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Theses and Dissertations Thesis Collection

1983

Bank-to-turn cruise missile terminal guidance and

control law comparison.

Watterson, Kent B.

Monterey, California. Naval Postgraduate School

http://hdl.handle.net/10945/19654

Dudley Knox Library, UPSMonterey, CA 93943

NAVAL POSTGRADUATE SCHOOL

Monterey, California

THESISBANK-TO-TURN CRUISE MISSILE TERMINALGUIDANCE AND CONTROL LAW COMPARISON

by

Kent B. Watterson

June 1983

Thesis Advisor: Marie D. Hewett

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Bank-To-Turn Cruise Missile TerminalGuidance and Control Law Comparison

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Master' s ThesisJune 1983

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7. AUTHORfaJ

Kent B. Watterson

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18. SUPPLEMENTARY NOTES

19. KEY WORDS (Continue on ravataa aid* It naeaaaarr and Idantlty by block numbar)

Bank-To-Turn Terminal HomingProportional NavigationGuidance and Control

L

20. ABSTRACT fContintf on ravaraa alda It nacaaaaty and Idantlty by block numbar)

This work consists of the development of the six degree of freedorrnon-linear model of a sea launched generic bank-to-turn cruisemissile attacking a medium sized combatant ship. Two guidanceand control schemes are compared in the terminal phase. Thefirst, or baseline guidance scheme (pop out maneuver), uses a50-foot altitude hold for an ingress phase, followed by a pop outmaneuver, and then an attack phase which uses proportional

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20. ABSTRACT (cont'd)

navigation in elevation and azimuth planes along with bank-to-turn maneuvering. The second scheme (sea skimmer) uses identicalingress and attack phases but eliminates the pop out maneuver.Miss distances for both schemes are compared while varyingmissile roll rate limit, ECM blinking frequency, and burnthrough ranges.

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2


Bank-To-Turn Cruise Missile TerminalGuidance and Control Law Comparison

by

Kent B. WattersonLieutenant Commander, United States Navy

B.S., Findlay College, 1969

Submitted in partial fulfillment of therequirements for the degree of

MASTER OF SCIENCE IN ENGINEERING SCIENCE

from the

NAVAL POSTGRADUATE SCHOOLJune 1983

f <"

Dudley Knox Library. NPS

Monterey, CA 93943

ABSTRACT

This work consists of the development of the six degree

of freedom non linear model of a sea launched generic bank-to-

turn cruise missile attacking a medium sized combatant ship.

Two guidance and control schemes are compared in the terminal

phase. The first, or baseline guidance scheme (pop out

maneuver), uses a 50-foot altitude hold for an ingress phase,

followed by a pop out maneuver, and then an attack phase which

uses proportional navigation in elevation and azimuth planes

along with bank-to-turn maneuvering. The second scheme

(sea skimmer) uses identical ingress and attack phases but

eliminates the pop out maneuver. Miss distances for both

schemes are compared while varying missile roll rate limit,

ECM blinking frequency, and burn through ranges.

k

TABLE OF CONTENTS

I. INTRODUCTION -. 17

II. DEVELOPMENT AND SIMULATION OF MISSILE DYNAMICS ... 20

A. MISSILE EQUATIONS OF MOTION 20

1 . Assumptions 21

2. Coordinate System 23

3. The Equations of Motion 23

4-. Trim Equations 26

5. Other Useful Relations 27

6. Definition of Controls and Control Limits . 27

B. AERODYNAMIC COEFFICIENT BUILD UP 29

1. Definition of Coefficients 29

2. Coefficient Build Up 30

C. MISSILE LINEARIZED EQUATIONS OF MOTION 32

1. Additional Assumptions for Linearized

Equations 32

2. Linearized Equations Summarized 33

3. Definition of Stability Derivatives .... 34-

4-. Trim Equations 36

III. DEVELOPMENT AND SIMULATION OF THE MISSILE AUTOPILOT. 37

A. AUTOPILOT INNER LOOP REQUIREMENTS AND DESIGNS . 37

1 . Description of the Autopilot Inner Loops . . 37

2. Normal Acceleration Command System 39

3. Bank Angle Command System 4-4-

4. Turn Coordinator 4-9

B. AUTOPILOT OUTER LOOP REQUIREMENTS AND DESIGNS . 57

1 . Description 57

2. Vertical Flight Path Angle System 57

3. Altitude Hold System 58

IV. DEVELOPMENT AND SIMULATION OF MISSILE GUIDANCE

SYSTEM 63

A. MISSION DESCRIPTION 64

1 . Baseline Guidance Scheme (Pop Out Maneuver) . 64

2. Alternate Guidance Scheme (Sea Skimmer) ... 65

3. Electronic Counter Measures (ECM) 65

4.. Glint 68

B. SEEKER EQUATIONS AND SIMULATION 68

1 . Line of Sight Rates 68

2. Missile Dynamic Filters 73

C. BASELINE GUIDANCE LAW DESIGN (POP OUT MANEUVER) . 73

1. Attack Phase 73

2. Ingress Phase 78

3. Offset Phase 80

4-. Pop Up Phase 81

5. Guidance Summary (Baseline) 83

D. ALTERNATE GUIDANCE LAW DESIGN (SEA SKIMMER) ... 83

1 . Ingress Phase 83

2. Attack Phase 84

3. Guidance Summary (Sea Skimmer) 85

V. PROBLEM SIMULATION, RESULTS, CONCLUSIONS AND

RECOMMENDATIONS 86

A. CSMP SIMULATION 86

B. RESULTS 88

6

c.

1

.

Analysis of Base Line Scheme Results . .

2. Analysis of Sea Skimmer Scheme Results .

3. Analysis of Glint Simulation Results . .

4-. Proportional Navigation Ratio Selection

CONCLUSIONS AND RECOMMENDATIONS

APPENDIX A

APPENDIX B

APPENDIX C

APPENDIX D

AERODYNAMIC COEFFICIENTS

STEADY STATE DATA

AUTOPILOT ROOT LOCUS PLOTS

BASELINE GUIDANCE LAW SIMULATION

APPENDIX E: SEA SKIMMER GUIDANCE LAW SIMULATION

LIST OF REFERENCES

INITIAL DISTRIBUTION LIST

92

129

129

132

132

134

1 51

153

166

176

187

1 88

LIST OF TABLES

2-1 Longitudinal Equations (Perturbed Variables) . ..". 34-

2-2 Lateral Directional Equations (Perturbed Variables). 34-

3-1 Autopilot Inner Loop Description 38

5-1 Baseline Guidance and Control Scheme 90

5-2 Sea Skimmer Guidance and Control Scheme 91

8

LIST OF FIGURES

2-1 Missile System Flow Diagram -. 22

2-2 Body Coordinate System Description 25

3-1 Normal Acceleration Command Autopilot Description . 4-0

3-2 Normal Acceleration Command Autopilot Block

Diagram (Analysis) 4-2

3-3 NAC Autopilot Response to 1 g Commanded Load Factor. 45

3-4- Bank Angle Command Autopilot Description 4.6

3-5 BAC Autopilot Response to 60 Commanded Bank Angle

(PHI Response) 50

3-6 BAC Autopilot Response to 60 Commanded Bank Angle

(Altitude Response) 51

3-7 Turn Coordinator Autopilot Description 52

3-8 Turn Coordinator Autopilot Block Diagram

(Analysis)

3-9 System Response to 8.6 Commanded Vertical Flight

Path Angle 59

3-10 System Response to 50 Ft. Altitude Hold Command . 62

4.-1 Baseline Guidance Scheme Flight Profile 66

4.-2 Sea Skimmer Guidance Scheme Flight Profile .... 67

4.-3 Seeker Simulation Flow Chart ." 69

4.-4- Azimuth Line of Sight Description 71

4.-5 Elevation Line of Sight Description 72

4.-6 Baseline Guidance Law Flow Diagram 74-

4.-7 Load Factor and Bank Angle Description 76

5-1a Baseline Response for No ECM, Roll Rate Limit

100 deg/sec (PHICMD, PHI vs TIME) 93

9

5-1 b Baseline Response for No ECM, Roll Rate Limit

100 deg/sec (PCMD, P vs TIME) 94

5-1 c Baseline Response for No ECM, Roll Rate Limit

100 deg/sec (RANGE, XRANGE, YRANGE, ZRANGE vs

TIME) 95

5-1 d Baseline Response for No ECM, Roll Rate Limit

100 deg/sec (L STAB, R STAB vs TIME) 96

5-1 e Baseline Response for No ECM, Roll Rate Limit

100 deg/sec (RUDDER vs TIME) 97

5-1 f Baseline Response for No ECM, Roll Rate Limit

100 deg/sec (YMISLE, HMISLE vs XMISLE) 98

5-2a Baseline Response for ECM 1 CPS, Roll Rate Limit

100 de'g/sec (PCMD,P vs TIME) 99

5-2b Baseline Response for ECM 1 CPS, Roll Rate Limit

100 deg/sec (PHICMD, PHI vs TIMS) 100

5-2c Baseline Response for ECM 1 CPS, Roll Rate Limit

100 deg/sec (RANGE, XRANGE, YRANGE, ZRANGE,

vs TIME) 101

5-2d Baseline Response for ECM 1 CPS, Roll Rate Limit


5-2e Baseline Response for ECM 1 CPS, Roll Rate Limit

100 deg/sec (RUDDER vs TIME) 1 03

5-2f Baseline Response for ECM 1 CPS, Roll Rate Limit

100 deg/sec (YMISLE, HMISLE vs XMISLE) 104.

5-3a Baseline Response for ECM 1 CPS, Roll Rate Limit

50 deg/sec (PCMD,P vs TIME) 105

10

5-3b Baseline Response for SCM 1 CPS, Roll Rate Limit

50 deg/sec (PHICMD.PHI vs TIME) 106

5-3c Baseline Response for ECM 1 CPS, Roll Rate Limit

50 deg/sec (RANGE, XRANGE, YRANGE, ZRANGS vs

TIME)'

1 07

5-3d Baseline Response for ECM 1 CPS, Roll Rate Limit


5-3e Baseline Response for ECM 1 CPS, Roll Rate Limit


5-3f Baseline Response for SCM 1 CPS, Roll Rate Limit

50 deg/sec (IMISLS, HMISLS, vs. XMISLE) 110

5-4-a Baseline Response for ECM 1 CPS, Roll Rate Limit


5-4-b Baseline Response for ECM 1 CPS, Roll Rate Limit

200 deg/sec (PHICMD,PHI vs TIME) 112

5-4-c Baseline Response for ECM 1 CPS, Roll Rate Limit


TIME) 113

5-Ad Baseline Response for ECM 1 CPS, Roll Rate Limit

200 deg/sec (L STAB, R STAB vs TIME) 11 4.

