L.M. Ryzhkov, D.I. Stepurenko. Comparison of the covariance analysis results of the QUEST and 39

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1 L.M. Ryzhkov, D.I. Stepureko. Comparso of the covarace aalyss results of the ES ad 39 UDC : (45) L. M. Ryzhkov, D. I. Stepureko COMPARISON OF HE COVARIANCE ANALYSIS RESULS OF HE ES AND HE RIAD MEHODS Faculty of Arcraft ad Space Systems Natoal echcal Uversty of Ukrae Kyv Polytechc Isttute Kyv, Ukrae E-mals: Abstract hs paper descrbes the aalytcal expresso of the atttude covarace matrx for the ES ad RIAD methods a dvdual case. Ifluece of measuremet model parameters o the accuracy of atttude determato va the methods method s aalyzed. A fluece of chages the accuracy of oe of the sesors o the atttude determato s cosdered. Idex terms Atttude determato method; ES; RIAD; measuremet model; covarace aalyss; atttude covarace matrx. I. INRODUCION Determstc atttude determato methods are based o the measuremet of two or more base drectos to some observed objects a sgle pot tme. hese drectos are kow the referece frame. I the body frame they are measured by the approprate sesors. he case whe oly two base vectors are measured s commo for such type of satelltes as mcrosatelltes. he two vectors are typcally the ut vector to the Su ad the Earth s magetc feld vector for coarse su-mag atttude determato or ut vectors to two stars tracked by two star trackers for fe atttude determato. RIAD was the frst method to obta three-axs atttude for spacecraft. Because ts smplcty t has become oe of the most popular oes []. I 965 Grace Wahba proposed a atttude determato problem []. he problem s to fd best estmate of the atttude matrx A based o a least squares crtero,. e. to fd the orthogoal matrx A wth determat + that mmzes the loss fucto L( A) a b Ar, Natoal Avato Uversty, 4 () where b vector measuremets the spacecraft body frame ad r vectors the referece frame; a s a set of postve weghts assged to each measuremet. It was prove that the loss fucto ca be rewrtte as wth L( A) tr AB, () a ad B ab r.. (3) he loss fucto wll be mmum whe the trace of the matrx product AB s maxmum, uder costrat A A I. (4) It ca be see that such problem formulato allows corporatg more that two measuremets for the estmato of a atttude matrx. Moreover, measuremets derved by meas of dfferet sesors are take to accout dfferet way through the coeffcets a. Almost all determstc methods solve the Wahba s problem oe way or aother [3]. he ES s oe of the most popular oe. It s wdely used dfferet msso [4]. I addto to atttude determato t s mportat to evaluate the accuracy of atttude determato. Covarace aalyss s a commo method used to solve ths problem. It allows studyg the relatoshp betwee errors measuremets ad error quattes derved from the measuremets. A atttude covarace matrx s a statstcal measure of the atttude estmato error. he results of covarace aalyss of cocered methods are preseted []. II. PROBLEM FORMULAION A accuracy of sesors s ot costat ad vares durg the flght uder the fluece of varous factors. he ma goal of the artcle s to aalyze the fluece of the measuremet model parameters o the atttude determato accuracy. he case whe oly two base vectors are measured s cosdered. he ES ad RIAD methods s used to atttude determato. III. HE SOLUIONS OF HE WAHBA'S PROBLEM (ES AND RIAD) May atttude determato methods whch solve Wahba s problem have bee developed, but the frst useful soluto of the problem for spacecraft atttude determato was provded by Daveport [3], [4]. he method developed by hm s kow as

