GUI DEDWEAPONCONTROL SYSTEMSSec ondEditionBYP. GARNELLPrincipal Lecturer. RoyalMilitaryCollegeof Science.Shrivenham. Swindon. Engl andBRASSEY' SDEFENCE PUBLlSHERSaMemberofthePergamonGroupLONDOi'iTORONTOINf\SHINGTOND.C.SYDi'iEY . PI\RISNEW YORKFRf\ i'iKFURTCONTENTSPreface t o tneSecond Eci ti onCHAPTERI THEACCURACYOFTARGET TRACKERS1. I Int roduct i On1.2 Some objecti ves wi t hfeedbac k1.3 Some general concepts onaccura cy1.4 A tracker ser vo1.5 Tracking acc uracyin t heabse nceof noi se1.6 The effect of t hermal noi se1. 7 Theeffect of other inputs and di sturbances1. 8 A self opti misi ngservoCHAPTER2MI SSILE SERVOS2. I Servo requirement s2.2 Sto red cold gas Servos2. 3 Ho t; gas se rvos2.4 Ramair servos2.5 Hydra ul i c ser vos2. 6 Ele ct r i c servos witn d.c . mot ors2.7 Ot her elect ric servos2. 8 Some tentat f ve concl usi onsCHAPTER 3MISSI LECONTROLMETHODS3.1 Introducti on3.2 Why notbybanking?3.3 Ro ll control3.4 Aerodynamiclate ral control3.5 Aerodynamic pol ar cont ro l ve rs us car tes t ancont rol3.6 Thrust vect or cont rol3.7 ftethods of t hr ust vect or i ngCHAPTER 4 AERODYNAMIC DERIVATIVESANDTRA"SFER FU"CTIONS1X11348131824272829303132333537394042474850 ,4. I4. 24.34. 44. S.4.6 4. 74.8Notation and conventi onsEuler ' s equations of mot ionf or a rigi dbodyTrajectoryconsiderati onsContro l surface Aerodynami c der i vat ivesAer odynamic transfer f unct i onsAl titude andspeedconversi onfact orsAerodynamicder i vat i ves wi th v5658-00,60616573745.1 Int roduct i on5.2 Elementary t heory of gyros copes5. 3 Free Or pos i t ion gyros5. 4 Rate or constrai nedgyros.'5. 5 Accel eromet ers5.6 Resolvers5. 7 Altimet er svi6. I 6 . 2.6. 36. 4. 6. 56. 66. 76 . 86. 96 .106.116.126.136. 146.156 .166 . 17Con t ent sCHAPTER5MISSILEI NSTRUMENTSCHAPTER6AUTOPILOT DESIGNIntroductionLat eral autopi lo t des ign objectivesA late ral aut opil ot us i ng one accelerometer andone rateAdi scussionof t he "i mportant "ae rodynamic derivativesThe two- accelerometer l ate ral autop ilotThe s ingl e ra te gyro la te ral autopilotThe effe ct of canardcontrols onlateral autopilot des ignA velocitycont r ol aut opil otLatera l autopilo t s and'dispersion at launchAutopi lo ts for ro11 con tro1The effe ct of rol l rate onl at er al autop ilot performanceAutopilots and achang ing environmentAzimuth control bygyros andaccelerometersHeight cont rol and seaskimming sys t emsVertical launchautopi lotsThe effect of fin servo saturationDigital autopilotsCHAPTER7 LINEOF SIGHTGUIDANCELOOPS_ lo.... _ ~ _76 S76 S80 S83 S86 S87 S87SSSSS9294qyr-c 971101121141141181211261301341361381431461497. 1 The effect of target andmissil e mot ion onmissil e "g" requi reme nts 1527.2 Types of LOSsys tems 1587.3 Kinema tic closure and stabi l ityof the guidance loop 1637. 4 The concept of f eedforwar cte rms 1667.5 Phas ing err or andorientation difficult ies 1707.6 The effect of a di gi t al computer ins ide t he guidance loop 1727.7 Some numer ical exampl es ont heesti mat ion of guida nce accura cy 1747.8 Some general 'conclus ions onaccuracy 182CHAPTER 8HOMINGHEADS ANDSOMEASSOCIATEDST ABI LITYPROBLEMS8.1 Int r oduct i on8.2 Homing head requirements8 .3 Some e1ect ro- mechani ca1arrangements8 .4 The eff ect of radom. aberration8 .5 Isola ted sight line andmissile compensation185185188192196: _ -L- JContent svi 'CHAPTER 9 PROPORTIONALNAVIGATIONANDHOMINGGUIDANCELOOPSS-9 . I Introduction198 59 . 2A parti cular case198)9. 3The ma the matical model202 39 . 4A summaryof previous wo rk205;9 . 5The effect of "amiss i l e headi nger ror20679. 6Mi ss distance dueto a targe t l at er alecce1erati on2227g.7Mis s di s t ancedue t oangular noise2299 . 8Missdueto gl int2319 . 9Three dime nsi onalhomin g2349 . 10Ani nte grat edformof propor t io nalnavi gat ion2359 . 11Other homing guid ance l aws237INDE X243- .'_,J9 , 19 . 29 . 39. 49. 59 . 69 . 79. 89 . 99 . 109 . 11INDEXCont entsCHAPTER 9PROPORTIONALNAVIGATIONANDHOMINGGUIDANCELOOPSIntroducti onA part i cular caseThe rna the ma t i ca1mode1A summa ry of previous workThe effect of ' ami ssil e headin9 er rorMiss di st ance dueto atarget late ral dccel erationMiss distance due t oangular noiseMi ss di s t ancd due t oglintThree dime ns iona l homin gAn integratedfor m of proporti onal navi gati onOth er homing guidance lawsvi 1198198202205206222229231234235237243l ~ _CHAPTER ITHE ACCURACY OF TARGETTRACKERS. ,l.l INT ROD UCTIONA guide dmts s i l e i s one wh i cn i s usuall y fired i n a Q.i re ct i on approxi ma te ly t owe rost he target ands ubsequently recei ves steeringcomman ds fromt he guidance system t oi mprove its accuracy. Inerti a1gui dance is oft en used in me di um and l ong r angemi s s i l es (ove r 40 kmsay) when t he in tention is to hi t a gi ven map reference . Thetechniques used in such sys tems are quite different from t hos e used in mos t s hortand me diumr ange sys t ems ; mor eove r t hey have been adequate ly describe de lsewhere(1) , (2), (3) . The guidance- cont ro l systems cove re din thi s boo ka re corrmandsystems and homi ng s ystems . The re is much i n conrnon between t hes e two systems ; f orins t ance one has to track t he target in bothsystems . Incormandsystems t het r acke r is usual lystationary or movi ngsladl y (e . g. t he t arget tracker could be ona ship). In homing systems the target tracker is in the mi ss il e and i n such aca se it i s t he relat ive movement of t ar get and mi s s i l e whi ch i s re levant . Thes pecia l t racki ng problems associatedwi th homing are cons i dered in chapters 8 and 9;.so in th is chap t erweassume that the t r acke r speed i s smal l eno ugh not toin f l uence t he ki nematics of t he engagement serious l y .1.2SOMEOBJECTIVES WITHFEEDBACKThi s bookis about t he accu racyof closed-loopsystems . ands ince s pace is three -dimens i onal we have t he problem ofeli minat ing t he separation of mi s s i l e and tar get(a) down -range(b) up- downand (c) left -rightThe propulsionsystemis , of course , aesi gnec to give t hemissi le suf fi ci ent rangeto reacht he target anddoes not concernus in t his book . with t his one provi so :thepropul sion systemcanaffect t he accuracyift he m i s s i l ~ speedis not constant.If t here is a component of acce le rati on U al ong t he mi s s i l e fore and aft axis and1zGui dedWeapun Cont rol Sys tems+feecbacktransfe r funct io nt here is some body in ci denceit f ol l ows t hat t he re mus t be a component ofU s in a _p erp en di cular c Ot he vel oci t y ve ct or and this will alte r t hedire ction 0-1: t he missile ve locity. "!Q s hall have t o r eturn t o t his later.Neve rthe l es s', it i s broadly correct t o s ay th at it i s the guidance andcontrolsy stems wh ichare designe d t o no di fy t he mi s s i l e ' s t r aj e cto ry; the s igna l s t o t hemi s s il e a re us ually i n t he f ormof up- down and left-right conmands . For manyreasons t he guidance and control systems are i mper f ect :(a) Duet o t he inertia of t he target tracke r t here mus t be a tendency for theest imation of the target direction to be i nerror. Moreove r due t o t he ma ss of themis sile t he re must always be a delay i n e f f ect i nga chang e i nits pos ition .