5-4-6 Baseline Response for ECM 1 CPS, Roll Rate Limit

200 deg/sec (RUDDER vs TIME) 11 5

5-4f Baseline Response for ECM 1 CPS, Roll Rate Limit


5-5a Sea Skimmer Response for No ECM, Roll Rate Limit


11

5- 5b Sea Skimmer Response for No ECM, Roll Rate Limit

100 deg/sec (PHICMD, PHI vs TIME) 118

5-5c Sea Skimmer Response for No ECM, Roll Rate Limit


TIME) 119

5-5d Sea Skimmer Response for No ECM, Roll Rate Limit


5-5e Sea Skimmer Response for No ECM, Roll Rate Limit


5-5f Sea Skimmer Response for No ECM, Roll Rate Limit


5-6a Sea Skimmer Response for ECM 1 CPS, Roll Rate Limit


5-6b Sea Skimmer Response for ECM 1 CPS, Roll Rate Limit

100 deg/sec (PHICMD, PHI vs TIME) 12^

5-6c Sea Skimmer Response for ECM 1 CPS, Roll Rate Limit


TIME) 125

5-6d Sea Skimmer Response for ECM 1 CPS, Roll Rate Limit


5-6e Sea Skimmer Response for ECM 1 CPS, Roll Rate Limit


5-6f Sea Skimmer Response for ECM 1 CPS, Roll Rate Limit

100 deg/sec (YMISLE, HMISLE vs X MISLE) 128

5-7a Baseline Response with Glint (SYT,THETAT vs

TIME) 130

5-7b Baseline Response with Glint (XT,YT vs TIME) ... 131

12

SYMBOLS AND ABBREVIATIONS

U Linear Velocity along body x axis

V Linear Velocity along body y axis

W Linear Velocity along body z axis

P Roll rate

Q Pitch rate

R law rate

X Aerodynamic force in body x direction

Y Aerodynamic force in body y direction

Z Aerodynamic force in body z direction

$ Bank angle

Pitch angle

^ Yaw angle

X-r, Earth coordinate of missile (Latitude)

Yth Earth coordinate of missile (-Longitude)

-Zr,=H„ Earth coordinate of missile (Altitude)

L. Rolling Moment about x axis (Aerodynamic)

M. Pitching moment about y axis (Aerodynamic)

N. Pitching moment about z axis (Aerodynamic)

V m Total missile velocity

T Thrust_ 2q Dynamic pressure (t p 'L )

a Angle of attack

S Side slip angle

13

Y Flight path angle (9 + «)

p Air density (0.002377) î^/ft3

S Wing area (12 ft.2

)

b Wing span (8.4-85 ft. )

c Mean aerodynamic chord (1.4.14 ft.)

n Elevator Deflection

£ Aileron Deflection

£ Rudder Deflection

n TLeft Stabilator Deflection

n D Right Stabilator DeflectionR

L Lift force

D Drag force

N Normal force

C Chord wise force

m Missile mass (68.38 slugs)

g Acceleration due to gravity (32.17 ft/sec )

ST Static coefficient

DYN Dynamic coefficient

n Normal load factor (body fixed axis system)

*

z "

n Lateral load factor (body fixed axis system)-'

n?

Vertical load factor (earth fixed axis system)"""

n~ Horizontal load factor (earth fixed axis system

u Perturbed linear velocity x axis

v Perturbed linear velocity y axis

w Perturbed linear velocity z axis

U

p Perturbed angular velocity about x axis

q Perturbed angular velocity about y axis

r Perturbed angular velocity about z axis

(J)Perturbed Euler angle

\/~/\

8 Perturbed Euler Angle-x-

i|) Perturbed Euler Angle ¥

* Laplace transforms of these variables are shown

in capital letters

SUBSCRIPTS

SS Steady State

E Earth

B Body

x,y,z In body fixed axes

X,Y,Z In earth fixed axes

T Due to thrust/Target

A Due to Aerodynamic Forces

xx,yy,zz About the axis specified

ST Steady State

DYN Dynamic

EL Eleration

AZ Azimuth

c Commanded

15

ACKNOWLEDGMENT

The author wishes to express his sincere appreciation

to Professor M. D. Hewett for his guidance, direction and

understanding in completing this work.

16

I. INTRODUCTION

The relative merits of bank-to-turn versus skid-to-turn

missiles have been argued for years. Bank-to-turn missiles

which must roll to a commanded bank angle before a lateral

acceleration can be commanded in the appropriate direction are

inherently slower to respond to target maneuvering than

skin-to-turn missiles of comparable lateral acceleration

capabilities. This is particularly true if the bank-to-turn

missile has limited roll authority or limited roll rate

capability. On the other hand, as discussed in Gonzalez,

{Ref. 1}, it is sometimes difficult to build a skid-to-turn

missile which meets certain operational goals and performance

criteria and also has sufficient control authority in both

the y and z directions to generate the large lateral accel-

erations required to perform skid-to-turn.

In this thesis the terminal guidance accuracy of a

bank-to-turn cruise missile is studied for a sea launched,

sea target cruise missile of conventional wing-tail config-

uration. It is assumed that roll authority is extremely

limited due to operational design restrictions which impose

the use of low authority ailerons or differential stabilator

(no ailerons) for rolling. Lateral acceleration is also

assumed to be extremely limited in the y direction due to

the presence of only a small vertical surface at the tail

(vertical stabilizer and rudder).

17

Two bank-to-turn guidance schemes are compared for

accuracy against a medium sized combatant ship employing

electronic countermeasures (an ECM blinker mounted aft on

the ship). The first baseline scheme employs a pop-out

maneuver consisting of a low altitude run in, azimuth offset,

pop-up and roll into the target using proportional navigation

in azimuth and elevation. The second, or sea-skimmer scheme,

employs a straight-in low altitude attack. Miss distances

for both schemes are compared while varying missile roll

rate limit, ECM blinking frequency and burn through ranges.

A CSMP III simulation was coded to perform the study.

Missile dynamics were represented by 6 degree of freedom

nonlinear equations of motion with table look-up aero-

dynamic coefficients. These coefficients are representative

of a generic cruise missile with a conventional wing- tail

configuration of limited roll authority. Inner loop

augmentation and autopilot modes were designed to improve

missile damping in all axes, provide commanded load factor,

commanded bank angle, and turn coordination. Autopilot

outer loops were designed to provide altitude hold, and

vertical flight path angle hold. Guidance loops were designed

to provide proportional navigation in elevation and azimuth.

Since this study involves the influence of flight

dynamic parameters (roll rate limits) on terminal guidance

accuracy, no extensive tracker modeling is employed in the

simulation. It is assumed that the seeker always tracks

18

the point target perfectly. The point target is, however,

moved by an ECM blinker and contaminated with a glint model.

In Chapter II the equations used to represent missile

dynamics are presented along with the methodology used to

build up aerodynamic forces and moments. In Chapter III

the development, design and simulation of the missile auto-

pilot modes are presented. In Chapter IV the design of the

guidance systems and a complete description of the two

guidance schemes is presented. In Chapter V the CSMP

simulation is presented followed by the results and conclusions

of this study.

19

II. DEVELOPMENT AND SIMULATION OF MISSILE DYNAMICS

In this chapter the linear and nonlinear mathematical

models are developed which are used to describe the flight

dynamics of a generic bank-to-turn cruise missile. The

linearized mathematical model of missile dynamics is used

to design the missile autopilot control laws and guidance

laws. The nonlinear mathematical model is used to accurately

represent missile dynamics in a CSMP computer simulation of

a sea launched generic cruise missile attacking a medium

sized combatant ship. Since the purpose of this investigation

is to perform a detailed evaluation of terminal control

laws, it was decided that a linear dynamic simulation would

not adequately represent missile motion; hence a full six

degree of freedom (6 DOF) nonlinear dynamic simulation was

encoded. The overall system that is modeled in this study

is depicted in block diagram format in Figure 2-1 and will

be developed in this thesis.

A. MISSILE EQUATIONS OF MOTION

The full nonlinear six degree of freedom rigid body,

dynamic equations of motion are used to represent the motion

of a generic bank-to-turn cruise missile. The aerodynamic

forces and moments are built up from representative tabular

coefficient data for the generic missile. These tabular

20

data are given in Appendix A in graphical form along with the

generic missile's physical and geometric characteristics.

1 . Assumptions

The assumptions used in the development of the

equations of motion are given below.

a. The earth is flat, does not rotate, and is fixed

in inertial space.

b. The mass of the missile is constant.

c. The missile is a rigid body.

d. The mass distribution of the missile is constant.

e. Engine angular momentum is neglected.

f. The missile is symmetric about the body's xy

plane. Therefore, the products of inertia I and I are

zero.

g. The engine thrust line is parallel to the missile

body x axis. Thus, the thrust components T and T areJ y z

zero

.

h. The density of the atmosphere is constant.

i. The engine thrust line passes through the missile

center of mass. Therefore, the moments due to thrust L™,

Mm and N m are zero.

j. The wind is calm.

k. Right aileron trailing edge down is positive

deflection. Negative elevator deflection yields nose up

pitching moment (positive).

21

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22

2. Coordinate System

An earth fixed coordinate system is established with

its origin fixed at the initial position of the missile. The

Xth axis points to true north, the Y^ axis points east, and

the Z.-, axis points toward the center of the earth. Altitude

(H) therefore equals the negative of Zn. The system is

considered to be an inertial system.

A stability axis coordinate system is used with

coordinates x, y, z fixed at the missile center of mass and

oriented such that the x axis lies along the vehicle's

forward velocity vector in steady state balanced cruise

flight at 0.75 Mach number. The y axis is chosen perpendicular

to the plane of symmetry and is oriented out the right wing.

The z axis is chosen perpendicular to the x axis in the down

direction and in the plane of symmetry.

3

.

The Equations of Motion

The following nonlinear equations as developed by

Hewett {Ref. 2} describe the motion of the missile in

6 degrees of freedom.

m (U-VR+WQ) = -mg sin 9 + X + T (1a)

m (V+TJR-WP) = mg sin cos 9 + Y (1b)

m (W-UQ+VP) = mg cos cos 9 + Z (1c)

PI -(R+PQ)I + RQ(I -I ) = L. (2a)xx xz ^ zz yy A

QI +PR(I -I ) + (P2-R

2)I = M. (2b)^ yy xx zz xy A

RI -PI +PQ(I -I )+QRI = N, (2c)zz xz yy xx xz A

23

I = P+(Q sin <£> + R cos $) tan 9 (3a)

8 = Q sin $ - R sin .(3b)

4> = (Q sin $ + R cos $ ) sec 9 (3c)

Equations (1a), (1b) and (1c) describe the transla-

tional motion of the missile. Equations (2a), (2b), and

(2c) represent the rotational motion. Equations (3a),

(3b) and (3c) are the Suler relations for bank, pitch and

yaw angles.

Equations ( 4-a ) , (4-b), and (4-c) describe the missile's

position referenced to the earth fixed system (X^, I™, Z^).

v = U cos ^cos 9+ V( cos l

i; sin9sin*-sin4; cos$ ) (4-a)

+ W( cos^ sin9cos<l>+ sin^sinO )

Y^ = U sin fcos 9+ V( sinYsinesinfc+cos^cosfc ) (4-b)

+ W{ sin ll'sin9 cos^-cos^sin^ )

Zxn = Usin9-Vcos9sin$-Wcos9 co3<*> (4-c)

Equations (5), (6) and (7) are also required.

Equations (5) and (6) yield angle of attack and side slip

angle respectively in terms of velocity components U, V and

W. Equation (7) represents the total velocity of the missile

« = arctan 7T = arcsm « ~ i (5)U(u

2+w

2)

5

S = arctan?

„ = arcsin —s * s i (6)IT+w (U +V +W )

a

v = (U2+V

2+W

2)* (7)

Figure 2-2 illustrates the positive directions and

locations of forces, moments and velocities in the body

coordinate system (x, y, z).

24

X X

5 ..T, V

Aerodynamic and Thrust Forces Acceleration of Gravity

Aerodynamic and Thrust Moments Linear and Rotational (Angular)Velocities

Figure 2-2 Body Coordinate System Descripti on

25

Equations (1a), (1b), (1c) and (2a), (2b), (2c) can be

rewritten in state format as follows:

& = -ffsin9+VR-WQ + — + —5 ^ m m

V = gsin$cosQ-UR+WP + —& m

W = gcos$cos6+UQ-VP + -zL° m

(8a)

(8b)

P = L.I +N.I -PQ(I -I -I )l„ -RQ(I2

-I I +I2

)A zz A xz yy xx zz xz ^ v zz yy zz xz

(I I -I2

)XX zz xz

Q =[m.-PR(I -I )-(P2-R

2)l^ A xx zz xz / I

yy

R = L.I +N.I -PQ(I I -I2

-I2

)-RQ(l +1 -I )IA xz A xx yy xx xx xz xx zz yy xz

(8c)

/

(9a)

(9b)

/

(I I -I2

)XX zz xz(9c)

Now integrating equations (3), (4-), (8) and (9) results

in U, V, W, P, Q, R, !>, 8, 'f , X„, Y„ and Z^ (-H). The

missile's velocities, angular rates, and angles and positions

have thus been described.

