ADJOINT ANALYSIS OF GUIDED PROJECTILE TERMINAL PHASE

Timo Sailaranta and Ari Siltavuori

Faculty of Engineering and Architecture

Aalto University, School of Science and Technology, Finland

NOTATION

a Acceleration

mC Pitching moment

coefficient qSd

M

mC Pitching moment

coefficient slope

mC

qmC Pitch damping moment

coefficient

VQd

Cm

2

mC Pitching moment control

derivate

mC

NC Normal force coefficient qS

N

NC Normal force coefficient

slope

NC

NC Normal force control

derivate

NC

d Diameter

Iy Moment of inertia

G Gain

l Length

L Laplace-transform

denotation

M Pitching moment

m Mass

n Load factor

q Kinetic pressure

2

2

1V

Angular velocity

s Laplace -s

S Reference area

4

2d

t Time

tgo Time-to-go

tF Flight time, final time

w Target lateral velocity

V Velocity

N Navigation gain, normal

force

α Angle of attack

Canard deflection angle

Damping ratio

Natural frequency

Time constant

ρ Air density

λ Seeker-head turning

angle

Miss distance

adjx Adjoint vector

Subscribts

AF Airframe

AP Autopilot

c Command, closing

SH Seeker-head

N Noise

T Target

o Initial value

Summary–Guided projectile terminal phase against target at ground level is investigated

using an adjoint simulation. A pseudo-optimal projectile navigation gain is looked for

against a target disturbing the projectile guidance. The use of counter-measures is

“modeled" as a suddenly detected target abrupt motion during the guidance terminal phase.

The miss distances obtained are studied and the projectile optimal navigation gain is

chosen based on the maximum tolerated miss distance.

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INTRODUCTION

Beside the time-forward direct simulations the adjoint technique is often utilized in guided

weapon end-game analysis. The method has particularly merit to quickly give performance

projections of linear time-variant systems. So far the method has not been used as widely as one

would expect based on its flexibility and application potential [1].

The objective of this paper is to study the capability of the adjoint method to predict the end-

game miss distance. An optimal system gain is chosen based on the miss distances obtained.

However, at first some generic guided projectile aerodynamic properties and other characteristics

etc are estimated to find out some representative adjoint simulation input data.

GENERIC GUIDED PROJECTILE

A generic guided projectile studied is depicted in Fig. 1 and its characteristics are listed in

Table 1.

Fig. 1. A generic canard-controlled guided projectile.

Table 1. Physical and geometrical data of a generic canard-controlled guided projectile.

Characteristics Value Characteristics Value

diameter, d 155 mm NC 12

length, l 1000 mm NC 3

mass center (CG)

(from the nose) 580 mm qmC -90

weight 50 kg mC -0.5

moment of inertia, Iy 3.25 kgm2 mC 5

wing span 350 mm Actuator dynamics ω = 100 rad/s; ζ = 0.7

canard control fin

span 300 mm Ref area, S

4

2d

A simple acceleration autopilot model was used to find out the projectile step command

response in order to choose realistic autopilot (AP) pole and projectile airframe (AF) quadratic

pole locations. The autopilot block diagram is depicted in Fig. 2.

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Fig. 2. Autopilot block diagram. The gain values solved are G1=1.1494, G2=0.0221, G3=7.1408

and G4=-0.0781 (Ma=0.9 at flight altitude 1000 m).

Fig. 3. Projectile response to 5 g lateral acceleration command (Ma=0.9 at flight altitude

1000 m). The equivalent time constant (63 % of command reached) without seeker-head

contributions is about 0.25 s.

The projectile command response including the autopilot and actuator lag at Mach number 0.9

and at 1000 m altitude is depicted in Fig. 3. Corresponding angle of attack history with

commanded and true fin deflections are depicted in Figures 4 and 5. The command update

frequency used was 100 Hz.

