A Design Configuration and Optimization for a Multi Rotor UAV

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1 UNCLSSIFIED/UNLIITED Desgn Confguon n Opmzon fo ul oo UV Els Cpello, Gogo Gugle, Fulv Quglo olecnco Tono DIS Coso Duc egl buzz 4 9 Tono (Ily) els.cpello@polo. -fulv.quglo@polo. - gogo.gugle@polo. STCT ul oo UVs offe ge poenl we nge of chllengng pplcons ue o he hgh mnoeuvbly n o he poenl o hove, ke off n fly n smll es. Neveheless, he esgn s n some wy ccl. The mn concen s he neque level of hnlng qules ue o he nnsc nsbles of hs ype of smll sze vehcles. specfc hwe wh conol mx s lso eque o sblze n mnoeuve he cf. non-mgnl us-o-wegh o s mnoy h mples eque szng of he powe oupu pove by he populson sysem, genelly compomsng he enunce n he pylo cpbles. In he ls yes, n he emp o ovecome hese ssues, sevel mul oo unmnne vehcles hve been evelope. The m of he pesen pojec s o cee compc, obus n hghly mnoeuvble uonomous UV. The qu-oo ELIS (nege mulioo fo Suvellnce pplcons) s conolle by chngng he oon spee of he moos. The oque n he yw econ s cncelle by spnnng wo of he popelles clockwse n he ohe wo nclockwse. The opml confguon hs been chosen n oe o ncese he cf sucul sffness n o enhnce he sbly n he conollbly of he vehcle. The mn-oos e suppoe by se of eqully spce compose l bs, ech of hem lnke wh cenl pylo cse. The esech seps pesene n hs ppe concen he nlyss of he vehcle confguon n he efnon of he mhemcl moel escbng he ynmc behvou.. n esuls of hs wok wll be epoe n wely scusse n he ppe.. INTODUCTION oy-wng UVs cn hve cpbles o pefom mssons h cn no be cheve wh fxe-wng UVs. oy-wng vehcles hve he poenl o be vey useful n eol monong n, especlly n he ls yes, /C helcopes e use fo el suvey. ny Unvesy esech goups hve evelope oy-wng sysems, even f oy he wespe esech n use s lmos eseve o mly ognzons. ecenly, he Unvesy of yln hs lso evelope wo oy- wng mco UVs []. Ths mco vehcle hs wo coune-ong coxl oos n weghs 4 gms. Smll oy-wng UVs wh VTOL n hove cpbles cn hve mny pplcons; hese UVs coul be especlly useful fo noo flgh o fo ubn mssons. Usng UVs fo econnssnce n hese suons s lso chllengng becuse of he sho lne of sgh n mny obscles. They e ble o fly n es wh obscles n poo quly GS sgnls. Incopong elble sem-uonomous o uonomous conol sysem n hese smll vehcles, he opeo oes no hve o consnly mono he plfom flgh pmees o locon. Howeve he on bo sofwe wll hve o be vey compc o f n he vlble memoy of he smll mcopocessos, TO--SCI- 3 - UNCLSSIFIED/UNLIITED

