, prediction error discrete is small, and prediction accuracy is high; P bigger the
better, equally error probability is small, fitting accuracy high. If the residual test, posterior
variance test can be passed, it can be predicted by their model, otherwise corrected residuals.
Table 2. Target prediction location
Target prediction Small error probability Posterior variance ratio Model accuracy
X 1 0.188 I
Y 1 0.107 I
Z 1 0.115 I
2.3. Track forecast results and analysis
The radius, velocity and range angle β of observation point D are known, the plane parallel
to the direction of velocity and passed point D can be approximated as missile flight plane, plane
2Q
that through geocentric O ，point D and perpendicular to the direction of the bullet speed,
plane 3Q
that through geocentric O ，placement B and perpendicular to the direction of the
bullet speed. Angle between plane 2Q
and plane 3Q
is the range angle β . Release impact
coordinate, shown in FIG 1.
O
β
B
v
1Q
2Q 3Q
D
Figure 2. Solving placement schematic
If plane 1Q is 1 1 1 0A x B y C z+ + = ， normal vector is 1 1 1 1n Ai B j C k= + + ， and
1 0 0 0OD
x y z
i j k
n r v x y z
v v v
 
 
= × =  
 
  ， , ,i j k are unit vector for each axis, the flat 1Q
can be solved by
the equation.

49
If plane 2Q is 2 2 2 0A x B y C z+ + = ， normal vector is 2 2 2 2n A i B j C k= + + ， and
2 1 0 0 0
1 1 1
OD
i j k
n r n x y z
A B C
 
 = × =  
  ,the flat 2Q
can be solved by the equation.
If plane 3Q
is 3 3 3 0A x B y C z+ + =
,normal vector is 3 3 3 3n A i B j C k= + +
， and
2 3 2 3
3 1
3
cos
0
1
n n n n
n n
n
β =

=
 = ，the flat 3Q can be solved by the equation, intersection of the plane 1Q ,
the plane 3Q
and the earth's surface is the placement of B,
( ), ,B x y z=
， and
1 1 1
3 3 3
2 2 2 2
0
0
A x B y C z
A x B y C z
x y z R
+ + =

+ + =
 + + =
Placement coordinates can be obtained by solve formulas, after forecast, Placement coordinates is
（-2423400，-2103400，,5488000），error radius 1000m，algorithm complexity
( )0 n
.
Figure 3. Forecasting model predicted target track
REFERENCES
[1] Li Feng, Jin Hongbin, Ma Jianchao. A New Method of Coordinate Transformation Under Multi-
Radar Data Processing System [J].Control & Measurement, 2007,23(4):303-305.
[2] Sun Fuming. Research on State Estimation and Data Association of Motion Targets [D]. University of
Science and Technology of China,2007.

50
[3] Song Yingchun. Research on Kalman Filter in Kinematic Positioning [D].Central South University,
2006.
[4] Liu Na. Study on Data Association Method in Multitarget Tracking of Ballistic Missile Denfense
Radar [D]. National University of Defense Technology, 2007.
[5] Feng Yang .The Research on Data Association in Multi-target Tracking [D].XiDian University,2008.
[6] Zhou Hongren. Maneuvering target "current" statistical model and adaptive tracking algorithm [J].
Journal of Aeronautics,1983,01:73-86
[7] Yao Jinjie. Research on Techniques of Target Localization based Stations[D].North University of
China,2011.
[8] Liu Gang. Multi target tracking algorithm research and Realization [D]. Northwestern Polytechnical
University,2003.
Authors
Qing-ping Yu is currently studying in Mechanical and Electronic Engineering from Shanghai
University of Engineering Science, China, where she is working toward the Master degree. Her
current research interests include ant colony algorithm, their design and develop in Embedded
system.
Xiao-Ming You received her M.S. degree in computer science from Wuhan University in 1993,
and her Ph.D. degree in computer science from East China University of Science and
Technology in 2007. Her research interests include swarm intelligent systems, distributed
parallel processing and evolutionary computing. She now works in Shangha i University of
Engineering Science as a professor.

Maneuvering target track prediction model

More Related Content

What's hot (20)

Viewers also liked (19)

Similar to Maneuvering target track prediction model (20)

Recently uploaded (20)

Maneuvering target track prediction model