si Piniples of Homing Guidne Neil F Plumbo, Ross A luwkmp, nd Justin M Lloyd his tile povides oneptul foundtion with espet to homing guidne upon whih the next sevel tiles e nhoed To this end, bsi geometi nd nottionl fmewok is fist estblished Then, the wellknown nd often-used popotionl nvigtion guidne onept is developed The mehniztion of popotionl nvigtion in guided missiles depends on sevel ftos, inluding the types of inetil nd tget sensos vilble on bod the missile Within this ontext, the line-of-sight eonstution poess (the olletion nd ohesttion of the inetil nd tget senso mesuements neessy to suppot homing guidne) is disussed Also, guided missiles typilly hve no diet ontol ove longitudinl eletion, nd they mneuve in the dietion speified by the guidne lw by poduing eletion noml to the missile body Theefoe, we disuss guidne ommnd pesevtion tehnique tht ddesses this lk of ontol The key hllenges ssoited with designing effetive homing guidne systems e disussed, followed by usoy disussion of midouse guidne fo ompleteness ske INTRODUCTION The key objetive of this tile is to povide bod oneptul foundtion with espet to homing guidne but with suffiient depth to dequtely suppot the tiles tht follow Duing guidne nlysis, it is typil to ssume tht the missile is on ne-ollision ouse with the tget The implitions of this nd othe ssumptions e disussed in the fist setion, Hndove Anlysis The next setion, Enggement Kinemtis, estblishes geometi foundtion fo nlysis tht is used in the subsequent setions of this nd the othe guidneelted tiles in this issue In ptiul, line-ofsight (LOS) oodinte fme is intodued upon whih the kinemti equtions of motion e developed; this oodinte fme suppots the subsequent deivtion of 25
N F PALUMO, R A LAUWKAMP, nd J M LLOYD popotionl nvigtion (PN) guidne Ove the pst 50 yes, PN hs poven both elible nd obust, theeby ontibuting to its ontinued use The Development of PN Guidne Lw setion pesents the deivtion of nd disusses this popul nd muh-used tehnique The subsequent setion, Mehniztion of PN, disusses how PN my be implemented in homing missile Key dives hee e the type of senso tht is used to detet nd tk the tget, how the senso is mounted to the missile, nd how the LOS (te) mesuement is developed The next setion, Rdome/Idome Design Requiements, biefly disusses the funtionl equiements imposed on moden missile domes This is followed by n expnded setion on Guidne System Design Chllenges Hee, the ontibutos to the degdtion of guidne system pefomne (dome eo nd othes) e disussed Then, Guidne Commnd Pesevtion pesents tehnique fo ommnding eletion ommnds pependiul to the missile body tht will effetively mneuve the missile in the desied guidne dietion Although this tile is minly onened with teminl homing onepts, the Midouse Guidne setion biefly disusses issues nd equiements ssoited with the midouse phse of flight Hndove Anlysis When teminl guidne onepts e being developed o elted systems nlysis is being pefomed, it typilly is ssumed tht the missile inteepto is on ollision ouse (o nely so) with the tget In elity, this is not usully the se Fo exmple, thee n be signifint unetinty in tget loliztion, ptiully ely in the enggement poess, tht peludes stisftion of ollision ouse ondition pio to teminl homing Moeove, the unpeditble ntue of futue tget mneuves (eg, tget booste stging events, enegy mngement steeing, oning of bllisti missile eenty vehile, o the weving o spiling mneuves of n nti-ship uise missile) n omplite the development of tgeting, lunh, nd/o (midouse) guidne solutions tht guntee ollision ouse onditions befoe initition of teminl homing In ddition, umultive eos e ssoited with missile nvigtion nd the effets of unmodeled missile dynmis tht ll dd togethe to omplite stisftion of ollision ouse ondition Fo simpliity, ll of these eos n be olletively egded s unetinties in the lotion of the tget with espet to the missile inteepto Thus, if the inteepto is lunhed (nd subsequently guided duing midouse) on the bsis of estimted (pedited) futue tget position, then, t the time of quisition by the onbod seeke, the tul tget position will be displed fom its pedited position Figue notionlly illusttes this ondition, whee is the LOS veto between the missile nd the pedited tget lotion; LOS = / is the unit veto long the LOS; v T nd v M e the veloity vetos of the tget nd missile, espetively; v R = v T v M is the eltive veloity veto; nd e is the displement eo between pedited nd tue tget position As disussed, idelly, the eltive veloity veto is long the LOS to the tue futue tget position (t the time of inteept) Howeve, Fig depits moe elisti ondition whee the eltive veloity is long the LOS to the estimted futue tget position in spe Fo this se, the inteepto will miss the tget unless it pplies oetive mneuves Figue illusttes tht the displement eo, e, n be deomposed into two omponents: one long ( e ) nd one pependiul ( e ) to the pedited tget LOS This deomposition is expessed in Eq : e e = = ( e : ) LOS LOS = # ( e # ) LOS LOS In this eqution, x y epesents the dot (sl) podut between the two vetos x nd y, nd x 3 y epesents the oss (veto) podut between the two vetos Note tht, beuse the eltive veloity veto, v, is long the LOS to the pedited tget lotion, the eo e will lte the time of inteept but does not ontibute to the finl miss distne Consequently, the miss distne tht must be emoved by the inteepto fte tnsition to teminl homing is ontined in e (ie, tget unetinty noml to the LOS) Homing missile guidne sttegies (guidne lws) ditte the mnne in whih the missile will guide to inteept, o endezvous with, the tget The feedbk ntue of homing guidne llows the guided missile (o, moe genelly, the pusue ) to tolete some Tget v T veloity Missile x Fixed tget fme z Atul tget position Line of sight v R x v M Missile veloity y z Pedited tget position Figue To ssist in simplifying the nlysis of hndove to teminl homing, ll of the ontibuting nvigtion nd enggement modeling eos e olletively egded s unetinties to the lotion of the tget with espet to the missile t hndove e e e () y 26
