Interception Planning System

1. Interception Planning System Omer Cohen ShiloAbramovicz With the guidance of: EliranAbutbuland Sharon Rabinovich

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

3. Problem Definition Finding an interception plan (a launch yaw and pitch) Which satisfies the following constraints: 1.The launch does not occur in the past 2.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 .

4. 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

5. 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.

6. Model Design- Forces - Drag Force A force that oppose the relative motion of an object through a fluid (a liquid or gas). -Velocity Vector -Drag Coeff -Cross-sectional area -Material Density -Gravitation

7. Motion Equations Ballistic Coefficient

8. Atmosisa Function [T a P rho]=atmosisa(height) The function gets the height above sea level And returns: -Temparture -Pressure -Air Density -Speed of sound

9. Atmosisa Function Uses the International Standard Atmosphere model This function uses another function, “atmosplase”, with constants, such as: and are calculated using the Ideal Gas Model.

10. Calculating β(ballistic coeff) We calculate βusing a linear interpolation

11. Euler’s Approximation Method A second order approximation method, used here to solve the motion equations. For a certain and the initial conditions :

12. RK4 - Approximation Method A second order approximation method, used here to solve the motion equations. For a certain and the initial conditions :

13. RK4 - Approximation Method 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 :

14. Comparing the Methods

15. Comparing the Methods

16. Tolerances-Temperature(3D)

17. Possible Solutions We gathered all the possible trajectories with:

18. Possible Solutions Each 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.

19. Refernces • Wikipedia- Runge-Kutte Method. • The International Standard Atmosphere

20. THE END!