An Analog Computer Study of the Effectiveness of Interceptor Commands Derived from a Prediction Equation of Second Order
COPY.230
NASA MEMO 10-7-58A
NASA
MEMORANDUM
I
AN ANALOG COMPUTER STUDY OF THE EFFECTIVENESS
O F INTERCEPTOR COMMANDS DERIVED FROM
A PREDICTION EQUATION O F
I
I
SECOND ORDER
By Brian F. Doolin and John D. McLean
Ames Research Center
Moffett Field, Calif.
I
Thls materlal conIniormatlon affecting the Natlonal Defense of the Unlwd States wlthln the rneanlng
Secs.I 9 3 and 794,the transmlSSlon or revelatlon of whlcb In any
manner to an unauthrlzed person 1s prohlblted ty law.
Of the espionage laws, Title 18, U.S.C.,
SPACE
ADMINISTRATION
I
WASHINGTON
I
llllliilllslllslsll I
L/
NATIONAL AERONAUTICS AND SPACE
AMVIINISTRATION
OOb3050
MEMORANDUM 10-7-58~
AN ANALOG COMPUTER STUDY OF THE EFFECTIVENESS
OIF
NTERCEPTOR
COMMANDS
DERNED FROM
A P€EDICTION EQUATION OF
SECOND ORDER*
By Brian F. Doolin and John
D. McLean
SUMMARY
The present report contains an attempt
t o improve the accuracy of an
automatic interceptor flying a lead-collision course against
a maneuvering
t a r g e t . For t h i s improvement, the prediction equations that provide the
i n t e r c e p t o r ' s guidance were modified by incorporating terms of second
order t o p r e d i c t t h e f u t u r e l o c a t i o n of a s t e a d i l y maneuvering t a r g e t .
"he i n t e r c e p t o r commands derived from t h e second-order prediction equations allow the interceptor to fly
a s t r a i g h t l i n e course against a t a r g e t
flying with constant acceleration.
The s t a b i l i t y of t h e system i s studied
by means of an analog computer. The system accuracy was evaluated i n terms
of rocket miss and i s comparedon t h i s basis with the performance of an
interceptor with commands derived from a f i r s t - o r d e r p r e d i c t i o n scheme.
The comparison covers cases of unlimited and l i m i t e d i n t e r c e p t o r a c c e l e r ation capability, constant and p u l s e a c c e l e r a t i o n t a r g e t maneuvers, and
v a r i a t i o n s i n r o c k e t speed.
i
!
INTRODUCTION
The present report i s p a r t of a f l i g h t and analog computer study of
t h e f i n a l a t t a c k phase of automatic interception currently being conducted
a t t h e Ames Aeronautical Laboratory of t h e NACA. The i n i t i a l work (ref. 1)
concerned improvements in tracking accuracy and system s t a b i l i t y of an
i n t e r c e p t o r f l y i n g a pursuit course.
The work was continued i n r e f e r ences 2 t o 4, where improvements i n system s t a b i l i t y of an i n t e r c e p t o r
f l y i n g a lead-collision
course
against
a nonmaneuvering t a r g e t were
reported.
I
I
2
-
1
I
.
I n a pursuitcourse,
where a n a i r p l a n e t r i e s t o
keep fixed guns
.
pointed a t a t a r g e t f o r a protracted period of time, the system accuracy
can be described i n terms of the tracking accuracy of the interceptor.
I n a c o l l i s i o n course, whether t h e i n t e r c e p t o r ' s r o c k e t s h i t
or miss t h e
t a r g e t depends on t h e i n t e r c e p t o r ' s heading and p o s i t i o n a t t h e s i n g l e
i n s t a n t of f i r i n g , and the tracking accuracy a t times other than firing
However,
time i s much less importantthan it i s i n a pursuit course.
although the tracking accuracy requirements of
a rocket-firing interceptor
on a collision course are low throughout most of an attack, the geometry
computing accuracy requirements are high.
Thecomponents of a predicted miss are calculated from measured and
computed geometric quantities.
The computation i s done by an attack computer which takes i n t o account the present range and bearing of the target
from t h e a t t a c k e r , and how t h e range and bearing change withtime.
On
t h e b a s i s of t h i s information and t h e knowledge of the distance and direct i o n t h a t t h e r o c k e t s w i l l t r a v e l between t h e t i m e t h e y a r e f i r e d and t h e
time they should h i t t h e t a r g e t , t h e computer predicts by howmuch t h e
rockets will miss t h e t a r g e t .
This predicted miss i s converted i n t o commands f o r t h e a u t o p i l o t s o t o modify t h e heading of t h e a i r p l a n e a s t o
reduce the predicted miss t o zero.
miss i s t hd
eesirec
dr i t e r i o n ,
Since
the
actual,
not
predicted,
rocket
the accuracy of a system depends not o n l y on how well the airplane follows
i t s commands, b u t a l s o on t h e q u a l i t y of prediction. Here t h e q u a l i t y of
i t s computation.
p r e d i c t i orne f e rttsohseo r t
assumptions
of
underlying
In current fire-control systems, for instance, first-order prediction
is
used; t h a t is, t h e miss i s predicted on t h e assumption t h a t t h e t a r g e t
w i l l continue t o maintain i t s presentheading u n t i l impact time. If t h e
t a r g e t maneuvers, t h e r o c k e t s a c t u a l l y will miss t h e t a r g e t even though
t h e a i m l a n e i s f l y i n g so as t o keep the predicted miss zero. It i s c l e a r
t h a t improvement i n radar and control system dynamics w i l l not substant i a l l y change t h i s r e s u l t .