2 4 ISSN Electrocs ad Cotrol Systems 4. 4(4): q-method. Daveport restated the problem () terms of the quatero q q, q 4, for whch A q q q I qq q q, (5) 4 4 where q3 q q q3 q, (6) q q s skew-symmetrc matrx. hus substtuto of the quatero stead of the rotato matrx leads to the ext form of the loss fucto g q q Kq (7), where K s the symmetrc 4 4 matrx gve by wth S I Z K Z (8) S B B, (9) Z B B, B B, B B, () tr B. () o maxmze the ga fucto (7), we take the dervatve wth respect to q, but sce the quatero elemets are ot depedet the costrat must also be satsfed. Addg the costrat q q to the ga fucto wth a Lagrage multpler yelds a ew ga fucto,. g q q Kq q q () Dfferetatg ths ga fucto shows that has a statoary value whe g q Kq q. (3) he largest egevalue of K maxmzes the ga fucto. he egevector correspodg to ths largest egevalue s the least-squares optmal estmate of the atttude. Q-method offers smple ad elegat soluto of the atttude estmato problem. But ths method you eed to compute egevalues ad egevectors of matrx K that s 4 4. At the tme of ts troducto t was a problem, because computg capabltes of oboard computers ad eve groud statos of motorg ad support were lmted. Shuster troduced a method whch estmates a atttude of a spacecraft less accurately the q-method, but t requres sgfcatly fewer computatos []. hs cotrbuted to ts wdespread use for atttude determato of spacecrafts ad other types of movg objects. I accordace to the method the optmal quatero s determed as follows x qopt, (4) x where ad x I tr BS S z v max, S (5) max tr B det, (6) tr adj S max tr B. (7) I the last expresso adj meas a adjot matrx. he more detaled descrpto of the method s gve [] ad [3]. he RIAD s the smplest determstc way to fd the atttude matrx []. I accordace wth t trads of orthoormal ut vectors are costructed the referece frame ad the body frame: r r v r, v, v3 v v, (8) r r b b w b, w, w3 w w.. (9) b b he based o costructed trads two matrces M ad M are composed: 3 M v v v, () 3 M w w w. () As compoets of matrces M ad M are orthogoal ut vectors so these matrces are orthogoal matrces. he atttude matrx based o these matrces ca be wrtte as show A MM MM (). It should be oted that as the frst vector of trad b (ad r respectvely) should be selected a vector that s measured body frame more precsely. Based o the atttude matrx A agles of rotato ca be calculated f a sequece of rotatos s kow. IV. MEASUREMEN MODEL AND COVARIANCE ANALYSIS he atttude determato error covarace matrx or smply atttude covarace matrx s defed as follows:

3 L.M. Ryzhkov, D.I. Stepureko. Comparso of the covarace aalyss results of the ES ad 4 where pxx pxy p xz P M pxy pyy pyz, pxz pyz p zz,,, 3 (3) (4) s the error agle vector that s defed as the set of agles (measured radas) of the small rotato carryg the true atttude matrx to the measured atttude matrx. he agles are defed the body frame. hs smplfes the calculato of the covarace matrx as compared wth the case whe atttude parameters are used. he covarace matrx P s postve defte ad symmetrc, P P whch meas that there are oly sx uque elemets the matrx. he atttude covarace matrx for the ES method s defed as follows [], [6]: P I bb. (5) he atttude covarace matrx for the RIAD method s defed as follows [6]: P w w w w w w, (6) R w where w s uormalzed vector w defed (9). It should be oted that practce the measured vectors b are used here stead of true values b be- * cause the last oes are ukow. he artcle [] otes that matrx P s atttude depedet whle the same matrx sgfcatly depeds o the atttude whe t s recorded the parameters of atttude whch descrbe oretato of the body frame wth respect to the referece frame. he measuremet model s a set of equatos used to aalyze a accuracy of atttude estmato derved by oe or aother method. We use the ES measuremet model (QMM) [6], [7] whch s qute commo. Accordg to the author of the model, t s pretty smple, but realstc. ES measuremet model s based o the assumpto that measured vectors b wth hgh prob- * ablty are cocetrated a small volume about the true drecto [6]. I accordace to QMM, * b Ar b, (7) b Ar, (8) where b - the vectors of measuremet errors of -th vector of referece drecto. As you ca see from (8), vectors b are the plae that s perpedcular to the true drecto A the body frame. hs r plae s a approxmato of the part of the sphere that cotas eds of vectors Ar ad b *. he dstrbuto of the compoets of the.th measuremet error vector perpedcular to the true are assumed to be Gaussa ad uformly dstrbuted phase about the true vector wth varace per axs. he vectors b have such propertes: M b, M b b I ( Ar )( Ar ), (9) (3) where s stadard devatos of vector compoet values. hese varaces of ut vectors ca be terpreted as agular varaces radas. Work [6] states that the error dstrbuto b s axally symmetrc about drecto Ar. hus QMM whch s descrbed by the equatos (7) (3) defes a coe that ca cota a measured * vector b wth some probablty. I accordace to the measuremet model the error vector rotates the true vector turg t to the measured vector. It should be oted that QMM s assumed that expected value of the measured vector s equal to zero,. e. systematc errors are abset. he weghts a are chose to be verse varaces [6]. e. a. (3) V. COMPARISON OF HE COVARIANCE ANALYSIS RESULS he square root of trace of the atttude covarace matrx s wdely used as a scalar characterstc of atttude determato accuracy. hus the atttude estmato error s characterzed by the value: tr P. (3) Our work cosders that the referece frame ad the body frame are cocdg. Referece vectors the referece frame drected alog the coordate axes for smplfcato. e. r,, ad r,,. Cosderg that the referece frame ad the body frame are cocdg, our smplfcato allows obtag a smple aalytc expresso for the atttude covarace matrx (5). It wll be as follows