ASIC PRINCIPLES OF HOMING GUIDANCE level of (senso) mesuement unetinties, eos in the ssumptions used to model the enggement (eg, unntiipted tget mneuve), nd eos in modeling missile pbility (eg, devition of tul missile speed of esponse to guidne ommnds fom the design ssumptions) Nevetheless, the seletion of guidne sttegy nd its subsequent mehniztion e uil design ftos tht n hve substntil impt on guided missile pefomne Key dives to guidne lw design inlude the type of tgeting senso to be used (pssive IR, tive o semi-tive RF, et), uy of the tgeting nd inetil mesuement unit (IMU) sensos, missile mneuvebility, nd, finlly yet impotnt, the types of tgets to be engged nd thei ssoited mneuvebility levels We will begin by developing bsi model of the enggement kinemtis This will ly the foundtion upon whih PN, one of the oldest nd most ommon homing missile guidne sttegies, is intodued Enggement Kinemtis: The Line-of-Sight Coodinte System The development pesented hee follows the one given in Ref In the sequel, the following nottion is used: X = n 3 m (ed n-by-m) mtix of sl elements x i,j, i = n, j = m; x = n 3 veto of sl elements x i, i = n; n 2 x = / i = x i = Euliden veto nom of x; x = x/ x = n 3 unit veto (eg, x = ) in the dietion of x; /t = time deivtive with espet to fixed (inetil) oodinte system; nd d/dt = time deivtive with espet to otting oodinte system Conside the enggement geomety shown in Fig 2, whee M nd T e the position vetos of the missile inteepto nd tget with espet to fixed oodinte fme of efeene (epesented by the tid { x, y, z }) Consequently, we define the eltive position veto of the tget with espet to the missile s shown in Eq 2: = T M (2) The eltive position veto n be witten s = R, whee R = is the tget missile nge, nd is the unit veto dieted long (we efe to s the LOS unit veto) Diffeentiting the eltive position veto, = R, with espet to the fixed oodinte system, we obtin the following expession fo eltive veloity v: v / = Ro R t t (3) Fom Eq 3, one n see tht the te of hnge of the eltive position veto (ie, eltive veloity) ompises two omponents: (i) hnge in s esult of hnge in length (Ṙ) nd (ii) hnge in dietion ( ottion) s esult of the te of hnge of the LOS unit veto LOS oodinte fme = n M Fixed oodinte fme x z n = = Missile t t LOS T y Tget Figue 2 In the LOS oodinte fme, the LOS te is pependiul to the LOS dietion nd ottion of the LOS tkes ple bout the We define this hnge in dietion by the veto n s given in Eq 4: n / t Consequently, seond unit veto, n, is defined to be in the dietion of n s shown: n (4) / t = = n (5) / t n Finlly, to omplete the definition of the (ighthnded) LOS oodinte system, thid unit veto,, is defined s the oss podut of the fist two: = 3 n (6) In genel, the ngul veloity of the LOS oodinte system with espet to n inetil efeene fme is given by o = o o nn o, whee the omponents of the ngul veloity e given in Eq 7: o = o : o n = o : n o = o : (7) Thus, upon eexmintion of Eq 4, we note tht the LOS te, n, n be expessed s shown in Eq 8: n = d dt o # (8) On the ight-hnd side of Eq 8, the expession d /dt epesents the time deivtive of the LOS unit veto with espet to otting oodinte fme, nd o is 27
N F PALUMO, R A LAUWKAMP, nd J M LLOYD the ngul veloity of the otting fme with espet to the inetil fme The fome omponent is equl to zeo (ie, the LOS unit veto is onstnt) Theefoe, the LOS te nd oesponding unit vetos simplify to the following expessions: n = o # o # (9) n = o # Thus, fom Eq 3, the eltive veloity expession is given s v = R o R( o # ) (0) The typil guided missile ontol vible is inteepto eletion Thus, tking the deivtive of Eq 0, nd using Eq 9, we obtin n expession fo the eltive eletion: v= Rp Ro R( o # ) R( # ) R # t o p t ` o t j = Rp 2Ro ( o # ) R( p # ) R[ o # ( o # )] ( ) Next, using the definition of o in Eq 7 nd the ft tht = [ 0 0] T, we expnd the tem o # / n in Eq, whih esults in the following eltion: n o # = det o o n o = o n o n (2) 0 0 Hee, det epesents tking the deteminnt of mtix Fom Eq 4, the dietion of n n hve no omponent long Thus, we onlude n = 0 nd obtin the esult in Eq 3: o # = o (3) n In Eq 3, o / n Using Eq 3, the othe osspodut tems in Eq yield the following esults: o # ( o # ) o o o p # = p 2 = n (4) Using Eqs 3 nd 4 in Eq, we obtin the desied expession fo eltive eletion: 2 2 t = T M 2 = ( R p R o ) (2R o o R p ) n ( R o o ) (5) Fom Eq 5, the omponents of eltive eletion in the LOS oodinte fme n be witten s shown in Eq 6: 2 ( T M) : = Rp Ro ( T M) : n = 2Ro o Rp ( ) : = Ro o T M (6) Development of PN Guidne Lw Mny guided missiles employ some vesion of PN s the guidne lw duing the teminl homing phse Sufe-to-i, i-to-i, nd i-to-sufe missile enggements, s well s spe pplitions (inluding endezvous), use PN in one fom o nothe s guidne lw 7 A mjo dvntge of PN, ontibuting to its longevity s fvoed guidne sheme ove the lst five dedes, is its eltive simpliity of implementtion The most bsi PN implementtions equie low levels of infomtion egding tget motion s omped with othe moe elbote shemes (some e disussed in the next tile in this issue, Moden Homing Missile Guidne Theoy nd Tehniques ), thus simplifying onbod senso equiements Moeove, it hs poven to be eltively elible nd obust As lso will be seen in the next tile in this issue, unde etin onditions nd (simplifying) ssumptions bout tget nd missile hteistis, the PN lw is n optiml guidne sttegy in the sense of minimizing the teminl miss distne In ode to develop the PN guidne sttegy, we fist look to the omponents of Eq 6 fo suffiient onditions to hieve n inteept Looking t the fist omponent, suffiient onditions fo n inteept e (i) the LOS te is zeo ( = 0), (ii) inteepto pbility to elete long the LOS is gete thn o equl to tget eletion long the LOS ( M T ), nd (iii) the initil nge te long the LOS is negtive (Ṙ(0) < 0) In this se, missile-to-tget nge (R) will deese linely (( T M ) = 0) o qudtilly (( T M ) < 0) with espet to time nd, eventully, pss though zeo Fom the disussion bove, the inteepto must elete suh s to null the LOS te ( ) We look to the seond omponent of Eq 6 to detemine how this n be done We fist define losing veloity s V Ṙ Note tht fo n ppohing tjetoy Ṙ < 0; thus V > 0 If, fo the moment, we tet losing veloity nd nge s onstnt, then tking the Lple tnsfom of the seond omponent of Eq 6 yields the following polynomil in s: ( ( s) ( s)) : = ( sr 2V ) o ( s) (7) T M n In Eq 7, s epesents the Lple tnsfom vible Thus, if we define inteepto eletion pependiul to the LOS to be M (s) n = (s), then, fom Eq 7, we n wite the tnsfe funtion fom tget ele- 28