(b ) Biases and disturbances such as t hrust mi salignmen ts , win ds, var i abl e fri cti onlevels , and biases in r ecei ve rs and i ns t r uments mus t i nt ro duce anelemen t ofinaccuracy .(c) Noi s e i nt he sys tems uchas t ne rme l noi se andgl i nt will reduce t he acc uracy.It is bot h convenient andsignificant t o su b-divide items (b) and (c) above i nt otho s e i nput s , disturbances or bi es cs (th ey canall be. re garded as "tnput s " t o t hesys t em) wh i chori gin ate at t he ta rget andt hese "inpu t s "whichor iginate anywheree l se i nthe sys t em. The si t uat i orl i s ill ust rat edi nFi g1.2-1.FI G. 1.2-1 Acloned-Loop system withmor e then one inputMos t of t he closed l oops wh i chHe are concernedwith have unity f ee dbac k-rlj t h afe edfoma r d trans fe r fun ct ion of G(s ) s ay . If an6benters s onewh e re in,.the fo rward pat hwe can r egard t he feed f oma r dpat h consist ing of t HO trans fe rf unct i ons Gl (s ) G2(s ) - =G(s ) . en is a noise i nput ori gi nat i ngat t he input . Inany closed-loop sys t emoutput = ,-- -'- _inputConside rin g8ias t he only input fer t he t i me bei ng(1.2 - i)\The Ac cura cyof Tar get Tr ack er s 3-+ 1asPe rhaps it is mo re rreaningful ifweto make 6e+ O. HenceGl (s) GZ (s )de f i ne 6e' 6i- 60as the error andwewis h(1.2-3)6e6i- ao60_ 1e;-' 6i, 1- e;-- G1(s ) G2(s ) + I (1.2-Z )and this clearly + 0 as G1(s) Gz (s ) - ee i .e . we re quire as high an open lo op gainas possible. Nowconsider t he effect of a di s turbance 6b. Idea ll y t he va l ue oft he output 60due t o 6bshoul dbe zero. In t his i nstance GZ (s) is the fee dforwardtransfer fu nction and Gl (s ) is t he f eedbacktrans feri .e.60_ 1 , GZ(s )eb- I G( ) +Gl(S ) GZ(s )"G2Tsi + 1sWi t hout fee dback aoleb ' GZ(s ) andtherefo re i t must al ways be less wi th fe edback .Theeffect of 6bon t he out put - 0 if Gl (s)He see t herefore th at t here qui rerrents to f ollowan input s i c:r. d rej e ct a dist urbance 6bin general do notconfl i ct ; we re qui re G1(s ) to be as large as pos s ible . In pract i ce we wi ll beli mited by stabi lity andeng i nee r i ngcons iderations .Howeve r , such is not t he case ifan input 6 or igin at es at the s ane point as the. nt rue in put 6. . Th e t r ansfer fun ction 6 / 6 is exact l y th e s arre , as t hat fo r 60/6,.1 0 nas given in equati on 1.2-1. For 90t o- 0 for any gi ven 6nthen G1(s) GZ(s ) mus tals o - 0 andthis is clearl y t ot al l y incompat i bl e wit h t he pri me tas kof fol l owi ngt he ta rget accurat e ly . All that can be done i s to des i gn t he sys t em trans ferfun ct ion (o r f ilte r ) so th at t he mi ss distan ce due to att empt i ngto followboththe ta rge t andt he'noi se i s a mi nimum.I.3SOME GENERALCONCEPTS 0:-;' ACCURACYCons i de r t he s i mplest mode l of anangular pos i t ion se r vo, s ay a r adar tracke r ,designed to f ol l owan input 9t . It consist s of an angul ar error det ec tor of gai nk1V/ r ad, an ampl ifi er of gain kZV/V and a mo tor , el e ct r i c or e lect ro - hydraulic,wh ichwi ll produce an antenna velocity of k3ra d/s / V as s ociatedwith a t i me lagdefin ed by a t ime constant T[sorre t i me s call ed a f ir s tt orde r lag or exponenti all ag ) . The l oopis cl os e dby implied integration s i nce vi e are defini ngt he out putas OUtput position not velocity , see Fig 1. 3- 1.k,Ts+ IFIG. 1.3-1 A simpte positionservoIse,ii\l!,\rGJided WeapcnCentral SystemsIt is cl ear t hat i f any coulomb f ri ct i onor exte rn al disturbance exi sts t he remus t be s ome mi s al ignment and t his C1r. be s r:ali if the sy stem gain is highenough. Consi der now -a r an? i nput (i .e. a constant ve loci t y i nput ) defined byet=ntwhe r e n has t he dimensions rac/ s . In t he stea dystate theservowi l l develop asteady s tate e rror 6ejust l arge enough t o dr i ve the antenna at an anqul ar- rate D.The error i n pos i t ion wi l l nOd nei the r increas e or decreas e and is given byn !. I, k - P 'I - n"" et-..1"'? ' 3- " ei' -6e= n/K (1.3- 1)where K::: kikZk3and i s us ual ly re ferred to as t he ve loci t y constant; it is t hesteadystate out put ve locity f or unit position mi salignment , in t he absence of anyot her loads , bias es or disturbances . The sys t em is call e da Type 1se rvo be causet here is one i ntegrati on i n t he iee -Suppose 60=0.3 rad and vie c;::. 11Q pztfo n:-: ance no HorSEthat de f i ned bywn= 150 and u :: 0.: . Si nce th e f in vc lcci ty i s :: w90:.: 45 rad /s , orabout 450 r-ev/ mi ncne hasC::-lS1 C2r tllic.:. ccaparat i ve ly l arge amo l t t ccethz t f i ni at e haveint hat tends t ounstabi lise the aut : pil at . SiGcei f it occurs,be ma in lyto2710", frequency glint noi s e jand bread-bandwhite therma l noise, it fo llows that anaccurate esti mate of noise levels is desirable before a servo spec ification is"ritten. Hence, maximumfin rate wi l l always be defined . The l oad ine rt ia will bequot ed of course in addition to the maxi mum aerodynami c hinge r.or.nt due to thecontrol surface centre of pressure not al ways coi nci di ngwith t he axi s of r otati on.Some t hr us t vector methods involve considerable coulomb fr t ct i on due to s eals .28 Gui oe c v-IE:dPOf; Contro l Systems. .-cc-'-.