A SPREAD ACCELERATION GUIDANCE SCHEME FOR
COMMAND GUIDED SURFACE-TO-AIR MISSILES
D. Ghose
Department of Electrical Engineering
Indian Institute of Science, Bangalore-560 012, INDIA
B. Dam
Department of Electronics and Telecommunications Engineering
Jadavpur University, Calcutta, INDIA
U. R. Prasad
Department of Computer Science and Automation
Indian Institute of Science, Bangalore-560 012, INDIA
ABSTRACT
maneuvers to get reflected in the LOS
rate and then null this rate in a
feedback mode, as is the policy in PN.
A
new guidance law for command
guided
surface-to-air
missiles
is
presented.
It attempts to reduce the
integral control effort while at the
same
time takes into
account
the
maneuvers of the target. A Monte-Carlo
simulation
of
a
missile-target
engagement in a n inclined plane
is
carried out to check the performance of
the new guidance law in comparison with
existing laws i n terms of
integral
control effort, interception time and
miss distance.
Both APN and MGS are again derived
from linearized kinematics though the
missile-target encounter geometry
is
essentially nonlinear. A guidance law
which uses the nonlinear kinematics is
the Maximum Latax Scheme (MLS), known as
MADDOG in [ 3 ] and hard turn guidance in
[ 4 ] . This requires the missile to pull
the maximum latax whenever a change in
course
is
required
by
it.
The
information on the change in course may
be obtained either from the LOS rate [ 4 ]
or b y predicting the future position of
the target [ 3 ] . This guidance law is
time
optimal but suffers from
the
disadvantage that the control effort and
the corresponding maneuver induced drag
are very high. There could be wastage
of control effort due to chattering as
well.
In this paper, a new guidance
law
for surface-to-air missiles
is
proposed
which reduces the
control
effort and eliminates chattering at the
cost
of
a
slight
increase
in
interception time. The basic idea is to
use a predictive guidance
mechanism
which projects the position of
the
target into the future (as in [ 3 ] ) and
then spread the missile latax uniformly
over
the
entire
time-to-go.
Recomputations
are done at
regular
intervals using fresh information on the
target.
1. INTRODUCTION
Several guidance laws have been
proposed in the literature for tactical
missiles, employing empirical, optimal
control or differential game concepts.
Some are based on general nonlinear
kinematics
while
some
others
use
linearized
encounter geometry.
The
classical proportional navigation (PN)
guidance law, which has been applied
quite extensively in tactical missiles,
i s a n empirical guidance law
which
derives justification from linearized
kinematics.
But PN suffers from the
drawback
that large miss
distances
result when the missiles are deployed
against maneuvering targets [ l ] .
This
is due to the missiles incapability in
generating
the required high
latax
during the terminal phase of engagement.
To alleviate this problem, rather large
values of the navigation constant are
needed during the initial stages when
the target starts maneuvering in order
to keep the missile latax requirements
down during the terminal phase. This is
done by measuring and then incorporating
the
target
acceleration
into
the
guidance law in a feedforward mode, as
in Augmented Proportional
Navigation
(APN) and Modern Guidance Scheme (MGS)
[ 1 , 2 ] , rather than allow for the target
The paper is organized a s follows :
Section 2 describes the new guidance law
and its comparison with existing laws.
Section 3 presents the missile-target
encounter model. Section 4 describes
the modeling of the radar
tracking
errors and the design of the shaping
filters. Section 5 presents the results
of the Monte-Carlo
simulation
runs.
Section 6 concludes the paper.
202
CH2759-9/89/0000-0202 $1 .OO
'1989 IEEE
2.
THE SPREAD
(SAG) LAW
ACCELERATION
GUIDANCE
fi
t
go’
which
The Spread Acceleration Guidance
(SAG) law uses a predictive guidance
concept which projects the position of
the target into the future and then
computes the minimum latax that would
enable the missile to reach the target.
Fig.1 shows a simplified model of a
missile-target encounter geometry in 2dimensions. The projection o f
target’s
position into the future needs, at the
current time tC, the estimation of the
A
t
go
missil:
time t
the m nimum latax a
which
used by the missile will enable
reach (xf, yf) is computed as
The
2(
when
it to
under.
p - Q.
radius of curvature of the
missile
f
-y ) 2 +
m
(xf-xm)2~1/2/2 Sin( dm/2).
