Development of On-Board Orbit Determination System for Low Earth Orbit (LEO) Satellite Using Global Navigation Satellite System (GNSS) Receiver

Transcription

1 Developmen of On-Boad Obi Deeminaion Sysem fo Low Eah Obi (LEO) Saellie Using Global Navigaion Saellie Sysem (GNSS) Receive Sandip Aghav, S. A. Gangal Depamen of Eleconic Science, Univesiy of Pune, Pune Mahaasha, India. Absac: In ode o mae he obi conol sysem auonomous, and educe he need fo gound inevenion hee is a need fo an on-boad availabiliy of coninuous and accuae nowledge of he saellie obi. In he pesen wo we popose o use on-boad GNSS eceive o compue he obi of he Low Eah Obi (LEO) saellie. hee ae si obial elemens defined fo a saellie, which ae o be specified o define he saellie obi. Wih he help of fou GPS saellies in he GNSS consellaion, on-boad Global navigaion Saellie Sysem (GNSS) eceive collecs he navigaional daa and calculaes is own posiion. he posiion infomaion in he GNSS conains he GNSS daa in he fom of Pseudoanges wih espec o ime. Fom his infomaion; posiion and velociy veco is calculaed as a funcion of ime. Obial elemens of he cuen posiion ae calculaed using he posiion and velociy vecos. Compaing heses obial elemens wih he efeence obi, one can ge he eos ou of ha and coec i fuhe. Measuemen noise and pocess noise models ae seleced fo simulaing acual scenaio. Fo obi esimaion, Eended Kalman file mehod is used. Runge-Kua mehod is used o popagae he efeence obi. Sofwae fo he obi inegaion is developed in MALAB. he effec of vaious zonal peubaions lie J, J, and J 4 wee esed. Fom his sudy i is obseved ha J is he main zonal paamee which affecs he sae veco moe. Howeve j and j4 ems ae included, he effec shows moe deflecions. Paamees J, and J 4 affec fo long em inegaion. In he pesen applicaion long em inegaion is no used. Equaion of moion wih J only is used fo obi inegaion. he sae ansiion mai is used o popagae covaiance mai. I is obseved fom ha, obi of he saellie is confined wih is obial plane bu when ula peubaion em j is inoduced, he saellie obi ges affeced due o gaviaion pull up because of eah obleness. Nomally disibued andom noise vecos ae geneaed. hese vecos ae fuhe used o simulae GPS measuemens. he main pupose of his wo is o sudy a ahe simple bu sill faily accuae algoihm o deemine he aificial saellie obi, in is eal ime and wih low compuaional buden, by using aw navigaion soluion povided by GPS eceive

2 . Inoducion: GNSS (Global Navigaion Saellie Sysem) is a saellie sysem which is used o pinpoin he geogaphic locaion of a use's eceive anywhee in he wold. wo GNSS sysems ae cuenly in opeaion: he Unied Saes' Global Posiioning Sysem (GPS) and he Russian Fedeaion's Global Obiing Navigaion Saellie Sysem (GLONASS). Fo he pesen applicaion, well esablished Global Posiioning Sysem (GPS) is used. his deemines he posiion, velociy and ime (PV) wih high pecision. he GPS sysem allows GPS eceive o deemine is posiion and ime a any place using daa signal fom a leas fou GPS saellies []. Using such sysem, i is poposed o compue, in nea eal ime, a sae veco composed of posiion, velociy, GPS eceive cloc bias and dif of a saellie equipped wih on-boad GPS eceive. his can be done by fileing he aw navigaion soluion povided by he eceive. Kalman File is used o esimae he sae veco based on such incoming obsevaions fom he eceive. he file dynamic model includes geopoenial (i.e. J, J J 4 ) sola adiaion pessue, and he peubaion due o he Sun and he Moon and he cloc bias is modelled as a andom wal pocess. he obsevaions includes he aw navigaion daa composed of posiion and ime bias ha ae compued sepwise by GPS eceive and povides insananeously he absolue posiion.. Mehodology: he obi deeminaion algoihm fo he on-boad obi deeminaion is developed. In he pesen wo, he Eended Kalman file is seleced o geneae he sae esimaes of he saellie obi. he dynamic model of he saellie obi is designed and simulaed. he peubaions lie j, J and j4 ae consideed fo simulaions. he calculaions of Jacobian mai, sae ansiion mai ae also caied ou. he sample aw navigaion daa is pocessed and posiion of he GPS saellie is also calculaed... Eended Kalman File (EKF) Algoihm: heoeical Bacgound: Equaion of moion fo saellie obi is of non-linea fom, and heefoe EKF is seleced fo obi esimaion. I can be consideed as wo sep pocedue; he fis being he ime updaes which pedics he sae veco a subsequen ime using he sysem dynamic model. Second sep is measuemen updae ha esimae he sae veco a he cuen ime based on he measuemen and he pio infomaion fom he fis sep. Sysem model and measuemen model ae especively [] f ( ( ), ) w( ) () z h ( ( )), ()

