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Transcript of A New Estimation Method for Burnout Point Parameters of Ballistic Missile Based on Closest Distance...
A New Estimation Method for Burnout Point
Parameters of Ballistic Missile Based on Closest
Distance Method
P. Ni, Y. L. Liu, and Z. H. He School of Air and Missile Defense, Air Force Engineering University Xi’an 710051, Shaanxi, China
Email: [email protected] {liuyonglanco1.hi, fuqiang66688}@163.com
Abstract—The estimation for burnout point parameters of
ballistic missile has not been solved systematically. Under
the condition without prior information, the paper starts from the change of motion rules caused by the changes of
force before and after burnout, and uses the combination of
track time cross-extrapolation and the closest distance
search to estimate burnout point parameters of target.
results show that the closest distance search method is
evidently better than the ordinary methods.
Index Terms—Ballistic Missile; Burnout Point; Closest
Distance; Range Query
I. INTRODUCTION
The ballistic missile has the characteristics of long range, high speed of reentry, high-accuracy and its small
RCS. Therefore, the probability of detecting the ballistic
missile is small so that it is difficult to intercept the
ballistic missile. With the spread of ballistic missile technology, countries around the world pay more and
more attention to the development of ballistic missile
defense technology. As the “eyes” of and missile defense technology, sensors, the validity and reliability of the
early warning is the premise and foundation of the anti-
missile combat. In the space early warning system, early
warning satellite detects ballistic missile trails by its infrared detector, in order to get the target infrared
radiation intensity and angle measurement information.
And then the whole ballistic parameters and impact point
of the missile can be estimated. But after burnout time, the motions of target rely on the inertia so that the
infrared detector can't continue to detect and track it. So,
the subsequent estimations of ballistic parameters and impact point depend entirely on the burnout point
parameters. In other words, the accurately and rapidly
estimation for burnout point parameters of ballistic
missile is the key step of the anti-missile operational planning and improvement of intercept probability.
At present, there has been no systematic dissertation on
estimating burnout point parameters of ballistic missile.
In Reference [1], the estimation on burnout point parameters is based on prior trajectory profile, which
means to use the maximum burnout time of prior, the last
time of sensor observing target, and the next possible observation time to estimate burnout time. Limited by the
objective conditions, the method has greater estimation
error in estimating burnout time. In Reference [2], the importance of burnout time is emphasized, but the last
observation time and half of the sampling time are
directly used as burnout time. Therefore, it has easy
processing method, while the error is difficult to be measured. In Reference [3], CA-EKF filtering algorithm
is used for tracking targets in powered phase, and CV-
EKF filtering algorithm is used in free phase. And interactive multi-model is used for tracking algorithm
alternation, and the probability update of the model is
used to achieve the estimation on self-adapting state of
targets. It avoids the solution of burnout points. In Reference [4], under the assumption that the burnout
point parameters is known, a mathematic model of impact
point estimation based on the state of burnout point is built up. But there is no analysis about estimation of
burnout point parameters. In Reference [5], based on the
detection information of early warning satellite, the
theory of double satellites location and ballistic trajectory and the transformation model is proved from
ECF_VVLH coordinate to ECF coordinate. And then the
tactical parameters evaluation of ballistic missile burnout
point with early warning satellite is given. It is based on the location information of powered phase to estimate the
burnout point parameters. In Reference [6], the burnout
time is determined by the last time when satellite detects target, sampling interval of the satellite and the priori
maximum burnout time in database. In conclusion, at this
stage, most of the estimation method for burnout point
parameters of ballistic missile is according to warning satellite information to extrapolate estimate.
For the articles about prediction of ballistic missile
[7~16], most articles suppose that burnout time and
burnout point parameters have been known, for making the subsequent process. In Reference [7], it is consider
that the trajectory prediction error estimation model is
determined by burnout point position error and radius growth rate of the guide area. It can be used to determine
the search projection area of the 3-D volume boundary.
