INITIAL STATE ESTIMATION FOR A GUN LAUNCHED PROJECTILE IN A SPATIALLY VARYING MAGNETIC FIELD

Transcription

1 NTAL STATE ESTMATON FOR A GUN LAUNCHED PROJECTLE N A SPATALLY VARYNG MAGNETC FELD by Fn Chawla A THESS subttd to Orgon Stat Unvrsty n partal fulfllnt of th rqurnts for th dgr of Mastr of Scnc Prsntd Jun 13, 006 Concnt Jun 007

2 AN ASTRACT OF THE THESS OF Fn Chawla for th dgr of Mastr of Scnc n Elctrcal and Coputr Engnrng prsntd on Jun 13, 006. Ttl: ntal Stat Estaton for a Gun Launchd Projctl n a Spatally Varyng Magntc Fld Abstract approvd: Mark F. Costllo Molly H. Shor Sart wapons pros to provd lap ahad capablty wth rgard to accuracy and ngagnt rang for du and larg calbr projctls. On of th ost crtcal coponnts of a sart wapon syst s ts snsor sut that provds poston, orntaton, and vlocty nforaton as th projctl fls down rang so that ffctv control acton can b takn n flght. Grat strds hav bn ad n cratng vry sall and ruggd nrtal Masurnt Unts (MU) usng MEMS acclrotrs and vbratng gyroscops. Howvr, all MU systs oprat by ntgratng acclrotr and gyroscop asurnts. Thus, thy ust b ntalzd at launch to produc suffcntly accurat poston and orntaton data. Du to nhrnt uncrtanty n shotto-shot launch condtons, for gun launchd projctls, ntal condtons cannot b adquatly spcfd by th frng platfor lk t can wth arcraft and ssls. Currntly, thr s no adquat thod to ntalz MU snsor suts on gun launchd untons.

3 Ths thss nvstgats a novl concpt for dtrnng th full stat of a projctl nar th uzzl of th gun. Th thodology rls on th gun syst nducng a known spatally varyng agntc fld n th vcnty of th uzzl of th gun. Usng radngs fro a clustr of agntotrs bddd wthn th projctl, th full stat of th projctl s dtrnd by solvng a nonlnar st of quatons.

4 Mastr of Scnc thss of Fn Chawla prsntd on Jun 13, 006. APPROVED: Co-Major Profssor, rprsntng Elctrcal and Coputr Engnrng Co-Major Profssor, rprsntng Elctrcal and Coputr Engnrng Drctor of th School of Elctrcal Engnrng and Coputr Scnc Dan of th Graduat School undrstand that y thss wll bco a part of th prannt collcton of Orgon Stat Unvrsty lbrars. My sgnatur blow authorzs rlas of y thss to any radr upon rqust. Fn Chawla, Author

5 ACKNOWLEDGEMENTS would lk to xprss y sncr grattud to Dr. Mark Costllo for hs patnc, gudanc, ncouragnt and support durng y graduat study. H provd to b an xtrly undrstandng prson who xtndd hs full support to y work and at th sa t was rfrshngly crtcal of t. would lk to thank Dr. Molly Shor who consdrd worthy nough to work undr hr gudanc, and wthout whos support t would not hav bn possbl to coplt ths projct. would lk to xtnd y grattud towards Dr. Robrt Hgdon and Dr. M Ln for sparng thr valuabl t and bng on y cott. a gratful to th offc staff of th Elctrcal and Coputr Engnrng, and Mchancal Engnrng Dpartnts. My spcal thanks to y faly and frnds for thr lov and support.

6 TALE OF CONTENTS Pag 1. NTRODUCTON OUTLNE ORGANZATON OF THE THESS.... ACKGROUND.3.1. SMART WEAPON SYSTEMS NERTAL NAVGATON FOR SMART WEAPONS NTAL STATE ESTMATON RGD ODY MODEL OF PROJECTLE REFERENCE FRAME RELATONSHPS PROJECTLE STATE VARALES MAGNETOMETER SENSOR READNGS PROJECTLE NTAL STATE ESTMATON Enhancd Nwton s Mthod Non Lnar Rgrsson MPLEMENTATON AND RESULTS EXPERMENTAL SETUP AND PROCEDURE STATE ESTMATON WTHOUT SENSOR NOSE ar Magnt Magntc Fld... 8

7 TALE OF CONTENTS (contnud) Pag 4... Rctangular Loop Magntc Fld STATE ESTMATON WTH SENSOR NOSE ar Magnt Magntc Fld Rctangular Loop Magntc Fld TRADE STUDES Varaton of Error wth Snsor Nos Lvl Varaton of Error wth Data urst Prod Varaton of Error wth Nubr of Magntotrs CONCLUSON DSCUSSON OF THE RESULTS SCOPE FOR FUTURE WORK... 50

8 LST OF FGURES Fgur Pag 3.1: PROJECTLE GEOMETRY WTH RESPECT TO NERTAL, ODY AND SENSOR REFERENCE FRAMES : EULER ANGLE GEOMETRY OF A PROJECTLE WTH RESPECT TO NERTAL FRAME : SELECTON OF A DATA URST OF A NONLNEAR SEQUENCE OF MAGNETOMETER DATA : ESTMATES OF FRST SX STATES FOR AR MAGNET MAGNETC FELD : ESTMATES OF LAST SX STATES FOR AR MAGNET MAGNETC FELD : ESTMATES OF THE PROJECTLE STATES FOR RECTANGULAR LOOP MAGNETC FELD : ESTMATES OF PROJECTLE STATES WTH 4 10 % SENSOR NOSE WTH AR MAGNET MAGNETC FELD ESTMATES OF PROJECTLE STATES WTH 0.1% SENSOR NOSE WTH RECTANGULAR LOOP MAGNETC FELD : VARATON OF ERROR N ESTMATE WTH SENSOR NOSE : VARATON OF ERROR N ESTMATON OF STATE VALUES AT LOW NOSE LEVELS 4 4.9: VARATON OF FNAL ERROR WTH DATA URST LENGTH WTH MAGNETOMETER READNGS CORRUPTED Y 0.03% NOSE : VARATON OF FNAL ERROR WTH DATA URST LENGTH WTH ACCURATE MAGNETOMETER SENSOR READNGS... 44

