optimized trajectory shaping guidance for an air-to-ground
TRANSCRIPT
Optimized Trajectory Shaping Guidance for an Air-to-Ground
Missile Launched from a GunshipCraig Phillips
Ernie OhlmeyerShane Sorenson
Overview
• Mission Scenario• Notional Munition Concept• Guidance Laws
– Generalized Vector Explicit Guidance– Composite Guidance
• Footprints• Trajectories• Conclusions
Next Generation Gunship Mission Scenario
VerticalImpactAngle
Munition launchedfrom port side of aircraft
Munition will be able to reachextended ranges and have360 degrees of coverage,which will help keep thegunship out of the range ofan enemy’s weapons.
Next Generation Gunship Mission Scenario
Trajectory shaping will help themunition reach these extendedranges, have 360 degrees of coverage, and meet MOUTrequirements
Munition must meet Military Operationson Urban Terrain (MOUT) requirements:• Be able to attack in urban canyons• Have low collateral damage
• Thrust: 1350 lb (Boost)163.9 lb (Sustain)
• Burn Time: 2 sec (Boost)15 sec (Sustain)
• Total Impulse: 5158.5 lb*s
Notional Gun-Launched Munition Concept
• Weight: 47.5 lb15 lb (Warhead)20 lb (Propellant)
• Length: 42.5 inches
• Diameter: 4.7 inches (120 mm)
• Angle of Attack (Max): 20 deg
• Achievable Acceleration: 35 g’s (Aerodynamic Limit)
• The optimum guidance consists of a PRONAV-like term with gain and a trajectory shaping term with gain .
• The Generalized Explicit Guidance solution defines a family of guidancelaws defined by the parameter . The parameter may be chosen toachieve specific performance objectives and to satisfy terminal constraints.
• In general, may be used to modify the curvature of the trajectory, whileenforcing a specified final position and final velocity orientation (impact angle).
Generalized Vector Explicit (GENEX) Guidance
1K2K
n
[ ]TyyKTyyyKT
u MfMMf )()(1212 &&& −+−−=∗
n n
Reference: Ohlmeyer, E. J., “Control of Terminal Engagement Geometry Using Generalized Vector Explicit Guidance,” Proceedings of American Control Conference, Vol. 1, 2003, pp. 396-401.
: Missile acceleration: Time-to-go: Missile position and velocity: Desired final missile position and velocity: Zero Effort Miss wrt a specified final position
)2)(1(2 ++−= nnK
)3)(2(1 ++= nnK
Define the gains as:Tu*
ff
MM
yyyy&
&
,,
ZEM = design parameter
ZEM
n
Effect of Varying Design Parameter, , in GENEX Guidance Lawn
0
5
10
15
20
25
30
0 2 4 6 8 10 12
Downrange (NMI)
Alti
tude
(kft)
0
50
100
150
200
0 5 10 15 20 25 30 35 40
Time (sec)
Ach
ieve
d A
ccel
erat
ion
(ft/s
^2)
Increased curvature as is increasedFor , large accelerations achievedearlier in flight to allow for smalleraccelerations at terminal (e.g. smaller angle of attack)
0=n1=n
2=n
n0>n
2=n
0=n
1=n
Composite Guidance Law
( ) ( )[ ]BiasN gtCuBtCa +−+= 1
( )
Bias
N
guBtC
a : normal acceleration commanded by guidance: time-dependent composite index: composite bias acceleration vector: GENEX control term: one g up bias term
Level flight commanded for secFor sec, For sec, For sec,
picked to be linear for simplicity. should be optimized for best results.
For , above equation becomes GENEX equation
5.0≤t
5.75.75.25.25.0
>≤<≤<
ttt ( )
( ) ( )ttCtC
2.05.10.1−=
=
( ) 0.0=tC( ) 0.0=tC
( )tC ( )tC
The bias acceleration vector B, when added to GENEX, provides higher gains early in flight, increases the curvature of the trajectory, and extendsthe kinematic reach of the munition.