This method reduces the burden of the radar in terms of
scanning and tracking, and relative increase the effectiveness of radar. In Reference [9], the modeling of
interception window is based on the ellipse trajectory
which is determined by the burnout point parameters. In
2504 JOURNAL OF NETWORKS, VOL. 9, NO. 9, SEPTEMBER 2014
© 2014 ACADEMY PUBLISHERdoi:10.4304/jnw.9.9.2504-2513
Reference [11], the analysis for burnout point effect on ballistic missile range is given. In Reference [16], a
model probability accumulation factor was defined,
which is used in estimating missile’s engine burnout time
and its corresponding state. Under the condition without prior information, this
paper starts from the motion track of ballistic missile
target, and compares the change of motion track caused by outside force of target before and after burnout to
estimate burnout point parameters. Different from the
method based on priori information adopted by the
majority of scholars, this paper is based on the theory that combining back-stepping trajectory of powered phase
with forward-stepping trajectory of free fight phase to
estimate the burnout point parameters accurately, which conforms to the realistic combat background.
II. IMPORTANCE OF BURNOUT POINT PARAMETERS
AND INFLUENCE ON IMPACT POINT
A. Importance of Burnout Point Parameters
Before estimating the burnout point, the importance of
burnout point needs to be made clear. High-orbit warning satellite only can use infrared sensor to detect the motion
state of missile in powered phase, and it can’t observe the
target after burnout. And the speed of target at burnout
time is the largest in powered phase. After that, warhead completely depends on gravity for flight, and its motion
rule follows elliptical orbit theory. Therefore, the
estimated parameters of burnout points have direct influence on orbital elements of ellipse trajectory, and the
orbital elements directly affect the estimation accuracy. It
means that the estimation of burnout time has great
influence on prediction of impact point, which can’t be avoided. Meanwhile, the state vector at burnout time
affects the establishment and accuracy of ellipse
trajectory in free phase. The motion state vectors of
burnout point include the position and speed of ballistic missile.
B. Relationship Between Burnout Point Parameters and
Impact Point
The position vector and velocity vector of burnout
point is ( , , )g g gX Y Z and ( , , )xg yg zgV V V . According to the
basic ellipse trajectory theory, we can get,
2 2 2
2 2
2
2
cos ( )
1 ( 2)cos
1
g g g
g
g g g g
g g
g
g
g
r vP
v r v re
u u
Pa
e
(1)
Further, the follow equations can be deduced:
arccos( )g g
g
g g
g g
g
g g
P rf
r e
r aE
a e
(2)
In the formula (1) and formula (2),
2 2 2
2 2 2
,
,
arccos(( ) / )
g g g g
g xg yg zg
g g xg g yg g zg g g
r X Y Z
v V V V
X V Y V Z V r v
By getting the first-order difference of three orbital
elements in formula (1), the total differential formula can be got as follows,
2 2 2 2 2 2
2 2 2
2 2
2 2 2
2
2 cos ( ) 2 cos ( ) sin(2 )
2( 1)cos ( 1)cos
1 ( 2)cos 1
g g g g g g g g g g g g
g g g g g g g
g g g
g g g
g g g g
g g g
g g g g g g g
g g
g
g g g g g
g
P P P r v r v r vP r v r v
r v
e e ee r v
r v
v v r v r v r
rv r v r v r
2 2
2 2 2
2 2
2
2 2 2
2 2
( 2)sin(2 )
( 2)cos 2 1 ( 2)cos
2 2 sin( )2
( 2) ( 2) ( 2)cos( )
g g g g
g
g g
g g g g g g g
g g
g g g g g g g
g g g g g g g
g g g g g g g g g
g
v r v r
vv r v r v r
a a a r v ra r v r v
r v v r v r v r
(3)
In the formula (3), the process of solving total
differential of ga is very complicated, and the derivation
is as follows (4). The process of solving total-differential of two orbit
