Transcript Empirical Virtual Sliding Target Guidance law

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
q - Line of site (LOS) angle
am - missile flight path angle
at - 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 motion
T D
 g sin a m
m
a  g cos a m
am  m
vm
vm 
xm  vm cos a m
ym  vm sin a m
ac  am cos(am  q )
Augmented Proportional Navigation
APN is the optimal guidance law for a non inertial system in the sense that
tf
2
dt
is minimal  acommanded
0


APN navigation: aM  N '  vcq  aT  ,
2
1
c



vc  closing velocity
Substituting into the guidance law:

1  D T

aM c  N ' vcq  
sin(a m  q )  g cos q  at , y cos q  at , x sin q  
2 m


vc  vm cos(a m  q )  vt cos(at  q )
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:
tgo estimated: t go 
r
vt cos a t  q   vm cos a m  q 
Predicted Intercept Point (PIP) of missile and target is calculated:
PIP   xt  vt cos at tgo , yt  vt sin at tgo 
Missile guided towards new VST location.
Target
Missile
VST
18000
16000
14000
12000
y [m]
D
VST slides towards PIP. Sliding velocity: vil  i
t go
20000
10000
vil
8000
6000
4000
2000
PIP
0
0
0.5
1
1.5
x [m]
2
2.5
3
4
x 10
VST Guidance Law – Cont’d
20000
20000
Target
Missile
VST
18000
16000
16000
14000
14000
12000
y [m]
12000
y [m]
Target
Missile
VST
18000
10000
8000
Di
8000
10000
Di
6000
6000
4000
4000
2000
2000
0
0.5
1
1.5
x [m]
PIP
0
PIP
0
2
2.5
3
0
0.5
1
4
1.5
x [m]
2
x 10
Stage 2: Missile guidance towards target:
– Missile guided towards target at lock-on range from target.
2.5
3
4
x 10
Parameter
Value
Diameter
300 [mm]
Length
4000 [mm]
Mass
165 [kg]
Propellant mass
75 [kg]
Burn time
40 [sec]
Thrust model:
4
3
Thrust Profile
x 10
2.5
2
thrust[N]
Missile Specifications:
Simulation model
1.5
1
0.5
0
0
5
10
15
20
25
time[sec]
30
35
40
45
50
Propellant mass rate of
Atmospheric conditions:
change:
 Tˆ  288.16  0.0065h
[K ]

 (( g / aR ) 1)

 Tˆ 
[kg / m3 ]
   1.225 


 288.16 
 ˆ
[K ]
 T  216.16

 ( g ( h 11000) / 288Tˆ )
[ kg / m3 ]
   0.3655e
h  11[km]
h  11[km]
8[kg / s ]

m   1[kg / s ]
 0[kg / s ]

0t 5
5  t  40
t  40
Simulation Model – cont’d
Angle of attack ≤ 30°
Di  L sin a i  LkCL  Lk
Drag:
1
 vm 2 SCD
2
CD  CD 0  CDi
mam
D
1
 vm 2 S
2
1
1
 vm 2 S
 vm 2 S
am  2
sin a i  4
km
km
CD0 profile:
CD0 - zero lift drag
coefficient
CDi - induced drag
coefficient
y
T
am
CDi  kCL  k
m2 am 2
2
a
am
D
mg
vm
x
1
2 
  vm S 
2

