SATELLITE ORBIT ESTIMATION USING ON-LINE NEURAL NETWORKS. Mahsa-Sadat Forghani, Mohammad Farrokhi

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1 SATELLITE ORBIT ESTIMATION USING ON-LINE NEURAL NETWORKS Mahsa-Sadat Foghai, Mohammad Faohi Depatmet of Electical Egieeig Cete of Ecellece fo Powe System Automatio ad Opeatio Ia Uivesity of Sciece ad Techology Teha IRAN Abstact: This pape pesets satellite obit estimatio usig atificial eual etwos. A multilaye Pecepto is used to estimate the positio of a low-eath obit satellite. The mai goal is to filte out oisy o icomplete data eceived fom sesos. The algoithm is applied to the CHAMP satellite. The same obit is estimated usig the eteded Kalma filte. Simulatio esults show supeio pefomace of the eual etwo as compaed to the eteded Kalma filte. Copyight IFAC Keywods: Satellite obit, Estimatio Neual etwos, Eteded Kalma Filte 1. INTRODUCTION Low-Eath-Obit (LEO) satellites cicle the eath i diffeet altitudes ad icliatios. The obit icliatio is the agle betwee the plate of obit ad the equato. The altitude of LEO satellites is a few huded ilometes above the eath suface (Sidi, ). Diffeet filtes, such as ecusive filtes, batch filtes, ad Kalma filtes have bee poposed i liteatue fo positio estimatio of LEO satellites (Mahy, 1; Psiai, ; Vegez et al. 4; Yoo et al., ). I this pape, Neual Netwos (NN) ae employed fo the positio estimatio of CHAMP LEO satellite. The eal data fo simulatios ae obtaied fom the followig websites: I ode to compae the pefomace of the poposed method, the Eteded Kalma Filte (EKF) is also used to estimate the positio of the same satellite. Simulatio esults show supeio pefomace of the NN as compaed to the EKF.. DYNAMIC OF SATELLITE ORBIT The law of plaetay movemets, which was discoveed by Keple about 4 yeas ago, is the basis of satellites otatio aoud the eath. Accodig to the basic piciples of these laws, if the mass of satellite is igoed as compaed to the mass of the eath, ad if the eath is assumed to be spheical, the accodig to the Newto s gavity law, the acceleatio of satellite ca be calculated as (Motebuc ad Gill, ) = (1) whee M is the mass of eath, G 1 = (6.6759±.85).1 mg s is the gavity costat, is the distace of satellite fom the cete of eath, ad is the uit vecto coectig the satellite to the cete of eath. Equatio (1) shows that the acceleatio of satellite is ivesely popotioal to the distace of satellite fom the cete of eath. I ode to descibe otatio

2 of the satellite aoud the eath, the followig idepedet paametes should be defied: R = [ a e i Ω ω M ] T () whee a, called the semi-majo ais ad measued i mete o feet, is a costat defiig the size of the obit, e, called the ecceticity, is a costat defiig the shape of the obit (=cicula, less tha 1=elliptical), i, called the icliatio is the agle betwee the equato ad the obit plae, Ω, called the ight ascesio of the ascedig ode, is the agle betwee the veal equio ad the poit whee the obit cosses the equatoial plae (poitig to the oth), ω, called the agumet of peigee, is the agle betwee the ascedig ode ad the obit's poit of closest appoach to the eath, ad M, called the tue aomaly, is the agle betwee the peigee ad the satellite i the obit plae. Two obital elemets, a ad e, defie the shape of obit, M defies the positio of satellite o the obit, ad thee othe elemets (i.e. i, Ω ad ω ) defie the diectio of obit i the space. These si elemets ae calculated i tems of the positio ad the velocity vecto (Tapley, 4) = [ y z y z ] T () whee, y ad z defie the positio i ECEF coodiate system. Both R ad ca be used to defie the positio of satellites. I this pape, is employed fo estimatio of LEO satellite positio. Liea diffeetial equatios lie = F. ae ot appopiate fo discete estimatio as i Kalma filte. Fo istace, tasitio mati i Kalma filte is cosideed geeally as = ϕ. -1, which ca be witte as =. (4) whee the tasitio mati ϕ is equal to... y z y y y... = y z z z z... y z which ca be calculated as d = = F. = d =F. dt dt d =F.dt (5) d =.dt =. dt F F (6) l l = l = F.(t t ) -1-1 = -1 F.(t t ) e. = e. F.(t t ) -1 F Sat Usig the epasio seies t 1 1 e F = I+ F t + F t + F t +!! ad omittig the oliea tems yields Φ = I+ F t (7) (8) Usig the equatio of motio of satellites i (1), the followig equatios ae obtaied: d dy dz =, = y, = z dt dt dt d = = dt dy = y = y dt dz = z = z dt (9) Calculatig the patial deivatives with espect to the positios ad speeds, the dyamic equatio of satellite movemets ae = y z (1 ) 5 5 y y y z (1 ) 5 5 z y z z (1 ) 5 5 (1) Whe the positio of satellite (i.e. the age, the azimuth ad the elevatio agles) is cotiually available, oe ca estimate the positio of satellite with espect to the eath tacig statio. The usig the positio of the eath-tacig statio the measued positios ca be calculated with espect to the eath cete. I this pape, the obital estimatio of the CHAMP satellite is cosideed. The obital specificatio of this satellite is show i Table 1. Table 1: Obit specificatio of CHAMP satellite Obit icliatio (i) Agumet of peigee ( ω ) Right Ascesio of Ascedig Node( Ω ) Semi-majo ais (a) Km Ecceticity (e). The closest distace fom the obit The fa distace fom the obit The time fo oe complete otatio 58 Km 64 Km 91.58mi