elements in formula (2) is similar to that of ga . The
above total-differential equations reflect the relationship
between orbital elements and burnout point parameters, which is expressed by matrix as follows (5),
JOURNAL OF NETWORKS, VOL. 9, NO. 9, SEPTEMBER 2014 2505
© 2014 ACADEMY PUBLISHER
2 2
22 2
2 2 2
( ) ( ) ( )
2 ( 1)cos ( )2 cos ( )
(1 ) (1 )
g g g
g g g g
g g g
g g g g g g g g g g g g
g g g
g g g g g g g g g g g g
g g g
g gg g g
g
g g
a a aa r v
r v
a P a e a P a e a P a er v
P r e r P v e v P e
v v rP
r vr
e e
2
22 2
2 2 2
2 2
2 2
2 2 2
22 ( 1)cos ( )
2 cos ( )
(1 ) (1 )
( 2)sin(2 )sin(2 )
(1 ) (1 )
g g g g
g gg g g
g
g g
g g g g
g gg g g
g
g g
v r v rP
r vv
e e
v r v rP
r v
e e
2
2 2 2
2 2
2 2 sin( )2
( 2) ( 2) ( 2)cos( )
g g g g
g g g
g g g g g g
g
r v rr v
v r v r v r
(4)
g g g
g g g
g g g
gg g g
g g gg g g
g g f g
g g g
g g g
g g gg
g g g
g g g
g g g
P P P
r v
e e eP
r ve r r
a a aa v A v
r vf
f f fE
r v
E E E
r v
(5)
15 partial derivatives in formula (6) can be achieved by
estimation of numerical calculation. Through simulating
calculation for N times, it can obtain:
1
1
1
1 1
1
1
1
g gN
g gN g gN
g gN f g gN
g gN g gN
g gN
P P
e e r r
a a A v v
f f
E E
(6)
If
1
1
1
1
1
g gN
g gN
g gN
g gN
g gN
P P
e e
a a
f f
E E
G ,1
1
1g gN
g gN
g gN
r r
v v
M ,
according to least square method, we can derive,
1( )T T
f
A GM MM (7)
It is the influence matrix of burnout point parameters on orbital elements of ellipse trajectory.
Through simulating for 50 times, the relation matrix
between burnout point parameters and orbital elements
can be got as follow,
0.1450 318.7839 8.6242( 5)
1.0426( 8) 4.6513( 5) 0.1357
0.5739 170.7860 5.5901
1.4195( 8) 5.3746( 5) 2.0121( 4)
1.1996( 8) 5.3684( 5) 0.1357
f
e
e e
e e e
e e
A
From the analytical expression of fA , we can see that
speed estimation error and angle estimation error of
burnout point have great influence on orbital elements P
and a , while have little influence on , ,e f E . And the
influence of angle estimation error on P is greater than
that of speed estimation error, and the influence of error
of speed estimation on a is greater than that of angle
estimation. Position estimation has little effect on orbital elements. In short, speed and angle estimation of burnout
point has great influence on P and a , which means
having important influence on the generation of elliptical
orbit of moving target. The impact point of ballistic missile is similarly
located on elliptical orbit, so orbit elements of the ellipse
have direct influence on prediction accuracy of impact point. And the relationship among burnout point, orbital
elements and impact point is as follows.
burnout point
parameters
orbital elements of
ellipse trajectory
prediction of
impact point
Indirect influence
Direct
influenceDirect
influence
Figure 1. Relationship among burnout point, orbital elements and
impact point
2506 JOURNAL OF NETWORKS, VOL. 9, NO. 9, SEPTEMBER 2014
© 2014 ACADEMY PUBLISHER
From Figure 1, we can see that burnout point parameters have great effect on prediction of impact point.
And for predictions of trajectory and impact point, the
starting time of free phase is an important estimator. So
the following burnout point parameters need to be determined, burnout time, location vector and velocity
vector of burnout point.
III. ACCURATE ESTIMATION OF BURNOUT POINT
PARAMETERS
In this section, the models of target motion are given in
order to describe state motions of ballistic target in the
powered phase and the free phase. And on this basis, the fundamentals and algorithm flow of the closest distance
method are given.