The total equivalent time constant is approximately 0.25 s (see Fig. 3) about which the

airframe contribution is 0.15 s if the actuator portion is taken to be negligible. With the damping

factor 0.5 chosen the resulting natural frequency ω is about 10 rad/s.

Fig. 4. Angle of attack time history.

0

1

2

3

4

5

6

0 0,2 0,4 0,6 0,8 1 1,2 1,4

Load

fac

tor

[g]

Time [s]

Response

Command

0

2

4

6

8

10

12

14

0 0,5 1 1,5

An

gle

of

atta

ck [

de

g]

Time [s]

G1

G2

s

G3

Airframe

a

q

Actuator

ac

G4

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Fig. 5. Fin deflection histories obtained.

ADJOINT SIMULATION MODEL

The baseline projectile linearized guidance loop at the background of the adjoint model is

depicted in Fig. 6. In this study a 5th

order guidance loop with three real poles and a quadratic

distribution models the guided projectile systems. The three real pole time constants in the loop

are for seeker-head lags (τSH for seeker-head and τN for noise filter) and for autopilot (τAP). The

time constants are 0.1 s for each of first order components. The projectile airframe (AF) inertia is

modeled with a second order response. The natural frequency ω and the damping ratio ζ are

those above-mentioned 10 rad/s and 0.5 respectively. The projectile systems total time constant

obtained is about τtot = 0.45 s.

Fig. 6. The time-forward missile guidance loop used in this study. The adjoint model is based on

this original system.

Projectile maneuvering capability was not limited for the sake of the adjoint system linearity.

No aerodynamic data was explicitly present in the simple loop of Fig. 6.

The standard proportional navigation algorithm was used in this study. The closing speed Vc

in the navigation formula is practically the same as the projectile velocity at the descending part

of trajectory. This is assumed typically to be about the speed of sound. Constant value 300 m/s

was used for the projectile velocity in the computations.

0

0,5

1

1,5

2

2,5

0 0,1 0,2 0,3 0,4 0,5

Fin

def

lect

ion

[d

eg]

Time [s]

Response

Command

s

1

s

1

goctV

1

SH

1

s

1

sN1

1

2

221

1

ss

sAP1

1

-

-

cNV

Ta

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The adjoint system can be time-varying and for example the navigation gain may change

during the simulation. However, the gain of the original system must be generated backwards for

the adjoint system [1].

In the traditional presentation with the inverted block-diagram signal flow the original system

output of interest (the miss distance) is seen to become an impulsive input to the adjoint system.

Correspondingly the original system input turns into an adjoint output [1]. However, the

traditional adjoint construction is not performed in this study. Instead of that the presentation

follows the text of Ref. [2] with the adjoint method derived in the general setting of state-space

models.

The block diagram of Fig. 6 system can be written in state space form

)()( tBtA uxx (1)

)(tCxy (2)

and we obtain by inspection

T

AF

AF

AP

N

SH

APAPc

NNSHgocNSH

SHgocSH

AF

AF

AP

N

SH

n

a

a

aNV

tV

tV

a

a

adt

d

0

0

0

0

0

0

1

0100000

20000

00/1/000

000/1/1/10

0000/1/10

0000001

0000000

2

122

2

1

(3)

The input to system is the target maneuvering nT(t) which is taken to be 0 in this study.

Variable tgo = tF - t is the time-to-go from the impulse initiation (= resolution of the target

movement) to the interception and tF is the final time or the time of flight. The seeker-head

turning angle and airframe are denoted as λ and AF.

The miss distance is wished as the result and the output is chosen to be

y= [0 1 0 0 0 0 0]

2

1

AF

AF

AP

N

SH

a

a

a

(4)

where the matrix C is [0 1 0 0 0 0 0].

The adjoint of the time-forward state-space model is

adjTadj

go

At

xx d

d (5)

Tadj C)0(x (6)

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The signal flow in the adjoint loop is physically meaningless. The results wished are obtained

by processing the outcomes suitably outside the loop. For example the miss distance ξ due to the

target constant lateral velocity w is obtained from

)()( 1 oFadj

F ttwxt (7)

The system is linear and some target complex maneuvering effect on the miss distance can be

obtained utilizing the superposition principle for simple maneuvers.