4-. Trim Equations

For straight and level flight at constant velocity

the following variables are all zero: U, V, ¥, P, Q, R,

|, 0, | t ZE

, Pss , Q ss

, Rss , $ ss , L , N , I cc , y Qa ,

SS•SS' 'SS

3r-n» ^cjq» M. Thus the trim equations reduce to:

Xss

+Tx

= mg sineSS

,- = -mg cos 9SS

26

xe ss

= uss

GOSli/ss

cos9ss-

vss

sin^ss

+wss

cos4;ss

siness

hss

= uss

sinvpss

cosess

+vss

coslFss

+wss

sin*ss

siness

taness

= Wss

/Uss

VTSS

= (USS

2+ W

SS2) ~

Since bank angle equals zero and the sura of flight

path angle and angle of attack equal the pitch angle, then

in steady state pitch angle equals angle of attack.

B = oc

3S SS

These equations are used to define the initial cruise condi-

tion of the missile which is low level straight and level

cruise flight at 0.75 Mach number. The precise initial

flight condition and initial target position is described in

Appendix 3.

5. Other Useful Relations

Components of load factor at the missile center of

mass in the y and -z directions are given below:

V,

n-Zmg

T(Q- «) (10a)

V,

n (0 + R)y mg g

6. Definition of Controls and Control Limits

(10b)

The control configuration considered in the thesis

is a conventional wing and tail with conventional rudder

(?), aileron (§) and elevator (n) control surfaces. In

addition, to cover a missile configuration which uses

27

differential horizontal stabilizer (stabilator) for roll

control as opposed to aileron, the quantities left stabilator

deflection ( ru ) and right stabilator deflection (n R ) are

defined and calculated.

Limits are imposed on rudder, elevator, and aileron

travel. Limits are not imposed on differential stabilator

travel but stabilator travel is presented for all simulations

to show the stabilator travel required if indeed that were

the control configuration. The aileron power coefficient

data presented in Appendix A reflects the limited control

authority in roll typical of a missile configured with

differential stabilator for roll control as opposed to

aileron.

The following control definitions and limits are

applied

:

a) Thrust (T) is a constant and is oriented along

the x axis.

T = T = constant

b) Rudder deflection ( £ ) is defined as positive

trailing edge left from the rear and is limited to

" -15° < x, < 15°

c) Elevator deflection (n) is defined as positive

trailing edge down (produces a nose down moment) and is

limited to r\

-. -O „ -O-1 5 < n < 1 5

28

d) Aileron deflection {£_) is defined as positive

with trailing edge of the right aileron up (produces a

positive (clockwise from the rear) roll) and is limited to

-15 * £ < 15°

e) Left horizontal stabilizer deflection (

n

T ) is

positive trailing edge down.

n L= n + £

f) Right horizontal stabilizer deflection (rip) is

positive trailing edge down.

n R= n - C

3. AERODYNAMIC COEFFICIENT BUILD UP

Aerodynamic forces and moments are built up from coeffi-

cient data in standard fashion using the following relations

D - C^ q 5 = n. M *—

'

I =

m - nM

L . = C-.qS'BA 1 -

-I, = C i 3 zA n

1 Deiimtion o: uoemcien.s

Each coefficient is partitioned into two parts; a

static term and a dynamic term as follows:

+ C„cd " :

:z: DIN

GL

= CL

+ CL

v y yX ST "jYN

C = cm m

+ C

5T mDYN

c, = c, + C-,±ST

±DYN

29

c = c + cnST

nDYN

2. Coefficient Build Up

The static terms are constructed from coefficient

data stored in tables (Appendix A) as functions of two or

three variables (either «, 0, M, n, £»?) as follows:

CL

= f (-, $,M) eUv*Wbasic

AC L __( n ) . f («,6, n ) £ ; r«M^Lgr

CD

= f (oc,3,M)basic

AC n (n) = f («, 3. n)UST

ACD (?) = f («,8.5)ST

AC- (5) = f (<x,3, c )UST

GY = f (-,3)basic

ACy (5) = f («,B.c)-L rimO 1

ACV (§) = f (<x,3,£)X ST

Cm

- f (-,3,M)basic

AC ( n ) = f (oc,3, n )mST

Cn

= f (-,3)basic

AC ( c ) = f («,3, c )nST

AC ( ? ) = f («,Bt5)nST

30

L»1

= f («,B)basic

(?) = f (*,8,S)'ST

C, (?) « f («.B.c)

The static coefficients are formed as:

CT

= GT

+ ACT (n)

LST L basic

L

C n = C^ +AC n (n) + AC n (€) + AC n (?)UST ^basic ST

U3T

UST

C v = Cy +AC (?)+ACy (C)XST

x basic X STXST

+ AC„ (n)c = cmST

mbasic mST

C n = C n + AC n ( S) + AC n (?)ST basic ST ST

C-, = C, +AC. (C) + AC (?)ST basic ST

±ST

The dynamic terms are constructed from dynamic

coefficient data stored in tables (Appendix A) as a function

of angle of attack and Mach number f (QC,M). They are

L * _ L D" D xoi i m « m no n nair q o orp

0.D rp

C-, , and C-j .

r p

The dynamic coefficients are formed as:

C = — (CL.« f C q)

^DYN 2V TL *

q

DDIN 2V

T

(Cn.« + G. q)D

q

b"LDYN 2V,

(Cv r + C Y p + Gy • £)I

31

C = -£- (C .« + C q)mDYN 2V

T

m«mq

C = — (C 3+ C r + C p)nDYN 2V

TnS

nr

nP

C, = — (C, r + C, p)XDYN 2V

Tr p

The total coefficient build up is summarized below.

CT

= CT

+act

(n) + — (C -+C, q)L L basic

LST 2V

TLoc L

q

Cn

= Cn

+AC (n)+AC (C)+AC (c)+ -57- (CD.-+C ft)

D Dbasic STUST

UST ^

;

T -uq

Cv = Cv +ACy (0+ACy (0+ — (Cy r+C p+C 6)1 Ybasic

XST

X ST 2VT

l?

x

fx3

C = C +AC (n)+ — ( C m."+C

m Om basicmST 2V

T

m«

mq

C = C +AC (T) +AC n(C)+ ?V" (C

n e+C

nr+C

n p)

oasic "ST oi 1 £ r p

c. = c, +ac, ( 5 )+aC 1 (c)+ -k—(c, r+C, p)1 Hasic "ST "ST 2V T r ~p

C. MISSILE LINEARIZED EQUATIONS OF MOTION

The linearized quations of motion used to design the

missile autopilot are presented in this section. The equations

are linearized about the cruise flight condition.

1 . Additional Assumptions for Linearized Equations

The additional assumptions required for linearized

equations are as follows:

a) Derivatives are given in stability axes which

are fixed for the missile low level cruise condition.

b) Standard small perturbation assumptions apply.

1) Perturbed angles are small.

32

2) Products of perturbed variables are neglected

3) There is no coupling between longitudinal

and lateral directional motions.

c) Thrust derivatives are neglected.

d) The steady state condition is chosen as low

level straight and level cruise.

2. Linearized Equations Summarized

The linear equations of motion in state variable

form are shown in Tables 2-1 and 2-2. The following

coefficients are defined as used in Table 2-2.

A =I Izz xz

G =

2I I -IZZ XX xz

I IXX xz

I I -I2

ZZ XX xz

B =XX

Ixz

"H -IZZ

u —

Ixz

3 . Definition of Stability Derivatives

The dimensional aerodynamic stability derivatives

used in the equations of motion are defined by the following

relations

.

a. Longitudinal Dimensional Stability Derivatives

xu " -^ss

s<V2CDss

)/mUss

1/«»

X a = qssS(C

L-C

D)/m ft/sec

o o *

KhXn

= -5SSSC

Dn

/m ft/sec2

ZU =-5*SS

S((V 2CLss

/mUS3

1/sec

33

X tS3

•8

. a IS]

n cr I

i S3 COCO • a COCO 2: t=>

CD +

-—

I

o

cv

Kl? k_p

>o^

>H

^~. ^ \

»^ kj^

^ sCO m QCO + +

Cd KS< ^3 hJV ^ * *

<c O

CD

I

cv

HPQ

Eh

en

CD

rH

Cti

Hcd

>

CD

Fn

-pin

CD

CO

co•H-PClj

crw

03

C•H

-P•H

aoHH

1

COCO

>-CO

oo

coCO

>-

a•HCO

I

•8

isi

1

COCO

cd

o"tsi

COCO

X tsi

cdX! tSJ

COCO

>?a •«•H CS!

CO 1

ho CO• 8 COS cd

CSI

•8 -t-

IS] CO •bI CO CS]

CO £3 1

CO —

'

COcd •8 CO

2 Cd

cr

-8IS!

•8 8 I

<] SI COI •8 COCO 2 cdCO -H

2£

•8

ISI

1 &•8

IS]

I

CO ES3 COCO •8 CO

Ira ST CD

cvI

cv

PQ<Eh

CO

CD

r-\

,QCd

•HFh

cd

>

T3CD

Jh

-P!h

CD

Ph

COrH

o•HPcd

cr

O•HP>OCD

Fh

•HQ

cd

CD

-Pcd

COCO

1=5

UJ1 UJ1

J 3P3 OH- -t-

W1 UJ1

a JI _.

<£1

+CO. -e- a, Fh

I

1 \

COCO

1=3

I

u>H

COCO

CD

1-3

pa

&H

a>H

COCO>- COCO CO

CD

bfl

CQ

COcai co

M ICD

PQ+

.J

o

.J aCO PQ QCO v- -t- -r

CD Oh

s 1-4

V '

< O

a

Ph

ca

QQ.

Q+-

QQ.

II

r~ )

{=> •8 •cr •CD

L_ 1

1. 1

II

J 1

• CQ •& • (X, .ul_ »

34.

-qss

S(CL

+CDss

)/ m

Z * = -q q cScC T/2mU

SSWL. SS

Zq

=-*SS

S ° CL

/2mUSSH

q

zn

=^ss sc

Ln

/m

MTJ

=^SS

S5'

(CM

lJ

+ 2CMss

)/lyy

USS

M

M

M

qssSCC

Myi

yy

q qq Sc2C m /2I U qqn oo m* yy bb

— -2.

q oc Sc C /2I U ccH SS m yy SS

ft/sec

ft/sec

ft/sec

ft/sec

1/ft sec

1 /2

1 /sec

1 /sec

1 /sec

M = q„ScC /In

H SS m^ yy1 /sec

b. Lateral Directional Dimensional Stability

Derivatives

Y Q= q QQ 3C Tr

/m

6S "So'-y,

Yp

=^3S

SbCy

/2mUSS

Y = q CQ SbC /2mUr HSS y SS

I F= q qqSC /mSS" y £

YC

=^SS SCy/

m'

3

2,L = q ss

Sb C, /2Ixx

Ussy

P

L = q Q(3 St> C, /2I TJ cqr nSS 1 xx SSr

ft/sec

ft/sec

2

ft/

ft/

sec

sec2

ft/sec

1 /2

1 /sec

1 /sec

1 /sec

2

35

iL h =*ss

Sb\ /l™

illr \ - ?S SSbC

l/l

XX

1 /sec'

1 /sec'

H6

= W^n,/ 1.!

% " ^SSSb2c

n/2I

zzUSS

Nr " ^SS

Sb Cn

/2Izz

aS3

1 /sec'

1 /sec

1 /sec

1 /sec'

1 /sec'

4. Trim Equations

For straight and level flight at constant velocity

the following variables are all zero: U = V=W=p=Q = R = 9 = y = <|>

=ZS = Q SS

= PSS

= RSS

=*SS

=LA

= MA_ =V„ =YSS =Y SS

= 6SS

= VSS

=

O OO OO

Wno = ôn =Coc!

=SS "3S ^SS

The trim equations reduce to:

Tzin=css

= Bg -(c + C « + CL n ss )q ss

S

O « p

Too3"sS =

(CD

+CD =SS

+CD "sS^SS8

o « n

o = (c + c ^„+c^ n q n)q qq s"cm m bo m oo ^boo « n

36

III. DEVELOPMENT AND SIMULATION OF THE MISSILE AUTOPILOT

The missile is assumed to be equipped with an autopilot

capable of providing closed loop control which consists of

the following:

a) normal acceleration (n )

£i

b) lateral acceleration (n )

c) bank angle ( <t>) .

