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Report No. RE-TR-63-6
C.3MANEUVERING REQUIREMENTS AND MINIMUM MISS DISTANCES FOR
HOMING MISSILES IN A RESTRICTED SET OF ANTI-TANK ENGAGEMENTS
17 May 1963
C= US ARMY MISSILE COMMAND,__ *R EDSTONE ARSENAL, ALABAMA
C=I
RSA FORM 1385, I JAN 63 PREVIOUS EDITION 15 OBSOLETE
Destruction Notice
Destroy; do not return.
17 May 1963 Report No. RE-TR-63-6
MAUVERING UIMm rS AND MINIMUM MISS DISTANCES FOR
HOMING MISSILES IN A RSTRICTED SET OF ANTI-TANK ENGAGMENTS
by
Gilbert C. Willems
DA Project Number: I-D-22-901-A-04Army Materiel Command Management Structure Code Number 5221.11.146
Correlation BranchElectromagetic Laboratory
Directorate of Research and DevelopmentU. S. Army Missile CommandRedstone Arsenal, Alabama
ABSTRACT
The homing trajectory equations are solved for a limited set ofanti-tank engagement parameters, resulting in quantitative data on re-quired missile maneuverability to overcome various perturbations. Thetheoretical minimum miss distances (noiseless) are calculated forcoasting terminal trajectories.
TABLE OF COBSENS
Page
I. INTRODUCTION. . . . . . . . . . . . .
II. MISSILE PERFORMANCE REQUIRMENMS FOR PROPORTIONALNAVIGATION .. .. .. . .. .. .. . .. .
A. A Reading Error .. .. .. ... . . .. 5
B. An Acceleration Bias . ........... ..... 6
C. A Maneuvering Target . . . . . . .. .. ... 7
D. An Initial Tracking Error . ............ 8
III, TH WOTITICAL MINIMUM MISS DISTANCE VERSUS COAST RANGE.. 8
A. Neutral Command Terminal Trajectory . . . ... . .. 9
B. Retention of Last Command . ...... ... 0. 9
IV. CONCLUSIONS. . . . . . . . . . . . . . . . . . . . . . 9
Iii
LIST OF ILLUSTRATIONS
Figure Page
I Definition of Angles ...... ............. 11
II Missile-Target Geometry .... . . . . . .. . . . . 12
III Computer Diagram for Trajectory Equations. . . . .... 13
IV Computer Diagram - Forcing Function for Case 2 .. 4
V Computer Diagram - Forcing Function for Case 3 ..... 15
VI Computer Diagram - Forcing Function for Case f .. . 16
LIST OF TABLES
Table
I Defi on and Sybols...............
II Miss Distances for Case A ........ .. . . . . . . . 18
III Miss Distances for Case B ......... .. 19
LIST OF GRAPHS
Graph
1 through 9' Z for a heading error ............... ..20-28
10 through 16: 2m for .an acceleration bias ........ .... 29-35
17 through 22: Zm for a maneuvering target ......... 36-41
23 through 28: 2m for an Initial tracking error. ...... 42-47
iv
I. IN4TRODUCTION
The trajectory equations that govern the kinematic behavior ofmissiles flying various proportional navigation trajectories are pre-sented with the goal of establishing missile maneuvering requirementsfor a specific set of engagement parawAtere. These parameters arethose that might be encountered in ground-to-ground tank engagementsat the 2 to 3 kilometer range. Analog simulation results, based on thetrajectory equations, show the maneuver capability required of a homingmissile to overcome various perturbations.
In additiou, a problem peculiar to seekers that track the targetimage contour is treated. This problem, that of guidance loss in theterminal tracking phase due to the increase in image size with decreas-ing range, results in a need for the missile to coast unguided for afraction of this trajectory. The theoretical miss distances due to thisphenomenon have been calculated and are also presented.
