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

2. 2 ProjectProject definitiondefinition Problem constraints Way of calculating forces Solving motion equations Forces equation Examples of simulations for different initial conditions Possible solution directions References

3. Faculty of Electrical Engineering Technion – Israel Institute of Technology 3 Developing an algorithm for updating the course of the Interceptor in the air due to changes in predictable trajectory of the interceptor or target.

4. Faculty of Electrical Engineering Technion – Israel Institute of Technology 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) 4

5. Faculty of Electrical Engineering Technion – Israel Institute of Technology 5  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. Faculty of Electrical Engineering Technion – Israel Institute of Technology In Aerodynamics, the drag force refers to forces that oppose the relative motion of an object through the air. Drag forces act in opposite direction to the velocity. 6

7. Faculty of Electrical Engineering Technion – Israel Institute of Technology 7

8. 8 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.

9. Faculty of Electrical Engineering Technion – Israel Institute of Technology 9 z y x Now, if we multiply the acceleration with m we will get the forces

10. Faculty of Electrical Engineering Technion – Israel Institute of Technology 10 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.

11. Faculty of Electrical Engineering Technion – Israel Institute of Technology 11 Therefore, the temperature is calculated with:

12. Faculty of Electrical Engineering Technion – Israel Institute of Technology 12

13. Faculty of Electrical Engineering Technion – Israel Institute of Technology 13 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

14. Faculty of Electrical Engineering Technion – Israel Institute of Technology 14 Mach 0.130 0.8 0.140.9 0.161 0.211.1 0.171.4 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

15. Faculty of Electrical Engineering Technion – Israel Institute of Technology 15 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

16. Faculty of Electrical Engineering Technion – Israel Institute of Technology 16

17. Faculty of Electrical Engineering Technion – Israel Institute of Technology 17 With the atmospheric model Without the atmospheric model R=8378m H=4231m R=8176m H=4192m i=85 X=11,627 Y=3342 i=85 X=11,627 Y=3342 i=85 X=12,170 Y=3208 i=85 X=12,170 Y=3208

18. Faculty of Electrical Engineering Technion – Israel Institute of Technology 18 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.

19. Faculty of Electrical Engineering Technion – Israel Institute of Technology 19 stunner target

20. Faculty of Electrical Engineering Technion – Israel Institute of Technology 20 stunner target

21. Faculty of Electrical Engineering Technion – Israel Institute of Technology 21 stunner target

22. Faculty of Electrical Engineering Technion – Israel Institute of Technology 22

23. Faculty of Electrical Engineering Technion – Israel Institute of Technology 23

24. Faculty of Electrical Engineering Technion – Israel Institute of Technology  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 24 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

25. Faculty of Electrical Engineering Technion – Israel Institute of Technology the stunner should hit the target in a limited Azimuth (relative to the ground ) in order to maximize the damage to the target 25 30° 60° This is the angle which the stunner can hit the target target

26. Faculty of Electrical Engineering Technion – Israel Institute of Technology 26 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 ”

27. Faculty of Electrical Engineering Technion – Israel Institute of Technology 27 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]

28. Faculty of Electrical Engineering Technion – Israel Institute of Technology 28 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-delta the distance from the target is smaller than we check I min

29. Faculty of Electrical Engineering Technion – Israel Institute of Technology 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

30. Faculty of Electrical Engineering Technion – Israel Institute of Technology 30 target stunner 10m 5m 8m In this case this point will be chosen

31. Faculty of Electrical Engineering Technion – Israel Institute of Technology 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. 31

32. Faculty of Electrical Engineering Technion – Israel Institute of Technology 32

33. Faculty of Electrical Engineering Technion – Israel Institute of Technology 33 stunner target

34. Faculty of Electrical Engineering Technion – Israel Institute of Technology 34 stunner target

35. Faculty of Electrical Engineering Technion – Israel Institute of Technology 35 stunner target

36. Faculty of Electrical Engineering Technion – Israel Institute of Technology 36 first hit point Hit point after the target change azimuth

37. Faculty of Electrical Engineering Technion – Israel Institute of Technology 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

38. Faculty of Electrical Engineering Technion – Israel Institute of Technology  Create a Data structure 2D MATRIX  Every cell is a struct of arrays 38 Offline work: speed Theta*10000

39. Faculty of Electrical Engineering Technion – Israel Institute of Technology 39 speed 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

40. Faculty of Electrical Engineering Technion – Israel Institute of Technology  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 Offline work:

41. Faculty of Electrical Engineering Technion – Israel Institute of Technology  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 online work:

42. Faculty of Electrical Engineering Technion – Israel Institute of Technology Example1:  For receiving z[time] for this Initial conditions.  And the same for x and y. 42 online work: Z[time]=table[|v|][theta][Z.z(time)]+delta z

43. Faculty of Electrical Engineering Technion – Israel Institute of Technology  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 online algorithm:

44. Faculty of Electrical Engineering Technion – Israel Institute of Technology  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:

45. Faculty of Electrical Engineering Technion – Israel Institute of Technology  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 online algorithm:

46. Faculty of Electrical Engineering Technion – Israel Institute of Technology  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

47. Faculty of Electrical Engineering Technion – Israel Institute of Technology  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

48. Faculty of Electrical Engineering Technion – Israel Institute of Technology  The international standard atmosphere (ISA)—Mustafa Cavcar, Turkey  Wikipedia (about runge kutta)  Gui missile flyout –taylor & francis group From MIT 48