Faculty of Electrical Engineering Technion – Israel Institute of Technology CONTROL AND ROBOTICS LABORATORY

Faculty of Electrical EngineeringTechnion – Israel Institute of Technology

Liraz AmarHagay Abramovsky

CONTROL AND ROBOTICS LABORATORY

Interceptor

and target route

update

Project supervisers:Eliran AbutbulSharon Rabinovich

2Presentation Layout

Project definition

Problem constraints

Way of calculating forces

Solving motion equations

Forces equation

Examples of simulations for different initial conditions

Possible solution directions

References


Project definition

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Developing an algorithm for updating the course of the Interceptor in the air due to changes in predictable trajectory of the interceptor or target.


Problem definition

Given interception scenario of interceptor and target. After launching , if update has received for the target / interceptor that caused changes in the predictable trajectory . we should find a way to update the interceptor trajectory to the new hit point.

The optimal hit will be only with these Certain conditions :*hit in a given space of time.*Minimum time interception.*Maximum hit speed. (energy)

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Problem Constraints

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Flight Ceil Height- this is an aerodynamic Restriction that affects maneuverability.

Minimum close velocity- Minimum Relative hit speed of one missile in the other in order to “hit to kill”.

Aspect Azimuth- the interceptor should hit the target in a limited Azimuth in order to damage the target.


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F is the force vector of drag

is the density of the air

v is the velocity of the missile relative to the air

allistic coefficient

D

D

Ais the referencearea

C is thedrag coefficient

is ab

1| |

2 DF A C v v

1| | 12 | |

2D

DD

D A C

m

F ma

A C v vFa v v

m m

Drag Force


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Gravitation, or gravity, is a natural phenomenon in which objects with mass attract one another .

In everyday life, gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped.

Gravitation g


Forces Equation

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1| |

2x xa v v 1111111111111111111111111111

1| |

2z za g v v 1111111111111111111111111111

1| |

2y ya v v 1111111111111111111111111111

z

y

x

Now, if we multiply the acceleration with m we will get the forces


Drag force calculation

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[T, a, P, ]=atmosisa(H0);

T temperature in kelvin

a speed of sound

P air pressures

- air densities

At first:

Atmosisa- implements the mathematical representation of the International Standard Atmosphere values for ambient temperature, pressure, density, and speed of sound for the input altitude.


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T temperature in kelvin

a speed of sound

P air pressures

- air densities

Explanation about calculating atmosisa:

0

( )6.5

1000

h mT T

Therefore, the temperature is calculated with :

Temperature modeling:


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00 0

5.25610

0

/ / ( )

( )

0.0065

(1 0.0065 )

hp

ph

d gdh at a small atmosphere element

p RT

dp gdh gdh

p RT RT

dp g dh

p R T h

hp p

T

1111

11

11

11

1111 11 11000

( )

11

"11":

226.32216.6511,000

:

a

p h

p h

gh h

RT

when the parameters with correspondto the values at the tropopausep hPT Kh m

in case of pressure above the tropopause

dp gdh

p RT

p p e

Pressure modeling


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Pressure modeling

p

RT

Since the pressure and standard temperature are known for a given altitude ,the standard density can easily be calculated from the perfect gas equation

R= real gas constant for air 287.04

2

2sec

m

k


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V velocityMach

a speed of sound

Mach

0.13 0

0.13 0.8

0.14 0.9

0.16 1

0.21 1.1

0.17 1.4

The Interpolation table we use in order to find the appropriate beta:

After those steps we have all the arguments we need to find the drag acceleration

Missiles with the same ballistic parameter ( beta )will have the same flight trajectory

DC


Solving motion equations

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For the velocity we use runge kutta 4-5 method:

1

2 1

3 2

4 3

i 1 i

1

( , )

( , )

1 1( , )

2 21 1

( , )2 2

( , )

1v v k1 2k2 2k3 k4

6

i i

i i

i i

i i

i i

dvf t v a

dtk f t v

k f t t v k t

k f t t v k t

k f t t v k t

t

t t t

numerical analysis, the Runge–Kutta methods are an important family of implicit and explicit iterative methods for the approximation of solutions ordinary differential equations


For the location equation we also used the Runge–Kutta method:

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1

2 1

3 2

4 3

i 1 i

1

( , ) ( )

( , )

1 1( , )

2 21 1

( , )2 2

( , )

1x x k1 2k2 2k3 k4

6

i i

i i

i i

i i

i i

dxf t x v t

dtk f t x

k f t t x k t

k f t t x k t

k f t t x k t

t

t t t


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With the atmospheric model

Without the atmospheric model

R=8378mH=4231m

R=8176mH=4192m

i=85X=11,62

7Y=3342

i=85X=12,17

0Y=3208


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Atmosisa- implements the mathematical representation of the International Standard Atmosphere values for ambient temperature, pressure, density, and speed of sound for the input geopotential altitude.The Atmosisa return the air density , we already know that as we go higher the air density become thinner .

From the graphs , we can see that in high velocity the affect of the Atmosisa is low. But on the other hand, for the low velocity there is a major different in ‘x.’

We assume that the different in the low velocity is because Thin air density allows the missile to go further in the ’x’ axisdue to Low resistance.


