when bad things happen to good missiles

8/13/2019 When Bad Things Happen to Good Missiles

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When Bad Things Happen To Good MissilesPaul Zarchan*

Charles Stark raper Laboratory, Inc.Cambridge, Massachusetts 02139

AbstractNormalized miss distance curves for a generic proportional navigation guidance system are developed to showhow a missile is influenced by late target resolution Rules of thumb are developed relating the necessary ratio ofthe time left for homing after resolu tion has taken place to the miss ile guidance system time consta nt Theinfluence of the finite acceleration capability of the interceptor on the resultant miss is also quantified Suggestionsfor alleviating the guidance system problem are presented

IntroductionIn endoatmospheric and exoatmospheric

engagements there may be an instantaneousapparent shift in target position which can causeunacceptable degradation in homing missileperformance. For example, in endoatmospherictactical radar homing missiles, the interceptor maybe guiding on the power centroid of two aircraftflying in close formation. When one of the aircraftfalls outside the missile seeker beam, the otheraircraft will be resolved. In this case it appears tothe pursuing interceptor that the target hasinstantaneously shifted from the location of thepower centroid to the location of the resolvedaircraft. In other words, there has been an apparentstep change in target position. Similarly. astrategic exoatmospheric missile may be homingon one of two closely spaced objects. After a whilediscrimination takes place and the interceptorsoftware may conclude that one object is a decoywhile the second object is the real target. In thiscase too, as far as the missile is concerned, itappears as if the target has instantaneously changedposition from first object or decoy to secondobject).

In both preceding examples the targetdisplacement disturbance occurs late in the flightwhich is the worst possible time from a missileguidance system point of view. Large missdistances may result because of insufficientremaining homing time and the finite accelerationcapability of the interceptor. This paper will*Principal Member of Technical StaffAssociate Fellow I MCopyright 1993 by the American Institute ofAeronautics and Astronautics, Inc.

develop normalized design curves for a genericproportional navigation guidance system to bothillustrate and quantify the magnitude of thepotential guidance system problem. Rules ofthumb will be developed relating the necessary ratioof the time left for homing after resolution hastaken place to the guidance system time constantand the miss due to the apparent shift in targetlocation. The influence of the finite accelerationcapability of the interceptor on the resultant misswill be quantified. Suggestions for alleviating theguidance system problem will also e presented.Important Closed-Form Solution

Figure 1 presents the simplest possibleproportional navigation homing loop. In thisperfect guidance system, models of the seeker.noise filter, guidance, and flight control systemshave been considered to be perfect and withoutdynamics. Such a block diagram is known as azero-lag guidance system or homing loop. Themiss distance will always be zero in a zero-lagproportional navigation homing loop.

Figure 1 Simplest Possible ProportionalNavigation Guidance Homing Loop


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In Fig. 1 missile acceleration nL is subtractedfrom target acceleration nT to form a relativeacceleration. After two integrations we haverelative position y, which at the end of the flight

t = t ~ )s the miss distance [MisS=y t~)]. Adivision by range or the closing velocity Vcmultiplied by the time to go until intercept t o)yields the geometric line-of-sight angle A h etime to go is defined as

where t is the current time and t~ is the total flighttime.The missile seeker, which is represented in Fig.1 as a perfect differentiator, attempts to track thetarget. Effectively the seeker takes the derivative ofthe geometric line-of-sight angle, thus providing ameasurement of the line-of-sight rate. Theguidance system noise filter must process the noisyline-of-sight rate output of the seeker in order toprovide a more accurate estimate of the linesf-sightrate. A guidance commandn is generated from thenoise filter output. From Fig. 1 we can see that ithas been assumed that the acceleration commandobeys the proportional navigation guidance law or

where N is a designer chosen constant known asthe effective navigation ratio. The flight controlsystem which is represented by unity gain in Fig.1) must cause the missile to maneuver in such away that the achieved acceleration n~ matches thedesired acceleration commandncLet us consider obtaining the closed-formso~utionl*~or the required missile trajectory andacceleration for the case in which there are noguidance system dynamics and there is an apparentstep in target displacement YTIC. n the absence ofother error sources such as target maneuver andheading error, the relative acceleration targetacceleration minus missile acceleration) can bepressed

Integrating the preceding differential equation onceyields

where C1 is the constant of integration.Substitution of the line-of-sight angle definitionfrom Fig. 1 nto the preceding expression yields thefollowing linear time-varying fmt-order differentialequ tion