A. The Models of Target Motion
1) Improved gravity turn model in the powered phase
Basis gravity turn model It is well known the ideal motion equation of ballistic
missile consists of 9 state variables: position vector,
velocity vector and acceleration vector. But, actually,
there are still some difficulties in calculating the complete solution of 9 state variables based on the measurements in
angle given by the sensors. Therefore, to the practical
situations, some reasonable assumptions would be used
and the constraints of the motion equation would also be designed in order to acquire more concise and more
accurate model.
Constraint 1: The motion trajectory of target approximately keeps in an orbital plane.
Constraint 2: The thrust, gravity and aerodynamic drag
in the forces acting on the target would be major
concerned .And other outside forces, which do not much affect the target, can be ignored, for their value is very
small comparing with the three outside forces (thrust,
gravity and aerodynamic drag). Constraint 3: During the most of later stage in the
powered phase, attack angle of the ballistic missile is
relatively small, approximately 0.In other words, the
direction of thrust is almost consistent with that of target’s speed, while the direction of aerodynamic drag is
in the opposite direction. So, it can be considered that the
bend of trajectory is absolutely due to the result of
independent action by gravity. According to the constraints above, we can get: the
ratio between the speed of target and the acceleration
generated by the resultant force of thrust and aerodynamic drag is invariable in the directions of three
axes [2] [17]. And this ratio is denoted the 7th state
variable 7x . The gravity turn model denoted by 7 state
variables is as follow (8).
In the formula (8), 1 2 3, ,x x x is the position component
of target, 4 5 6, ,x x x is the speed component of target,
is the coefficient of the earth’s gravity (398613.52
km3/s2), and r is the distance between target and the
earth’s center. Improved gravity turn model
Based on the three constraints of basis gravity turn
model, a new constraint is added,
1 4
2 5
3 6
3
4 7 4 1
3
5 7 5 2
3
6 7 6 3
7
/
/
/
/ ( / )
/ ( / )
/ ( / )
/ 0
dx dt x
dx dt x
dx dt x
dx dt x x x r
dx dt x x x r
dx dt x x x r
dx dt
(8)
Constraint 4: In the powered phase, the fuel of ballistic
missile is consumed at a uniform speed in unit time.
According this constraint, the mass of fuel
consumption in unit time is tm, which means that the
change rate of the absolute mass of ballistic missile.
If the thrust generated by the fuel tm in unit time is
tF , according to Newton’s third law, the correspond
acceleration ( )t ta can be got as follow,
( ) / ( )t tt m t a F (9)
Based on constrain 4, the initial mass of ballistic
missile is 0m , and we can get,
0( ) tm t m m t (10)
Combing formula (9) and formula (10), ( )t ta can be
computed by formula (11),
0 0
/( )
( / )
t t t
t vTo
t t Id
mt
m m t m m t t t
F Fa L (11)
In the formula (11), /t tF m , it is means the using
efficiency of fuel. On the basis of constraint 4, the can
remains approximately a constant value for the same
ballistic missile, and it is irrelevant to the flight time of
ballistic missile. 0 /To
Id tt m m , it is means the time when
the mass of target runs out under ideal conditions. But,
actually, the mass of target would not run out at the time
of burnout point. So To
Idt t ( t is the any time during
flying in the powered phase). vL means the unit
directional vector of target’s speed. Because of the
directions of thrust and target’s speed are parallel, the
directions of speed can substitute that of thrust.
The direction of acceleration ( )da t generated by
aerodynamic drag is contrary to that of target’s speed,
and it can be calculated according to formula (12).
2 2( )1 ( )( )
2 ( ) 2 ( )
d d
d v vTo
t Id
v C A h C A v ha t L L
m t m t t
(12)
In the formula (12), A is the cross sectional area in the
normal direction of the motion of ballistic missile. dC is
drag coefficient, which is the function about the speed of
ballistic missile v and A . ( )h is an atmospheric
density function that relates to the altitude h .
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It can be seen from formula (11) and formula (12) that
there is the same variation form 1
( )To
Idt t in the formulas
of calculating ( )ta t and ( )da t , which reflects the
change of target’s mass. Therefore, based on the basis
gravity turn model, the 8th state variable 8x is given as
follow,
8
1To
Id
xt t
(13)
2
8 82
1/
( )To
Id
dx dt xt t
(14)
It is a reasonable assumption that speed is a continuous
state variable which cannot change abruptly while the acceleration is a discontinuous state variable which would
change abruptly because of the changes of outside forces.
