1. Empirical Virtual Sliding Target Guidance law Presented by: Jonathan Hexner Itay Kroul Supervisor: Dr. Mark Moulin

2. Introduction A new guidance law for long range surface to air missiles is tested. Guidance law is empirical based on aerodynamic considerations. Idea: missile achieves a high altitude during boost phase, allowing low drag during pursuit of target. Altitude is achieved using a virtual sliding target (VST), initialized at a high altitude sliding towards target. Basic guidance scheme used to guide the missile towards VST and real target is proportional navigation (PN).

3. 2D Missile Engagement model Legend: T – Thrust m – missile mass g – gravity D – Drag  - Line of site (LOS) angle  m - missile flight path angle  t - target flight path angle a c - commanded acceleration perpendicular to LOS a m - missile acceleration perpendicular to missile body. v m - missile velocity. v t - target velocity. a t - target acceleration Equations of motion

4. Augmented Proportional Navigation APN is the optimal guidance law for a non inertial system in the sense that is minimal APN navigation: APN navigation: Substituting into the guidance law:

5. VST Guidance law - detailed Stage 1: Missile guidance towards VST: –Boost Phase: missile guided towards stationary point. –Midcourse Phase: missile guided towards virtual target, which slides towards target. Guidance cycle: t go estimated: Predicted Intercept Point (PIP) of missile and target is calculated: Predicted Intercept Point (PIP) of missile and target is calculated: VST slides towards PIP. Sliding velocity: VST slides towards PIP. Sliding velocity: Missile guided towards new VST location.

6. VST Guidance Law – Cont’d Stage 2: Missile guidance towards target: Stage 2: Missile guidance towards target: – Missile guided towards target at lock-on range from target.

7. Simulation model Thrust model: Missile Specifications: ParameterValueDiameter 300 [mm] Length 4000 [mm] Mass 165 [kg] Propellant mass 75 [kg] Burn time 40 [sec] Atmospheric conditions: Propellant mass rate of change:

8. Simulation Model – cont’d Drag: C D0 - zero lift drag coefficient C Di - induced drag coefficient S - wetted surface area. Angle of attack ≤ 30° D T y x mg C D0 profile:

9. Non maneuvering target example

10. VST testing VST compared with PN in several nominal scenarios: –Approaching & Receding Non maneuvering target. –Approaching & Receding maneuvering target (a t >0, a t 0, a t <0). Different VST 0 tested. Parameters tested: –Interception time –Velocity at lock on – correlates with launch boundary envelope Missile initial conditions constant: – v m0 = 100 [m/sec] –  m0 = 10° y x

11. Simulation (1) – Non Maneuvering Receding target Target parameters: Velocity at lock on (m/sec) Intercept time [sec] Initial position of VST [m] Guidance law 332.26456.83(1000,15000)VST 350.58951.996(3000,15000)VST 347.50347.22(5000,15000)VST 343.02242.763(7000,15000)VST 405.85437.736(5000,5000)VST 337.59738.393(5000,10000)VST 332.76655.471(5000,20000)VST 525.478620.129---PN VST 0

12. Simulation (2) – Non Maneuvering Approaching target Target parameters: Guidance law Initial position of VST [m] Intercept time [sec] Velocity at lock on [m/sec] VST(1000,15000) 53.458350.540 VST(3000,15000) 51.7343.993 VST(5000,15000) 50.668338.770 VST(7000,15000) 49.856335.624 VST(5000,5000) 46.361335.062 VST(5000,10000) 48.277328.102 VST(5000,20000)MISS --- PN--- 42.86332.2483 VST 0

13. Simulation (3) – Maneuvering Receding Target Target parameters: Velocity at lock on (m/sec) Intercept time [sec] Initial position of VST [m] Guidance law 345.473160.5550 (1000,7000)VST 333.514552.6740 (3000, 7000)VST 319.200844.5480 (5000, 7000)VST 390.109842.8970 (7000, 7000)VST 332.264 82.0860 (1000,20000)VST 350.589 73.4880 (3000, 20000)VST 347.503 66.3290 (5000, 20000)VST 343.022 60.3530 (7000, 20000)VST 477.818533.5770 ---PN VST 0

14. Simulation (4) – Maneuvering Approaching Target Target parameters: Velocity at lock on (m/sec) Intercept time [sec] Initial position of VST [m] Guidance law 326.785452.3530 (1000,7000)VST 331.195551.6790 (3000, 7000)VST 327.785450.3970 (5000, 7000)VST 323.481149.0220 (7000, 7000)VST --- MISS MISS (1000,20000)VST 330.7044 55.4170 (3000, 20000)VST 336.6116 54.1190 (5000, 20000)VST 337.5600 53.1520 (7000, 20000)VST 317.557446.4070 ---PN VST 0

15. Simulation (5) – Maneuvering Receding Target Target parameters: Velocity at lock on (m/sec) Intercept time [sec] Initial position of VST [m] Guidance law 345.323354.8800 (1000,15000)VST 329.941552.1730 (3000, 15000)VST 328.124842.3700 (5000, 15000)VST 331.793837.7060 (7000, 15000)VST 336.4938 60.7280 (1000,20000)VST 348.6227 57.2540 (3000, 20000)VST --- MISS (5000, 20000)VST 326.6120 44.4850 (7000, 20000)VST 338.152331.9570 ---PN VST 0

16. Simulation (6) – Maneuvering Approaching Target Target parameters: Velocity at lock on (m/sec) Intercept time [sec] Initial position of VST [m] Guidance law 331.251951.6190 (1000,15000)VST 328.038450.3630 (3000, 15000)VST 326.742449.4300 (5000, 15000)VST 326.849648.6750 (7000, 15000)VST 299.3433 56.1490 (1000,20000)VST --- MISS (3000, 20000)VST 339.0638 51.4640 (5000, 20000)VST 334.4351 50.5240 (7000, 20000)VST 332.542444.0450 ---PN VST 0

17. Non Linear sliding velocity Recall: Non linear: –Initially faster slide: v inlf = v il Fe ft F>0,f 0,f<0 –Initially slower slide: v inls = v il S(e st -1) S>0,s>0 v inls = v il S(e st -1) S>0,s>0 Approaching target example (VST 0 = [1km,15km])VST Velocity at lock on [m/sec] Intercept time [sec] Linear slide 341.027351.5280 Non-Linear slide initially fast 325.915849.0130 Non-Linear slide initially slow 124.914267.1050 Initially faster => lower altitude Initially slower => higher altitude Very unstable

18. Summarizing results Unsuccessful choice of VST 0 : –Low missile velocity at lock on –Missile misses target Successful choice of VST 0 : –High missile velocity at lock on (increased launch boundary)

19. Summary & Conclusions VST guidance law was tested using various target scenarios with different VST 0 positions. Results show similar behavior for maneuvering and non- maneuvering targets: –Increased velocity at lock-on for approaching target. –Increased intercept time. Main advantage: simple implementation. Drawbacks: lacks analytic basis, not robust to VST 0 position.

20. Questions???