Empirical Virtual Sliding Empirical Virtual Sliding Target Guidance lawTarget Guidance law

Presented by:Presented by:

Jonathan HexnerJonathan Hexner

Itay KroulItay Kroul

Supervisor:Supervisor:

Dr. Mark MoulinDr. Mark Moulin

IntroductionIntroduction

A new guidance law for long range surface to air A new guidance law for long range surface to air

missiles is tested.missiles is tested.

Guidance law is empirical based on aerodynamic Guidance law is empirical based on aerodynamic

considerations.considerations.

Idea:Idea: missile achieves a high altitude during boost missile achieves a high altitude during boost

phase, allowing low drag during pursuit of target.phase, allowing low drag during pursuit of target.

Altitude is achieved using a virtual sliding target (VST), Altitude is achieved using a virtual sliding target (VST),

initialized at a high altitude sliding towards target.initialized at a high altitude sliding towards target.

Basic guidance scheme used to guide the missile Basic guidance scheme used to guide the missile

towards VST and real target is proportional navigation towards VST and real target is proportional navigation

(PN).(PN).

2D Missile Engagement model2D 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

ac - commanded acceleration

perpendicular to LOS

am - missile acceleration

perpendicular to missile

body.

vm - missile velocity.

vt - target velocity.

at - target acceleration

Equations of motionEquations of motion

sin

cos

cos

sin

m m

m mm

m

m m m

m m m

T Dv g

ma g

v

x v

y v

cos( )c m ma a

Augmented Proportional Augmented Proportional NavigationNavigation

APN is the optimal guidance law for a non inertial system in the sense that APN is the optimal guidance law for a non inertial system in the sense that

is minimal is minimal

APN navigation: APN navigation:

Substituting into the guidance law: Substituting into the guidance law:

2

0

ft

commandeda dt1

' , closing velocity2cM c T ca N v a v

, ,

1' sin( ) cos cos sin

2cM c m t y t x

D Ta N v g a a

m

cos( ) cos( )c m m t tv v v

VST Guidance law - detailedVST Guidance law - detailed

Stage 1: Missile guidance towards VST:Stage 1: Missile guidance towards VST:

– Boost Phase:Boost Phase: missile guided towards stationary point. missile guided towards stationary point.

– Midcourse Phase:Midcourse Phase: missile guided towards virtual target, which slides towards missile guided towards virtual target, which slides towards

target. Guidance cycle:target. Guidance cycle:

ttgogo estimated: 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.Missile guided towards new VST location.

cos , sint t t go t t t goPIP x v t y v t

iil

go

Dv

t

0 0.5 1 1.5 2 2.5 3

x 104

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

x [m]

y [m

]

Target

Missile

VST

PIP

vil

cos cosgot t m m

rt

v v

VST Guidance Law – Cont’dVST Guidance Law – Cont’d

0 0.5 1 1.5 2 2.5 3

x 104

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

x [m]

y [m

]

Target

MissileVST

PIP

Di

Stage 2: Missile guidance towards target:Stage 2: Missile guidance towards target:

– Missile guided towards target at lock-on range from target.Missile guided towards target at lock-on range from target.

0 0.5 1 1.5 2 2.5 3

x 104

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

x [m]

y [m

]

Target

Missile

VST

Di

PIP

Simulation modelSimulation modelThrust model:Thrust model:

0 5 10 15 20 25 30 35 40 45 50

0

0.5

1

1.5

2

2.5

3x 10

4

time[sec]

thru

st[

N]

Thrust Profile

Missile Specifications:Missile Specifications:

ParameterParameterValueValue

DiameterDiameter300 [mm]300 [mm]

LengthLength4000 4000

[mm][mm]

MassMass165 [kg]165 [kg]

Propellant Propellant

massmass75 [kg]75 [kg]

Burn timeBurn time40 [sec]40 [sec]

Atmospheric conditions:Atmospheric conditions:

(( / ) 1)

3

ˆ( ( 11000) / 288 ) 3

ˆ 288.16 0.0065 [ ]

11[ ]ˆ1.225 [ / ]

288.16

ˆ 216.16 [ ]11[ ]

0.3655 [ / ]

g aR

g h T

T h K

h kmTkg m

T Kh km

e kg m

8[ / ] 0 5

1[ / ] 5 40

0[ / ] 40

kg s t

m kg s t

kg s t

Propellant mass rate of Propellant mass rate of

change:change:

Simulation Model – cont’dSimulation Model – cont’dDrag:Drag:

2

0

1

2 m D

D D Di

D v SC

C C C

CCD0D0 - zero lift drag - zero lift drag coefficientcoefficient

CCDiDi - induced drag - induced drag

coefficientcoefficient2 2

22

21

2

mDi L

m

m aC kC k

v S

S - wetted surface area.S - wetted surface area.