ASIC PRINCIPLES OF HOMING GUIDANCE tion (pependiul to the LOS) to the oesponding LOS te: o ( s) = ( ( s) : ( sr 2V ) T n (8) Refeing to Eq 8, to ensue stble system, we equie > V This leds to wht is known s the tue PN guidne lw shown in Eq 9: = NV o, N > 2 (9) M : n As is le fom the pevious development, tue PN ommnds missile eletion noml to the LOS This is, pehps, moe obvious by efeing bk to Eq 3 nd ewiting Eq 9 s shown in Eq 20: = NV o #, N > 2 (20) M Hee, M epesents ommnded missile eletion noml to the LOS Ahieved missile eletion is physilly elized though eodynmi ontol sufe defletions, ontol thuste opetion, o ombintion of both Thus, Eq 20 lso emphsizes the ft tht the development of PN ssumes no-lg missile esponse (ie, the missile is ssumed to espond instntly to, nd hieve pefetly, the guidne ommnd) We will simply mention nothe vint of PN, efeed to pue popotionl nvigtion (PPN) The nmes given to these vints e somewht bity, but these nmes hve stuk The genel theedimensionl vesion of PPN n be expessed s shown in Eq 2 M = k o # v (2) Hee, k is the nvigtion gin Clely, PPN ommnds missile eletion noml to the missile veloity veto, v M In view of the impotne of the PN lw in missile guidne nd spe pplitions, onsideble nlytil study hs been onduted egding the behvio of missile guided unde PN Sine the diffeentil equtions govening PN motion, even when onsideing kinemtis only, e highly nonline, only limited suess hs been hieved in solving these equtions nlytilly Mehniztion of PN Hee, we disuss how PN n be mehnized in homing missile Centl to this topi is the type of senso tht is used to detet nd tk the tget: whethe it is pssive (eg, IR), semi-tive, o tive (eg, RF o lse) senso nd, s impotnt, how it is mounted to the missile M As disussed, the onventionl implementtion of PN equies losing veloity nd LOS te infomtion to podue the guidne (eletion) ommnds If we ssume (fo lity) tht the enggement is pln, then we n ewite Eq 20 s shown below, whee l is the LOS te in n inetil fme of efeene: M = NV l (22) Thus, implementtion of the PN guidne lw in thee dimensions dittes the neessity to mesue LOS te in two senso instument xes tht e mutully pependiul to the senso boesight (ne-oinident with the mesued LOS to the tget) As mentioned peviously, the wy in whih losing veloity (V ) nd LOS te ( l) infomtion is obtined to mehnize PN guidne (Eq 22) is funtion of the type of tget senso tht is used nd how it is mounted to the missile body Aquiing losing veloity infomtion depends pimily on the tget senso type Given n (onbod) tive o semi-tive RF system, fo exmple, the obseved Dopple fequeny of the tget etun n be used to develop good estimte of losing veloity, V In othe implementtions, missile-to-tget nge o losing veloity n be peiodilly up-linked to the missile to filitte PN guidne The wy in whih LOS te ( l) infomtion is deived depends on the type of tget senso tht is used nd how it is mounted to the missile Fo exmple, spestbilized senso (ould be RF, IR, o lse) is mounted on gimbled pltfom to inese the field of egd of the senso nd to isolte it fom missile body motion Convesely, tking systems tht do not equie lge field of egd o tht employ n IR fol plne y, fo exmple, e fixed to the body (stpdown systems) Hee, we will onside spe-stbilized systems efoe intoduing the detils on how to deive LOS te, we biefly disuss spe-stbilized tget senso systems Vious spe-stbilized designs e possible, but typil design is one in whih two mutully pependiul gimbls e employed long with te gyos used fo pltfom stbiliztion nd LOS/LOS te eonstution (Typilly, these systems ely on the missile utopilot fo oll stbiliztion) Suh gimbled pltfoms use sevomoto in eh xis to ommodte seeke pointing Hene, we will define spe-stbilized seeke to be omposed of the tget senso (ntenn/enegyolleting devie nd eeive), gimbls (nd ssoited sevomotos), gyos, nd the neessy ontol eletonis The neessy seeke funtions e s follows: (i) tk the tget ontinuously fte quisition, (ii) povide mesue of the LOS ngle () o LOS ngul te ( l), (iii) stbilize the seeke ginst signifint missile body te motion (pithing nd ywing te) tht my be muh lge thn the LOS te to be mesued, 29
N F PALUMO, R A LAUWKAMP, nd J M LLOYD m Mesued line of sight bse m (Tue) line of sight Seeke enteline Missile body enteline Inetil efeene mesues the tking eo ( m ) with espet to seeke oodintes (Pimy ontibutos to tking eo e disussed lte) The mesued tking eo, in tun, is used by the tking system (the seeke tk loop) to dive the seeke dish ngle (vi sevomoto toquing of the gimbls) suh s to minimize the tking eo, theeby keeping the tget in the field of view Consequently, the seeke dish te,, is ppoximtely equl to the inetil LOS te The tnsfe funtion of LOS te to seeke dish te n be ppoximted by the following fist-ode tnsfe funtion: Figue 3 Illusttion of the two-dimensionl definitions of the vious ngles often used when nlyzing the LOS eonstution poess nd (iv) mesue the losing veloity, if possible 8 Item iv is possible to hieve with d systems but is diffiult with IR systems Seeke pointing is omplished by n oute gimbl ottion b O bout the missile body y xis (pith), followed by n inne gimbl ottion b I bout the subsequent z xis These two ottions e used to define the oodinte tnsfomtion fom the missile body fme () to the seeke oodinte fme (S), s shown in Eq 23: C S os bo os b I sinbi sinbo os bi = > os bo sinbi os bi sinbo sinbi H (23) sinb 0 os b O In ode to outline possible ppohes to deive LOS te fo guidne puposes, we will efe to Fig 3, whee we hve defined the following ngul quntities: is the inetil ngle to the missile body enteline; is the inetil ngle to the seeke enteline (inetil dish ngle); b is the gimbl ngle (ngle between seeke boesight nd missile enteline); is the tue tking eo (epsilon) between the LOS nd seeke enteline; bse is petubtion to the tue epsilon used by dome eftion of the RF enegy o idome distotion of IR enegy s it psses though the mteil; m is the mesued epsilon; l is the tue inetil LOS ngle; nd l m is the mesued, o eonstuted, inetil LOS ngle Tking of tget equies the ontinuous pointing of the senso bem t the tget As illustted in Fig 4, the eeive LOS O o o = s s (24) In this eltionship, t s is the seeke tk-loop time onstnt Thus, the seeke dish te will lg the LOS te The uy to whih the seeke is stbilized ples fundmentl limittions on the homing peision of the missile One possible LOS te estimtion sheme is shown in Fig 5, whih illusttes simplified blok digm ompising the seeke, guidne ompute, flight ontol system, nd body te eodynmi tnsfe funtion In Fig 5, the Lple opeto is indited by s Fo simpliity, the flight ontol system (ie, the ombined epesenttion of the ontol sufe tutos, eodynmis, nd utopilot) is expessed s the tnsfe funtion epesented by G FC (s) The guidne system is epesented s simplified LOS te guidne filte followed by PN guidne lw The ombined guidne system tnsfe funtion is shown in Eq 25, whee t f is the guidne filte time onstnt: Noise Seeke Gimbl ngle /s /s Gimbl sevo Dish te o m Reeive NV = s Gimbl ontol f s Rte gyo ody ttitude te (25) Tking eo Figue 4 Simplified pln model of gimbled seeke tk loop (without dome effets) In this onfigution, the ommnded dish te is popotionl to the tking eo m 30
