46 ISSN Electronics and Control Systems N 3(53): 46-50
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1 46 ISSN Elctocs ad Cotol Systms 7 N (5): 46-5 UDC 57 (45) DOI: 87/ L M Ryzhkov AIUDE DEERMINAION BASED ON AXIS AND ANGLE ROAION COMPUING Acaft ad Spac Systms Dpatmt Natoal chcal Uvsty of Uka Iho Skosky Kyv Polytchc Isttut Kyv Uka E-mal: lv_yzhkov@amblu Abstact h algothm of atttud dtmato o th bas of axs ad agl otato computg s suggstd Was assumd th kow th pojctos of th omalzd vctos th fc ad th body coodat systms h poblm s to dtm th ut vcto ad th agl of otato of th coodat systm coctd to th body latv to th fc coodat systm Compaso wth algothm QUES s fulflld It was show that th accuacy of poposd algothm s quvalt to th accuacy of algothm QUES Idx ms Atttud dtmato I INRODUCION h poblm of atttud dtmato o th bass of fomato about o-paalll vctos s aalyzd h us of gomtc latoshps s a ffctv mas of gd body otato dtmato basd o th masumt of vctos [] [] h chaactstc fatu s that you do ot d to us a matx (as th RIAD [4] [5] algothm) o quato (as th QUES [6] [7] algothm) algbas h poposd algothm uss gomtc latos whch tak plac wh th body otats III PROBLEM SOLUION I Fgu th agl of otato of th pla whch cotas th axs of otato ad th vcto s dcatd A ut vcto chaactzs th dcto of otato of ths pla h ut vcto of otato of a body b s qual b II PROBLEM SAEMEN Dvlopg th appoach poposd [] [] w cosd th poblm of dtmg th otato of a body o th bass of th us of gomtc latos W wll assum th kow pojcto of omalzd vctos th fc ( o ) ad th body ( ) coodat systms h poblm s to dtm th ut vcto ad th agl of otato of th coodat systm coctd to th body latv to th fc coodat systm hat s th vctos a fxd th pojctos a vaabls th two spcfd coodat systms Chag th poblm statmt W assum that th vaabls a vctos that s w assum that th vctos otat aoud th axs of otato (Fg ) th oppost dcto wth spct to th dcto of otato of th body-coctd coodat systm latv to th fc coodat systm ajctos of th ds of vctos a ccls wth cts o th axs of otato Ulk most kow algothms w wll spaatly dtm th axs of otato th agl of otato ad th dcto of otato [] [] Natoal Avato Uvsty 7 Fg Vctos h bass of th algothm s th us of vctos a whch a paalll plas ad a o ppdcula to th vcto hs vctos cospod to latos a = () W us ths fomula to fd th vcto It s sstal that fom ths fomula w ca fd a vcto that s dctd alog th axs of otato but whch dos ot dtm th dcto of otato that s = hfo th dfto of th dcto of otato must b pfomd addtoally
2 LM Ryzhkov Atttud Dtmato Basd o Axs ad Agl Rotato Computg 47 akg to accout th os of masumt of vctos th poblm wll b solvd o th bass of th mthod of wghtd last squas that s th vcto wll b dtmd fom th codto of mmzg th loss fucto l a a a a a a G () wh a = a wghts G a a a o-gatv Cosd th codto that th vcto must b ut ad assum such a loss fucto l : () G l wh s th Lagag multpl Cosd th poblm of mmzg th loss fucto Lt's wt dow l G hat s th codto of th mmum s G h you ca wt (4) = (5) l G = (6) hat s w a tstd th mmum valu of th paamt hus th mmzato poblm s quvalt to fdg a vcto as a gvcto of a matx whch cospods to th mmum gvalu of th matx I th Matlab vomt th gvalus of a matx ca b foud by fucto g( G ) I th absc of masumt os w hav G that s m hfo fo small os masumts of vctos ca b assumd I ths cas th gvalu of matx ca b foud by such tchqu h gvalus of th matx a th oots of th chaactstc quato G I f f f Gv that w a tstd w wll accpt f m f h f m (7) f wh f t G t G f dt( G ) Nxt you ca fd a vcto as soluto of quato G I m Cosd th qusto of fdg th agl of otato ad dtmg th vcto o do ths w dcompos th vctos o ad to two mutually ppdcula compots (Fg ) u u = ( ) E o o o m u E o o o o m E wh E E I E wh Fg Agl o otato Not that th s a plac th latos E I E E Nxt w wll fd E E o E E o E E o m m wh = E o Rwt ths xpsso wh = E o o o = I E o (8) (9) cos () o