According to this, 7th state variable is modified as shown in formula (15), and the modulus of acceleration
generated by the resultant force of thrust and
aerodynamic drag.
70
0
8
70
7 8
( ) ( ) ( ) ( )/ lim
( ) ( )1
( ) ( )lim ( ) ( )
1( 1)1
lim
t d t d
t
t d
t d
t dt
t
a t t a t t a t a tdx dt
t
a t t a t t
a t a ta t a t
t
x tx
t
x x
(15)
Accordingly, the motion state equation is improved as follow,
1 4
2 5
3 6
4 1
4 7 32 2 2
4 5 6
5 2
5 7 32 2 2
4 5 6
6 3
6 7 32 2 2
4 5 6
7 7 8
2
8 8
/
/
/
/
/
/
/
/
dx dt x
dx dt x
dx dt x
x xdx dt x
rx x x
x xdx dt x
rx x x
x xdx dt x
rx x x
dx dt x x
dx dt x
(16)
This is the improved gravity turn model in powered
phase. If the target state at reference time rt is given, the
position and speed of target at any time can be got by
using numerical integration. 2) Ellipse trajectory method in free flight phase
The basic idea of ellipse trajectory method is to
determine the elliptical orbit parameters of ballistic
missile and the corresponding position of ballistic missile on the ellipse trajectory, based on motion parameters of
target. And indeed, the ballistic trajectory is a space curve,
which veers to the right in the northern hemisphere and to
the left in the southern hemisphere [18]. However, these errors are so small that can be ignored. So the ballistic
trajectory can be assumed to be a plane curve.
Traditional computing method of ballistic trajectory is based on the relation between ECI (Earth Centered
Inertial, ECI) and perifocus coordinate system, and the
ballistic trajectory is computed on the ballistic plane. The
process of computing is approximately divided into three parts:
Part one: Obtaining the basic information about
position and speed of ballistic missile This is a process of information entry and conversion.
This article assumes that the information about azimuth,
angle of altitude and distance of ballistic missile can be
detected by radars. For the detection data inevitably include noises, the data should be preprocessed to
eliminate the outliers. Through data pretreatment and
coordinate transformation, the information about position
and speed of ballistic missile can be got. Part two: the parameters about initial state of ballistic
missile can be calculated. (The initial time is time 0t )
(1) At time 0t , the modulus
0r , speed 0v and the
obliquity angle of speed 0 of target are Calculated as
follows,
2 2 2
0 GO GO GOr X Y Z (17)
2 2 2
0 0 0 0XG YG ZGv V V V (18)
0 0 0 0 0 0
0
0 0
arccos( )G XG G YG G ZGX V Y V Z V
r v
(19)
In these formulas, 0 0 0, ,G G GX Y Z and
0 0 0, ,XG YG ZGV V V
are the position component and speed component of the
target in ECI, which can be got by coordinate
transformation the between radar measuring coordinate
system and geocentric rectangular coordinate system. 0
is the angle between the position vector of target and the
speed vector of target, and its value range is in the interval [0, ] .
(2) The orbital elements of ellipse trajectory can be
calculated: orbital parameter (semi-parameter) P ,
eccentricity ratio e and semi-major axis a .
2
0 0cosP r V (20)
2
01 ( 2)cose V V (21)
2/ (1 )a P e (22)
In these formulas, 2
0 0( ) /V v r is the energy
parameter, is the gravitational constant.
2508 JOURNAL OF NETWORKS, VOL. 9, NO. 9, SEPTEMBER 2014
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(3) The polar angle 0f and argument of perigee 0E at
time 0t are calculated,
0
0
0
arccos( )P r
fr e
(23)
0
0 arccos( )r a
Eae
(24)
Part three: according to the initial state parameters of ballistic missile, the state parameters of target at any time
are computed.