9 LST OF FGURES (contnud) Fgur Pag 4.11: VARATON OF FNAL ERROR WTH NUMER OF MAGNETOMETER SENSORS... 46

10 NTAL STATE ESTMATON FOR A GUN LAUNCHED PROJECTLE N A SPATALLY VARYNG MAGNETC FELD 1. NTRODUCTON 1.1. Outln Gun launchd projctls provd a sgnfcant challng for th probl of navgaton. Unlk arcrafts and ssls, whos ntal condtons can b spcfd by th frng platfor, th ntal condtons for gun launchd projctls cannot b adquatly spcfd, du to th nhrnt uncrtanty n shot-to-shot launchng condtons. Sart wapons or Prcson Gudd Muntons ar wapons that can b ad or drctd aganst a sngl targt, rlyng on thr own gudanc systs. On of th ost crtcal coponnts of a sart wapon syst s ts snsor sut that provds poston, orntaton, and vlocty nforaton as th projctl fls down rang so that ffctv control acton can b takn n flght. Grat strds hav bn ad n cratng vry sall and ruggd nrtal Masurnt Unts (MU) usng MEMS

11 acclrotrs and vbratng gyroscops. Howvr, all MU systs oprat by ntgratng acclrotr and gyroscop asurnts. Thus, thy ust b ntalzd at launch to produc suffcntly accurat poston and orntaton data. Currntly, thr s no adquat thod to ntalz MU snsor suts on gun launchd untons. Ths thss nvstgats a novl concpt for dtrnng th full stat of a projctl nar th uzzl of th gun. Th thodology rls on th gun syst nducng a known spatally varyng agntc fld n th vcnty of th uzzl of th gun. Usng radngs fro a clustr of agntotrs bddd wthn th projctl, th full stat of th projctl s dtrnd by solvng a nonlnar st of quatons. 1.. Organzaton of th Thss Chaptr provds th background of ths thss. t brfly xplans th otvaton bhnd th ntal stat staton probl. Chaptr 3 dscrbs th athatcal odl of th syst and dvlops th rlatonshp btwn snsor radngs and th projctl stat. Chaptr 4 prsnts th rsults of th MATLA sulaton for varous dffrnt condtons. Chaptr 5 dscusss th rsults of th sulatons and prsnts th scop for futur work n th ntal stat staton probl.

12 3. ACKGROUND.1. Sart Wapon Systs Sart Wapons or Prcson Gudd Muntons provd a lap-ahad capablty n trs of accuracy and ngagnt rang for du and larg calbr projctls. Thy ar wapons that can b ad at a targt basd on thr own gudanc systs. Sart wapons can b launchd fro arcrafts, shps, subarns, land vhcls or vn by ndvdual soldrs on ground [6]. Typcally, sart wapons hav two coponnts th Gudanc Unt and th Control Unt. Th gudanc unt conssts of a snsor whch snss nrgy orgnatng fro th sourc or targt dstnaton. Th control unt controls th flght of th projctl fro th sourc to th targt... nrtal Navgaton for Sart Wapons nrtal navgaton s th ost accptd soluton for gudanc of sart wapons n ltary navgaton applcatons, snc t dos not nd any xtrnal ad or rfrncs. t can work anywhr and can oprat autonoous, wthout usng antnnas or producng sgnaturs. Thus, t can not b dsturbd or anpulatd by xtrnal sourcs.

13 4 Howvr, rrors n nrtal navgaton grow wth t and nrtal navgaton snsors rqur a sgnfcant calbraton ffort bfor opraton. t has bn shown that a cobnaton of GPS rcvrs and MU snsors can b usd to dtrn accurat poston, vlocty and atttud nforaton of a projctl n flght, f th pr-launch calbraton of MU snsors s don accuratly [10]. ut, th ntalzaton of nrtal navgaton snsors can not b don bfor launch, snc th launch shock changs snsor rrors unprdctably. Thus, nrtal snsor calbraton has to b don n flght [9]. nrtal navgaton s basd on nrtal snsors. Thr ar any typs of nrtal snsors for us wth sart wapons, such as acclrotrs, gyroscops, agntotrs and nrtal asurnt unt snsors []. Acclrotrs asur th acclraton of a pont rlatv to th ground. Gyroscops asur th angular vlocty vctor of th projctl wth rspct to th ground. Magntotrs asur th dot product btwn th agntc fld vctor and th snstv axs of th agntotr at a partcular pont. Thy do not asur any rgd projctl odl stats drctly, and rqur procssng to obtan usful snsor fdback data. Th nrtal Masurnt Unt Snsors ar ult-snsors that utlz thr orthogonal acclrotrs and thr orthogonal gyroscops. Thy us projctl knatc dffrntal quatons to obtan stat stats. Howvr, snc stats ar obtand

14 5 by a nurcal soluton of a nonlnar dffrntal quaton, th ntal condtons ust b suppld to th snsor. Magntotr snsors hav bn usd n navgaton snc cnturs. Advancs n tchnology hav ld to sold stat lctronc copasss, basd on th orgnal agntc copasss usd by salors. Elctronc copasss offr any advantags ovr convntonal ndl typ or gballd agntc copasss, such as shock and vbraton rsstanc, lctronc copnsaton for stray fld ffcts, and drct ntrfac to lctronc navgaton systs [3]. Thy can b usd to sns th strngth and drcton of agntc fld gnratd not only fro th Earth, but also fro prannt agnts, agntzd agnts, and flds gnratd for lctrc currnts. Thus, agntc snsors ar bng usd wth any navgaton control systs.