A modification must be made t o t h e p r e d i c t i o n .
Modifying the prediction equation has not been t h e u s u a l method
adopted i n designing interceptors for use against maneuvering t a r g e t s .
Since i n f i r s t - o r d e r p r e d i c t i o n a s t e a d y p r e d i c t i o n l a g i s introduced by
a s t e a d i l y maneuvering t a r g e t , t h e c o n t r o l systems approach has suggested
that various amounts of integration be added t o t h e a u t o p i l o t commands.
Experience, however, i n d i c a t e s t h a t t h e a d d i t i o n of even varying
amounts
of integration-decreases the interceptor system s t a b i l i t y without satisf a c t o r i l y improving t h e chancesof h i t t i n g t h e t a r g e t .
Another method
i s t o attempt input differentiation (e.g.,
of reducing the steady error
problems
s e e r e f . 5 ) which -can be applied successfully in pursuit-course
where s i m i l a r f i n a l geometry recurs from run t o run, and t h e problem
depends much less on time.This
method, however, w i l l not be successful
when a p p l i e d t o a collision course unless the system gains a r e _scheduled
_ .... -..
,, ..
..
i n a r a t i o n a l manner.
*.
- . " .". .1-
"?q
1
a
d
3
The simplest way t o schedule the gains i s t o determine t h e i r
But the airborne
dependence on t h e geometryof t h e p a r t i c u l a r a t t a c k .
determination of t h i s dependence i s j u s t what i s accomplished i n t h e nonmaneuvering case by the first-order prediction
of miss. In t h e sameway,
a higher order of prediction can be made i n t h e computation of miss t h a t
rill t a k e i n t o a c c o m t t a r g e t maneuvers with reasonable success.
The present report summarizes some workdone along these lines.
A
second-order equation of prediction i s derived, and autopilot
command
equationsareobtained
from it. The c h a r a c t e r i s t i c s of t h e r e s u l t i n g
path of t h e i n t e r c e p t o r (which i s a s t r a i g h t l i n e , r e g a r d l e s s
of t h e target acceleration, as long
as it i s constant) are compared with those of
the path of t h e i n t e r c e p t o r r e s u l t i n g from f i r s t - o r d e r p r e d i c t i o n .
The
accuracy of two i n t e r c e p t o r systems which d i f f e r o n l y i n t h e i r p r e d i c t i o n
and command equations i s compared on t h e b a s i s of "actual" miss b y r e s u l t s
ofanalogsimulation.
It i s shown t h a t , i n c o n t r a s t t o t h e
systemunder
f i r s t - o r d e r guidance, t h e second-order system can be designed t o perform
s u c c e s s f u l l ya g a i n s tt a r g e t si ns t e a d y
g turning maneuvers. Comparisons
between f i r s t - and second-order predictions also are
made t o show the
e f f e c t of limiting the attacker's maneuverability,
and the e f f e c t of
increasing the average speed of the attacker's
armament.
'The e f f e c t s of additive input noise on the operation of the
system
havebeenignored
in this study.
The present report primarily specifies
the geometric dependence of the various terms of t h e a t t a c k computation,
a dependence t h a t will be common t o a l l systems that t r y t o accomplish
t h e same task. Furthermore, it seems d e s i r a b l e t o know whether or not a
conceptual scheme w i l l work and what a r e i t s inherent limitations apart
from considerations of noise before optimization of a system i s attempted.
The answers to these questions can be ascertained only by
such a study as
i s contained herein.
NOTATION
A
azimuth position angle between radar antenna and a i r p l a n e
(sketch(b)),radians
as superscripts o r s u b s c r i p t s e i t h e r d i f f e r e n t i a t e between
attacker, earth, or target coordinate systems o r distinguish
anattackerproperty
from t h a t of t h e t a r g e t ( a s
Va i s
attacker speed)
1
L-
.
E
elevation position angle between radar antenna and a i r p l a n e
(sketch ( b ) ) , radians
F
distance traveled by the rocket relative to the attacker,
ft
4
g
32.2 f t / s e c 2
M
miss, or distance between rocket
and
target
p,q,rangularvelocity
a t impact time, ft
components of theairplane,radians/sec
R
present
distance
S
Laplace
transform
variable,
l/sec
T
time-to-go,
duration
sec
tf
rocketraveltime
tm
time between thebeginning
sec
V
speed of a t t a c k e r or t a r g eftt,/ s e c
x,Y
positionparametersofattacker
system ( s k e t c h ( c ) ) , f t
a
angle of a t t a c k of attacker
airplane,
radians
7
a t t a c k e r 'vs e l o c i t y
direction
angle
w i t rhe s p e ctto
reference (sketch ( c ) ) , radians
At
duration
of
acceleration
pulse
0
t a r g evt e l o c i t yd i r e c t i o na n g l ew i t hr e s p e ctto
ence(sketch(c)),radians
5
heading
angle
of
radar
antenna
with
respect
( s k e t c h( c ) ) ,r a d i a n s
R
r a t e of r o t a t i o n of radar
antenna
coordinates,
radians/sec
l , 2 , 3l a b e l s
(-1
between t a r g e t and attacker, f t
of time from t h ep r e s e n tu n t i l
impact time,
or timeofiring,sec
of a t a r g e t maneuver andimpact
time,
or t a r g e t i n anearthreference
an e a r t h
of t a r g e t maneuver, sec
an e a r t hr e f e r t o anearthreference
ofanyright-hand
t r i a d of unit vectors(assubscripts,
t h e components of a vector associated with the pertinent unit
vectors)
vector
quantity
ANALYSIS
The simulation described i n reference 2 was used i n analog computer
runs of t h e F-86D control-surface
tie-in (CSTI) system against-a target
* , -.:~...,.- .:;-. :
?
q
#
" .
