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Weste rn Kentuck y Universit y ~4~.

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AN APPLICATION OF OPTiMAL CONTROLTHEORY TO AN ANTI -TANK WEAPON SYSTEM

u-I r~ 6f ~\Ll~f l

PrirLiple Researchers : Dr. Randy J. York -Dr. Daniel C. St.ClairDepartment of Mathematicsand Computer Science

Research Assistant Peter SislerA

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SECUMITY C L A S S I F IC A T I O N OF THIS PAGE (When Data Enterer!)

REPORT DOCUMENTATION PAGE BEFORE FORMT~~~~EPORT NUMBER 2. GOVT ACCESSION NO. 3. RECIPIENT S C A T A L O G NUMBER

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. T I T L E f e n d .SuA.iI1~ ) .--.~-.~~~~~~ . ~~~~~_~_____ .4~~~~~~~~~~ F nBrenT~~~~~rQr rn cpyERED_An App lication of Op timal Control Theoc~~to / / Final re~~~~. ~~~~~~~~~~~~

an AntiTank Weapon System s J n~~~~~~~~~~~~~~~ P~~~T NUMBER1~ R a n dy J / Y o r k /

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~~~~~~~~FORMIN G ORG~~~~ ZATI O N NAME AND ADDRESS 10. PRO EL~~~ EN~~~PR ~~~ECT . TASKA R E A & WOR K UNIT NUMBERSWestern Kentucky UniversityBowling Green , Kentucky 42101 K

II. CONTROL LING OFFICE NAME AND ADDRESS

Commander ; U.S. Army Missile Research and ~~~~~~ne d78~

Developmen t Command 13.ATTN : DRDMITCN , Redstone Arsenal~ AL ~~c~gpg 16514. MONITOR & ADDRESS(I1 dif ferent from Cont rol ling Office) 15. SECURITY CLASS. (of thia repo rt)i~~f.1.1 J.... / J-f . Unclassified

I ISa . ~I~ Ck~~~~IIICAT ION/D OWNG RA D I NG16. DISTRIBUTION STATEMENT (of thu Report)

Approved for public release; distribution unlimited.

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Il. DISTRIB UTION S T A T E M E N T (of the abetted entered I r . Block 20, if differen t from Report)

tO . SUPPLEMENTARY NOT ES

19. KEY WORDS (Continue err rever e. aide if nec...ary end Identify by block number)

Opt imal control law formulation\ Optimal control law implementation\Terrninal guidanceknci_Tank weapon system

20. ABST~~~iC1 (Camtthu. ,,v.r.. at~~ ft n.c...atv ~~d fd.nhify by block number)

~~~ The probl em of imp lementing a threestate optimal control law in ai

an~~.tank weapon system was examined. The weapon system was representedby a detailed six degree of freedom simulation.The der ivation of the control law is included along with a four

state simulation that was used to model tt1e 6DOF simulation . The procedureused to implement the contro l law is given in Chapter III , and the methodof obtaining optimal control parameters appears in Appendix C. The questioiof sensitivity is also examined using both simulations.

DD ~~~ 1473 EOPT~ON OF I NOV 65 IS ODSOLETE- ,- / / i Un c las sj ..~~ ed~ i / /~ 6 SECUP I V C LA S c f o~ cj I THIS PAGE: (WIr.n Del. Ent.r. ~t)

_ L~~~14

June 1, 1978

FINAL REPORT

forWestern Kentucky University Contract # DA.AK 4077C0052(Redstone Arsenal Contract #~~~N-~TB-3-3)

IAN APPLICATION OF OPTIMAT~ CONTROLTHEORY TO AN ANTI-TANK WEAPON SYSTEM

-

71 ~Mt. ito 1Harold Pastrick , Contract Supervisor ~~~~~~~~~ . Guidance and Control DirectorateResearch and Development LaboratoryU. S. Army Missile Command

01 IIRedstone Arsenal , Alabama 35809 _~~~~~ -~~~~t ~~~~ i~. w IWA.

IfromDr. Randy J. York and Dr. Daniel C. St. Clair .Depar tment of Mathematics and Computer ScienceWestern Kentucky UniversityBowling Green , Kentucky 42101Peter Sisler , research ass is tant

ABSTRACT

The problem of implementing a threestate optimal controllaw in a antitank weapon system was examined . The weapon systemwas represented by a detailed six degree of freedom simulation.

The derivation of the control law is included along witha fourstate simulation that was used to model the 6DOF simulation .The procedure used to implement the control law is given in Chapter III,and the method of obtaining optimal control parameters appears inAppendix C. The question of sensitivity is also examined using bothsimulations. Final conclusions and recommendations appear in Chapter V.

ACKNOWLEDGEMENT S

Dr. Harold Pastrick , contract supervisor , of the U.S. ArmyMissile Command , has been most helpful with his suggestions anddemands. We would also like to acknowledge the assistance ofDr. Thomas Madron and his Research and Computer Services group atWestern who have patiently answered our many questions and requests.This contract , with its security requirement , has posed specialproblems, and we would like to thank Dr. Glenn Crumb and his assistantsat the Office of Grants and Contracts for their aid.

This report could not have een prepared without the help ofDebra Wheeler, who not only was most careful in the typing of thismanuscript , but who was able to spot and correct oversights on ourpart.

Someone has to be last, but we hope that no affront is taken.We would like to especially thank our wives for their patience andunderstanding they showed during this contractual period and thepreparation of this final report.

IdiLJ . ~A . -- ~~~~~ . - - .~~~-~~~ --~ .-------~~~~~~~ .--. ----- .

______________________________________________________________________________________ - - .- _- . -.- -_ .

CONTENTS

Chapter Page

I. Introduction.

1.1 Backgr ound 11.2 Imp lementation of the Control Law 2

II. Implemen tation of the Six Degree of Freedom Simulationon the IBM 370 System.

2.1 Incompatability of IBM and CDC FORTRAN 32.2 Disk Storage and CJS 32.3 Selection of Comp iler 42.4 Simulation Simplification for Developmental Work. . . 42.5 Add itional Capability Added 4

2.5.1 Roll, Yaw Control 42.5.2 Plotting Capability 5

2.6 General Review of the 6DOF Simulation 52.6.1 Airframe Model 52.6.2 RotatIon 72.6.3 Translation 72.6.4 Aerodynamic Forces and Moments 82.6.5 Target Model 92.6.6 Guidance and Control Models 102.6.7 Gyroscope Model 102.6.8 Guidance Filter Model 112.6.9 Ac tuator Model 11

III. Control Law Imp lemen tation Ideal Case3.1 Brief Summary of Control Law 123.2 General Outline of Approach 143.3 Determination of Contro l Parameters y and 8 163.4 6DOF Simulation Implementation 17

3.4.1 Modifica tion of Control Module Cl 183.4.la Implementation of the Con trol Law. . . 183.4.lb Control Module Cl Listing 18

3.4.2 Evaluation of Control Gains GPIT , CYAW 273.4.3 Control Parameter Improvement 27

3.5 Sensitivity Analysis 273.5.1 Sensitivity for the FourState Simulation. 273.5.2 Sensitivity with the 6DOF Simulation 29

IV. Control Law Imp lementation Practical Case4.1 Disadvan tages of the Ti~reeState Controller 364.2 A Controller Using A , X 364.3 Implementation of the A , A , 0 Controller 374.4 EstimatIon of the Timetogo 38

d

Page

V. Conc\~usions and Recommendations for Future Study 39

VI. Simplified 6DOF Simulation 40

References 124

Appendix

A. Optimal Terminal Guidance With Constraintsat Final Time Al

B. FourState Simulation Listing BlC. Determination of Optimal Control Parameters:

An Application of Mathematical Programming . . ClD. HookeJeeves Algorithm Listing DlE. Derivation of the ThreeState Control Law El

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--~~~~..---~~~~~-

___________________________ - -

LIST OF ILLUSTRATIONS

Figure Page

2.1 Modularization of the 6DOF Simulation 62.2 Block Diagram of Pitch or Yaw Guidance Channel 72.3 Ideal Seeker Characteristics 103.1 Geometry of Tactical Missile Target Position 123.2 Hooke Jeeves Algorithm for n = 2 173.3 Original Feedback Control System 193.4 Missile Roll Stabilized Position 183.5 Initial Implementation of the 3State Controller . . . 283.6 RZE vs. RXE 343.7 THETA vs. TIME 354.1 Missile Target Geometry 36

H

LIST OF TABLES

Table Page

3.1 Definition of Variables 123.2 New Variables for Subroutine Cl 20

3.3 Performance for Various Initial Conditions (Baseline I) . . 303.4 Performance for Various Initial Conditions (Baseline II)~ . 313.5 Performance Dependence on GPIT 32

- 3.6 Performance Dependence on y and 8 323 7 6DOF Simulation Runs for Various Initial Conditions 33

I,,

_ _ _ _ _ _ _ _ _ _ _ A

- - ~~~~.__

Chapter 1

INTRODUCTION

1.1 Background

Recent intelli gence suggests that the impenetrable natureof heavy armor nay be susceptible to missile attacks at a relativelyhigh angle of impa ct , with respect to the horizon. In many modes ofdirect encounter , the torget may not be reachable with a body p itchattitude angle of the proper magnitude. There are several poss iblereasons fur this condition , including lack of energy (fuel), lack oft ime to ni.inel1v ~r into the more desirable attitude , or lack of controlin f o rm ation by appropriate sensurs to command tile response. Thiscondition has been recognized for some t ime at the Missile Researchand Developm ent Command , and consequently there have been attemptsto modif y trajectory shapes by a variety of pred etermined controllaws. ilowover , therL has been a certain lack of robustness in thesolutions obtained over the entire range of conditions anticipated.This situation motivated a search for optimal solutions to the guidanceproblem and a study of tiadeoffs among the suboptimal candidates thatwere deemed feasible.

Terminal guidance schemes for tactical missiles may be basedon a classical approach , such as a prop or tional nav igation and gu idancelaw [3 ,4 ] , or on a modern control theoretic approach [1 ,5,6]. In thelatter , a control. law is derived in terms of timevary ing feedbackgains when formulated as a linear quadratic control problem . A sub

f r .~ optimal terminal guidance system for reentry vehicles , der ived us ingthe modern approach , was the basis for the initial work of thisprobl em.

Kim and Grider ~2] studied a suboptimal terminal gu idancesystem tor a reentry vehicle by p lacing a constraint on the bodyattitude angle at impact. Their problem was oriented to a longrange ,highalt itude mission. Their scenario was formulated as a linearquadratic control problem with certain key assumptions. The angle ofattack of the reentry vehicle was assumed to be small and thus wasneglected . Furthermore , the autopilot response was assumed to beinstantaneous , i.e., ~ith no lag time attributed to the transfer ofinput commands to output reaction.

Thes e condi tions have been studied in an ex tension of theirearlier work where a formulation is given for a system that has finitetime delay. In fact , the increase and decre ase in time delay hasinteresting ramificaLions on the solution. The angleofattackassumption is investigated , and , although not solved analytically inclosed form , the system is derived [9 1 .

There is more than just a passing academic interest in thisproblem . As suggested previously, the ant iarmor role of several Armyweapon systems very well may be enhanced by this techn ique. Thereduc tion to a prac t ical imp lementation or mechanization is the aim

of this contract .

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1.2 Imp lementation of the Control Law

In order to investigate the problems arising from the prac ticalimplementation of such an optimal control law on a real system , a de tailedsix degree of freedom (6DOF) digital simulation of such a system wasneeded. Such simulations are so complex that to implement the controllaw from just a programming point of view would require that the simulationwould have to be in modular form; otherwise , the programming d i f f icultieswould obscure the implementation difficulties . Such a simulation of asemiact ive laserguided airtoground missile was provided to the authorsby Dr. Harold L. Pastrick of the Army Missile Command , the contractsupervisor. Unfortunatel y , none was ava ilable tha t would work on theIBM37O system at the University of Kentucky that would be used seesec t ion 2 .1 for a d iscussion of the considerable difficulties encountered .

The detailed 6DOF simulation would give an accurate representation of the real system and how it would respond with the new controller.Al though a simp ler fourstate simulation would be used as a guide inimplement ing the con trol law , it assumed only a point mass representation

of the missile. The 6DOF simulation had wind tunnel test data used inthe calculation of aerodynamic forces and moments , and cou ld be used toexamine performance under more realistic conditions.

The original work [2] assumed no lag in the autop ilot , ano therpossible source of trouble. A more complex control law based on a firstorder autop ilot lag model had been derived [9] and was available forimplementation if needed. Perhaps the most crucial assumption of allto be tes ted , was that of no hard constraint on the controller as ispresent on the missile in the tail fin stops. The only constrain t inthe formulation of the control problem was a sof t cons traint presentin the cost function J to be minimized , Equation (3.3).

The appro ach used was to imp lemen t the con trol law with theset of state variables given in [2]: missile attitude angle e , projec tedmissiletotarget distance on ground Yd.and ra te id. Perfect knowledgeof the states were assumed . Since id~ Td cannot be measured , the setof state variables was then altered to include the line of sightangle A and rate ) . In this setting, all var iables were to be assumedavailable at first , and then r ealistic measurements from the seekerfor A ,~ was to be used , along with a gyro measured attitude angleSee section 1 .2 for a more complete outline of the approaches used , alongwith other sections of Chapters III and IV.

Conclusions and recommendations for future study appear inChapter V.

2

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Chapter II

IMPLEMENTATION OF THE 6DOF SIMULATIONON THE IBM37O SYSTEM

2.1 Incompatability of the IBM, CDC FORTRAN s

One might be inclined to think that all FORTRAN s arecompa tabl e, but such is certainly not the case. Each computingsys tem, whether it be that of IBM or CDC, has certain uniquefeatures of which a programmer can take advantage. The code ofthe 6DOF simulation took extensive advantage of such featuresand made imp lementation on the IBM37O system quite difficult.All incoinpatabilities between the two FORTRAN s had to be detectedand resolveda process which required several months of effort .The f inal goal of the altering of the FORTRAN code was to producea simulation that woul.d run on the IBM37O as well as the CDC6600,that is, the code was not to take advantage of the special featuresof either system.

Some of the programming inconsistencies were easy tospot and correct. Among theue were the use of * for titles inFORMAT statements , variable names that were too long, restrictionson handling alphanumeric character data, restric tions on combiningthe initialization of data for arrays and specifying their length ,the number of lines on which a statement can be continued , restrictionson the indexing of parameters in DO loops.

Other programming incompatabilities were more difficult todetect and correct. The plotting capability in the simulation madeextensive use of in house plotting subroutines and had to be bypassed.A very troublesome programming practice was that of not initializingvariables when their initial value was to be 0 . The CDC machinewill zero out its memory with the start of any new job. The IBMcomputer does not. The garbage in those particular memory locationswill he used in the executation phase without any warning to the user.The way this problem was finally solved was to run the program witha WATFIV comp iler which will flag uninitialized variables.

Some subroutines such as those involvad with linear inter-polation ~~t une or two variables , made such extensive use of specialCDC array manipulation features that they had to be rewritten and ,consequent ly, retested for accuracy .

2.2 Disk Storage and CJS

The simulation was of such length that disk storage wasessential. Once the simulation was placed on disk , the problem wasstill how to implement the numerous corrections decided upon . Tothis end , it. was decided to use a new IBM editing creation called CJS,Conversational Job System which was a subset of a larger packagecalled CMS, Conversational Monitor System. EspecIally desirablefeature s Included were the ability to search for phrases , to deleteor add lines , rear range whole sections of code , to make global

L

changes as well as local ones, and to create executive programs toserve a wide variet y of needs.