GENEX Footprints
-30
-20
-10
0
10
20
30
40
50
-40 -30 -20 -10 0 10 20 30 40
Downrange (NMI)
Cro
ss ra
nge
(NM
I)
-30
-20
-10
0
10
20
30
40
50
-40 -30 -20 -10 0 10 20 30 40
Downrange (NMI)
Cro
ss ra
nge
(NM
I)
25,000 ft Launch 10,000 ft Launch
Muzzle Velocity = 1100 fpsGunship Velocity = 405 fpsLaunch Angle = 0 deg
0=n
1=n
2=n
GENEX less effective with drop in launch altitude
Gunship Directionof Flight
Hit Criteria:• less than 5 ft Miss Distance• within 10 deg of vertical impact angle
-40
-30
-20
-10
0
10
20
30
40
50
60
-50 -40 -30 -20 -10 0 10 20 30 40 50
Downrange (NMI)
Cro
ss ra
nge
(NM
I)
GENEX
Bias = 2
Bias = 3
Bias = 4
Bias = 5
Bias = 6
Bias = 7
Bias = 8
Bias = 9
Bias = 10
Bias = 11
Bias = 12
Bias = 13
-40
-30
-20
-10
0
10
20
30
40
50
60
-50 -40 -30 -20 -10 0 10 20 30 40 50
Downrange (NMI)
Cro
ss ra
nge
(NM
I)
GENEXBias = Bias = Bias = Bias = Bias = Bias = Bias = Bias = Bias = Bias = Bias = Bias = Bias =
Composite Guidance Footprints
25,000 ft Launch2=n
10,000 ft Launch2=n
Composite Guidance increases the footprintsignificantly compared to GENEX only• 10-25 NMI farther for 25 kft launch• 5-10 NMI farther for 10 kft launch
Like GENEX, Composite Guidance becomes less effective with dropin launch altitude
Gunship Directionof Flight
Trajectories
0
0.5
1
1.5
2
-7-6-5
-4-3-2-10
123
0
5
10
15
20
25
30
35
40
45
50
Crossrange (NMI)Downrange (NMI)
Alti
tude
(kft)
GENEXBias = 1Bias = 2Bias = 3Bias = 4Bias = 5
Target: Cross range = -7 NMI25,000 ft Launch
2=n
Gunship Directionof Flight
GENEXBias = 1Bias = 2Bias = 3Bias = 4Bias = 5
All cases hit thetarget with required
impact angle
In all cases, themunition passes well
in front of thegunship’s flight path
Velocities and Accelerations
0
500
1000
1500
2000
2500
3000
0 20 40 60 80 100 120 140 160
Time (sec)
Velo
city
(fps
) GENEXBias = 1Bias = 2Bias = 3Bias = 4Bias = 5
0
200
400
600
800
1000
1200
0 20 40 60 80 100 120 140 160
Time (sec)
Ach
ieve
d A
ccel
erat
ion
(ft/s
^2)
GENEXBias = 1Bias = 2Bias = 3Bias = 4Bias = 5
Target: Cross range = -7 NMI25,000 ft Launch
2=n
The 5 g Bias case gives a 40% reduction in max accelerationand still achieves approximately the same impact velocity
GENEX
GENEX
Bias = 1, 2, 3, 4Bias = 5 Bias = 5
Target: Downrange = -28 NMI, Cross range = -15 NMI25,000 ft Launch
Trajectories
2=n
Only the 5 g Biashits the targetwith the requiredimpact angle
GENEXBias = 1Bias = 2Bias = 3Bias = 4Bias = 5Gunship Direction
of Flight
The 4 g Bias hits the target,but falls well short of therequired impact angle
Velocities and Accelerations
0
500
1000
1500
2000
2500
3000
0 50 100 150 200 250 300 350
Time (sec)
Velo
city
(fps
) GENEXBias = 1Bias = 2Bias = 3Bias = 4Bias = 5
0
100
200
300
400
500
600
700
0 100 200 300 400
Time (sec)
Ach
ieve
d A
ccel
erat
ion
(ft/s
^2)
GENEXBias = 1Bias = 2Bias = 3Bias = 4Bias = 5
Target: Downrange = -28 NMI, Cross range = -15 NMI25,000 ft Launch
2=n
The munition expends too much energy to reach the target.Only the 5 g Bias case, with its increased curvature, is ableto reach the target with the required impact angle.