2

4

diamaterS

Angle of attack ≤ 30°Angle of attack ≤ 30°

2

2 2

sin1

21 1

2 4sin

mi i L

m

m m

m i

maD L LkC Lk

v S

v S v Sa

km km

ma

D

T

mv

y

xm

mg

CCD0D0 profile: profile:

Non maneuvering target exampleNon maneuvering target example

0 0.5 1 1.5 2 2.5 3

x 104

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

x

y

Receding Target

PN

TargetVST guidance

Virtual Target

0 0.5 1 1.5 2 2.5 3

x 104

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

x

y

Approaching Target

PN

TargetVST guidance

Virtual Target

VST testingVST testingVST compared with PN in several nominal scenarios:VST compared with PN in several nominal scenarios:

– Approaching & Receding Non maneuvering target.Approaching & Receding Non maneuvering target.

– Approaching & Receding maneuvering target (aApproaching & Receding maneuvering target (att>0, a>0, att<0).<0).

Different VSTDifferent VST00 tested. tested.

Parameters tested:Parameters tested:

– Interception timeInterception time

– Velocity at lock on – correlates with launch boundary envelopeVelocity at lock on – correlates with launch boundary envelope

Missile initial conditions constant:Missile initial conditions constant:

– vvm0m0 = 100 [m/sec] = 100 [m/sec]

– m0m0 = 10° = 10°

mv

y

xm

Simulation (1)Simulation (1) – Non Maneuvering Receding – Non Maneuvering Receding targettarget

Target parameters:Target parameters:

0 0.5 1 1.5 2 2.5

x 104

0

1000

2000

3000

4000

5000

6000

X [meter]

Y [

mete

r]

Receding Target

(1000,15000)

(3000,15000)

(5000,15000)

(7000,15000)

, ,

2

200[ / sec], 0[ / sec]

0[ / sec ]

t x t y

t

v m v m

a m

0 0.5 1 1.5 2 2.5

x 104

0

1000

2000

3000

4000

5000

6000

X [meter]

Y [

met

er]

Receding Target

(5000,5000)

(5000,10000)

(5000,15000)

(5000,20000)

Guidance law

Initial position of VST [m]

Intercept time

]sec[

Velocity at lock on

)m/sec(

VST)1000,15000(56.83332.264

VST)3000,15000(51.996350.589

VST)5000,15000(47.22347.503

VST)7000,15000(42.763343.022

VST)5000,5000(37.736405.854

VST)5000,10000(38.393337.597

VST)5000,20000(55.471332.766

PN---20.129525.4786

VST0

Simulation (2) – Non Maneuvering Approaching Simulation (2) – Non Maneuvering Approaching targettarget

Target parameters:Target parameters:

0 0.5 1 1.5 2 2.5

x 104

0

1000

2000

3000

4000

5000

6000

X [meter]

Y [

met

er]

Approaching Target

(5000,5000)

(5000,10000)

(5000,15000), ,

2

200[ / sec], 0[ / sec]

0[ / sec ]

t x t y

t

v m v m

a m

0 0.5 1 1.5 2 2.5

x 104

0

1000

2000

3000

4000

5000

6000

X [meter]

Y [

met

er]

Approaching Target

(1000,15000)

(3000,15000)

(5000,15000)

(7000,15000)

TargetPN

Guidance law

Initial position of VST [m]

Intercept time

]sec[

Velocity at lock on

]m/sec[

VST)1000,15000(53.458 350.540

VST)3000,15000(51.7 343.993

VST)5000,15000(50.668 338.770

VST)7000,15000(49.856 335.624

VST)5000,5000(46.361 335.062

VST)5000,10000(48.277 328.102

VST)5000,20000(MISS---

PN---42.86 332.2483

VST0

Simulation (3) – Maneuvering Simulation (3) – Maneuvering Receding TargetReceding Target

Target parameters:Target parameters:

0 0.5 1 1.5 2 2.5

x 104

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

x [m]

y [

m]

Manuvering Target - Receding

Vtx = 200 [m/sec], Vt0

y = 200 [m/sec], at

y = -4[m/sec2], Vm

0 = 100 [m/sec]