15 6 3. NTAL STATE ESTMATON Ths chaptr suggsts a soluton to th probl of ntal stat staton. t dscrbs th rgd body odl for a projctl and dvlops th quatons of oton for a agntotr snsor syst. t thn suggsts a soluton to th ntal stat staton probl usng th Enhancd Nwton Mthod and Nonlnar Rgrsson Rgd ody Modl of Projctl Th dfnton of th poston and orntaton of a 6 Dgr Of Frdo (DOF) projctl wth snsors s add by thr an rfrnc fras dfnd as follows -fra: Th ground s usd as an nrtal rfrnc fra. t s fxd to th surfac of arth and stuatd such that ponts down. J ar n th plan of ground and K -fra: Th body fra s locatd at th projctl ass cntr. t s fxd such that ponts out of th nos of th projctl and handd syst. J K for a rght S-fra: Th snsor fra s fxd on th rgd body and algnd wth a snsor. Th snsor fra s dfnd such that th outputs of th th snsor ar along S, J, K. S S

16 7 Fgur 3.1 shows th gotry of a projctl wth rspct to ths thr rfrnc fras. nrtal Fra s fxd to surfac of th arth, ody Fra s cntrd at th Cntr of Gravty of th projctl and Snsor Fra s cntrd at th Cntr of Gravty of th snsor, whch s placd on th projctl. Snsor Fra (S) ody Fra () nrtal Fra () -z x y Fgur 3.1: Projctl Gotry wth rspct to nrtal, ody and Snsor Rfrnc Fras Th rgd body odl of a projctl can b dfnd by usng coponnts of th poston vctor of ass cntr n nrtal rfrnc fra ( x, y, z ) and body orntaton Eulr angls ( φ, θ, ψ ). Fgur 3. shows th gotry of th projctl wth rspct to

17 8 orntaton of th projctl n th nrtal fra, wth Eulr roll (φ ), ptch (θ ) and yaw (ψ ) angls. φ φ ψ J K Fgur 3.: Eulr Angl Gotry of a Projctl wth rspct to nrtal Fra 3.. Rfrnc Fra Rlatonshps Th nrtal Rfrnc Fra and ody Rfrnc Fra ar rlatd by a squnc of thr sngl-axs ody Fxd Rotatons. Startng wth th nrtal fra and rotatng through angl ψ about axs K gnrats ntrdat Fra 1. Rotatng ths fra through an angl θ about th axs J 1 gnrats th No Roll Fra (NR). Rotatng NR

18 9 through an angl φ about th axs NR gnrats th ody Fra (). Ths rotatons can b xprssd athatcally as shown n quatons (3..1) (3..3). 1 cosψ snψ 0 J 1 snψ cosψ 0 = J K K NR cosθ 0 snθ 1 J NR = J1 K NR snθ 0 cosθ K NR J 0 cosφ snφ = J NR K 0 snφ cosφ K NR (3..1) (3..) (3..3) Th thr quatons can b cobnd togthr to gnrat th quaton for th transforaton btwn th nrtal Fra and th ody Fra shown n quaton (3..4) c θcϕ cθ sϕ sθ J = sφ sθ cϕ cφ sψ sφ sθ sϕ + cφ cϕ sφ cθ J = [ T ] J K c s c + s s c s s s c c c K K φ θ ϕ φ ψ φ θ ϕ φ ψ φ θ (3..4)

19 10 Slarly, th transforaton btwn th ody Fra and Snsor Fra s gvn by quaton (3..5) J K S S S = [ T ] S J K (3..5) 3.3. Projctl Stat Varabls A rgd projctl has sx dgrs of frdo. Snc ach dgr of frdo gnrats a scond ordr dffrntal quaton, th odl rqurs 1 stat varabls. Th full stat of th projctl s gvn by quaton (3.3.1). x y z φ θ ψ ξ = u v w p q r (3.3.1) whr, x, y, z = Coponnts of poston vctor of th ass cntr n an nrtal fra.

20 11 φ, θ, ψ = Eulr roll, ptch and yaw angls. u, v, w = Coponnts of vlocty vctor of th ass cntr n an nrtal fra. p, q, r = Coponnts of angular vlocty of th syst n th body rfrnc fra. Th vlocty vctor dfnton for a rgd body s gvn n quatons (3.3.) and (3.3.3). V = x + yj + zk (3.3.) CG / V = u + vj + wk CG / (3.3.3) Coparng, quatons (3.3.) and quaton (3.3.3) usng quaton (3..4), w can wrt quaton (3.3.4) c c s s c c s c s c + s s x θ ψ φ θ ψ φ ψ φ θ ψ φ ψ u y = c s s s s + c c c s s s c v θ ψ φ θ ψ φ ψ φ θ ψ φ ψ z w s s c c c θ φ θ φ θ (3.3.4) Slarly th dfnton of angular vlocty for a rgd body s gvn by quatons (3.3.5) and (3.3.6). ω = ψ K + θ J + φ (3.3.5) / 1 NR ω = p + qj + rk / (3.3.6) Coparng quatons (3.3.5) and (3.3.6) usng quaton (3..4), w can wrt quaton (3.3.7)

21 1 1 s t c t φ φ θ φ θ p θ = 0 c φ s φ q ψ 0 s / c c / c r φ θ φ θ (3.3.7) Equatons (3.3.4) and (3.3.7) ar th Knatc Dffrntal Equatons of th projctl Magntotr Snsor Radngs Nar th uzzl of th gun, t s assud that a agntc fld s gnratd that s a functon of th spatal poston of th snsor. t s ost asly dfnd n th ground fra (-fra). A = ) K (3.4.1) x ( ro A ) + y ( ro A ) J + z ( ro A whr, A s th agntc fld at pont A n spac. Th agntc fld xprncd by a agntotr ovng n spac s gvn by S = ( r ) = ( x, y, z ) + ( x, y, z ) J + ( x, y, z ) K (3.4.) S O A x y For an dal agntotr, th coponnts of th agntc fld ar rcordd n th S rfrnc fra. S x S y = ( x, y, z ) + ( x, y, z ) J + ( x, y, z ) K (3.4.3) Equatng xprssons of S fro quatons (3.4.) and (3.4.3) n fra S ylds z S z S

