Research and Simulation of satellite orbit modeling Wenle Yuan, Xuanmin Lu, Jinjie Cao, Wensheng Luo

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1 4th Iteratioal Coferece o Electrical & Electroics Egieerig ad Computer Sciece (ICEEECS 2016) Research ad Simulatio of satellite orbit modelig Wele Yua, Xuami Lu, Jijie Cao, Wesheg Luo (School of Electroics ad Iformatio, Northwester Polytechical Uiversity, Xi a) Keywords: Beidou satellite avigatio system; Beidou satellite orbit modelig; observatio data simulatio; trajectory simulatio; geometric accuracy factor Abstract. Chia's Beidou satellite avigatio system has completed the regioal etwork i Asia Pacific, ad successfully applied i surveyig ad mappig, telecommuicatios, trasportatio, disaster relief ad public safety ad other areas, resultig i sigificat ecoomic ad social beefits. With the wide applicatio of Beidou avigatio system, Beidou avigatio products are more ad more, how to simulate the satellite orbit o the groud, has become a major problem of Beidou avigatio product performace testig eeds to be solved. I order to solve this problem, through the aalysis of the Beidou satellite orbit simulatio ad modelig, ad usig the Chebyshev polyomial fittig algorithm of satellite orbit. Fially, the operatio results of simulatio module is aalyzed i detail, the simulatio results are cosistet with the Beidou satellite related research data, verify the validity of the desiged system. Therefore, the coclusio: Beidou satellite orbit modelig successfully. 1. Itroductio I the simulatio of the Beidou satellite costellatio, eed to kow the istataeous satellite positio ad velocity. Typically, the istataeous satellite positio ad velocity is through the orbit satellite avigatio message issued the parameters accordig to the orbit model solutio obtaied, so the modelig ad Simulatio of satellite orbit satellite costellatio simulatio is a key lik. This sectio will itroduce the orbit model, satellite orbit simulatio, Chebyshev polyomial fittig algorithm of satellite orbit. 2. Track model aalysis Similar to the shape of the earth rotatio ellipsoid, strictly speakig, is a ellipsoid of irregular shape, ad the earth's mass distributio is ot uiform, therefore, to the earth satellite gravity is ot always toward the earth. The motio of the satellite i space, except by the earth's gravitatioal field perturbatio, but also by the the other celestial gravitatio, especially the perturbatio ad the solar radiatio pressure ad other factors, the orbit is very complicated, it is difficult to make accurate mathematical methods by satellite. I all of the perturbatio effect of the earth's gravitatioal field, the maximum perturbatio, so the simulatio of satellite orbit parameters cosiderig the gravitatioal field of the earth perturbatio. O the basis of Kepler's Law: the motio of the satellite orbit by a elliptical geocetric plae, the ellipse with a focus o the coicidece. Ideal elliptical orbit ca use the followig 6 orbit parameters (Kepler Track parameter): Track log axis a orbital eccetricity e The size ad shape of the elliptical orbit of the satellite are determied by the log axis ad the eccetricity of the orbit. icliatio of orbit i Track the dihedral agle ad the equatorial plae; right ascesio of ascedig ode Ω The earth's equatorial plae, the agle betwee the veral equiox ad the ascedig ode; argumet of perigee ω The ceter of the earth's orbit plae, ear the itersectio poit betwee the locatio ad the ceter of the earth; The orbital icliatio, RAAN, ear poit distace of three parameters to determie the Copyright 2016, the Authors. Published by Atlatis Press. This is a ope access article uder the CC BY-NC licese ( 301