PN

TargetVST0 = [1 20] km

VST0 = [3 20] km

VST0 = [5 20] km

VST0 = [7 20] km

, ,

2

200[ / sec], ( 0) 200[ / sec]

4[ / sec ]

t x t y

t

v m v t m

a m

0 0.5 1 1.5 2 2.5

x 104

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

x [m]

y [

m]

Manuvering Target - Receding

Vtx = 200 [m/sec], Vt0

y = 200 [m/sec], at

y = -4[m/sec2], Vm

0 = 100 [m/sec]

PN

TargetVST0 = [1 7] km

VST0 = [3 7] km

VST0 = [5 7] km

VST0 = [7 7] km

Guidance law

Initial position of VST [m]

Intercept time

]sec[

Velocity at lock on

)m/sec(

VST)1000,7000(60.5550 345.4731

VST)3000 ,7000(52.6740 333.5145

VST)5000 ,7000(44.5480 319.2008

VST)7000 ,7000(42.8970 390.1098

VST)1000,20000(82.0860 332.264

VST)3000 ,20000(73.4880 350.589

VST)5000 ,20000(66.3290 347.503

VST)7000 ,20000(60.3530 343.022

PN---33.5770 477.8185

VST0

Simulation (4) – Maneuvering Simulation (4) – Maneuvering Approaching TargetApproaching Target

Target parameters:Target parameters:

0 0.5 1 1.5 2 2.5

x 104

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

x [m]

y [

m]

Manuvering Target - Aproaching

Vtx = -200 [m/sec], Vt0

y = 200 [m/sec], at

y = -4[m/sec2], Vm

0 = 100 [m/sec]

PN

TargetVST0 = [1 7] km

VST0 = [3 7] km

VST0 = [5 7] km

VST0 = [7 7] km , ,

2

200[ / sec], ( 0) 200[ / sec]

4[ / sec ]

t x t y

t

v m v t m

a m

Guidance law

Initial position of VST [m]

Intercept time

]sec[

Velocity at lock on

)m/sec(

VST)1000,7000(52.3530 326.7854

VST)3000 ,7000(51.6790 331.1955

VST)5000 ,7000(50.3970 327.7854

VST)7000 ,7000(49.0220 323.4811

VST)1000,20000( MISSMISS---

VST)3000 ,20000(55.4170 330.7044

VST)5000 ,20000(54.1190 336.6116

VST)7000 ,20000(53.1520 337.5600

PN---46.4070 317.5574 0 0.5 1 1.5 2 2.5

x 104

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

x [m]

y [

m]

Manuvering Target - Aproaching

Vtx = -200 [m/sec], Vt0

y = 200 [m/sec], at

y = -4[m/sec2], Vm

0 = 100 [m/sec]

PN

TargetVST

0 = [1 20] km

VST0 = [3 20] km

VST0 = [5 20] km

VST0 = [7 20] km

VST0

Simulation (5) – Maneuvering Receding Simulation (5) – Maneuvering Receding TargetTarget

Target parameters:Target parameters:

0 0.5 1 1.5 2 2.5

x 104

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

x [m]

y [

m]

Manuvering Target - Receding

Vtx = 200 [m/sec], Vt0

y = -200 [m/sec], at

y = 4[m/sec2], Vm

0 = 100 [m/sec]

PN

TargetVST

0 = [1 15] km

VST0 = [3 15] km

VST0 = [5 15] km

VST0 = [7 15] km

, ,

2

200[ / sec], ( 0) 200[ / sec]

4[ / sec ]

t x t y

t

v m v t m

a m

Guidance law

Initial position of VST [m]

Intercept time

]sec[

Velocity at lock on

)m/sec(

VST)1000,15000(54.8800 345.3233

VST)3000 ,15000(52.1730 329.9415

VST)5000 ,15000(42.3700 328.1248

VST)7000 ,15000(37.7060 331.7938

VST)1000,20000(60.7280 336.4938

VST)3000 ,20000(57.2540 348.6227

VST)5000 ,20000(MISS---

VST)7000 ,20000(44.4850 326.6120

PN---31.9570 338.1523 0 0.5 1 1.5 2 2.5

x 104

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

x [m]

y [

m]

Manuvering Target - Receding

Vtx = 200 [m/sec], Vt0

y = -200 [m/sec], at

y = 4[m/sec2], Vm

0 = 100 [m/sec]