2 UNCLSSIFIED/UNLIITED Desgn Confguon n Opmzon fo ul oo UV bu poweful enough o pove conol wh senso of lme quly. These vehcles cn be vey chllengng. ecenly, qu-oo s oy wng UV h hs been he subjec of sevel ecen esech pojecs. Smll qu-oos hve mny excng poenl mssons nclung flgh noos n n ubn es. Howeve, he evelopmen of he conol sysems neee o fly. The mos known qu-oo s he Dgnflye [], commecl pouc fom C Toys; hs vehcle s flown usng n /C nsme n s onbo eleconcs. The plo cn conol he hole seng fo he fou moos n he yw, pch n oll es of he plfom. The ecen veson of Dgnflye nclues fou nfe he sensos o llow he qu-oo o level self whle s beng flown ouoos. nohe plfom s he EDS Quocope, use s esbe fo evelopng mco vehcle flgh conol [3]. The Quocope s cpble of mnue flgh wh sngle chge of s lhum bees. The vehcle s lengh 65 cm, weghs bou hlf klogm n s eecble fuselge cn be soe n bckpck. The elecc moos llow hs UV o opee quely. Une evelopmen n usl he X-4 flye [4] hs fme lengh of 7 cm n weghs kg wh lmos cm mee oos. The fs flgh esng ws conuce usng uck bey n ehe cho o pove powe o he plfom. Howeve, hese ess wee no successful n he us mgn of he X-4 flye ws no lge enough o llow conollble flgh. The nex gols fo mpovemen e o esgn new X-4 flye h woul be cpble o pouce moe hus wh weless sel lnk n cme sysem. esech ems n some Unveses e evelopng qu-oo sysem conol, sng fom commecl vlble moel. Fo exmple, esech em fom Fnce employe he commecl Dgnflye o suy s sblzon [5]. In he sme econ, esech goup he Unvesy of ennsylvn s evelopng qu-oo usng he commecl moel HX-4. Due o he wegh lmons, no GS o onl cceleomees coul be plce on he plfom. he Conel Unvesy wo qu-oo pojecs hve been pefome. The gol of he fs pojec ws o evelop meho o esme he ue of vehcle by usng n offbo vson sysem n hee onbo gyoscopes [6]. The secon pojec ws concene on he fou hus poucng uns n sucue of qu-oo. These wo es wee especlly mpon snce hs qu-oo ws heve (6. kg) hn he pevously menone esgns. he eospce Engneeng Depmen (DIS) of olecnco Tono, he Flgh echncs Goup s wokng on he evelopmen of qu-oo fo he eol monong n he mp pojec fo e zone n/o gey e. The esech s oene o nlyze how wok n how mneuve hs kn of plfom. In oe o bee unesn hs poblem, conseng he ynmc equons n he pefomnce, yng o mpove he confguon, we hve o elze he cf. The esech pncple scope s o opmze he uomc conol of mul qu-oos n fomon flgh, povng hese plfoms wh mn uoplo.. ODEL DESCITION Ths ppe s focuse on he esgn n he mhemcl moel of fou-oo flyng vehcle. quoo s mechnclly smple n s conolle only chngng he spee of oon of he fou vng moos. The oque n he yw econ s cncelle by spnnng wo of he oos clockwse n he ohe wo n-clockwse. The ue conol n oll, pch n yw econ s obne by vyng he oonl 3 - TO--SCI- UNCLSSIFIED/UNLIITED

3 UNCLSSIFIED/UNLIITED Desgn Confguon n Opmzon fo ul oo UV spee, whch elmnes he mechncl complexy of pch lnkge. So he fou oo uns o no eque cyclc n collecve pch commns. The ol hus s conolle wh he smulneous von of ll oo spee. The opml confguon hs been chosen ccong o mnmze he cf wegh n o opmze he conol segy of he plfom, kng no ccoun he cf sucul sffness. symmec cucfom lyou wh pephel populson uns ws selece n oe o smplfy he blnce of he sepe huses. We cn pck ou beween wo opons; one confguon hs he fou oos connece o he cenl fuselge by fou sepe compose bs. In he secon cse, he mn-oos cn be suppoe by se of eqully spce compose l bs, enfoce by sque fme, ech of hem lnke wh he cenl pylo cse (Fg.). Fgue : Secon qu-oo confguon The secon confguon s penlze by he sucul wegh, bu s efnely less pone o vbons n benng, povng hghe sffness. In fc, he osonl sffness of close secon s subsnlly bgge hn hs of n opene fme secon. To vo smll sucue flexbly, he secon confguon s chosen. The sucue ws eque o be smple, ugge n emounble. The gols fo Vs hs o mke sysem h cn be use by sngle opeo n cn sy n bckpck [7]. In oe o mnmze he wegh, gphe bs e consee fo he sucue consucon n snwch of fbe glss n ex fo he cenl boy. The hub hs o be esgne o ensue he coec locon n oenon of he sus on ssembly n coul be smple une-ove clmpng sysem o pove ugge emounble p. Fo he pooype pojec he elecc moos n he oos ws souce fom commecl C equpmen, fo esons of smplcy n pccly. The use o g oos smplfes he eoynmc moellng, even f s possble h he exenl subnces e ncese. In oe o ncese he sfey vehcle, kng no ccoun h he popelles e no poece, n exenl cbon fbe wh ecngul pofle s ope. The chcescs of he efeence cf e pesene n Tb.. Tble : cf chcescs TO--SCI- 3-3 UNCLSSIFIED/UNLIITED