2 relative positio ad directio of satellite orbital plae ad the earth. true aomaly f The agle betwee the earth's orbit ad ear the earth's orbit. The positio of the satellite i the orbit plae is determied by the true ear poit agle. The schematic diagram of Kepler's orbital parameters is show i Figure 1: Z 卫星真近点角近地点轨道面倾角地心赤道近地点角距春分点轨道升交点赤经 X 升交点 Y Fig. 1 Schematic diagram of Kepler track parameters Figure 2-2, the istataeous positio of the satellite i the orbital plae is described by the ear poit agle, i additio to the real poit of the ear poit, there are commoly used ear poit agle ad ear the poit of peace. 3. Beidou satellite orbit simulatio The motio orbit satellite by satellite ephemeris i orbit log radius, eccetricity, orbit icliatio, RAAN, ear poit distace determied. The simulatio o the orbit of the satellite, accordig to the 5 parameters, calculate the space positio of the satellite orbit. The satellite orbit simulatio, through the followig 5 steps. to establish the coordiate system The earth is the origi, establishmet of space right i Cartesia coordiates, the X axis is semi axis poitig logitude 180 directio, The axis of the Y axis is poitig to the orth pole, ad the Z axis is determied by the X axis ad the Y axis. Create Ellipse I the equatorial plae of the earth, the semi major axis of the ellipse, which is a ellipse, is created a ; Heart rate e. Place the earth i oe of the focal poits of the ellipse. settig the icliatio agle of the track The rotatio matrix is obtaied accordig to the defiitio of the icliatio of the orbit Ri Lift the ellipse aroud the X axis i Ri = 0 cos i si i 0 si i cos i 1 set up the ascesio poit of ascesio Accordig to the defiitio of the ascedig ode, the rotatio matrix is obtaied cos Ω 0 si Ω 2 RΩ = si Ω 0 cos Ω set the argumet of perigee Accordig to the defiitio of the AP, get the rotatio matrix Rω, the earth moves to a focal poit of the ellipse, ad through the ceter of the earth ad perpedicular to the orbital plae trajectory as 302

3 the rotatio axis, the rotatio agle aroud the axis orbit ω cos ω + Bx2 (1 cos ω ) Bx B y (1 cos ω ) Bz si ω Bx Bz (1 cos ω ) + B y si ω Rω = Bx By (1 cos ω ) + Bz si ω By Bz (1 cos ω ) Bx si ω (3) cos ω + By2 (1 cos ω ) Bx Bz (1 cos ω ) B y si ω B y Bz (1 cos ω ) + Bx si ω cos ω + Bz2 (1 cos ω ) Amog: B = ARi RΩ A = [0 1 0] 4 4. The Chebyshev fittig algorithm of satellite orbit The update cycle of Beidou satellite ephemeris for a hour, so there will be some ephemeris error, thus calculated the Beidou satellite istataeous positio ad istataeous speed will have a certai error. So, the eed for simulatio of satellite orbit approximatio through a certai algorithm, commoly used algorithm Lagrage differece algorithm ad Chebyshev fittig algorithm. I this paper, usig the Chebyshev fittig algorithm. [ ] Assumed i time iterval t 0, t 0 + t, usig the Chebyshev polyomial approximatio, amog t 0 Is the startig time, t is Iterval time.first, the coversio of time iterval: 2 (5) t = (t t 0 ) 1, t [t 0, t 0 + t ] t The istataeous positio ad istataeous velocity of the satellite ca be expressed as the compoet of the X axis: x(t ) = C xi Ti (t ) i =0 (6) x(t ) = C xi Ti (t ) i =1 Amog: Chebyshev polyomial order; C xi The Chebyshev polyomial coefficiets. The calculatio method of the compoet alog the Y directio ad the Z directio is similar to, the X directio Ti (τ ) ad Ti (τ ) the expressios are as follows: T0 (τ ) = 1 T1 (τ ) = τ T (τ ) = 2T 1 (τ ) T 2 (τ ) τ 1, 2 T1, (τ ) = τ, T2, (τ ) = 4ττ, T, (τ ) = 2 τt, 1 (τ ) T, 2 (τ ) From equatio (5) coefficiets by least squares to Chebyshev polyomials: (9) C x = ( B T B) 1 B T X amog 8 : 303

T0 (τ 1 ) T1 (τ 1 ) T (τ ) T (τ ) 1 2 B= 0 2 T0 (τ m ) T1 (τ m ) X = [ x1 x2 x3 T2 (τ 1 ) T (τ 1 ) T2 (τ 2 ) T (τ 2 ) T2 (τ m ) T (τ m ) x ]T 10 11 Obtaied coefficiet C x we ca use type (6) to

