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: nipeng198509@163.com {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

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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),

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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

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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.

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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.

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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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