3 . ORBIT ESTIMATION USING EXTENDED KALMAN FILTER Sice the elatioship betwee the measuemets of the satellite positio is oliealy elated to the state of the system, this violates the liea assumptio of the Kalma filte. The Eteded Kalma Filte (EKF) is a ad hoc techique to povide a way to use the stadad Kalma filte o o-liea pocess o measuemet models esultig i sub-optimal estimates. The measuemet model ad pocess model ae lieaized about the mea ad covaiace at evey iteatio ad the the stadad Kalma filte is applied to the lieaized models. I the liea discete Kalma filte, the state of the system ca be updated with a staightfowad mati multiplicatio F ( + 1, ). Similaly, covetig fom the state to measuemet space is accomplished with aothe mati multiplicatio C ( ). Both of these matices ae appoimated i the EKF usig a fist ode Taylo epasio. To accomplish this, the Jacobia mati of both the pocess model ad the measuemet model eed to be calculated. Sice the pocess model is aleady liea, the calculatio is tivial, p + v t py + vy t pz + vz t F (, ) = (11) v v y vz 1 t 1 t F ( ) 1 t (1) = But, the Jacobia mati of the oliea measuemet model is otivial p + py + pz p y C( ) = acta( ) p p accos( z ) p + + py pz p C( ) p y = pz + p z p pz + p y p pz + pz p y pz + pz p z (1) (14) These lieaized matices ae icopoated ito the EKF usig the followig equatios. The Kalma gai is calculated as (Hayi 1) H H G f ( ) = K( ) C ( )( C( ) K( ) C ( ) + Q ( )) (15) whee Q 1, Q, F ad C ae the covaiace pocess-oise, the covaiace measuemet oise, the lieaized state tasitio ad the lieaized measuemet matices, espectively. The estimatio eo is equal to α ( ) = y ( ) C ( ˆ ( Y )) (16) whee y ( ) is the vecto cotaiig the actual measuemets ad C(, ) is the measuemet model fuctio at time. The ew state estimates ae calculated as ˆ( Y ) = ˆ( Y 1) + G ( ) α( ) (17) The covaiace eo mati is equal to K ( ) = ( I G f ( ) C( )) K( ) (18) The the et estimates of states ae calculated as ˆ( + 1Y ) = F( ( Y )) (19) Ad the eo covaiace fo the et iteatio is pedicted, ( 1, ) ( 1, ) ( H K + = F + K ) F ( + 1, ) + Q1( ) () whee F ( + 1, ) is the Jacobia mati evaluated at the cuet state estimate. Figs. 1-5 show the simulatio esults fo the EKF fo estimatig the positio of the CHAMP satellite i oe complete otatio aoud the eath. The measuemet oise is 7% of the actual values. The, y ad z vaiables ae show i m, measued fom the suface of the eath ad i the diectio of the eath cete. Net, it is assumed that thee eists pacet loss i data. That is, i the time iteval5< t < 6mi, o data is eceived fo estimatio. Fig. 4 shows the estimated positio i this case. I additio Fig. 5 shows the oisy measued data ad the estimated states i D. Estimatio eo of (m) Estim atio eo of y (m) Estim atio eo of z(m) Fig.1. Estimatio eos alog the, y ad z ais usig EKF f