22 13 x y z = [ T ][ T ] S x y z (3.4.4) x y = z x [ T ] T + [ T ][ T ] S y z S x y z (3.4.5) Lk all snsors, ral agntotrs xprnc rrors such as nos, bas, cross axs snstvty and scal factor. Thus a ral agntotr output taks th for ~ ~ ~ x y z n = n n x y z b + b b x y z s + c c xx yx zx c s c xy yy zy c c s xz yz zz x y z (3.4.6) Whch, fro quaton (3.4.4) s quvalnt to ~ ~ ~ x y z n = n n x y z b + b b x y z s + c c xx yx zx c s c xy yy zy c c s xz yz zz [ T ][ T ] S x y z (3.4.7) And,

23 14 ~ ~ ~ x nx sxx y = n y + c yx czx nz z c s c xy yy zy c c s xz yz zz x [ T ] T + [ T ][ T ] S y z S x y z (3.4.8) W shall now gnrat an xprsson for T. Fro quaton (3..4), w hav T J = [ T ] J J = [ T ] J K K K K (3.4.9) Also, not that d dt dj dt dk dt = T 11 + T 1 J + T 13 = T1 + T J + T3 K = T31 + T3 J + T33 K K d dt dj T T J T [ T ] J = dt = K K dk dt (3.4.10) Usng th fra drvatv rlatonshp,

24 15 d dt d = + ω / x = ( p + qj + dt d dt rk ) x = rj qk (3.4.11) Slarly, dj dt dk dt = r + pk (3.4.1) = q pj (3.4.13) Thrfor, d dt dj dt dt dk 0 = r q r 0 p q p J 0 K (3.4.14) Equatng xprssons (3.4.10) and (3.4.14), T 0 = r q r q 0 p p 0 [ ] T (3.4.15) Snc x, y and z ar functons of quantts x, y and z,

25 = x x x x z z y y x x (3.4.16) + + = y y y y z z y y x x (3.4.17) + + = z z z z z z y y x x (3.4.18) [ ] = = E x x x y y y x x x z y x z y x J z y x x x x x x x z y x (3.4.19) Th poston of th th agntotr s rlatd to th ass cntr poston by th followng quaton [ ] T x x SL y y T L WL z z = + (3.4.0) whr, CG O K z yj x r + + = CG K WL J L SL r + + = Takng a drvatv of ths xprsson ylds

26 17 x x SL T y = y + T L WL z z (3.4.1) Usng rotatonal knatc quaton (3.3.4) and quaton (3.4.15) w gt x u 0 r q SL T y [ T ] v r 0 p = + L w q p 0 WL z (3.4.) R-xprssng th dal agntotr quatons, x y z = [ T ][ T ] S x y z (3.4.3) x 0 r q x u 0 r q SL T y [ T S ] r 0 p [ T ] y [ TS ][ T ][ JE ][ T ] v r 0 p = + + L q p 0 w q p 0 WL z z (3.4.4) Substtutng quaton (3.4.3) nto quaton (3.4.4),

27 18 x 0 r q x u 0 r q SL T T y [ T S ] r 0 p [ TS ] y [ TS ][ T ][ JE ][ T ] v r 0 p = + + L q p 0 w q p 0 WL z z (3.4.5) Thus th full stat of th rgd projctl s contand n th agntotr and agntotr drvatv xprssons. Th translatonal vlocty coponnts ( u, v, w ) and th angular rat coponnts ( p, q, r ) ar prsnt n th t drvatvs of th agntotr radngs. Thus, n ordr for th translatonal vlocty coponnts to b contand n x, y and z th agntc fld around th uzzl of th gun ust b spatally varyng. f th agntc fld s constant n th ara around th gun uzzl, thn th atrx J E s zro and th dpndnc of x, y and z on u, v and w s lnatd. Howvr, th angular rat coponnts p, q and r appar twc; onc wth th agntc fld coponnt havng a spatally varyng natur, and onc wthout t. Thus p, q and r ar prsnt n x, y and z vn whn th agntc fld s spatally constant.

28 Projctl ntal Stat Estaton A spatally varyng and known agntc fld s st up nar th gun uzzl. Ths agntc fld s sapld by th agntotrs that pass through th fld. t s assud that an array of agntotrs ountd on th projctl rcords a fnt sapl of asurnts n th ara around th uzzl of th gun. Th probl s splt nto two parts statng th frst sx stats of th projctl, and thn usng Projctl Knatc Equatons to fnd th last sx stats Enhancd Nwton s Mthod To solv th frst part of ths probl, consdr th agntotr quaton gvn n quaton (3.4.3). Gvn th agntotr data at so t, and an stat of th stats, th rsdual of ths quatons s coputd. f f f x y z = x y z [ T ][ T (, θ, ϕ) ] S x φ y (3.5.1) z whr f x, f y, f z, ar th quaton rsduals. f N tr-axal agntotrs ar ountd on th projctl, th rsdual vctor s dfnd n th followng annr. F = { F } (3.5.)