Since a linear fust-order differential equation ofthe form

has the solution

we can solve the relative trajectory differentialequation exactly. An instantaneous step in targetdisplacement means that the initial condition on thefirst state y is the value of the displacement or

Under these circumstances, after much algebra, wefind that the closed-form solution for the relativeseparation between missile and target y and themissile acceleration nc due to a step in targetdisplacement are given by

t N -2N ( 1 - -t~ tic

From the relative separation expressionwe can see that the miss distance y(tF) ina zero-lag guidance system is alwayszero In addition, from the acceleration formulawe observe that the magnitude of the initial missile


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acceleration is proportional to the size of the targetdisplacement and inversely proportional to thesquare of the flight time. Doubling the apparenttarget displacement will double the initial missileacceleration, whereas doubling the flight time ortime remaining after the apparent targetdisplacement occurred will quarter the initialmissile acceleration. Of course, if the accelerationrequired by the preceding formula is not availablethere may be a significant miss distance.

The closed-form solution for the missileacceleration response due to a step in targetdisplacement is displayed in normalized form inFig. 2. We can see that higher effective navigationratios increase the acceleration requirements at thebeginning of flight (when resolution occurs) andreduce the requirements towards the end of theflight. Figure indicates that the missileacceleration is always monotonically decreasing asthe flight progresses (i.e., more acceleration isneeded at the beginning of the flight than at the endof the flight). From a system sizing point of view,the designer usually wants to ensure that theacceleration capability of the missile is adequate atthe beginning of flight so that saturation can beavoided. For a fixed missile acceleration capability,Fig. shows how requirements are placed on theminimum guidance or flight time required afterfinal resolution (or the time remaining after theapparent step in target displacement occurs) andmaximum allowable target displacement.

Figure Normalized Missile Acceleration DueToStep In Target Displacement

In order to illustrate the use of the normalizedacceleration curves of Fig. 2 let us consider anumerical example involving the classical multipletarget problem.2 Figure 3 shows two aircraftflying in formation being pursued by a missile.Initially both aircraft are close enough so that themissile with seeker beamwidth BW homes on the

power centroid of the two aircraft. At the pointwhere one of the aircraft falls outside the seekerbeam resolution takes place and it appears to themissile that the aircraft has been instantaneouslydisplaced a distance YTIC If the missile andaircraft are traveling at constant speed with closingvelocity Vc the missile will be a distance of V c t ~from the power centroid at the point of resolution.As before, t~ is the time remaining for guidanceafter ker resolution.

< 7\ Power Centroid

Figure 3 Multiple Target GeometryFrom trigonometry we can see that the seekerbeamwidth is related to the aircraft displacementaccording to

Using the small angle approximation and solvingfor the effective time remaining for guidance afterseeker resolution we get

If a seeker has a beamwidth of 1 rad (nearly 6 deg),the two targets are separated by 400 ft and theclosing velocity is 4000 ft/sec, the time remainingfor guidance after resolution will be 1 sex or

Assuming that the missile effective navigationratio is 3. we can see from either the formula formissile acceleration or the normalized accelerationcurves of Fig. 2 that the maximum accelerationoccurs at the time of resolution and is given by


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n 3 2c = - = - = 6 tlsec2MAX 2 21This means that the missile will require nearly 2 gof acceleration to hit the resolved target when theaircraft formation spacing is 4 ft, the missileseeker beamwidth is approximately 6 deg and theclosing velocity is 4000 ft/sec.Single Time onstant Guidance System

We have observed from the closed-formsolutions that if the missile has sufficieniacceleration capability and there are no guidancesystem dynamics, the apparent shift in targetlocation will not cause any miss distance.Guidance system dynamics will of coursecontribute to the miss distance. If all of theguidance system dynamics can be lumped togetheras a single time constant T then the originalguidance system block diagram of Fig. 1 ismodified and is redrawn in Fig. 4.

Guidanca SynemDynamicsFigure 4 Single-Lag Guidance System

In this interceptor guidance system model, thesingle time constant represents the combineddynamics of the seeker, noise filter, and flightcontrol system. Closed-form solutions for themiss distance due to a step in target displacementcan be derived for a single time constant guidancesystem using the adjoint technique.1*2*3The missdistances for different effective navigation ratios fora single time constant guidance system can beshown to be

Miss 3 4-X 2 x= e 1 - 4 x + 3 x -TIC N -5 3 24

where

The ratio of the flight time to the guidance systemtime constantx is often refmed to as the number ofguidance time constants. The various closed-formsolutions for the miss distance are displayed innormalized form in Fig. 5 We can see that if theratio of the flight time to the guidance system timeconstant is greater than 5 then the miss distance isvirtually zero. Therefore we can say that for asingle time constant guidance system the miss iseffectively zero after 5 guidance time constants.