3 48 ISSN Elctocs ad Cotol Systms 7 N (5): 46-5 Us last-squa stmato ad spcfy th lost fucto f o () wh cos f Usg th codto w fd cos () o o Lt's cosd mo dtal th choc of vctos oі od to futh fd th vcto h poposd tchqu s basd o th fact that th vcto s ppdcula to th pla whch th vctos a a locatd hs mas that at last two vctos a must b o-paalll ( ths cas thy wll fom ths pla) Fom Fg w s that ths pla s fomd by th compots m oі ad m і of ths vctos Lt's llustat ths compots fo two vctos (Fg ) Fg Aalyss of vctos Fom ths fgu w s that th vctos a ad a wll b paalll wh th vctos m o ad m o wll b o l It ca also b fomulatd as follows: th vctos a ad a wll b paalll wh th vctos o o ad th axs of otato wll b th o pla hfo t s advsabl to t a addtoal vcto ad cosd th systm at last of th o o o vctos o o o I ths cas you ca always fom a pla whch th a vctos a hs sult ca also b obtad algbacally Assum that th vctos a o ad a o a paalll h you ca wt o k o wh k th umb W us th matx of th dctoal coss R ad wt R I k R I that s Dz wh o o z o o D R I k Lt's wt ths xpsso th fom Dz z wh h th vcto z ca b cosdd as a gvcto z of th matx D whch cospods to gvalu Gv that th gvcto of th matx D s qual to th gvcto of th matx R w wt z v (v s th umb) h o k o v hs mas that th vctos o o ad th axs of otato a th o pla h last stp solvg th poblm s to dtm th dcto of otato cosdg that th fst stag w foud ot a vcto but a vcto As ca b s fom Fg th vcto a cocds th dcto wth th coss poduct v o Gv that o of th vctos a ca b zo ( th cas wh th cospodg vcto і cocds wth th axs of otato) th dcto of th vcto wll b dtmd such way z () wh z sg v a v a v a Not that ths cas th dcto of otato ca b foud aoth way amly though a fucto s sg h postv valu of th agl cospods = ad th gatv valu of th agl cospods = that s = sg (s ) (4) Usg Fg wt m m M s o o m (5) wh mo z mo y M o moz mo x mo m a mo y mo x omalzd vctos h agl wll b sought fom th codto of mmzg th loss fucto s M om s M m s M m o o s s M om m M om om (6)
4 LM Ryzhkov Atttud Dtmato Basd o Axs ad Agl Rotato Computg 49 wh m m M o M o o mmz th fucto of losss w fd s M om hat s sg(s ) sg M o m (7) I th cas vctos ad a quvalt Quato of tu ca b dfd as q cos s o wh accos o Esstally th poposd algothm fuctos fo ay agl of otato Fo th th vctos slctd ths way w wll aalyz th fluc of th masumt o of th vcto o th accuacy of th otato dtmato ad compa t wth th accuacy of th QUES algothm h vcto of o by a sz 5 wll fom ppdcula to th vcto wth vaabl dcto wth a stp a pla ppdcula to th vcto h w valu of vcto s omalzd Fo calculatos w wll accpt 6 o ' o ' h sults of th calculatos fo th poposd algothm ad QUES algothm a pstd Fgs 4 ad 5 spctvly Eos of algothm (dg) ptch oll -4 4 Agl of o vcto tu (dg) Fg 5 Eos of agls dtmato I Fgus 6 ad 7 a pstd sults of computatos fo aoth agls ad vctos Eos of algothm (dg) o [ ]' o [ ]' ptch oll - 4 Agl of o vcto tu (dg) Fg 6 4 Eos of agls dtmato Eos of algothm (dg) ptch oll -4 4 Agl of o vcto tu (dg) Eos of algothm (dg) - ptch oll -4 4 Agl of o vcto tu (dg) Fg 4 Eos of agls dtmato Fg 7 Eos of agls dtmato