Bias = 5
Bias = 5
GENEX
Conclusions
• Composite Guidance, a blending of GENEX guidance with a bias acceleration vector, provides a larger footprint than GENEX alone
• Composite Guidance extends the effectiveness of GENEX guidance
– Allows the munition to reach 15-25 NMI farther on the starboard side and 10 NMI farther behind the gunship for a 25 kft altitude launch
– Allows the munition to reach 5-10 NMI farther on the starboard side and 5 NMI farther behind the gunship for a 10 kft altitude launch
• Able to meet strict Military Operations on Urban Terrain (MOUT) requirements
– Munition comes in vertically to allow an attack in urban canyons, simplifying mission planning
– Small miss distance for low collateral damage
Backup Slides
Generalized Vector Explicit Guidance (GENEX)
ubXAX +=& ( ) 00 XtX = ( ) 0== ff XtX
Define a set of linear state equations and boundary conditions as:
where is the state and is the control (missile acceleration).
Select a cost function of the form:
( )∫ ∫==0 0
0 0
2
,2
T T
n dTTuLdTTuJ
X u
where is the time-to-go, and is an integer .The above equation comprises a family of cost functionsparameterized by the index .
ttT f −= n 0≥
n
Reference: Ohlmeyer, E. J., “Control of Terminal Engagement Geometry Using Generalized Vector Explicit Guidance,” Proceedings of American Control Conference, Vol. 1, 2003, pp. 396-401.
= zero effort miss with respect to a specified final position= difference between current velocity and desired final velocity= desired final missile position and velocity
Generalized Vector Explicit Guidance (GENEX)
Mf
MMf
yyX
TyyyzX&&
&
−==
−−==
ν2
1Define the states as:
subject to the terminal conditions and at .0=z 0=ν 0=T
ff y,y &vz
)2)(1(2 ++−= nnK
)3)(2(1 ++= nnKDefine the gains as:
Reference: Ohlmeyer, E. J., “Control of Terminal Engagement Geometry Using Generalized Vector Explicit Guidance,” Proceedings of American Control Conference, Vol. 1, 2003, pp. 396-401.
( ) ( ) ( ) ( ) ( ) ( ) ( )ttattattata N 332211 φφφ ++=
Treat Guidance Command as composed of a set of basis functions
Increasing the number of degrees of freedom in the Basis Set increases flexibility and thus allows performance to equal or surpass lower order forms
In comparison with Open-Loop Trajectory Optimization, GENEX forms have shown excellent match with the mid-trajectory regions
GENEX matching in regions of transition at initiation and termination of trajectories may be improved
Thus, use GENEX as one of the basis functions
Add another function that is active only for initial phase (ramp)
Select Bias as simplest form for initial investigations
Composite Guidance Rationale
Target: Downrange = 20 NMI25,000 ft Launch
Trajectories
2=n
Launch
Velocities and Accelerations
500
1000
1500
2000
2500
3000
0 20 40 60 80 100 120 140
Time (sec)
Velo
city
(fps
) GENEXBias = 1Bias = 2Bias = 3Bias = 4Bias = 5
0
100
200
300
400
500
600
700
800
0 20 40 60 80 100 120 140
Time (sec)
Ach
ieve
d A
ccel
erat
ion
(ft/s
^2)
GENEXBias = 1Bias = 2Bias = 3Bias = 4Bias = 5
Target: Downrange = 20 NMI25,000 ft Launch
2=n
Composite Guidance allows for smaller accelerationsto hit target with approximately the same impact velocity
GENEX