Figure 5 First-Order Normalized MissFor A StepIn Target Displacement

If we consider the same example of the previoussection where there was no miss distance, theimportance of guidance system dynamics and thenormalized curves of Fig. 5 can be demonstrated.For a 200 f t target displacement equivalent to 400f t aircraft separation), 1sec of effective flight timeand an overall guidance system time constant of 5


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see (T=.5), the number of guidance time constantsis 2 (t~/ T=1/.5=2)and the corresponding missdistance for different effective navigation ratios cane computed from Fig. 5 as

Miss = 14 200 = 28ftN =3Miss =- .05 200=- l o f tN =4Miss = .05 200= lof tN =5

Thus we can see that a non-zero guidance systemtime constant causes miss distance positivemiss distance means that the missile is between thetarget and power centroid while a negative missdistance means the missile is above the target.Therefore, if the effective navigation ratio is 3, -28ft of miss means that the missile missed the targetit was homing on by 28 ft and the second target by428 ft (400 28 = 428). We can see from Fig. 5that if the missile time constant can be halved or ifthe seeker beamwidth can be halved then thenumber of guidance time constants is doubled to 4(tFD=1/.25=2/.5=4) and the miss will be reduced.The formulas and curves indicate that large ratios offlight time to guidance system time constant yieldsmall or near zero miss distances whereas smallratios of flight time to guidance system timeconstantcan yield large miss distances.Higher Order Guidance System Dynamics

We have just seen that guidance systemdynamics will contribute to the miss distance. Wechose a single time constant representation of theguidance system because closed-form solutions forthe miss distance could be derived. The single timeconstant approximation to a missile guidancesystem is useful because the resultant closed-formsolutions suggest normalization factors for themiss distance. However the single time constantrepresentation of the guidance system also seriouslyunderestimates the miss distance. A much betterand convenient representation of a missile guidancesystem transfer function is a canonic fifth-orderbinomial given by3*4

where T is the total guidance system time constant,n~ is the achieved missile acceleration and is theline-of-sight angle. In this generic interceptorguidance system model, one time constantrepresents Ihe seeker, another represents the noisefilter, and the other three time constants representthe flight control system dynamics (aerodynamicsplus autopilot). It is easy to show that with thiscanonic guidance system model the overall guidancesystem time constant is simply the sum of the fiveindividual time constants or T. The fifth-orderbinomial missile homing loop is shown in blockdiagram form in Fig. 6.

Gvldance Systembynamla

Figure 6 Fifth-Order Binomial Guidance SystemNormalized miss distance curves for different

effective navigation ratios are presented in Fig.for the fifth-order binomial guidance system in thecase where the missile has infinite accelerationcapability. By comparing Figs. 5 and 7 we canconclude that in general the miss distances for thehigher order system are much larger. In addition,Fig. also shows that the ratio of the flight timet~ (or time remaining after the apparent step intarget displacement has occurred) to the guidancesystem time constant T must now be greater than10 (for the single time constant system the numberof guidance time constants had to be greater than 5)for there to be negligible miss distance. If thenumber of guidance time constants is less than 10it is really a matter of luck as to how large themiss distance will be. Luck is involved becausethe point at which resolution occurs for a specificengagement is random. In addition, we can seefrom Fig. 7 that the final miss may even be larger


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than the original displacement (i.e., cases wherenormalized miss is less than -1).

Figure 7 Fifth-Order Normalized Miss For A StepIn Target DisplacementIf we consider the same example of the previoussection, where the m iss distances were smaller, the

importance of higher order guidance systemdynamics c n be seen to be even more important.For a 200 ft targe t displacem ent (equivalent to 400ft aircraft separation), 1 sec of effective flight time(tF=l) and an overall guidance system timeconstant of .5 sec (T=.5), the number of guidancetime constants is 2 (tF/T=1/.5= 2) and thecorresponding miss distance for different effectivenavigation ratios can be computed from Fig. 7 as