-'--tc c. -,techniques and apply t oall these rvo s to be descri bed.. ActuationSTOIUC!h en t heerror is zero the springs act as brakes lo ckin gt he out put shaft .Int hi s des ign the maximumpoweroutput of th eroto r was reducedt oless t han60%of t he maximumpower demand. Alternatively , the spa ce .evai l eble for t he servocomponent s may suggest that the capstans be dispens edwit handt he ma i npowe rampl i f i cati onbe provided by magnet i c particle clutches alone ; the hea t dissipa tioni n t he clutches may nowbe a limiting f act or . However , th e fa ct t hat one cheapmotor can bes hare dbet wsent wo Orfo ur servos is aconsi derabl e advantage . Howdosol enoids for direct actuat ion of control surfaces compare with othe r forms ofe le ctri c actuation? Sure Iy f or sne e r si mplicity, cheapness and re l i abi l i ty t he rei s noequal in t he fieldof el ect ro-me chani cel energyccnversivn. Howeve r , on a...--------35pm'ler/weigrlt bas i s t hey comp a re unfavourao l y wi th,"" ,-.... '-. mo to r s . f ht funuauent a1- i-I- j- I",- ,reason whythere is a differenc e is t hat al t hough bot h devices utilise the forcegeneratedbetween t wo s urfaces due t oelect ric currents andelectro- magnetic fieldswe use these forces over and over again in a rotary machi ne thus enabling l a rgepcwers to be converte d. In a sol enoi dth e f orc e i s a la rge attracti ve onewhich i sused once on ly in a wor ki ngst rok e ; t he fo r ce a l so va r i es considerably with thechanging air gap . Howeve r , f or s ma l l mi s s il es and very s hort times of flight thesi mplicity and compactness of direct sol enoi d ope rati onwill certainlybe attractive.The use of solenoids in homi ng head actuation is re marked upon in Chapter 8. Andfi nallywenot e t hat although al l mi ss il esneed some formof electric powersupply , those usingelectric ac tuati onwill need much more . I t is difficult t oprevide the perfect power suppl yfor amis s i le wi th aflight tiwe of 20oreven 10 seconds as ideally Vie ,IDu1d l ike a si mple cheap .!i'attery which pro vides anear -constant vol t a ge andwhi ch i s capabl e cf gi vi ng t he who 1e of i t s s to re denergy i nt his ve rJ s hort t i me . thermal batt eries conta in an el ect r ol y t e whi chator di na ry tempe r at ures is an i ne rt soli d.pcwer is r equi re d a che mical heatso ur ce Hi th in t he battery is ign:ted an,j tn i s ne i t s th e el ect r ol y te . Act i va ti ont i rop-s are us ually l ess t han a s ec ond. They hcve ex cellent shelf l ife andreliabili tybut s uf f er from c ons i ce rab I e vol t eqe dro op as dis.:harge cont in ues. I f "':he fi ightt ime is l es s t han 40seconds or so some of th,: cct t ery capaci t y has to bewa s tc c.The re app ea r s to be no hope at t he presEnt of prcvi oir. q'a t he r ma l bat te ry wh i ch i sefficient for ext remely s hort di scharge t i res I alt hough of course t hey are wi de l yused in sys tems "li t h ma ximumf l i cnt ti rr,es of SC'J 12 seconcs . Such s ys t.ems de notusually use e lectric ac t uat i on. The ai te rn et ive to the t hermal ba t t ery i s t nes11 ver-zinc batterywhe r e t he pot es s i urn hydro xi ce e l ectrolyte is fo r ced in to thecells j ust before us e. These batteries are be s t used wh en the di s cha rc e ti me i sseveral minut es .2.850 :11 TE-" TATI VECO NCLUSIONS_ 1Hissile ser vos a re ra re l y cheap, especial ly for t he hi ghperformance 5ur ra ce-t o-ai rand air-to-ai r mi s s i l es whi ch require such a la rge banc!.idt h and hi sn ma ximum finrate. The a ut hor is of the opin ion t hat it i s not pos s i bl e to t ne c..... leight of prope llant is not large and an increase of say 5- 10%i nt hepropellant car r i edto make upfor t he thrust loss es may not impose an overallweight penalty . Iloving nozzles are vervefficient but reguire more nowerfu lservos._ Two sVlive1 nozzles can also pro vide roll control. At the present t ime--they are cons i de re dat aii could be us edonly when REF ERENC ESi. FARRARD.J . Tne Bloodhoun d J .R. Ac . Soc Jan 1959 .2. BESTD. Some probl ems of polar missi le cont rol J .R. Ac. Soc Aug196 0.3. LLOYDR. Areviewof thrust ve ct or control systems fo r tacti cal mi ss ilesAlAAj SAE 14th Joint Propulsion Conference July 1978.inject i on methods , ifdeflecti on an gles areMissile Contr oT Methods

(,{55\.,. .

1.1NOTATIONANDCONVENTIONS55t an_ s cos (4 .1 -4 )('L 1-5)(4.1 -6)(4. 1- 1)(4. 1-2 )(4 .1 -3), .":., . -:of inertia about 0 are de f i ned as :'.-.F ;: em xy;3.: incidence i nt heplane .A; in cidence plane angle .6: total incidence, s ucht nat :and6= t3r es i n hCP.APTER 4AERODYNAMIC DERIVATIVES ANDAERODYNAMIC T RANSFER FUNCTIONSTheyaY, Qil ne isplaneana __plane. Tne fo11m'li09

angles are defined :a: incidence in t he pi tch pl ane.The prcc uct s of in ert ia ar e de r ineo as:0 = E5m yz .- = EomxzFi g. 4. 1-1. Themoment sA= Eom (i + / )B = Z 15m(zZ+ l)c=...z om(i -i- l )angu1a r ve l ociti es , and t he oft hese quantities are s he:..mi nzaxis, cane dt heya';'{ axi s , dmm\'ords i n the plane of syrare t ry to formar i qht handedorthogorlcl sy s tem \'l i:h the othf:'f t',':o.,Table iL"; -1 de fi nes the f orces an d riOiT.-2n ts ect i nq on t he mi s sile, the 1 i nee r J,nClThe reference a xis system standardis ed in the guidedwee pons industry is centredonthe e.g. and fixed, in t hebodyas fo l l ows :xaxis , calledt he ro ll a xis, f or wa r ds , al ong the axis of synrre t ry if oneexists , but in any case in the pl aneof symre t ry .y axis, calledt hepitchaxis: 30dt othe right if viewingthemissi ie from behi nd.!'=---_. __ .- - .. _- - - _.. _----.--- - - - - _. __._----------_.__ : ,1Aerodynamic Derivatives andAerodynamic Transfer Funct ions57zrzBvyqyPitch axi sx...., :t.LxupTABLE 4 1-1 'WTATI ONRoll axiseachaxi sabout eachaxi sncme nt s of i ne rt i a about of InertiaI Angular ratesIComponent of missileI velocity along each axi sr Component of force actingI On missile a l ong ea ch axisI noments acting On mi s s il e II !,......,.......,.......,.__.,.....,._....,.........,. -+-JI . , i L C "C_ ID, E . c. i -F:-:-j__ _ :--LThe rae son i','hy U, t he mi s s il e ve l oc ity al ongx axi s is cen c t e c oy a capi t a l, Iletter is to t hat it i s 2. large pos i t i ve quan t i ty changing at mos t onl y\rrIIiiIThe angula r rates and comp onents of velocity along thetendt obemuchsma ll er quant i t i es wnichcaobeha ve la rge r r ates of change.IS CENTRE OFMISSILtci{NOTE: 0./BMq.,af ew per cent per secon d.pitch and yawaxes howe ver,posi ti ve or ne gat i ve and canIt.--- - - - ,58 Guioed iieapon Con t ro l Sys tems4. 