'
I
5
".
, ,".
5
executingsteady
g maneuvers i n elevation. The runs i n d i c a t e dt h a t
l a r g e misses could be expected
i n a t t a c k i n g a t a r g e t so maneuvering.
Furthermore, it was apparent that the dynamics of t h e radar and t h e i n t e r ceptor control system, i f modified according t o references 2 and 4, had
r e l a t i v e l y l i t t l e e f f e c t on t h e magnitude of the misses which was due
fundamentally t o improper p r e d i c t i o n i n t h e a t t a c k computer. To improve
t h e s e r e s u l t s , new second-order prediction equations
were derived 17hich
reduce t o t h e p r e v i o u s commandswhen t h e t a r g e t makes no maneuver, but
vhich enable the attacker t o f l y a n - e f f e c t i v e l y s t r a i g h t - l i n e i n t e r c e p t i o n
against a t a r g e t i n a steady g maneuver.
Themeaning of the old prediction equations
inspection of sketch ( a ) .
can be understood a f t e r
Sketch ( a )
The q u a n t i t i e s V a and & represent the velocities
of t h e a t t a c k e r and
t a r g e t . The d i s t a n c e r e l a t i v e t o t h e a t t a c k e r t r a v e l l e d
by a rocket i s
designatedby
H. The r e l a t i v e p o s i t i o n of t h e t a r g e t from t h e a t t a c k e r
at anytime i s representedby
R. If t h e a t t a c k e r and t a r g e t each
- fly
i n a ' s t r a i g h t l i n e , t h e n i n a time T, t h e y t r a v e l a distance VaT and VtT,
respectively. From t h e diagram, then TaT + H.+ E = E + VtT, where E,
t h e miss, closes the vector-polygon.
Taking a =
Va, one can m-ite
theequation
+ P = + TR.
a
vt -
The new prediction equations were derived after it was recognized t h a t
t h e miss equations mechanized i n t h e p r e s e n t E-4 system can be considered
a Taylor's expansion of the separation of t h e t a r g e t and t h e i n t e r c e p t o r
aroundimpact time.Second-orderpredictionequations
are obtainedsimply
6
-
by incorporating a term of higher order i n time-to-go, T. Instead of
M + P = E + Tg, the equation becomes a +
= R + T? + (T2/2)$.
The
first and second rates of range of
t a r g e t from attacker are represented
by
and E. The quantity T, t h e time-to-go, i s thelengthoftime
from
"now" until rocket .impact.
2
..
Thenew equation, when expressed in radar coordinates, i s given by
the following set of equations which a r e d e r i v e d i n appendix A.
M1 = R
T2
- F cos A cos E + T (1+ 22 A
R(Qz2 + Qg)
at) k - 2
M2=FsinA+R
E 2)
(
1+--
-% = F cos A s i n E +
R
R2&
+ T2
- RQlS22
2
~ax
) R2Q2 - T22 Rill&
(
l
+2
The r e l a t i o n s h i p between airframe and radar coordinates i s i n d i c a t e d i n
sketch ( b ) . The radarcoordinate system, with unit vectors T, P,
is
obtained from the airplane
coordinate
system
(with unit vectors i a , 2a, 3 a ) by first r o t a t i n g through the angle A. about t h e 3a direction,
thenbyrotatingthroughtheangle
E about
the 2 direction.
3,
'
3a
Sketch ( b )
M1 = R
-M3
When a l l motion of t h e t a r g e t and a t t a c k e r
i s constrained t o t h e same v e r t i c a l plane, the
azimuth component of miss, M2, 'becomes zero.
The other
equations,
the
time
and elevation
components of miss, become
( E 2) k - 2
- F COS E + T 1 + - -
( - -at
= F s i n E + - 13-
T'
R
2 ').R2i22
T2 RQz2
J
7
k
The significance of the various terms i n equations (1)becomesmore
evident i f t h e equations are expressed i n a fixed coordinate system.
With t h e d e f i n i t i o n s i n d i c a t e d i n sketch (c),
t h e equations take t h e form
Mle = -F cos(y-s)+x cos k+y s i n E+T(k cos E+?
T2
(2 cos k+j; s i n E-)
2
M2e = -F sin(y-()+T($ cos 5-k s i n E )
sin E) +
1
+
J
where x = xt-x,,
y = yt-ya.
The q u a n t i t i e s Va, xa, ya,and
7 specify
a t t a c k e r speed, position, and heading. The q u a n t i t i e s V t , x t ,y t ,
and 8
s p e c i f yt h et a r g e t
speed, position, and heading. %e angle 5 s p e c i f i e s
t h e heading of t h e radar which i s mounted i n t h e a t t a c k e r and points
toward t h e t a r g e t .
