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