4 UNCLSSIFIED/UNLIITED Desgn Confguon n Opmzon fo ul oo UV Dmenson opelle sze Sucul wegh Tol wegh ylo wegh ushless engne Sze Wegh Spee conolle xmum Wegh 65x65 mm 54x mm 8 g 3 g 3/35 g 3x 37mm 55 g 4 g Two popelles oe n-clockwse n he ohe wo clockwse; n oe o mnoeuve he cf s necessy o conol he oo spee. In pcul, conol mx mus be esgne o oppounely obne he ese combnon of ue. In hoveng, ll popelles mus oe wh he sme oonl spee. s consequence, he hus of ll oos mus be equl n he oques hve he sme moulus bu he oppose econ. Fgue : Hove conol Fo he fow flgh, conseng he fs oo hs he splcemen econ, he oon s me oun he pch xs. The hus on he oo 3 s ncese nse he hus on he oo s ecese o mnn he equlbum of he ol oque. To move he plfom n he gh lel se, he oon s me oun he oll xs n he hus on oo s ncese conempoly he hus on oo 4 s ecese. The yw s obne ncesng he hus on he even oos n ecesng he hus on he o ones, n oe o mnn consn he lue. 3. THETICL ODEL In oe o nlyze he ynmc behvou of mul oo UV n o evlue he flgh sbly n quly, we conse g ble moel, neglecng he ynmc couplng beween he g boy n he sucue flexbles. Ths s ue o he smll vehcle mensons n becuse we on suy he eomechncs sbly. Tkng no ccoun he eoynmcs behvou, ech ble s ppe n n fel ue o he combnon of he oonl spee, he ggng spee n he nflow (ue o he lf ble con). 3-4 TO--SCI- UNCLSSIFIED/UNLIITED

5 UNCLSSIFIED/UNLIITED Desgn Confguon n Opmzon fo ul oo UV Conseng he moel of smll fou oo, he nflow ynmcs cn be moelle s unfom one, n hs wy: σ 64 ( 3.) λ + ϑ σ whee σ s he soly o, s he lf slope (usully 5.7) n ϑ. 75 s he ble pch 75% us [8]. The ws s ege lne, kng no ccoun he smple mechnsm of hs qu-oo: ϑ ϑ + ϑ ( 3.) w whee s he l locon on he ble, mesue fom he cene of oon o he ble p, ϑ s he commn ws n whch we conse he elnce on he ngul velocy von n ϑ w s he lne ws e. The mhemcl moel of he fou oo sysem nclues he fuselge (he cenl boy), he fou pncpl oos n he nflow of he pncpl oos. The fuselge ynmcs s moelle s sx.o.f. g boy, conseng he Eule non lne equons. The bles n fuselge eoynmcs s consee lne. The foces n momens of ne fo he ble nfnesml elemens e: F m F whee s he cceleon of he pon wh n nfnesml mss m n s he -O veco n whch O s he hnge pon. Fs of ll, s possble o moel he ble s:. hn lye wh he mss eqully sbue, kng no ccoun he ws ngle. In hs cse he ws ngle s consee n he poson veco of pon ; so n he me evve we hve o conse lso hese vons, ponng ou he couplng effecs.. hn b wh he mss eqully sbue. In hs cse he couplng effecs of feheng moon e neglece becuse he ws ngle s consee consn n me. In hs cse, we ssume h he ble s moelle s hn b. We conse he followng efeence fmes [9]:. boy efeence fme (X,Y,Z ): he ogn s n he boy cene of gvy; hs s he efeence fme ele o he cenl boy;. oonl efeence fme (X,Y,Z ): hs efeence fme oes wh he ble n X les n he sme plne of he hub n he econ s ele o he zmuh ngle. 3. ble locl efeence fme (X,Y,Z ): hs efeence fme oes wh he ble n X s concen wh he longunl ble xs n Y s pepencul o X n he oon sense. These efeence fmes e ffeen f we conse he nclockwse oos o he ohe wo. Ths s ue becuse usully he oon s posve when hs econ s nclockwse, so he efeence fmes e gh-hne. Tkng no ccoun h evey oons mus be gh-hne (fo n geemen). We cn obseve he efeence fmes n he followng fgues. m ( 3.3) TO--SCI- 3-5 UNCLSSIFIED/UNLIITED