(1) The argument of perigee tE at any time t can be
calculated by using Newton iteration method [19],
according to formula (25).
0 0 03
3
( ) sin
( ) sin
p
p t t
t t E e Ea
t t E e Ea
(25)
(2) According to formula (26), the modulus tr , polar
angle tf , speed
tv , and the obliquity angle of speed t
can be given at the time t ,
2 2
2
(1 cos )
arccos( )
1 cos
1 cos
sinarctan( )
1
t t
t
t
t
t
t
t
t
t
r a e E
P rf
r e
e Ev
a e E
e E
e
(26)
(3) The information about target state in ECI can be
calculated.
In the formula (27), 0t t t is the time-span of
calculating, 0t tf f f is the difference of polar angles.
3
0 0 0 0 0
3
0 0 0 0 0
3
0 0 0 0 0
[1 (1 cos( )) ] [ ( sin( ))]
[1 (1 cos( )) ] [ ( sin( ))]
[1 (1 cos( )) ] [ ( sin( ))]
t t t t X
t t t t Y
t t t t Z
a aX E E X t E E E E V
R
a aY E E Y t E E E E V
R
a aZ E E Z t E E E E V
R
(27)
(4) The position component in earth centered fixed
coordinate can be calculated, and tZ remains unchanged.
cos( ) sin( )
sin( ) cos( )
Gt t t
Gt t t
Gt t
X X t Y t
Y X t Y t
Z Z
(28)
In the formula (28), is the rotational angular
velocity of the earth.
In conclusion, the whole process of ellipse trajectory method in free flight phase is shown in figure 2,
Input the information about
the position and the
speed of target
at time
( , , )x y z( , , )x y zv v v
0t
Initialize
and obtain
0 0 0, ,r v
Calculate the orbital elements
of ellipse trajectory
0 0, , , ,P e a f E
Calculate the argument of
perigee at any time by
using Newton iteration methodtE
Calculate
at time, , ,t t t tr f v
t
t
Calculate the information
about target state in ECI
at time t
Calculate the position and
speed of target in earth
centered fixed coordinate
at time t
Figure 2. The calculated algorithm flow chart of basis ellipse
trajectory method
B. The Theory of Closest Distance Method
Before burnout, there are three outside forces on ballistic missile, gravity, thrust and atmospheric drag.
The resultant force direction of thrust and atmospheric
drag parallels the velocity direction of target, and the acceleration generated is equivalent to tangential
acceleration. After burnout, ballistic missile only receives
gravity. And it is evident that the outside force of target
changes greatly before and after burnout, which makes motion track change. Before and after burnout, the
movement characteristic of target obeys different
movement rules. It is evident that burnout point is not
only located on motion track of powered phase of missile, but also is located on motion track of free phase of
missile. It means that burnout point meets two different
movement characteristics. The measurement errors of satellite and early warning radar sensor are different in
practical measurement. If the measured data is used for
extrapolated smooth, the condition without intersection
may exist, as shown in the condition of A C and LE in
figure 3.
B
C
AA
G
C
qV
E
G
W
L
Figure 3. Motion tracks of targets before and after burnout
The following conclusion can be got.
Conclusion: if motion models of target in powered phase
and in free phase are accurate, burnout time is the time when the track point of ballistic missile in powered phase
is the closest to that in free phase. It is also applied to the
condition that there is intersection on the tracks of two phases.