In t h e two-dimensionalsystemunderconsideration,
t h e system consists of t h e time-to-go, or simply, the timechannel and
t h e e l e v a t i o n channel. The timechanneldetermines
the proper instant
at which t o f i r e the rockets.
The elevation channel determines the proper
normal acceleration of t h e i n t e r c e p t o r .
The miss equations (1)or ( 2 ) are not the appropriate expressions
f o r u s e as commands t o t h e a i r p l a n e - a u t o p i l o t
system. Experience showed
That portion of
t h a t a systemusing them a s commands w i l l be unstable.
t h e r e l a t i v e a c c e l e r a t i o n due t o t h e i n t e r c e p t o r ' s own motion must be
removed from them and t h e remaining signals used as t h e commands:
,-
.
I
i
I
.
8
S l e = -F cos(y-{)+x COS E+y s i n 6+T(2 COS
T2 (%cosE+j;tsin
6)
2
T2
= Mle + - (jraCOS 6+jiasin 6 )
2
S2e = -F sin(y-E)+T(j, COS 6-f s i n 6 )
T*
= M2e + 2
+ T2
2
COS E-jrasin E )
When written in the radar coordinates, these equations
become
T2
SI = R-F C O S ( E " U ) + T ~+~ [E-RR2+Vaf sin(E+a)]
2
S2 = F sin(E+a) +
R
+5
~ax) R2R + T2
2 Vai. cos (E-)
(
l
.
1
(4)
When t h e i n t e r c e p t o r performs according t o t h e s e command equations, the
miss equations are said to be nulled through the
a c t i o n of the "outer-loop
geometry. ''
The r e s u l t s of numerical calculations
made using equations ( 4 ) a r e
comshown in figures 1 and 2. They can be contrasted with the results
puted with the
same equations minus t h e terms i n T2, which are shown i n
at
f i g u r e s 3 and 4. The computation assumes t h a t t h e i n t e r c e p t o r t u r n s
a r a t e p r o p o r t i o n a l t o t h e e l e v a t i o n command without any dynamic e f f e c t s ;
t h a t is, the airplane-autopilot loop
i s assumed perfect. The speed of
the interceptor, Va, i s taken t o be 1,000 f e e t p e r second; t h a t of t h e
a t t h e r a t e of
t a r g e t , 8 0 0 - f e e t p e r second. The target begins turning
0.05 radian per second a t the beginning of the calculation and i s a t t h a t
time 4,000 f e e t d i r e c t l y aheadof theinterceptor.
The value of F i s
1,500 f e e t .
when flying against
Figure 1 shows the path taken by the interceptor
a target flying the course
*shown i n t h e f i g u r e .
The i n t e r c e p t o r t r a j e c t o r y i s nearly a s t r a i g h t l i n e aimed a t an impact point predicted
immedia t e l y by i t s second-order a t t a c k computer. The pathflownby
the
interceptor with a f i r s t - o r d e r a t t a c k computer i s curved ( f i g . 3). I n
t h e l a t t e r case, since the impact point predicted a t any time l i e s along
a t t h a t i n s t a n t , t h e impact point keepschanging
theatarget's flight path
i t s position. The interceptor,therefore, mustmaneuver continually.
In figures 1 and 3, t h e l i n e s connecting t h e two f l i g h t p a t h s
representtheinterceptor'sline
of s i g h t t o t h e t a r g e t .
The sequence
i
9
of l i n e s i n each figure provides a t i m e h i s t o r y of the angle between
t h i s l i n e and the interceptor's path.
A comparison of the behavior of
t h i s l e a d a n g l e i n t h e two f i g u r e s shows t h a t it varies considerably i n
the case of second-order prediction,
andremains nearly constant i n t h e
case of first-order prediction.
(4)
Figures 2 and 4 show t h e t i m e h i s t o r i e s of t h e terms of equations
from t h e computation. The terms of t h e f i r s t - o r d e r command system ( f i g . 4)
a r e s e e n t o b e much smoother. The magnitude of t h e a c c e l e r a t i o n terms,
- (E-RQ"+V,T s i n E ) and T
T"
R2Q+Vaf cos
, i nf i g u r e s2 ( a ) and(b)
2
however,shows t h a t t h e y a r e n o t n e g l i g i b l e ,
a fact graphically illust r a t e d b y d i f f e r e n c e i n i n t e r c e p t o r f l i g h t p a t h s i n f i g u r e s 1 and 3.
Figures 2( c) and (a) show the important influence ofeachof
t h e components of the acceleration terms; none of them can be neglected without
a serious modification of t h e shapes of the acceleration terms.
,
TEST EQUIPMENT AND PROCEDURE
I n t h e previous section, a p o s s i b l e s e t of second-order command
equations was obtained whose use should increase the effectiveness of an
i n t e r c e p t o r a t t a c k i n g a t a r g e t which i s turning a t a steady rate. The
analysis, however, neglected a l l those transient dynamic e f f e c t s w i t h
which the designer of an actual system must cope. Since experience has
indicated that results of studying a dynamic problem on an electronic
analog computer agree quite well with results obtained
i n f l i g h t , and
s i n c e t h e methods of mechanizing t h e p r e d i c t i o n e q u a t i o n f o r an analog
computer p a r a l l e l t h o s e a v a i l a b l e t o t h e d e s i g n e r
of airborne hardware,
it i s u s e f u l t o i n v e s t i g a t e t h e complete dynamic system on a n e l e c t r o n i c
analog computer. The remainderof t h i s r e p o r t i s concerned with an anal o g s i m u l a t i o n t o determine i t s s t a b i l i t y and i t s effectiveness under
variedconditions.