~~~~~~~~~~ -- .~-~~ ~- -.-~~~~~~ --~--

To reduce execution time, all subroutines which were toremain unchanged were stored on disk in obj ect f orm to lessen

compilation time.

2.3 Selection of Compiler

The %.~ATFIV FORTRAN compiler was chosen for the initialphase of the work because of its rather extensive collection ofsyntax error messages. Once the syntax errors were removed , theGlevel compiler was used because of its speed in compiling. Whenfinally the point was reached were code changes were few, theHlevel compiler was incorporated because it optimizes the executionof the code. ln order to do this, it does take longer to compile.

2.4 Simulation Simplification for Developmental Work

For the developmental phase , a shorter, less comp lexsimulation was desired , one that would be less expensive to run.The original 6DOF simulation provided capabilities that were neitherneeded nor desired. A variety of seeker modules , actua tors , andautopilots were available. The simulation was set up to run MonteCarlo sets and to give various statistical analyses for the results.There was in add it ion a plotting capability which would be quitedesirable but was unusable due to the utilization of subroutinesfound only on the CDC system.

To reduce compilation and run time, it was decided topare down tile simulation so that the resulting shorter form of thesimulation would contain only the modules and subrou tines necessaryfor the development work. To this end , some 27 subroutines weredeleted. They were C2, C2I, NORMAL , RANNUM , MCARLO , AERROR, TERROR,CEPAS, CEPP , NORM, KSTEST, TABLE, PPLOT, XLOC, G2, G2I, S4, S41, CS,C5I, RESET, PLOT4, S2, S3, PLOT2 , PLOTN, and SUBL1.

A listing of the shortened form of the 6DOF simulationappears in Chapter VI.

2.5 Add itional Capability Added

2.5.1 Roll , Yaw Control

Since the initial work in [2] assumed motion in one planeonly, the capability of controlling roll and yaw motion was added .The simulation already had a roll control feature , but this had tobe altered somewhat when the yaw control was added. The user setsswitch OPTN3 equal to 0 or 1, depend ing on whether he wishes toallow the missile to roll or not , respec tively. If no roll isselected , then by setting another switch OPTNYW to 0 or 1, he can

_ _ _ _ _ _ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~-- -~~~~~~~~~~~~

- _aU~~ J~~~

allow the missile to leave the pitch plane or fo rce it to f l y in it.

The coding of this alteration of subroutine D2 follows:

IF (OPTN3.LE.O.) GO TO 45

IF (OPTNYW.LE.0) GO TO 55

GO TO 65

45 WPD CRAD*FNXBA/FMIX

55 WRD = (CRAD *FMBBA+(FMIX_FMIY)*WP*WQ/CRAD) /FMIg

65 WQD = (CRAD*FMYBA+ ( FM I~~~FMIX) *

WP* WR/CR A D )/ FNI Y

2 .5 .2 P lo t t i ng Capabi l i ty

The original plotting capability was replaced with oneprovided by Dr. St. Clair which would generate line printer plots.Any variable in the Carray could be plotted against time, and anytwo variables in the Carray could be plotted against each other.Since the graphs are done on the line printer , they are ra thercrude in appearance , but they do provide some insight intohow variables are changing with respect to time.

The variables to be plotted are stored on disk. Althoughthe plotting subroutines that are used on the IBN37O cannot be usedon the CDC6600, any program using a package compatable with the CDC6600 could be used to read off these stored variables.

2.6 General Review of the 6DOF Simulation The modularization of the 6DOF digital simulation of a

semiactive laserguided airtoground missile is given in Figure 2.1.A block diagram fo r one channel (p i tch or yaw ) is shown in Fi gure 2 .2 ,and a br ief descr ip t ion of the various components follows . For a moredetailed examination , refer to [111 and [12].

2.6.1 Airframe Model

The a i r f r ame model included t rans la t iona l and rotat ionaldynamics and generation of aerodynamic forces and moments. Airframedynam ics were simulated in 6DOF with the missile orientation repre-sented by three Euler angles. Maneuvering control was achieved bymeans by four surfaces in cruciform configuration hinged about fixedstabilizing fins.

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GUWANCE ~~~~[AMP

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ACTUATOR

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TURNING ILOS RATE I AIRFRAMEGEOMETR Y u 1 AERO

Figure 2.2. Block diagram of pitch or yaw guidance channel.

2.6.2 Rotation

1X~ B = M~~ (2.1)

I~ Q~ = MYA + IX)P BR B (2 .2)-~~~~~~

I~ = MZA - ~~Y X~~B~B (2.3)where ~~~ ~~~

RB are rotational rate components about body axes X8, ~~R ZB respectively ; ~x and 1y are moments of inertia about XB and TB~ anaM~~ , MYA , MZA are components of aerodynamic moment about the body axes.

By resolving body rotational rates into the inertial andintermediate axis systems, the following equations for Euler angles areob tained :

= (Q Bcos I~E R~sinq~~)/ costp~ (2.4)

= ~~~~~~~~

+ R Bcos~~E (2 .5)

= ~B E

SIfl*E . (2.6)

2 . 6 . 3 Translation

It is advantageous to express the translation equations of motio n in terms of inert ial axes giving

mX = F~~ (2.7)

mY FYA (2.8)

mZ FZA mg (2.9)

- ~~~~. - -_-~-~rn -

_ -- - - - - -- _ - _ - - - - - - ------ -

- . ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -~-~~-~-- - .~~ ----- ~~~~~~~~~~ _ _ -~~ ~~~~~~~~~~~~~~~~~~~~~~

where m is the mass of the missile ; Fy~, FYA, FZA are the aerodynamicforces in the inertial axes ; and g is acceleration due to gravity.Aerodynamic forces are obtained in inertial axis form by transformingbody axis components according to the following matrix equation :

F~~ F xFYA = [TI F~ (2.10)

FZA Fzwhere Fx, F~ , Fz are the body axis aerodynamic force components. Thebodyearth transformation matrix has elements defined in [8].

2.6.4 Aerodynamic Forces and Moments

For both missile models , aerodynamic forces and moments areexpressed in terms of body axes. The missile trajectory is dividedinto two sections; unguided flight before the target laser spot isvisible (the preacquisition phase) and the guided portion of thetrajectory after the seeker has acquired the target spot (the postacquisition phase).

Aerodynamic force and moment components for both flight phasesare as follows:

F x = qSC~~ (2.11)

F~ = qSC~ (2.12)

Fz = ~SC~ (2.13)

= qSD(C~ + DPBCP/2V) (2.14)

MYA = qSD(C~~~ + ~~BCMQ~l2~7) (2.15)

MZA = qSD(_C~~0 + DRBCNR /2V) (2.16)

where q is the dynamic pressure, S is a reference area (crosssectionalarea of the missile), D is a reference length (diameter of the missile),and V is the total speed of the missile relative to the atmosphere.Dynamic pressure is given by

q = pV2/2 (2.17)

where p is atmospheric density and is a function of altitude.

L

Aerodynamic coefficients and derivatives referenced in Equations(2.11) through (2.16) are obtained from wind tunnel tests and otherestimates are expressed in graphical form as functions of angle of attack,Mach number and , during the postacquisition phase, control vane angles.Coefficients and derivatives are defined for both models by differingfunctions for the two flight phases. Details of the coefficients forboth models are presented in [8].

- -- - - _ .

~~ _ _ _ _

Angle of attack components (m and ~) may be defined by the bodyaxis components of velocity. However, for digital computation , there issome advantage in expressing a and ~ in terms of integral equations usingwind axis coordinates. Thus, using equations for & and ~ derived in [8]and making small angle approximations , the equations for angle of attackcomponents are:

= (FZB aFXB)/ mV 8P B + ~B (2.18)

= (FYB I~~XB) 1mV + uP R (2.19)

where the body fo rce components (F XB, FYB, F ZB) include aerody namicfo rces and gravity force resolved into body axis components.

2.6.5 Target Model

The target model included target position as a funct ion oftime and the calculation of missiletarget miss distance at intercept.Ta rget motion was specified deterininistically as either constan t acceleration or constant velocity from an initial starting point. However, theeffect of jitter in the designating laser beam direction was intro

- duced by applying a random disturbance to the calculated target position .

Target to missile displacements are

AX = X~ X (2.20)

~m (2.21)

AZ = Z~ Z (2.22)

where X~, Y~ , Z~ are target position coordinates and X , T , Z aremissile position coordinates . The gyro platform is charac~eriied bytwo Euler angles ~~~~ P51) which define the orientation of the platformaxes relative to inertial axes parallel to X, Y , Z. Target to missiledisplacements in digital seeker axes are given by

AX5 AX

AT5

= [5] ~~ (2.23)

AZ5 AZ

where the t ransformation matrix [S] has elements as shown in [8] .

From the target to missile displacements in seeker coordinates,the boresight error angles are defined as:

BEPSZ = tan 1 (AZ5/AX5) (2.24)

BEPSY = tan 1 (AY 5/ ~~~ + AZ~) . (2.25)

-~~~ - .~~~~~~~~ - . - ---_-~~~~~~~~~~~~~~~~~~~~~-

-_ -~ ~~~~~~~~~~~~~~~ ~~~~~~~ ~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~ - - - -

2.6.6 Guidance and Contro l Models

The guidance and contro l models included the seeker detectorand gyro , guidance filters , and vane actuators . The laser detector ismounted on a 2DOF gyrostabilized platform in the nose of the missile,and the laser beam is viewed through a fixed lens. After the gyro hasbeen uncaged during the terminal homing phase of the missile flight,gimbal torquing signals are generated to null the spot displacementfrom the center of the detector and the resulting gimbal angles andrates relative to the missile are used to generate guidance and sta-bilizing signals for input to the guidance filters. Detector outputIs from a sampleandhold unit which operates at a fixed period .

A proportional navigation guidance scheme is employed; the guid-ance filters are leadlag networks in each of the pitch and yaw channelswhich serve to decouple the missile natural frequency in pitch and yawfrom the control system.

The ideal characteristics of the seeker transfer function areshown in Figure 2.3.

I I _ _ _ _ _ _ FOV LIMIT 2 1 1 2 FOV LIMIT

j : LOS ANGL E

Figure 2.3. Ideal seeker characteristics.

2.6.7 Gyroscope Model

For man y of the studies using the digital simulation , it isdesirable to model a gyro which has no dynamic time constants. Thesimplest model possible is one represen ted by a perfect integration ,i.e., 1/s in the Laplace notation.

As is well known [8], for example, the differential equationrelating output axis motion to input torque or input rate is given by

JO + BO + KO = H 4 c

10L ~~~~ - - ~~~~~ -.-- ~. -- - .

2where J ~ ftlb/(rad/sec ), B -~ ftlb/(rad/sec), K -~- ft/rad , andH -s- ftlb/(rad/sec).

2.6.8 Guidance Filter Model

The guidance filter or compensation network is of the leadlag type with transfer function (T3s + l)/(T4s + 1), where T 3 is thelead time constant and is the lag time constant. The input to theguidance filter comes from two sources , the sampleandhold output ofthe seeker and the damping network output.

2.6.9 Actuator Model

A firstorder actuator was used in the simulation with transferfunction AA/ (s + AA) where AA is the actuator model constant. The inputto the actuator is from the guidance filter, and the output drives thecanards. The yaw control canards are on a common shaft, as are thepitch canards.

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Chap ter III

CONTROL LAW IMPLEMENTATION IDEAL CASE

3.1 Brief Summary of Control Law -

In the paper Terminal Guidance for Impact Attitude AngleCons trained Fli ght Trajectories [2], a simp lified model of a missile ta rget scenario is used to produce a three s tate control law that willresult in the missile impacting vertically on the target , Figure 3.1

deplicts the geometry of the terminal guidance phase , and Table 3.1defines the variables introduced.