iss = 1.44 00 288 tN =5

Note tha t t hese miss d i s t ances a re anorder of magnitude greater than those oft h e s i n g l e t i m e c o n s t a n t g u i d a n c esystem Thus we can see that the miss distancesc n be enorm ous and in fac t (for cases in which theeffective navigation ratio was 4 or 5) even greaterthan the original apparen t target displacement. Forexam ple, if the effec tive navigation ratio is 4, -238ft of miss means that the missile missed theresolved target (target it was homing on) by 238 ftand the unresolved target (second target) by 638 ft400 238 = 638). If the missile time constantc n be halved or if the seeker beamwidth can behalved then the number of guidance time constants

is doubled to 4 (tF/T= 1/ .25=2/.5=4) thusconsiderably reducing th miss or

Miss = 6 00= 3 ftN =4

Therefore we can see that the ratio of the flighttime remaining after resolution has occurred to theguidance system time constant is critical indetermining the expected miss distanceAccelerat ion Saturat ion

We have observed in the previous two sectionsthat both the guidance system dynamics andeffective navigation ra tio play an important role indetermining the miss distance due to a step in targetdisplacement. The finite acceleration capability ofthe interceptor is also important in determining themiss distance. Normalized miss distance curves canalso be developed when missile accelerationsaturation effects are considered. In this case it ishypothesized that miss distance normalizationfactors remain unchanged but new curves have to bedeveloped for the nondimensional ratio

where ~ L I Ms the value of the acceleration limit.For a fixed level of target displacement andguidance system time constant, a missile withmore limited acceleration capability (smalleracceleration limit ~ L I M )as a sm aller ratio. Usingthe preceding ratio and the normalization factors formiss due to a step in target displacement, we c nderive normalized miss distance curves by themethod of brute force. In other words, we cangenerate normalized miss distance curves bysimulating all of the possibilities. We can theninfer performance by making extrapolations fromthe normalized miss distance ~ u r v e s . ~ , ~f coursedetailed checks have been made to ensure that thenormalization factors are correct. Figure 8 through10 presents the normalized miss distances due to a


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step target displacement for effective navigationratios ranging from three to five respectively. Asexpected, we can see that less missile accelerationcapability (ratio smaller) requires more guidancesystem time constants to move from the powercentroid (normalized miss is one) to the resolvedtarget (normalized miss is zero). On the otherhand, we can also see that missiles with lessacceleration capability have less of a tendency toovershoot the target (less negative normalized miss)since there is less energy available. In otherwords, if the number of guidance timeconstants is less than ten, it is possiblefor a missile with less accelerationcapability to have a smaller missdistance than a missile with a largeracceleration capability We can also see bycomparing the normalized curves of Figs. 8-10 thatlarger effective navigation ratios tend to increase thenumber of guidance time constants required for themiss to approach zero (i.e., approximately sixguidance time constants are required for N =3 andten guidance time constants are required for N=5 .

Figure 8Normalized Miss Due To Saturation andTarget Displacement For n Effective Navigation

Ratio of 3

0 2 4 6 8 1 0T

Figure 9 Normalized Miss Due To Saturation andTarget Displacement For An Effective NavigationRatio of 4

5th Order Syslern, H 3

Figure 10Normalized Miss Due To Saturation andTarget Displacement For An Effective Navigation

Ratio of 5In order to demonstrate the use of the normalizedcurves of Figs 8-10 let us again consider the same

example of the previous section where there wasan apparent 200 ft step target displacement(equivalent to 400 ft aircraft separation), 1 sec ofeffective flight time and an overall guidance systemtime constant of .5 sec (T=.5). In this case thenumber of guidance time constants is still(tF/T=1/.5=2)and the corresponding miss distancefor an effective navigation ratio of 3 and differentvalues of missile acceleration limits can be readfrom the normalized curves of Fig. 8. Forexam le, if the missile acceleration limit is 320i/sec 9.9 g then the ratio can be computed as

5n T 2atio = LIM - 5 32 5 = 2

TIC 2


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Therefore for 2 guidance time constants thenormalized miss corresponding to a ratio of .2 canbe read from Fig. 8 s .6. We can calculate theactual miss distance for this example to be

Increasing the missile acceleration limit to 800ft/sec2 (24.8 g) inc reases the ratio to 5 or5n IM 5 5 2Ratio = = 5

Therefore for 2 guidance time constants thenormalized miss corresponding to a ratio of 5 canbe read from Fig. 8 as 0 In this case the actualmiss distance is zero or

Increasing the missile acceleration limit again to1600 ft/sec2 (49.7 g) increases the ratio to 1or5n TLIM 5 16 5Ratio = = 1