The present section describes the simulation
as set
up on an Electronic Associates analog computer.
Simulation of Automatic Interceptors
The simulation i s a modification of the simulation of the F-86D CSTI
system described i n reference 2. The simplified block diagram (fig. 5 ) ,
adapted from t h i s r e f e r e n c e , i n d i c a t e s that t h e system can be divided
i n t of i v ep a r t s :
radar, a t t a c k computer, attackcoupler,airplaneautopilot loop, and geometry. The radar, being mounted on t h e i n t e r c e p t o r
and receiving reflected signals
from t h e t a r g e t , measures t h e range, range
rate, position, and a n g u l a r r a t e of t h e l i n e of sight from t h e i n t e r c e p t o r
t ta ro gtehte
and provides
computer. The a t t a c k
10
'
,
.
computer i s an analog device which operates on the radar-furnished
q u a n t i t i e s t o compute a predicted value of miss by means of those missprediction
equations
described
in
the
previous
section
on Analysis. It
i s t h e f u n c t i o n of the attack coupler to process the predicted
miss values
provided by the computer i n t o a form t h a t t h e a u t o p i l o t can use as commands
It computes t h e normal acceleration
todrivetheairplanecontrols.
required andbank-angle e r r o r of t h e i n t e r c e p t o r .
The airplane-autopilot
loop, by properly reacting
t o t h e c o u p l e r commands, banks and accelerates
t o b r i n g t h e p r e d i c t e d miss t o zero. The box marked "geometry" i n t h e
computer i s concerned,
diagramof f i g u r e 5, a s far as the electronic analog
contains that assortment of operations on t h e motionof t h e two airplanes
which provides the information on t h e i r a b s o l u t e and r e l a t i v e p o s i t i o n s
r e l e v a n t t o t h e problem.
The iresent study required two b a s i c changes i n t h e s i m u l a t i o n
described i n reference 2: t h e a t t a c k computer was expanded t o include
the operations necessary to obtain a second-order prediction of
miss;and
t h e geometry vas diminished so a s t o f i t t h e t o t a l problem on t h e two
analog-computerconsolesavailable.Forgeometricsimplicity,the
motions
of t h e i n t e r c e p t o r and t a r g e t were confined t o a single plane containing
t h e v e r t i c a l (i.e. a t a i l chase). Although t h i s c o n s t r a i n t i s d r a s t i c ,
it does not invalidate the conclusions
of t h e r e p o r t f o r two reasons.
I n t h e first place, equations (1)of the previous section show what i s
needed a n a l y t i c a l l y f o r t h e e x t e n s i o n of t h e miss p r e d i c t i o n t o t h r e e
dimensions, so that the extension may be made i n a straightforward, though
physicallycomplicated,fashion.Inthe
secondplace,preliminaryunreported t r i a l s i n d i c a t e d t h a t t h e most serious prob1,ems of s t a b i l i t y
p e c u l i a r t o t h e second-order prediction were encountered during attack
6 i l l u s t r a t e s i n schematic
from t h e nose or t a i l of t h e t a r g e t . F i g u r e
form t h e geometry mechanized f o r t h e problem.
,
The geometric constraint decreased the requirements of the .other
four boxes shown i n f i g u r e 5 . The block diagramof f i g u r e 7 depicts the
attack coupler and the airplane-autopilot loop.
The constraintreduces
as
the required channels fromazimuthand elevation to elevation alone,
far a s a i r p l a n e performance i s concerned. I n t h e a t t a c k computer, only
two channels a r e needed, t h e e l e v a t i o n channel,and the time channel.
These channels a r e i l l u s t r a t e d i n f i g u r e s 8 and 9. By comparison of t h e
show
mechanization of first- and second-order prediction, these figures
the increase in operations
needed f o r second-order miss prediction. The
radar simulation can also be simplified. Since not only the geometric
reduction but especially the imprcvements reported i n reference 2 have
removed t h e radar a s a possibte source of f l i g h t p a t h i n s t a b i l i t y , a radar
t r a n s f e r f u n c t i o n of u n i t y was used i n t h e p r e s e n t work f o r both the
first- and second-order systems.
Simulation of Miss
The equations and method for obtaining the quantities from which t h e
distance of miss was calculated are described in appendix
B. A number
of l i m i t a t i o n s on t h e a c t u a l motion of rockets was made which simplified
t h e computation without invalidating the system performance comparisons
These limitationsfollow.
A singleaverage
described i n t h i ss t u d y .
rocket i s f i r e d i n any pass on t h e t a r g e t , and f l i e s i n a straight l i n e
with a known average velocity along the direction tangent to the path
of
theinterceptor at firingtime.
I t s distance from t h e t a r g e t i s evaluated
exactly 1.5 seconds a f t e r it has been f i r e d . This distance i s the value
of miss use2 i n t h i s r e p o r t .