6 UNCLSSIFIED/UNLIITED Desgn Confguon n Opmzon fo ul oo UV Fgue 3: The efeence fmes on he n-clockwse oos Fgue 4: The efeence fmes on he clockwse oos Sng fom he pevous emks, he oo equons of moon ffe n funcon of he spn econ. In fc, he nsfomon mces beween he oonl efeence fme n he ble locl one n beween he oonl fme n he boy one e ffeen fo he fou oos. Fs, we conse he nclockwse oos n, n el, he poson veco of genec pon, nse he ble, n he efeence fme F. ssumng he ble s hn b, he poson veco n he fme F cn be wen s: whee s he pon coone long X. To obn he veco n he efeence fme F, wo nsfomon mces e pple. ( 3.4) 3-6 TO--SCI- UNCLSSIFIED/UNLIITED

7 UNCLSSIFIED/UNLIITED Desgn Confguon n Opmzon fo ul oo UV genec veco cn be wen s: v β Tkng no ccoun he followng nsfomon mces [ ] [ ] v ( 3.5) [ ] β [ ] cosβ sn β cos sn sn β cosβ sn cos ( 3.6) he genel veco s: cosβ cos cosβsn sn β v sn cos v sn βcos sn βsn cosβ ( 3.7) s nce n he fgues bove, he ze xes Z n Z e upws. The me evve ove he ble of he veco poson s: & + ω ( 3.8) The veco ω epesens he ngul velocy of he efeence fme. The fs componens e he ngul velocy expesse n he efeence fme F n he h one s he hnge ngul velocy expesse n he sme fme. We hve he followng mx expesson β& sn β cos & cosβsn p cos cos e β + V cos q sn & + cos cos sn sn β& β & β Ω sn βcos ( 3.9) Fnlly, he cceleon n he pon cn be expesse s: & V + ω V ( 3.) Tkng no ccoun h fo hs vehcle he lg n he flp moons e consn, ll he me evves of hese wo componens e null. So he pevous expessons cn be smplfe. The ngul velocy cn be clcule sng fom he boy ngul velocy. genec veco n he efeence fme F, n funcon of he boy xes, cn be expesse s: TO--SCI- 3-7 UNCLSSIFIED/UNLIITED

8 UNCLSSIFIED/UNLIITED Desgn Confguon n Opmzon fo ul oo UV [ v] [ ] [ v] ( 3.) whee he nsfomon mx s obne wh wo oons (3). The fs one s oun he Z xs of n ngle ( ) n he secon oon s of n ngle ( +8 ) oun he Y xs. Conseng he ule of he oon mces, we obne he followng esuls: [ ] [ ] cos sn cos sn sn cos sn cos cos sn cos Ω sn Ω sn Ω cos Ω sn cos ( 3.) In he specfc cse of he ngul velocy, we hve: p cos Ω q sn Ω sn Ω cos Ω p q p q p cos Ω + q sn Ω p sn Ω + q cos Ω ( 3.3) The me evves e: p& q& p& cos Ω + pωsn Ω + q& sn Ω + qω cos Ω psn & Ω + pω cos Ω + q& cos Ω qωsn Ω ( 3.4) We e conseng he m conons whee he lne n ngul veloces e zeo (sey se conon). In oe o obn he eoynmc n nel cons, we hve o nsfom he veloces n he cceleons n he ble locl efeence fme. s s befoe, genec veco cn be nsfome n he followng wy [ ] [ ] v ( 3.5) v β So he veloces n he cceleons becomes V T {[ ] β [ ] } T {[ ] [ ] } β V ( 3.6) The nel momen long he xs X s zeo ue o he sysem moel (moelle s hn b). Inegng long he ble sng fom he hnge pon, we obn he foce n momen esulns. The negon mus be o long he non-mensonl l coone x. Fo hs eson, he nfnesml mss m mus be ewen s he mss pe un lengh η; so we obn he followng elonshp: 3-8 TO--SCI- UNCLSSIFIED/UNLIITED