~~~~~~~~ACT ICAL MISSI LEA L COS O

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~~~~~~

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VER TICAL I __________ TARGE TREFERENCE !Figure 3.1. Geometry of Tactical Missile Target Positions.

Table 3.1. Definition of Variables

Variable Definition

Missile position variable projected on the ground (ft)

Target position variable (ft)

Position variable from missile to target projected onthe ground

~~d =

____________________ ~- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ . V. - ~__ ~~

__ _ __-~ -- _-

Table 3.1. Continued

Variable Definition

Time derivative of ~d

(ft/sec)2AL Lateral acceleration of the missile (ft/sec )

0 Body attitude angle of the missile (deg)

The following set of state variables was chosen for themodeling process.

Y d

-

X . (3 .1)-

AL

0

Under the following assumptions :

1) the ang le of a t tack is small and can thus be neglected , and

2) the a u t o p i l o t has zero lag ,

U = C~~~(t)~~~~ + C2 (t )? d + c3 (t ) 0 (3 .2 )

a control law of the form can be obtained which will minimize theperformance index

J = Y~~~( t~~) + Y0 (t f) + ~ftfu2(t)dt (3.3)

where the s t a t e variables are subject to the following dynamics:

= ~d

(3 .4)

= A L cos 0 ( 3 . 5 )

AL = w1A 1 + K1u (3.6)

= Ka u . (3.7)

P e r f o rmance is considered to he acceptable when the following constra intsare satisfied:

IY d (t f )i < 5 feet (3.8)

< 5 degr ees . (3.9)

~~~~~~~~

The time varying coefficients appearing in the controllaw are given by:

2 2c1 = [~g(tft) gyK~ (t f t ) / 2 1/ A (3.10)2 2 3

= [ ~ g(tft) gyK~ (t f t ) / 2 ] / A (3.11)2 3

C3 = [~~~~Ka + yK~g ( t f t ) / 6 ] / A (3.12)

where-hKi . -g - (b = cosO in the linearized form) (3.13)

and 2 2 2 3+ Y~ Ka (t f t ) + ~g (tf t) /3 +

2 2 4 (3.14)yg Ka (tf t) /12,

t time , t f = time of impact.

A derivation of this control law is given in Appendix E.

The assumption of no lag in the autopil ot has been removedby Pas trick and York , Optimal Terminal Guidance with Cons traints atFinal Time s [9]. This work discusses a control law more general innature which allows the control law of [2] as a limiting case.Appendix A contains a detailed examination of this more realistic ,and conseq uen tly, more complex control law.

3.2 General Outline of Approach

A simp l i f i e d s imulat ion us ing the dynamics as modeled insection 3.2 was developed (Appendix !~) . Whenever feasable , new ideaswere tried out on the simplified simulation first to test their merit.This approach is not onl y phi losop hica l ly sound , but there was a verypractical reason for it as wellmoney , or rather the lack thereof.The large scale simulation received from Redstone Arsenal requiredlarge amounts ol memory and was r a t h e r expensive to run (@ $20 fo r acomp lete flight).

The smaller four state simulation was made to resemble thelarge 6DOF one as closely as possible. The original constant velocitywas replaced with a time varying velocity profile taken from a 6DOFsimulation run. Thc parameters K1, Ka~ w1that appear in the fourstate dynamL.s were reexamined and evaluated to be sure that theyreflected the dynamics of the actua l weapon system . It was found thatthe parameter values given in [2] of a typical tactical missile didreflect the dynamics in our case as well. One discrepancy was thatour autopilot lag was more accurately represented by

w 9.8 (3.15)1

instead of

-

w~= :

________ _____________________ - -~

- - - -,

-

but it was decided to use the smaller value in the developmentalwork as this represents more lag in autopilot. Since our smallscale simulation ignores aerodynamic forces and moments , it wasfelt that this increased lag might make our s imple simulationeven more representative of the actual system.

The 6DOF simulation was altered as well. Since theinitial work in [2] assumed flight in one p lane only, it wasdecided to firs t try to implement just the pitch control, andthen later a yaw control. Consequently , the roll and yaw controlparameters were set so that the missile was allowed to neitherroll or yaw. Once the control law was properly implemented in thepitch channel, then symmetry would suggest that it could be implementedin the same manner in the yaw channel.

Once the contro l law was performing properly with the fourstate simulation , then it would be used on the 6DOF simulation . Theapproach used in implementing the contro l law was to assume that allof the states Td Td and 0 were known and could be measured exactly,and that it was possible to generate the time vary ing coefficie,. -sc1, C2 , c~ without error. It was felt that for the initial implemen-tation that it was better to leave the choice of states as given in[2], although there is in fact no way to measure the range and rangerate states of and 1d For this reason , this imp lementation isreferred to as the ideal case. For the practical case , an alternativeset of state variables was used.

The control parameter values appearing in [2] were optimizedfo r a ver tical impac t angle, but the initial geome try was that of afor tyfive degree triangle (10,000 f rom the target 10,000 up).When using their control parameter values of

y = 3823 and 13 6 . 9 4 E 04 (3.17)

with our low profile trajectory (200 x 2600), the m issile in thesimp le simulation would overfly the target. The same type of trajectorywas obtained on the 6DOF simulation . Basically , the trouble seemedto be that the missile could not turn in the time allowed. The presenceof aerod ynamic forces and moments made this problem even more acute.

I t was decided to f i r s t t ry to achieve an a t t i t u d e angle thatwould be less demanding on the control system , such as

0 ( t f ) = 45 degrees. (3.18)

Append ix A shows how it is possible to use the same control law toachieve an arbitrary angle. These initial runs also suggested thatthe geometry for the terminal guidance phase needed to be alteredto allow more height with which to work. Thus , it was decided thatthe trajector y would need to consist of two phases : a preprogranmiedp itch maneuvi~r to gain altitude followed by a terminal guidance phase.

~

i-_ _

rr - _____________________ ~~~~~~~~~~~~~~~~

Initial conditions for the terminal guidance phase werechosen to be:

= 1000, Td = 5000, 0 = 85 . ( 3 . 1 9 )

The same assumptions were made as in [2] of no angle of attack and nolag in the autopilot. It was decided to implement the 3state controllerbecause of its relatively simple form. Should lag become a problem ,the 4state controller discussed in Appendix A could be tried. Thebasic problem then was to find the optimal values of the two controlparameters r and 8 that will minimize the cost functional J given asEquation (3.3). This problem was solved using the Hooke JeevesAlgorithm as discussed in section 3.3.

With the following parameter values ,

y = 5525.51 and 8 = .19E06 (3.20)

a miss distance of 3.3 was achieved with an attitude angle at impactof 45.270. With these control parameter values , the control law wasused with the 6DOF simulation . Section 3.4 discusses how the control

law was implemented. In summary , thoug h, a miss distance of 18 feetand an attitude angle of 37 degrees was obtained before efforts weremade to improve on the control parameter values, see section 3.4.Finally, performance sensiti vi ty to initial conditions is discussedin section 3.5.

~ ~~~- 3.3 Determination of Control Parameters y and 13

Two control parameters appear in the cost functional to bemm i.nized:

J = Y~2 ( t~~) + i[0(t~ )_45]

2 + ~J~ tu

2(t)dt . (3.21)

The problem was to determine optimal values for y- and 13 so that thefollowing success criteria could be achieved :

h d (t f ) I < 5 feet (3.22)

< 5 degrees . (3.23)

The approach used to solve this problem was to view it asthe mathem atical programm ing problem of minimizing

= Yd2(t f) + [ 0(t f )_ 4 5 ]

2 (3 .24 )

where the t i n ~~t Ion evaluation was achieved through a computer run ofthe fourstate simulation program with specified values for y and 8.(see Appendix C for a discussion of mathematical programming ingeneral and this problem in particular ).

- ~~~~~~~~~-~~ -~~~~~_ - --- - - -- - - - - ---- -

__________________________ - -

To make the mathematical programming problem mathematicallyfeas ible, it was assumed that F(y,13) varied in a continuous manner withrespect to its two arguments. Since F(y,13) could not be expressedin any closed form, there was no derivative information available.Af ter some thought it was decided to use the Hooke Jeeves Algorithmto tackle this problem as it is rather straight forward , has almostguaranteed convergence, and was readily available.

This particular algorithm will make two explora tory searchmoves , one in each coordinate direction, to reduce F(y,13), and thenwill follow this with a pattern search (in the direction of thediagonal).

02

U1

Figure 3.2. Hooke Jeeves Algorithm for n = 2

Using this algori thm , the optimal control parai~eter valueswere found to be

y = 5525.508 13 = l9OEO6 (3.25)which yielded the following results:

Yd ( t f ) = 3.3 and 0(tf) = 45.27 . (3.26)

3.4 6DOF S imulation Imp lementation

The first task in implementing the threestate optimal controllaw was to modify the control module Cl. This modification is givenin section 3.4.1.

The second task was to evaluate the two gains GPIT, GYAWthat appear in the control module , see section 3.4.2.

~

~~~ . ~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~

The last task was to improve upon the values of - -~ and 3given by the Hooke Jeeves Al g o r i t h m working w i t h the f o u r s t a t esimulation which was being used to approximate the 6DOF simulation.This improvement is discussed in section 3.4.3.

3.4.1 Modification of Control Module Cl

3. 4.la. Imp lementat ion of the Control Law

Figure 3.3 gives the original feedback control systembased on proportional navigation. As discussed in section 3.1,it was decided to initially control pitch motion only. The rollstabilization channel was left in tact, as was the cross couplingof the pitch and yaw control channels. This cross coupling isnecessary since this particular missile when it is roll stabilizedflies with the following tail fin configuration:

NO.1

~~~~~~~~~~ ::. 3~~~~~~~~~~ NO.2

LOOKING FORWA RD . -O

Figure 3.4. Missile Roll Stabilized Posi t ion

All f o u r ta i l f ins are required to execute a pi tch maneuver , aswould also be true for a yaw or roll maneuver.

The initial command signal to the tail fin occurs at thefirst summation in the pitch (upward) and yaw (lower) controlchannels. This signal is the difference between the commandedattitude angle and the sensed attitude angle (the feedback signal).The new commanded attitude angle was given by the control law

u = cl(t)Yd + C 2 (~ ) c~d + c3(t)0 . (3 .27 )

The guidance filters in the p itch and yaw channels were replaced bygains GPIT and GYAW for the initial implemen tation. The new autopilotis given in Fi gure 3.5.