3 ... ime updae: In he ime updae sep of an EKF, is fuhe divided ino wo pas, sae veco and sae eo covaiance mai updae wih ime. In his sep, he sae veco and sae eo covaiance mai ae popagaed fo he fied ime ineval. ( ) =[ y z v v y v z ], a pioi sae esimae. Whee is he ime ineval beween measuemen (i.e - - ), and f ( ), F () ˆ whee F is a Jacobian mai of he sysem dynamic equaion and w() pocess noise.... Sae veco updae: is a Equaion of moion of pue Kepleian obi is given by; (4) Whee is he disance beween saellie o cene of he Eah and Gaviaional consan. is he Equaion of moion is ond ode diffeenial equaions of he fom; d d,, ; d d Using 4 h ode Runge-Kua mehod, posiion and velociy vecos as a funcion of ime is calculaed by inegaing he acceleaion componens; Velociy veco is v() v ()i v ()j v () and posiion veco is () = ()i + y ()j + z () hei magniudes ae given by; y z v ( ) v v y v z and ( ) y z he, y and z componens of Equaion of moion (Eqn. 4) wih em J ae given by, y y Re z J (4a) (4b)

4 z z Re J, I F z (4c) o updae he sae veco, equaion 4a, 4b and 4c ae numeically inegaed using Runge-Kua 4 h ode fied sep size mehod.... Sae Eo Covaiance Mai updae: he Sae Eo Covaiance Mai is popagaed using following equaion, Whee, P is he ansiion /, P, Q () mai appoimaion, which is compued using a fis-ode aylo seies epansion: P P / is a pioi Sae Eo Covaiance Mai is updaed/pediced sae eo covaiance mai, covaiance mai. Q pocess noise... Measuemen updae: In his sae, he acual measuemens ae colleced fom measuing insumen and modelled in measuemen model given by; z h ( ( )), (6) he measuemen mai fo he same is calculaed fom elaion [], h ( ), H (7) ˆ / H is measuemen infomaion mai. Wih abovemenioned paamees Kalman Gain K is calculaed using he elaion K P / H H he Kalman Gain K is used as feedbac fo coecing sae esimaes, R is measuemen noise covaiance noise mai and new esimae ( ˆ ) is obained fom equaion given below; P / H R

5 ˆ K z h ˆ /, And new pediced eo covaiance ( P ) is obained fom equaion given below; P I K H P / I K H K R K 4.. Eended Kalman File (EKF) Implemenaion: o implemen he EKF, some iniial assumpions ae needed o fulfil he file s equiemen. hey ae spel ou in following paagaphs. Flow cha fo EKF based Obi esimaion is shown in Fig:. SAR ACQUIRE A PRIORI SAE AND COVARIANCE ESIMAES A SE =, i.e Iniializaion =+ ACQUIRE A MEMBER OF OBSERVAION VECOR PROPAGAE SAE VECOR O, CALCULAE SAE RANSIION MARIX F (, + ) CALCULAE EXPEXED MEASUREMEN X AND PARIAL DERIVAIVES OF X WIH RESPEC O X - ( ) PROPAGAE SAE NOISE COVARIANCE MARIX Q(, ) PROPAGAE ERROR COVARIANCE MARIX P - ( ) CALCULAE GAIN MARIX K UPDAE X* - O BECOME h SAE ESIMAE UPDAE ERROR COVARIANCE MARIX P N LAS OBSERVAION? PROPAGAE X() O ANY IME OF HE INRES Y END Fig:. Flow cha fo EKF based Obi esimaion

