A spreader acceleration guidance scheme for command guided ... - Core

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


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

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