Interception Planning SystemOmer Cohen

Shilo Abramovicz

With the guidance of:Eliran Abutbul and Sharon Rabinovich

Project Definition

Designing an algorithm for intercepting ballistic missiles with a ballistic interceptor, based on target and interceptor model.

Problem Definition

Finding an interception plan (a launch yaw and pitch)Which satisfies the following constraints:1.The launch does not occur in the past2.The maximum height of the interceptor doesn’t cross a certain height.3. The interceptor’s velocity at the interception point must be larger then the user’s demand.4. The aspect of the interception must be close enough to .

90

Problem Definition

From the feasible solutions we choose the one that maximize the following objective function:

(w1, w2, w3)- user’s input.

w1*IcpVel+w2*RelativeVel+w3*IcpAccel

Development Steps

• Building a model of ballistic missile trajectory.

• Finding all the feasible interception plans under the given constraints

• Choosing the optimal plan according the objective function.

Model Design- Forces1 | |2 dF A C v v

-Material DensityA -Cross-sectional area

dC -Drag Coeffv -Velocity Vector

-Gravitation

-Drag ForceA force that oppose the relative motion of an object through a fluid (a liquid or gas).

Motion EquationsFam

1 | |21 | |21 | |2

xx x

yy y

zz z

dv v vdtdv

v vdtdv g v vdt

dA Cm

Ballistic Coefficient

x

y

z

Atmosisa Function[T a P rho=]atmosisa(height)

T [ ]Ka -Speed of sound

secm

-Air DensityP -Pressure

[ ]pascal2

kgm

-2000 0 2000 4000 6000 8000 10000 12000295

300

305

310

315

320

325

330

335

340

345

height [m]

Speed of Sound Vs. Height

a [m

/sec

]

-Temparture

The function gets the height above sea level And returns:

Atmosisa FunctionUses the International Standard Atmosphere model

This function uses another function, “atmosplase”, with constants, such as:

0 288.159.80665

11000troposphere

T Kgh m

a and are calculated using the Ideal Gas Model.

Calculating β (ballistic coeff)

We calculate β using a linear interpolationβ Mach

0.13 00.13 0.80.14 0.90.16 10.21 1.10.17 1.4

velocity VMachsound velocity a

Euler’s Approximation Method

t (0) , (0)o ov v r r

(( 1) ) ( ) ( )

(( 1) ) ( ) ( )

dv n t v n t v n t tdtdr n t r n t r n t tdt

(*) dr vdt

A second order approximation method, used here to solve the motion equations. For a certain and the initial conditions :

RK4 - Approximation Method

t (0) , (0)o ov v r r

1 2 3 4

1 2 1

3 2 4 3

1(( 1) ) ( ) ( 2 2 )6

1( ) ( ( ) )2

1( ( ) ) ( ( ) )2

v n t v n t t k k k k

d dk v n t k v n t t kdt dtd dk v n t t k k v n t t kdt dt

A second order approximation method, used here to solve the motion equations. For a certain and the initial conditions :

RK4 - Approximation Method

1(( ) )2

v n t

1(( 1) ) ( ) [ ( ) (( 1) ))]2

r n t r n t t v n t v n t

Using this method for propagating the location requires the calculation of the velocity at half the time, such as:

Which complex the calculation difficulty.

Therefore, we used the following approximation :

Comparing the Methods

6524 6526 6528 6530 6532 6534 6536 6538 654040

45

50

55

60

65

70

75

80

85

X: 6529Y: 55.81

Endcourse

X: 6533Y: 53.8

X: 6534Y: 53.68

X: 6535Y: 53.87

3990.3666 3990.3666 3990.3666 3990.3666 3990.3666 3990.3666

3274

3275

3276

3277

3278

3279

3280

X: 3990Y: 3280

Midcourse

X: 3990Y: 3278

X: 3990Y: 3276

X: 3990Y: 3276

0 1000 2000 3000 4000 5000 6000 7000-500

0

500

1000

1500

2000

2500

3000

3500

RK4 t=0.05

Euler t=0.05

Euler t=0.01

RK4 t=0.01

Comparing the Methods

0 5000 10000 15000-1000

0

1000

2000

3000

4000

5000

6000

7000

1.428 1.4285 1.429 1.4295 1.43 1.4305 1.431 1.4315

x 104

139.303

139.304

139.305

139.306

139.307

139.308

139.309

139.31

139.311

139.312

X: 1.429e+004Y: 139.3

X: 1.429e+004Y: 139.3

X: 1.43e+004Y: 139.3

X: 1.43e+004Y: 139.3

Endcourse

RK4 t=0.05

Euler t=0.05

Euler t=0.01

RK4 t=0.01

Tolerances-Temperature(3D)

0500

10001500

20002500

30003500

40004500 0

5001000

15002000

25003000

35004000

45000

500

1000

1500

2000

2500

3000

3500

X: 4637Y: 4637Z: 41.3

X: 4618Y: 4618Z: 44.35

X: 4600Y: 4600Z: 47.63

Default

Default+7o K

Default-7o K

Possible Solutions

0 1000 2000 3000 4000 5000 6000 7000

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

5500

All possiable Trajectories

We gathered all the possible trajectories with:

0.005

700seco

radmv

Possible SolutionsEach point in the space can be achieved with two different launch pitches

Suggestions:

• Using two tables- one for the lower impact angle and the other for the larger.

• fit every relevant paremeter (pitch angle, impact angle, impact velocity, etc.) to a fifth degree polynomial.

• fitting using ANN.

Refernces

• Wikipedia- Runge-Kutte Method.

• The International Standard Atmosphere

THE END!