2EULER'SEQUATlO!';SOFMOTIONFORA RIGIDBODY(4 .2 -6 )Lr1Nm(V+q'd-rv) = Xm( v + rU Prl)ym (w- qVpv) ZAp(B C) 0( 1'2 2E(pq r- ) F{r-p qr -r-q ) - -e Bq (C A) E( p2r2) F( qr p) O(pq-rp -r- -r Cr- (A B) F(q2pZ)O(rp q) E(ar pq + -t-I r.ere an:: six eq ua t ions of mo t ion for a oooy with six degrees of fr ee dom: t n reef orce equat ions and t hre e momen t e quat i ons . The eq uati ons a re somewha t si mple r ifthe mass i s cons tant andthe r adii of gyr at i on changes lowly (1) , in whichcaset he "standard" Euler eq ua t i ons can be used . During boos t t he mass f at e o f ch angecan De rela tively fas t Gut the r adii of gyration usually cha nge s l owl y . In t ntgre a t major i ty of des l gn ca lcula ti ons the standa rd equa ti ons are used . They a r e:(4.2-1)( 4. 2- 2 )(4 .2-3)(4 .2 -4)(4 .2 -5)The first equation does not ree l l y c.mcern cs , ... e ar e in t er e s t e o in t.ne ecce l ere t i onperpendicular to the velocity vector as tm s wiil resul t i na change i n tne ve t oc i tydire ction . In any case in order to determine t ne cnariqe in t r.e f o rv.e ro speed 'o"i t:need to know:he meqn i tude ofo ro pul s ive eno orag t orc es . l"tO" consider' e\..jvctlon4 . 2-2 . The term- mp...' is Sdyl ng tns t t rie r e 1S 0. force inY c i r e c t i o n duE: toi nc i de nce i n pi tch ( 0. :: ""/ U) and roll motion. In QUi et "fI o r ds : ne pi tch ing mo tionof t hemissile is coupl e d to the ya'f!ingmotion on ac count of roll ra te . The t e r mmpv in equat i on4 . 2-3 is also saying that yawi ng mo tion ind UCeS fo r ce s i nthE: pi tcnplane i f roll ing mo tion is present. This is mos t unoes i r ab l e as \"I Erequire t ne s etwo "channels" t obe compl e t e ly uncouple d. I dea ll y r udder mo veme nt s should producef o r ces and momen t s i T:! t heyar,' plane and resul t i n ya'wing motion only; e l e vato rsshou ld res ul t in a manoeuvre i n t he pi tch plane . Cr'oss vcucpli nq Jetheen t he planesmust contribute t osysteminaccuracy. l o reduce t nes e unCles itdc l e effects t hede si gner t r i e s t okee p roll rates as low as pos s tc l e , and in :. si mplifi ed an alysison e usually neglects t he t e rmsenc p", t f r-oll r-a t cs are expe c t e o t o De sma lland inc i de nce (vane wa r e propor t ione t t o in c i oence ) is not l arg E: . Neve r t nel e s s":.,.:t he re i s e vi de nce t hat 10" roll ra tes (say 1-4ca n nave a s i gnifi cant effecton s ystemaccuracy . If p = 20 say ( a ppr o xlma t e l y 3revs/ s), U: 300and " =0. 1?thenw=30 , and t he p(oduct ir, Equat ioll 4 ,2 -2 i s 600 any a ut o-pilot or guidan ce f eedback tne r oll r a t eon tne pitcning motiona spi r al motion of ampl itude 600/p2 =1 .5mradius . Ahi ghEI- 1"01 1 rate proauces alarger cross -coupledUut t e nds t o be atttfluated I n positl0n aue tothe higher frequen cy . If an autop ilot wit h accelera tion fe edDac lLA'EOU T(;l C EENO COvERFIL LE.Revv cOUTERCOVERi i86 Guided WeaponControl Systems1"i1i'f Jj.jI,I~ \1'\, iIiI!'IIi!1IjiI,,,,,i\;,, ,,.!. \~ !i!\\\idifferential t r ans f onr.e r often calledan "E type" pick-o f f. I f one requires are al ly accurat e rate gyro a small angul ar movemen t is essenti al s i nce anydispl ace -ment of t he spi naxis will res ult int he sens i t ive axis bei ngdisplacedby t hesame arnountvandhence cross coupling. fromrat es i n a plane at right angles t ot henomi nal one will occur ,Ve ry good re so lution and lineari ty can be obt ain edwith rate gyros but to reducebias over a wide r ange of operating t empe rat ure s t omuch less t han 2%of ful lsca le calls for muchs ki l l onthe par t of t he instrument des i gner (1) . Rat egyros , li ke positi on gyros, car. be blas t startedor electrically dri ven. Ifbl as t started andall owe dt ocoast a conti nuous l oss of gainwith t imemust beacce pted but ift he i ns trument ' is requi red fo r aut opilot dampings ome 20%los s ofgain wo ul dnot be critical. On e gas fired rate gyro weighing onl y 130grams costsabout ~ 1 0 0 . Conve rs ely .a re all y high per formance mi ni at ure twoaxis rate gyrowh i ch t ogether wi th anelect ronic module provi des t wo d. c. out put s cor r es pondi ngt o r at e input s about twoorthogonal axes is made by Brit ish Aerospace andwei ghs56 gr ams plus 36 grams fo r t he electroni cs . Th i s sub mi ni at ure. transducer hasawi de dynami c r angewithexcel lent re s oluti onand l ow bi as . The operati onoft he se nsor depends onthe gyroscopi c f orces act i ngon f iuid re volving about anaxis at high speed. Sucha package would be i dea l for stabilisation purposes ince r ta i nhomi ngheads .5.5ACCELEROMET ERSThere are three type s of linear accele rometer in conrron us e :(a) Spr i ng- mass acce le romet e rs usually calle dse ismic mass accelerometers(b) Pie zo-el ectri c acce l e romet e rs(c) Forc e- balance accelerometersType tb) exhibit an electric cha r ge ac r oss t wo f aces propor t i onal to th e i mpressedf orce andhence,accel erat i onbut need a special charge ampl i f i er i f lowfrequencyaccelerations are to be recor ded. Type (c l i s real l y a more accurate spring-massac cele r ometer and is used when great accuracy is requi re d. Type (a ) is t het ype mos t often emp loyed jri t actical mi ss il es andconsists of a mass suspended in'. ,a case by a l ow hyst eresi s spring ; fl uid damping is generally used . Theforce andhence deflect ion of t he spring mus t be proporti onal tothe accelerat ion and t hedes ign is suchthat t here i s one sens i t ive axis onlywith ve ry li ttl e cr oss -coupli ng , althoughthere are s ome desi gns witha more complicated suspension andt wo pick- offs so that t woorthogonal accelerat i ons canbe measu re d, However, ast he sys t emis a conventional sp r ing-mass arrangement t here is a secondor der lagi nvol vedas wi th t he rate gyro ; a. c. pi ck-offs s i milar t ot hose us edwit h r ategyr os measure dis placement with re spect t o the case . Un dampednat ur al f req uencies- I- II -II-I-II-II-.I-II-.I-I-JI11lJ.... . ....." - .-..-< .. I=87=== 55 i l e Ins tr uaentsi n the range 80-100 Hz are typical. Again l inearityand high res ol ut i on arecomparat ivelyeasy to achieve , but to guarantee a bias mu ch less t han about =2%of f ull sca le deflection under conditions of vibrat ion , rough usa ge andwi dera nges in t emperature is not easy . Acce lerometers for t actical mi s s iles ar e li kelyto cos t from60-250 depending on speci f i cat ion . Someaxi s accelerometerdetail s an:; shown in Fi g5.5 -1 . 11:1....lJ!15.6 RESOLVERSInsi t uat i ons where the guidance is on the ground and the mi ss i le is not rol1-pos i't ion stab il is ed it is nece ssary to reso l ve the two gui dance corronands (i . e ."