29 0 whr, F fx 1 f y1 f z1 f x f y = fz f xn f yn f zn Th goal of th staton procdur s to fnd th stat vctor X = [ x y z φ θ ψ ] such that F s suffcntly clos to zro. A nonlnar last squars procdur s usd for ths purpos whr a rt functon dfnd blow s nzd. J 1 T = F WF (3.5.3) Not that W s a postv dfnt sytrc atrx. To nz th rt functon, t s approxatd as a quadratc functon n th ntal projctl stat vctor. whr, J J 1 T J X ) J 0 + xo ( X X o ) + ( X X o ) xo ( X X o ) (3.5.4) X X (

30 1 ) ( ) ( 1 o o T o X WF X F J = J J J J X x y ψ = Now, WF X F F J X F X J T T = = (3.5.5) And, J J J x x y x r x J J J J x y y y r y X J J J x y r ψ ψ ψ = (3.5.6) + = X F W X F WF X F x WF X F x WF X F x X J T T N T T 1 (3.5.7) Solvng for X to forc X J =0, 0 ) ( = + xo X o X X J X J Or,

31 { X} { X } 1 J J = 0 X X xo xo (3.5.8) Snc our nonlnar quatons ar not prfctly odld by a quadratc, w tratvly prfor ths opraton untl convrgnc. { X } { X } J J = X X (3.5.9) Furthror, th nonlnar quatons can b so poorly odld by a quadratc that a nw pont ay actually gnrat a hghr valu of cost. To gt around ths, w us th Enhancd Nwton s Mthod, whr w altr th pur Nwton s Mthod wth a ln sarch scalar paratr. { X } { X } J J = α X X (3.5.10) whr, α s a ln sarch paratr and th tr drcton. J X 1 J X gvs th sarch Th ln sarch paratr s dtrnd by a back stppng procdur n whch α s assud to b 1, at ach traton, to coput th nxt stat. Ths nw stat s usd to calculat th nw cost functon J 1. f J1 βj 0, thn α = 1 s accptd and th

32 3 procss s rpatd. On th othr hand, f J1 > βj 0, thn α s st to α / k, and J s coputd. f J βj 0 thnα = 1 s accptd and th procss s rpatd Non Lnar Rgrsson To solv th scond part of th probl, w fnd th drvatvs of th stats of th frst sx stats usng Non Lnar Rgrsson, and us th Projctl Knatc Equatons to fnd th translatonal and angular vlocty coponnts of th stat. For fndng th drvatv of an statd stat, consdr ts stat ovr svral nstancs of t. For xapl, lt us consdr th stat of stat x ovr p nstancs of t. That s, th stat vctor [ x1 x x p ] corrsponds to t nstants [ ]. W assu that th data s ft to an th ordr polynoal, such that t1 t t p th followng quatons hold. a + a t + a t + + a t = x a + a t + a t + + a t = x 0 1 a + a t + a t + + a t = x 0 1 p p p p Ths can b wrttn n atrx for as follows

33 4 a 0 1 t1 t1 t 1 1 a 1 1 t x t t a = 1 t x p t p t p p a x (3.5.11) Lt, 1 t1 t1 t 1 1 t t t = 1 t p t p t p a0 a 1 A = a a X x1 x = x p Th quaton can b rwrttn as A = X (3.5.1) Equaton (3.5.1) can b solvd for th vctor A to fnd th coffcnts of th nonlnar quatons. Snc s a rctangular atrx, w wll hav to us th Last Man Squar soluton of quaton (3.5.1) to solv for A.

34 5 Onc th curv through a stat s found, w can fnd ts drvatv as follows a + a t + + a t = x a + a t + + a t = x 1 1 a + a t + + a t = x p p p 1 t t a x t x t a = 1 1 t x p t p a p (3.5.13) Stats u, v, w can b found by usng drvatvs of stats of stats x, y, z and th Projctl Knatc Equaton (3.3.4). Slarly, stats p, q, r can b found by usng th drvatvs of statd stats φ, θ, ψ and Projctl Knatc Equaton (3.3.6).

35 6 4. MPLEMENTATON AND RESULTS Havng suggstd th soluton to th ntal Stat Estaton probl, w now provd th prcal vrfcaton of th hypothss. Ths chaptr prsnts th rsults of th sulatons and trad studs prford for dffrnt snsor gotrs and agntc flds. Th sulatons wr prford usng MATLAv Exprntal Stup and Procdur n ordr to stat th ntal stat of th projctl, agntotr array snsor data has to b obtand. Accordng to th thodology dscrbd n Chaptr 3, snsor data for ach snsor n th array s rqurd ovr a prod of t. Th prod of t ovr whch ths data s obtand, should b chosn carfully. t should b short nough to nsur lnar varaton of data, but long nough to gnrat nough nubr of sapls for fndng stat drvatvs. For xapl, fgur 4.1 shows th varaton of snsor radngs of a sngl agntotr wth t. W would lk to slct a squnc of as any ponts fro ths data st, as possbl; snc a gratr nubr of ponts would gnrat or accurat stat drvatv stats. Howvr, f w pck th coplt data st, nonlnarts n th data would prvnt us fro statng stat drvatvs

36 7 accuratly. Thus, w pck th data ponts nclosd by th rd rctangl n th fgur, to slct a prod whch s alost lnar and gnrats nough data ponts. Fgur 4.1: Slcton of a data burst of a nonlnar squnc of agntotr data Aftr an approprat slcton of agntotr data s ad, th frst sx stats of th projctl ar statd at ach data pont, usng th Enhancd Nwton s Mthod as dscrbd n Scton Aftr th frst sx stats ar statd at ach t nstant ovr th prod of data burst, th nxt sx stats ar dtrnd usng Nonlnar Rgrsson, as dscrbd n Scton 3.5..

37 8 Thus, an stat of th full stat s obtand by startng fro agntotr snsor radngs. 4.. Stat Estaton Wthout Snsor Nos Whn thr s no snsor nos, that s, ach agntotr n th snsor array asurs th agntc fld accuratly, th thodology works xtrly wll. Th followng sctons show th rsults for a agntc fld gnratd by a currnt flowng through a a bar agnt and a rctangular currnt carryng loop ar Magnt Magntc Fld Ths scton prsnts th rsults of stat staton whn th projctl s travlng n a agntc fld gnratd by a bar agnt placd on th top of th uzzl of th launchng gun. Th agntc fld radngs ar obtand ovr 5 t stps by a snsor array havng fv agntotrs placd randoly on th projctl. Th avrag of th fnal rror ovr th 5 t stps s found to b of th % actual valus. Th graphcal rsults ar as shown n fgurs 4. and 4.3.

