Faculty of Electrical Engineering Technion – Israel Institute of Technology

1. CONTROL AND ROBOTICS LABORATORY Faculty of Electrical EngineeringTechnion – Israel Institute of Technology Interceptor and target route  update LirazAmar HagayAbramovsky Project supervisers: EliranAbutbul Sharon Rabinovich

2. Presentation Layout

3. Project definition Developing an algorithm for updating the courseof the Interceptor in the air due to changes in predictable trajectory of the interceptor or target.

4. 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)

5.  Problem Constraints • 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.

6. Drag Force

7. Gravitation 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.

8. z Forces Equation y x Now, if we multiply the acceleration with m we will get the forces

9. Drag force calculation 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.

10. Explanation about calculating atmosisa: Temperature modeling: Therefore, the temperature is calculated with:

11. Pressure modeling

12. Pressure modeling 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

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

14. Solving motion equations For the velocity we use runge kutta 4-5 method:  numerical analysis, the Runge–Kutta methods  are an important family of implicit and explicit iterative methods for the approximation of solutions ordinary differential equations

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

16. With the atmospheric model i=85 X=12,170 Y=3208 R=8378m H=4231m Without the atmospheric model i=85 X=11,627 Y=3342 R=8176m H=4192m

17. Explanation to the graphs: 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’ axis due to Low resistance.

18. Examples of simulations for different initial conditions stunner target

19. Second simulation: stunner target

20. Third simulation: target stunner

23. Cost function this function helps us to decide which of the hit points is the best choice • 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 Cost =alpha*t+beta*|vstunner-vtarget|+gama*vhit_stunner • Alpha, beta, gamaare normalized factors that we decided on • according to the importance of the Relevant Multiply. • Each point has its own cost calculated by the costfunction. Point with the highest cost  is the better hit point

24. 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 30° 60° This is the angle  which thestunnercanhit the target 30° target

25. Possible solution directions • 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 sec • for every theta we calculate the best hitting point using the algorithm called “desert lion ”

26. Desert lion algorithm 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]

27. Demonstration for a specific theta: (timeMin+ timeMax)/2 First time we check Time + delta Time-delta If for (time+ delta) the distance from the target is shorter than we check If for Time-delta the distance from the target is smaller than we check I min I max I max I min

28. 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.

29. Demonstration for all the possible theta target In this case this point will be chosen 10m 5m 8m stunner

30. 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.

32. target stunner

33. target stunner

34. target stunner

35. first hit point Hit point after the target change azimuth

36. 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]

37. Other possible solution Offline work: • Create a Data structure 2D MATRIX • Every cell is a struct of arrays Theta*10000 speed

38. Theta*10000 The 2D Matrix In every cell there is a struct with 8 array speed time X.x X.v X.a Y.y Y.v Y.a Z.z Z.v Z.a

39. Other possible solution (continue) Offline work: • 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).

40. Other possible solution (continue) online 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.

41. Other possible solution (continue) online work: Example1: • For receiving z[time] for this Initial conditions. • And the same for x and y. Z[time]=table[|v|][theta][Z.z(time)]+delta z

42. Other possible solution (continue) online algorithm: • 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.

43. Other possible solution (continue) 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. Other possible solution (continue) 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.

45. 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.

46. 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.

47. references • The international standard atmosphere (ISA)—Mustafa Cavcar, Turkey • Wikipedia (about runge kutta) • Gui missile flyout –taylor & francis group From MIT