6 4... Iniializaion: Fo he sysem sae veco popagaion, he sysem model (i.e. equaion a) is numeically inegaed using Runge-Kua 4 h Ode mehod. o sa he Kalman file, i equies iniial sae esimae and iniial sae eo covaiance mai. he sae veco ( ( ) ) of he sysem model is given by; ( ) =[ y z v v y v z ] 6 ; hese include he si dimensional saellie posiion and velociy veco, he hee dimensional. he file is saed using he eceive inenal single-poin navigaion soluion. Fuhemoe, a diagonal apioi covaiance mai P wih sandad deviaions (s) of m (posiion),. m/s (velociy), is assumed. Wih hese assumpions he apioi covaiance mai P is given by; P =diag [s s y s z s v s vy s vz ] 66 ; Wihin he ime-updae phase of he EKF, he saellie sae veco is numeically popagaed fom he laes sae esimae, while he emaining paamees ae assumed o be consan beween he epoch s i- and i Popagaion of Sae and Sae Eo Covaiance mai: he sae ansiion mai (F ) is used o popagae covaiance mai is given by,, I F Whee, J y z y y zy z yz z F J X X (8) I X X 6 X 6 (9) Whee, J [] is Jacobian coefficien.

7 o cope wih deficiencies of he employed popagaion model, a fied diagonal pocess noise mai Q is consideed in he ime updae of he covaiance mai. P /, P, Q Repesenaive pocess noise values used in he pacical applicaions ae ( - m) (posiion), ( - 6 m/s) (Velociy). Wih hese seings, he Kalman file memoy is dominaed by he acceleaion pocess noise, which eflecs he epeced unceainy of he dynamical model.. GPS daa pocessing: hee ae wo ypes of GPS codes ae ansmied by he GPS saellie vehicle. One of hem (P-code) povides pecise posiioning wih an accuacy of appoimaely ens of mees. his code can only be used by a eceive wih access o he encypion ey. his code is only fo miliay uses. he ond code is available o any commecial use. his code is nown as Coase/Acquisiion (C/A) code and his code is discussed in his oveview. Each saellie ansmis wo caie signals. One is cened a 7. 4 MHz (nown as L caie) uses Phase Shif eying (PSK) o modulae boh C/A and P-code ono he caie. he ohe signal (nown as he L caie) is cened a 7.6 MHz and uses PSK o modulae P-code ono he caie. L caie is he signal used by he commecial eceives. I is modulaed wih. MHz Pseudo-Random noise (PRN) code which is unique o each saellie. Each GPS saellie ansmis he Navigaion Message hough C/A code. he C/A codes fom a leas fou GPS saellies ae equied o calculae he use eceive posiion in Eah Cened Eah Fied (ECEF) coodinae sysem [4]... GPS Navigaion Soluion: I is saed ealie ha minimum fou GPS saellie ae mus be in view fo he eceive o deemine o -dimenional posiion. his is because hee ae fou unnowns in he se of fou navigaion equaions. heefoe, o solve fo use posiion and ime, we need o solve he following simulaneous equaions. 4 ( X ) ( Y y) ( Z z) c. ( X ) ( Y y) ( Z z) c. ( X ) ( Y y) ( Z z) c. ( X ) ( Y y) ( Z z) c (a) (b) (c) (d)