RESULTS ANT) DISCUSSION
The considerations i n t h i s section are divided into two parts. I n t o
t h e f i r s t f a l l the considerations about mechanizing t h e p r e d i c t i o n and
command equations s o as t o i n s u r e s t a b i l i t y and smoothness of t h e i n t e r ceptoroperation.
The p r a c t i c a l system, u n l i k e t h e t h e o r e t i c a l
one studied
i n t h e Analysis sectioqhas certain transfer
f'unctions t h a t a r e more or
less fixed.
Furthermore, i n t a k i n g t h e d e r i v a t i v e n e c e s s a r y f o r t h e p r e dictions, new transfer functions must a r i s e . Thus, whatmust be done f o r
s t a b i l i t y , what can be done t o improve s t a b i l i t y , and what can be done t o
improve the response are questions considered
first.
Once a s t a b l e and reasonably fast system has been secured, the next
question i s t h a t of i t s adequacy as a predictor system. In t h e examinawill be
t i o n of this question, first- and second-order prediction systems
compared. The b a s i s of comparison will be the distance by
which t h e
rockets miss a maneuvering t a r g e t .
Stability
A s has been mentioned i n t h e s e c t i o n on Analysis, the primary factor
i n a c h i e v i n g s t a b i l i t y i n t h e system i s t h e removal of t h e ownship component of maneuvering acceleration (Vaf) from the prediction equations before
submitting them a s c,ommands t o t h e a i r p l a n e ' s a u t o p i l o t and time servo.
Figures 10 and ll i n d i c a t e t h e behavior of t h e system under changes i n
t h e amount of VaY i n t h e command. On t h e t e s t s from which thesetime
f l e w against a t a r g e t inih i s t o r i e s of Vaf were taken, the interceptor
t i a l l y 6,000 f e e t ahead. The interceptor speed was 1,000 f e e t p e r second;
t a r g e t speed was 800 f e e t p e r second. A t 20 seconds t o go (before pred i c t e d impact of rocket and
t a r g e t ) , t h e t a r g e t p i t c h e d up a t t h e rate
of 0.06 radian per second, which corresponds t o a maneuvering acceleration
12
The t i m e h i s t o r i e s of f i g u r e 10 i n d i c a t e t h a t t h e p r e s e n c e i n t h e
commands of a component of ownship acceleration normal t o t h e r a d a r l i n e of-sight (the prediction equations are written in radar coordinates) acts
i n t h e s e n s e of a negative feedback. The l e s s i t s removal, t h e g r e a t e r
t h e feedback. The sequenceoftime
histories in the figure
shows t h a t
progressive removal of amounts of t h e feedback increases the effective
forward gain of the airplane
system as far a s t h e f i r s t peak of t h e
response i s concerned. The period of t h e o s c i l l a t i o n s i n f i g u r e lO(a)
shows t h e e f f e c t of t h e i n t e r c e p t o r - t a r g e t geometry on the period of t h e
system i n t h i s case. On t h e b a s i s of t h e t e s t s from which these time
h i s t o r i e s were taken, it appeared that the best response occurs if'about
30 percent of the
ownship acceleration (corresponding t o K = 0.7 i n
f i g . 8 ( a ) ) i s l e f t i n t h e e l e v a t i o n channel. With t h i s amount offeedback,
the response i s as shown i n f i g u r e 1 O ( c ) .
During t h e runs from which t h e time h i s t o r i e s of f i g u r e 10 were taken,
as much omship acceleration as possible wasremoved from the time channel.
Leaving any ownship a c c e l e r a t i o n i n t h e t i m e channel has a deleterious
11 shows a sequenceoftime
e f f e c t on t h e s t a b i l i t y of t h e system.Figure
ownship accelerah i s t o r i e s of Vay during runs withprogressivelyless
t i o n remaining i n t h e time channel (corresponding t o changing the value
I n t h i s run, 30 percent of t h e ownship
of K from 0 t o 1 i n f i g . 9( a ) )
acceleration component was l e f t i n t h e e l e v a t i o n channel. But t h i s time,
t h e e f f e c t on s t a b i l i t y wasmore severe.
.
Adjustment of computer l a g s . - Once t h e b a s i c s t a b i l i t y of t h e system
has been secured by proper
removal of olamship accelerations, the choice
of the various lags in the time
and elevation computer loops can be invest i g a t e d . The transfer functions of the differentiations
govern the values
of o t h e r l a g s t o be inserted. The value of 1 second f o r t h e timeconstant
of derivative process yielding
E (shotm i n f i g s . 8 and 9 ) was chosen
because t h e l a r g e s t v a l u e of t h e e f f e c t i v e numerator time constant,
T/2,
i s 10 seconds. A lead-to-lag ratio of 1O:l i s usually considered a reasonable compromise between responsespeed andinduced noise. Attempts t o
v a r y t h i s timeconstantas
a function of T from 1 second t o a small
value not only led to considerable
complexity, but also provided l i t t l e
success. The otherterms i n t h e command equations were putthrough l a g s
t o match them t o t h e d i f f e r e n t i a t e d s i g n a l .
I n the time channel, t h e
matched s i g n a l s a r e R2, Vaf s i n E, and F cos E. The range, R, v a r i e s
slowly enough t h a t t h e l a g i s unnecessary. I n t h e e l e v a t i o n channel,
t h e matched s i g n a l s a r e Vay cos E and.(F/T)sin E. A s shown i n f i g u r e 8(a), a t i m e constant of 2 seconds proved t o be a b e t t e r choice f o r
the latter quantity.