ASIC PRINCIPLES OF HOMING GUIDANCE Noise LOS Seeke Gimbl ngle /s /s Reeive LOS te pikoff m s LOS te estimte ^ f s NV Guidne PN filte Guidne ompute M Commnded eletion Gimbl sevo Dish te Gimbl ontol Rte gyo Flight ontol system ody ttitude te G FC (s) A s v m Ahieved eletion ody te eodynmis Figue 5 Simplified pln model of tditionl LOS te eonstution ppoh tht dietly supplies n LOS te mesuement to the guidne ompute Note tht the LOS te pikoff is ssumed to be popotionl to the boesight eo mesuement The mesued LOS te is subsequently filteed to mitigte mesuement noise nd then pplied to the PN homing guidne lw Moeove, the tnsfe funtion fom ommnded eletion (fom the guidne lw) to missile body te (ċ) is ppoximted by the following eodynmi tnsfe funtion, whee t A is the tuning te time onstnt nd v m is missile veloity: o A s = v m (26) In this ppoh, the ft tht the LOS te is embedded in the tking eo ( m ) is exploited As illustted, LOS te estimte is deived by ppopitely filteing the eeive tking eo sled by the seeke tkloop time onstnt Othe ppohes n be used to deive LOS te fo homing guidne puposes; these e genelly efeed to s eithe LOS eonstution o LOS te eonstution We next outline thee ltentive tehniques (two LOS eonstutions nd one LOS te eonstution) LOS Reonstution As shown in Fig 3, LOS eonstution woks to onstut mesued LOS, l m, in n inetil fme of efeene The mesued LOS then is filteed (vi n ppopite guidne filte) to deive n estimte of LOS te fo guidne puposes Two diffeent LOS eonstution ppohes e s follows: 9 Integte the seeke gyo output nd sum it with the mesued tking eo A blok digm of this ppoh is shown in Fig 6 Mthemtilly, this ppoh n be expessed s Eq 27: l m = m # dt (27) 2 Integte the output of the missile body te gyo, obtined fom the missile IMU, nd sum the integted IMU gyo output togethe with the seeke gimbl ngle nd the mesued tking eo We illustte this ppoh in the blok digm shown in Fig 7 This ppoh is expessed mthemtilly s shown in Eq 28: l m = m b # ċ dt (28) Fom Fig 3, it is le tht the two onepts e lgebilly equivlent in the bsene of noise nd ssuming pefet instuments (no gyo bises, dift, et); howeve, in ptie, this is not the se Moeove, we note tht, in genel, the guidne filtes fo the two ppohes e not neessily the sme (eg, fo the simplified guidne filtes shown in the figues, the filte time onstnts, epesented by t f, e not neessily the sme) The fundmentls of guidne filteing will be disussed in the ompnion tile in this issue Guidne Filte Fundmentls It is shown in Ref 9 tht the two LOS eonstution ppohes n yield signifintly diffeent esults when noise nd impefet instuments e used How these diffeenes mnifest themselves depends on the qulity of the mesuements nd instuments tht e used LOS Rte Reonstution A guidne signl lso n be geneted by diffeentiting the tking eo nd dding it to the seeke te gyo output Fo ptil puposes, tking the deivtive of the tking eo is om- 3
N F PALUMO, R A LAUWKAMP, nd J M LLOYD Noise LOS Seeke Gimbl ngle /s /s Gimbl sevo Dish te Reeive Gimbl ontol Rte gyo m LOS pikoff s /s m LOS te estimte ^ s f s NV Guidne PN filte Guidne ompute Flight ontol system ody ttitude te G FC (s) M A s v m Commnded eletion Ahieved eletion ody te eodynmis Figue 6 LOS eonstution ppoh wheein the seeke gyo output is integted nd summed with the mesued tking eo to fom n LOS mesuement The mesued LOS is subsequently pssed to the guidne ompute, whee it is filteed to extt n LOS te nd then pplied to the PN guidne lw Noise LOS Seeke Gimbl ngle /s /s Gimbl sevo Dish te Reeive Gimbl ontol Rte gyo m LOS pikoff LOS te estimte ^ m s M f s NV Guidne PN Commnded s filte eletion Guidne ompute ody ttitude Flight ontol G FC (s) system /s ody ttitude te A s v m Ahieved eletion ody te eodynmis Figue 7 LOS eonstution ppoh 2 wheein the output of the missile body te gyo (vi the missile IMU) is integted nd summed togethe with the seeke gimbl ngle nd the mesued tking eo to develop n LOS mesuement The mesued LOS is subsequently pssed to the guidne ompute, whee it is filteed to extt n LOS te nd then pplied to the PN guidne lw plished by using led netwok suh s tht given below: s t s D (29) Figue 8 illusttes this ppoh Hee, t D is the time onstnt of the led netwok As shown, the eonstuted LOS te is filteed by n ppopite guidne filte (hee epesented by simplified lg filte with time onstnt t f ) to deive LOS te estimte This method my led to exessive mplifition of the eeive noise s esult of the diffeentition poess Howeve, in Ref 0, it is shown tht LOS te eonstution n wok well if the missile tuning te (ie, esponsiveness to ommnds) is vey fst Rdome/Idome Design Requiements In endotmosphei enggements, dome (o idome) is equied in ode to potet the onbod seeke fom the elements Fo exotmosphei vehiles, dome/idome is not neessily equied The key dome equiements e summized below 8 : 32
ASIC PRINCIPLES OF HOMING GUIDANCE Noise LOS Seeke Gimbl ngle /s /s Gimbl sevo Dish te LOS te pikoff m s m D s s Gimbl ontol Reeive Rte gyo m LOS te estimte ^ f s NV Guidne PN filte Guidne ompute Flight ontol system ody ttitude te M G FC (s) A s v m Commnded eletion Ahieved eletion ody te eodynmis Figue 8 Simplified pln model of n ltente LOS te eonstution ppoh tht n dietly supply n LOS te mesuement to the guidne ompute Note tht the LOS te eonstution diffeenes between this figue nd Fig 5 As befoe, the mesued LOS te is subsequently filteed to mitigte mesuement noise nd then pplied to the PN homing guidne lw It must onvey the enegy with minimum loss 2 It must onvey the enegy with minimum distotion, ptiully ngul distotion beuse this etes psiti feedbk loop tht n hve signifint negtive impt on guidne pefomne (disussed in moe detil below) 3 It must hve minimum eodynmi dg 4 It must hve stisftoy physil popeties, suh s suffiient stength, esistne to theml shok (fom pid eodynmi heting), esistne to in eosion t high speeds, nd minimum wte bsoption /2 L/D /2 Idel eletomgnetilly 5 L/D 5 Idel eodynmilly L/D 3 Compomise dome Figue 9 Thee possible dome shpes e illustted Fo minimum ngul distotion, hemispheil shpe (o hypehemispheil shpe s in gound-bsed d) would be idel eletomgnetilly (uppe left), but the dg penlty is exessive Fom n eodynmi pespetive, the lowe left dome shpe is pefeble, but it tends to hve signifint ngul distotion hteistis The tngent-ogive shpe (on the ight) is typil ompomise design Nevetheless, some missiles use muh blunte dome designs despite the dg penlty L/D, lift-to-dg tio 3 As n exmple, Fig 9 illusttes thee oneivble dome shpes The tngent-ogive shpe (on the ight) is typil ompomise design Guidne System Design Chllenges Thee e signifint numbe of hllenges to designing guidne systems: the design must povide the desied pefomne while emining obust to multitude of eo soues, limited ontol system bndwidth, nd inheent system nonlineities Some of these hllenges e summized below The oot-men-sque finl miss distne fom ll deleteious noise soues must be minimized 2 Guidne system stbility must be mintined in the psiti feedbk loop (s mentioned bove, this is used by ngul distotion of the dome/idome) Rdome ngul distotion, in ptiul, is key ontibuto to finl miss distne but is onsideed septely fom othe noise soues s its impt on guidne system stbility is substntil 3 System nonlineities must be voided s muh s possible; eg, seeke gimbl ngle nd gyo te stutions must be voided, s should ommnding missile eletion beyond wht is physilly elizble by the missile Contibutos to Finl Miss Distne With espet to guidne hllenge item bove, thee e numbe of ontibutos to finl miss distne (othe thn ngul distotion of the dome/idome) Fo exmple, in eithe n RF o IR guidne system, mesuement noise fom the vious instuments (onbod 33