."up-down" and "l eft - r i ght " ) into freely r olli ng miss il e axes . Suppose t heguidance demand up-down in '11and the missile wa s l aunchedwith t he e l evat orshori zont al but has since rolled anti -clockwis e through anI f theelevat or servos r e cei ve a corrrnand proportional to Vlcosand't he r udde r servosreceive a command proportional to - Vlsin the missile wil l manoeuvre always assuming that the aerodynamics are linear and the cont ro l surface movementsare proportional to the input cor.mand. The induction res ol ver consi s t s of a roto rand a stator, each with twowindings whose electri'cal axes are at 900to each othe r .The secondary voltages which result are proportional to t he si ne and cosine oft he shaft angle . We therefore hold the rotor st at ionary i nspace by means of aroll gyro and al l owthe stator to rotate with the missile . The l eft - righ t guidances ignal i s appl ied to the other primarywinding . The res ul t inq r elat ive voltagesand forces fromthis arrangement are showndiagramatically in Fi g5. 6-1. The re arecoupl i ng factors between pr-imary and secondarywindinqs whi ch are normally t hes ame for eachwinding .An alter nat ive resol ve r using d.c . is shownin Fig 5. 6- 2. If the up-down guidancecorrmand is V1and the ler't -right corrrnand is V2then V1, V2- Vl , - V2are appl i e dt o th e point s indicated. 'Twowipers at right angles will pi ck of f linear functionsof '11and V2wh ich are well approximated to sines and cos ines i f th e values ofresist ances are carefully chosen . Using eight tapping points as il1 ust r at ed an:;ason ab1e approximation is obtainedif the voltage at 450is 0.75 t imes thevoltage at 0 . Fig 5.6 -3 shows one approximation to a'cosine for wiper angl esfrom0t o 360 fo r given values of R2/Rland RiR1 '5.7 ALTIMETERSRadar altimeters' are used in aircraft to indicate height above the ground or sea .A barometric altimeter indicates height above sea level or someother s el ect edelevation . If amissile is required to flyat a given above the ground fora distance of 20or 30 kma simple barometric capsule or even a pie zo electricG.W.(".$.-I,, '\ -v, : ' ...\88- -t1 ; 55 i 1e Ins t ruments.. ..... . l, .. .. . " .. 1. ':;' "". ~ _ - - Z . .8S-,Ot "OO", 10"-FI G. 5.5-190 Gu ided Cont rol .-+V . R.m]'1---11FI zR] R, R, R]- ,RzR,Rz Vz- VzR,RJR]RzWWt #R] .V, RJFIG. 5.6-2 Rinc resoZvel"WIPERvOLTAGEl ' - -------- -- - ---- - - - - - -75 r--t _::"'_':=FIG. 5.6-3ReL-acitJecs J. [unc zi onofwi pe l' englepres sure transducer shoul dbe accurate enoug nto indicate height provi cingtnis isnot less than about 100 m. Belowt his height t hey are not su itable due to localsmall variations i n and t heir l imi t at i on of dis cri minati onand accuracy.Radar altimeters are discussed by Hovaness i an (2) . Bot n FH/Cw alia pcl seds ystemscan no wT; 1 as some high f re quency noise will almo st certainly be present . Theraaxi mua amount of cha se advance'" OCCurS when1"' "maxwT ; l/icgi vi ng(1 , )/ Z-(6. 2- 1) t an ; - '0.me l.For a;0.3"max;31a 0.2.;~ 2 max_ _0a;0.1~ m a x00If .:l =. 0. 2 , t he network is often referredto as a "5 to 1 pnase aovence network"and if 0. ; 0.1 as a "10 t o1 phase advance network". The poin t that i s beingstresse d at the moment i s that amplitUde s multiply andphases aad.\':e sh ai l poi nt out in t he next chapter that one can-do a li ttle be tter tnan th i swith a nore complextype of 'phas e advance transfer function but the fun dane nt a 1problem still remains : if one requires rore than about 600phas e aovance One nasto use SEveral networks inseries andthe deterioration insignal -to-noise ratiotends to be catastrophic . Hence , although the numberis sonewhat arbi trary,desi gne rs do tend to limit the armunt of phase advance that is going t o Dere quiredto about 600. This means that if one is goi ng to aesign a guiaance loopI,GUldt:'u \"Jt:c pon Contro l Sys temsi-ntn c. minimumof 45 pn ase margin , the tota l phase lag permi ss ible from t hemi s si l e se rvo and t he aerodynami cs at gui dance l oop unity gain cross -over fre quency\'d11 be 15 . Hence the s ervo must be very muchfaster and li kewi se t he weat he rcockfrequen cy s houl dbe much f as t e r (say by a facto r of fi ve or mo re) than the gui dancel oop undamp ednatural frequency - i . e . t he ope n- l oop unity gain crOSS-overfrequency . Thi s may not be a pract i cal pr opos i ti on for an open loop mis s il e cont r olsystemespecial ly at the l owe r end of t he mi s s i l e speed r ange and with a smal lstati c margin .To increas e t heweat hercock dampingWe have seent hat thewe athercockmode i s very un deraamped, es pecial l ywithal arg e stat ic margin andat high al ti tudes . This has s ev e ral unues ir abl e effects.Fi r st l y, a badly dampedoscillatory mode res ul ts in e l ar ge r .m. s . out put t ob roacband noise. The r .m.s . incidence is unneces sar il y large and "his resul ts i n asignificant re duct i on i n r ange due t o t he i nduced dr ag. The accu r acy of themi s si l e wiil als obe degraded somewhat. A s udden in cre ase i nsi gna l wnich coul cocc urafte rat emporary s i gnal fade will result 111 al arge ove rshoot both ini ncidence and in ac hi e vedlateral g. Thi s might cause s ta ll ing and in any case itwouldmean t hat t he airframe woul chave t obe s t resse dt o s t andnearly t wi ce themaximumdesignedsteadystat e g.To reducrq croee-rcoupiiriq between. pi zci: tind yea.; moz -ior:If the missile has twoaxes of symmetry andt he re is no roll rate tne re s noul obenocros s- coupl i ngbetween the pitch and yah mo tion . HO'r'1 eVEr many mi ss il es areal l owe dt o ro ll f r eel y and equ ation 4.2-2 shows t hat t here i s a t erm produci ngacce l era t ion along t he y aX1Sdue to roll ra te and in ci den ce in pi t ch . Likewi seequat ion 4. 2- 3s hows there is.a t erm producing ac ce le rati on along t he z a xi s duet o ro ll rate and incidence in yaw. Simi la rl y t he re are i nduced morren t s in pi t chand yawdueto roll, rate and angular mot i oni nt he otne r pl ane as t he moment ofinerti a abou t t he roll axis i s usually an order of magnit ude less t nan t hatabout t he pitch or ya\'1 axis; equat ions 4.2 -5 anc 4. 