(3)
But since,
R~
= Vi/am,
P
Similarly, if t
<
%o -
-
then the
Though the SAG law a s presented
above is for a 2-dimensional case, it
can be extended to the 3-dimensional
case by using the concept of maneuver
plane. The maneuver plane, within each
iteration cycle, i s defined as the plane
passing through the current position of
the missile and the future predicted
position of the target, and containing
the velocity vector of the missile.
Thus, the missile must maneuver in this
plane in order to reach the target and
hence the equations given above for a 2dimensional case will be sufficient to
compute
the
latax
required.
The
direction in which the latax has to be
applied
will be determined by
the
orientation
of the maneuver
plane.
Obviously, in each iteration cycle, the
maneuver plane changes its orientation
a s the predicted position of the target
changes.
(2)
trajectory is given by, Rm = [(y
&
is taken proportional to
The above procedure is repeated at
regular intervals to account for any
changes in target behavior.
But the
missile latax i s kept constant within
this time interval.
tan-’
=
quantity
i s used for the missile latax command.
Then the total angle of turn the missile
has to execute is
Hm
is increased b y a
-
Let,
g=
*t g o
go
go’
would overshoot the target at
So, %
is decreased b y a
go’
go
A
Tv
- t ). A
quantity proportional to (t
go
go
fresh computation of the future position
of the target is made and Eqns.(l)
(6)
are usad. This Ateration is continued
till
t
and t
come
close.
A
go
go
convergence limit is set to serve a s a
termination condition for the iteration,
The value of a m obtained in this manner
position,
velocity and
acceleration
(both longitudinal and lateral) of the
target. These quantities are assumed to
be
constantA over
the
time-to-go
estimated as t
Now, the position of
go’
the target at time (t + 2 ) is found.
c
go
This is shown as (xf , yf) in Fig.1.
Then,
).
so
(4)
substituting (2) and (3) we obtain,
2
am = 2Vm Sin( j3
[(Yf-Ym)
2
- dm )
+ (Xf-xm) 1 ll2
which gives the latax to be used by
missile in the SAG law.
The time-to-go for the missile
now calculated as,
yv
t
= 2
Rm
(
g -
It would be interesting to compare
analytically the SAG law with existing
guidance laws. It is found that when
the kinematics are linearized the SAG
law
is similar to APN with N ’ = 2.
This i s shown as follows. The predicted
position of the target in a linearized
kinematic model is given b y
/
Jm)/Vm.
(5)
the
is
(6)
Similarly, the future position of the
missile
with current
velocity
and
is,
acceleration a applied over t
go
f.
> t
then the missile will not
go’
go
reach the target in the predicted time
If
203
=
Average
number
crossings per second.
For interception to occur (7) and
must be the same and this yields
(8)
of
zero
The missile target engagement model
i s governed b y a number of
system
differential equations which are written
with reference to Fig.1.
The target
equations are,
Xt =
The expression within brackets is the
zero
effort miss [ l ] .
Thus,
the
=
2 is
equivalence with APN with N'
clear.
But the important thing to be
noted here is that the SAG Law (unlike
APN) is actually designed taking into
the
account the nonlinear nature of
encounter geometry.
Vt c o s
yt = Vt Sin
it =
at/Vt
-
Jt *
3t
The missile equations are,
3 . THE MISSILE-TARGET ENCOUNTER MODEI
The engagement plane i s assumed to
be inclined at a certain angle to the
horizontal as shown in Fig.2.
Thus,
though the actual simulation is done for
a two-dimensional case the effect of
altitude
variations
can
also
be
incorporated. The advantage it has over
a vertical engagement plane is that one
can have more realistic target maneuvers
as
a target aircraft
very
seldom
maneuvers for a long time in a vertical
plane.
= [ ( pt)k
a
=
achieved latax of the missile,
=
commanded latax of the missile.
SAG scheme described in the previous
section. Its computation is based upon
measurement o f certain system variables.
These measurements are corrupted
by
noise and thus the input to the system
i s also noisy. The effect of these
noises
will be taken into
account
through shaping filters.
4
RADAR TRACKING
FILTERS
ERRORS
AND
SHAPING
The radar measures the position,
velocity, latax, etc of the two vehicles
and uses them to generate commands to
the autopilot. These commands are not
sent continuously, a s this would place a
heavy burden on the radar system, but at
discrete instants of time.
The time
period between any two such commands is
called the guidance cycle time.