38 Stat No 1 Stat No Varaton of x Varaton of y Varaton of thta T Axs Stat No 3 Varaton of z T Axs Stat No 5 Varaton of ph Varaton of ps T Axs Stat No Actual valu of stat Estatd valu of stat T Axs x 10-4 Stat No 6 T Axs T Axs Fgur 4.: Estats of frst sx stats for ar Magnt agntc fld

39 30 Stat no 7 Stat no 8 Varaton of u Varaton of v T Axs Stat no 9 T Axs Stat no 10 Varaton of w Varaton of p Varaton of q T Axs Stat no 11 Varaton of r T Axs Stat no 1 T Axs T Axs Fgur 4.3: Estats of last sx stats for ar Magnt agntc fld

40 Rctangular Loop Magntc Fld Ths scton prsnts th rsults of stat staton whn th projctl s travlng n a agntc fld gnratd by currnt flowng through a 5c by 3c rctangular loop conductor, cold around th uzzl of th launchng gun. Th agntc fld radngs ar obtand ovr 5 t stps by a snsor array havng fv agntotrs placd randoly on th projctl. Th avrag of th fnal rror ovr th 5 t stps s agan found to b %. Th graphcal rsults ar as shown n fgur 4.4. Stat No 1 Stat No Varaton of x Varaton of y T Axs Stat No 3 Varaton of z T Axs Varaton of ph T Axs Stat No Actual valu of stat Estatd valu of stat T Axs x 10-4

41 3 Stat No 5 Stat No 6 Varaton of thta Varaton of ps T Axs Stat no 7 T Axs Stat no 8 Varaton of u Varaton of v T Axs Stat no 9 T Axs Stat no 10 Varaton of w Varaton of p T Axs T Axs

42 33 Stat no 11 Stat no 1 Varaton of q Varaton of r T Axs T Axs Fgur 4.4: Estats of th projctl stats for Rctangular Loop agntc fld Thus, both bar agnt and rctangular loop gnratd flds show xactly sa rsults whn snsor data s prfct Stat Estaton wth Snsor Nos Whn snsor nos s prsnt staton of projctl stat s dpndnt upon th prcntag of nos prsnt n th snsor radngs. Ths scton prsnts th rsults of stat staton wth rctangular loop and bar agnt gnratd agntc flds whn snsor nos s prsnt ar Magnt Magntc Fld Th staton rror s drctly dpndnt on th aount of nos addd to th snsor radngs. Ths scton prsnts th staton rsults for a projctl travlng n a

43 34 agntc fld gnratd by a bar agnt placd on th top of th uzzl of th launchng gun. Th agntc fld radngs ar obtand ovr 5 t stps by a snsor array havng fv traxal agntotrs placd n a rng around th projctl. Th snsor radngs ar corruptd by zro an, wht gaussan nos whch s approxatly 4 10 % of th snsor radngs. Ths s found to b th hghst lvl of nos that ths syst could tolrat, wthn rasonabl rror bounds. Th avrag of th fnal rror ovr th 5 t stps s found to b 0.05%, whch s consdrably hghr than th no nos cas. Th graphcal rsults ar as shown n fgur 4.5. Stat No 1 Stat No Varaton of x Varaton of y T Axs T Axs

44 Stat No 3 Stat No 4 Varaton of z Varaton of ph T Axs Stat No 5 T Axs Stat No 7 Varaton of thta Varaton of ps T Axs Stat No 6 T Axs Stat No 8 Varaton of u Varaton of v T Axs T Axs

45 36 Stat No 9 Stat No 10 Varaton of w Varaton of p T Axs Stat No 11 T Axs Stat No 1 Varaton of q Varaton of r T Axs T Axs 4 Fgur 4.5: Estats of Projctl Stats wth 10 % snsor nos wth ar Magnt agntc fld Rctangular Loop Magntc Fld Ths scton prsnts th staton rsults for a projctl travlng n a agntc fld gnratd by currnt flowng through a 5c by 3c rctangular loop conductor, cold around th uzzl of th launchng gun. Th agntc fld radngs ar obtand ovr 5 t stps by a snsor array havng 11 rngs, ach havng 36 agntotrs,

46 37 placd around th projctl, and 3 rngs, ach havng 4 agntotrs, bddd nsd th projctl. Th snsor radngs ar corruptd by zro an, wht gaussan nos whch s approxatly 0.1% of th snsor radngs. Ths s found to b th hghst lvl of nos that ths gotry could tolrat, wthn rasonabl rror bounds. Th avrag of th fnal rror ovr th 30 t stps s approxatly found to b 0.05%. Thus, th stat rsults for a squar loop gnratd agntc fld ar consdrably bttr than thos for a bar agnt gnratd fld n th cas of nosy snsor radngs. Th graphcal rsults ar as shown n fgur 4.5. Stat No 1 Stat No Varaton of x Varaton of y T Axs T Axs

47 Stat No 3 Stat No 4 Varaton of z Varaton of ph Varaton of thta T Axs Stat No 5 Varaton of ps T Axs Stat No 6 T Axs Stat no 7 T Axs Stat no 8 Varaton of u Varaton of v T Axs T Axs

48 39.05 Stat no 9 Actual valu of stat Estatd valu of stat Stat no 10 Varaton of w Varaton of p 1.85 Varaton of q T Axs x 10-4 Stat no 11 Varaton of r T Axs Stat no 1 T Axs T Axs Fgur 4.6 Estats of Projctl Stats wth 0.1% snsor nos wth Rctangular Loop agntc fld Snc rctangular loop agntc fld gnrats bttr stat stats as copard to bar agnt agntc fld, w wll us th rctangular loop agntc fld for all rsults hncforth, unlss spcfd. Th agntc fld radngs wll b obtand fro a snsor array havng 11 rngs, ach havng 36 traxal agntotrs, placd around th projctl, and 3 rngs, ach havng 4 traxal agntotrs, bddd nsd th projctl.