2-6 refer . These cros s - coupli nge ffects canbe recarded as andanu ciosed-loopsys t emwi l l De vcons iderab l y less s ens iti VE to anydisturbance thananopen-loop one.Toaoei et: in qazher-iriqIna command systemthe missile is usually l auncne ds one oi s t ance off tne line ofsi ght andtoe- in and s upere l evat i on are added s uch tnat i f t he r e ere no ui s turnence s ,the mi ss ilewill fly in to t he beam or LOS ; in ot he r woros the mi ss il e nas to flyopen-loopfor some time . Thrust mi s al i gnme nt , bi ases andcro s s winds all cont ributeto dispersion . A cl ose d- l oopmi s s il e control sys tem (i. e . an autopilot) , i f itbehaves like any other 'c lose d-Jocpsyst emshoul dbe reasonably resistant to outs i cedi s turbances . To i mprove gui dance accuracy t hesystems engin eerwill wan t t he,.II,:i----- ---_.-.._-Auto;:>i iot Des i go.. 97nar r owe s t g ui da nce beampos s i bl e . Toe majority o f supersonic CLOS and Deamri de r s would st and a very s mall chance of gathering if t hey possessed no eutopil ot .Neve r the l es s t here are a nurrce r of t ube- Laoncheomiss il es which carry a beaconanduse aninf ra- redse nsor f or tracking the mi ss i l e. This sensor is moun tedal ongs i de the launching tube , and t he refore the mi s s il e is actually launched inth e beam. Some of t hese missiles donct have anautopilot .Be fore proceeding to c det ail eddiscus sion of sor.e part i cular autopilots it maybe helpful t o classify t hem as s hown in Fig 6 .2 -1.AUrDPILO TS, -':- .IROLLAU TOPIL OTSIROLL POSIT/ONSTA B/ LISA TlON[F;OLL RATESTAB/LISA TlONISPECIALAu TOPILOTSiROLLPOS/TlOIlOEMANDLATER':"L AUTOPIL OTSFOR CONVENTIONA LCLOS, BEAM RlD/NGAND /-IOM/NG SY TEMSITw OACCELEROME TERSMANUALGUIDANCESYSTEMSISEA SKIMMINGAND HEIGHTLOCKSYS/ENSONEACCELEROME TERONE RATE GYROVERTICALLAUNCHSYSTEMSAZ IMUTh CONTR L8Y INERTIA LMEANS ONE RATEGYRO6.3 AL 'l.T ERALAUT OPI LOTUSIN GONEACCELEROMETERA;-';DO;-';E RAT EGYROAn a r r anqe rnent whereby an acce l e r ometer prov i ce s tn e me i n reecnack an a a. r a t e gyr' Ois used to act as a c:a:7: ;J er i s CO:iIO-::-o n in many ni qn per ro rmance conmenc ana nami ngrri s s i l es . Identical autopilots are used to control the pi tcn 5.n " .jS2 2-r- 1)(:-:: 0One sees immediately that an increase in the value of "2 mus t assis t in promotings t abi l i t y. This is just not so for the other coefficients a3and al . Now t hevalue of alis largely dependan t on the rate gyro gain kg' It is eas y t o showonan anal ogue s i mul at or that too large a value of rate gyro ga in cause s instab il i ty .Si mil ar ly i f the servo damping is re ally exc ess ive (inc r eas i ng t he val ue of a3)equat i on 6.3- 6 i s warn i ng uS that in s tabi l ity can a r ise due to t his al s o .We wi ll nowsee how t he aut opi lo t re spo nse varies as the mi s s ile s pee dchanges .I nves t i gat i onof wi nd tunne l re sults of some typ i cal r athe r s lender l owas pect rat i osupersonic missi le configurations shows that to a goodf irs t approxi ma t i on thenormal f orce fo r a given inciden ce is proport ional to and is onlyweaklydependan t on i1achnumber in the s peed range under considerat ion. I f therefore t hes peed increases by a factor of iZ " ewi 11 as sume that y , n and n i ncreas e by a:-U 5 00 . t:--I'\I I- 1_--"I------ ':)j' ,_. _'\: ._-.1_ .=:==-= ==----=-=::rLJJ1- ==IftII.,II iI

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jj;.!I , ,Il\lr\ 1L,.[PRECESSING RATE oCPERMAHEHT MAGNET SIZE11 SERVO MOI ORSMINIATURERAIE GYROS fROMHORTlIROPORELLI OTT IYPE IGRHAPPROX. wmll Of PART S SIIOWH H kgSERVOERROR fORMAXIMUMTRACKINGRhlf . 0 1'HOMIHAL SERVOOAHOYl IOlH . >10HI--_..-------_.__.__._- - -A-

1[}=O\f /' C' _ _ .< ' ,----...,." J@AlIMUIK10101 0\1g [\!ilH_AA.II' IIH tRAHSfORMERRE MOVEOVIE W Yl IIH GYROCHOl( REMOVED. __ . -" ---_.. ----FIG8.Mar'c oni Spac e and Defence Syst em::; Ltdhoming head using d . c . mot ors andr-at e gy r o8ti on of t he sigh t li ne rate .It 'dill be noted that the si gh! line r..t E: S crtas seen D}' t he !1omir'9andexcep t when eri e l ooks s traight ahe edt.he 0.) ' ::' axes ':-i i i: no t cou.c i de',dt h t hemi s sil e axe s . IFt he oute r gili1ba l is mounte diti t :-( i-t s exi s or rot at ten i ii th eerror s i qna l vl1th the ra te gyro signal t h?. t t ne re s ul t arr t i:; the best repre sent a-Homi ng neec s (In 'J .. ;;,i':: Associated St eoi l i t y Ptoblc:!ls 191between the mo t cr rotc!' and t he bccy =. voltage wi l i bs genEr ated i nthe mot orwindings . Thedesi gnpri nciple is akin tc t hat shown in Fi g1. 4- 1 in that onerelies on hi gh ga i n inner ra te loops us-ing ra ce gy r osto de coupl ehe adf rombody motion. In SOi:1e large homing heads hydr aul i cjacks are used to move t heheedand th e me chani cal coupiing problemcanbecorae me r e se vere. The additi onaltechniques of "rni s s i l e cornpens at i on" an d "I ine of sight i s ol a t ion" a re di s cus s e d It canbearguedthatrat e gyroout put s are nct the s ignal st o repr esen t sight li ne rates as i n additi on t ney conta irl t hesignalsi"/h i ch we do not want to knoweo out . Ingen i ous s cheme s ha ve been de vised t o mix theIIII\\ I\II !I,ahE.::d aos i t i on, the re sol ut i ons 'nhich are are,!ii\8mz-6h?ccs P, -c:. f1 C iJ B. co.:;- -=.:;;. In-;if:"l7' ,j" r.y 'i":where s uf f i ce s in anc h refe r- t o :;t; c-'::This ri::qi...i rcs t wor esol vers e nd tfh=se a n.: 5.;';;,. ;pos t t tr, n 'in F i g 8.3-2. Iue r e i s af:..: r the r compl i cet ion in t he gyro i'if:2 t: !. ers i ons i ncc th e s ceady pr ecess ionrate abo ut the outer gimba l ax i s 15l; j -:h.: ,;OS1rI (;- of arl l::jle o f r otati on oft he t orque axi s , To obt ain an aCCUI"'?, r. eof tr.e pre cess t onal ra t e one has t oat.tenuate the si gnal prcpcrt ior.a l t ocur rent "-J y cos 0. ,somesystems wh i chexpect t ohave J.CU not CG t ni s resolution ,if t he angl es a and 6are small snOrl S t he accuracy ofhomin g is not si gni fi cant l y affected.A'1d f inall y ) heMdoes _homing eye ecquir-e c. e n ? There are t "G di s ti nctme t bcds. If t here i s ::. s urve i l lence ra dar ha s acqu i r e d a target as e re s ul tof an ang ul ar s earch and has s toredits angu ljr posi t ion t he homing head ca n Det hat t he s ur-vai l l ence.radar and mi ss i I ehave to lJ2al i cned. The al te rns t i vemet hod is t o make t he hea d sea rch rc :" 6. t,;;'get . The two precess ing mo tc rs (i r e fe c"withsignal s 90vof',ii t.il oth?r cers inc the head to des