2
S - wetted surface area.
diamater 2
S 
4
Non maneuvering target example
Receding Target
Approaching Target
20000
20000
PN
Target
VST guidance
Virtual Target
18000
16000
16000
14000
14000
12000
12000
10000
y
y
PN
Target
VST guidance
Virtual Target
18000
8000
10000
8000
6000
6000
4000
4000
2000
2000
0
0
0
0.5
1
1.5
x
2
2.5
3
4
x 10
0
0.5
1
1.5
x
2
2.5
3
4
x 10
VST testing
VST compared with PN in several nominal scenarios:
– Approaching & Receding Non maneuvering target.
– Approaching & Receding maneuvering target (at>0, at<0).
Different VST0 tested.
Parameters tested:
– Interception time
– Velocity at lock on – correlates with launch boundary envelope
Missile initial conditions constant:
y
vm
– vm0 = 100 [m/sec]
– am0 = 10°
am
x
Simulation (1) – Non Maneuvering Receding target
Target parameters:
Receding Target
6000
(1000,15000)
(3000,15000)
(5000,15000)
(7000,15000)
5000
vt , x  200[m / sec], vt , y  0[m / sec]
at  0[m / sec2 ]
Y [meter]
4000
3000
2000
Guidance
law
Initial position
of VST [m]
Intercept
time
[sec]
Velocity at
lock on
(m/sec)
1000
VST
(1000,15000)
56.83
332.264
VST
(3000,15000)
51.996
350.589
VST
(5000,15000)
47.22
347.503
VST
(7000,15000)
42.763
343.022
VST
(5000,5000)
37.736
405.854
VST
(5000,10000)
38.393
337.597
VST
(5000,20000)
55.471
332.766
PN
---
20.129
525.4786
0
0
0.5
1
1.5
2
2.5
X [meter]
4
x 10
VST0
Receding Target
6000
(5000,5000)
(5000,10000)
(5000,15000)
(5000,20000)
5000
Y [meter]
4000
3000
2000
1000
0
0
0.5
1
1.5
X [meter]
2
2.5
4
x 10
Simulation (2) – Non Maneuvering Approaching target
Target parameters:
Approaching Target
6000
vt , x  200[m / sec], vt , y  0[m / sec]
(5000,5000)
(5000,10000)
(5000,15000)
5000
at  0[m / sec2 ]
Y [meter]
4000
3000
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
2000
1000
0
0
0.5
1
1.5
2
2.5
X [meter]
4
x 10
Approaching Target
6000
(1000,15000)
(3000,15000)
(5000,15000)
(7000,15000)
5000
4000
Y [meter]
VST0
3000
2000
1000
Target
PN
0
0
0.5
1
1.5
X [meter]
2
2.5
4
x 10
Simulation (3) – Maneuvering Receding Target
Manuvering Target - Receding
Target parameters:
Vt x = 200 [m/sec], Vt0y = 200 [m/sec], at y = -4[m/sec 2], Vm0 = 100 [m/sec]
20000
16000
PN
Target
VST0 = [1 20] km
14000
VST0 = [3 20] km
18000
at  4[m / sec2 ]
VST0 = [5 20] km
12000
y [m]
vt , x  200[m / sec], vt , y (t  0)  200[m / sec]
VST0 = [7 20] km
10000
8000
6000
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
4000
2000
0
0
0.5
1
1.5
2
x [m]
2.5
4
x 10
Manuvering Target - Receding
Vt x = 200 [m/sec], Vt0y = 200 [m/sec], at y = -4[m/sec 2], Vm0 = 100 [m/sec]
20000
PN
18000
Target
VST0 = [1 7] km
16000
VST0 = [3 7] km
14000
VST0 = [5 7] km
12000
y [m]
VST0
VST0 = [7 7] km
10000
8000
6000
4000
2000
0
0
0.5
1
1.5
x [m]
2
2.5
4
x 10
Simulation (4) – Maneuvering Approaching Target
Manuvering Target - Aproaching
Vt x = -200 [m/sec], Vt0y = 200 [m/sec], at y = -4[m/sec 2], Vm0 = 100 [m/sec]
20000
PN
18000
Target
VST0 = [1 7] km
16000
VST0 = [3 7] km
14000
vt , x  200[m / sec], vt , y (t  0)  200[m / sec]
VST0 = [5 7] km
12000
y [m]
Target parameters:
VST0 = [7 7] km
at  4[m / sec2 ]
10000
8000
6000
4000
2000
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)
MISS
---
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
0.5
1
1.5
2
x [m]
2.5
4
