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Calhoun: The NPS Institutional Archive
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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Naval Postgraduate SchoolMonterey, California 9394-0
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Naval Postgraduate SchoolMonterey, California 9394-0
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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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Approved for public release, distribution unlimited
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
100 deg/sec (L STAB, R STAB vs TIME) 102
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
50 deg/sec (L STAB, R STAB vs TIME) 108
5-3e Baseline Response for ECM 1 CPS, Roll Rate Limit
50 deg/sec (RUDDER vs TIME) 109
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
200 deg/sec (PCMD,P vs TIME) 111
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
200 deg/sec (RANGE, XRANGE, YRANGE, ZRANGE vs
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
200 deg/sec (YMISLE, HMISLE vs XMISLE) 116
5-5a Sea Skimmer Response for No ECM, Roll Rate Limit
100 deg/sec (PCMD,P vs TIME) 117
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
100 deg/sec (RANGE, XRANGE, YRANGE, ZRANGE vs
TIME) 119
5-5d Sea Skimmer Response for No ECM, Roll Rate Limit
100 deg/sec (L STAB, R STAB vs TIME) 120
5-5e Sea Skimmer Response for No ECM, Roll Rate Limit
100 deg/sec (RUDDER vs TIME) 121
5-5f Sea Skimmer Response for No ECM, Roll Rate Limit
100 deg/sec (YMISLE, HMISLE vs XMISLE) 122
5-6a Sea Skimmer Response for ECM 1 CPS, Roll Rate Limit
100 deg/sec (PCMD,P vs TIME) 123
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
100 deg/sec (RANGE, XRANGE, YRANGE, ZRANGE vs
TIME) 125
5-6d Sea Skimmer Response for ECM 1 CPS, Roll Rate Limit
100 deg/sec (L STAB, R STAB vs TIME) 126
5-6e Sea Skimmer Response for ECM 1 CPS, Roll Rate Limit
100 deg/sec (RUDDER vs TIME) 127
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) ^i^/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
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rH
Cti
Hcd
>
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Fn
-pin
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co•H-PClj
crw
03
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oo
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isi
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cd
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cdX! tSJ
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CO 1
ho CO• 8 COS cd
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CO £3 1
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cr
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cr
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cd
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<£1
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pa
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bfl
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a
Ph
ca
QQ.
Q+-
QQ.
II
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{=> •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 = ^on =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
<U
o
I
CD
ttO
•H
09 '8 OH '2 02 "2
H13H102 'S 00 -S 08 '2 00*2 08 "J OS't
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—
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JLS
00- 0- S2'0- QS'O-
45
ca
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ca fl
H >5S Q
oo -4-
st +1 ca
cven«
> oU OCD rHca
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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
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a _
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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
—
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1 . r~~-
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O cv
m
m
-—v •
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—
\
—
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+-
CO
in
1— cv-<* •
• o r^-** + cdT— cv W
1 CO
oo -<fr
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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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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 ^^AZEsin(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
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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
b. Limits on Guidance Commands
In the ingress phase the following limits are
applied to commanded variables:
-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
b. Limits on Guidance Commands
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.
b. Limits on Guidance Commands
In the pop up phase the following limits are
applied to commanded variables:
-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
5-3 (a-f) ECM (1Hz), 50°/sec roll rate
5-4 (a-f) ECM (1Hz), 200°/sec roll rate
SEA SKIMMER SCHEME
5-5 (a-f) NO ECM, 100°/sec roll rate
5-6 (a-f) ECM (1Hz), 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— -
: •
- £^mm
Side Slip Characteristics
« vs S vs C^
^basic
13^
BtelC
t . I .
Side Slip Characteristic
a: VS 6 VS C-,
'basic135
SAUC
':::::--:::r:„::::.-M
:
, » i' — '—
'
• • 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
n^aa '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. .
!
Directional Control Characteristics
vs £ vs ACn, (?)'ST
uo
Ac* (tf
t•
' •I~. '
(pg.^
Directional Control Characteristics
<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
Drag Characteristics
nvs ACD (
n)
U5
.3
o o<6~
COO.-rr:O ;.
r
::Q.:_0:;:::
^7;*rr
:tr: .:_•_- :rr::.
Drag Characteristics
K vs A CUST
U6
Drag Characteristics
£ vs ACD (9
U7
O.Oftl
Lateral Control Characteristics
* vs K' vs AC (£)
1 ^8
AC- If)
• - • * •
Lateral Control Characteristics
vs £ vs ACnST
(?)
U9
Ac^s)
Lateral Control Characteristics
<* 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
ooȣX
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
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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
Department of Electrical EngineeringNaval Postgraduate SchoolMonterey, California 9394-0
5. Professor G. Thaler, Code 62Tr 1
Department of Electrical EngineeringNaval Postgraduate SchoolMonterey, California 9394-0
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
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ThesisW2825 Watterson
2017«*5
c.l Bank-to-turn cruisemissile terminal guid- C
ance and control lawcomparison.