Post on 29-Jan-2016
description
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???