N F PALUMO, R A LAUWKAMP, nd J M LLOYD seeke, IMU, et) will tend to inese the finl miss distne In d-guided system, high illumintion powe on the tget is desied in ode to edue the neessy eeive gin nd, onsequently, the intenlly geneted noise Similly, pssive IR seeke is designed to hve mximum petue size to mximize inoming powe Theefoe, ntenn/petue size usully is mde mximum within the onstint of missile body dimete to mximize powe eeption nd minimize ngul bem width, theeby leding to less noise The bility fo ny given tget senso to esolve losely sped objets lso is limited, ontibuting to dditionl miss distne Fo exmple, missile d ntenn usully hs eltively wide bemwidth, nd so it is unble to esolve losely sped tgets in ngle until vey lte in the endgme of the enggement In this sot of mbiguous sitution, the homing missile n inu lge finl miss distne Some of the key ontibutos to finl miss distne e intodued below 8, : Seeke eeive nge-dependent ngle noise, d This noise is funtion of the stength of the tget etun powe nd, hene, the signl-to-noise tio Rnge-dependent noise powe vies invesely with nge to tget (R) s /R 2 fo semi-tive d nd s /R 4 fo tive d Fo IR systems, ngedependent ngle noise typilly is not onsideed to be pimy ontibuto 2 Seeke nge-independent ngle noise, s i This noise is independent of tget etun powe nd is used by numbe of intenl soues like signl poessing nd quntiztion effets, gimbl sevo dive eos, nd othe eletil noise 3 Sintilltion (nd glint) noise, s glint These e oupled effets nd e used by tget efletions tht vy in mplitude nd phse ove time (The effet is muh like tht of sunlight glinting fom the shiny sufes of n utomobile) Sintilltion/glint noise n be vey sevee nd is funtion of the physil dimensions nd motion of the tget 4 Clutte nd multipth noise, s These noise effets beome impotnt t low ltitudes, ove eithe the lnd o se This noise is used by the unwnted stteing (fowd nd bkwd) of d etuns fom, fo exmple, se-sufe wves 5 Impefet seeke stbiliztion used by seeke gyo nd gimbl sle ftos, gyo dift, nd missile body bending (flexing) Impefet seeke stbiliztion n intodue psiti feedbk loops simil to tht used by dome/idome distotion The net effet n be viewed s induing bis o petubtion on the mesued LOS te Extly how these eos ffet guidne pefomne will be funtion of the wy in whih LOS eonstution is mehnized 9 6 Initil heding eos t the stt of homing Heding eo n be defined s the ngle between the tul missile veloity veto t the stt of teminl homing nd the veloity veto tht would be neessy to put the missile on n inteept ouse with the tget Heding eo n be viewed s n initil ondition distubne to the guidne system t the stt of teminl homing 7 Tget eletion (mneuve) pependiul to the LOS A tget n mneuve fo ny numbe of esons (to void detetion, to ent neessy ouse oetions, to evde pusuing inteepto missile, et) Whteve the eson, fom the viewpoint of the guided missile, tget mneuve n hve stohsti qulity to it nd, in some instnes, it n indue vey lge finl miss distne Relevnt tget mneuve (eletion) levels nd oesponding hteistis (hoizontl weve, oksew, hdtun, et) e key dives to the design of missile guidne systems Tget mneuve genelly is onsideed (potentilly signifint) distubne the thn noise to the guidne system Items 5 e stohsti noise soues, nd the emining items typilly e efeed to s non-noise ontibutos to finl miss distne Rdome/Idome Reftion Eo The ogivl shpe of domes helps to edue dg but is signifint soue of touble to guidne system pefomne Idome shpes tend to be less poblemti thn dome shpes, but thei effet on guidne system pefomne still must be onsideed The fundmentl issue is tht s dition psses though the dome, it is efted by etin ngle whih, in tun, depends on the seeke look ngle, b As illustted in Fig 0, the dome distots the boesight eo mesuement by n mount bse The mgnitude of eftion duing homing depends on mny ftos, inluding dome shpe, fineness tio, thikness, mteil, tempetue, opeting fequeny, nd poliztion of the tget etun signl Hene, it is vey diffiult to ompenste pefetly fo dome eo pioi Typilly, the speifition of dome hteistis involves the dome/idome slope (see Fig 0), whih is lol popety of the dome nd is defined s shown in Eq 30: 2bse / (30) 2 Although is not onstnt ove the entie dome, it n be egded s onstnt ove smll nge of seeke look ngles, b The most optimisti sitution is one in whih the designe would be ble to speify the mnuftuing tolenes nd, thus, the limits on the llowble vitions of 34
ASIC PRINCIPLES OF HOMING GUIDANCE Rdome bse (dome eo) bse Appent LOS Tue LOS Seeke enteline (Seeke gimbl ngle) Missile body xis bse (slope) Using the ft tht the seeke dish ngle n be witten s = b, we obtin the following expession fo the deivtive of the mesued tking eo: m = l ( )ḃ ċ (33) Using Eq 33, we n modify the blok digm shown in Fig 5 to inlude the dome distotion effet (vi Eq 33) The esult is shown in Fig Clely, the boesight eo etes nothe signl pth in the missile guidne loop To obtin n ltente view of this psiti loop nd how it n ffet guidne pefomne, we evisit Eq 32 Refeing to Fig 3, we note tht the seeke gimbl ngle n be expessed s b = nd tht the mesued LOS n be witten s l m = m Also noting tht it is typil fo <<, we n ppoximte Eq 32 s l m l ċ (34) Figue 0 This illusttion highlights the ft tht the missile dome (o idome) distots the boesight eo mesuement The boesight eo distotion is funtion of look ngle Refeing one gin to Fig 3, it follows tht we n wite the following ngul eltionship fo the mesued tking eo: m = bse (3) Diffeentiting Eq 3 nd using the definition of dome slope shown in Eq 30, we obtin the following expession fo m : m = l bse = l ḃ (32) Continuing, we lso equie tnsfe funtion desiptions of ll omponents epesented in Fig 5 An ppoximte epesenttion of the seeke blok illustted in Fig 5 is given by the tnsfe funtion G s (s) below: s ( s) Gs( s) = ( s) s s s (35) Dietly fom Fig 5, the ombined guidne system (filte nd lw) tnsfe funtion n be expessed s M( s) NV Gg ( s) = NV G ( s) s ( s) F = s f (36) Next, we ssume the tnsfe funtion fom ommnded to hieved missile eletion, G FC (s), n be LOS Noise Seeke LOS te pikoff Reeive m s Gimbl bse ngle oesight eo /s /s Rdome slope Gimbl sevo Gimbl ontol Dish te Rte gyo LOS te estimte ^ f s NV Guidne PN filte Guidne ompute Flight ontol system ody ttitude te G FC (s) M A s v m Commnded eletion Ahieved eletion ody te eodynmis Figue Simplified pln model of gimbled seeke tk loop with the inlusion of dome/idome distotion effets on the mesued boesight eo 35