3.4.lb. Control Module Cl Listing

The new variables that have been introduced are given inTable 3.2

_

_______________ ______ 2~~~ _.._~~~~ -c:~ ~~~~~~_ ~~~~~~~~~~~~~~~~

~-

t c i~~~

_ _ _ _ F

3, .~; ~ ~1u I-

.~~~~~~

, ~~~~~~

8

1

______________ ________________________________ - - -

- . -- -

~~~~~~~~~~~~~~~~~~~~~~~~~ --

~~~ ~~~~~~~~~~~~~~~~~~~ .~~~~~~~~~~--~

--- - -~ .- - ~~~~~~~~~~~~~~~~~~~~~~~

Table 3.2. New Var iables for Subroutine Cl

Location inName Symbol C Array Definition

BETA 13 2917 Control parameter

BDELTC( I ) cj 856 Commanded ta i l fi n posi t ion , i = 1,2 ,3,4

BJJ 881 P i t ch si gnal a f t e r coupling

BJS 537 PItch signal before coupling

BKK 882 Yaw signal a f t e r coupling

BKS 545 Yaw signal bef ore coupling

BJJSSS 987 Pitch signal before limiter

~

BKKSSS 988 Yaw signal be fo re l imiter

BTHT 350 Attitude pitch angle (measured from horizon)

BXX Roll signal before network

BXXSSS 989 Roll signal after limiter

CTHTA(T) c (t ) Control law coe f f i c i en t for 0

CXREL(T) c (t) Control law c o e f f i c i e n t for Y1 d

CXRELD(T) c ( t ) Control law coefficient for ~2 dDELTA(T) A Denominator for control coefficients

G g -bK1/w1GAMMA 2916 Control parame ter

CPIT 2921 Gain in pi tch channel

CYAW 2922 Gain in yaw channelKA K 2918 Proportionality constant appearing ina dynam ics

K1 2919 Constant appearing in autopilot lag model

I t 2000 Time

TGOX Est imate of t imetogo in x d i rect ion

TGOY Estimate of timetogo in z direction

11k - ~~~ .--- --- --

- ,,,~~-, - .= = ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ L.~~~_ 2 . ~~~~~~~~~~~~~~ ~~~~~~~~~~~ ____~-- -_ r _ i~ . . _ ~~~~ _ - - __ -~ ~~~~ -

Table 3.2. Continued

Location inName Symbol C Array Description

TGO Mm (TGOX, TCOY)IF t

f 2906 Estimate of time of impact

Wl w1 2920 Au top ilot lag cons tant

XRE~. ~d 2904 Relative xdirection missile ta rge t

position (projected on the ground)

XRELD ~d

2905 Time derivative of XREL

YREL 2902 Relative ydirection missile target position (projected on the ground)

YRELD 2903 Time deriva tive of YREL

?

_ _ -- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - ---~~~~~~~~~~~~~~~- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~

~~~~. -~~~

- -

SUBROU f1NE ClCOMMON C ( 3 8 3 0 ) , G R A P H ( 3 O O ,~ 4 )DIMENSION BDELTC (~4) ,VA R (1O l )C

C * * I N P U T D A T AEQUIVALENCE (CC 860 ) , TD Y )E Q U I V A L E N C E ( C ( d6 - l ) ,GBIA S )E Q U I V A L E N C t L ( C ( ~ 6 2 ) , GN )EQUIVALENCE (C ( 361 3 ) ,W N2 )EQ JlVA L~ NCE (C( d6L4 ),WN1 )EQUIVALENCE ( C ( 865),WLE Q U I V A L E N C E ( C ( 866 ),WLXX 1 )EQUIVALENCE (CC ~6T) ,WLXX 2 )EQUIVALENCE (C( d68),W L J K 1E Q U I V A L E N C E ( C ( 8 6 9 ) , W L J K 2EQUIVA LENCE (C( 870),H JK )EQUIVALEN CE (C( ~(l) ,WXX )EQUIVAL ENCi. (C( 812) ,DXX )EQ UIVAL ~ NCE (C( ~37$) ,WJK )EQ UIV i~L:J4CE (C( 17N) ,DJK )EQ U IVA LENCE (C( 875),GXX )EQU IV~ LEN.E (C( 8!6), GJK )

- - EQU IVi LENC E (C( d77),RESE Q U I - .AL ENC E ( C ( 878 ),QDN )E Q U I V O L E N C E ( C ( 879 ) , Q U P )E Q U I V A L E N C E ( C ( 39 0 ) ,HXXEQ UiVA L E.~CE ( C ( 89 2 ) , Q B I A S

)EQU IVA LENC E. (C ( d 9 3 ) , R B I A S )EQUIV A LENCE ( C C ~ 9 9) , O P T C 1 )E Q U iV A L E NC E (C ( 9L17) ,GNS )EQUIV ALENCE (C( 9~48),~4S1 )EQUIV A LENCE (C( ~49),WS2 )C

C * * I N P U T S F R O M OT HER N1 - J D U L E SE Q U I V A ~. EN C E ( C ( i 7 ) , SP II I )EQU I V A L E N C E ( C C 8 7) , STHT )E Q U I V A L E N C E ~C( 97 ) ,SPSI )EQC1VA L~JCt. ~C( 353),BPH1 )EC U IV A L E N C o ( C ( 35~4 ) ,BTF12 )EQ UI1iA L E N C E (C ( j 5 5) ,.B PS1 )EQ U V A L E N C E ( C C ~4 O 3 ) ,~~LAMQEQ U I VA L E N C L ( C C L4 O 1) ,~~LAMR )E Q U I V A L E N C E ( C C ~6 l ) ,CAG E )L Q U IV A L~~NC E C( 1 46 2 ) , T K R Z )EQUIV A LENCE ( C C 4 6 3 ) , T K R YEQU IV I - L EN C E ( C ( 1 2 3 3 ) ,BDR )EQUIVALENCE ( C ( l 7 ~4 7 ) ,W H )EQ U IV ALL ~-C E ( C ( 1 7 1 4 3 ) ,WQ )EQUIVALE N CE ( C ( 1 7 3 9 ) , WP

C C * * I NPUT ~, F R C ;~ M A I N P 8 O G R A NE Q U I V A l E N C E ( C ( 2 u 0 0 ) , T )

CC~~~~~ ST A TE V A R I A B L E J U T P U T S

EQ UIVALENCE (C( ~OO) ,nLQSDD)EQUIVALENCE (C( 803),~ LQSP )EQU iVA LENCE (CC ~3O~ ),~~LQ3D )E Q U I V A L E N C E (C( 807),~~LQS )EQUIVALENCE (CC dOS) ,WLQSSD)

L -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - ~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

~~~~~~~~~~~~~~~~~~~~~ -

EQUIVALENCE (CC 811 ) ,WL QSS )EQUIVALENCE ( C C 8 12 ) , W L R S D D )E Q U I V A L E N C E (CC ~3l5 ),W L R S P )EQUIVA LENC E ( 8 l6 ) ,WL RSD )EQUIVALENCE ..C C 81 9 ) ,WLRS )EQUIVALENCE (C( 820),W L R S S D )EQUIVALENCE (CC 823),WLRSS )EQUIVALENCE (C( 32~4) ,BLQSSD)EQUIVALENCE CC C 827),BLQSS )EQUIVALENCE (CC 828),BLRSSD)EQUIVALENCE (CC 831) ,BLRSSEQUIVALENCE (C( 832 ) , B J J S D D )EQUIVALENCE CC C 835 ) , B J J S P )EQUIVALENCE (CC 836) ,BJJSD )EQUIVALENCE (CC 8 3 9) , BJJS )EQUIVALENCE CCC 8 L 0 ) , B K K S D D )EQUIVALENCE CC . 8 4 3 ) , B K K S PEQUIV ALENCE (CC 8~~4 ) ,BKKSD )E Q U I V A L E N C E ( C C ~~~7) , BKKS )E Q U I V A L E N C E (CC 8~-~8),B X X S D D )EQUIVALENCE (C( 851) ,I3XXSP )EQUIVALENCE C C ( 852),BXXSD )EQUIVALENCE (CC 855),BXXS )E Q U I V A L E N C E C C C 9~~1) ,BJSSD ) , (C( 93~4 ) ,BJ SS )EQUIVAL ENCE (CC 935 ) , BKSSD ) , (C( 938),BKSSEQUIVALE N CE (CC 950),SN P2 ) , ( C ( 953),SNP1 ) ,(c ( 956),SNPO )EQUIVALENCE (CC 957),SN Q2 ), (C( 960),SNQ1 ),(C( 963),SN~~O )EQUIVALENCE (CC 96L4 ),S N R 2 ) , (C( 967),SNR1 ),(C( 970),SNRO )EQUIVALENCE (C( ~~ 7 l ) , B P C 2 ) , ( C ( 9P 4 ) , B P C 1 ) , ( C ( 977),BPCO )EQUIVALENCE (C(903) ,H13P) , (C(904 ) ,t-11 3M)EQUIVALENCE ( C ( 9 0 5 ) , H 2 i 4 P ) , ( C ( 9 0 6 ) , H 2 L I M )EQU iVALENCE (CC9O7) ,CDRFT1) ,(C(908) ,CDRFT2 )EQUIVALENCE ( C (9 0 9) , C D R F T Y )EQUIVAL ENCE (C (9814 ),CDRFTX)EQUIVA LENCE (CC 1 6ib ) ,A N G X )EQUIVAL ENCE ( C ( ~~7 B ) , B D R F T D ) , ( C ( 9 8 l ) , BDR F T )EQU IVA LEU CL (C(965) ,NLMT1 ) ,(C(986) ,NLMT2)E QU I V A L E N C E ( C ( 9 8 7 ) , BJJSSS ) , ( C ( 9 8 8 ) , BKKSSS )EQUIVALENCE (C (989) ,BXXSSS )EQUI VALENC E ( C ( 9 9 0 ) , B JJSSL ) , ( C ( 9 9 1) , BKKSSL )~Q 2 i L.EN C L C ( 5~~O) ,BJSDD) ,(C(5314) , B J S D ) , ( C(537 ) , BJS )E Q U I V A L E N C E C C C~~ja),BKSDD) ,(C(5 L~2 ) ,BKSD) ,(C(5 145),BKS)EQUIVA LEN CE (C(546) ,BXSDD) ,(C(550) ,BXSD ) ,(C(553) ,BXS )EQUIVALENCE ( C ( 5 j 3 ) ,C J S D ) ,(C(5 L 11) ,CKSD ) ,(C(549) ,CXSD)EQUIVA LEN EE (C (9142) ,W L 2 )EQU IVA LEN L:~. (C(943) ,D J 2 )EQUIVAL ENCE ( C ( 9 4 4 ) , w J 2 )EQ U IVALE N~ E C C ( 9 L4 5 ) ,DX2 )EQU l .AL iWC~ ( C ( ~~ ft ) ,wX2)

L

EQUI VALENCE (C(2965) ,VA R(l ))CC~ ~OU T P Uj AEQU IVALL~~CL ( CC 856 ) ,N D E L T C ( l ) )

CMO T HER UUTPU ~ 3EQ U I V A L L N C E ( C ( 880 ) ,BPHIS )EQU i V A L E N C E ( C ( 5 1 8 ) ,B13 SS )EQU I V A L E N C E ( C ( 5 H ) , 02 14SS )E Q U I V A L E N C E ( C C 8d 1) ,BJJE Q U I V A L E N C E ( C C d82 ) ,i3K K

_ _ _ _ _ -- ---- - _

- - ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~ - --~ --~~~~~

EQUIVALENCE CC ( 8 3 3) , I3XXSS )EQUIVALE NCE (CC 884),BJJSS )EQUIVALENCE (C( 8~~5) , BK KSS )EQUIVALENCE (CC 886),BTHTS )EQUIVALENCE C C C d o 7 ) ,BPSIS )

C**AD D IT IONAL INPUT FOR NEW CONTROL LAWEQUIVALENCE ( C (2 0 0 0 ) ,T )EQUIVALENCE ( C C 350 ) ,BTHTE Q J V A L E N C E ( C C j ) 1) , B P SI )EQUIVALENCE ( C ( 1 6 1 5 ) ,RXE )E Q U I V A L E NC E ( C ( 1 6 0 3 ) , VX E )EQU IVALENCE ( C ( 1 6 1 9 ) ,RYE )EQUIVALENCE ( C ( 16 2 3 ) ,RZE )EQUIVALENCE ( C ( l l l ) ,VZEEQU iVALENCE C C C 1 6 O 7 ) ,VYEEQUIVALENCE ( C C 1 6 5 1 ) ,RTXE )EQUIVALENCE ( C ( 16 6 0 ) ,VTXEEQUIVALENCE ( C ( 1 6 5 5 ) , R T Y E )EQUIVALENCE ( C ( l 6 6 1 ) ,VTYE )EQUIVALENCE ( C ( 2 9 l 6 ) ,G A M M A )EQUIVALENCE C C ( 2 9 1 7 ) ,BETA )EQUIVALENCE ( C ( 2 ~j l 8 ) , K A )EQU IVALE NCE ( C ( 2 9 19 ) ,K 1)EQUIVALENCE ( C (~~92 O ) ,~~l)EQUIVALENCE (C (2921) ,GPIT)EQU IVA LENC E (C (2922) ,GYAW)

C**AU DITIONAL OUTPUTSEQUIVALENC E CC(29014 ),XREL )EQU IVAL ENC E. (C(2905) ,XRELD )E .AJIVA LENCE (CC2;02) ,Y R E L )EQUI VALENC E. (C(2-~O3),Y R E L D )EQ UIVA LENCE (C (2906 ) , TF)

CNE AL K A ,~~1DA T A B / . 7 O 1 l /D A T A P I D A/ 1 .5 / 0 / U b/

CU E [ . T A ( f ) B E i A 4 * 2 ~~~~~ A M M A * B E T A* K A * * 2 * ( T F _ T ) + B E T A * G * *2 * (T F _ T ) * * 3 / 3 . +

(GA ~ N1~ * * * 2 * KA * * 2 * C rr~~T)~~)/l2.C r H r~~~r )~~~_ B E T A * ~~\ M M A * K A + ( G A M M A * K A * G * * 2 * ( T F _ T ) * * 3 ) / 6 . ) / D E L T A ( T )C X r ~r . L (T ) (_ B E T A * 3 * ( T F_ T ) _ ( G * G A M M A * K A * * 2 * ( T F _ T ) * * 2 ) / 2 . ) / D E L T A ( T )CXR:-.Lu(T )~ ~.