4 Measumet Data Estimated positio Measumet Data 4 4 Estimated Positio 1 1 z(m) -1 z(m) y(m) -4 (m) Fig.. Positio estimatio of the CHAMP satellite i D, usig the EKF y(m) (m) Fig. 5. Positio estimatio i D, whe thee is data pacet loss 1 Azimuth agle(deg) Elevatio agle(deg) Rage(m) estimated positio Measumet Data Time (mi) Fig.. Positio estimatio i pola coodiates Estimatio eo of (m) Estimatio eo of y(m) Estimatio eo of z(m) Fig. 4. Estimatio eo usig the EKF with pacet loss i data 4. ORBIT ESTIMATION USING NEURAL NETWORKS It is a well-ow fact that eual etwos ae capable of estimatio. Diffeet eual etwos ae employed fo estimatio i egieeig applicatios. I this pape, a multilaye pecepto with eo bacpopagatio algoithm is used fo estimatio of satellite positio. The goal of this algoithm, which is based o the gadiet descet method, is to miimize the istataeous estimatio eo (Hayi 1999) T ( ) ( ) E( ) = y ( ) y( ) = F( ; w ) y( ) (1) o whee y ( ) is the output of the eual etwo ad o y( ) is the desied output at th iteatio step, espectively. The vecto w cotais the adjustable weights (syaptic ad bias) of the etwo. The etwo employed i this pape, has thee layes: the iput laye, which cotais the iput odes, with seve iputs, the hidde laye, which cotais euos with oliea activatio fuctio (sigmoidal fuctios), ad the output laye, which cotais euos with oliea activatio fuctio (sigmoidal fuctios) ad povides the estimated output of the etwo. Fo satellite estimatio i this pape, thee ae 7 iputs, 5 euos i the hidde laye, ad 1 output i the eual etwos. Hece, thee eual etwos ae used to estimate, y ad z, espectively. The iputs to the eual etwos ae [ yd ( i + 1) yd ( i) yd ( i ) yd ( i ) ym( i ) ym ( i ) ym( i )], whee y d is the desied obit ad the y m is the measuemet data. I ode to avoid satuatio i the euos duig taiig of the etwo, the iputs ad outputs ae omalized betwee zeo ad oe (Hayi 1999). Weights ae iitialized adomly usig small umbes. The leaig ate is equal to.9. Adaptatios of weights ae caied out o-lie. That is, thee is o stop i taiig of the etwo. Figs. 6-9 show the simulatio esults. As these Figues show, the eual etwo ca estimate the positio of the satellite much bette tha the EKF, eve whe thee is pacet loss i data.

5 Estimatio eo of z(m) Estimatio eo of y(m) Estimatio eo of z(m) Fig. 6. Estimatio eos alog the, y ad z ais, usig eual etwo Estimatio eo of (m) Estimatio eo of y(m) Estimatio eo of z(m) Fig. 9. Estimatio eo usig the eual etwo, whe thee is pacet loss i data. z(m) y(m) measumet Data Estimated Positio -4 - (m) Fig 7. Positio estimatio of the CHAMP satellite i D, usig the eual etwo age(m) Azimuth agle(deg) Elevatio agle(deg) Estimated positio Measumet Data Fig. 8. Positio estimatio i pola coodiates 5. CONCLUSION I this pape, the eteded Kalma filte ad eual etwos wee employed fo estimatio of LEO satellites obit. I the case of the eteded Kalma filte, the iitial values of states must be defied popely; othewise states of the filte ca divege, yieldig istability. O the othe had, this poblem does ot eist i eual etwo, sice iitial values of weights ae selected adomly. The RMSE 1 fo estimatig the age alog the, y ad z ais usig the eteded Kalma filte ae 1, 11 ad 7. Km, espectively, ad fo the eual etwo these eos ae.94,.9 ad.76 Km, espectively. I othe wods, the o-lie taied eual etwo could estimate the obital positio of the satellite much bette the eteded Kalma filte. Moeove, both methods could cope well with data loss. Nevetheless, the eual etwo still shows less estimatio eo i this case. Oe impotat fact fo the eual etwo is that, it must be taied o-lie with o stoppig i the taiig. REFERENCES Hayi S. (1999) Neual Netwos, A Compehesive Foudatio Petice Hall, New Jecy. Hayi S. (1) Adaptive Filte Theoy, 4 th editio Petice Hall, New Jecy. Mahy El. (1). Efficiet satellite obit detemiatio algoithm. Poceedig of the Eighteeth Natioal Radio Sciece Cofeece, 1. Motebuc, O., E. Gill (). Satellite Obits - Methods, Models ad Applicatios, Spige Velag, New Yo. Psiai, M. L. (). Satellite obit detemiatio usig a sigle-chael global positioig system eceive, Joual of Guidace, Cotol ad Dyamics, 1 (5), pp Root Mea Squae Eo

6 Sidi M. J. (). Spacecaft Dyamics ad Cotol, Cambidge Uivesity Pess, Cambidge, UK. Tapley, B. D., B. E. Schutz, ad G. H. Bo (4) Statistical Obit Detemiatio Elsevie. Vegez, P., L. Saute, ad S. Dahle (4). A Impoved Kalma Filte fo Satellite Obit Pedictios. The Joual of the Astoautical Scieces, 5 (). Yoo J. C., B. S. Lee, ad K. H. Choi (). Spacecaft obit detemiatio usig GPS avigatio solutios, Aeospace Sciece ad Techology, (4), pp