N F PALUMO, R A LAUWKAMP, nd J M LLOYD ppoximted by the following fist-ode lg epesenttion: G FC ( s) M M ( s) = ( s) t FC s (37) Finlly, ell tht the missile eodynmi tnsfe funtion fom eletion to body te n be ppoximted s shown in Eq 26 nd epeted hee fo onveniene: G s( s) ( s) = ( s) A M A s v m (38) Using Eqs 34 38, Fig 5 n be dwn s illustted in Fig 2 Refeing to this blok digm, the losed-loop tnsfe funtion fom LOS te to missile eletion n be witten s M( s) GR( s) s l ( s) NVGF( s) GFC( s) Gs( s) /s (39) = NVGA( s) GF( s) GFC( s) Gs( s) sv Stti Effets Without eftion ( = 0), the stedy-stte gin of the system in Eq 39 is given by g lim G ( s) = NV, m ss s" 0 R = 0 whee ell tht N > 2 On the othe hnd, if 0, then the stedy-stte gin of the system is given s NV gss lim GR( s) = / Nu ( ) V 0! 0 NV (40) s" v In Eq 40, we hve defined the esulting effetive nvigtion tio s Ñ() = N/( NV /v M ) Suppose the nvigtion tio of the system is N = 3 (fo the = 0 se) Then, exmining Ñ(), if is positive nd (eltively) lge nd the tio V /v M ineses beyond unity (s is often the se, ptiully fo ne hed-on LOS m Seeke dynmis s s s s M Rdome slope Guidne filte f s s enggements), then the effetive nvigtion tio Ñ 2 Rell fom the disussion of PN tht nvigtion gin of 2 o less will led to guidne instbility This ft is illustted in Fig 3 fo N = 3 nd thee diffeent tios of V /v M Convesely, when is negtive, the effetive nvigtion tio Ñ will inese, esulting in dditionl noise thoughput, whih lso n hve detimentl effets on guidne pefomne Dynmi Effets It is shown in Ref 4 tht the guidne system tnsfe funtion given by Eq 39 nd illustted in Fig 2 n be ppoximted futhe by the following simplified tnsfe funtion: G R M( s) NV ( s), s( s) s R (4) In Eq 4, = NV /v M, nd the ppoximte ovell time onstnt, t R, is given by the omposite expession in Eq 42: R ( ) = s f FC A (42) Refeing to Eqs 4 nd 42, when the dynmi pessue is low (eg, high ltitudes nd/o low missile veloity), the eodynmi time onstnt t A tends to tke on eltively lge vlues (the missile tuning te beomes sluggish) Fo positive vlues of, this effet is exebted nd the missile guidne loop beomes even moe sluggish On the othe hnd, when the dynmi pessue is low, lge negtive vlues of oupled with lge t A n indue guidne instbility As n exmple, onside Fig 4, whih illusttes the deleteious effets of dome-indued boesight eo on homing guidne pefomne This exmple uses PN guidne to guide the missile ginst non-mneuveing tget No othe noise soues e onsideed othe thn dome boesight eo Sevel uves e shown, eh epesenting n utopilot esponse pbility given by the time onstnt t = t FC Clely, moe sluggish time onstnt exebtes the effets ˆ NV M M FC s /s Guidne lw s A s v m ody te eodynmis Flight ontol Figue 2 Altente simplified blok digm of (line) missile guidne nd ontol system tht ounts fo seeke dynmis, dome distotion, guidne filte effets, nd utopilot/ifme esponse hteistis of dome eo slope on homing pefomne Guidne System Nonlineities In guided missile, thee e limits to the egion of line opetion Fo exmple, the ltel eletion tht guided missile n ttin is limited in one of two wys: (i) fo low-ltitude inteepts, stutul onsidetions will limit the mximum eletion levels, nd (ii) fo high-ltitude inteepts, the 36
V /v M 035 030 025 020 05 00 005 0 0 oundy = 07 00 Nvigtion tio: N = 3 Effetive nvigtion tio: Ñ () = N/( NV /v M ) Unstble egion: Ñ ( ) N/( NV /v M ) 2 002 003 V /v M = 3 V /v M = 2 V /v M = Stble egion: Ñ ( ) N/( NV /v M ) > 2 004 005 006 Rdome slope (ds) Figue 3 Fo nvigtion tio of 3, the effet of positive dome boesight eo slope () fo inesing tios of losing veloity ove missile veloity is shown To mintin guidne stbility, the effetive nvigtion tio must emin gete thn 2 Closest point of ppoh (ft) 0 9 8 7 6 5 4 3 2 0 00 eletion limit is moe esult of mximum ngle-ofttk limittions In eithe se, the missile guidne nd ontol system design must tke the eletion limits into ount The most signifint implition is tht the ommnded eletion fom the guidne system must be hd-limited so s not to exeed the eletion limits of the missile, whih lso implies 007 = 05 s = 03 s = 0 s = 04 s = 02 s 005 0 005 Rdome eo slope (%) 00 05 Figue 4 This plot illusttes the miss pefomne of PN vesus dome boesight eo slope The esults wee deived by using pln Simulink model ssuming no eo soues side fom dome eo Eh uve epesents the esults using n utopilot with speified time onstnt t A lge time onstnt indites lge lg in the missile esponse to eletion ommnds 008 009 00 ASIC PRINCIPLES OF HOMING GUIDANCE tht the flight ontol system (utopilot) must be designed to podue minimum oveshoot to n eletion step ommnd If/when the guidne ommnd stutes, the missile guidne loop is essentilly opened If guidne ommnd stution pesists long enough, nd if this pesistent stution ous ne inteept, signifint finl miss distne n esult Missile seeke systems lso e pone to nonline stution, whih n hve signifint deleteious effets on ovell guidne pefomne Fo exmple, on gimbled pltfoms, the seeke gimbl ngle n stute unde etin stessing onditions (eg, while pusuing highly mneuveing tget) If this stution ous, the guidne loop hs effetively beome open nd, if this ous ne inteept, signifint finl miss distne n ou Guidne Commnd Pesevtion Thee-dimensionl guidne lws usully genete guidne ommnd veto without onsidetion to how the missile inteepto will effetute the speified mneuve Fo exmple, it is typil fo guided missile to (i) hve no diet ontol ove its longitudinl eletion (eg, xil thust is not typilly thottleble) nd (ii) mneuve in the guidne dietion (speified by the guidne lw) by poduing eletion noml to the missile body Theefoe, the question ntully ises s to how thee-dimensionl guidne ommnd, speified by the guidne lw without onsidetion of the foementioned onstints, n be hieved by mneuveing the missile pependiul to the missile enteline We shll efe to this poess s guidne ommnd pesevtion In Ref, two guidne ommnd pesevtion tehniques e developed The fist ppoh, efeed to hee s GCP, is futhe elboted in Ref 2 The seond guidne ommnd pesevtion tehnique (GCP2) is minimum nom solution As suh, GCP2 ould (theoetilly) povide svings in ommnded eletion (o fuel use in exotmosphei pplitions) s omped with the fist method The GCP tehnique is disussed hee GCP Tehnique To desibe this guidne ommnd pesevtion tehnique, we efe to Fig 5 nd mke the following 37