In addition, an outer closed loop control, which serves

as the outer autopilot, is as follows:

a) altitude (H)

b) vertical flight path angle (y)

These two outer loops are employed as required during certain

phases of the attack mission.

It is assumed that the missile possesses accelerometer

sensors in the y and z body axes located at the missile

center of gravity, rate gyros and rate integrating gyros

about the x, y, and z axes, and a radar altimeter. Sensor

noise is neglected in this simulation.

A. AUTOPILOT INNER LOOP REQUIREMENTS AND DESIGNS

1 . Description of the Autopilot Inner Loops

The autopilot is assumed to employ three inner loops

as depicted in Table 3-1

.

37

TABLE 3-1

Autopilot Inner Loop Description

LOOP

1 NormalAccelerationCommand

COMMANDED CONTROLVARIABLE

n (normal nz

acceleration) Elevator

CONSTRAINTS

15° K- n * 15°

-2 ^ N S4.

z

2 BankAngleCommand

*

(Bank Angle) Aileron

-15o <

5 * 15°

<D * 60°*

op S 50,100,20(T/sec

3 Turn n (Lateral z,

Coordinator '

Acceleration) Rudder

15°-15

N =y

c-

Applies in certain mission phases only

38

2. Normal Acceleration Command System

The purpose of normal acceleration command (NAC)

system is to provide a vertical load factor (n ) response toz

a commanded load factor (n ). An accelerometer is used tozc

provide the primary feedback and a rate gyro is used to

provide inner loop pitch rate feedback for improved damping.

a. Block Diagram

The block diagram of the normal acceleration

command autopilot is shown in Figure 3-1 . Limiter a allows

the commanded acceleration to a range from -2.0 to + 4..0 g's.

Limiter b allows the elevator control to vary from -1 5

degrees to + 15 degrees.

b. Assumptions

1

)

The rate gyro and accelerometer dynamic lags

are negligible.

2) The accelerometer is mounted at the center

of gravity. Therefore the moment arm is zero and the

feedback load factor is totally n . (C=0)Li

3) The fix servo can be represented by a first

order lag.

4) The missile dynamics can be represented by

a short period approximation.

5) Commanded normal acceleration is limited from

-2.0 to +4.0 g's.

c. Design

The following transfer function was developed

from equation (11):

39

CO

CD OH •HH SCO cdCO CI

H >ss Q

o•H-PPh•H*H

OCO

CD

Q-POrH•H0,O-P

cd

ssooS3

o•H-Pcd

in

CD

HCD

OO

ESh

o

I

CD

Sh

=5

he•H&4

40

q(s) _ W znM « )s + ^VW

V fT1{s2-(M_+Z_ +M-J s + (Z.M^-M^)

V

Tk "

~v

"q'li" ltx/ " '

v "a"q 'ioc/

}

TVT

From this transfer function and equation (10a)

V,

n (Q-i)

g

the following to n transfer function was obtained° z

Mz(s) _ V

T-Z

rl

s24-(M

qZ

rl

+ ZnM^)s + (M

ocZn-Z

a:M

rl)

Q(s) g (V„M +Z M. )s+(M Z -Z M )i f| f] °c ac f| « n

Utilizing the characteristics listed in Appendix A

for this cruise missile at 0.75 Mach number, the above transfer

functions become

Q(s) 3-26.99(s+1.009) and

n(s)

Nz^ -0.08(s2+0.1985s-323.9)

s~+1 .289s+27.13 Q(s) (s+1.009)

Evaluating the characteristic polynomial of

the Q(s)/n(s) transfer function yields the natural frequency

of 0..8 29 Hz and a damping ratio of 0.1237. The resulting

block diagram for the normal acceleration command inner

loop autopilot for analysis purposes is shown in Figure 3-2.

( 1 ) Root Locus Evaluation .

(a) Pitch Rate Loop. The transfer function

for the pitch rate loop is:

41

oT— cv+ +co CQ—

'

ONc- cvCV •

1 T

+cv

ca

ooJQ)

-Pcti

O

•HPL,

•Hca

rH

scti

Sh

cti

•HQ

OOrHPQ

-POH•HCUO-P

<C

ccti

SEOo

o•H-PCti

?H

CD

rHCD

OO<

Cti

s

O

CVI

a)

!m

So•H

42

Q(s) _1080 (s+1

)

E(s) ( s3+4-1 .29s

2+(78.7+1080KR1 ) s+ ( 1 084+1 08 0KR1 )

Appendix C contains the root locus plot of

the pitch rate loop for ranging KR1 . A KR1 of 0.28 is

chosen which yields the following characteristics:

s = -5.600 ±3.8U j

<j>= 55.53°

?= 0.82

u> = 1 .08 Hn z

(b) Normal Acceleration Command Loop.

Using a KR1 of 0.28 in the pitch rate loop, and locating the

acceleration at the missile center of mass (C=0), the normal

acceleration command loop becomes:

AT Q- 86.4-KA1 (s2+ 0.2s-329

s(s 3U1 .29s"+381 .1s-1 386.4-)<c

N-^

-86.UA1 (s +0.2s-329)Nz(s)

=

N (s) s(s3+^1 .29s + 381 .1 s-1 386.^)-86.^KA1 (s

2+0.2s-329)

z c

Appendix C contains the root locus plot of

the normal acceleration command system for varying KA1 . A

KA1 of 0.05 is chosen which yields the following charac-

teristics :

s = -1 .981 ± 3.317 j

30.85°

0.513

0.61 5Hn

43

With a KR1 of 0.28 and a KA1 of 0.05 the

system transfer function relating n to n is:zc

z

N ( s ) -4.32(s2-0.2s-329)

z v

N (s) s4U1 . 29s J+376. 78s -1387. 26s-U21 .28

zG

The normal acceleration command autopilot

has zero steady state error for a step input as it is a

type 1 system. The system response to a 1 g commanded load

factor is shown in Figure 3-3.

3 . Bank Angle Command System

The purpose of the bank angle command (BAC) auto-

pilot is to command missile bank angle. Limits are applied

to commanded roll rate and in certain mission phases

commanded bank angle. A rate integrating gyro feeds back

bank angle and a roll rate gyro provides feedback as a

roll rate damper.

a. Block Diagram

The bank angle command autopilot is presented

in block diagram in Figure 3-4-. Limiter a allows the

commanded bank angle to vary from to 60 degrees and is

active in certain flight phases. Limiter b limits the roll

rate to either 50, 100, or 200 degrees/second. Limiter

c limits aileron deflections to range from -15 to +15

degrees

.

b. Assumptions

1 ) The rate gyro and displacement dynamic lags

are negligible.

U

S Q < + X

L_| oLJ 13 (X

||LU o_l —j

l_l rvi

CC Zoo(_)

cr i—u. Ill UJ_j _i x >-s in k in

02 -t

OTi'G 02'E

2ft 96 '0 38 '0

auoiZN08 '0

00 "'S 08 '2 09-

2 OH 'Z

bdlb02-2

2i-

00-2 08 'I

QQ-Q 02'0- 0ft- 0- 09"0- 08-0- 00V-

31302M- 0tTI- 09-T-

uo-pocd

o

a>

sEoo

o-p

CDi

o

CO

CD

03

-poH•H

O-P

5S Q

oo -4-

st +1 ca

cven«

> oU OCD rHca

CD

C -P•H Cti

fin

a. pcj

cd

oI—

I

EhP-.Mc£OCOwQEhOI—

I

PhoEh

<Q

Oo

P3

I

CD

U

bo•H

O

46

2) The fin servo can be represented by a first

order lag.

3) The missile dynamics can be represented by

a roll mode approximation.

4) There are limits on commanded bank angle and

commanded roll rate as noted above.

c. Design

Equation (12) can be written to find the

, , « - . . g ( s) <fr ( s) and f(s)lateral transfer functions =-*—r, *-j—K —?

—

<r ,

where 5 = 5 or £. By allowing a roll approximation the

transfer function relating aileron and bank angle becomes:

<S>(s) L .

5(a) s(s-L )

P

The root at the origin indicates the missile does not have

bank angle stability without autopilot stabilization.

Evaluation of the transfer function coefficients results

<D(a) =-116

and p(s) =in the transfer functions

-116

s + 6.2

by Pmax

c (s) s(st6.2) C

The maximum missile roll rate can be approximated

= 280.6 degrees/secondK .max

L

(11 6) (1 5)

6.2

The block diagram of the bank angle command

system is:

9 ~Y_J KD v +r> KR2-40 -116

s + 6.2

P 1

s

e >Jrs 3+40

f ^

Roll Rat e Loop

i.n

(1 ) Root Locus Evaluation .

(a) Roll Rate Loop. The transfer function

for the roll rate loop is:

P(s) _ 4640 KR2

E(s) s2+4-6.2s + 2-48 + 4.64-0KR2

Appendix C contains the root locus plot

of the roll rate loop for varying KR2. A gain KR2 of 0.10

yields the following characteristics.

s = -23.10 ± 13.36 s

4) = 59.96°

S = 0.8656

co = 4.247H7n u

(b) Bank Angle Command Loop. With a KR2

of 0.10 the bank angle command loop becomes:

d> ") 4.6 4KD

s(s2+ 46.2s+712)

* -4

f-

->

The transfer function is:

ds) = 46^KD

d) (s) s3+46.2s

2+712s+464KD

The root locus plot is shown in Appendix C

for varying KD. Choosing a KD gain of 10.8 yields the

following characteristics:

48

s = -9.98 ± 9.58J

<j) = 45.056°

5 = 0.718

4i= 2 ' 19 H

z

Using gains of 0.10 for the inner loop and

10.8 for the outer loop of the bank angle command system,

the system transfer function G(s) becomes

o (s) = 5011 .2

<J>(s) s

3+ 46.2s

2+712s+5011 .2

c

and the system is a type 1 system for a step input. The

transient reponse is demonstrated in Figures 3-5 and 3-6

for a commanded bank angle of 60 degrees.

4.. Turn Coordinator

The turn coordinator (TC) is a lateral autopilot

which orovides a body directional load factor, n , to ajy

commanded load factor, n . Like the normal accelerationyJ c

command autopilot it employs an inner loop yaw rate damper.

The outer loop uses unity accelerometer feedback and has

the capability of moment arm feedback if the accelerometer

is located at a different position than the missile center

of mass.

a. Block Diagram

The block diagram representing the turn coordina^

tor autopilot is shown in Figure 3-7. Limiter a limits

the rudder deflection( £ ) from -15 to +15 degrees.

49

a e

o z aII O <x

a _

,

o o _la. X >- zd ^<n a. a_ 2 az cr

oCJ

oo "rig-

9S 00-3ti 00

-0ti Q0

-

2E 0OTi2IHd

oo-gt oo -tf

CD

CQ

aoPhCQ

(D

OS

I—

I

CD

tuO

<

S

CD

E£oo

oo

o-p

CD

CQ

coOhCQ

CD

BJ

-POrH•HPhO-p

DO'Oi 00"0g 00'OS 00- 0h 00

-

0£ 00-02 OOTJtd

OO'O BO'01- o

gfO 21'Q 30 '0 WTO 00*0 H'D-QbOIJLN

8Q-0- 21*0- gi'o-I

og-o"

og-o

00-

00 -Q

og-o- 02M- 08M- 0fT2-any

00- £-

og-o- 0Z*1- OB'1- 3H-2-lld

0Q-E-

09- £-

og'E-

02T-

02Tl-

CD

Ih

•H

50

00-09: 00- 0M 00'02t OO'OOl 00-08 00"09 OD'Qli

,QIx H1U03X00-02 oo -o°

00'Ohl oQ-ozt oo-oot ocoa ao-09 oo-on 00-02

sOtx H1UU3JL

OO'O 00 "02-

OO'fiS OOMS 00-811 OQ-Sli oo-2^ oo-ee3anilb

00-9E OQ-EE 00*06

oo"Of>8 00-SE8 oo-osa 00-S2B oo- o28 do'sis oq-018 oo-soa oo-ooa

n

CD

CO

o

CO

CD

cc

CD

-p•H-p

CD

rHbo

PQ

T3

PJ

cd

esoooo

o-p

CD

CQ

so

to

CD

on

-poH•H

o-p

oPQ

I

en

CD

u

•H

0O-QE 00*02 oo-or OQ-0 00-QT- 00-02-A

00 "0E- 00 "Qh- oo -as-

51

CQ

CD OH •HH Em cti

CO cH >>S Q

o•H-PPh•H!h

OCQ

CD

Q-POrH•Hao-p

<

O-Pnj

•H

?h

OOOau

EH

CD

w•H

52

b. Assumptions

1 ) The rate gyro and acceleroraeter dynamic

lags are negligible.