cr ibe of':el li pse. I f the incoliiing energy 12'-,i (;1 r e3.chcs a ce rt a i n lE; ve1 t he s e si gna ls a r einh ibited ; the(.ead s hould t hen \' lock on" .192 GuidedWe aponCont rol Sys t ems8.4THEEFFECTOFRADOMEABERRATIONConsider a homi ns missile witht he t arget tracker i nside aprotective cover cal ledaradome, t heext er nal shape of whi chis det ermi nedmainly byaerodynamic considera-tions . The r adome causes anangular di s pl acement of the tracking aer ia l polardiag ramandthe angular difference betweenthe apparent s ight - line andthe act ualsi ght - l i ne is called radome aber r at i on. For agiven radome thi s aberrat ion i s afunction of the angle of look6-, see Figs 8.4-1 and8.4-2 . If the abberat ionmis not constant anymi ss i le motionwil l change th eangle of l ookandapparentsight- linerates vlill begenera t ed, whichin t urn wi ll affect mi ss i l eatt itude .Sma ll diameter mis s i l es ar eat adi sadvantage her esi nceaber rat i oni s a functi onof wavelength-to-diameter . We have to ask(a) Can th is apparent di sh-bodycouplingunstabilise t hegui dance loop?(b) Can it degrade t hesys t em performance?(c) Howcanone specifywhat sor t of r adome aberrat i oncanbet ol erat ed?For anyvalue of 6 -theaberrati onangle canbe defi nedbyin(8 .4 -1)wher e c is not necessar i ly a cons t ant andy is t he aberrat i onslope at tha t poi nt.The valueof y is al sonot aconstant, but the f ol l owi nganalys is is jus t i f i ed Onthe grounds t hat i t is often apprOXimately constant over a rangeof ang l e . Hencethe angle 6 ' as see nby t he homing eye i s given byO'OO+Ob o+c+y(oL h =1 .0>L h =0. 5>L h =0 .25a =3. 510 ////30 WnhT.\v.\\.\\ ,.\\.'\\ /, /'J226FIG9 .5-7Niss Di.s zan ce Due Toa HeadinaErrorBi quadrati c Systemw - w ~ ; 0. 5Yo. Lat axLimitsna - .nh' a .'r I i!.c215 Gui danceLcops =1.0 =0. 5 =0. 25a25Propor t iona ; ':avi g,- t i:J iljIIMhwilhU dm+'Eu. rt ' -._..',- I, i1 ; "i\I, \Ji \

"II .. IMh Wr.h1(\\.'\urn'1J E

I ',Ii \I \:V' \! \I ,i i. I \II I \t / ..... !,t :. ,20 WnhT I\;:1\7J,\ I! II \/fI\J/I \'h =1 .0Uh =O. 5uh =0. 25.--a = 3.5Mh Wnh0-- --- UmlJiE\\,.:--30 WnhT/20.\ \\ -./.JQ=4.53f-2'\Ii\',MhWnhr>,"\. urn lJiEI' \' \__ I\./ . \ I \ , .0\10/, I30WnhT\ I ' ......... /....:J\ !\ ',J, /"--.:FIG 9 . 5-9 Miss v.. : s t c:r.ce DueTo a Headi ngError System w5w= 0 .5No Latax. r..a nit aC", hII- I- I_JHw= 2.7 , from fig 9 .5 -7U .1'ill egi ven byProportional Navigation and Homing Guidance Loops 217engagement ti me is halved the n in s ituat i on ( b) wehave a t ransient whi ch occu pi es Jone hal f of the t i me t hat obt ai ns in situation (a) ; t heir normal i s at i on frequenci esdiffer by a fa ctor of 2 t o 1. Put in another \25or t hereabou ts fo,' e qual dynamiclags, and i f"nhT >15if t he autopilot is muchfaster t han t he homing head . Ne verthe less witha ; 4.5 andpoordampingof the domi nant lag the sys tem t akes alongt ime t osettle and t hi s effect is "are pr onounce dwhent he lags are equal. In genera l wh i chis ent he si deis to light damping but the responsecan be r a ther sl uggis h 2andthe "g"l imi t er s must beset t oa value i nexcess of 2Um A number of computer runshave been done with this in min d; i neachs uccessive r unt he value set ont heli miters was reduced. Eachseries of runswas vm en, as aresult ofprolonged limi ting , the mi ss di s tancewas seen to increase suddenly andappreciaDly .Clearly t he longer t heti me of t he engagement t he mo re near ly does t hecriticalvalue of li mits approach 2UIT. I f t heachi evab le lateral accelerat ion f . ism e magi venby:f " ; C Ume m(9 .5-16)J.- , ,.-"h; 0. 25"h;0. 5i-l h;1.02.5 3.5 4.5 2.5 3.5 4.5 2.5 3.5 4.510 2.7 2.6 2.5 2.8 2.7 2.6 3.0 2.8 2. 715 2.5 2.4 . 2.3 2.5 2.5 2.4 2.7 2.5 2.3. "2.4 2.3 20 2. 4 2.3 2.3 2.4 2.4 2.3 2.530 2.3 2.2 2.2 2.3 2. 3 2.2 2.3 2.2 2.2thenTable 9 .5 -1 give s tne minimumvalue of c for aoumDer of pos s ibl e systempar amet er s suchthat li mit ingwi ll not contribute to mi ss distance .TABLE 0 51I' INIMUMVAl Uf SOFc"a ; 0. 5 and "'na; 2 "'nhin ali casesA note of cauti on " henconsi der i ng transient ef f ect s , whe ther it is i nconnecti onwith miss distance or mini mum acce l era t ion re quireme nts . It is usually quiteaccurate toassu me a constant f igure t ore pre sent t he homing headdynami cs , butth is cannot beso f or t here sponse of t he aut opil ot . If t hemi ss i l e is l aunchedfrom rest it maytake t"o or t hree seconds t o attai nfull speed. Ou r ingthe earl ypar t of th i s boosting per iod i t "ill have anappreciablysl ower response t hanat full.221~ '"Proport i onal NavigationandHoming Gu ida nceLoopsspeed due to t hevery 101' aerodynami cgain . However fas t we make t he homingheadres ponse, t he act ual t i me tha t has t oel apse bef ore t heauto pi lo t can respondquickly must , inpracti ce , l i ~ i t the absolut eminimumhomi ng ti me for agiven misst oa heading error . Mo r eover , f i gur es ouoted in Tabl e9. 5-1 are for w 2w h', na n -.These values wi l l certainlybeoptimistic i f the aut opil ot has a lowbandwi dth andtakes along t ime inatta ining it . In this r2spect , air to "ai r mi s si l es have asl i ghtly bet ter t r ansi ent per formance , si nce th eyare l aunched wi t ha non - zerovelo city . Further insight ont he kinematics of homing canbeobtainedbyconsideringthe f light path of a lag free systemwith a heading error ~ E i nterce pting asta t io nary t arget at Towi t ha set to 2. Ref er r i ngbackto Fig9.3-1 andequati ons9.3-2 and9.3- 3 wecan, wr i t e..o - ( z" - .-(9 . 5-17)For a headi ngerror Zt' 0fe r all val ues of t for a st at i onaryte rqet . Hence- zmUr(I - t )(9 . 5- 18)r.This is rea dily diff erentia t edwi th re spect to t ime to gi ve(Put ting t o andnotingU (T - .' ( i ) - zU -) -6r m m rC.dU (T -rz 0m(9 . 