49 Trad Studs Ths scton dscrbs how th accuracy of th staton vars accordng to changs n varous factors such as, nos lvls, prod of data burst, and snsor gotry. Snc th rctangular loop prfors bttr n nosy condtons, th trad studs ar prford usng th rctangular loop agntc fld Varaton of Error wth Snsor Nos Lvl Th staton accuracy dcrass as snsor nos lvls ar ncrasd. Th fnal rror of staton ncrass lnarly wth snsor nos. Ths can b sn fro Fgur 4.7, whch shows th avrag of th fnal rror ovr 5 t nstants for svral nos lvls. Th fnal rror plottd s also calculatd as a prcntag of th actual stat radngs. Nos s asurd as a prcntag of xact agntotr snsor data.

50 41! " # $ & % $ Fgur 4.7: Varaton of Error n Estat wth Snsor Nos Snc th rror varaton at low nos lvls s not vry clar fro th fgur, Fgur 4.8 has bn ncludd to show th stady ncras of th rror wth nos prcntag at xtrly low nos lvls.

51 4! " # $ & % $ Fgur 4.8: Varaton of Error n Estaton of Stat Vals at Low Nos Lvls Varaton of Error wth Data urst Prod Data burst prod plays a vry portant rol n dtrnng th accuracy of th stat stat. Th prod of data burst rqurd s dpndnt upon th snsor nos lvls. n low nos condtons, a lowr data burst prod gvs a or accurat rsult than a hgh prod, snc ncrasng th nubr of t stps also ncrass th nonlnarts n th snsor radngs and stat valus. Howvr, at hgh nos lvls, a longr prod of data burst works bttr snc staton of th last sx stats by usng nonlnar rgrsson (dscrbd n scton 3.5.) s or accurat wth hghr nubr of data ponts. Fttng a curv through th frst sx stats also acts as a crud for of fltrng

52 43 out th nos, and ths fltrng ffct ovrshadows th ll-ffcts of nonlnarts to a crtan xtnt. Ths can b sn or clarly fro a coparson of fgurs 4.9 and Fgur 4.9 shows th varaton of fnal rror wth rspct to data burst prod n th cas of agntotr radngs corruptd by 0.03% nos. Fgur 4.10 shows th sa rsults for snsor radngs wthout any nos. t can b sn fro th two fgurs that whl th rror falls n th frst cas, t ncrass stadly n th scond.! " ' $" ( ) & ( ) ' $" Fgur 4.9: Varaton of Fnal Error wth Data urst Lngth wth agntotr radngs corruptd by 0.03% Nos

53 44 t can b sn fro Fgur 4.9 that as th nubr of data ponts s ncrasd, th fnal rror n staton dcrass. Ths fall n th rror s alost quadratc. Howvr, aftr data burst lngth of 5, th rror bgns to ncras agan. Ths s bcaus aftr ths pont th postv ffcts of th crud fltrng dscrbd arlr can no longr ovrshadow th ffcts of ncrasng nonlnarts n th data.! " ' $" ( ) & ( ) ' $" Fgur 4.10: Varaton of Fnal Error wth Data urst Lngth wth accurat agntotr snsor radngs

54 45 Th rsults n Fgur 4.10 ar n a total contrast to thos n Fgur 4.9. Whn thr s no snsor nos, th fnal rror ncrass stadly wth th nubr of data ponts, bcaus thr s no ffct of fltrng, snc no snsor nos s prsnt Varaton of Error wth Nubr of Magntotrs Th nubr of agntotrs and th arrangnt of snsors on th projctl also hav an ffct on th accuracy of th stat stat whn snsor data s nosy. Ths s bcaus crtan gotrs gnrat a bttr sgnal to nos rato copard to othrs. n gnral, ncrasng th nubr of agntotrs n th snsor array lads to an ncras n accuracy of th thod. Ths can b sn or clarly fro Fgur 4.11, whch shows th rsults for th stat staton wth approxatly 0.03% snsor nos.

55 46 &, % * -$... & % * + $ Fgur 4.11: Varaton of Fnal Error wth Nubr of Magntotr Snsors Fgur 4.11 prsnts th rsults for snsor gotry wth n rngs, ach havng traxal agntotrs fxd sytrcally on th projctl. Th rsults show that as th nubr of rngs (n) s ncrasd, th rror dcrass. Also, ncrasng th nubr of agntotrs pr rng () causs th rror to dcras stadly.

56 47 5. CONCLUSON Ths chaptr dscusss th rsults prsntd n Chaptr 4 and th utlty of th agntotr snsor syst for statng ntal stat of gun launchd projctls. t also prsnts th furthr provnts that can b ad to th dsgn and scop for futur work rlatd to ths syst Dscusson of th Rsults Th rsults shown n Chaptr 4 ndcat that th suggstd soluton to th ntal stat staton probl works vry wll for a no nos cas, rrspctv of th snsor gotry, nubr of agntotrs and th agntc fld usd. Howvr, th rsults show a clar dpndnc on ths dsgn paratrs whn snsor nos n ncrasd. As th nos prcntag of snsor radngs s ncrasd, th bhavor of th syst changs wth th agntc fld usd for th staton. Whl a bar agnt gnratd agntc fld syst can not wthstand snsor nos hghr than %, th axu snsor nos that th syst can wthstand wth a rctangular loop gnratd fld s approxatly 0.1%. Th rctangular loop agntc fld, thus, gvs us practcally usabl rsults, snc agntotr snsors ar known to b vry accurat and nos lvls ar not xpctd to xcd ths valu.