4 4 ISSN Electrocs ad Cotrol Systems 4. 4(4): P dag,,. herefore from the formula (3) we obta (33),. (34) For the RIAD,. (35) R Values of QMM parameters vary the rage from. to.5, step.5. he the surfaces ad wll be as show o Fgs ad., deg R least accurate sesor s slght. But usg of the ES of tself do ot crease sgfcatly the atttude determato accuracy for the case of two vectors. Numercal values of atttude determato error for the ES ad RIAD were also calculated. For each par of values ad N tests were performed. he atttude determato error s measured by usg the rotato agle betwee the true ad estmated atttudes: arcocs ˆ ˆ tr A A. (36) he results of calculato accordace to the (36) for both cocered methods match wth the correspodg results derved aalytcally. If a su sesor ad a magetometer are used to obta atttude, the. he dfferece betwee the values of the error calculated from (35) ad (34) s show o Fg Fg.. Atttude error derved from the ES covarace matrx, deg.5.6.4, rad..., rad.3.4.5, deg Fg.. Atttude error derved from the RIAD covarace matrx he accuracy of measuremet of the frst vector (9) s more mportat for the RIAD because of the trad s costructed o the bass of ths vector. he advatage of the ES appears whe more tha two sesors are used. Herewth the fluece of the Fg. 3. Atttude error dfferece betwee the RIAD ad the ES methods CONCLUSION Usage of the covarace aalyss allows estmatg a probable error of atttude estmato caused by some types of sesor errors. he used measuremet model allows takg to cosderato whte ose error of the measured vectors. he paper cosders the case of measuremet of two referece vectors, whch s the smplest, but very commo, especally for mcrosatelltes. he surfaces aalytcal expresso whch descrbes the depedece of the atttude determato error o the measuremet model parameters were obtaed for ths case. he ES has a lttle advatage over the RIAD whe the accuracy of the frst sesor (more accurate oe) s ot hgh. Whe the accuracy of the vector that s used to costruct the trple of the auxlary RIAD vectors s decreased the advatage of the ES s creased.