_ 3E fA * .3* (TF_T)**2_C*GAMMA*KA**2* (TF_T)**3/2.)/DELTA (T)CCJUIDA~-.:E CI~ ~AL Si4P1NGC 1G UI~~A~~ .z. S~~IT C r L N ~iQSL)

~Lk~~f) 4N2*(4NU* (WLAMQ WL Q S ) 2. *WLQSD) WLRS ) 2 . *W L R S D )

3N C~~LQ3U /WL + WLQS ) + Q B I A S + G B I A S+ R B I A S

IF ( Ti< R Z . T . O . .A NA . T.GT.TDY) GO TO 40.

+ -~L3.L 1~S + QDNI F ( CA G E .~ . 0. .AND. T.GT. TDY) WQC WLAMQ + Q B I A S + GBIAS4 IF ( T K R Y .GT . 0 . ) GO TO 5~L H S D D 0.W R C R B I A 3IF ( CA G E .CT. 0 . ) WRC WLAMR + RBIAS

5 C UNr INUE

24

~LQSSD WN1 (WQC - WLQSS )WL RSSD W N 1 * ( W R C - W L R S S )I F (W N 1 .GT . 0 . ) GO TO 3WL QSS = W QC~4 L R S S W R C3 BLQSSD W LQSSB L R S S D = W L R S S

CC RA Tt ~Y R O D Y NA M ICS AND LIMITI NG

B D R F T D = ( C D R F T 1 * A N G X + C D R F T 2 )B T H T S = B T H 2 + B D R F T

BPSIS=_BPS 1 +CDRFTY*I3DRFTBPHIS=_ BPH 1+CDRFT X * BD R FTB P H I S D = WP (W Q C O S D ( B P H 1) _ W R * S I N D (_ B P H l ) )

* *SIND (.8p31)/COSD (..Bp31)BPi-IS = BPHISD / ~4 L X X 2 + B P H I S6 IFCGNS .LE. 0 .) GO TO 8SNP2 WSlWS2(GNSSPHI~~SNP0) (WS 1 +WS2)SNP1SN Q2 ~Sl*WS2* (GNS*STHT_SNQO) (WS 1 WS2)*SNQ1SNR2 WS1*WS2*(GNS*SPSI_SNRO) (WS1+WS2)*SNR1BP H S = BPHS S W P 1BTHTS BTHT S - SN Q 1BP S IS BPSIS - SN R 1

8 CONTINUEBXX BP HSB TSS = BTHTSBPS S B P S I S

C* * * * * S P E C IA L CASE PROGRAMMED FLIGHTI F ( O P T C 1 . L E . 0 . ) GO TO 9BL QSSD = Q B I A SBLRSSD = 0 .BDC 0.BT:O.BP=O.IF(T .UE .1.0000 .AN D. T.LE.3.2000)BDC= 5.IF (T.GE.4.2 .AND. T.LE.6.-4000)BT= 5.IFCT.GE .7.6000 .AND. T .LE .9.9000)BDC=5 .IF (T.GE. 11.0 00 .AND. T.LE. 13. 100)BP:5.IF(T.ur. .15.200 .AND. T.LE .17.300)BDC ~.IF(T .GE .18.320 .AND. T.LE.20.560)BT = 5 .IF(T.GE.2 1 .545 .AND. T .LE .23.675)BDC=5.IF (T.GE.24.((O .AND. T.LE .26.910)BP=5 .C(520)~~BPCOs-bPB TbPC2 18. 18*20. o(CBDCBPCO ) ( 18. 18+20.57 )*BPC 1BXX UXX + BPCO13 rS3~ BTSS-BTBP SS~ BPSSBPC

C9 CO NTINUE

C***NE~ O P T I M A L C O N I R O L FOR P I T C H , YAW SIGNALS1/W iX REL = R TX E RXEYR EL =R TY E R~~EXREL D zVT X EV X EY R ELD~~V T Y E V Y EC ****E 3TI~1ATE OF TIME-ID-GOT3 x = A 8 S ( X R E L / x R E L D )

71 TGUZ~~A B S ( R i E / V Z E )88 C O N T I N U E

~

rn -

~

.--

~

_ _ ._~ _ _ __~t _.___ _~ _

ri -~~

--- -

~~

-

~~

- -

~

- - ~~~~~~~~~~~~~~~~~~~~~~~ . -~~~ _L~~~~~~~~~~~~ rT~~~r-- -~~~~~~~~~~ -- ~~~~~~~~ --~~~~~~~~- - -- - -~~~~~~~~~~~ - - ~~ UII~~

TGO: AMIN 1 (TGOX ,TGOZ )TF = T + TGOBTHTR=BTHT/57 .29578BJS=CTHTA(T) (P102+BTHTR)+CXREL (T)XREL+CXRELD (T)XRELDBJS=-BJS

BKS :0 .B J J = B K S - E 3 J SBKK=BKSBJSBJJSSS= GP IT * BJJBKKSSS=G YAW BKK

CCGUIDA NCE SIGNAL SHAPING AND LIMITING

BXXSD = BXXSPBXXSDD = WXX* (WXX* (BXX BXXS ) - 2.DXXBXXSD)BXSD=CXSDB X X S S = G X X * ( ( B X X S D D + ( W L X X 1 + W L X X 2 ) * B X X S D ) / ( W L X X 1* W L X X 2 ) + B X X S )BXXSSS = BXS

CSIGNAL LIMITING10 BJJSSL=BJJSSS

B KK S S L B~~KSSSI F ( B J J S S L . G T . H i 3P ) B J J S S L: H 1 3P1F (BJJSSL. LT . Hi 3M) BJJSSL:H 1 3MIF ( B K K S S L . G T . H 2 4 P ) B K K S S L = H 2 14 PIF( BKKSSL. LT . H2~4M) B K K S S L = H 2 4 MB13SS=BJJSSLB24SS:BKKSSL

CCCOMMANDS TO ACTUATORS

k BD E LTC ( 1 )=B13S..+BXXSSS13DELTC(2)zB2A4SS+BXXSSSBD E L T C ( 3 ) z B 13 S S B X X S S SB DELTC ( 4 ) :B2 14 3 3 - I3 XXSS SRETURNE N D

___________________ - -- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -

- -

3 . 4 .2 E v a l u a t i o n of C o n t r o l Gains ~t lT , GYAL

Two new c o n s t a n t s , GPIT and GYAW , tru gains in the p itchand yaw con trol channel s, were introduced in the control module Cito regula te the amplitude of the tail fin command signal. As ranbe seen f r o m Figure 3. 5 , the signal is fed thr ough a l i m i t e r a f t e rthe multiplication by the gain .

B y working w i t h f o u r s t a t e s imula t ion , opt imal valueswere ob ta ined fo r the control parameters , ~ and ~ (see section 3 .3 ) .Under the assumption that the fourstate simulation approximated the6DOF s imulat ion , it should be possible to obtain a reasonable missdis tance and impact ang le wi th the more comp lex simulation by usingthe prede termined o p t i m a l values of the control parameters . Severalruns were made w i th v a ry i n g values of the gains GPIT and CYAW. Theamplitude of the control signal had to be such tha t the tail f i n swere not at the stops throughout most of the flight , but would behard over at the end to achieve the desired attitude angle at impact.A miss d i s t ance of 18 fee t and an a t t i t ud e impact ang le of 37 degreeswas achieved wi th the fo l lowing gain va lues :

GPIT = GYA W = .000243

3. 4 .3 Control Parameter Improvement

The values of the contro l parameters y and ~ were altered(using the Hooke Jeeves Algorithm) to improve performance in the600F s i m u l a t i o n . Rather than a t tempt to incorporate the 6DOF simulatlon as a function subprogram to evaluate F(Y ,E), it was decidedto bypass the a n t i c ipated programming d i f f i c u l t y by j u s t running thesimulation any time y or B was changed and F(y,B) needed to be re-evaluated for the HJ Algorithm. Although the iterations thusperforr -ied were slow , overall it seemed to be the most straigh tforwardway to proceed .

3.5 S e n s i t i v i ty Anal ysis3.5.1 Sens i t iv i ty fo r the FourState Simulation

When the 3state control law was used on the 6DOF simulat ion,a small change in lowering the missile ve loc i ty f rom 1095 f t/ s e c to1090 f t/ s e c resu l t ed in the miss d is tance j umping to 264 . Clearly,the ques t i on of s ens i t i v i ty had to be i n v e s t i g a t e d f i r s t w i th thesimpler f o u r s t a t e simulation and then with the 6DOF simulation.

It was no t iced t ha t the i n i t i a l condi t ions for the t e rmina lgu idance p hase g iven by Equat ion (3.19) had been poorl y chosen in thesense that the target was out of the field of view , and so an al terna teset was used :

1000 , Y 1 = 5000 , P = 89 . (3 .29)

The v e l o c i ty was left the0same at 1095 ft/sec and the initial attitudeang le was increased to 89 since the missile would be flying approximatelyhorizontal after the initial preprogrammed pitch maneuver.

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ _ _

______________ _____ - z-

- rs ~~-0 0 0 00 0 0 0

t _

I I

~1)

00 l..i

(J) &) (1 (fl (f)U C --) > cs~~~~ 00

4J

_ V~1

:~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

L~LL1 I

L~] L~ L~J

_ ___ __ :L__ _~~ _ ._~_ _

- - : :~~ :~~~~-~~~~ - ~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~-- -~~~~~~ r~~~~~~~~~ : - -

- - -

-

Vary ing each of the ini t ia l condit ions ~ d

H~~, 0 , andV~1(t 0) separately, Tables 3.3a , 3.3b , 3.3c , 3.3d were- -generatedusing the fourstate simulat ion. As can be seen from them , decreases

in the acqu isiti on range are tolerable , but when it is increased to6500 the simulation becomes unstable. The heigh t can be increasedwithout det r iment , but when lowered to 800 instability occurs. Theini tial a t t i t u d e ang le can onl y be decreased by two degrees. Perfor-mance is most sensitive to decreases in velocitya decrease of5 ft/sec of velocity from 1095 ft/sec to 1090 ft/sec causes instability.Recall that the performance obtained with the 6DOF simulation considerably worsened as well when the velocity dropped 5 ft/sec.

The above information suggests that a worst case scenarioabou t which to opt imize the control parameters mi ght be:

= 1000 ~d

= 5000 , 0 = 87.5 , V M = 1080 ft/sec . (3.30)

An initial attitude angle of 87.5 would hopefully allow a largerwindow for acceptable initial attitude angles. A lower velocity waschosen as it would be more demanding on the control system . Using theHooke Jeeves Al gor i thm , the optimal control parameters turned out tobe

y = 5 4 6 0.9 9 2 , 3 .l8929lE06 (3.31)

with a r e su l t ing p e r f o rmance of

Y d (t f ) = .915 , e(t~ ) = 45 .012

. (3 .32)

The f o u r s t a t e s imulat ion was ran in doub leprec i s ion to min imize thee f f e c t of roundof f error on the resul t s . Varying in i t ia l condi t ionsone at a t ime , Tables 3. 4a , 3. 4h , 3 .4c , 3.4d were produced .

The chang es in performan ce when acquisi tion range , heigh t , orinitial attitude angle are altered agree qualitatively with the previousresults. Yd can be decreased and Ht can be increased wi th perfor manceremaining acceptable. The ingerval of0acceptable initial attitudeangles would be from about 86 to 89.5 . Tl~e one very curious coqiplete Areversal of s ens i t iv i ty is in the var iable .Yd. Previously, whenlessened , performance worsened; now , when ~d

increases , performanceworsens. With these control parameters , the fourstate simulationpredicts that the missile should not acquire the target until i ts ve loc i tydecreases below 1080 ft/sec.