END-GAME GEOMETRY

The target is at ground and is disturbing the terminal phase of the projectile flight. The

projectile approaches the target about from above (see Fig. 7). Only the final seconds of flight

are investigated.

The projectile resolves the true target motion at some point of the terminal phase. The target

movement consists of constant lateral velocity 10 m/s to East associated with oscillating

longitudinal (North-South) velocity (peak value 10 m/s). The target possess either cosine-

distributed or sinusoidal velocity oscillation with angular velocity 1 rad/s. The positive directions

are to East and North (right and forward respectively, see Fig. 7).

The projectile pitch and yaw guidance loops are identical and are studied separately. In

practice this is to combine the adjoint outcomes of the same loop using the Pythagoras formula

in order to get 2D miss distance results. Once the adjoint vector xadj

(or “error tracks”) is

obtained it will be readily available for studies to find out various maneuver combination’s effect

on the miss distance.

Fig. 7. The end game geometry studied. The projectile approaches about from above and the

target is located at the Origin at ground level 0 m. The true target abrupt motion is detected at

some point of the terminal phase. In the adjoint simulations the sudden movement will take place

at all tgo -values (all distances) in one run.

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The sinusoidal maneuver can be represented as an impulse through a second-order shaping

network [1] since

22))sin((

s

wtwL (8)

The corresponding Laplace transform of cosine-distributed maneuver is

22))cos((

s

wstwL (9)

where w and ω are the target velocity and the angular velocity. The miss distances wished are

computed by integrating the error trackadj

x2 through the networks.

The target evasive manoeuvre presented is perhaps not realistic but it is hoped to be

illustrative in the context. The target paths on the ground seen from above are depicted in Fig. 8.

Some slant trajectory angle effect could obviously be introduced in the value of longitudinal

velocity oscillation amplitude.

Fig. 8. The target end game manoeuvres as a result of velocity oscillation for 0...10 s motion.

The projectile is approaching the coordinate system Origin from about above. The target motion

from the Origin is detected during the terminal phase.

RESULTS

At first in Fig. 9 is depicted the linearized time-forward simulation and the reversed adjoint

system results obtained in case of a pure target lateral velocity 10 m/s. In case the navigation

ratio N was 3. The projectile-target initial distance was varied in the time-forward runs to find

out the miss distances as a function of flight time tf. The corresponding adjoint tgo-graph was

obtained in a single computation. The miss distance is simply the lateral velocity times the

readily available error track adj

x1 . The results are seen to match perfectly.

-15

-10

-5

0

5

10

15

20

25

0 20 40 60 80 100

No

rth

[m

]

East [m]

Cosine oscillation

Sinusoidal oscillation

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Fig. 9. The miss distances in case of a pure target lateral velocity as a function of the flight time

or time-to-go.

Secondly the adjoint results are compared with the ones obtained using nonlinear time-

forward based method [3]. The total miss distances obtained are depicted in Figures 10 and 11.

The agreement of results is seen to be at least fair.

Fig. 10. The miss distances in case of target constant lateral velocity and cosine distributed

longitudinal velocity. The results are presented as a function of the time-to-go and the adjoint

and nonlinear results are compared (N=3).

-4

-2

0

2

4

6

8

10

0 1 2 3 4 5 6

Mis

s d

ista

nce

[m

]

Flight time or time-to-go [s]

Adjoint

150 m

300 m

450 m

600 m

900 m

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7

Mis

s D

ista

nce

[m

]

tgo [s]

Adjoint

Nonlinear

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Fig. 11. The miss distances in case of target constant lateral velocity and sinusoidal longitudinal

velocity as a function of the time-to-go. The results are presented as a function of the time-to-go

and the adjoint and nonlinear results are compared (N=3).