I. MISSILE PERFORUNCE HQUIlMNTS FOR PROPORTIONAL NAVIGATION
In proportional navigation, the missile turning rate (t ) is madeproportional to the line-of-sight (LOS) rate (b):
N is the proportionality (navigation) constant. Figure I shows thatthe normal velocity component of the missile is given by: y w V sinyVyf'or small angleA; therefore, the normal accelexation is given by:
y =V, + Vy WVj for a constant velocity missile.Thus:
One practical means of attaining proportional navigation is tocarry onboard the missile a tracking device capable of measuring theLOS rate and using this information to command a lateral accelerationproportional to this rate. A means of measuring 6 can be explainedwith the aid of Figure I. A gimballed seeker attempts to track thetarget, i. e. reduce the error angle 9 to zero. This angle exists be-cause the tracker cannot have an infinitely fast response and will thuslag behind the LOS by an amount determined by the tracking loop timeconstant (T ). Thus the instantaneous value of the anglic is givenby:
From Figure I it is obvious that a a a--IL, and thus:
" _" +rrS-" = rSI +rs I+rs I+rs
Therefore, the instantaneous tracker error angle, a rather easilymeasured quantity, can be used to effect proportional navigation if wemultiply it by some gain factor X and use the resulting value to com-mand a missile acceleration:
bITS
where am = missile lateral acceleration in g's.
The above equation will be used in the derivation of the homing tra-jectory differential equations. These equations are greatly simplifiedif we use the constant-bearing (collision) course (Reference 1 and 2)as a reference and consider the proportional course as perturbationsabout this reference. Such a reference frame is shown in Figure II,and the associated symbols are identified in Table I. From the figure,we obtain:
Zm = -0mg (2)
i. e. the commanded acceleration of the missile normal to the referencecourse is directed so that it tends to eliminate Z,, thus the negative-ign. Coblingii equations 1 and 2:
tm=_ gXTS
or, converting the whole equation to time domain:
3 r'Zm +Zm 2 -gAT&Integrating above:
rZm+ m = -gXra+C (3)
C is the constant of integration (initial conditions) and is given by:
C: T2M10, + 2m1 , +gXr(C] 0,)
where o+ denotes value of variable at t = o
Equation 3 can then be written:
2
vZm+Zm gxTO-+T 2mj0 . +Zm..1 + +g). (0.I04) (4i)Equation 4 is then the trajectory differential equation and can besolved if a can be determined. SinceOr is generally small, the sim-plifying approximation a- o tan a- can be used. From Figure II it isalso clear that the tangent of o' is given by:
tan a- a- (T+2)-( -) orRt
(XmSin8 +Zmcosb) - (Xt sin_+Ztcos')Rt
since Rt w R (to - t) where:range rate m V0 - closin velocity
toi n nominal homin-time = Rn
- elapsed flight time
Introducing above value of Cr into equation 4 and substitutingVC (to - t) for R, we obtain the complete trajectory differential
T2.m + 2M' --- (Xmsinb+Zmcos-Xtsinf-ZtCOSI)+T2m1+ZPLVC0 T°-T) +gXT((tot1)0+
Transposing Zm cos 8 to the left side, we obtain:
(m gVTcos g) -t_ (Xmsin8-) sin*-Zfcog*) (5)rr+ m kv0 (to-t)Zm V0 (to-t) mifB tnhtC8)(5
gXrcoS3 + r2mJO +imJ0, +gkr (a fe.)The term V. is almost constant for small deviations from acollision course and is generally defined as the effective navigationconstant N'. A more common expression for N' is given by:
N'. Vm NcoSVc
The above two expressions can be shown to be equivalent if the follow-ing equality can be demonstrated:
g.r =VmN
This can be done as follows:
It has been shown that the seeker measures r& but with a lag(I+rs) term. Thus, since the basic error isr6- , then from equation1, the basic commanded acceleration is:
am * Xro (6)
From the basic navigation law, we have shown:
.NVm."NVt6 or am(7)
Equating the right sides of equations 6 and 7, we obtain:
)r=NVm/g or )rg=NVm
Dividing and multiplying the right side of equation 5 by cos 8 andsubstituting N', we obtain.