57 48 Th dffrnc n th rsults for th two knds of agntc flds can b attrbutd to th spatal varaton th flds. Fgurs and show th spatal varaton of th bar agnt and squar loop gnratd flds. t can b sn fro th fgurs that th spatal varaton of th squar loop fld s uch or than th spatal varaton of th bar agnt fld. Thus, t can b concludd that th spatal varaton of th agntc fld n whch th projctl travls plays a vry portant rol n th accuracy of th stat. Spatal Varaton of th total Magntc Fld n th X-Y Plan 0.5 Spatal Varaton of total Magntc Fld n th X-Z Plan Dstanc n y drcton Dstanc n z drcton Dstanc n x drcton Dstanc n x drcton Spatal Varaton of total Magntc Fld n th Y-Z Plan Dstanc n z drcton Dstanc n y drcton Fgur 5.1: Plot of Magntc Fld gnratd by a ar Magnt

58 49 Spatal Varaton of th total Magntc Fld n th X-Y Plan 0.5 Spatal Varaton of total Magntc Fld n th X-Z Plan Dstanc n y drcton Dstanc n z drcton Dstanc n x drcton Dstanc n x drcton Spatal Varaton of th total Magntc Fld n th Y-Z Plan Dstanc n z drcton Dstanc n y drcton Fgur 5.: Plot of Magntc Fld gnratd by a Rctangular Loop Th nubr of agntotrs usd, and th snsor gotry also hav an ffct on th staton as snsor nos ncrass. Ths s bcaus spcfc arrangnts of th snsors on th projctl body hlp n canclng out th ffcts of snsor nos, and ncrasng th Sgnal to Nos Rato (SNR). Howvr, th agntotr gotry that nsurs good rsults at 0.1% snsor nos for a rctangular loop agntc fld dos not s to b practcally ralzabl. Ths s bcaus th nubr of agntotrs (468) s too hgh for a practcal ralzaton.

59 50 Morovr, apart fro th snsor nos lvls, th rsults of th staton ar hghly dpndnt upon th data burst prod and th nubr of snsors also. t s ntrstng to s how nos lvl affcts th prod of data burst rqurd for an accurat stat. n hgh nos cass, ost of th rror n staton cos fro th rror n th last sx stats, and ths can b ovrco by usng a largr data burst prod. Howvr, as can b sn fro th rsults shown n Fgur 4.9, rrors du to nonlnarts start to st n f th data burst prod s ncrasd byond a crtan lngth. 5.. Scop for Futur Work Most of th rror n th stat staton procss cos fro th us of nonlnar rgrsson n statng th last sx stats fro th frst sx stats. Thus, th staton accuracy can b provd by usng or accurat sgnal procssng algorths for fndng th drvatv of th stats of th frst sx stats. Th ffcts of agntc fld proprts on th stat can b furthr nvstgatd and agntc flds that gnrat or accurat stats can b found. Also, by studyng th ffcts of snsor arrangnts on sgnal to nos rato, practcally ralzabl snsor gotrs, wth fwr agntotrs, that gv bttr stat stats could b found.

60 51 Mor thods to stat substs of th stat vctor could also b nvstgatd. For xapl, a subst of th snsor sut could b swtchd on and off, statng a part of th stat vctor ach t, and fnally th rsults could b cobnd to obtan a quckr and or accurat stat of th coplt stat. A agntotr snsor array could also b cobnd wth othr typs of snsors such as acclrotrs and gyroscops to nvstgat th provnts n spd and accuracy of staton.

61 5 LOGRAPHY 1. A. alogh, C.M. Carr, M. H. Acuna, M. W. Dunlop, T. J. k, P. rown, K. H. Fornacon, E. Gorgscu, K. H. Glassr, J. Harrs, G. Musann, T. Oddy and K. Schwngnschuh, Th Clustr Magntc Fld nvstgaton : n-flght Prforanc and ntal Rsults, Europan Gophyscal Socty, Nl M. arbour, John M. Elwll and Roy H. Sttrlund, nrtal nstrunts Whr to Now?, n AAA Gudanc, Navgaton and Control Confrnc, 199, p M. J. Caruso, Applcaton of Magntorsstv Snsors n Navgaton Systs, Snsors and Actuators 1997, SAE SP-10 (15-1), Fbruary Mark Costllo and Thanat Jtprapha, Dtrnng Angular Vlocty and Angular Acclraton of Projctls Usng Traxal Acclraton Masurnts, Journal of Spaccrafts and Rockts, Vol. 39, No. 1 (73-89), Jrry H. Gnsbrg, Advancd Engnrng Dynacs, Cabrdg Unvrsty Prss, Scond Edton. 6. Rchard P. Hallon, Prcson Gudd Muntons and th Nw Era of Warfar, Ar Powr Studs Cntr Workng Paprs.

62 53 7. Wlla H. Hayt, Jr., Engnrng Elctroagntcs, Tata McGraw-Hll Publshng Copany Ltd, Nw Dlh, Ffth Edton. 8. Gary L. Katulka, Mcro-lctrochancal Systs and Tst Rsults of SC MEMS for Hgh-g Launch Applcatons, Snsors, 00, Procdngs of EEE. 9. Thoas Kuhn, Aspcts of Pur and Satllt Add nrtal Navgaton for Gun Launchd Muntons, Poston Locaton and Navgaton Syposu, EEE, Roy R. Mnor and Davd W. Row, Utlzaton of GPS/MEMS-MU for Masurnt of Dynacs for Rang Tstng of Mssls and Rockts, Poston, Locaton and Navgaton Syposu, EEE Wlla H. Prss, Saul A. Tukolsky, Wlla T. Vttrlng and ran P. Flannry, Nurcal Rcps n C Th Art of Scntfc Coputng, Cabrdg Unvrsty Prss, Scond Edton. 1. Lloyd N. Trfthn, Davd au, Nurcal Lnar Algbra, Socty for ndustral and Appld Mathatcs.