Elevation dead zone.- Test runs w i t h t h e a t t a c k computer arranged as
described above showed adequate s t a b i l i t y a g a i n s t a s t e p t a r g e t a c c e l e r a t i o n . However,when
the target did not
maneuver, t h e i n t e r c e p t o r had a
tendency t o wander, with an amplitude of
normal acceleration which was
small at long and short interceptor ranges
and l a r g e r a t intermediate
ranges. Insertions of
a small dead zone 2 f t / s e c 2 wide i n t h e e l e v a t i o n
acceleration terms removed t h i s tendency. The e f f e c t of t h e dead zone i s
a small u n c e r t a i n t y i n t h e p r e d i c t e d normal r e l a t i v e v e l o c i t y , correspondi n g t o t h e n o i s e l e v e l of t h e e l e c t r o n i c computer elements as amplified
by the process of d i f f e r e n t i a t i o n . No such noise problem arose i n t h e
timechannel.
I n f a c t , it was found p o s s i b l e t o i n c r e a s e t h e g a i n
of t h e
a c c e l e r a t i o n t e r m i n t h i s channel from T2/2 t o T2.
i
Miss Evaluation
After the various parts of t h e a t t a c k computer had been adjusted i n
t h e manner just described, it was desired $0 compare t h e performanceof
t h e second-order prediction system
with t h a t of t h e f i r s t - o r d e r system
by means of the rocket miss. There a r e four s e r i e s of tests i n t h i s
evaluation program. I n t h e f i r s t t h r e e s e r i e s , t h e t a r g e t a i r p l a n e p e r formed a s t e p a c c e l e r a t i o n maneuver at some time during the
In t h e
last series, the target started pitching
upward a t some time during the
run then, a f t e r v a r i o u s f i x e d i n t e r v a l s of time, resumed s t e a d y s t r a i g h t
f l i g h t ( a t a constantangleofclimb).
Such a maneuver corresponds t o a
pulse target acceleration.
run.
In t h e first andsecond s e r i e s of tests, t h e t a r g e t ' s normal
acceleration change was s e t a t 1, 1.5, and 2 g ' s ( c o r r e s p o n ~ n g t o
heading-change r a t e s of 0.04, 0.06 and 0.08 radian per second).
In t h e
first s e r i e s , t h e limits t h a t e x i s t i n t h e u s u a l a c c e l e r a t i o n
command
system'selevationchannel
were removed. The r e s u l t s of t h i s s e r i e s t h e n
e s t a b l i s h e s t h e c a p a b i l i t y of t h e second-order system i n comparison with
t h e f i r s t - o r d e r system.
,
1
-T '
Figure 1 2 shows t h e r e s u l t s of t h i s s e r i e s of runs. The ordinate in
t h e f i g u r e i s t h e e l e v a t i o n miss per g of t a r g e t maneuvering acceleration.
The abscissa indicates the length
of t i m e t h e t a r g e t maneuver l a s t e d , from
t h e time it began u n t i l t h e end of t h e run. The run ended 1.5 seconds
after rocket firing time.
Rocket firing time occurs when t h e time-to-go,
output
ofhe
time servo, i s 1.5 seconds. The distance of
given by the
the rockets from t h e t a r g e t 1.5 seconds a f t e r f i r i n g t i m e i s t a k e n t o b e
the rocket miss, and i s resolved into an elevation component and a time
component of miss. Thus, an elevation component of miss p l o t t e d a t
t m = 8 seconds i s the rocket miss a f t e r a run i n which t h e t a r g e t
maneuvered during the l a s t 8 seconds.
!
14
i
The misses r e s u l t i n g from runs against a l l t h r e e magnitudes of t a r g e t
acceleration were s u f f i c i e n t l yp r o p o r t i o n a tl ot h e
magnitude
of
t h e normal
acceleration change t h a t t h e r e s u l t s ofeach command system defined the
was used. Figure 12(a)
s i n g l e curve shownwhen t h e o r d i n a t e i n t h e f i g u r e
shows t h e e r r o r t o beexpectedof
a f i r s t - o r d e r system. The improvement
t o be expected by using a second-order system i s shown by comparison of
t h e curves i n f i g u r e s 1 2 ( a ) and (b). The i n i t i a l r i s e i n t h e curves, up
can climb during
the
t o 1.5 seconds, i s due t o t h e d i s t a n c e t h e t a r g e t
time the rockets are flying. A maneuver begun during this period has
no
e f f e c t on t h e i n t e r c e p t o r , which has already fired i t s rockets. The miss
curves keep rising during the time the interceptor
becomes aware of t h e
maneuver and begins t o respond. The curves reach a maxim when t h e i n t e r ceptorbegins t o outclimb t h e t a r g e t . S i n c e i n t h e
second-ordersystem
t h e a c c e l e r a t i o n commanded of t h e i n t e r c e p t o r does not cease u n t i l t h e
i n t e r c e p t o r i s headed t o a rocket impact point predicted on t h e b a s i s t h a t
t h e t a r g e t will continue t o maneuver a t i t s present rate, the misscurve
drops t o a small value after about 5 secondsof maneuvering. I n t h e firstorder system ( f i g . 1 2 ( a ) ) , however, since the interceptor tends
to point
to t h e t a r g e t ' s f l i g h t p a t h , t h e
miss
t o an impact point along the tangent
remains p r o p o r t i o n a l t o t h e r a t e a t which t h i s impact point i s changing.