9 UNCLSSIFIED/UNLIITED Desgn Confguon n Opmzon fo ul oo UV m ηx m η xx ( 3.7) The fnl expessons e: F η η η p X p Y p Z x FX x FY FZ x ( 3.8) η η p Z p Y xx xx X Y Z ( 3.9) In oe o obn he foce n momen esulns, we hve o clcule he spee wh espec o one ble pon, n he efeence fme F. The flux spee wh espec o he ble s veco sum of hee ems:. he spee ue o he zmuh, lg n flp moons.. he vehcle ggng spee, h s expesse n boy xes, so hs spee mus o be nsfome n he efeence fme F, conseng he nsfomon mx [ ]. 3. he oo nflow s expesse n he non-ong xes (concen wh he boy xes); we conse gn he mx [ ]. Ths ls em s clcule n he followng wy: W Ω λ D ( 3.) whee D s he cuo sc us n λ s he nflow coeffcen. s s befoe he nflow s conse unfom. The flux spee wh espec o he ble, conseng ll he componens, s he ffeence beween ll he ems. These componens mus be nsfome n he efeence fme F, n oe o obn he l, pllel n pepencul componens of he elve spee long he nfnesml ble n he neghbohoo of he pon. We hve o clcule he nfnesml lf n g, usng he momenum-ble elemen heoy. We conse he followng foce n spee sbuon. TO--SCI- 3-9 UNCLSSIFIED/UNLIITED

10 UNCLSSIFIED/UNLIITED Desgn Confguon n Opmzon fo ul oo UV Fgue 5: omenum ble elemen heoy fo n-clockwse oos The nfnesml lf n g n he focl momen c on he eoynmc cene. In he equlbum he momen s consee zeo, nse he lf n g foces e: L ρu C S ρ L D U C S D ρ ρ ( U + U ) T C ( U + U ) T C L D S S ( 3.) The nfnesml sufce cn be expesse s funcon of he ble cho n of he non-mensonl l coone x, n fc we hve: S c x. The lf n g componens mus o be clcule n he efeence fme F, so we hve o conse he vlue wh espec o he Y n Z xes. epenence fom he zmuh ngle n he ble oon. The eoynmc n nel foces mus be nsfome n he efeence fme F. F F F X Y Z F F F X Y Z cosβcos sn sn β cos cosβcos sn sn βcos cosβsn cos sn βsn cosβsn cos sn βsn sn β FY cosβ F Z sn β FX FY cosβ FZ ( 3.) ( 3.3) s you cn see, hese foces e epenen by he zmuh ngle. In he sme wy, we cn clcule he eoynmc n nel momens fo he efeence fme F. These foces n momens mus be nsfome n boy xes. In oe o obn he foces n momens gven o he hub by ll he bles, we hve o clcule he vege cons n we hve o mulply conseng he ble numbe. The eoynmc n nel 3 - TO--SCI- UNCLSSIFIED/UNLIITED