5-19)u 'v U~m Em:u.r:' -r ko,,,hereR i s the initial ranqe . Inothe r wor ds t he initia l s i ght I i ncrate i s notoaffectedbythe actual value of the relat ive velocity. Fi g9.5- 11 clearl y shawst hat for aconst ant fli ght pat hrate (or l at er al accel e,' at ion) t he total change inFIG9.5-11 A homing:rajectoryuit h ~ ~ 2 U/ 3m'f act orf l ight pa thangle f romthe launchto intercept io ni s twic e the total change insight line angle ( ~ . ) . Now cons ider a movi ngtarget andassume saytha t Utand i t s velocity i s in the directiGnMT. Int his case U is reducedby ao 0 rof 2:3 andthe time of engagement has been, i ncr easedby3:2. But fo r the sameheadin gerr or the i ni t ia l si ght li ne ra te wi l l bet he same, Since now the inter -cepti onwill beat a point "ell tc the r ight of '0we donot needs ucha l arge222GuiaedWeapon Control Systemsdemand if tne accelerati c.m is to re main cons tant t hroughout tr.e t raje c t ory .Co nverse l y i f t hetarget i s comi ng t owards us sucht hat U is i ncr ease dwe naverless s pace (or ti me) i nwh ich to change t hemi ss ile f l ight path; hence thecommandt ot hemi ssil emust be greater . This clea rl ydemons t rat es that if t hedemand t othe autop i lot is propor t io nal to sight line rat e times th e r elativevelocity thent hegeneral shapeof the t raj ectory as afunction of time isinvariant .And finally we should note anoth er f ormof "headi ngerror " . flhatever t he initi alconditi ons, t he homing headi tself mus t be poi nti ngvery near ly at t he target;certai nl ywith in onehalf of its beamdidth . If t here is adish poi nt i ng,error at t = 0t hi s " ill i nt roducea transi ent i ntot hetrajectory . It is as teppo sition input t ot hehominghead . Theeff ect s of th i s i nput are not showngra phical ly but t heyar eapproxi mately twice t hat f or a headi ng error . Thetransie nt takes exact l ythe same t i me t odi e awayas that for a heading error;however , s ince t he dishpoint ing error i s at mo st'l ikeiytobe twoor t hreedegree s, the ef fect On mi ss dis tance must be smal l except f or extremelyshor tengageme nts .9.6MISS DISTANCE DUE TOA TARGETLATERALACCELERATIONInt his case it has beenass umed t hat t he perturbation from the const ant bear ingt r aje ct ory is due to t he t ar get executinganevasive manoeuvre equal t oaconstantgt urn T se conds before i nt ercept i on. I f the target la teral accel era t i onis ftt henthe component per pendicular to the original signt l ine is ftcos o' andt hecomp onent of veloc i ty perpendicular t o t heor iginal s i ght l i ne is ftcos '0t. Whenconsi der,jng headi ng errors we said t hat there was ani nput velocityequal toUcos . Since t hesystem res ponds to relative mo t ionit mat t ers not wn etherm e - cowe regar dth is di s'turbance as amissile l atera l veloci ty or t arget lateral veloci t y .Hence , t he differenti al equ at i onmus t beof t hesame f ormas equat i on9.5-2:Hence :cos 2,fmft0{ (l ,{1- ( 1-

(9.6 -2 )--ZJJcos (l - i n it is seen t hatif a - 00as t - T. If t he ef fect of t he cosi nemThe soluti onto this equation isf t cos ' ( T - t )T1,'0t i T)" -z=(a - I )(a - 2){(a -1) t IT-1 -t- ( 1-JmThis canbe differenti at edt" ic e t oyi eldz:zm= ftcoso{ a }{l -( 1-t IT)c - 2)a-:-z'(v+_a_ ) Z T- t m= ftcos t (9.6-1 )

c-223cte rms is neglig-ible (e .g . head onor tan cnas e ) i t is also se enth at if a =3thei ni ti al mi ss il e acceleration is zer oe?Jer; in a Zc.q-free s ys t emandt he ter mi nall ata x is three ti mes the targe t i a t ax ; i f v; = 4te rm-inal l ata x is tw i ce th at ofthe t a r get . The limiting case foY' 0. = is fo r the effe ct i ve mi s s il e latax t oequal the t ar get lataxt hr oughout the engagemen t .cFi g9 .6- 1s hows mi ss il e lat era l accel era tion as a fu nctionof non-di mens ional t i metiTfor wnh T= 20andfor a bi quadrati c re presenta tionof the system; the full l i nein eac hfigure shows the r esponse of a lag free 'sys t emas given i nequat ion 9.6- 2.The t r ajec tory th erefore i s quite different from t nat duet oaheading er r or . Ifa > 2t hen themissile lata xdecre.ses as the engagement proceeds for a headingcccr:--error ; in t he caSe of a target lateral accelerat i onthe mi ss ile latax increasesfromzerotc a ma ximumat i mpact. In the l atter case li mi t i ngwi l l occ ur , if it -occurs at all , near andat the end of the engagement . - "If t his occur s f or a secondor so the effect onsystemaccuracy canbe di sa strous .Themiss distance H_due to a t ar ge t latera l acceleration commencedat Tsecor. dsabefore impa ct i s shown i nFigs 9 .6-2 to9.6 - 4. It wa s though t des i r able t oputSOIT: 2 "i"imits On demanded latEral eccelcret i on. For a;:; 2. 5t he l ag- free effect i ve __termi nal miss ile l at axis five ti mes the targe t effec ti veands i ;nul at io nhassho-,n tna t i na sys temwi th dynamicthe limits are set to about 25% inexces s of thi s then limiti ngdoes not r es ult in anyseri ous degj'adation of accurdcy.keep engagement ti lT12 outdi vi di ng by T2t i me cue t ot he homing"jaqsbecome re 1at i ve1yanda =4.5 the 1i mi tsThe ma in reason for this addi t i onal mi ss il e l a tz xbeing necessary i s the " Tost"headand aut opi l ot dynanr i c lags ; as "", i t hhe ali ing errors th es eless significant fo r longer engagemen t t imes . For a= 3.5have been s et to l ower val ues'but i nboth cas es th ey ar e25% i nexcess of the respective terminal l ag-free value . Si ncesetting th e li mi t sappre ciably belowthese wo uld have resul ted in la rge mi ss distances i t wasconsider ed better to set themat the minimumaccept abl e val ue f or anaccuratei ntercepti ont obe poss i bl e . Setting these limit s anyhigher woul dhave had onl ya very minor effec t onthe shape of t hese graphs. The j ust i fi cat i on f or t henormali sation of the miss is as follows. If t he ta rs et manoeuvre corrrnences at Tseconds bef ore nominal i mpact t hen ithas r:.ano2uvred a dis tanceof 0.5 ftcos oT2perpendi cul ar to theagain ignorin gany si gni f ica nt di s t or tionof th e or ig ina l interception geometry. Soth e normali sed mi ss could Ma/ftcos2.0T. (Hadt he figure of 0.5 beenincl udedt he or di nat es wouldhave been plot t edt wice as la r ge.) For anygiv enpoi nt ona graph 0.)01/ i s def ined andin or der t o2of the ordt nates one; has to rr:ult i pl y by l0nhin steadofIt i s seen fromthe figures t hat u ..; 2. . 5is net goudf or intercept ing manoeuvringespecially if the main l ag i nthe sys tern is VI(;l1 damped . : The response isc.c, 224GuidedWeapon Control Systems6543232 35 6 7 89 '0tIT

=20 " =0 .5 Nocecce Lera t i on' 0' 0\\88 9t I T77669tiTdue toa target Latex lin; i ts55.,4332a 35//>. ///. /!/././.,.,2lateral,w..1'nn.21m cos 0FIG9 .6-1 MissiLeI,IiIIIi!,'1;!ii1\:I,jI!!:,., j \ P; ::1i"0 ,J 1.,; ;:!i:"-,___ c* (,.2 0Propor t iona l Navigationilnd Homing Loops 11 h =5Iln = 25225(c15.1m =5 25 It cos cos 4> IcC:r-It cos c:cos tVft cos1>ocos (\J leo0. =351m =225";.Ia "= 0 .5= w ""/ -, ", "r ," '\--"\ \ ,/ \ \\ -, / \ /"\ \/ /\