JOURNAL OF NETWORKS, VOL. 9, NO. 9, SEPTEMBER 2014 2509
© 2014 ACADEMY PUBLISHER
The
improved
gravity
turn
model
Ellipse
trajectory
Get the first and second
target data which radar
detects
Estimate the information
about the position and speed
of target
Forward-stepping trajectory
in free flight phase
lg wgt t t
the track of free phase is
extrapolated forward
get position coordinate
( )l tS
Back-stepping trajectory in
powered phase
lg wgt t t
The track of powered phase
is extrapolated backward
get position coordinate
w gt
Interval Search
Set step length1t
Set step length2t
Using alterative method
to calculate min 1( )S i
Get the time interval where
the min exist
1 1[ ( -1) , ( 1) ]wg wgt i t t i t
1 1 1( ) ( ) ( )w wg l wgS i t i t t i t S SSet
Set2 1 2 1 2( ) ( ( 1) ) ( ( 1) )w wg l wgS j t i t j t t i t j t S S
coarse search
elaborate search
1( )S i
Using alterative method
to calculate min 2( )S i
Get the target data which
Satellite detects at time
Set the time-length of search
Set the time-length of search
( )w tS
determ
ine th
e short in
terval
Meet
the actual accuracy
requirements
No
Yes
Search process ends
Output burnout time
Figure 4. Algorithm flow chart
As shown in Figure 3, point W is the position that
satellite detects the ballistic missile at the last time, and
point B is the first location that radar detects the ballistic missile. If ballistic missile keeps outside forces
unchanged after burnout ( )G G , and there are still three
outside forces, the target must move along tangent qV of
velocity. The arc ( )GC G C in the figure 3 means that the
track of powered phase is extrapolated for WC . Similarly,
the track of free phase is extrapolated, such as arc BE . We can see that at burnout time, the trajectory distance of
two phases is the closest. And it can be proved by
formula derivation which is not discussed because of
complicated process. Through the above conclusion, relative distance can be
used for search, so the burnout time can be obtained. The
step length of search is related to accuracy of requirement
results. And the solution process is as follows.
C. The Algorithm Flow
Assuming that the time is corrected, the last time that
satellite detects the target is wgt , and the time that early-
warning radar begins to detect the target is lgt , so the
burnout time gjt is in the interval lg[ , ]wgt t . Do search on
the interval. In order to save time, coarse search is firstly
made, which means that the span of step length is bigger.
After determining the short interval, according to required accuracy, small step length is settled for
elaborate search. As shown in the figure 4, the whole
searching process is given, (1) The track of powered phase is extrapolated
backward, and the length of extrapolation time is
lg wgt t t , so the position coordinate of track point in
the interval wS t can be got. Similarly, the track of free
phase is extrapolated forward for the same length, and we
can get position coordinate lS t .
(2)If step length is settled as 1t ,
1 1 1( ) ( ) ( )w wg l wgS i t i t t i t S S
the minimum of 1( )S i can be solved, and the short
interval 1 1[ ( -1) , ( 1) ]wg wgt i t t i t where the
minimum locates also can be achieved.
(3)If the step length is settled as 2t ,
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1 2
2
1 2
( ( 1) )( )
( ( 1) )
w wg
l wg
t i t j tS j
t i t j t
S
S Similarly, the
minimum of 2 ( )S j can be solved, and the shorter time
interval can be got. If the result meets the actual accuracy requirements, the burnout time is the mean of two
endpoints of the interval. Or more elaborated search
process is made until the time interval meeting accuracy
requirement is achieved. (4) Search process ends.
Now we can get burnout time, and can achieve the
position vector and velocity vector of burnout points. The process is called the closest distance search method. As
position vector and velocity vector are changing
continuously, the calculation results of tracks in two
phases should be the same under the condition without errors. Under the condition that there are errors, the
selection of results needs to be considered carefully.
When satellite exchanges data with early-warning radar, there is error inevitably existing. Besides burnout point is
at the end of powered phase and has great effect on
accuracy of elliptical orbital elements, so selecting the
burnout point of track in powered phase as the burnout point of the whole trajectory is more persuasive.
IV. SIMULATION EXPERIMENT
A. Simulation Environment
If the observation data of ballistic missile in powered
phase and in free phase is achieved by different sensors,
and the motion model of ballistic missile in powered phase conforms to the constraint gravity turn model, the
track in free phase is a part of elliptical orbit. The last
time that early-warning satellite achieves data is wgt , the
beginning time that early-warning radar achieves data is
lgt , and the blind zone time between them is 10s. The
burnout time is any time of blind zone time. In simulation
experiments, the actual burnout time is 3gl wgt t . The
closest distance search method is used to make accurate
estimation on burnout time.