3.5.2 Sensitivity with the 6DOF Simulation

The choice of GPIT , GYAW for the 6DOF simulation imp lemen tationturned out to be a very delicate one for the new set of initial conditions. Two comp le tely different types of trajectories were observedt o r var ious values of GPIT , depending on whether it was above or belowsome value close to .0004975. When GP1T was less, the missile wouldove r f l y the ta rget and the s imulat ion would te rminate when the maximumflight time parameter was exceeded. For GPIT greater than this number ,

- - T~!III~

Table 3.3. Performance for Various Initial Conditions (Baseline I)

ITable I.C. I.C. Yd(tf) 0(t f)45 Baseline I

Variable Value (ft) (deg) (denoted by * )

3.3a Yd(tO) 3000. 1.61 5.31 H~ (t 0) = 1000( f t ) 3500. 4 .63 0 . 2 4

4000. 2.83 5.03 0(t0) = 894500. 3.09 0.485000. 2.80 0.00 Yd(tO) = 5000 5500. 17.85 l27.tO6000. 4.83 2 .39 VM(t O) = 1095 f t/sec6500. Uns table

3.3b H t (t 0) 800. Uns table -y = 5535.43( f t ) 900. 3.05 3 . 3 5

l000.~ 2.80 0.00 8 = .18939E61100. 5.13 1.041200. 2 . 2 9 6 . 6 71300. 4.77 0.351400. 4.37 0.581500. 4.97 4.121600. 2 . 7 9 11.78

3.3c ( t 0) 8 5. Unstable (deg) 86. 5.95 58.70

87. 1.49 2 . 9 988. 4 .07 3 . 4 889.* 2.80 0.0090. 2.47 2.64

3.3d VM (tO) 1090. Unstable (ft/sec) 1095. 2.80 0.00

1100. 4.14 0.321105. 5.65 3.79

_____ _____ -~ --- ~~~~ -. - --- ~~-~~~~~ -

Table 3.4. Performance for Various In i t i a l Condit ions(Baseline II)

Table I.C. i.C. Yd (tf) U( t f)45 Baseline IIVariable Value Cft) (deg) (denoted by 5 * )

3.4a Y (t 0) 3000. 2.41 4.24 H~ (t0) = 1000~ f t ) 3500. 4 .13 -0.07

4000. 4 .96 0.69 9(t 0) = 87.5 4500. 12 .9 8 6 . 5 75000. * 0 .92 0.01 Y d (t O) = 50005500. 16 .24 0.5156000. Unstable V M(t O) = 1080 ft/sec

3.4b Ht(t0) 700. Uns table -y = 5460.99( f t ) 800. 17 .45 33.40

900. 124 .35 10.97 8 = .l8929E6 1000.~ 0.92 0.01

1100. 4 .33 0 .781200. 5.57 4.181300 . 3.84 0.811400. 3.30 3.861500. 3.9 0 . 0 81600. 1.7 2 5.12

3 .4c 0 ( t 0) 85.5 Unstable (deg) 86.5 k 5 .22 0.5487.5 0 .92 0.0188.5 1.23 9.7989.5 5.49 0.179 0 . 5 Uns table

3 .4d VM ( t O ) 1065. 4 .82 0.05(ft/sec) 1070. 5.84 0.87

1075. 0 .46 2.051080. * 0 .92 0.011085. U n s t a b l e

_____________ ~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~ ~~~--~~ ~~~~~~~~~~~~~ ~~~~~~~~~ ~~~-- ~~~~~~~~~~~~~~~~~~~~~~~~~~ - -- -

l

the missile would impact considerably short of the target. Given theseven digit accuracy of the computer , it was not possible to obtainreasonable performance. See Table 4.1 for a closer examination ofthis behavior. (The control parameters y and B were set at 5460.99and .l8929lE6 , respectively.)

Table 3.5 . Performance Dependence on GPIT

GPIT R X E ( f t ) R Z E ( f t ) 0 (t f ) (deg)

.000 5 1210. 1.1 -28.6

.000 499 1144. .8 2 7 . 5

.000498 945. .4 46.1

.000497 530. 5 67. t>t

.000495 529. 591. t> tx

The control parameters were then altered in the hope thatperformance could be improved and sensitivity reduced. With

GPIT = .497843 (3.33)

and the missile impacting 510.6 sho r t of the ta rge t , severalruns wer e made with various values of the control parameters y,8.As can be seen from Table 4.2 , small changes can resul t in a

/

~ total ly d i f f e r e n t t r a j e c t o r y .

Table 3.6 . Performance Dependence on y and B

y R X E ( f t ) R Z E ( f t ) 8 ( tf

) (deg)

5460.996 .l895E6 510.6 .5 625461. . l895E 6 536. 380. t~ tmax5460.996 . l9 l OE 6 118. 178. 27 . 95460.996 .1920E 6 536. 408. t~ tmax

Desp ite the small miss dis tance and excellen t impac t angleachieved w i t h the f o u r s t a t e s imulat ion , it proved to be impossibleto achieve reasonable performance through an appropriate choice oft he gain p laced in the p i t c h and yaw channels. Since vary ing thecontrol para mete rs was unsuccessf ul as well , it was decided to re turnto the alternate set of initial conditions for which the missile was14 above cro wd w i th an attitude angle of 38.10 below the horizon.

32_ _ - - -- ---- - - - ------ ~~-- - ----- ~~~~~~~ - - - - - - - - * ----~~ -,- ----~~-

_______________________________ _____ - -~~.=-~~~~~ -. ~~~~~~----- -- =-~~~~_~~~~~~~~=--- ~ _~~~~~~~

Initially, the control parameters were

y = 5525.508 , B = .l9E6 , CPIT = .243E3 . (3.34)

Using the Hooke Jeeves Algorithm , the miss distance was reducedto 10.4 with an attitude angle of 38 .2 when the con trol para-meters were 5525.45 and .l6OE6 , respectively.

Different runs were made on the 6DOF simulation with variousinitial conditions. Table 3.7 contains the results~ The f i r s t linerepresents the baseline case.

Table 3.7. 6DOF Simulation Runs for Various Initial Conditions

Yd (tO) H~ (t0) 0(t0) VM(tO ) RXE RZE -

0( t f )

5000. 1500. 5. 1090.856 .3 10.4 38.14500 . 1500. - 5. 1090.856 1.6 6 8 . 9 44 .5500. 1 500. 5 , 1090.856 1065.3 1.2 77.55000. 1400. 5. 1090.856 690.7 .5 4 7 .25000. 1600. 5. 1090.856 116,7 1.5 89.75000. 1500. 4. 1090.856 .7 150.4 32.45000. 1500. 6. 1090.856 495.1 .9 41.8 5000. 1500. 5. 1085.856 .9 127.1 35.55000. 1500. 5. 1095.856 448.7 1.2 43.4

/ ~~~ -

As can be seen from Table 3.7, performance is very sensitiveto any change in the ini t ia l condition . Such sensitivity was notobserved with the fourstate simulation . This is probably due to thefact that in the simpler simulation there was no hard constraint 8nthe control ler u ; whereas , in the 600F simula tion a bound of 10existed for tail fin disp lacement.

Figures 3.6 and 3.7 represent the baseline trajectory.

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - A - -

__________________ - - - ~=s-~-~ ~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -~

-: ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ --~ ~=~------- ~~~~~~~~~~~~~~~~~~~~~~~~ - ---i --

0I II ~~

* I + (V~* *

*

* *** c,J

* 0** *S

* 0~* I** * I I*

* * (4* 0** 5

U

* * a..** a-.

* I** *5 0r

** +0

S ~rl*

S3 - I5 *

* Y~* 0

** l:r~ * 5 + 0r 0

*** I5 *

_

~~~~~~~~~~~~~

I::

_ _

- -i - - ~~~r~~~~~~ -

-~~- -~~ ~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

0

* I +~~~ * * * I 0 * * * I* I

* * I* I5 * 5 I 0

* * * I 0 * 5 *5 - 5 -4 1 ~~5 * 4- I + 05 * 4 -~~

* 5 I5 * I a- I* U S I 0

* 5 I 05 * 1

* I ~~U I + 0 0 * I C\J

5 * I* a-.5 * I

* 5 * I5 * I 0

* * I 0* 5

* I* 1 + 0

* I 0* I* I 0* I

* I* I 0

* I 0* I* I ~~* 1 + 0 ~~

5 * I 0* I 0 * I

* 0* I 0

5 * I* I L.~* 1 + 0

I ~3~

I ~-r1~ * I- - - - - * I 0 N-* I 0

* I* I ~~* 1 + 0

* 5 I 05 * I I I 0

* IS I* I 0* I 0

* I* I ~~5 * 1 + 0U _ I I (\J

5 * I5 * I -0* 5 I

5 * I5 * I 0

* I 0* 5 * I

* 5 * I (..~* * 1 + 0* I 0* 5 I

* 5 * I Lt1* * I* * IN- * S . I 0

5 * 5 I 05 5 I

* 5 I ~~* I + 00

0 .- .- ~- .-0 0 0 0 0

Lz3O U. N-0 0 a..

U- .- ,-I I I

Li ;: - - - - -

-, ~ r ~~~~r~~~~~~~ -~~c - - -

Chap ter IV

CONTROL LAW IMPLEMENTATION PRACTiCAL CASE

4.1 Disadvantages of the ThreeState Controller

Only one of the three states used to control the missilecan be measured , the attitude angle 0. Both range

~d and range

rate are not available as there is no radar or other such measuringdevices present. The laser guided missile can detect the reflectedlaser beam and would have A (the time derivative of the lineofsightangle A ) approx imated by the seeker. This alternate set of statevariables (A ,A ,e) would seem to be more attractive than the set

because of the possibility of using actual physical measurements.

4.2 A Controller Using A , A

Figure 4.1 gives the relationshi p between the various state- variables.

MISSILE_II ~~~~~~~~~~~~~~~~~ T Q ~~MU SSI UTARGET

Figu re 4 .1. Missile Target Geometry

Using F igu re 4 ,1, the follow ing equation is obtained.

= Httan (8+ X ) . (4.1)

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

- -_- T~~~~~~~ ~--- ~~~~~~~~~~ -~~~~ ~~~~~~~~~~~~

--- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~

--~~~~

- -

T)ifferentiating,

= H~ sec

2 ( +~

) (~

+~

) + t a n ( 9 + A ) H~

. (4.2)

Substituting, the control law becomes of the form

u = f ( ~~,A ,G ,8h~~,H~ ) 4.3)

where

f ( A ,~~,0,~~,H ,H ) = L1H~ tan (+ ) + c2 (H~ sec2 ( +

~) (O +A ) ( 4 . 4 )

+ t a n ( +~ ) H ) + c30

4 . 3 i m p l e men t a t i o n ot the A , \ , 0 C o n t r o l l e r

i n i t i a l l y, all s t a t es were assumed to be known. Underthis assumption , the following performance was obtained:

= **** , P(tf) = **** (4.5)

As fo r i m p l e m e n t i n g the two pa i r s of va r iab les ( A , andone of ea ch pair will be measured phys icall y , and the other memberwill be appr ox imated by some appr opr ia te signal processingeitheran integration or differentiation network.

~ %~_ As for the last pair , neither H~ nor Ht can be measureddirectly. rt possible, Oowever , to approximate H~ , if the a t t i t u d eand veloci t \ of t h e m i ssile are known , by using

= V~~~cos 0 . (4 .6)

If the initia l height is known , Equation (4.6) could be used toupda te t he , s t i m a t e of H~~. The only way t h a t t h i s migh t be donewould be in the osign (If tne preprogrammed pitch maneuver.

There seems to he no way to avoid the need for range andra:~ t~ rate information. Equation (4.6) seems to come as close aspossible to drawing upon variables that can be measured. The velocityV~1 is not rne, o~i red but does behave nicel y throughout the flight , almos ta linear function of t. Perhaps a profile of V,1 could be stored foror generated during the flight.

Note: 1h~ ~~~~~ *~~~ appearing in Equa t ion ( 4 . 5 ) have been inser tedm- m n - o the eode h ad some errors in it, These numbers shouldbe close to those previously obtained since the two formulations are mathematically equ ivalent. This approach wasnot followed up due to the sensitivit y problem discussedin Chapter III . It was felt that it was importan t tooutl ine the approach to be used .

LL~~~ _ _ - -~~~~~~~- -~~~~~~ - - ~~~~~~-- -- - ~~~~~~~~~ -~~~~~~~~

- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ _~~~~~ fl.__ __ I~~ _ -

4.4 Estimation of Timetogo

The problem of es t imat ing timetogo that is needed forthe control law remains even if all other problems are resolved . Afixed estimate could be used , but it would have to vary depending onthe initial conditions.

A second approach migh t be to use H and ~t

but since thesetwo variables are being estimated , this metho~ probabl y would no t faretoo well.

A third alternate, perhaps, is to use the seeker, If theintensity of the reflected laser beam can be measured , along wi ththe rate at which the intensity is varying , then an estimate of

timetogo can be obtained,

Intensity I is inversely proportional to the square of thedistance f rom the missi le to the target , the slant range R. Thus

- - I = k/R 2 . (4.7)

Differentiating with respect to t ime y ields ,

dI/d t = -2kR 3dR/dt . (4.8)

Hence ,

timetogo = dR/ dt

=

dI/ d t (4.9)

~----

~~~~~~ -- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - ~~ --i --~~ --

Chapter V

CONCLUSIONS ANt ) RE C0~~1ENDATI0NS FOR FUTURE STUDY

Even assuming p e r f e c t knowled ge of the s t a t e s ~d ~ d ~~the best pe r fo rmance that could be obtained wi th the 6DOF simulation

was

RZE = 10.4 , 0(t~~) = 38.1 . (5.1)

When one considers tha t the center of mass of the t a rge t is about5 above ground , the performance is very close to satisfying thec r i t e r i a fo r success . When actual measurements are used , one canonly expect performance to deteriorate.