2) An accelerometer is mounted at the center

of gravity. Therefore the moment arm is zero. C=0

3) The fin servo can be represented by a first

order lag.

4.) The missile dynamics can be represented by

a dutch roll approximation.

c. Design

The dutch roll approximation involves side slip

angle (3 , and yaw angle ¥ . Equation (12) with (J> neglected

(assuming a small C-, results in the following set of3

equations

:

SVT - *

6

-N3

e(vT-i

r :

s2-sN

6(s)

C (s)

'D(s)

5(a)

— <»

Y5

-

N^

extracted

g(a)

The following transfer functions can be

Y s+(Y N r -VTN r -Y N )

L r g T ^ c r2

?(s) VTs -(Y

g+V

TNr)s+(Y

gNr+V

TN -Y

rNB

)

IU1C(s)

VTNcs + (Y^Y

gNc

)

s{ VTs2-(Y

3+ V

TNr)s+(Y

3Nr+V

TNr Y

rNe

) }

53

R(S) _ 'tV +<WW?

(s) VTs2-Y

s+V

TNr)s+(Y

3Nr+V

TNB-Y

rN3

)

Using the equation (10b) in Chapter I the

transfer function for yaw rate to n is

Ny(s) _ Vj_ Y

;

,s2-f(Y

rN

;;

-YgNr)s+(Y

g;

Ng-Y

3Nc

)

R(s) g VjN^s+CT^-TgN )

Substituting coefficient data obtained from Appendix A

results in the transfer functions as:

R(s) = -U.5(s+0.11 )

5(3) s2+ 0.2^5s +U.5

N (s) -O.U(s 2-0.01 s-20.9

R (s) s+0.11

The resulting turn coordinator block diagram

for analysis purposes is given in Figure 3-8.

( 1 ) Root Locus Evaluation .

(a) Yaw Rate Loop. The loop transfer func-

tion is

R(s) ,580(s+0.11

E(s) s3+4.0.25s

2+(24.3+580KR3)s+(580 + 63.8KR3)

Appendix C shows the root locus plot of

the yaw rate loop for varying KR3. Selecting a gain KR3

of 0.4-0 results in the following characteristics:

s = -3.581 ± 2.339J

<t>= 56.85°

r = 0.8372

54

—•

ocv

1

CO

T

o r-• ^

—

o1

•

Ocv •+

CO COV

-cf

1

—

I ™1

1 . r~~-

nCO

O cv

m

m

-—v •

T— »tf^

—

\

—

• +o CO

+-

CO

in

1— cv-<* •

• o r^-** + cdT— cv W

1 CO

oo -<fr

-<f +CO

L

CO

•HCO

Hcd

<

suj

in

cd

•HQJ*i

Oo

m-po

•H

o

<

OPcd

a•H

OOO

Eh

COi

en

CD

U

M•H

55

u = 0.6807Hn z

cj)= 56.85 degrees

(b) Turn Coordinator Loop. With a KR3 of

0.4.0 the turn coordinator loop becomes:

Ny, <> -81 .2KA2(s

2-0.01 s-20.9)

s(s 3+^0.25s2+2 56.3s+605. 52)

-r

The transfer function is:

N (s) -81 ,2KA2(s2-0. 01 s-20.9)

N, (s) s^+4.0. 25s 3+(2U. 7-81 . 2KA2 )

s

2+

(

604. 2+81 .2KA2)s +

C1697KA2

The root locus plot for varying KA2 is shown

in Appendix C. Selecting a gain KA2 of 0.25 yields character-

istics as follows:

s = -2.206± 2.807J

<$>= 38.16°

5 = 0.618

00 = 0.5682Hz

The turn coordinator system transfer function

G(s) becomes:

N (s)y

-32.48(s -0.01 s-20.9)

N (s) s4+40. 253^ + 212.23^+604.53 + 678.

8

^c

The system is type 1 and therefore has zero steady state

error to a step input. Figure 2-5a shows the response

56

of the turn coordinator for a step aileron input. Note that

an n of zero results in a coordinated turn.

B. AUTOPILOT OUTER LOOP REQUIREMENTS AND DESIGNS

1

.

Description

The outer loop auto pilot has two functional loops

consisting of an altitude hold and a vertical flight path

angle hold.

2. Vertical Flight Path Angle System

The purpose of the vertical flight path angle hold

loop or gamma command loop is to provide missile reponse to

a command of 8.6 degrees during a specified phase in the

terminal profile.

a. Block Diagram

o-Y+

KY

VT

N (s)

N (T Ntel Y

b. Assumption

1) Bank angle is zero.

c. Design

Starting with equation (10a) n = ^r(Q-i) = ^T*

zg g

Y ( S ) £Tthe transfer function »l / \= ttt \ can be derived. Using

N (s) VT(s) s

the inner loop design previously for load factor the vertical

flight path angle block diagram becomes:

57

Y -i. KY

2-9.5CU(s +0.2s-329)

s^+41 .29s 3 +371 .6s2+138^s+3127

X.

The transfer function

Y(s) -9.504-K (s +0.2s-329

y(s) .

' 7 \ is:

Y (s) s5U1 .29s^+37. 6s

3+(138^-9.50^K )

s

2+ (31 27-1 . 9K )s+3127K

Y

(1) Root-Locus Evaluation . Appendix C shows the

root locus plot of the vertical flight path angle loop for

varying gain K . A gain Ky of 1.0 yields the following

characteristics

:

Y

s = 1 .2725 ± 2.5407J

(j)= 26.6

?= 0.U78

co = 0.4-5H™n Z

The system transfer function G(s) with a

gain of 1.0 becomes:

y(s) _ -9.50^(s2+0.2s-329)

Y (s) sp+^1 .29s 4+371 .6s 3 +1374..5s

2+3125.1s + 3127

c

Figure 3-9 depicts the system response to

a commanded vertical flight path angle of 8.6 degrees.

3. Altitude Hold System

The purpose of the altitude hold autopilot is to

hold altitude at 50 feet during specified flight phases.

a. Block Diagram

H +'

"V1

y(s) . H(s) H V

r y (s)c

Y(s)

i

—

s

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b. Assumptions

1) Vertical rate of climb (or desent)

subtends a small angle ( y is less than 20 ), therefore

H = VTsin Y = V

TY.

2) Bank angle equals zero.

c. Design

Let altitude command and altitude be designated

Hr and H respectfully. The design is based upon the

following two relations:

H = KH(H C-H)

Yc .Jjj(H c

-H)

From the vertical flight path angle development,

Y(s) £IPTsT V

Ts

sH(s) = Vty(s)

sH(s) y Y (s)N (s) T NTs)

(Divide both sides by a (s))n

V m a.

VTs

H(s)

Y(s)

v r

H

The altitude hold block diagram becomes:

K TH

-9.50^(s2-0.2s-329)

5+41 .29s^+371 .6s3+137^.5s

2+ 312 5.1s + 3127

H

The transfer function H(s)/H„(s) is:

60

H(s) .

H (s)

2-9.504KH(s +0.2s-329)

s°+41 .29s 5+371 . 6s4'+1374.5s 3+(312 5.1-9.504KH)s

2+

(3127-1 .9KH)s+3127KH

(1) Root Locus Evaluation . Appendix C contains

the root locus plot of the altitude hold loop for a varying

gain KH. Again KH of 0.3 results in characteristics:

s = -1 . ^04-5 ± 2.424J

d)= 30.08°

r = 0.501

co = 0.4-45 Hzn

The altitude hold loop transfer function

G(s) with KH of 0.3 is:

H(s) 2.8512(s2+0.2s-329)

Hc(s) s

6+41 .29s 5+3 71 . 6s ^+1 374. 5s

3+31 22. 2 5s

2+31 26. 4s+938 .

1

and Figure 3-10 shows the response to a y _ of 50 feet.

61

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62

IV. DEVELOPMENT AND SIMULATION OF MISSILE GUIDANCE SYSTEM

The purpose of this study is to evaluate the effect of

limited roll rate and various guidance laws on the terminal

performance of a bank-to-turn cruise missile using propor-

tional navigation in the terminal (attack) phase against a

medium to large combatant ship. A baseline guidance and

control scheme is designed and studied which uses a pop

out maneuver, proportional navigation, and bank-to-turn

maneuvering in the attack phase. The trajectory parameters

and navigation constants are optimized for best performance

against a moving ship utilizing blinking countermeasures

.

The missile is assumed to have limited roll performance

due to the use of differential stabilizer for roll control

vice ailerons. The terminal performance of this baseline

system is evaluated for varying roll rate limits, electronic

countermeasure blinking rates and burn through ranges.

A second control scheme is also designed and evaluated

for the attack phase. This mode uses a sea skimming scheme

in which a 50 foot altitude is maintained and proportional

navigation is used in azimuth during ingress until a specified

range is reached at which time proportional navigation is

used in both azimuth and elevation to attack the target.

63

A. MISSION DESCRIPTION

1 . Baseline Guidance Scheme (Pop Out Maneuver)

The mission is divided into four phases:

a) ingress

b) offset

c) popup

d) attack

The mission commences with target ship located 25,000 feet

north of the missile and moving east at 21 knots. The

missile initial heading is north.

The ingress phase commences with the missile flying

at an altitude of 50 feet on altitude hold and utilizing

proportional navigation in azimuth to home in on the

target. At a given range the offset phase is initiated with

a turn to the right of approximately 12 degrees followed by

a commanded bank angle of zero degrees. No proportional

navigation is used in this phase. At another given range

a pop up maneuver is initiated which generates a low angle

climb in preparation for the attack phase. During the pop up

maneuver, proportional navigation is used to begin turning

the missile toward the target. Finally the missile enters

the attack phase at a specified range where proportional

navigation is used in both vertical and azimuth planes.

In this phase the bank angle is unlimited but roll rates are

limited and used as one of the parameters in studying miss

distances. The second parameter under study is the electronic

64

counter measure blinking rate frequency which is initiated

at the start at the start of the attack phase and allowed to

remain activated until a specified burn through range which

is the third miss distance study parameter. Figure 4-1 is

a geometric depiction of the missile flight path.

2. Alternate Guidance Scheme (Sea Skimmer)

The mission for this guidance scheme is divided into

two phases:

a. ingress

b. attack

This mission begins with the same initial scenario as the

baseline scheme, however, the pop out maneuvers (turn and

pop up phases) is eliminated.

The missile flies at 50 foot altitude during the

ingress phase and utilizes proportional navigation in

azimuth until a range of 3000 feet at which point the

missile enters the attack phase and dives in on the target

using proportional navigation in both azimuth and elevation.

This low altitude approach essentially resembles a sea

skimming effect. Figure 4-2 shows the geometric flight

profile of this scheme.

3. Electronic Counter Measures (SCM)

It is assumed that the target has the ECM capability

of simultaneously shifting the aim point of the missile 75

feet aft and 10 feet vertically. The simulation is tested

for SCM blinking frequencies of 0.5» 1.0, and 2.0 cycles

65

X

AH

Figure 4.-1 Baseline Guidance Scheme Flight Profile

66

^

H

Y->

Figure 4.-2 Sea Skimmer Guidance Scheme Flight Profile

67

per second. Blinking is commenced during the attack phase

and continues until jammer burn through occurs when the

target signal power can be seen over jammer power. Although

3800 feet (as calculated using Hosington, {Ref. 3}) is

considered a typical burn through range the missile is

tested for burn through ranges from 3800 feet down to

4.00 feet.