Explanation to the graphs:


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Examples of simulations for different initial conditions

stu

nn

er

targ

et


Second simulation:

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stu

nn

e

r

targ

et


Third simulation:

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stunne

r

targ

et


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5%with

higher

:with


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5%with

higher

with


Cost function

Minimum time until hit -t Maximum relative velocity on hit time |vstunner-vtarget| Maximum stunner velocity on hit time- maximum hit speed of the

stunner. Vhit _stunner

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Cost =alpha*t+beta*|vstunner-vtarget|+gama*vhit_stunner

Alpha, beta, gama are normalized factors that we decided on according to the importance of the Relevant Multiply .

Each point has its own cost calculated by the cost function. Point with the highest cost is the better hit point

this function helps us to decide which of the hit points is the best choice


Stunner’s Azimuth during hit time the stunner should hit the target in a limited Azimuth (relative to the ground ) in order to maximize the damage to the target

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30°

30°60°This is the angle which

the stunner can hit the target

target


Possible solution directions

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A way to find Intercept Algorithm is to• look at the moment that the target

change azimuth.• Calculate the new route of the target.For the stunner we run on theta from the current theta to zero with delta of 0.1 secfor every theta we calculate the best hitting point using the algorithm called “desert lion ”


Desert lion algorithm

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We choose one spot on the middle of the stunner’s course, check the previews and the next with the chosen theta and for a specific time we calculate the distance from the target. In addition, We also calculate for point[time+delta] and for point[time – delta].if the next point distance is shorter then the previews point.we keep searching for the minimum distance [time , max time ],Else we keep searching for the minimum distance [min time , time]

Faculty of Electrical EngineeringTechnion – Israel Institute of TechnologyDemonstration for a specific

theta:

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I min I max

First time we check

(timeMin+ timeMax/)2

Time + delta

Time-delta

If for )time+ delta( the distance from the target is shorter than we check

I max

If for Time-deltathe distance from the target is smaller

than we check

I min


for all the possible theta

In that way we find for every theta the time in the route with the minimum distance from the target. From all the thetas, we take the theta that give us the minimum distance to the target.

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Demonstration for all the possible theta

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target

stunner

10m

5m

8m

In this case this point will

be chosen


for all the possible theta

After we got the theta with the best result.we look again in the range of [theta-delta, theta+delta] with smaller resolution of theta (we divide delta in 10) . Until we find the best results and as long as delta>0.0001.

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stu

nn

er

targ

et


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stu

nn

er

targ

et


35

stu

nne

r

targ

et


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first hit point

Hit point after the target change azimuth


Other possible solution

We can find stunner route by making an offline table. Launch speed- is an absolute velocity in the launch point (0,0,0,)Theta- is the launch angle. Assumption : (1) launch speed [1-700]

(2) theta[0.01- /2]

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Other possible solution

Create a Data structure 2D MATRIX Every cell is a struct of arrays

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Offline work:sp

eed

Theta*10000


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sp

eed

Theta*10000

X.x

X.a

Z.a

Z.v

Z.z

Y.y

Y.v

Y.a

X.v

The 2D Matrix

In every cell there is a struct with 8 array

time


Other possible solution (continue)

create routes with different Launch angles for theta[0.01-pi/2] and different launch speed.

∆theta=0.0001 rad During calculating the route we save

parameters to the relevant cell. We save for every direction: position, velocity, Acceleration. (according to the relevant launch speed and theta).

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Offline work:



Given a start point (X◦, Y◦, Z◦) , velocity|V| and theta.

We define: delta z=Z◦ delta y=Y◦ delta x=X◦ For every start point we shift the

matrix according to the given start point. With the relevant delta.

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online work:



Example1: For receiving z[time] for this Initial

conditions.

And the same for x and y.

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online work:

Z[time]=table[|v|][theta][Z.z(time)]+delta z



For a given velocity we should find the new theta. We are using the offline table.

We will start by calculating Theta=(theta_min+theta_max)/2

For a specific theta We will make “desert lion” on the time, and find the point with the minimum distance , and the time it’s happens.

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online algorithm:



After we found the time for the specific theta that gives minimum distance from the target.

We check the target in the same time. If the target is higher we return the

algorithm for[theta, theta_max ] Else we return the algorithm for [theta_min , theta] And so on until theta_max=theta_min. In that way we found theta that gives

the best results(minimum distance).44

online algorithm:



After we found the best theta we will use a dynamic algorithm.

In the range of [theta_best-delta, theta_best+theta]Note : explain how we find delta. We will make the first algorithm (lion

desert on time and on theta)and get a better result.

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online algorithm:


The problem in this Algorithm The problem in this Algorithm is that

we have an offline table and every cell save parameters for a specific altitude.

when we do the shifting of the table. We do not consider the beta that changes during the change in altitude.

Beta depends on mach, Mach depends on speed of sound, and Speed of sound depends on altitude.

And that is why beta depends on altitude.

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Solving the problem We solve the problem by using the

previous algorithm. After we find theta_best We do dynamic algorithm in the range [theta_best-delta, theta_best+theta] this dynamic algorithm calculate theta and beta in real time. In that way beta suit the current altitude.

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references The international standard atmosphere

(ISA)—Mustafa Cavcar, Turkey Wikipedia (about runge kutta) Gui missile flyout –taylor & francis

groupFrom MIT

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Documents

Faculty of Electrical Engineering Technion – Israel Institute of Technology CONTROL AND ROBOTICS LABORATORY