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 EquationsF

am

1

2

1

21

2

xx

yy

zz

dvv v

dtdv

v vdtdv

g v vdt

dA C

m

Ballistic Coefficient

x

y

z

Atmosisa Function

[T a P rho=]atmosisa(height)

T [ ]Ka -Speed of sound

sec

m

-Air Density

P -Pressure

[ ]pascal

2

kg

m

-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 Function

Uses the International Standard Atmosphere model

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

0 288.15

9.80665

11000troposphere

T K

g

h m

a and are calculated using the Ideal Gas Model.

Calculating β (ballistic coeff)

We calculate β using a linear interpolation

Cd Mach

0.13 0

0.13 0.8

0.14 0.9

0.16 1

0.21 1.1

0.17 1.4

velocity VMach

sound 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 t

dtd

r 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 )

61

( ) ( ( ) )2

1( ( ) ) ( ( ) )

2

v n t v n t t k k k k

d dk v n t k v n t t k

dt dtd d

k 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(( ) )

2v n t

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

2r 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

0 5000 10000 150000

1000

2000

3000

4000

5000

6000

X: 1.467e+004Y: 1384

RK4 vs Euler

X: 1.468e+004Y: 1376X: 1.466e+004Y: 1368

Euler T=0.001

RK4 T=0.05

Euler T=0.05

1.462 1.464 1.466 1.468 1.47 1.472

x 104

1320

1340

1360

1380

1400

1420

1440

1460

1480

X: 1.467e+004Y: 1384

RK4 vs Euler

X: 1.468e+004Y: 1376

X: 1.466e+004Y: 1368

Euler T=0.001

RK4 T=0.05

Euler T=0.05

Comparing the Methods

0 1 2 3 4 5 6 7

x 104

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

X: 6.741e+004Y: 520.6

X: 6.739e+004Y: 509.8X: 6.737e+004Y: 513.9

RK4 vs Euler (horizontal)

Euler T =0.001

RK4 T =0.05

Euler T =0.05

6.736 6.737 6.738 6.739 6.74 6.741 6.742

x 104

508

510

512

514

516

518

520

522

524

526

X: 6.741e+004Y: 520.6

X: 6.739e+004Y: 509.8

X: 6.737e+004Y: 513.9

RK4 vs Euler (horizontal)

Euler T =0.001

RK4 T =0.05

Euler T =0.05

Tolerances-Temperature

0 2000 4000 6000 8000 10000 12000 14000 160000

1000

2000

3000

4000

5000

6000

Range[m]

Hei

ght[

m]

Temperature tolerance

288 K

298 K278 K

1.525 1.53 1.535 1.54 1.545 1.55

x 104

50

100

150

200

250

300

X: 1.544e+004Y: 139.2

Range[m]

Hei

ght[m

]

Temperature tolerance

X: 1.537e+004Y: 108.7

X: 1.532e+004Y: 82.43

288 K

298 K278 K

Creating The TableWe’ll Us two tables- one for the lower impact angle and the other for the larger.

0.005

700seco

rad

mv

0 2000 4000 6000 8000 10000 12000 14000 16000-2000

0

2000

4000

6000

8000

10000

12000

Range[m]

Heig

ht[

m]

Table creating Trajectoris

Solution

Demonstration of the relation between the angle and the range and height paremeters:

Finding Optimal Solution• Developing the target’s trajectory

• Projecting each point to a 2D plane – z axis stays the same xy transform to Range.

• Performing the “best” interpolation from table data.

• Checking if the constraints are being satisfied.

• Calculating the target function and replacing the current solution if necessary.

Finding Optimal Solution

-50000

500010000

1500020000

0

50

100

150

200-1000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

x[m]

scenario height limit

y[m]

z[m

]

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

Suggestions:

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

• fitting using ANN.

Surface Fitting

surface fitting was performed for each table parameter resulting a Two variable, five degree polynomial.

The fitting is based on MMSE.

Instead of performing the interpolation, the height and range will beInserted into to polynomial and that will give us the wanted parameter.

Surface Fitting

Surface Fitting

Artificial Neural Network

Using Matlab's Neural Network Fitting Tool it is possible tocreate a neural network that is a close fit to the table.

The table cells are given to the tool and it trains a suitable Neural network.

In order to achieve better results this method will consume to muchtime and memory.

Artificial Neural Network

Graphic User Interface

Refernces• http://en.wikipedia.org/wiki/Drag_%28physics%29 http://en.wikipedia.org/wiki/Drag_coefficient http://en.wikipedia.org/wiki/Drag_equation• The International Standard Atmosphere (ISA) • http://www.learnartificialneuralnetworks.com/ a tutorial about ANN• http://mathworld.wolfram.com/Runge-KuttaMethod.html-RK4 method• http://www3.ee.technion.ac.il/labs/eelabs/Upload/Projects/Enrichment /winter2011/Graphics%20and%20GUI%20using%20Matlab.pdf

THE END!