11 esulns e obne n hs wy: 3.4 N L F Z F Y F X Z Y X Z Y X 3.5 N L F Z F Y F X Z Y X Z Y X s s befoe, hese emk cn be evlue only fo he coune-clockwse oos. In genel, he ssumpons ope fo he ohe wo oos hol ye ue, bu he nsfomon mces vy n he followng wy. Fo exmple, genec veco cn be wen s: [ ] [ ] 3.6 v v β Fo clockwse oon, o obn veco fom he efeence fme F o he fme F we hve o conse wo oons (oon oe: 3):. fs oon oun he Y xs of n ngle β ;. secon oon oun Z xs of n ngle. So, he nsfomon mces e: [ ] [ ] 3.7 cos sn sn cos cos sn sn cos β β β β β Fo clockwse oo only one oon s necessy n oe o lgn he oonl efeence fme F n he boy xes, kng no ccoun oon oe 3. nee, s suffcen sngle oon oun he Z xs of n ngle. s fo he n-clockwse oo, n oe o obn he foce n momen esulns, we hve o clcule he flux spee wh espec o he ble s sum of hee ems:. he spee ue o he zmuh, lg n flp moons.. he vehcle ggng spee, h s expesse n boy xes. Desgn Confguon n Opmzon fo ul oo UV TO--SCI- 3 - UNCLSSIFIED/UNLIITED UNCLSSIFIED/UNLIITED

12 UNCLSSIFIED/UNLIITED Desgn Confguon n Opmzon fo ul oo UV 3. he oo nflow s expesse n he non-ong xes (concen wh he boy xes); we conse. Ths ls em s clcule s s befoe (Eqs. 3. n 3.). gn he mx [ ] We hve o clcule he nfnesml lf n g, usng he momenum-ble elemen heoy. Due o he ffeen efeence fmes, we conse he followng foce n spee sbuon. Fgue 6: omenum ble elemen heoy fo he clockwse oos The nfnesml lf n g n he focl momen c on he eoynmc cene. The lf n g componens mus o be clcule n he efeence fme F, so we hve o conse he vlue wh espec o he Y n Z xes. Conseng he foce sbuon n Fg., he nfnesml eoynmc foce s: F F F Y Z Lsn D cos χ χ L cos χ + D sn χ ( 3.8) In oe o evlue he complee mhemcl moel fo qu-oo, we hve o conse he fuselge equons of moon []. The g boy equon e: whee + Q + K g ( 3.9) Q mvg s he momenum of momenum veco, Vg U î + V ĵ + W kˆ s he cene of 3 - TO--SCI- UNCLSSIFIED/UNLIITED

13 UNCLSSIFIED/UNLIITED K gvy spee n g [ ] g e In pcul, I ω Desgn Confguon n Opmzon fo ul oo UV s he ngul momenum. K g I I XX ZX I YY I I ZZ XZ ω e I I XX p I XZ I q YY I p ZX ZZ ( 3.3) wh ω e s he vehcle ngul velocy n he boy efeence fme. The me evve of he wo momenum e: & Q Q + ω Q & K K + ω K g g g ( 3.3) The ngul velocy ω s he velocy of he boy fme wh espec o n nel one; hs velocy s concen wh he vehcle ngul velocy, so ω ω p î + q ĵ + kˆ. The nel esuln cn be evlue clculng he me evve of he Q n coss pouc nce n he bove equon. So we obn: e K g vecos n he mu& & Q mv& mw& î ω Q p mu m U Q m V m W ĵ q mv (& + qw V) (& + U pw) ( & + pv qu) kˆ m qw m U mw m pv ( V) ( pw) ( qu) ( 3.3) The me evve of nel veco K g n he coss pouc e clcule s: The veco K g I I I XX YY ZZ p& I q& + I & I ZX & + q( I I ) pqi ZZ YY ZX ( p ) + p( I I ) XX ZZ p& + pq( I I ) + I q s elve o he cve foces cng on he fuselge XZ ZX YY XX XZ ( 3.33) TO--SCI- 3-3 UNCLSSIFIED/UNLIITED