5 L.M. Ryzhkov, D.I. Stepureko. Comparso of the covarace aalyss results of the ES ad 43 REFERENCES [] Shuster, M. D.; Oh, S. D. 98. hree-axs Atttude Determato from Vector Observatos. Joural of Gudace ad Cotrol, vol. 4, o., Jauary-February 98. pp [] Wahba, G Problem 65-: A Least Squares Estmate of Spacecraft Atttude. SIAM Revew, vol. 7, o. 3, July p. [3] Markley, F. L., Mortar, M.. Quatero Atttude Estmato Usg Vector Measuremets. he Joural of the Astroautcal Sceces, vol. 48, os. ad 3, Aprl-September. pp [4] Brasoveau, D.; Hashmall, J.; Baker D. Spacecraft atttude determato accuracy from msso experece, NASA echcal Memoradum 463, 994, 4 p. [5] Keat, J Aalyss of Least-Squares Atttude Determato Route DOAOP, Computer Sceces Corporato Report CSC/M-77/634, February 977. [6] Shuster, M. D Maxmum Lkelhood Estmato of Spacecraft Atttude, he Joural of the Astroautcal Sceces, vol. 37, o., Jauary-March 989, pp [7] Cheg, Y.; Shuster, M. D. ES ad he At-ES: Good ad Evl Atttude Estmato, he Joural of the Astroautcal Sceces, vol. 53, o. 3, 5, pp Receved October 4. Ryzhkov Lev. Doctor of Sceces Egeerg. Professor. he Arcraft Cotrol Systems Departmet, Isttute of Ar Navagto, Natoal Avato Uversty, Kyv, Ukrae. Educato: Kev Polytechc Isttute, Kyv, Ukrae (97). Research area: avgatoal strumets ad systems. Publcatos: 9. E-mal: lev_ryzhkov@rambler.ru Stepureko Dmytro. Graduate studet. Departmet of Arcraft Cotrol Devces ad Systems, Natoal echcal Uversty of Ukrae Kyv Polytechc Isttute, Kyv, Ukrae. Educato: Kyv Polytechc Isttute, Kyv, Ukrae (6). Research area: arcraft atttude determato methods. Publcatos: 4. E-mal: d_stepureko@ukr.et Л. М. Рижков, Д. І. Степуренко. Порівняння результатів коваріаційного аналізу методів ES та RIAD Представлено аналітичні вирази для коваріаційної матриці орієнтації, отримані для окремого випадку для методів ES та RIAD. Проаналізовано вплив параметрів моделі вимірювань на точність визначення орієнтації за вказаними методами. Розглянуто питання впливу зміни точності одного з вимірювачів на точність визначення орієнтації. Ключові слова: метод визначення орієнтації; ES; RIAD; модель вимірювань; коваріаційний аналіз; коваріаційна матриця орієнтації. Рижков Лев Михайлович. Доктор технічних наук. Професор. Кафедра систем управління літальних апаратів, інститут аеронавігації, Національний аваційний університет, Київ, Україна. Освіта: Київський політехнічний інститут, Київ, Україна (97). Напрям наукової діяльності: навігаційні прилади та системи. Кількість публікацій: 9. E-mal: lev_ryzhkov@rambler.ru Степуренко Дмитро Iванович. Здобувач. Кафедра приладів та систем керування літальними апаратами, Національний технічний університет України («Київський політехнічний інститут»), Київ, Україна. Освіта: Київський політехнічний інститут, Київ, Україна (6). Напрям наукової діяльності: методи визначення орієнтації літальних апаратів Кількість публікацій: 4. E-mal: d_stepureko@ukr.et

6 44 ISSN Electrocs ad Cotrol Systems 4. 4(4): Л. М. Рыжков, Д. И. Степуренко. Сравнение результатов ковариационного анализа метод ES и RIAD Представлены аналитические выражения для ковариационной матрицы ориентации, полученные для частного случая для методов ES и RIAD. Проанализировано влияние параметров модели измерений на точность определения ориентации с помощью указанных методов. Рассмотрен вопрос влияния изменения точности одного из измерителей на точность определения ориентации. Ключевые слова: метод определения ориентации; ES; RIAD; модель измерений; ковариационный анализ; ковариационная матрица ориентации. Рыжков Лев Михайлович. Доктор технических наук. Профессор. Кафедра систем управления летательных аппаратов, институт аэронавигации, Национальный авиационный университет, Киев, Украина. Образование: Киевский политехнический институт, Киев, Украина (97). Направление научной деятельности: навигационные приборы и системы. Количество публикаций: 9. E-mal: Степуренко Дмитрий Иванович. Соискатель. Кафедра приборов и систем управления летательными аппаратами, Национальный технический университет Украины («Киевский политехнический институт»), Киев, Украина. Образование: Киевский политехнический институт, Киев, Украина (6). Направление научной деятельности: методы определения ориентации летательных аппаратов Количество публикаций: 4. E-mal:

L.M. Ryzhkov Attitude Determination Based on Geometric Relations 17

L.M. Ryzhkov Attitude Determination Based on Geometric Relations 17 LM Ryzhkv Atttude Determat Based Gemetrc Relats 7 UDC 6975 (45) AIUDE DEERMINAION BASED ON GEOMERIC RELAIONS Faculty f Arcraft ad Space Systems Natal echcal Uversty f Ukrae Ihr Skrsky Kyv Plytechc Isttute