x 10
Manuvering Target - Aproaching
Vt x = -200 [m/sec], Vt0y = 200 [m/sec], at y = -4[m/sec 2], Vm0 = 100 [m/sec]
20000
PN
18000
Target
VST0 = [1 20] km
16000
VST0 = [3 20] km
14000
VST0 = [5 20] km
y [m]
12000
VST0 = [7 20] km
10000
8000
6000
VST0
4000
2000
0
0
0.5
1
1.5
x [m]
2
2.5
4
x 10
Simulation (5) – Maneuvering Receding Target
Manuvering Target - Receding
Vt x = 200 [m/sec], Vt0y = -200 [m/sec], at y = 4[m/sec 2], Vm0 = 100 [m/sec]
20000
PN
18000
Target
VST0 = [1 15] km
16000
VST0 = [3 15] km
14000
vt , x  200[m / sec], vt , y (t  0)  200[m / sec]
VST0 = [5 15] km
12000
y [m]
Target parameters:
VST0 = [7 15] km
at  4[m / sec2 ]
10000
8000
6000
4000
2000
0
0
0.5
1
1.5
2
x [m]
2.5
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
4
x 10
Manuvering Target - Receding
Vt x = 200 [m/sec], Vt0y = -200 [m/sec], at y = 4[m/sec 2], Vm0 = 100 [m/sec]
20000
PN
18000
Target
VST0 = [1 20] km
16000
VST0
VST0 = [3 20] km
14000
VST0 = [5 20] km
12000
y [m]
Guidance
law
VST0 = [7 20] km
10000
8000
6000
4000
2000
0
0
0.5
1
1.5
x [m]
2
2.5
4
x 10
Simulation (6) – Maneuvering Approaching Target
Manuvering Target - Aproaching
Vt x = -200 [m/sec], Vt0y = -200 [m/sec], at y = 4[m/sec 2], Vm0 = 100 [m/sec]
Target parameters:
20000
16000
PN
Target
VST0 = [1 15] km
14000
VST0 = [3 15] km
18000
at  4[m / sec2 ]
VST0 = [5 15] km
12000
y [m]
vt , x  200[m / sec], vt , y (t  0)  200[m / sec]
VST0 = [7 15] km
10000
8000
6000
4000
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
2000
0
0
0.5
1
1.5
2
x [m]
2.5
4
x 10
Manuvering Target - Aproaching
Vt x = -200 [m/sec], Vt0y = -200 [m/sec], at y = 4[m/sec 2], Vm0 = 100 [m/sec]
20000
PN
18000
Target
VST0 = [1 20] km
16000
VST0 = [3 20] km
14000
VST0 = [5 20] km
12000
y [m]
VST0
VST0 = [7 20] km
10000
8000
6000
4000
2000
0
0
0.5
1
1.5
x [m]
2
2.5
4
x 10
Non Linear sliding velocity
Recall:
vil 
Non linear:
Initially faster => lower
altitude
– Initially faster slide:
Di
t go
vinlf = vilFeft
F>0,f<0
Initially slower => higher
altitude
– Initially slower slide:
vinls = vilS(est -1)
S>0,s>0
Very unstable
Approaching target example
(VST0 = [1km,15km])
Approaching target example
10000
Target
Linear Slide
NL Slow (S=0.3, s=0.055)
NL Fast (F=2, f=-0.01)
9000
8000
VST
Velocity at
lock on [m/sec]
Intercept
time [sec]
Linear slide
341.0273
51.5280
Non-Linear slide
initially fast
325.9158
49.0130
Non-Linear slide
initially slow
124.9142
67.1050
7000
y [m]
6000
5000
4000
3000
2000
1000
0
-1000
0
0.5
1
1.5
x [m]
2
2.5
3
4
x 10
Summarizing results
4
3
x 10
PN
Target
VST guidance
2.5
Unsuccessful choice of VST0:
y
2
– Low missile velocity at lock on
1.5
– Missile misses target
1
0.5
0
0
0.5
1
1.5
x
2
2.5
3
4
x 10
4
3
x 10
Successful choice of VST0:
PN
Target
VST guidance
virtual target
2.5
– High missile velocity at lock on
(increased launch boundary)
y
2
1.5
1
0.5
0
0
0.5
1
1.5
x
2
2.5
3
4
x 10
Summary & Conclusions
VST guidance law was tested using various target scenarios
with different VST0 positions.
Results show similar behavior for maneuvering and nonmaneuvering targets:
– Increased velocity at lock-on for approaching target.
– Increased intercept time.
Main advantage: simple implementation.
Drawbacks: lacks analytic basis, not robust to VST0 position.
Questions???