r~rn ±±m N' Zm -N' 8 i t CgT'm Tn, +N oi" - '- mtanB-Xt sin* zCsto- t cos8- Z Cos a
(8)
The preceding trajectory equation is a linear differential equa-tion (time varying coefficients). Since the principle of superposi-tion applies, the equation can be solved for certain specific conditions of interest one at a time and the results linearly combined. Ofparticular interest is the solution of Zm for the following condi!-tions:
1. An initial missile heading error, i. e. the missile is notpointed at the target at launch.
2. An acceleration bias. This can be caused by thrust malalign-ment, improper gravity correction, etc.
3. A maneuvering target.
4
4. An initial tracking error, such as would be caused by improperseeker pointing at launch.
Given typical values for the above four conditions, the solution ofequation 8 for 2m yields the important specification of requiredmissile maneuverability. The analog computer studies to be discussedare concerned with obtaining this specification. Since no one specificmissile is under consideration, a range of representative parametershave been selected so that the data may be applicable to a wide rangeof vehicles. The general environment considered is that of a ground-to-ground engagement at ranges of 6000 and 10,000 feet. Since in this casegravity acceleration acts normal to the flight path, the results obtain-ed are valid for either pitch or yaw. This is due to the fact that forthe ground-to-ground case an external "g" bias would be required toovercome the gravitational acceleration, and is thus not part of theclosed loop dynamics. If the navigtion law were to be used to over-come the effect of gravity, an excessive portion of the missile's ma-neuvering capability would have to be employed; additionally, theground clearance problem would become overwhelming unless the missilewere to be launched into a high initial trajectory. This could be doneby initiating guidance after launch, and the problem would be reducedto one of overcoming an acceleration bias, a case treated herein.
A. Case I - An initial heading error
From Figure II it is evident that Zmjo =VmSin ro;$VmveWith all other initial conditions = 0, equation 8 becomes:
N'ZmT i + 2" + to-t -VM o
or, since to -t "R/Va
Ti i o t (9)
A computer program for equation 9 is shown in Figure III. Thescale fators ahown are only one set of several used. The scale changeswere necessitated by the dynamic range of the equations and by noiseproblems. Graphs I through 9 show plots ofZ M versus range for a setof representative conditions. (0 in the graphs refer a to Y0 ).Several interesting observations can be made from these plots. The in-dividual plots on Graphs I an= 2 show the effect of changing r only.The magnitude of the required 2m oes not change with T ; however, therange at which the peak v~lue of LM occurs does change considerably.This is also evident from Graphs 4 and 5 where the initial conditionsare the same as for I and 2, except for initial range. The longer
5
range results in a decrease in required peak acceleration.
Graphs 3 and 6 through 9 show the effect on the required Z of in-creased N' and velocity. It is evident that an increase in either para-meter results in an increase in missile maneuver requirements. The im-plication is that missile flight time and navigation ratio are thecritical parsLters of interest, This is evident from equation 9. Ifwe replace Vc7Ry /to-t, Jerger (Reference 1) has shown that for agiven navigation constant and initial conditions, the acceleration re-quirements are dependent solely on the ratioto/T . This term is de-noted "control system stiffness." For example, in the first set ofplots in graph l,to/T - 12. The Zm values would be equally validfor R a 12,000 feet and V = 2000 ft/sec since t./Twould remain unchang-ed. Another useful generalization can be stated as follows. Forto/T >Z 0 , the system does not deviate considerably from a perfectsystem (to/r'=nC ).
B. Case 2 - Au acceleration bias
If an acceleration bias exists in the missile, an additional fix-ed acceleration will be commanded in addition to that called for by thenavigation law. Equation I then becomes:
am = ab + I+TS
The term a will appear after one ltegration as abt in equation 3 andas -at on the right side of equation 8.