It must be L pt in mind that the above was assuming t~atall states could be measured exactly. When actual measurements areused for some of the variables and estimates for others , the performance~iili degrade. If the threestate controller is to work , these errorsshould be qui te small.

The other observat ion tha t needs to be made is that t h ep e r f o r m a n c e is extremel y sensi t ive to changes in the initialconditions when the target is acquired. Even if the miss distanceand attitude angle at impact were zero , this sensitivity would de fea tonv practical imp lementation of the control law. Small changes inthe control parameters also resulted in large changes in performance;that is, the assumption of continuity that was made in determiningthe control parameters is not valid for the actual system asrepresented by the 6DOF s imula t ion .

The onl y conclusion tha t can be reached is that the threes t a t e c o n t r o l l e r cannot be s a t i s f a c t o r i ly imp lemented in the 6DOFs imu la t ion . The main exp lanat ion the au tho r s f ee l is that there isa hard cons t ra in t on the controller in the DOF simulation , but notin the f o u r s t a t e s imu la t ion . This , more than the assumptions ofneg 1ec ~ing the angle of a t t a c k , or no lag in the au top ilot , d ic ta testhat the imp lernLrit ation will not prove to be sa t i s f a c t o r y . Thecontrol parameters can be a d j u s t e d so that a successful flight isachieved fo r most set i - of i n i t i a l condi t ions , but the extreme sensitivitymakes th is imp lementa t ion of no value.

The authors feel that it would be worthwhile to reformulateand solve t h e c o n t r o l problem so th a t a hard cons t ra in t is present ont h e controLie r. t-uch a controller (which will be of the bangbangv a r i e t y ) shoui h be imp lemented , assuming again that all 3tates canbe ncasured , to verif y whether or not the presence of the hard con-strain t on the t a i l fin deflection is the main culpr it for thesensitivity problem. If successfully implemented , then the problemof practi ca l implement ation of .;uchi a control law could be examined .

_ _ _ _

-- ~~ _~~~:L~~~_ ~~~~~~~~~~~~~~~~~ .- - ~~~~~~~~~~~~~~~~~~~~ - T

- - - - - ~~~~~~~~ - - - ~~~~~~~~~ - -

Chap ter VI

SIMPLIFIED 6DOF SIMULATION

This listing is complete except for the omission of subrout ines COSD , SIND , ATAND which are the same as the FORTRAN librarysubroutines but work with degrees instead of radians.

The old subroutine DUMMY has been relabeled TIMEV . TIMEV was placed as an entry poin t in DIJMMY bu t the presence of argumen tscaused a syntax error which was most easily correc ted by renamingsubrout ine DUMMY.

Since the altered simulation does not have the randomerror disturbance capability, the main program could be shortenedby omi tt ing the second half which is the statistic al analysis par tfor Monte Carlo sets.

__________________________ - ~~~~~~_ -~~~~~~~~~ ~~~~~~ - - ~~- - -~~~~~~~ ~~~ -- -~~~~~~~~~~~ ._~_~~~~~~:i-~~;::i: ~~~~~~~~~~~ .- --- - - - - - - - - - - - - ~~~ - -~~

CC*****DIMODS TO BE USED WITH FORTRAN AMRK INTEGRATION ROUTI NEC

COMMON C(3830) ,GRAPH (3OO ,~4)R E A L KA ,K1COMMON/CE PASS/X ( 1 0 0) , Y( 10 0)EQUIVALENCE (C(2662) ,H:-~IN ) , ( C ( 2 6 6 3 ) ,HMAX ) , ( C ( 2 b 6 ~4 ) , D E R ( 1 ) ) ,C (C(2561 ),N ), ( C ( 2 5 6 2 ) , I P L ( 1 ) ) , (C(2965) ,VAR (1)) ,

C ( C ( 2 ( J J ) , T ), ( C ( 2 0 1 1) ,KSTEP ) , ( C ( 2 0 1 0 ) ,STEP ),C ( C ( 2 0 1 2 ) ,LSTEP ), (C(2008) ,PLOTNO ) , ( C ( 2 0 0 9) ,NOPLOT ),C (C(2023) ,OPOINT) , (C(2025) ,TIME (1)), ( C ( 2 3 2 5 ) , VLA I3 LE ( 1 , 1 ) ) ,C (C(3167) ,N OO U T ), ( C ( 2 0 2 2 ) , O P T N 1 O ) , ( C ( 2 0 0 6 ) , R E P P L T ) ,C (C(2d65) ,EU (1)) , (C(2765) ,E L ( 1 ) ) , (C(2007) ,PTLESS)EQUIVA LENC E (C(1971) ,RITE ), (C(1972) ,RKUTTA )E QU I V A L E N C E ( C ( 1 j73),KASE ) , (C(197~4 ) ,NJ ) , (C (1975) ,NPTEQUIVALENCE (C (3512) , I S G C T ) ,(C(3721) ,I T C T) ,(C(3511) ,RNSTRT)EQUIVALENCE (C( 21) ,IbVNSW )Ec~jIVALENC E (C( 22), IPLOT)EQUIVALENCE ( C ( 2 3 ) ,X LA M BD )EQUIVALENCE (C( 2~4 ) , K S S I G )EQUIVALENCE (C( 25),CEPSIG (1))EQUIVALENCE (C(3 00),R M I S S )UIMLNSION RMISST(lOO)EQUIVALENCE (C(1000) ,RMISST(l) )~.QUIVALE NCE (C(301) , L)E~~U I V A L E N C E ( C (3 0 2) , RYE)EQUI\1ALE NCE(C( 19) ,P S I Z E )EQUIV AL ENCE (C (303) , RZF)E QU I V A L E N C E ( C ( ~~1) ,L C E P )EQ-~ I V A L E N C E ( C ( 3 3 2 5 ) , N C A S E ) , ( C ( 6 2 5 ) , IE3L)EQUI VA LENCE ( C ( 2 6 6 2 ) ,D E R S V )E QU IV A L E N C E (C(3000) ,V S D ( 1 ) )E.QU IVALENCE (C(3 010) ,VMtI AN (1))r IJ IV A Lt N Ct . (C(302 0), I M V N D X ( I ) )L - ~JIVA LENCE (C(3 030), I M V C T )~EAL KSSIGL~~FEG ER C E P S I U0ll4LN~~lO N CE P S I G ( 6 )D I M E N S I O N TIME(300)D I M E N S I O N V L A D L E ( 2 , 15) , IPL (lOO) , DER(101)DIMENS iON V A R( 1 O 1 ) , EL (100) , EU (100)DIMENSIO N iNV ~ D X ( 1 O) , V M E A N ( 1 0 ) , V S D ( l O )E Q u I v A L ~ ; N c E ( C ( 1 b 3 0 ) , hN )EU U IVALENCE (C(1981) ,R N T )~,IU IVALIJ ~CE (C(1982) ,PLOTNI4)t QUI VA L ENCE (C(1983) ,PLCTU2)ELU IVALENCE (C(19c ~4) ,NPLOT )C~ -,,JtvAL E .~CE (C(2020) ,LCONV)EQ U I V A L E N C t C ( 1 6 5 1 ) , RTXE) ,(C(1655) ,RTYE) ,(C (1659) ,RTZE) ,

. ( C ( h t , 1 5) ,l~c X E ) ,( C ( l 6 1 9 ) ,R Y E ) ,( C ( 16 2 3 ) , RZ E )I N T : - ; E R OPO INT~;~ rE~~h~R - JPT~ X t E k:~A L A U X i U~3C * ~ ~~ ~~~~~~~~~~~ * * * *d $ * * *0 0 0 0 t ~KU CHANGE**************1)0 1 1 1 1 I 1 , 3d3 0

1 1 1 1 C(I) 0.0C * * * * * h $ * * * * * * I * * * * * * * $ * * $ * * * W KU C I IA N G E* * * * * * * * * * * * * *

I o C 1 :0

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ISGCT~~1ITCT:OITCT ~ 1MOJL~~-1I TSNDX 0

CCC THIS CALL TO S U B R O U T I N E R A N N U M IS TO PERMIT USE OFC DIFFERENT RANDOM N U M B E R GENERATOR STARTERS ( I R N S T ) .C IF SUBROUTINE NORMAL .13 CALLED WITHOUT FIRST CALLINGC RANNUM , THE RANDOM NUMBER SEQUENCE WILL ALWAYSC BE STARTED WITH THE SAME NUMBER (ENTERED AS A DATAC STATEMENT IN SUBROUTINE NORMAL ) , WHICH WILL RESULTC IN THE SAME SEQUENCE ALWAYS BEING GENERATED.

L~~~~1NP 0CALL COUNTV

1000 CAL i~ Z E R O1001 I F ( P L O T N O . L E . O . ) G O T O 7I F ( R E P P L T . G T . 0. )GOTO(

CC REPPLT 0. USE NEW NO. 11,7 ( DISCARD OLD)C 1. USE OLD PLUS THOSE ADDEDC 1 . USE NEW NO. 7 ( DISCARD OLD)

IF ( RE PPLT .GT . 1 . 0 ) NOOUT 0N PLOT :O

7 CALL OINPT 18 C O N T I N U E

CALL RA N NU M ( 1 . ,RNSTRT ,DU M )I F ( IS G C T .G T .0 ) CALL MCARLO (DUM , 1 , IDUb-1 )C (1976) 1.K A SE :0L S T E P STEPN PLOT11:PLOTNLIN P L O T2~~PL O TN2NO PLOT P L O T N ONO P LOT :N PL OT

1002 CALL SUBL 11003 C A L L AUXI10011 CALL SUBL21005 CmJ~4TINUE

D E R ( 101 )~~DER ( 1)IF(DEH( 1 ) . G T . D E R S V ) D E R ( 1 ) :DERSVC ( 1 -~7 6 Y 1.1006 C A L L A U X S U3

1007 N J ~~N 1CALL A M R K ( A U X S U B )1 008 CONTINUE1009 CONTINUE

H DE LX ~RTXE R XER D E L Y : R T Y E R Y ERDELZ R T Z E R Z EIF(RDELZ .LT. 0. .OR. RDELX .LT . 0 .) LCONV :2CALL SUBL3IF ( KSTEP .EQ. 1 ) GO TO 1007

CCCCC

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CC*****************SAVE MISS DISTANCE FOR EACH RUN OF THE MONTE CARLOC R U N SET********************************************CCC

I F( L C E P . N E . 1) GO TO 20LCEP:ON P N P + 1X ( N P ) z RY FY ( I J P ) : HZFR~1ISST ( NP) : R M I S SGO TO 21

20 CONTINUEWRITE (6 ,805)

805 FOr~MA T ( 1 H 0 ,/ / /)W R I T E ( 6 ,22) L22 F O R M A T ( 1 H O , 1 1 X ,2 bH*$* $*$ WARNING RUN NUMBER ,I3,2 371-1 DID NOT INTERSECT TARGET PLANE*$*$*$ ,/ ,3 // ,26X ,38H$*$*THIS RUN DROPPED FROM DATA SET*$*$)WRITE( ,805)

21 CONTINUEL :L + 1

CCCC~~ ~~ C O N T R O L P A R A M E T E R O U T P U T

P R I N T -466 , C ( 2 9 1 6 ) , C ( 2 9 1 7 ) ,C (2918),C(2919),C(2920),C(2921)1166 FORMAT(1UO , GAMMA : ,E15.6,11X , BETA : ,E15.6,/ ,

1 KA : ,F 10,6 ,~4X , Ki ,F1O .6 ,4X , W i : ,F10.6 ,/,2 GP I T ,F 1O .6)THOFF C (350)115~4RITE(6 , 12 1 2) C (35O) , THOFF1 ~~~ F O H M A T ( O T H E T A MISSILE IN DEGREES : ,E15.7