14 UNCLSSIFIED/UNLIITED Desgn Confguon n Opmzon fo ul oo UV + + g ( 3.34) whee X î + Y ĵ + Z kˆ s he eoynmc n nel esuln of he oo foces exchnge wh he fuselge, X î + Y ĵ + Z kˆ s he eoynmc fuselge esuln n mg î + mg ĵ + mg kˆ s he wegh fuselge esuln. g X Y In nlogy, he cve momens on he fuselge e Z + ( 3.35) whee L î + ĵ + N kˆ s he nspo momen esuln, L î + ĵ + N kˆ s he eoynmc fuselge esuln. The gvy veco, known n he vecl xes, mus be nsfome n he boy fme kng no ccoun he nsfomon mx cos Ψ cos Θ sn Ψ sn Φ + cos Ψ sn Θ cos Φ sn Ψ cos Θ [ ] sn Ψ cos Φ + cos Ψ sn Θsn Φ cos Ψ cos Φ + sn Ψ sn Θsn Φ cos Θsn Φ ( 3.36) T V cos Ψ sn Φ + sn Ψ sn Θ cos Φ sn Θ cos cos Θ Φ In concluson we cn we he foce n momen equons. X + X Y + Y Z + Z m V m W (& + qw V) (& + U pw) ( & + pv qu) m U L + L I + I N + N I XX ZZ p& I YY q& + I & I XZ ZX & + q ZX + mg sn Θ mg cos Θsn Φ mg cosθ cosφ ( I I ) ( p ) + p( I I ) p& + pq ZZ ( I I ) YY YY XX pqi XX + I XZ ZX ZZ q ( 3.37) ( 3.38) In oe o complee he equon sysem, we hve o evcue he coelon beween he Eule ngles n he vehcle ngul velocy. We hve o conse he knemc equons: p Φ& Ψ& sn Θ q Θ& cos Φ + Ψ& cos Θsn Φ Θ& sn Φ + Ψ& cos Θ cos Φ ( 3.39) 4. NLYSIS OF THE HOVE CONDITION We woul lke o nlyze he hove conon; so we hve o smplfy he equons. In hove, he vehcle spees e zeo n, s consequence, ll he fuselge eoynmc effecs e zeo. So we hve: 3-4 TO--SCI- UNCLSSIFIED/UNLIITED

15 UNCLSSIFIED/UNLIITED Desgn Confguon n Opmzon fo ul oo UV U V W X Y Z L N ( 4.) In hs cse he un e s zeo so he heng s consn: Ψ &. s consequence, he ngul veloces e null, bu, n he genel suon, he ue ngle n he oll one e ffeen fom zeo. The foce n momen equons e funcon of he ngul oon Ω. In genel conons he fou ngul oon e ffeen, bu n hoveng hese become Ω Ω3 Ω Ω 4, so we hve o conse hs smplfcon n he equons of moon. Ths moel s mplemene n lb Smulnk pogm, n oe o evlue he ynmc sbly of he sysem. Fgue 7: oel mpleon n Smulnk We mpose n nl oo ngul oon, n oe o vefy he plfom sbly. Whou conolle, he sysem vege fe we pmees oscllons. So he sysem mus be conolle. TO--SCI- 3-5 UNCLSSIFIED/UNLIITED

16 UNCLSSIFIED/UNLIITED Desgn Confguon n Opmzon fo ul oo UV Von of Eule ngles.5 φ [] [s].5 θ [] [s] Fgue 8: Von of he Eule ngles Von of ngul veloces p [] [s] q [] [s] 5 [] [s] Fgue 9: Von of he ngul spee 3-6 TO--SCI- UNCLSSIFIED/UNLIITED

17 UNCLSSIFIED/UNLIITED Desgn Confguon n Opmzon fo ul oo UV The npu conol of ou sysem s he oo ngul oon, so, n oe o sblze he plfom, cn be necessy o conol he von of he oo ngul spee. Conseng he conol lws of he qu-oo, he von of he ngul spees s conolle by he chnge of he ngul veloces p,q,. We hve mpose s nl conon h he ffeenl von sum of he fouh ble oons s zeo. The plfom mus be sble fe gven subnce, movng bck o he ue nl conons. Wh l n eo meho we hve o efne he gns, coele o he ngul veloces. We mpose peubon of fve egees on he oll ngle. To mp he oscllon n o euce he sey se eo, D conolle s necessy. The followng esuls hve been obne. Von of Eule ngles φ [] [s]..5 θ [] [s].. [] [s] Fgue : Von of Eule ngles wh D conolle TO--SCI- 3-7 UNCLSSIFIED/UNLIITED