It is worth of noting that the projectile peak accelerations obtained at small tgo-values are

fairly high (up to 10…15 g) and may exceed the true maneuvering capability available. The

projectile acceleration has not so far been limited in this study since it is not possible in the

adjoint analysis. The third work phase was to carry out the nonlinear computations once more

with g-limit 5 for both channels separately. The limitation effect is depicted in Fig. 12 for the

case with cosine distributed longitudinal velocity. The pattern is still recognizable even though

the new results are seen to be considerably larger when tgo < 3 s.

Fig. 12. The miss distances in case of target constant lateral velocity and cosine distributed

longitudinal velocity. The results are presented as a function of the time-to-go. The adjoint and

nonlinear results also with g-limit effects included are compared (N=3).

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7

Mis

s D

ista

nce

[m

]

tgo [s]

Adjoint

Nonlinear

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7

Mis

s D

ista

nce

[m

]

tgo [s]

Adjoint

Nonlinear

Nonlinear+ 5 g limit

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Fig. 13. The miss distances obtained for the N=1 case. The gain is too small to make the

projectile maneuver enough to hit the target.

Fig. 14. The miss distances obtained for the N=2 case.

Fig. 15. The miss distances obtained for the N=3 case. The gain gives most hit opportunities.

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7

Mis

s D

ista

nce

[m

]

tgo [s]

Lat Velocity+Long Cos Velocity

Lat Velocity+Long Sin Velocity

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7

Mis

s D

ista

nce

[m

]

tgo [s]

Lat Velocity+Long Cos VelocityLat Velocity+Long Sin Velocity

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7

Mis

s D

ista

nce

[m

]

tgo [s]

Lat Velocity+Long Cos Velocity

Lat Velocity+Long Sin Velocity

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Fig. 16. The miss distances obtained for the N=4 case.

The obtained miss distances with the navigation gain as a variable are depicted in Figures

13…16. The results are presented as a function of time-to-go tgo. Only four runs were needed for

the four different navigation gains N (1, 2, 3 and 4) considered to obtain the results presented.

The total miss distances caused by the constant lateral and oscillating longitudinal velocity were

obtained using the Pythagoras formula utilizing the same error track for the both channels

separately.

The hit-criterion in this paper is defined to be 3 meters or less. The tgo –windows to hit for

different navigation gains N (1, 2, 3 and 4) were compared and based on this very limited study it

seems that the case with N=3 gives most hit opportunities.

CONCLUDING REMARKS

The adjoint method was used to obtain the miss distances against a target located at ground.

The navigation gain N was varied and the pseudo-optimal value was found to be 3 for the

weapon systems and end-game case studied. With some simplifications and assumptions done on

mind the method proves to be capable to produce the projectile performance projections quickly.

The case studied, expanded from the presentation of Ref. [4] including now ie the two channels

guidance loop and the second order projectile response, is still very simplistic. The method

flexibility allows investigating far more complex systems.

REFERENCES

[1] Paul Zarchan, 1997, Tactical and Strategic Missile Guidance. AIAA Progress in Astronautics and Aeronautics,

176.

[2] Martin Weiss, 2005, Adjoint Method for Missile Performance Analysis on State-Space Models. AIAA Journal

of Guidance, Control and Dynamics, 28(2).

[3] Timo Sailaranta and Ari Siltavuori, 2009, A Simplified Missile Model Against Maneuvering Target.

Proceedings of NSCM-22: the 22nd

Nordic Seminar on Computational Mechanics, October 22-23, Aalborg,

Denmark.

[4] Timo Sailaranta and Ari Siltavuori, 2010, Adjoint Simulation of Guided Projectile Terminal Phase. Proceedings

of NSCM-23:23rd

Nordic Seminar on Computational Mechanics, October 21-22, Stockholm, Sweden.

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7

Mis

s D

ista

nce

[m

]

tgo [s]

Lat Velocity+Long Cos VelocityLat Velocity+Long Sin Velocity

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