If we assume no perturbations and no initial heading error, equa-tion 8 reduces to:
to -t 0
Since 20+ =- ab and(to-t):R/V, the above
equation reduces to:
TZ, + 2m+ Zm _ +r) (10)
The right side (forcing function) of the above equation is the only termthat differs from equation 9. Thus, with the one exception, the samqcomputer program ney be used. Mechanization of the term -am(t + rJis shown in Figure IV. Plots of the values of Zin required to overcome
6
bias errors of 1/, 1/2, and 1 "g" are shown in Graphs 10-16.
C. Case 3 & maneuvering target
The case for a maneuvering target can be considered as follows:Let I nZmJ i Xm 02-m ,,.o+ =° + = t
and Z t= 1 ot 2 (constant acceleration target).
Then equation 8 can be written:
N'Vc m N'Vcos , at t']
?ZmITm± 2 mj 2Rcos 8
If we consider the most severe case, i.e. the initial missile-target collision course is along a straight line (head on engagement)and the direction of target maneuver is normal to this, the above equa-tion can be written:
rtM+ M+ N'Vc Zm I N'V at t2R 2R
This equation is again identical to the other cases considered exceptfor the foiLi16 function. Thus the computer program on Figure III canagain be used with a suitable modification for the term of the rightside. This is shown in Figure V. Plots of the required missile accile-ration for two magnitudes of target maneuver (4 ft/sec 2 and 8 ft/sec')and a representative set of engagement parameters are shown in Graphs17 through 22. From these graphs, it is evident that the missile maneu-ver requirements increase considerably as the target is approached.
D. Case 4 - An initial tracking error
If the missile borne seeker exhibits a tracking error a* att = 0, due to imperfect aiming, it will deviate from its correct courseand restoring commands are generated. The effect of Cc can be eval-uated by solving equation 8 for the case where it is the only perturba-tion. It has been shown (Reference 1) that the trajectory equationforcing functiorr for this case is given by: -VcN Io/COs 8
Thus the trajectory equation becomes:
rZm+Zm +N'VZm/R = V4N'eO/cos8
The mechanization of the forcing function is simplyf properly scaledstep input and is shown in Figure VI. The plots of Lm for various para-meters of interest are shown in Graphs 23 through 29.
7
The plots iadicate that very large initial accelerations arerequired; however, we should keep in mind that the values of E o employedare also quite large. Initial tracking error much smaller than 10 - 30should be realizable without excessive difficulty.
IIL THEORETICAL MINIMUM MISS DISTANCE VERSUS COAST RANGE
The employment of an image contour tracker as the error sensingelement in a homing missile presents a multitude of peculiar problems.One of these is related to the angle subtended by the target in thefield-of-view and is of interest due to the following reasons:
1. At maximum range, the field-of-view of the optics must be smallenough to yield a target image large enough to exceed theresolution limitations of the TV seeker.
2. As the missile nears the target, the image will subtend a pro-portionally larger portion.of the field-of-view, until even-tually the image to field area ratio becomes unity and track-ing information is lost. The missile must thenboast unguidedto the target. For hardware presently under development, thefield-of-view limitation is approximately 20 for an initialrange of 2 kilometers. Considering a 7 1/2 x 7 1/2 foot tar-get, the missile must coast for approximately the last 250feet. For a 3 kilometer initial range, the coast distance be-comes approximately 500 feet.
Miss distance calculations are extremely complex avd involve manyprobabilistic and design factors. Our concern here is to determine themiss distance due only to target maneuver during the missile coastperiod when trajectory corrections cannot be effected. Two alternativesare discussed.