18 UNCLSSIFIED/UNLIITED Desgn Confguon n Opmzon fo ul oo UV Von of ngul veloces 5 p [] [s].5 q [] [s].6.4 [] [s] Fgue : Von of he ngul veloces wh D conolle fe he gn unng, sng fom n bsolue oo ngul oon n hoveng of /s, he fou esulng ngul spees e: Ω Ω Ω Ω / s / s / s / s The mplemenon of he moel wh n negve gn nceses he sysem oscllon, whou mpovng he se me of he sysem n he sey se eo. The equemens fo hs kn of plfom e: he se me hs o be less hn 5 secons, he oveshoo une he 5% n he sey se eo une he %. In oe o obne hese esuls n opml egulon hs o be mplemene. 5. CONCLUSIONS In hs ppe qu-oo pooype hs been esgne usng commecl n COTS componens. The elecc n sucul ps e efne opmzng he plfom sffness n povng ugge sysem, no neglecng he ol cf wegh. The exploon of he mhemcl moel of he vehcle s lso oulne, kng no ccoun he ynmc moel of he fuselge n of he fou oos. Some esuls e obne nlyzng he hove conons. We cn emonse h popoonl conolle on Eule ngles oes no gunee sble conons. 3-8 TO--SCI- UNCLSSIFIED/UNLIITED

19 UNCLSSIFIED/UNLIITED Desgn Confguon n Opmzon fo ul oo UV The esponse s ynmclly sble n, fe peubon, he cf s movng bck o he m conons. s fuue esech he esponse cn be opmze n he close-loop sysem cn be sympoclly sble, hs s possble wh complee gn mx K n n LQ conolle. In fc he LQ conolles e lso nheenly obus wh espec o pocess unceny. The nex sep nclues he consucon of pooype oocf, n oe o vle he sbly chcescs n emoe conol. Wh he conolle bo we cn emonse f he gn mx K obne s compble o f nohe gn unng s eque. he sme me, he moel of he emoe conolle hs o be mplemene n he complee block gm, so s o epouce s fhfully s possble he el sysem. In hs wy, he moel npu s he el one.. efeences [] Conoy, J.K., Smuel,.D., nes, D.J., Developmen of V Conol n Nvgon Sysem, I Infoech@eospce Confeence, lngon (Vgn), 6-9 Sep 5. [] ckeow,., oellng he Dgnflye fou-oo helcope, 4 IEEE Inenonl Confeence on obocs n uomon, New Olens (L, US), 6 pl-y 4. [3] nng,.k., Sssen, S., Flgh Conol of co el Vechcles, I Gunce Nvgon n Conol Confeence n Exhb, ovence (hoe Isln), 6-9 ug. 4. [4] ouns,., hony,., Geshm, J., Coke,., obes, J., Tows Dynmclly-Fvouble Qu-oo el obos, uslsn Confeence on obocs n uomon, Cnbe (usl), 6-8 Dec. 4. [5] Csllo,., Dzul,., Lozno,., el-tme Sblzon n Tckng of Fou-oo n oocf, IEEE Tnscons on Conol Sysems Technology, volume, pges 5 6, July 4. [6] El,. G., D ne,., el-tme ue Esmon Technques pple o Fou oo Helcope, 43 IEEE Confeence on Decson n Conol, hms, 4-7 Dec. 4. [7] Offce of he Secey of Defense, Unmnne cf Sysems omp: 5-3, 5 [8] Johnson, W., Helcope Theoy, Dove ublcons nc, 98 [9] Tlbo,.D., Tnlng,.E., Decke, W.., Chen,.T.N., mhemcl moel of sngle mn oo helcope fo ploe smulon, Nonl eonucs n Spce mnson, NS Techncl emonum 848, Sepembe 98 [] ouhngou,., zouz, N., oellng of flexble quoo helcope, SE 6 Inenonl Desgn Engneeng Techncl Confeence & Compues n Infomon n Engneeng Confeence, Sepembe -3 6, hlelph, ennsylvn, US TO--SCI- 3-9 UNCLSSIFIED/UNLIITED

20 UNCLSSIFIED/UNLIITED Desgn Confguon n Opmzon fo ul oo UV 3 - TO--SCI- UNCLSSIFIED/UNLIITED