8 ,,, 4 ; ae he pseudo-anges o each of he saellies. A pseudo-ange is a measuemen of he disance beween he saellie and he eceive. X i, Yi, Z i ; fo i=,,,4 ae he coodinaes of he saellies in he Eah Cened Eah Fied, WGS-84 coodinae efeence fame,, y z ae he eceive WGS-84 coodinaes, c=.88* +8 (speed of ligh) m/s is he eceive cloc offse fom GPS ime (saellie ime) By lineazing equaions a, b, c and d, one can ge he obsevaion veco (i.e. eceive posiion, y, z ) and cloc bias. GPS saellie sends daa hough navigaion message in fames o GPS eceive o in solving above menioned simulaneous equaions. hese daa ae in spheical coodinaes and equied o ansfomed ino Caesian ones... GPS saellie Navigaion message: he navigaion message includes Almanac daa, ephemeis daa, iming daa, ionospheic delay daa and healh daa of he saellie. he infomaion in he navigaion message has basic five fames. Each fame is subdivided ino five -bi sub-fames and has wods of bi. Ou of above menioned fames, he Saellie Ephemeis daa, ionospheic daa fame and saellie iming daa fame ae of inees fo he pesen wo. A deailed descipion of all infomaion conained in he navigaion message is beyond he scope of his e. he sub-fame of navigaion message conains he ephemeis daa, which is used o deemine he pecise saellie posiion and velociy equied by he navigaion soluion. his ephemeis daa is valid ove a elaively sho peiod of ime (seveal ous), and applies only o he saellie ansmiing i. he componens of he ephemeis daa [] ae lised in able.

9 able: Componens of Ephemeis daa Name Descipion Unis M Mean anomaly a efeence ime semicicle n Mean moion diffeence fom compued value semicicle/s e Ecceniciy dimensionless a Squae oo of semimajo ais / m Longiude of ascending node of obial plane a weely semicicle epoch i Inclinaion angle a efeence ime semicicle Agumen of peigee semicicle Rae of igh ascension semicicle/s IDO Rae of inclinaion angle semicicle/s C Ampliude of he cosine hamonic coecion em of he ad uc agumen of he laiude C Ampliude of he cosine hamonic coecion em of he ad us agumen of he laiude C Ampliude of he cosine hamonic coecion em of he m c obi adius C Ampliude of he sine hamonic coecion em of he m s obi adius C Ampliude of he cosine hamonic coecion em of he m ic angle of inclinaion C Ampliude of he cosine hamonic coecion em of he ad is angle of inclinaion e Ephemeis efeence ime s IODE Issue of daa, ephemeis dimensionless.. Calculaion of ECEF coodinaes of he GPS saellie fom Ephemeis daa: As shown in able:, he ephemeis daa will be eaced fom navigaion message and hen fuhe used o compue he GPS saellie posiion in he fom of Eah Cened Eah Fied (ECEF) coodinae fame using following algoihm []. a ( a) ; Semimajo ais () n / a ; Compued mean moion, ad/s ()

10 M E esin E ; Keple s equaion of eccenic anomaly () f E cos E cos ; ue anomaly fom cosine (4) e ecos E cos f cos ; Eccenic anomaly fom cosine () ecos f Accodingly, he ond hamonics ems of he vaious quaniies ae also deemined. Wih he help of above menioned algoihm he GPS saellie posiion in ECEF coodinae fame is deemined using equaions. X Y cos y cosi sin ; ECEF X coodinae (6) cos y cosi cos ; ECEF Y coodinae (7) Z y sin i ; ECEF Z coodinae (8) he sofwae fo he same is developed in MALAB. 6.. Resuls and discussion: In his ion, he esuls of ime Updae Sep of Kalman file and GPS daa pocessing ae given Sae Veco Popagaion: As shown in he fig. pue Kepleian Obi is inegaed fo a peiod of =86,4. Sep size = is seleced fo obi inegaion. Sae veco used fo obi inegaion is popagaed using following iniial condiions, = [49.4; 984.4;.98]; v = [ ; ; ]; I is obseved fom fig. ha, obi of he saellie is confined wih is obial plane i.e. cenal gaviaional field. he equaion of moion is inegaed wih J. he equaions given in 4a, 4b, and 4c ae inegaed o ge he J peubed obi. As can be seen fom fig when ula peubaion em J is inoduced, he saellie obi ges deviaed fom i cenal gaviaion field. he main deviaion fom cenal gaviaional field is caused by dynamic flaening of he eah. In he Geodeic Refeence Sysem 98 (GRS 8 ), nomal field of he flaening coefficien is epesened by em J =.8. Similaly J and J ae given as, J = -. and J 4 = Afe adding J and J 4 ems, he obi shows addiional deviaion as shown in fig. 4. Nomally disibued andom noise veco ae geneaed. hey ae fuhe used as GPS measuemens. Fig.. shows he noise disibuion of he GPS measuemens ove a peiod of ime