The results of simulatio aalysis are demostrated ad aalyzed The parameters of the simulatio aalysis fuctio iclude the positio of the observatio statio, the observatio time, the calculatio time

4 T0 (τ 1 ) T1 (τ 1 ) T (τ ) T (τ ) 1 2 B= 0 2 T0 (τ m ) T1 (τ m ) X = [ x1 x2 x3 T2 (τ 1 ) T (τ 1 ) T2 (τ 2 ) T (τ 2 ) T2 (τ m ) T (τ m ) x ]T Obtaied coefficiet C x we ca use type (6) to calculate the correspodig time iterval at ay time the positio ad velocity of the satellite. The simulatio usig 13 order Chebyshev polyomial fittig. 5.The results of simulatio aalysis are demostrated ad aalyzed The parameters of the simulatio aalysis fuctio iclude the positio of the observatio statio, the observatio time, the calculatio time iterval, the height of the cut-off agle, ad the iterface is show i Figure 2: Figure 2 Cofiguratio Iterface of Simulatio aalysis The iitial value settig of the simulatio aalysis module: the locatio of the observatio statio 110 E 40 N 0, observatio time period (September 20, miutes 0 secods to 0 ( ) miutes, 0 miutes, 5 miutes, secods), the time iterval of 60 secods, the height of the cut-off poit for the followig simulatio results are based o the above set coditios. Visible satellite umber curve drawig ad result aalysis: Figure 3 shows the umber of satellite curves Ca be see from Figure 3, the umber of visible satellites are available i 6, oly a very short period of time was 6, the rest are i 8 or so, ca be up to 11. The simulatio results are cosistet with the Beidou satellite research data show that the accuracy of the simulatio system. 304

5 6. Summary The key techology of the Beidou satellite orbit calculatio model ad Chebyshev polyomial fittig algorithm. Usig VC6.0 ad MATLAB as a combiatio of architecture, desig ad completio of the two regioal Beidou satellite costellatio simulatio ad visible satellite costellatio simulatio, GUI desig is completed by MFC, ad with MATLAB star Zuo Jiamo, matlab will evetually be compiled ito.c files so as to achieve the combiatio of the two. Through the research work i this paper, we have made some achievemets i the research ad implemetatio of the test system of the Beidou avigatio products, ad it has importat referece value for the costructio of the Beidou satellite avigatio system. Refereces [1] [2] [3] [4]ZHANG P F XU C D HU C S et al. Time scales ad trasformatios amog satellite avigatio systems [J]. LNEE [5] HAN C H YANG Y X CAI Z W. BeiDou Navigatio Satellite System ad its time scales [J].Metrologia S213-S218. [6] Hu Qia. Satellite avigatio simulatio system user trace ad gesture simulatio techology research [d]. Beijig: Beijig Istitute of Techology, 2012 [7] Gaa Yua, Tao Zhag. Usceted Kalma Filterig for Ultra-tightly Coupled GPS/INS Itegratio[J]. IEEE Iteratioal Coferece o Mechatroics ad Automatio. 2009, 18(1): [8] CHEN Ya CHEN Hui DING Qiaju Aalysis ad simulatio o coverage of No 2 BeiD0u satellite avigatio system[j] Commuicatio Techology (6) (i Chiese) [9] Yag Yuaxi, Li Jilog, Wag Aibig, Xu Juyi, He Haibo, Guo Hairog, She Jufei, Dai. Basic avigatio ad positioig performace of the Beidou satellite avigatio system [j]. Chiese Joural of sciece, 2014, 44 (1): [10] Dua Yupeg, Wei Zogkag. A improved method for GPS pseudo rage avigatio, [j]. avigatio ad cotrol, 2012, 11 (3). 305

The Mathematical Model and the Simulation Modelling Algoritm of the Multitiered Mechanical System

The Mathematical Model and the Simulation Modelling Algoritm of the Multitiered Mechanical System The Mathematical Model ad the Simulatio Modellig Algoritm of the Multitiered Mechaical System Demi Aatoliy, Kovalev Iva Dept. of Optical Digital Systems ad Techologies, The St. Petersburg Natioal Research