&. Glint

This study models glint with a gaussian function

that gives the aim point a random shift effect with a

variance of 25 feet in the Y axis and 4- feet in both the

X and Z axes. Since glint is imposed as a fluctuation in

target coordinates and target coordinates are used to

calculate the target line of sight angles then glint is

modeled as inversely proportional to range as required.

For ease in simulation glint is imposed only during the

attack phase. Results for runs with glint included are

presented in Chapter IV.

B. SEEKER EQUATIONS AND SIMULATION

A flow diagram of the seeker simulation is presented

in Figure 4.-3.

1 . Line of Sight Rates

It is assumed that the missile seeker has the ability

to sense target range and target line of sight rates in

azimuth and elevation (aA7 and o-, T ) in the body coordinate

system out to a range of 25»000 feet.

68

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• D • o

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• O .D

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69

These line of sight rates are generated in simulation

by a two step procedure. First, the line of sight rates

in earth referenced azimuth and elevation ( a .„ and o nT )

are constructed by using the following relations:

* m = tanY Y

-1XT - E

XT " X

E

8 _= tan-1 -H

((XT-X

E )

2+(Y

T-Y

s )

2)

i

AZ

\Z

oSL

9T-9

# •

e _eT

Figures 4-4- and 4-5 depict the geometry represented by

the above equations.

Second, the body line of sight rates are calculated from

the following transformation.. • .

o.„ = - ov, T sin<|>+ aA7 cos 9cos <$>

AA p Hi J_i AL

a = a cos'HfJ cos8 s in(|>zjLt, aL AZ

Once the body rates have been simulated ( a. „ , a,-,. )

they are transformed to earth horizontal and vertical rates

through the transformation T , , defined by:

&SL

=^L

B

COS ^ÂZEsin(j)

• • •

JAZ

= V sin4>faAZ n

cos4)o B

70

XT

X,

/N

Y, I,

*

QAZ

=*T

-*

Figure 4.-4. Azimuth Line of Sight Description

71

1\

H

°EL=9

T-e

T (shown negative)

>2, Tr 2<

(X +Y

Figure 4-5 Elevation Line of Sight Description

72

This transformation includes the assumption that is small

throughout the mission. The earth horizontal and vertical

line of sight rates (Cgr and a . „ ) are then used in the

appropriate guidance laws for proportional navigation.

2. Missile Dynamic Filters

In order to filter out missile dynamics from the

seeker line of sight rates, a low pass filter, 10/(S+10),

is used at the input to the seeker on 9 and y.

C. BASELINE GUIDANCE LAW DESIGN (POPOUT MANEUVER)

A flow diagram of the baseline guidance scheme is

shown in Figure 4.-6. Switch positions are a function of

mission phase. Each phase of the mission is described

below beginning with the attack phase.

1 . Attack Phase

Proportional navigation is used in azimuth and

elevation to construct guidance commands to the missile

autopilot commencing at a range of 9100 feet in the manner

described below.

a. Proportional Navigation Constants and Guidance

Commands

The earth horizontal and vertical line of sight

rates (o".7

and ct„t

) which are generated from the body rates

a A7 and a^ T by the transformation T are multiplied by

a navigation constant and V^/g to generate commanded lateral

accelerations in the earth vertical and horizontal planes.

73

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1

E-h hoE-i bjO >> 1

Cxi hi< Wr< l<

(S3 kJ< H• D . D

-e-

E-t

CQISJ en<U ^

• O

CO

o•H

-p cd Fh

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bo (D MIn £ CD

cd •H CDEh fc^ CO

Eh M

o

00

scd

Fh

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cd

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oH

cd

CD

ofl

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74.

A navigation constant of 4-.0 in each plane was found to

yield best results. Values from 3.0 to 4-« 5 were evaluated

with results of that evaluation shown in Chapter V. A one

g bias is applied to the vertical commanded lateral accelera^

tion to compensate for acceleration due to gravity. Thus

two lateral acceleration commands (n^ and n„ ) are computedc c

in the earth horizontal and vertical planes.

nT

= AAZ ~i~ °AZ (13)

vT

nZ

= XEL °EL+ 1 (U)

c g

These commanded lateral accelerations are subjected to a

further transformation (T«) in order to generate commanded

bank angle and normal load factor which are required by the

missile autooilot.

1

n?

*c

= tan'1 —

nzc

n = n^ cos^+n sintj)Z la LC C C

Figure 4.-7 depicts the geometric relationship described by

the above two equations. Commanded bank angle ( <tu ) and

normal load factor (n ) are used as inputs to the appropriatezc

autopilot outer command loops to provide closed loop bank

angle and load factor responses.

75

/

^ Earth

Body-

Body Coordinate System

Earth Coordinate System

Figure 4.-7 Load Factor and Bank Angle

Description

76-

b. Limits on Guidance Commands

In the attack phase, the following limits are

applied to commanded variables:

-21 n < 4.

zc

| * c I - i max

Missile performance is evaluated for maximum roll rates of

50, 100 and 200 degrees per second.

c. Bank Angle Anomoly

In the attack mode there is no limit on

commanded bank angle. Thus a computational scheme is

required to insure that the missile always rolls the

shortest way to a new commanded bank angle. This is accom-

plished by continuously calculating the quantity:

A (*) =| $ c

(t) - <>(t)|

If A(t) is less than 180 degrees then <J> (t) is sent to the

bank angle autopilot. If A(t) is greater than 180 degrees

then a modified bank angle command ( <$> ) is sent to the autoCM

pilot in accordance with the following logic:

<J) = 0^+360° for 4> <0CM

C c

d)=

d)-360° for <$> >0

cM c c

d. Switch Positions

The switches shown in figure 4-6 are in the

following positions for the attack phase:

SW1 Either position

77

SW2 Position 2

SW3 Position 1

SW4 Closed

e. Attack Phase Summary

In the attack phase proportional navigation is

used in both azimuth and elevation to generate bank angle

and normal load factor commands to the missile autopilot.

The phase is commenced at a range of 9100 feet with missile

offset in heading from the target at an altitude of approxi-

mately 250 feet. Roll rate is limited to <b . Bank angle' max &

is not limited. Normal load factor is limited from -2 to

i g*s.

2. Ingress Phase

The ingress phase commences with the beginning of

the problem simulation when the missile is 25,000 feet from

the target and initially heading north. The missile's Mach

number is 0.75 and the ship's velocity is 21 knots to the

east.

a. Altitude Hold Methodology and Guidance Commands

The missile is required to maintain altitude of

50 feet above sea level, during this phase. It accomplishes

this by comparing present altitude with 50 feet and multiplyingKHthe error by —rr— to establish a flight path angle command.

Ktt is assigned a value of 0.3. This calculated flight path

angle command is compared with the present flight path angle

78

vT

and the difference or error is multiplied by K TIandJ n g

to give an elevation line of sight rate, a ^ . A value of

0.3 is assigned for K T_,. Thus n„ is computed as follows:c

Vn7

= K„ ~- ( Y - Y) + 1

Zc

H g c

The lateral acceleration command n„ is computedc

exactly as described in the attack phase with bank angle and

roll rates limited.

The second transformation for the ingress phase

(T„) remains the same form as the attack phase.

<j) = tan nc y°c

nzc

n = n_ cos<tH-n Tr sin<t)z Z Yc c c


In the ingress phase the following limits are


-2 1 n < 4zc

o < 4)

c_ max

4) <Uc '

—' max

The maximum bank angle allowed is 60 degrees and roll rate

is limited to 50,100 or 200 degrees per second,

c. Switch Positions

The switch positions as shown in Figure 4-6 are

as follows during the ingress phase:

79

SW1 , SW2, SW3 Position 1

SW4 Closed

3. Offset Phase

The missile initiates the offset phase at 17,800 feet

The offset phase is subdivided into two sub-phases -- turn

and level.

a. Turn and Level Sub-Phases

The missile maintains an altitude of 50 feet

throughout the offset phase in the same manner as described

during the ingress phase. Thus

nz

= KYVT (v -y) + 1

C ' Gg

The missile performs the coordinated turn by

using a commanded bank angle and computing a normal load

factor as follows:

*c

= 60°nZC

n =zC COS

(J)

At a range of 1 5800 feet from the target a turn of approxi-

mately 12 degrees is accomplished. The missile then performs

a straight and level maneuver by commanding zero bank angle

along with the same normal load factor as follows:

nzC

nz cos oc *

80


For the offset phase normal load factor is limited

as follows:

-2 < n '4~ z —c

Commanded bank angle is fixed at 60 degrees during the turn

sub-phase and degrees during the level sub-phase.

c. Switch Positions

The switch position, as shown in Figure 4.-6, are

as follows during the offset phase:

1

)

Turn sub-phase

SW1 , SW2 Position 1

SW3 Position 3

SW4. Open

2) Level sub-phase

SW1 , SW2 Position 1

SW3 Position 2

SW4- Open

4.. Pop Up Phase

The missile enters the pop up phase when the range

becomes 10,200 feet. The purpose of this phase is to

pop up the missile to an altitude of approximately 250 feet

so that a dive into the target will give acceptable deck

penetration.

a. Vertical Flight Path Angle Methodology and

Guidance Commands

The missile achieves the pop up effect by

commanding a vertical flight path angle of 8.6 degrees.

81

This commanded climb angle is compared with the actual y andvT

the error is multiplied by K and — to give an elevation

line of sight rate. A value of 1.0 is assigned for Ky

in this simulation. Thus n„ is computed as follows:c

V r

nZ = K

y f climb

The lateral acceleration commanded n is computedy,

exactly as described in the ingress phase.

The second transformation T^ is as follows:n

4> = tanc

-1y,

n r

n = n 7 cos(J)+nv sin<j)

c c c

The missile is free to start rolling into the

target using proportional navigation during the pop up

phase and thus aids in the attack phase which follows the

pop up phase.


In the pop up phase the following limits are


-2 1 n ± Kzc

4) |•< |cj)

c max

max

82

The maximum bank angle is limited to 60 degrees

and the maximum roll rate is limited to 50, 100 or 200 degrees

per second.

c. Switch Positions

The switch positions as shown in Figure 4.-6 are

as follows during the pop up phase:

SW1 Position 2

SW2, SW3 Position 1

SW4- Closed

5 . Guidance Summary (Baseline)

The following matrix indicates the horizontal and

vertical controls during each phase of missile flight.

PHASE AZIMUTH CONTROL ELEVATION CONTROL

Altitude Hold

Altitude Hold

Flight Path Angle Hold

PN

"^Proportional Navigation

D. ALTERNATE GUIDANCE LAW DESIGN SEA-SKIMMER

The sea skimmer scheme differs only from the baseline

guidance scheme in that the offset and popup phases are

eliminated. In essence this scheme is composed of an ingress

phase and an attack phase.

1 . Ingress Phase

The ingress phase commences with problem initialization

which locates the missile 25,000 feet from the target with

33

ingress PN*

Offset <b Command

Popup PN

Attack PN

identical initial conditions as the baseline guidance scheme.

The vertical load factor is computed asV r

n r = K„ (Yc

- Y) + 1

The lateral acceleration command is computed as

n1

_ Av r

AZ AZ

The second transformation (T„) is as follows:

n.

<t>

c= tan'

1

n r

and n = n_ cos<j)+nv sine))

c c c

Limits on guidance commands and switch positions

for the ingress phase are identical to the ingress phase

for the baseline scheme.

2. Attack Phase

The attack phase commences at 3000 feet range and

uses proportional navigation in both elevation and azimuth,

The missile dives into the target from 50 foot altitude.

The bank angle command and normal load factor are

computed exactly as in the baseline scheme attack phase

and are:

^c

= tan-1

n„

n r

84

n = n„ cos(J)+n„ sm<t>z Z ic c c

Limits on guidance commands and switch positions

are also identical to the attack phase in the baseline

guidance scheme.