The parameter setting is as follows. An intermediate-range is taken as an example. The firing range changes
with the difference of burnout points. In the simulation,
the firing range of ballistic missile is 2500km. The position when ballistic missile flies for 45s is (3.78×106,
3.78×106, 3.48×106)m, and the velocity is (530,
510,5101)m/s. The elevation of earl warning satellite is
3.579417×107 m, the longitude is10°, the latitude is 0°, and the scanning period is T=1s. The time-span that early
warning satellite detects the target in powered phase is
reference time from 45s to 120s. The position of radar
station is (2.5×106, 5.6×106, 1.753×106)m, and the altitude is 200m. The time of target data in the free phase
detected by ground-based radar is after 130s. There is
blind zone between 120s and 130s, and the burnout point of ballistic missile is in blind zone of the time. As
mentioned above, the actual burnout time is120s+3s, the
scanning period of early warning satellite is 1s, and the
scanning period of radar is 0.2s. The range error of radar is 20m, and the angle error is 0.01°.
And the projections of ballistic trajectory along each
axis are as follows,
Figure 5. The projections of ballistic trajectory along each axis
B. Comparison and Analysis
The simulation results are as follows,
Figure 6. Relative Distance between back-stepping trajectory of
powered phase and forward-stepping trajectory of free fight phase
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TABLE I. COMPARISON RESULTS OF THE CLOSEST DISTANCE SEARCH METHOD, GENERAL METHODS AND ACTUAL VALUES
Methods
Estimation parameters The closest distance search method General methods Actual values
Burnout time (s) 122.8 120.5 123.0
Position of burnout points (103km) (3.899,3.892,3.597) (3.893,3.886,3.591) (3.899,3.892,3.597)
Relative error rate of positions (%) (0.013,0.012,0.014) (0.163,0.152,0.175) ——
Velocity of burnout point (km/s) (2.595,2.423,2.579) (2.479,2.315,2.463) (2.605,2.432,2.589)
Relative error rate of velocity (%) (0.398,0.395,0.401) (4.853,4.817,4.889) ——
Impact time (s) 704.8 716.6 703.7
Position of impact point(103km) (2.679,2.610,2.544) (2.464,2.399,2.339) (2.699,2.629,2.562)
Relative error rate of position (%) (0.720,0.722,0.718) (8.709,8.724,8.692) ——
(Note: the burnout time of general methods is the half of the last time of satellites detecting targets and scanning period of satellites.)
From Figure 5, we can see that when the closest distance of back-stepping trajectory in powered phase and
forward-stepping trajectory in free fight phase is close to
123s, the time is the burnout time of ballistic missile. From the results in Table 1, we can see that the
burnout time of the closest distance search method is
122.8s, that of general methods is 120.5s, and the actual
burnout time is 123s, from which we can see that the difference for the burnout time between the closest
distance search method and actual burnout time is 0.2s,
and the error rate is 0.2/10×100%=2%. The difference for
the burnout time between general search methods and actual values is 2.5s, and the error rate is
2.5/10×100%=25%. The calculation is used to achieve
burnout point data, and the impact time and position of impact point is calculated. And the results are shown in
Table 1. The impact time deduced with using burnout
time which is determined by closest distance method is
704.8s, which has 1.1s difference from actual values. While, the impact time deduced with using burnout time
which is determined by general methods is 716.6s, which
has 12.9s difference from actual values. Through the
comparison on the closest distance search method and general methods, actual values, there is a difference of
degree of magnitude in the position, velocity, error rate of
impact position determined by two methods. And the values of error rate are shown in the Table 1. Therefore
we can get that the closest distance search method is
evidently better than general methods.
V. CONCLUSION
There is no systematic analysis on estimation of
burnout point of ballistic missile. The paper starts from
the outside forces situation of target before and after burnout, and considers the influence of motion track of
target. The motion of target in different phases obeys
different motion rules. Combining track time
extrapolation with the closest distance search method, burnout time can be determine, further the burnout point
parameters can also be achieved. Simulation experiment
results indicate that the closest distance search method is
evidently better than general methods for estimating burnout point parameters and indirectly determining
impact point parameters.
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