- ,/ , VARIATION FROM DESIRED 145 DEGREE IMPACT: ,E 15 .7 ,/ / )CC * * * * * * * * * W K U PLOTTING RO U T I N E* * * * * * * * *C

UML L ~KPLOTCC * * * * * * * * * E N D WKU PLOTTING R O U T I N E * * * * *CCCC

CALL PROCEAi~ (O PTN1 O .GT. 0.) CALL DUMPO

C A L L R E S E T1F (L$TEP. EQ . 5. OR . LSTEP. EL .7 . O R . N O P L O T . E Q . O ) G O T O SC A L L T I M E V ( D E L T )WHITE (6, 96 ) DELT

96 FOXMAT( 1H , 17iISTAHT PLOTTING AT ,F111 .7)L E -~~ ~~ PT : PT LESSOPJINT :OPOINiLESSPT

CCHLL P L O T 1 1 ( G R A P H ,O POINT ,V LAB LE ,T IME ,NPLOT L4 ,NPLOT2 ,NOPLOT )U,~~L PLOT2UAL.L PLOTNCALL TIMEV(DELT)W R I T E (6 , 97 )DELT

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97 FO~ MAT( 1H ,18HPLOTTING ENDED AT ,F i4.7)5 GO TO (1000 , 1001 , 1 002 , 1003, 10011,1005 ,1006 , 1007, 1008 ,1009, 1 0 1 0 ) ,1 L~~TEP1010 CONTINUE

CCC~~*** MEAN VARIANCE AND STANDARD DEVIATIONCCC

W R I T E ( , 100)100 FORMAT( 1H1 , 13X ,36HMEAN VARIANCE AND STANDARD DEVIATION! ,

1 7X ,1OHCLOCAT ION ,9X ,8H M EAN VA R , 19X ,7HSTD DEV/)DO 120 I:1 ,IMVCTILOC : IMVNDX ( I )WRITE(6 , 1 0 2) ILOC ,VM EAN (I),VSD(I)

102 F O R M A T ( 1 O X , I5 , 8X , E 1 5. 8 , 1 1 X , E 15 . 8 )120 CONTINUE

CCCCCC************MONTE C ARLO AND CEP LOGIC F O L L OW S * * * * * * * * * * * * * * * * * * * ** * * * * *CCC

IF(NP. LT. NCASE. AND. L.LE. (NCAsE +5 ) )W R I T E ( 6 , 807)I F ( N P . L T . NC A S E .A N D . L . L E . ( N CA S E 5 ) ) G 0 TO 8

807 FORMAT(1 H1 ,3(/) , 39H THIS RUN ADDED DUE TO BREAKLOCK ,3(/))J :0.4HITE(6 , 800)

800 F O R M A T ( i H l , 96X ,1O1-I YM I SS ,1OHZMISS , 1OHMISS DIST I ,1 6X , i 2~~ 2 m 3 )00 801 I:1 ,NPJ J +1WRITE(6 ,8O2) X(I) , Y ( I ) ,RMI SST (I)

802 FORMAT( 6X , 1OH 1 , 10l-i 1 ,1OH 1 , 1OH 11 , 101-1 1 , 1OH 1 , 1OEI 1 , 1OI-l 1 , 1OH 12 , 11 -l 1 ,F9 .5 ,2H 1 ,F 9 .5 ,2H 1 ,F8.5 ,2H 1 )

WRIT E (6 , 8 0 3)803 FORMAT( X , 12 3 H

2 3 )IF (J.GT.30) WRITE (6 ,8 0 0)IF (J.GT.jO) J : 0

801 CONTINUEIF (IBL .LE. 0)GO TO 8011L :L 1X I dL~~IB LXL:L~ A T 10: XI B L / XL

W R I T E ( 6 ,8 O 6) IB L ,L ,RAT IO8-~u FORMAT( 111 1 , 1 5(/ ) ,1X , 1O ( 1 1 H* B R EA K L O C K * )/, 1X , 1 1I1*B REAKLOC K* , 3 X , 16HTHIS RUN SET H A D , 111, 2 S H B R E A K L O C K

.FLIGHTS OUT OF ,111,23FIG IVING A PROPORTION OF . , F6 . 11,11X ,11H BREAKLOC

.K/ , 1X , 9HBREAK LOCK , 3 X ,17HTHIS RUN SET HAD ,I4,214HBREAKLOCK FL

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.IGIITS OUT OF ,I11 ,2 3 H G I V I N G A PROPORTION OF ,F 6 .4 ,L4X , 9HBREAKLOCK ,/ ,1X ,11H*BREAKLOCK* ,88X , 11H*BREAKLO CK* ,/ ,1X , liH *BREAKLOCK* ,88X ,11H*BREAKLOCK* ,/1X ,iO( 11H*BREA KLOCK* ))8u11 CONTINUE

CALL CEPAS (NP, I BV N SW ,IPL OT ,XLAMBD ,KSS IG ,CEPSIG ,PSIZE)11668 COLITINUE

CALL EXITSTOPEND

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SUBROUTINE AMRK (A UXSUB )COMMON C(3830)DIMENSiON CSAV( iOO ) , IPL (iOO)REAL K1 (100), K2 (iOO ) , 1(3(100), KLl (iO0)EQUIVALENCE (C(2000), T)EQUIVALENCE (C(26614), DELT)EQUIVALENCE (C(19711 ), N J )EQUIVALENCE (C(2562) , IPL (1))EQUIVALENCE (C(1976) ,XNDRK)XNDRK : 1 .DO 1 I : 1 ,NJJ : IPL(I)

CC****STORE INITIAL VALUES

C S A V ( I ) C ( J + 3 )CC*** COMPUTE K 1

K 1( I ) D E L T * C ( J )1 C(J-s-3) : C S A V ( I ) + . * K 1( I)

T : T + . 5 * D E L TC ALL ~UXSUBC

C*** C~.M P U T E K2DO 2 I : 1 , NJJ IPL (I )K2(I ) DELT*C(J)

2 C(J~~3) CSAV(I) + .5*K2 (I)C A L L A U X SU B

CC*** COMPUTE K3

DO 3 I 1 ,NJ- .- - J IPL (I)

1K 3 (I) DELT*C(J)C(J+3) : CSAV(I) + K3 (I)T T + 5*DELTCALL AUXSUB

CC*** COMPUTE 1(11

DO 11 I : 1 , N JJ z IPL( I)K4 (I) DELT*C(J)

L4 C(J 3) (.3AV(I) + ( K 1 ( I ) + 2 . * ( K 2 ( I ) + K 3 ( I ) ) + K L I ( I ) ) / 6 .X N D R K : 1 .CA LL A U X S U BRET U R NEND

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S U B R O U T I N E A U X ICO;IMO N C ( 3 8 3 0 )EQUIVALENCE ( C (2 36 1 ) ,NOMUD ) ,( C (2 36 2 ) ,X M O D N O ( 1) ) ,( C ( 2 5 6 1) , N)DIMENSION XM O DNO (99 )N : 1DO 1 I:1,N O MO DL :XMODNO(I)GO TO (1 ,2,3,~4 ,5,6, 7 , 8, 9 , 10 ,11 , 12 , 13, 14 ,i5 ,16 ,17 ,18 ,19,2O ,2 1 ,22 ,231 ,214 ,25 ,26 ,27 ,28,29,30 ,31 ,32 ,33,34 ,35 ,36 ,37),L

2 CALL A llGO TO 1

3 CALL A2 IGO TO 1

14 CALL A31GO TO 1

5 CALL A111GO TO 1

6 CALL A5IGO TO 1

7 CALL CUGO TO 1

8 CALL C21GO TO 1

9 CALL C31GO TO 1

10 CALL C141GO TO 1

I i CALL C51GO TO 1

12 CALL C61GO TO 1

13 CALL C71GO T O 1

111 CALL C81GO TO 1

15 CALL C91GO TO 1

16 CALL C1OIGO TO 1

1T CALL DliGO TO 1

18 CALL D21GO TO 1

19 CALL D3IU0 TO 1

20 CALL DIIIGO TO 1

21 CALL D5 1GO TO 1

22 CALL Gi lGO TO 1

23 CALL G2IGO TO 1

211 CALL G31GO TO 1

25 CALL G~4IGO TO 126 CALL G51

GO TO 1

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~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - _ _ _

~ __.~~~~~.. ~~~~~~~~~~~ _ ~~~~~~~~~~~~~~~~~~

27 CALL G6IGO TO 1

28 CALL 811GO TO 1

29 CALL S2IGO TO 1

30 CALL S3IGO TO 1

3 1 CALL SII IGO TO 1

32 CALL S5IGO TO 1

33 CALL S IGO TO 1

314 CALL S7IGO TO 1

35 CALL S8IGO TO 1

36 CALL S9IGO TO 1

37 CALL S J O I1 CONTINUE

RETURNEND

.4-!

_ _ _ _ _ _ ~~-~~~--~~. 8

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SUBROUTINE AUXSU L3COMMON C (3830)EQUIVALENCE (C(2003 ),T),(C(2361) ,N O M O D ) , (C(2362) ,XMODNO (1 ))EQUIVALENCE (C(256l) ,N ) , (C(2562),IPL(1)), (C(2664 ),DER(l))EQUIVALENCE (C(2965) ,VAR( 1 ) )EQUIVALENCE (C(2020) ,L C O N V )DIMENSION DER (101) , VAR( 1O 1 ) , IPL (100)DIMENSION XMODNO (99)EXTERNAL AlEXTERNAL A2E X T E R N A L A .~DC 1 I:1 ,NO MO DIF ( LCO NV. EQ. 2 ) R E T U R NL :XMODNO (I)GO TO (1 ,2,3, 14 ,5,6,7,8, 9 , 10 ,1l , 12 , 13, 114 ,15 ,16 , 17, 18 ,19, 20 ,21 ,22 ,123, 2L1 ,25 ,26 ,27,28 ,29,30 ,31 ,32 ,33,3 Z4 ,35 , 36 ,37),L

2 CALL A lGO TO 1

3 CALL A2GO TO 1

4 CALL A 3GO TO 1- . 5 CALL A IIGO T O 1

6 C A L L A SGO TO 1

7 CALL ClGO TO 1

8 C4~LL C2GO TO 1

9 CALL C3GO TO 1

10 CALL C11GO TO 1

1 1 CALL C5UU TO 1

12 C A L L C 6GO TO 1

13 CALL C!GO TO 1

111 CALL CdG~ : TO 115 CALL C9GO TO 1

16 C A L L C1 0GO TO 1

17 CALL DlGO TO 1

18 CALL D2GO TU 1

19 CALL D3GO TO 1

20 CALL D 11GO TO 1

21 CALL D5GO TO i

22 C A L L o iGO TO 1

23 CALL G2

~~~ L.

___________ ~~~~~~~~~~~~~~~~~~~~~~~~ ------- - -~~- - - ~~~~~~~~~~~~~~~~~~~~~~~~~~~

GO TO 1211 CALL G3

GO TO 125 CALL G11

GO TO 126 CALL G5

00 10 127 CALL 06

GO TO 128 CALL S i

GO TO 129 CALL S2

GO TO I30 CALL S3

GO TO 1.11 CA LL SLI

GO TO 132 CALL S5

GO TO 133 C E L L S6

0 0 10 1314 CALL S7

GO TO 135 CALL S8

GO TO 136 CALL S9

GO T O 137 CALL S1O

1 CONTINUER E T U R NEND

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Bl~O U T I N E AlCOMMON C (3830CC**TABLE LOOKUP FOR AER O COEF

COMMON* / N C 1Z / N C 1 ( 2 ) /NC2Z / NC2 ( 14 ) / N C 3 Z / N C 3 ( 1 1 ) / NC5Z/N C 5 ( 14 )*/CA1 /VM1 (l5 ) /CA2/BA2(6) ,VM2(6) /CA3/L3A3 (7),VM3 (5)* / C A 5 / L 3 A 5 ( 7 ) ,VNI5 (3)/CZPFZ/CZPF (-~5) /CZ2FZ/CZ2F (35) /CMPFZ/CMPF(35) /CM2FZ/CM2F(35)*/CYL4FL /CYL4F(36) /CNLIFZ/CN4F(36) /CLLIFZ/CLIIF(21) / C L 2 F Z / C L 2 F ( 2 1)*/CZDFZ/ CZD F( 3 5 ) /CMDFZ /CMDF(35)*/CM QFZ./CMQF (36) /CLPFZ/CLPF (36) /CLDFZ/CLDF(21)*/CXOFZ/CXOF~ l i )COMMON /CMOFZ/ CViOF (6) /CAIIZ/CAI4 (6)COMMON /NC6Z/NC6(4) /C