The
guidance command is held constant during
this time period.
where,
k
autopilot time constant,
The other variables are explained
in Fig.1. The input to the system is
through am which is obtained from the
e- pt]/k!,
P(k)
=
Probability of
occurring in t seconds,
=
a
The target is initially assumed to
approach the missile launch point along
the OX' axis (Fig.2), and then turn away
from it at the moment o f missile launch.
The time optimal evasive maneuver of the
target consists of a circular maneuver
and then a straight line flight when the
LOS becomes tangential to the
turn
circle of the target. But in order to
increase miss distance the target should
employ random telegraph maneuver
at
short ranges [5]. Both these features
have been incorporated in this study to
model target behavior. The target is
assumed to maneuver in the time optimal
mode until the time-to-go is below a
certain pre-assigned
value.
Then it
starts a random telegraph maneuver (i.e
random sign switching between two limit
values),
which is characterized by the
Poisson distribution,
P(k)
5
changes
204
Again evaluating (19) and assuming unity
power spectral density for the input
white noise we have,
The various measurements,
based
upon which the calculations for latax
are performed, are corrupted by noise.
In order to simulate these we assume
certain characteristics for these noises
and then design appropriate
shaping
filters. the measurements of position,
speed and acceleration are assumed to be
affected by noise processes exhibiting
exponential
cosine
autocorrelation
functions.
Measurement of angles, on
the other hand, are assumed to
be
affected
by
noise
processes
with
exponential autocorrelation functions.
Now, these filters are used to generate
the noise processes for position, speed,
acceleration
and flight path
angle
estimates.
5 . SIMULATION RESULTS
To obtain these noise processes for
a Monte Carlo simulation of a missile
target engagement, white noise is passed
through
an
appropriately
designed
shaping filter. Let Gii(w) be the power
The simulation was
using the following data,
7,
spectral density of the input white
noise and G (w) be that of the output.
out
= 0 . 1 sec,
Guidance cycle time = 1 . 0 sec,
Vm = 800 m/sec (average),
00
If the transfer function of the
is HCsIthen,
carried
filter
Missile latax = 0-15 g,
Vt = 550 m/sec (average),
Target latax = 0 - 6 g ,
Initial slant range of target = 20 k m ,
Engagement
plane
angle
with
the
But the power spectral density of the
output can also be obtained b y taking
the
Fourier
transform
of
its
autocorrelation function R nn( 7 ) as,
horizontal = 3 0 ° ,
Initial altitude of target = 10 km,
Navigation constant N ’ = 3 (PN and APN),
Error in position estimates = 100 m,
Error in elevation estimates = 3 mrad,
Error in velocity estimates = 40 m/sec,
Error in acceleration estimates = l g ,
Error in flight path angle estimates =
Now, if the power spectral density
of the input white noise is unity and
the
output exhibits an
exponential
cosine
autocorrelation
function
represented by,
0.5’
To design the shaping filters we
assume
that
the
value
of
the
7 = 1.0 sec
autocorrelation function at
drops to 1 0 % of its value at
7 = 0.
Assuming, c = 1 we obtain,
(20)
A>O, k>O,
k = 1.686.
then, evaluating ( 1 9 ) and comparing with
( 1 8 ) we obtain,
12 A R T
I
s2
The error for position and velocity
estimates
are
given
in
spherical
coordinates.
They are first converted
to Cartesian co-ordinates b y assuming a
1/2
+
2ks
+
(k2
+
c2)
of
30’.
Thus
nominal
angle
parameters of the shaping filters
be,
2
A = 7725 m
(x - coordinate),
]
Similarly,
if
the
autocorrelation
function of the output is exponential in
nature then we have,
A = 3175 m
A
=
(22)
A
205
=
2
2
1 6 0 0 m /sec2
2
96.2 m /sec4
(y
-
the
will
coordinate),
(velocity),
(acceleration).
For t h e f l i g h t path estimates w e
that
assume
estimation.
B u t t h i s was f o u n d
minimal d u r i n g t h e simulations.
w1 = 2 B 2 / n
9
to
be
T h i s study has been c a r r i e d o u t f o r
a p a r t i c u l a r kind
of
evasive target
maneuver.