3. Guidance Summary (Sea Skimmer)

The following matrix shows the horizontal and vertical

controls during each phase of the sea skimmer guidance

scheme

.

PHASE AZIMUTH CONTROL VERTICAL CONTROL

Ingress PN Altitude Hold

Attack PN PN

85

V. PROBLEM SIMULATION, RESULTS, CONCLUSIONS AND RECOMMENDATIONS

This chapter describes the computer simulation special

language used and presents the results of test runs in which

roll rate, ECM blinking rate and burn through range are all

varied in both the baseline guidance scheme and the sea

skimming guidance scheme. From these results optimum para-

meters are chosen to yield minimum miss distance for each

guidance scheme. Miss distance ranges are also presented

for the baseline scheme with Glint imposed.

A. CSMP SIMULATION

The Continuous System Modeling Program III (CSMP III),

an IBM developed system, is the modeling technique chosen

for this study. CSMP, as discussed in Speclchart and

Green {Ref. 4-) » is compa table with the J?or t ran IV language,

has six selectable integration methods (fourth order Runge-

Kutta is used for this simulation), and has multiple special

function codes that can solve a large variety of control

engineering problems.

The special functions used for this simulation are:

a. FUN GEN - This function extracts the value of a

function of one variable from a stored table in memory. It

is used for aerodynamic coefficients.

b. TWOVAR - This function extracts the value of a

function of two variables from a stored table in memory. It

is used for aerodynamic coefficients.

86

c. INSW - Y = INSW (X1, X2, X3) assigns Y = X2 for

X <0, Y = X3

for X1

> 0. It is used to accomplish ECM

blinking action.

d. GAUSS - This function is a random number generator

with normal distribution. It is used to realize glint.

e. REALPL - This function simulates a first order

lag 1/(tS++1) filter. It is used in autopilot and seeker

equations as low pass filters.

f. DEBUG - This is a computer program aid that evaluates

all output parameters and variables at requested times.

In addition to the special functions, trigonometric,

integral, derivative and square root functions are used to

perform necessary steps with one invocation. CSMP also has

the ability to perform calculations in either a sort section,

where the order of computation is not considered, or a

nosort section, where ordered sequential fortran statements

are required. The program is structured as shows below:

MISSILE SIMULATION PROGRAM

INITIAL

CONSTANTSAERODYNAMIC COEFFICIENT TABLES

DYNAMICNOSORT

MISSION PHASE LOGICGUIDANCE EQUATIONSECM (BLINKING) EQUATIONS

SORT

6 DOF MISSILE DYNAMIC EQUATIONS

87

INNER LOOP AUTOPILOTSSEEKER EQUATIONS

TERMINAL

OUTPUT GRAPHS

The CSMP initially sets up the two and three variable

coefficients in tabular form in memory. The run commences

with the missile in steady state flight with steady state

variables established. These steady state variables are

used to enter the tables in memory and extract coefficients

which are used to calculate aerodynamic derivatives, forces,

moments, and other relations. From the steady state flight

condition small perturbations are made and new table entries

produced. This process is repeated for all phases of flight

B. RESULTS

Both the baseline guidance scheme and the sea skim

guidance scheme were tested under the same conditions and

quantitative results obtained. All scenarios started from

the following initial conditions:

1. Target position - 25,000 feet north of the missile.

2. Target speed - east at 21 knots.

3. Missile speed - north at 0.75 Mach number

4-. Missile altitude - 50 feet above sea level.

The simulation was run for 31 seconds. Miss distance data

was obtained for varying roll rate limits, ECM blinking

frequencies, and burn through ranges. The following matrix

of parameters was tested:

88

Burn through range (feet)

20001 5001200800^00

Roll rate limit (degrees per second)

20010050

ECM blinking frequency (Hertz)

2

1

0.5

The resulting miss distances are shown in Tables 5-1

and 5-2 for all runs. Figure 5-1 through figure 5-6 are

the graphical results of simulation runs. Each run is

presented in 6 consecutive figures a through f and depict

the following.

a. Commanded roll rate (PCMD) and roll rate (P) vs

time

.

b. Commanded bank angle (PHICMD) and bank angle (PHI)

vs time.

c. Range, X Range, Y Range and Z Range vs time.

d. Left and right stabilizers (LSTAB and RSTAB) vs

time

.

e. Rudder position vs time.

f. Map of YMISLE and HMISLE vs XMISLE.

The following matrix indicates the test conditions used

to generate each figure:

89

TABLE 5-1

Baseline Guidance and Control Scheme

Miss Distance (feet)

ECMBlinking

Rate

RollRateLimit 400

Burn

800

through Ran

1200

ges

1 500 2000

. 5 cps 50 19.9 15.7 16 16 6.28

100 17 11 .3 6.95 6.95 18.16

200 27 30 29 29 26.1

1 cps

2 CDS

50 24 23 11.3 9.8 10.7

100 12.6 13.7 12.5 11 6.9

200 5.4 3 4.9 2.9 6.57

50 25 20 8.9 6.6 4.34

1 00 23.7 15.7 2.2 1 .69 6.8

200 13.3 7.9 1 .5 4.2 6.81

cps 50

100

200

2.5

4.4

4.5

90

TABLE 5-2

Sea-Skimmer Guidance and Control Scheme

Miss Distance (feet)

SCM Roll Burnthrough RangesBlinking Rate

Rates Limit 400 800 1200 1 500 2000

.5 50 43 37.7 37.7 37.7 2.8

100 72 61 61 61 1 .56

200 44.64 28.7 37.7 28.7 1 .77

50 54 54 38.62 28.8 1 .48

100 52 52 20 20.6 3.9

200 65 63 26 26 3.09

50 34-6 31 .20 37.7 37.7 2.87

100 32.9 27.9 61 61 1 .56

200 43.5 37.35 28.8 28.8 1 .77

50 3.9

100 2.7

200 2.6

91

BASE GUIDANCE SCHEME

5-1 (a-f) NO ECM, 100°/sec roll rate

5-2 (a-f) ECM (1Hz), 100°/sec roll rate



SEA SKIMMER SCHEME

5-5 (a-f) NO ECM, 100°/sec roll rate


1 . Analysis of Baseline Scheme Results

a. Varying Burn Through Range

Miss distances generally decrease as the burn

through range increases. For runs beyond 1500 feet burn

through ranges, the miss distances are close enough to be

considered as hits since most of the miss distance is in the

I component of range which has the largest offset (75 feet)

applied during ECM.

b. Varying ECM Blinking Frequency

The miss distance decreases as blinking rate

increases for the same burn through range. Miss distances

for burn through ranges beyond 1200 feet are particularly

good for all three roll rates.

c. Varying Roll Rate Limit

Varying roll rate limit results in the largest

variation in miss distance. For 200 degree per second

roll rate limit and 0.5 Hertz ECM blinking frequency the

92

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miss distance remains high for all burn through ranges out

to 2000 feet. But, for the remaining blinking frequencies

and burn through ranges the miss distances improve signifi-

cantly for increasing roll rate limits. A miss distance of

7.9 feet for a burn through range of 800 feet and 2 Hertz

blinking frequency is excellent.

2. Analysis of Sea Skimmer Scheme Results

The results for the sea skimmer scheme are quite

different than the baseline scheme.

Burn through range seems to be the single most

controlling variable. Miss distances for a burn through

range of 1 500 feet are marginally satisfactory for all roll

rate limits "and ECM blinking rates. However, at a burn

through range of 2000 feet, miss distances for varying roll

rate limits and ECM blinking frequencies are extremely low

and considered quite good.

Varying the ECM blinking rate produces little effect

on the miss distances. However, a unique roll rate limit

of 100 degrees per second produces miss distances double the

miss distances for roll rate limits of 50 or 200 degrees

per second.

3

.

Analysis of Glint Simulation Results

Figure 5-7 shows the random introduction of glint

in the target position and line of sight rate. The miss

distance for the run in this figure is 1 5 feet, but analysis

from other runs show that glint increases the miss distance

30 to 50 percent.

129

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4.. Proportional Navigation Ratio Selection

The following matrix shows the result of miss distance

in a no ECM environment for varying proportional navigation

constant (PN).

PN Miss Distance (feet)

3 10.6

3.5 6.9

4. 4. k

4.5 3.2

Although a proportional navigation constant (PN)

of ^.5 results in the smallest miss distance excessive over

shoot to step inputs results in this proportional navigation

constant as being an unsatisfactory choice. A PN of 4-.0 was

chosen based upon satisfactory system responses and the

small miss distance achieved.

C. CONCLUSIONS AND RECOMMENDATIONS

Based upon the results of this study it can be concluded

that miss distance is a function of roil rate limit, ECM

blinking rate, burn through range, and the guidance scheme

chosen. The guidance scheme chosen is most significant

of these factors.

For the baseline guidance and control scheme, roll rate

limit is the most significant variable affecting miss

distance. One exception to this is the 0.5 Hertz blinking

132

^ ^ "--l '"•> T> "*

rate set of runs which shows a miss distance improvement

when roll rate is increased from 5- to *Sj degrees per seccr.d,

but yields poor miss distances for 20C degrees per second.

The most significant variable affecting miss distance

for the sea-skimmer scheme is burn through range. Nevertheless,

a 1 O0-degree-per-second roll rate limit appears to be

unacceptable at all burn through ranges simulated.

In summary, given that the targe* :cr.-.r:ls IT" blinking

frequency and burr, through range, the optimum selections

are

:

Baseline Scheme - r.cll rate limit = 2~" degrees re:

Sea-Skimmer - Roll rare limit = 5C ;r 223 degrees per

second

The following additional studies are recommended:

a. Further modeling with glint.

b. Variation of the direction and ragr.irude cf the ISM

shift.

c. Alteration cf the decision variable used in the

simulation to switch phases from range :; line of sight

rare .

d. Variations in flight path geometry.

e. Comparison of control laws using skid-to-turn with

control laws usin- tank-ro-rurn maneuvering.

133

APPENDIX A

;AERODYNAMIC: •COEFFICIENTS- --- -o.iu— -

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'

• • A

v:--::::!;:::: :::— »::::-::::: A::: :::.:. .;.: .:::: :.::.::::

i^~~~: :~'^-~:-~~- :~~ '*b-

:: .:::::::;:-:::::•.::::::•;:::.!::

;--: T :.;;-;

* i ' i i i .i . i .—»-—»»...- .i . .i ... .»»,,. . * ttwit^-Ttr^ * —-~ .« — - ^-^..» , . —_— * • -- 1

Side Slip Characteristics

<* vs 3 vs Cnbasic

136

a Bfrsic

ciroSwft

Lift Characteristics

« VS CLbasic

137

nâa 'tvis xtfuj .1 |>o

Lift Characteristics

n vs ACLST

(n)

138

*Cn~<?)

~&Q:i:::

;

::r.i;-;-L:.:::: ;'i::Li:;:i^;:j}::;T^!:f;; :: ~^^:T::^-^:

Directional Control Characteristics

ocvs £ vs ACTom (C)"ST

139

AC^(5)•:

1 f

t

to

• 1

• • 1 4 f • • < ....1

'I

1 .I I , ...

| . . ....;-:---- v: - -

:——

'

j.- .. j i . :•..:

1• ' I ...

r;...

^...... .... .__._ ^ ._. .._

:-

*'""'' r"

i'' i'--:

... !

!*'.".;

:

: ' ! : ',

'

'

|. . . . . :

1 -—t

-~"*~ ]""'}"' "~"r--' ,

r -]"-' '" -

!' : " '

T

: ;

i

-• - : - -;•

;--- : '

-•; - r:- d-o* '

«

5 » — ro•,.:;.,.; * i ;

I ; j

••:•: ..*

--;--^-—~^- s ;-f-...-\

' ' J

t

• «*ti*=$ .'" -«- 8>o ~^.o

j . i . .i

. . .

. - . . t

1, • .

rrt-- -—-. <i ,„*- --

r~~~~7~ ~F~T'~ '~~" ."' !"". '

:•-•:.:. . , ! i . .