A. Neutral Command Terminal Trajectory
In this scheme, when loss of track occurs, the control sur-faces return to neutral and result in a straight line missile trajectory.The miss distance can then be approximated by the target travel o1rm42,to the collision course during the missile coast period. These valuesare tabulated in Table I for two values of target acceleration andinitial range, and for several values of missile velocity. The missdistance (m) is given by:
2m = 1/2 atts
where at = target acceleration normal to LOSts = coast flight time
B. Retention of Last Co mand Prior to Loss of Track
If the control system is locked in a certain position at some
time t, the miss distance for maneuvering target can be found by study-ing an equivalent case,that of acceleration saturation. This occurswhen the engagement dynamics due to target maneuver are such that thenavigation law calls for a command that exceeds the missile's capability.The two cases are analogour, and the miss distance can be calculated byassuming that acceleration saturation occurs when thp seeker losestrack. The derivation of m is given in Reference I for the case whereTju 0:
m s at (to - t s ) N' x to2
2 to
where to R/Vc
ts =to - R/Vc
Miss distances were calculated for the same parameters as CaseA and the results are shown in Table I. It is evident that the valuesof m are negligible. It is true that the r - 0 assumption is not quitevalid; however, an useful approximation is obtained. Additionally, onegeneral important conclusion can be made, i.e. it is at least theoreti-cally possible to hit a moving tahk-sized target with a missile thatemploys the contrast controut tracker (CCT) under consideration
IV. CONCLUSIONS
It has been shuwn that for a perfet system, the miss distances re-sulting from terminal coasting into maneuvering targets capable of tank-type accelerations are quite insignificant. This of course, is basedon the assumption that the :.saile retains its last correct commandprior to loss of track. If the control system is returned to its neu-tral positions, the miss distances increase accordingly.
Quantitative data on the missile performance requirements to over-come several important perturbations has been obtained. A sufficientvariety of parameters has been employed in an attempt to cover the pro-bable values for a CCT-type system. It has been shown that the dataobtained can be extended to other than the cases shown, provided that-the missile velocity and engagement range are such that to remainsunchanged. In addition, the equations and analog programs for eachcase have been presented, and thus fulLher data could be obtainedrapidly if necessary.
Given a specific set of missile and attack parameters, the propergraphs may be consulted to determine the mnaximum acceleration requiredto overcome given disturbances. For example, if Vm = 1000 ft/sec,R0 - 6000 feet,toy=12andr = 0.5. What missile acceleration is re-quired to overome an initial leading error of 50? Graph 3 indicatesthat 50 ft/sec at a range of 4800 feet is required.
9
REFERENCES
1. Jerger, Joseph J., Systei, Preliminary Desigtn, D. Van Nostrand Co.,1960.
2. Puckett, A. E., and Ramo, S.,Guided Missile Engineering,, McGraw-.Hill, 1959.
3. Locke, A. S., Guidance, D. Van Nos~rand Co., 1955.
10
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Figure 11:. Missile-Target Geometry
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TABLE I
Definitions and Symbols for Figure II
R 0 Initial range
Rt Range at time t for reference collision course
Rt Actual range at time t
kl, k2 , k3 ,lk4 Auxiliary lines, parallel to Ro
Ao Bo C0 Collision triangle for constant bearing course
a, b1 Reference missile and target positions at time ta2, b2 Actual (perturbed) missile & target positions at
time t
i, 2, 3, 4 Auxiliary lines perpendicular to Po
Xm, xt Longitudinal missile and target perturbations
Zmb Z t Traverse missile and targct perturbations
8 Missile lead angle
1Target aspect angle
Line of sight perturbation angle
rO Initial LOS angle from reference
17
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0 o - - -
5 0
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00
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o 0
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47
17 May 1963 Report No. RE-TR-63-6
APPROVED:
PJACOB ZAROVI IVChief ,Corre\ktion Branch 'Electromagnetic LaboratoryDR&D, Army Missile Command
OROM4. WEDirector, Electromagnetic LabDR&D, Army Missile Command
48
DISTRIBUTION
CopyU.S. Army Missile Command
Distribution List A
for Technical Reports 1-98
AMSMI-RMr. McDaniel 99
AMSMI-RFE
Mr. Salonimer (6) 101-105
AMSMI-RG 106
AMSMI-RL 107
AMSMI-RS 108
AMSMI-RR 109
AMSMI-RE 110AMSMI-RES (4) in-114
AMSMI-REO 115
AMSMI-REE 116
AMSMI-RER 117
A1@MI -RB 118-122
AMNI-RAP 123
49
