# Omer Cohen Shilo Abramovicz With the guidance of: Eliran Abutbul and Sharon Rabinovich.

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Omer Cohen Shilo Abramovicz With the guidance of: Eliran Abutbul and Sharon Rabinovich

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

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.

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

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.

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

Ballistic Coefficient

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

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

We calculate β using a linear interpolation CdMach 0.130 0.8 0.140.9 0.161 0.211.1 0.171.4

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

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

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 :

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

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

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.

Each 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 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 be Inserted into to polynomial and that will give us the wanted parameter.

Using Matlab's Neural Network Fitting Tool it is possible to create 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 much time and memory.

http://en.wikipedia.org/wiki/Drag_%28physics%29 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://www.learnartificialneuralnetworks.com/ http://mathworld.wolfram.com/Runge-KuttaMethod.html-RK4 method http://mathworld.wolfram.com/Runge-KuttaMethod.html http://www3.ee.technion.ac.il/labs/eelabs/Upload/Projects/Enrichment /winter2011/Graphics%20and%20GUI%20using%20Matlab.pdf

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