11 (=86,4 ). he sandad deviaion s = m in posiion veco of GPS measuemen is assumed. 4 4 z[m] z[m] y[km] [Km] y[km] [Km]. 4 Fig:. Pue Kepleian Obi Inegaion Fig:. Obi Inegaion wih J ime Vs n oise in Km z[m] 4 y in Km ime Vs y n oise y[km] [Km]. 4 z in Km ime Vs z n oise

12 ime Vs Semi-majo ais ecceniciy Vs ime ime Vs Inclinaion Angle Semi-majo ais(a) in Km ecceniciy(e) Inclinaion Angle(i) in degees Righ Assenion of Ascending Node(OMG) in degees Righ Assenion of Ascending Node Vs ime Agumen of peigee(omg) in degees Agumen of peigee Vs ime ue Anomoly(v) in degees ue Anomoly Vs ime Fig: 4. Obi Inegaion J, J and J 4 Fig:. Simulaed nomally disibued noise veco Fig: 6: Obial elemens in Pue Kepleian Obi

13 689 ime Vs Semi-majo ais 9. - ecceniciy Vs ime 8.47 ime Vs Inclinaion Angle 8.47 Semi-majo ais(a) in Km ecceniciy(e) Inclinaion Angle(i) in degees Righ Assenion of Ascending Node(OMG) in degees Righ Assenion of Ascending Node Vs ime Agumen of peigee(omg) in degees Agumen of peigee Vs ime ue Anomoly(v) in degees ue Anomoly Vs ime Fig: 7. Peubaion of he Obial elemens due o main hamonic J. 688 ime Vs Semi-majo ais.7 ecceniciy Vs ime 8.6 ime Vs Inclinaion Angle Semi-majo ais(a) in Km ecceniciy(e) Inclinaion Angle(i) in degees Righ Assenion of Ascending Node(OMG) in degees Righ Assenion of Ascending Node Vs ime Agumen of peigee(omg) in degees Agumen of peigee Vs ime ue Anomoly(v) in degees ue Anomoly Vs ime Fig: 8. Peubaion of he Obial elemens due o main hamonic J J and J 4. In Fig: 6, i shows he obial elemens in case of pue Kepleian obis. In case of pue Kepleian obi, Eah is assumed o be a spheical, so he only cenal gaviaion foce play vial ole in obi inegaion. Fom fig.6 i obseved

14 ha, he obial elemens ae consan ecep ue anomaly ( ), because hee is no ohe foce which causes he change in obial elemens. he obained obial elemens ae given below; able :. Vaiaions in obial elemens in diffeen cases. Obial Pue Kepleian Obi J J, J, J 4 Elemens Ma Min Ma Min Ma Min a Km e I deg OMG deg omg deg Vaiaions in he obial elemens due o ula peubaions ae shown in Fig. 7 and Fig: 8. able: also shows he vaiaions in obial elemens poduced in diffeen cases.. Equaion of he saellie is inegaed fo peiod =86,4 wih fied sep size=. As shown in Fig: 7, egession of he ascending node unde he J peubaion is obseved. In case of J peubed obi, he ascending node and agumen of peigee ehibis significan linea vaiaion. he maimum and minimum values ae given in able:. he moion of he ascending node occus because of he added aacion of he Eah s equaoial bulge, which inoduces foce componens owads he equao. he ascending node egesses fo diec obis ( deg < i < 9 deg) and advances fo eogade obis (9 deg < i < 8 deg). Sofwae fo he obi inegaion is developed in MALAB. he effec of vaious zonal peubaions lie J, J, and J 4 wee esed. Fom his sudy i is obseved ha J is he main zonal paamee which affecs he sae veco moe. Ohe paamees lie J, and J 4 ae affecs fo long em inegaion. In he pesen applicaion long em inegaion is no equied. In his case equaion of moion wih J is used fo obi inegaion Sae Eo Covaiance Popagaion: o eecue his sep, a diagonal pioi sae eo covaiance is assumed as follows; P... Whee s posiion = m and s velociy =. m/s