•r: :] •::::•; : ,::;;}:;: ;.:.:.:» :. , . . , . ....

4 ^. jr»-'O'OS "

1

: ' '

- ... , . ...(

:.....' ...... -Oli-[:[

i

•;.' [

1

-. ' ... :::.;..•.

:

. ..it I. .

!


vs £ vs ACn, (?)'ST

uo

Ac* (tf

t•

' •I~. '

(pg.^


<xvs £ vs AC-, (5 )

U1

Pitching Moment Characteristics

i vs C.M basic

U2

'•+'

r :::::H----

• -•*•"

1 • - • • »- • -1 1 -.

- JL

fi\+r-":ii::--\]\:^ --•-.- •;: ; ~::: ;:;-;. :~iHHJ.'.'.".'..".'. .'!"" r. \z~..'. ~.z^'-~m, ' — . . ; . .

.

—rt-r-r

1 ; .

:ir

:i :..::

— j-

IE

..— ..*...

::r::i1 r < T-: .:.:. : ...": :- ....':. :::: .- ::: .:..: : .:

,

•- i -

: :~;i:;^ ;„•; ••

• : : •: ;f; • ;• •

. __;- — ;/ . i ••;;;*; ;"••;;

—— •—-

•• -— '-/•[ i - -M ' — -

j

f*""

- Qg _....,-.- .i i |

•

:::. .::::;::::;:;::::••::;:::::. ::n-:":::. yr?:: r:.":.::::1

, rvj

—

.-!:••;• ''•••;:"•:;

' ;•.:;.'-—;.;—...— •• • _;.—.:

1

.

^NcUa3^4^3 " £*ls ; >uai'

::lh

•1 :f.

Pitching Moment Characteristics

n vs ACm.

(n )

U3

BO

o —

"

;.^J

::!:: h!iiuO;i!ia

Drag Characteristics

'lvs C

basic D basic

1U


nvs ACD (

n)

U5

.3

o o<6~

COO.-rr:O ;.

r

::Q.:_0:;:::

^7;*rr

:tr: .:_•_- :rr::.


K vs A CUST

U6


£ vs ACD (9

U7

O.Oftl

Lateral Control Characteristics

* vs K' vs AC (£)

1 ^8

AC- If)

• - • * •


vs £ vs ACnST

(?)

U9

Ac^s)


<* vs £ vs AC, ( £)1ST

1 50

APPENDIX B

STEADY STATE DATA

W = 2200 LBS (At end of cruise flight) - i><n> * »,

«

= 68.38 Slugs z U-n* 'tt.llM ty * Wl.(, l*frm

S =12 feet2

b = 8.4.85 Feet

"c = 1 .41 4- Feet

I = 27.8 Slug feet2

xx &

I = 1 507 Slug feet2

yy

I = 1 512 Slug feet2

zz 6

I 11.7 Slug feet2

>/* },

x z. n L I W C

^ 5-

^ ^ W<-^-<P^ •

T =T = 600 LBSx

M = 0.75

V T = 839 Feet/SecXSS

^SS=

* P SLV t = 837 LBS/feet2

oS

CT

= = 0.219SS q SS

S

C n = 0.024-2USS

<^

ss= 6

SS= 2.65° = 0.04625 Radian

SSJSS

bb= 243.3 LBS

cosass

WS3

= VT

sln°SS

= 38.79 Feet/sec

1 51

U np = V™ cos«Qq = 838.1 Feet/secSS T

ssbb

-0.06mbasic

SS

mSTSS

H = -1.0° = 0.01 7A5 Radian

1 52

APPENDIX C

AUTOPILOT ROOT LOCUS PLOTS

REAL AXIS (UNITS PER INCH) = 5.0000IMflG AXIS (UNITS PER INCH) - 5.0000

ROOT LOCUS OF PITCH RATE LOOP OF NORMRLCOMMAND FOR INNER LOOP AUJOPILOT DESIGN

ACCEL

-SO.OQ -45. 00mum it i i|i I I i i I | i 1-

-40.00 -35.00 -30.00 -25.00 -20.00 -15.00RERL AXIS

III' " IK-10.00 -S.00 5.00

153

REAL AXIS (UNITS PER INCH) = S.OOOOIMRG AXIS (UNITS PER INCH) - 5.0000

ROOT LOCUS OF NORMAL ACCELERATIONFOR INNER LOOP AUTOPILOT DESIGN

COMMAND

oo

II I t l| I I I I I ) I I I I I I I l|l IIIIIIIIH

-50.00 -45.00 -40.00 -3S.00m-30.00

i» m i»iiiiiiiiin iiiimn

|H im ii M i nn i | i i i

i

n-20.00 -15.00 -10.00 -5.00

cr

C3(J)

-25.00REAL AXIS

5.00

154

REAL AXIS (UNITS PER INCH) « 1.0000IMRG AXIS (UNITS PER INCH) - 1.0000

ROOT LOCUS OF NORMAL ACCELERATIONCOMMANO FOR INNER LOOP AUTOPILOT DESIGN

155

REAL AXIS (UNITS PER INCH) = 5.0000IMAG AXIS (UNITS PER INCH) - 5.0000

ROOT LOCUS OF ROLL RATE LOOP OF BANK ANGLECOMMAND FOR INNER LOOP AUTOPILOT DESIGN

Qa

+

+

+

+

+

+

+

+

+

a-into

Xcc

Ice

*

+

+

+

K il l I I l|l I l I—I 1- 1 1 I 1 |) I I I I II 1

1

1

00 -40.00 -35.00 -30.00 -25.90 -20.00 -IS. 00 -10.00REAL AXIS

-50.00 3lF -5.00 ,fib 5.00

156

REAL AXIS (UNITS PER INCH) = 5.0000IMBG AXIS (UNITS PER INCH) - 5.0000

ROOT LOCUS OF BANK ANGLECOMMANDFOR INNER LOOP AUTOPILOT DESIGN

oo

1

_oin

aaJiia"

oaJO»

+

o +o

s +

+

+a° *_o<n +

+

•a+ aanvt

+ w-^

+CC

+°o+ °a:

++

+4- o

+ a+ jn

+

\ 4- ._

\ +• aa\ r JO+ pa

+ ++

+

+

*

ao"ui

oo

-50.00 -4S.00 -HO. 00 -35.00 -30.00 -25.00 -20.00REAL AXIS

-15.00

+

-10.00 -5.00 .fib

ooJU>

1

5.00

++

157

REAL AXIS (UNITS PER INCH) = 5.0000IMRG AXIS (UNITS PER INCH) - 5.0000

ROOT LOCUS OF TRW RATE LOOP OF TURNCOORDINATOR FOR INNER LOOf5 RUTOPILOT DESIGN

p 1^ < 11 » j

-50.00 J4S.00 -*40.00 -35.00 -30.00 -25.00 -30.00REAL AXIS

15.00 -10.00 -5100

+oi fib S.QO

158

REAL AXIS (UNITS PER INCH) = 1.0000IMAG AXIS (UNITS PER INCH) - 1.0000

ROOT LOCUS OF TAW RATE LOOP OF TURNCOORDINATOR FOR INNER LOOP AUTOPILOT DESIGN

aaJ3

+r-10. 00 -9.00 -13.00 -'7.00 -6.00 -^.00 -4.00

REAL AXIS

T-l

—

-3.00-ti—i—i i lit ) 1 1 nun— i

-2.00 -1.00

oo»£X

ft) l.ao

159

REAL AXIS (UNITS PER INCH) = 5.00001MHG AXIS (UNITS PER 1NCHJ - 5.0000

ROOT LOCUS OF TURN COORDINRTORFOR INNER LOOP AUTOPILOT OESIGN

iI i i t I |i i i I I i i i) 1 1 m in im

-50. 00 -45.00 -40.00 -3S.00 -30.00 -25.00 -20.00REAL AXIS

-15.00 -10.00—I I I

-5.00

tr

La 3:

oo-O

,9b 5.00

160

REAL AXIS (UNITS PER INCH) = 1.0000IMflG AXIS (UNITS PER INCH) - 1.0000

ROOT LOCUS OF TURN COORDINATORFOR INNER LOOP AUTOPILOT DESIGN

•10.00 -9.00 -8.00 -7.00-i 1 * —m in i ii n)iii i i

-6.00 -5.00 -4.00 -3.00REAL AXIS

1 1 i i i i i h i 1 1 niiMm-2.00 -I. 00

o. tn

x<x +

r, +2cr +

.:*=

a 1. 00

161

REAL AXIS (UNITS PER INCH) = 5.0000IMRG AXIS (UNITS PER INCH) » 5.0000

ROOT LOCUS OF FLIGHT PATH ANGLECOMMAND FOR OUTER LOOP AUTOPILOT DESIGN

-50. 00 -US. 00 -40.00 -35.00 -30.00 -25.00 -20.00 -15.00REAL AXIS

5.00

162

REAL AXIS (UNITS PER INCH) = 1.0000IMflG PXIStUNITS PER INCH) - 1.0000

ROOT LOCUS OF FLIGHT PATH ANGLECOMMAND FOR OUTER LOOP AUTOPILOT DESIGN

b.oo

+

+

++

++

+

++

+

+

+

\

aa~a

aa"«»

aO>»

8"io

oa

- -at

XCE

aa

aa

ae

-10.00 -9. 00 -a. ao -7.00 -6.00 -5.00 -4.00RE£L AXIS

+

-3.00 -2.00 -1. 00 ,6b l'.oo

oa

+*

163

HEAL RXIS IUNIT5 PER INCH) = 5.0000IMAG AXIS (UNITS PER INCH) - 5.0000

ROOT LOCUS OF ALTITUDE HOLOCOMMANDFOR OUTER LOOP AUTOPILOT DESIGN

-bo. 00 -as. oo -40.00 -35.00-l-30.00 -25.00 -20.00

REAL AXIS-15.00 -L0.00

-into

x<X

5.00

164-

REAL AXIS (UNITS PER INCH) = 1.0000IMRG AXIS (UNITS PER INCH) - 1.0000

ROOT LOCUS OF ALTITUDE HOLDCOMMANDFOR OUTER LOOP AUTOPILOT DESIGN

165

APPENDIX D

BASELINE GUIDANCE LAW" SIMULATION

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APPENDIX E

SEA SKIMMER GUIDANCE LAW SIMULATION

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LIST OF REFERENCES

1. Gonzalez, J., New Methods in the Terminal Guidance- andControl of Tactical Missiles , AGARD Lecture SeriesNo. 101, 1979.

2. Hewett, M.D., Guidance and Control Systems Course(Class Notes) , 1982.

3. Hoisington, D.B., Introduction to Electronic WarfareNotes, 1980.

J+. Speclchart, F.H., and Green, W.L. , A Guide to UsingCSMP-The Continuous System Modeling Program , Prentice-Hall, 1976.

187

INITIAL DISTRIBUTION LIST

No. Copies

1 . Library, Code 01 k.2 2Naval Postgraduate SchoolMonterey, California 9394-0

2. Department Chairman, Code 62 1

Department of Electrical EngineeringNaval Postgraduate SchoolMonterey, California 9394-0

3. Professor M.D. Hewett, Code 67Hj 5

Department of AeronauticsNaval Postgraduate SchoolMonterey, California 9394-0

4.. Professor H. A. Titus, Code 62Ts 1


5. Professor G. Thaler, Code 62Tr 1


6. Director 1

Attention: CDR Richard EessendenJCM 531Joint Cruise Missile ProgramsBldg. NC-1Washington, D.C. 20360

7. Professor D. J. Collins, Code 67Co 1

Department of AeronauticsNaval Postgraduate SchoolMonterey, California 9394-0

8. LCDR Kent B. Watterson 5

Naval Sea Systems Command (SEA-62Y1F)Department of the NavyWashington, D.C. 20362

9. LT Daniel A. Forkel 1

Naval Postgraduate SchoolSMC 2088Monterey, Calfiornia 9394-0

1 88

No. Copies

10. Defense Technical Information CenterCameron StationAlexandria, Virginia 2231 4-

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