N F PALUMO, R A LAUWKAMP, nd J M LLOYD = os M Missile eletion body definitions: is unit veto long the missile tget LOS, x is unit veto long the missile enteline (missile longitudinl xis), is the guidne lw eletion ommnd pependiul to the LOS, nd M = [ Mx My Mz ] T is the hieved missile eletion veto ompising omponents in the x/y/z xes Two omponent dietions seve s bsis fo the development of GCP: the dietion of the LOS ( ) nd the ommnded guidne dietion ( = / ) Assuming pefet inteepto esponse to ommnded eletion (ie, no lg fom ommnded to hieved eletion), the (guidne peseved) missile eletion n be expessed s follows: M = k k 2 (43) The design pmetes, k nd k 2, e detemined subjet to the following onstints: : = : = M x Mx M (44) Equtions 44 nlytilly embody the following onstints: (i) the missile does not ontol longitudinl eletion nd (ii) the omponent of missile eletion long the guidne dietion,, must be tht speified by the guidne lw ( ) Using Eqs 43 nd 44 nd given = 0 (ie, the guidne ommnd is pependiul to the LOS), n expession fo ommnded missile eletion tht will peseve the guidne ommnd is given in Eq 45: GL Mx : x = e o : (45) x x LOS Missile enteline M PN guidne ommnd LOS M Ahieved missile eletion Figue 5 GCP The LOS veto is shown long with the ommnded eletion fom PN, whih is pependiul to the LOS Also, the missile enteline (dietion of the missile nose) nd hieved missile eletion veto e supeimposed The guidne pesevtion poblem is to hieve missile eletion tht hs the omponent long the guidne dietion s speified by the guidne lw Eqution 45 is deived independently fom ny ptiul oodinte system It is esy to mehnize the expession in Eq 45 suh tht the omputtions e ied out in the missile body fme () We fist note tht missile eletion esolved into the missile body fme n M Mx My T Mz be expessed s = [ ] nd tht the body-efeened unit veto long the longitudinl xis of the missile n be witten s x = [ 0 0] T The quntities (LOS unit veto) nd (guidne lw ommnd) usully e defined with espet to guidne efeene fme To eflet this ft, we expess them s G = [ 0 0] T nd G = [0 G y G z ] T Hee, the supesipt denotes the fme of efeene with whih the quntities e uently expessed; in this se, the guidne efeene fme (G) Thus, we must esolve ll guidne fme quntities into the missile body fme though the oodinte tnsfomtion C G = G (i, j), i, j =, 2, 3; fo exmple, = C G [ 0 0] T [ G (, ) G (2, ) G (3, )] T Note tht if we fix the guidne fme t the stt of teminl homing (t = 0), we obtin the tnsfomtion CG I = C I (C S t = 0 ) The inetil-to-body tnsfomtion, C I, omes fom the IMU, nd C S my be omputed vi the seeke gimbls (see Eq 23) Subsequently, C G = C I (C G I ) T If neessy, the guidne fme my be updted fom instnt to instnt, but this dds omputtionl omplexity Given tht C G is vilble, the following mehniztion of Eq 45 is possible: GC Mx : x = f p : x R G V R (, ) S W S Mx x / f ( 2, ) ps G W S G(, ) S SG ( 3, ) W S T X T x y z V W W W W X (46) In Eq 46, [ x y z ] T is the guidne ommnd veto expessed in the missile body fme This mehniztion will genete guidne ommnds in the missile body fme tht stisfy the onstints in Eq 44 MIDCOURSE GUIDANCE So f, this tile hs been hiefly onened with the equiements of teminl homing, wheein tget mesuements e povided by one o moe onbod teminl sensos nd minimizing miss distne t inteept is the pimy objetive Fo ompleteness ske, we now will biefly disuss issues nd equiements ssoited with the midouse phse of flight Duing the midouse guidne phse of multimode missile (see Fig 6), tget tking is pefomed by n extenl senso to suppot the enggement 6 Extenl tking elxes the equiements fo the onbod senso to point t the tget nd detet it t lge nges 38
ASIC PRINCIPLES OF HOMING GUIDANCE oost (Inetil) Midouse (Guided) Teminl (Guided) Onbod inetil guidne poessing Sfe lunh/seption oost to flight speed Estblish flight pth Aive t pe-lulted point t end of boost (EO) EO In-flight tget updtes Hndove Seeke quisition Pedited inteept point (PIP) Endgme Fuzing Off-bod tget tking Onbod o off-bod guidne poessing One o moe dditionl booste stges Mintin desied ouse ing the missile lose to the tget Tjetoy shping/enegy mngement Cn be tive endo though exo Onbod seeke/guidne poessing Requies high degee of uy nd fst ommnd esponse Cn equie mneuveing to mximum pbility to inteept fst-moving, evsive tgets Ative endo o exo, typilly not both Pe-lunh Tgeting Tgeting Enggebility Enggebility detemintion detemintion Missile Missile initiliztion initiliztion Figue 6 Missile guidne phses The wepon ontol system fist deides whethe the tget is enggeble If so, lunh solution is omputed nd the missile is initilized, lunhed, nd boosted to flight speed Inetil guidne typilly is used duing the boost phse of flight duing whih the missile is boosted to flight speed nd oughly estblishes flight pth to inteept the tget Midouse guidne is n intemedite flight phse wheeby the missile eeives infomtion fom n extenl soue to ommodte guidne to the tget Duing the midouse phse, the missile must guide to ome within some esonble poximity of the tget nd must povide desible eltive geomety ginst tget when seeke lok-on is hieved (just pio to teminl homing) The teminl phse is the lst nd, genelly, the most itil phse of flight Depending on the missile pbility nd the mission, it n begin nywhee fom tens of seonds down to few seonds befoe inteept The pupose of the teminl phse is to emove the esidul eos umulted duing the pevious phses nd, ultimtely, to edue the finl distne between the inteepto nd tget below some speified level Howeve, the uy of sensos genelly degdes with distne It is theefoe unlikely tht stndoff senso will hve suffiient uy in tking both the tget nd missile to guide missile lose enough fo inteept The objetives duing midouse guidne e insted to guide the missile to fvoble geomety with espet to the tget fo both quisition by the onbod senso nd hndove to teminl homing Duing the midouse guidne phse, the wepon ontol must povide infomtion to the missile bout the thet This infomtion my be nothing moe thn estimtes of the thet kinemti sttes o my inlude pedited inteept point (PIP) Peditions of the futue tget tjetoy, whethe lulted on bod the missile o by the wepon ontol system, e bsed on ssumptions of wht mneuves the thet is likely to do nd wht mneuves e possible fo the thet to pefom These ssumptions typilly e sensitive to the type of thet ssumed but my inlude booste pofiles, eodynmi mneuvebility, o dg oeffiient(s) The PIP is theefoe highly pone to eos Osionl in-flight tget updtes my be needed to impove the uy of the PIP befoe hndove to teminl homing A fom of PN might be used duing the midouse guidne phse but genelly esults in exessive slowdown, howeve, s esult of the dded tmosphei dg geneted by mneuves Insted, it is desible to use guidne lw tht will mximize missile veloity duing the endgme suh tht missile mneuvebility will be mximized when lled fo duing stessing endgme mneuves Lin desibes n ppoh tht pplies optiml ontol theoy to deive effiient, nlytil solutions fo guidne lw tht ppoximtely mximizes the teminl speed while minimizing the miss distne The guidne ommnds e expessed in the fom given in Eq 47: K V 2 K2 sin( ) os( ) V 2 = sin( ) (47) R R Hee, V is the missile speed, R is the nge to the PIP, is the ngle diffeene between the uent nd desied finl veloity, nd s is the heding eo The fist tem shpes the tjetoy to hieve desied ppoh ngle to the inteept, nd the seond tem minimizes miss The gins K nd K 2 e time-vying 39