The e f f e c t of limiting.- Figure
13 i l l u s t r a t e s t h e e f f e c t of l i m i t i n g
limits used i n t h e s e t e s t s
theinterceptor'sacceleration
command.The
r e s t r i c t e d t h e t o t a l a c c e l e r a t i o n of t h e i n t e r c e p t o r t o s t a y between +3g
and -1g. F i g u r e l 3 ( a ) shows t h a t s i n c e t h e commanded acceleration of t h e
do n o t a f f e c t t h e
f i r s t - o r d e r system i s r e l a t i v e l y mild, these limits
i n t e r c e p t o r ' s performance u n t i l t h e t a r g e t ' s acceleration approaches t h e
l i m i t magnitude. Since the interceptor cannothead off a t a r g e t which
has a maneuvering acceleration equal to the incremental acceleration
allowed t h e i n t e r c e p t o r , t h e miss increases with maneuver duration.
-
U
This same e f f e c t i s n o t i c e a b l e i n f i g u r e l3(b) f o r t h e c a s e of t h e
2g s t e p t a r g e t a c c e l e r a t i o n .
For 1 and l . 5 . g ' ~of target acceleration,
t h e miss curves r e t u r n more slowly toward zero under conditions
of limited
accelerationcapability.Ifunlimited,inthe
1.5gcase,
theinterceptor
attempts t o p u l l a maximum of 7 g ' s when t h e maneuver begins a t long range.
This i s a peak, however, which remains above t h e allowed incremental value
of 2 f o r o n l y about 1 second.
Effect of rocket speed.- It was noted i n t h e d i s c u s s i o n i f f i g u r e 1 2
t h a t t h e miss curves rose during
t h e first 1.5 seconds because during t h i s
time of rocket
f l i g h t t h e i n t e r c e p t o r had no power t o c o r r e c t t h e r o c k e t ' s
flightpath.
Reducing t h i s f l i g h t time, Tfhich corresponds to increasing
the rocket average speed, reduces t h e t i m e a v a i l a b l e f o r t h e t a r g e t t o
evade the interceptor.
Consequently, it reduces t h e misses f o r both firstandsecond-ordersystems,
as i n d i c a t e d i n f i g u r e 14. Sincehere,as
in
a l l the other tests, the value
of F, the distance traveled by the rocket
r e l a t i vtteohien t e r c e p t o r ,
i s f i x e d a t 1500 f e e t , a time of f l i g h t
t, = 0.75 corresponds t o an average rocket speed
of 2000 f e e t p e r second
6.
with respect t o t h e i n t e r c e p t o r ; tf = 1.00 corresponds t o an average
rocket speed of 1500 f e e t p e r second. A l l t h e runs which established
t h e curves sholm were made against a l.5g step target acceleration with
no l i m i t on t h e a c c e l e r a t i o n command.
Pulse maneuvers.- I n t h e f i n a l s e r i e s of t e s t s , t h e i n t e r c e p t o r f l e w
During these tests, t h e a c c e l e r a t i o n
against a pulse target acceleration.
command was not limited, and the rocket flight time
was r e s t o r e d t o
1.5 seconds.Figure
15 compares results of t h e first- and second-order
At = 4-1/3, 6-1/2, and 8-2/3 seconds. The
systems forpulsewidths
curves of f i g u r e 1 2 a r e added t o r e p r e s e n t t h e l i m i t i n g c a s e
of wide
change
pulses (At + m ) . The curves i n the figure indicate the trend with
i n A t . The new curvesfollowthosefor
a s t e p a c c e l e r a t i o n until t h e
abscissa i s about 1.5 seconds longer than the pulse width. This time
The curves f o r t h e f i r s t - o r d e r
duration i s due t o r o c k e t f l i g h t t i m e .
system ( f i g . l 5 ( a ) ) drop t o aboutzero, as they should, since the target
i s not maneuvering f o r some time before the
end of t h e run. A f t e r a l l ,
i n t h i s system, when t h e t a r g e t s t o p s a c c e l e r a t i n g , t h e i n t e r c e p t o r h a s
only t o s t o p a c c e l e r a t i n g t o o , for under these conditions of no maneuver,
i t s predicted impact pointhasstopped
moving. In t h e second-ordercase,
on t h e o t h e r hand ( f i g . l 5 ( b ) ) , t h e i n t e r c e p t o r h a s
developed a l a r g e
Between t h e
lead angle on t h e t a r g e t t o b r i n g it t o t h e p r e d i c t e d p o i n t .
time the maneuver has stopped and the time the interceptor has corrected
i t s heading t o a new point, a s i z a b l e missoccurs.This
miss i s l a r g e s t
for the smallest pulse width because for this case the difference in
heading can become l a r g e s t f o r , a l t h o u g h t h e i n t e r c e p t o r p r e d i c t s t h e
same impact point as f o r maneuvers of longer duration, the target
changes
i t s heading l e a s t i n t h e s h o r t e s t maneuver. To a second-ordersystem, a
maneuver l a s t i n g i n t h e neighborhoodof 3 or &.seconds i s t h e most serious
because t h e d i f f e r e n c e i n headingcanbe
made g r e a t e s t . A s A t becomes
smalle