PN

TargetVST

0 = [1 20] km

VST0 = [3 20] km

VST0 = [5 20] km

VST0 = [7 20] km

VST0

Simulation (6) – Maneuvering Approaching Simulation (6) – Maneuvering Approaching TargetTarget

Target parameters:Target parameters:

0 0.5 1 1.5 2 2.5

x 104

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

x [m]

y [m

]

Manuvering Target - Aproaching

Vtx = -200 [m/sec], Vt0

y = -200 [m/sec], at

y = 4[m/sec2], Vm

0 = 100 [m/sec]

PN

TargetVST0 = [1 15] km

VST0 = [3 15] km

VST0 = [5 15] km

VST0 = [7 15] km

, ,

2

200[ / sec], ( 0) 200[ / sec]

4[ / sec ]

t x t y

t

v m v t m

a m

Guidance law

Initial position of VST [m]

Intercept time

]sec[

Velocity at lock on

)m/sec(

VST)1000,15000(51.6190 331.2519

VST)3000 ,15000(50.3630 328.0384

VST)5000 ,15000(49.4300 326.7424

VST)7000 ,15000(48.6750 326.8496

VST)1000,20000(56.1490 299.3433

VST)3000 ,20000(MISS---

VST)5000 ,20000(51.4640 339.0638

VST)7000 ,20000(50.5240 334.4351

PN---44.0450 332.5424 0 0.5 1 1.5 2 2.5

x 104

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

x [m]

y [m

]

Manuvering Target - Aproaching

Vtx = -200 [m/sec], Vt0

y = -200 [m/sec], at

y = 4[m/sec2], Vm

0 = 100 [m/sec]

PN

TargetVST0 = [1 20] km

VST0 = [3 20] km

VST0 = [5 20] km

VST0 = [7 20] km

VST0

Non Linear sliding velocityNon Linear sliding velocity

Recall:Recall: i

ilgo

Dv

t

Non linear:Non linear: – Initially faster slide:Initially faster slide:

vvinlfinlf = v = vililFeFeftft F>0,f<0 F>0,f<0

– Initially slower slide:Initially slower slide:

vvinlsinls = v = vililS(eS(estst -1) S>0,s>0 -1) S>0,s>0Approaching target example (VST0 = [1km,15km])

0 0.5 1 1.5 2 2.5 3

x 104

-1000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

x [m]

y [m

]

Approaching target example

Target

Linear Slide

NL Slow (S=0.3, s=0.055)

NL Fast (F=2, f=-0.01)VSTVSTVelocity at Velocity at

lock on lock on

[m/sec][m/sec]

Intercept Intercept

time [sec]time [sec]

Linear slideLinear slide341.0273341.027351.528051.5280

Non-Linear Non-Linear

slide initially slide initially

fastfast

325.9158325.915849.013049.0130

Non-Linear Non-Linear

slide initially slide initially

slowslow

124.9142124.914267.105067.1050

Initially faster => lower Initially faster => lower

altitudealtitude

Initially slower => higher Initially slower => higher

altitudealtitude

Very unstableVery unstable

Summarizing resultsSummarizing results

Unsuccessful choice of VSTUnsuccessful choice of VST00::

– Low missile velocity at lock onLow missile velocity at lock on

– Missile misses targetMissile misses target

0 0.5 1 1.5 2 2.5 3

x 104

0

0.5

1

1.5

2

2.5

3x 10

4

x

y

PN

Target

VST guidance

Successful choice of VSTSuccessful choice of VST00::

– High missile velocity at lock on High missile velocity at lock on (increased launch boundary)(increased launch boundary)

0 0.5 1 1.5 2 2.5 3

x 104

0

0.5

1

1.5

2

2.5

3x 10

4

x

y

PN

Target

VST guidance

virtual target

Summary & ConclusionsSummary & Conclusions

VST guidance law was tested using various target VST guidance law was tested using various target

scenarios with different VSTscenarios with different VST00 positions. positions.

Results show similar behavior for maneuvering and non-Results show similar behavior for maneuvering and non-

maneuvering targets:maneuvering targets:

– Increased velocity at lock-on for approaching target.Increased velocity at lock-on for approaching target.

– Increased intercept time.Increased intercept time.

Main advantage: simple implementation.Main advantage: simple implementation.

Drawbacks: lacks analytic basis, not robust to VSTDrawbacks: lacks analytic basis, not robust to VST00

position.position.

Questions???Questions???