15 Updaed Sae Eo Covaiance Mai is given by, P /, P, Q, and ( e F / ) Wih efeence o iniial condiions, Jacobian Mai is calculaed fo single sep. he esuls ae given below; F /.E - 6.8E - 6.E - 9.8E E E -.E E - -.8E - 6 And Sae ansiion Mai is calculaed fo he same condiions, he esul of he same is given below; /. 9.E - 7.E -.E - 6.8E - 6.E E E -.8E E E - 6.7E - 4.9E E E - -.8E - 6.E - 7.E -. 9.E E -.E - 7.6E - 9.E E -.E -.6E - 6.7E - 6.7E Similaly fom sae ansiion mai, new updaed sae eo covaiance mai is calculaed. Wih hese calculaion, he wo elaed o ime updae sep of he Kalman File is compleed. 6.. GPS daa pocessing: he sofwae code fo he calculaion of GPS saellie posiion in ECEF coodinae sysem fom navigaion message is developed in MALAB envionmen. he ECEF coodinae fo one he sample ephemeis ae compued and he value of he same ae given below. X = ; Y = ; Z = ; Above esuls ae validaed wih sample daa. Conclusion: In his wo, he Eended Kalman File (EKF) algoihm is developed and some pa of i is implemened and esed. he sae veco is popagaed using fied sep size obi inegaion. Saellie obi dynamic model wih J J and J 4 ae inegaed. Wih his epeimen, i is obseved ha obi inegaion wih J only is

16 sufficien o use fo pesen applicaion. he sae ansiion mai of he pue Kepleian Equaions is also compued. he ECEF posiion is calculaed by pocessing GPS navigaion message daa. Wih his wo, we concluded ha fo nea eal ime obi deeminaion applicaions, he use of simplified dynamic model wih J peubaion and suiable obi inegaion echnique educes compuaional buden fom he hadwae. Acnowledgemen: he auhos han he Indian Space Reseach Oganizaion-Univesiy of Pune Space echnology Cell, Pune fo he financial suppo. he auhos also han D. Pamod ale Refeences: [] Painson B. W, Spile J. Global Posiioning Sysem: heoy and Applicaions, AIAA, Vol., 996 (Pogess in Asonauics and Aeonauica 6). [] Fundamenals of Kalman Fileing: A Pacical Appoach, Second Ediion, by Paul Zachan, AIAA,. [] Real ime mulisaellie obi deeminaion fo consellaion mainenance, Poceedings of COBEM 7. [4] NAVSAR GPS use equipmen inoducion, US Govenmen, chape 7. [] Global Posiioning Sysem, Ineial Navigaion and Inegaion, Mohinde Gewal, A John Wiley and Sons Inc, Publicaion, page.no 7. Pape Refeence No.: PN-88 ile of he pape : Developmen of On-boad obi deeminaion sysem fo Low Eah Obi (LEO) saellie Using Global Navigaion Saellie Sysem (GNSS) Receive Name of he Pesene: Auho (s) Affiliaion: Mailing Addess: M. Sandip Aghav Senio Reseach Fellow and PhD Candidae Depamen of Eleconic Science, Univesiy of Pune, Mahaasha, India 4 7

17 Addess: elephone numbe (s) : , Mobile no.: Fa numbe (s) : Bio-daa i) Pesenly woing as a Senio Reseach Fellow (SRF) and a PhD suden in he Depamen of Eleconic Science, Univesiy of Pune on pojec Auonomous Saellie Navigaion Sysem fo Low Eah Obi (LEO) saellie using Global Navigaion Saellie (GNSS) funded by Indian Space Reseach Oganizaion (ISRO). ii) Woed as a Junio Reseach Fellow (JRF) in he Depamen of Eleconic Science, Univesiy of Pune on pojec Developmen of Gound Saion fo ANUSA funded by Indian Space Reseach Oganizaion (ISRO). iii) Pos Gaduaed fom Depamen of Eleconic Science, Univesiy of Pune in MSc. wih Eleconic Science as a Special subjec.