It would be i n s t r u c t i v e t o
o b t a i n similar r e s u l t s f o r other kinds
of
t a r g e t maneuvers and s t u d y
the
p e r f o r m a n c e o f t h e new g u i d a n c e l a w .
and so,
w1 = 4 . 8 4
T h e a b o v e d a t a was u s e d t o c a r r y
out
Monte C a r l o s i m u l a t i o n s
of
a
missile-target
engagements with
the
m i s s i l e u s i n g PN,
APN, MGS a n d SAG
guidance
laws.
T h e c o m p a r i s o n i s made
with
respect
t o the total
control
e f f o r t , miss d i s t a n c e a n d i n t e r c e p t i o n
time.
T h e s e a r e shown i n F i g s .
3-5.
The
r e s u l t s are obtained
from
50
s i m u l a t i o n r u n s and p l o t t e d w i t h 95%
confidence interval.
ACKNOWLEDGEMENTS
The a u t h o r s
wish t o acknowledge
h e l p r e c e i v e d f r o m Mr. P r a h l a d a a n d
K.N.Swamy.
the
Dr.
REFERENCES
1. F . W . N e s l i n e
and P.Zarchan,
A New
Look a t C l a s s i c a l V s Modern Homing
Missile G u i d a n c e , J o u r n a l o f G u i d a n c e
and C o n t r o l , Vo1.4, No.1,
pp.78-85,
1981.
T h e SAG l a w s h o w s s l i g h t l y b e t t e r
control
effort
than
PN
and
a
c o n s i d e r a b l e i m p r o v e m e n t o v e r MGS a n d
APN ( F i g . 3 ) .
I n terms o f miss d i s t a n c e
SAG s h o w s c o n s i d e r a b l e i m p r o v e m e n t o v e r
PN a n d c o m p a r a b l e r e s u l t s t o APN a n d MGS
(Fig.4).
SAG s h o w s c o m p a r a b l e v a l u e s t o
MGS a n d APN, a n d s l i g h t i m p r o v e m e n t o v e r
PN
i n terms o f
interception
times
(Fig. 5).
2. R.G.Cottre1,
Optimal
Intercept
Guidance f o r Short-Range
Tactical
M i s s i l e s , A I A A J o u r n a l , Vo1.9,
No.7,
pp.1414-1415,
1971.
3. J . G . R e i d , T . L . F u r l o u g h a n d J.T.Young,
Maximum A c c e l e r a t i o n D e s i g n ,
Digital
Optimal Guidance, A I A A Guidance and
C o n t r o l Conference, 1981.
6 . CONCLUSIONS
T h i s p a p e r p r o p o s e s a new g u i d a n c e
the
scheme which t r i e s t o minimize
c o n t r o l e f f o r t by s p r e a d i n g t h e l a t a x
p u l l e d by t h e m i s s i l e o v e r t h e e n t i r e
time-to-go.
In the process, it achieves
some i m p r o v e m e n t s i n i n t e r c e p t i o n times
and m i s s d i s t a n c e s . T h e method s u f f e r s
from t h e obvious drawback of h a v i n g t o
estimate t h e t a r g e t a c c e l e r a t i o n
(which
i s a l s o common t o APN a n d MGS),
unlike
It a l s o needs
t h e PN g u i d a n c e l a w .
somewhat h i g h e r time f o r c o m p u t a t i o n s
( u n l i k e PN).
This is the reason
for
p r o p o s i n g t h i s g u i d a n c e l a w i n a command
g u i d a n c e mode s i n c e p o w e r f u l
ground
based
c o m p u t e r s w i l l be a v a i l a b l e t o
carry out t h e i t e r a t i o n s .
During t h e
i t e r a t i o n s t o determine time-to-go,
it
may so happen t h a t t h e i t e r a t i o n s do n o t
converge.
T h i s c a n be
taken
into
a c c o u n t by k e e p i n g a l i m i t o n
the
maximum n u m b e r o f
iterations to
be
executed per guidance cycle.
T h i s would
i n j e c t some e r r o r
i n t h e time-to-go
4 . M.Guelman
and
J.Shinar,
Optimal
Guidance law i n a P l a n e , J o u r n a l of
Guidance and C o n t r o l ,
Vo1.7,
No.4,
pp.471-476,
1984.
5. F.Imado
a n d S.Miwa, F i g h t e r E v a s i v e
Proportional
Maneuvers
Against
Navigation
Missile,
Journal
of
A i r c r a f t , Vo1.23, No.11,
pp.825-830,
1986. .pa
206