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

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).

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

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:

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.

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.

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:

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:

Non maneuvering target example

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

Simulation (1) – Non Maneuvering Receding target Target parameters: Velocity at lock on (m/sec) Intercept time [sec] Initial position of VST [m] Guidance law (1000,15000)VST (3000,15000)VST (5000,15000)VST (7000,15000)VST (5000,5000)VST (5000,10000)VST (5000,20000)VST PN VST 0

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) VST(3000,15000) VST(5000,15000) VST(7000,15000) VST(5000,5000) VST(5000,10000) VST(5000,20000)MISS --- PN VST 0

Simulation (3) – Maneuvering Receding Target Target parameters: Velocity at lock on (m/sec) Intercept time [sec] Initial position of VST [m] Guidance law (1000,7000)VST (3000, 7000)VST (5000, 7000)VST (7000, 7000)VST (1000,20000)VST (3000, 20000)VST (5000, 20000)VST (7000, 20000)VST PN VST 0

Simulation (4) – Maneuvering Approaching Target Target parameters: Velocity at lock on (m/sec) Intercept time [sec] Initial position of VST [m] Guidance law (1000,7000)VST (3000, 7000)VST (5000, 7000)VST (7000, 7000)VST --- MISS MISS (1000,20000)VST (3000, 20000)VST (5000, 20000)VST (7000, 20000)VST PN VST 0

Simulation (5) – Maneuvering Receding Target Target parameters: Velocity at lock on (m/sec) Intercept time [sec] Initial position of VST [m] Guidance law (1000,15000)VST (3000, 15000)VST (5000, 15000)VST (7000, 15000)VST (1000,20000)VST (3000, 20000)VST --- MISS (5000, 20000)VST (7000, 20000)VST PN VST 0

Simulation (6) – Maneuvering Approaching Target Target parameters: Velocity at lock on (m/sec) Intercept time [sec] Initial position of VST [m] Guidance law (1000,15000)VST (3000, 15000)VST (5000, 15000)VST (7000, 15000)VST (1000,20000)VST --- MISS (3000, 20000)VST (5000, 20000)VST (7000, 20000)VST PN VST 0

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 Non-Linear slide initially fast Non-Linear slide initially slow Initially faster => lower altitude Initially slower => higher altitude Very unstable

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)

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.

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