5 5 ISSN Elctocs ad Cotol Systms 7 N (5): 46-5 W s that th poposd algothm ad QUES algothm a pcsly quvalt IV CONCLUSIONS h us of gomtc latoshps s a ffctv mas of gd body otato dtmato basd h chaactstc fatu s smplcty ad vsblty h poposd algothm fuctos fo ay agl of a body otato h accuacy of poposd algothm s quvalt to th accuacy of algothm QUES REFERENCES [] L M Ryzhkov Gomtc atttud dtmato usg vcto masumts Elctocs ad Cotol Systms o 4(46) pp 7 5 [] L M Ryzhkov Atttud dtmato basd o gomtc latos Elctocs ad Cotol Systms o (49) pp 7 6 [] Gog t al / J Zhjag Uv-Sc A (Appl Phys & Eg) o () pp [4] L M Ryzhkov ad DI Stpuko Last Squas Usg to Impov th RIAD Algothm d Itatoal Cofc "Mthods ad Systms of Navgato ad Moto Cotol" pp 6 8 [5] S ayg ad M D Shust h May RIAD Algothms Pap AAS- 7-4 AAS / AIAA Spac Flght Mchacs Mtg Sdoa Azoa Jauay 8 Fbuay 7 Pocdgs: Advacs th Astoautcal Sccs vol 7 pp [6] F L Makly ad M Mota Quato Atttud Estmato Usg Vcto Masumts h Joual of th Astoautcal Sccs vol 48 o pp 59 8 [7] Y Chg ad M D Shust QUES ad th At-QUES: Good ad Evl Atttud Estmato h Joual of th Astoautcal Sccs vol 5 o pp Rcvd May 7 Ryzhkov Lv Docto of Egg Scc Pofsso Acaft ad Spac Systms Dpatmt Natoal chcal Uvsty of Uka Iho Skosky Kyv Polytchc Isttut Kyv Uka Educato: Kyv Poltchc Isttut Kyv Uka (97) Rsach tsts: avgato dvcs ad systms Publcatos: 5 E-mal: lv_yzhkov@amblu Л М Рижков Визначення орієнтації на базі обчислення осі та кута повороту Запропоновано алгоритм визначення орієнтації на базі обчислення осі та кута повороту Вважаються відомими проекції нормованих векторів в опорній та зв язаній з тілом системах координат Задача полягає в знаходженні одиничного вектора та кута повороту зв язаної з тілом системи координат відносно опорної системи координат Виконано порівняння з алгоритмом QUES Показано що точність запропонованого алгоритму еквівалентна точності алгоритму QUES Ключові слова: орієнтація визначення Рижков Лев Михайлович Доктор технічних наук Професор Кафедра приладів та систем керування літальними апаратами Національний технічний університет України «Київський політехнічний інститут ім Ігоря Сікорського» Київ Україна Освіта: Київський політехнічний інститут Київ Україна (97) Напрям наукової діяльності: навігаційні прилади та системи Кількість публікацій: 5 E-mal: lv_yzhkov@amblu Л М Рыжков Определение ориентации на основе вычисления оси и угла поворота Предложен алгоритм определения ориентации на основе вычисления оси и угла поворота Считаются известными проекции нормированных векторов в опорной и в связанной с телом системах координат Задача состоит в определении единичного вектора и оси поворота связанной системы координат относительно опорной системы координат Выполнено сравнение с алгоритмом QUES Показано что точность предложенного алгоритма эквивалентна точности алгоритма QUES Ключевые слова: ориентация определение Рыжков Лев Михайлович Доктор технических наук Профессор Кафедра приборов и систем управления летательными аппаратами Национальный технический университет Украины «Киевский политехнический институт им Игоря Сикорского» Киев Украина Образование: Киевский политехнический институт Киев Украина (97) Направление научной деятельности: навигационные приборы и системы Количество публикаций: 5 E-mal: lv_yzhkov@amblu
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