Therefore for 2 guidance time constants thenormalized miss corresponding to a ratio of 1 canbe read from Fig. 8 as .6. The actual missdistance for this exam ple is simply

Finally, if the missile had infinite accelerationcapability the ratio would be infinity. For 2guidance time constants the normalized misscorrespondingto infinite ratio can be read from Fig.8 as 3. Therefore the actual miss distance for amissile with infinite acceleration capabilityincreases to

Miss , = .8 200 = 160FtThus we can see that a missile with moreacceleration capability does not necessarily have

bctter miss distance performance against a step intarget displacementAlleviating The Problem W ith Time oGo Information

So far we have seen that the apparent step intarget displacement causes a transient problem inthe missile guidance system such that the finalmiss distance may be larger than the initialapparent target displacement. In other words, itmight have been better if the missile continued tohome on the power centroidIf both time to go information and intelligenton-board missile software are available, it ispossible to soften the effects of the apparen t step intarget displacement. For example, intelligentmissile software may conclud e there has been anapparent shift in target displacement but thesoftware does not have to permit the interceptor torespond to the apparent step in target displacementimmediately. Instead, a time-varying bias can beadded to the line-of-sight angle in order for themissile to think that the target displacement iseither a ramp or a parabola. In this case the rampor parabola reaches the full displacement value atthe end of the flight (not the start of resolution). Inother words we can lie to the guidance system sothat the interceptor thinks the displacement isactually of the form

tYRAMP TIC t F

where t=0 denotes the beginning of resolution andt~ is the end of the flight.The effectiveness of softening the apparen t stepchange in target displacement can be observed if weexamine again the case in which there is a 200 f tapparent target displacement for a 5 sec guidancesystem time constant and an effective navigationratio of 3 in a fifth-order binomial proportionalnavigation guidance system. Figure 11shows howthe miss distance varies for the strategies in whichwe let the guidance system s either a step (thiswas the only case considered until now), ram p orparabola in apparent target displacement. Tosimplify matters we have assumed that the


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interceptor has infinite acceleraticn capability. Wecan see from Fig. 1 1 that when th apparent targetdisplacement is a step the missile overshoots thetarget (negative miss) for flight times longer than.6 sec and less than 1.8 sec. Approximately 4 secof flight time 8 guidance time constants) arerequired for the miss to stay near zero. When thedisplacement disturbance takes the form of a ramp,it takes more flight time for the missile toovershoot the target (approximately 1.2 sec butnow only 3.5 sec of flight time 7 guidance timeconstants) are required for the miss to stay nearzero. A parabolic target displacement neverovershoots the target and only 3 sec of flight time(6 guidance time constants) are required for themissto stay near zero.

1 2 3 5flight Time Sec)

Figure 1 1 There Is Less Overshoot For HigherOrder Target Displacements

If small miss distances rather than near zeromiss distances are required, then the minimumrequired flight time for the parabolic displacementcan also be reduced by increasing the effectivenavigation ratio. We can see from Fig. 12 that alltarget displacement miss distance responses arefaster when the effective navigation ratio isincreased but the parabolic displacement has theleast overshoot.

1 2 4 6ight Time Sec)

Figure 12 Increasing Navigation Ratio ReducesFlight Time Requirements For Parabolic Target

DisplacementSummary

This paper presents normalized miss distancecurves showing the designer how to calculate themiss distance due to an apparent step in targetdisplacement. The importance of guidance systemdynamics and missile acceleration saturation effectsare illustrated with additional design curves andexamples. A method for softening the targetdisplacement effects on the guidance system isintroduced.

eferences1 Bennett, R. R., and Mathews, W. E.,Analytical Determination of Miss Distance ForLinear in Navigation Systems, HughesAircraft Co., Culver City, CA, TN-260,March1952.2 Travers, P., Interceptor Dynamics.unpublished lecture notes, Raytheon Co., circa197 1.3 Zarchan, P., Tactical and Strategic MissileGuidance, Volume 124,Progress In Aeronauticsand Astronautics, published by AIM, Washington,DC. 19904 Nesline., F. W., and Zarchan, P., MissDistance Dynamics in Homing Missiles,Proceedings of AIAA Guidance and ControlConference, AIM, New York, Aug. 1984.5 Shinar, J. and Steinberg. D., Analysis ofOptimal Evasive Maneuvers Based On ALinearizedTwo-Dimensional Model, Journal ofAircrafi, Vol14,Aug. 1977,pp. 795-802.

when bad things happen to good missiles

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