N F PALUMO, R A LAUWKAMP, nd J M LLOYD nd depend on flight onditions, booste ssumptions, nd othe ftos A itil omponent of implementing this lw is omputing n ute estimte of time-to-go when the missile is still thusting, whih depends on the omponent of missile eletion long the LOS The guidne lultions n be mde on bod the missile o off bod, with eletion ommnds pssed fom the wepon ontol system to the missile whee they e onveted to body oodintes Fo exotmosphei flight, dg is no longe n issue, but long flight times will esult in uved tjetoies s esult of gvity Fom the uent position, the Lmbet solution (see Ref 7 fo simple synopsis) defines the neessy uent veloity to eh teminl inteept point t given time A guidne lw n be wpped ound the Lmbet solution to pogessively stee the missile veloity veto to lign with the Lmbet veloity Then eithe thust temintion o slowdown mneuve n be used to mth the mgnitude of the veloity of the Lmbet solution when the missile booste buns out CLOSING REMARKS The key objetive of this tile ws to povide eltively bod oneptul foundtion with espet to homing guidne but lso of suffiient depth to dequtely suppot the tiles tht follow Fist, we disussed hndove nlysis nd emphsized tht the displement eo between pedited nd tue tget position, e, n be deomposed into two omponents: one long ( e ) nd one pependiul to ( e ) the pedited LOS Thus, beuse the eltive veloity is long the LOS to the pedited tget lotion, the eo long this dietion ltes the time of inteept but does not ontibute to the finl miss distne It is the eo pependiul to the LOS tht must be emoved by the inteepto fte tnsition to teminl homing to effet n inteept Next, we developed lssil fom of PN nd noted tht pimy dvntge of PN, ontibuting to its longevity s fvoed guidne sheme ove the lst five dedes, is its eltive simpliity of implementtion In ft, the most bsi PN implementtions equie low levels of infomtion egding tget motion s omped with othe, moe elbote shemes, thus simplifying onbod senso equiements Moeove, it hs poven to be eltively elible nd obust This ptiul (nd somewht unique) tetment of PN ws tken fom 980 APL memondum witten by Aln J Pue We lso disussed how PN n be mehnized fo guided missile pplitions, with fous on LOS eonstution nd guidne ommnd pesevtion With espet to homing guidne, we itemized the pimy ontibutos to guidne pefomne degdtion tht n ultimtely led to uneptble miss distne The onbod missile seeke hs limited effetive nge beyond whih tget tking is not possible To suppot enggements tht initilly e beyond suh nge, midouse guidne is used to bing the missile within the effetive nge of the seeke Thus, in ontst to teminl homing, duing the midouse guidne phse of flight, the tget is tked by n extenl senso nd infomtion is uplinked to the missile The key objetives duing midouse guidne e to guide the missile to fvoble geomety with espet to the tget fo both quisition by the onbod missile tgeting senso nd to povide eptble hndove to teminl homing Mny of the teminl homing onepts disussed hee nd in the subsequent tiles on moden guidne nd guidne filteing in this issue lso e pplible to developing midouse guidne poliies Thus, minly fo ompleteness, we biefly intodued the poblem of midouse guidne REFERENCES Pue, A J, Popotionl Nvigtion nd n Optiml-Aim Guidne Tehnique, Tehnil Memondum FC(2)80-U-024, JHU/APL, Luel, MD (7 My 980) 2 en-ashe, J Z, nd Yesh, I, Advnes in Missile Guidne Theoy, Amein Institute of Aeonutis nd Astonutis, Reston, VA (998) 3 Loke, A S, Piniples of Guided Missile Design, D Vn Nostnd Compny, Pineton, NJ (955) 4 Shneydo, N A, Missile Guidne nd Pusuit: Kinemtis, Dynmis nd Contol, Howood Publishing, Chiheste, Englnd (998) 5 Shukl, U S, nd Mhpt, P R, The Popotionl Nvigtion Dilemm Pue o Tue? IEEE Tns Aeosp Eleton Syst 26(2), 382 392 (M 990) 6 Witte, R W, nd MDonld, R L, Stndd Missile: Guidne System Development, Johns Hopkins APL Teh Dig 2(4), 289 298 (98) 7 Zhn, P, Ttil nd Sttegi Missile Guidne, 4th Ed, Amein Institute of Aeonutis nd Astonutis, Reston, VA (997) 8 Stlld, D V, Clssil nd Moden Guidne of Homing Inteepto Missiles, Rytheon Repot D985005, pesented t n MIT Dept of Aeonutis nd Astonutis semin (Ap 968) 9 Mille, P W, Anlysis of Line-of-Sight Reonstution Appohes fo SM-2 lok IVA RRFD, Tehnil Memondum FE(94)U-2-45, JHU/APL, Luel, MD (22 De 994) 0 Nesline, F W, nd Zhn, P, Line-of-Sight Reonstution fo Fste Homing Guidne, AIAA J Guid Contol Dyn 8(), 3 8 (Jn Feb 985) Lin, C F, Moden Nvigtion, Guidne, nd Contol Poessing, Pentie Hll, Englewood Cliffs, NJ (99) 2 Abedo, J L, Shot Rnge Anti-Ai Wfe Enggement Simultion, Tehnil Memondum FE(86)U-3-03, JHU/APL, Luel, MD (0 Nov 986) 40
ASIC PRINCIPLES OF HOMING GUIDANCE The Authos Neil F Plumbo is membe of APL s Pinipl Pofessionl Stff nd is the Goup Supeviso of the Guidne, Nvigtion, nd Contol Goup within the Ai nd Missile Defense Deptment (AMDD) He joined APL in 993 fte hving eeived PhD in eletil engineeing fom Temple Univesity tht sme ye His inteests inlude ontol nd estimtion theoy, fult-tolent estutuble ontol systems, nd neuo-fuzzy infeene systems D Plumbo lso is letue fo the JHU Whiting Shool s Engineeing fo Pofessionls pogm He is membe of the Institute of Eletil nd Eletonis Enginees nd the Amein Institute of Aeonutis nd Astonutis Ross A luwkmp eeived SE degee fom Clvin College in 99 nd n MSE degee fom the Univesity of Illinois in 996; both degees e in eletil engineeing He is pusuing PhD fom the Univesity of Illinois M luwkmp joined APL in My 2000 nd uently is the supeviso of the Advned Conepts nd Simultion Tehniques Setion in the Guidne, Nvigtion, nd Contol Goup of AMDD His inteests inlude dynmi gmes, nonline ontol, nd numeil methods fo ontol He is membe of the Institute of Eletil nd Eletonis Enginees nd the Amein Institute of Aeonutis nd Astonutis Justin M Lloyd is membe of the APL Senio Pofessionl Stff in the Guidne, Nvigtion, nd Contol Goup of AMDD He holds S in mehnil engineeing fom Noth Colin Stte Univesity nd n MS in mehnil engineeing fom Vigini Polytehni Institute nd Stte Univesity Cuently, M Lloyd is pusuing his PhD in eletil engineeing t The Johns Hopkins Univesity He joined APL in 2004 nd onduts wok in optimiztion; dvned missile guidne, nvigtion, nd ontol; nd integted ontolle design Fo futhe infomtion on the wok epoted hee, ontt Neil Plumbo His emil ddess is neilplumbo@ Neil F Plumbo Ross A luwkmp Justin M Lloyd jhupledu The Johns Hopkins APL Tehnil Digest n be essed eletonilly t wwwjhupledu/tehdigest 4