A 16-parameter GEO Broadcast Ephemeris

Transcription

1 A 16-parameter GEO Broadcast Ephemeris ZHANG Zhongkai 1, DU Lan 1, LIU Li 2, HE Feng 2, LU Yu 1, ZHOU Peiyuan 1 1. Zhengzhou Institute of Surveying and Mapping, Zhengzhou 451, China 2. Beijing Global Information Application and Development Center, Beijing 194, China Abstract: Currently, the broadcast ephemerides used in GEOs are same as those of the MEOs and IGSOs in the BeiDou navigation constellation. However, a trade-off strategy, i.e. an orbital inclination of 5 rotation, is needed in the fitting algorithm to solve the ephemeris parameters as well as the user satellite position computation for GEOs. Based on the standard broadcast ephemerides, the representations of both the orbit and its perturbation were revised according to the second class of nonsingular orbital elements. In this research, a 16-parameter broadcast ephemeris is presented specifically for GEOs,and user satellite position computation formulas were derived correspondingly. Fit simulations show that the root of mean squares (RMS) of user range error (URE) with two hour and three hour data sets are better than.5 m and.1 m, respectively. Keywords: GEO satellites; broadcast ephemeris parameters; nonsingular orbital elements; perturbation; user range error(ure) 1. Introduction ; Precise satellite orbit and clock bias forecasts need to be fit to provide navigation, position, and time services and the broadcast ephemeris formed and provided to the customers[1-3]. he Beidou2 (also called as Compass) consists of GEO/IGSO/MEO satellites forming a mixed constellation. A unified form however, for the broadcast ephemeris is needed [4-5] for customer convenience. Currently, the broadcast ephemerides used in GEOs are same as those of the MEOs and IGSOs in the BeiDou navigation constellation. Nevertheless, the singular problem occurs when the GPS-like broadcast ephemeris is applied directly in GEOs, and leads to failures during fitting [6]. A trade-off strategy, i.e. an orbital inclination of 5 rotation, is needed in the fitting algorithm to solve the ephemeris parameters as well as user satellite position computation for GEOs [7-1]. However, it requires at least a four hour fitting period with complicated computations [11]. For small-eccentricity and small-inclination GEOs, there usually exist two types of orbit elements to describe the GEOs: second-type and synchronous elements. [12-13]. A 17-parameter broadcast ephemeris based on synchronous elements for GEOs satellites was designed in [14]. As the satellites west-east drafts influences are taken into account in the design, the stability of a fitting algorithm and fitting precision are improved. However, synchronous elements are usually used to describe GEOs in an

2 Earth-Centered Earth-Fixed frame (short as ECEF ), while a 17-parameter broadcast ephemeris is different, with a GPS-like broadcast ephemeris in element number. So, the 17-parameter type broadcast ephemeris is not quite general. A 16-parameter set for a broadcast ephemeris was designed. A broadcast ephemeris based on second-type elements can avoid singular problems related to a small eccentricity and small inclination. Additionally, the required fitting data period can be decreased to two hours, while the long-term influences of the west-east drafts are weakened. 2.Second-type Non-singular Elements Several different types of orbital elements have been used for satellite orbital representation. wo frequently used orbital elements are Kepler, first-type non-singular,, and second-type non-singular elements [15]. Among them, second-type non-singular elements can be used to describe the small-eccentricity and small-inclination GEOs. they can be used in the design of new GEO broadcast ephemeris. he Kepler elements ( aei,,, Ωω,, M) stand for orbit axis, eccentricity, inclination, longitude of ascending node, argument of the perigee and mean anomaly. Set second-type non-singular elements as ( a,,, M ) ei, and the relationships with Kepler elements are as following: M ( ex, ey) ( ecos ϖ, esinϖ) ( ix, iy) ( sin icos Ω,sin isin Ω ) e = = i = = (1) = Ω + ω+ M where ϖ =Ω+ ω, e and i are the eccentricity vector and the inclination vector respectively, and M denotes the mean longitude. he fast angular variable of mean motion M is then measured naturally from the vernal equinox, instead of the ascending node or the perigee. Similar to the true eccentric and mean anomaly of the Kepler elements, there are true and eccentric longitudes corresponding to mean longitude M for the non-singular elements:

3 f E = Ω+ ω + f = Ω+ ω + E (2) where f, E and M are the true eccentric and mean anomaly and f and E represent the true and eccentric longitude respectively. 3.Design of Ephemeris parameters Under various perturbations, orbit changes reflect on the changes of orbit elements. According to the length of periods, element variables include: long-term variables, long-cycle variables and short-cycle variables. Classical orbit analysis can describe the orbit in first-order analysis solutions and part of second-order analysis solutions. However, it is too complicated to use the classical orbit analysis solutions to describe an orbit.of a few hours duration. he GPS standard broadcast ephemeris uses 16 parameters to describe the short-term changes of the orbit (as table 1). For change in a orbit of a few hours duration, the methods to describe of GPS broadcast ephemeris are: 1) combine the long-term parts and long-cycle parts as the long-term parts, and keep them as parameters rate forms; 2) project the short-term parts on the radial, along-track direction and the right-handed triad (radial tangential normal, RN coordinate), and keep the main parts of the combination. hese methods describe orbit changes with only a few parameters and at the same time use simple analysis formulations calculate the position vector of perturbed satellites [16]. For the design of GEO broadcast ephemeris, we improved the second-class non-singularity elements and designed the 16-parameter GEO broadcast ephemeris, as follows: toe, A, ex, ey, ix, iy, M, Δn, ixdot, iydot, x = (3) Crc, Crs, Cλc, Cλs, CNc, C Ns Where t oe Ephemeris reference epoch e, e x y A Root mean square of orbit semi-major axis wo-dimensional components of eccentricity vector i, i wo-dimensional components of inclination vector at t oe x y Δ n M Mean longitude at t oe Deviation of the average angular velocity and calculated values

4 i dot, i dot wo-dimensional components of inclination vector rates x rc rs y C, C Short-cycle correction coefficients of geocentric distance C, C Short-cycle correction coefficients of true longitude λc Nc λs C, C Short-cycle correction coefficients of normal distance Ns ab. 1 comparisons between the two types of GEO ephemerides GPS-like standard ephemeris Design of GEO ephemeris Reference epoch t t oe oe Orbit elements ( Aei,,, Ωe, ω, M) ( Ae, x, ey, ix, iy, M ) Long-term correction coefficients Short-term correction coefficients ( Δ n, idot, Ω& ) ( Δ n, i dot, i dot) ( Crc, Crs, Cuc, Cus, Cic, C is) x ( Crc, Crs, Cλc, Cλs, CNc, C Ns) URE fitting accuracy <.1 m RMS (root-mean-square) <.5 m RMS Fitting period 4 hours 2 hours Efficient period 2 hours 2 hours y In table 1, the design components of the GEO ephemeris and GPS-like standard ephemeris are listed. he greatest point of difference is in the fundamental parameters set. he GEO ephemeris chooses the second-class non-singularity elements and the GPS-like standard ephemeris choose the classical Kepler elements as the fundamental parameters. For the design of GEO ephemeris, 1) orbital plane vectors and vectors rates in Earth Center Inertial Coordinate (ECI) replace the inclination, right ascension of the ascending node (RAAN) and their rates; 2) true longitude f = Ω + u replaces the argumentu = ω + f. 4.Calculation of GEO satellites ephemeris Set an assuming equinoctial point on the orbit, to make argument from assuming equinoctial point to the ascending node to be Ω. Define orbit coordinate O-PQW:earth mass central point as the origin point O, OP intends the equinoctial direction, OW along the direction of orbital angular momentum, OQ as the right-handed trail. Set θg as the Greenwich mean sidereal time of the starting epoch t oe, we can calculate satellite position vectors of the time t in Earth-Centered Earth-Fixed coordinate (ECEF) with ephemeris parameters of epoch. he processor is as following: 1) Calculate semi-major axis and mean motion velocity

5 a = n = ( A) 2 μ a 3 (4) Where μ is the Earth gravitational constant. 2) Calculate the mean longitude, partial longitude and true longitude of time t ( )( ) M = M + n+δn t t (5) oe Calculate the partial longitude by iterating generalized Kepler equation E e sin E + e cos E = M (6) x y he relationship between partial longitude and true longitude is Where sin f a e = sin E e + e sin E + e cos E cos f a = cos E e e e sin E + e cos E ( ) x y x y r β ( ) y x x y r β ( 1 xcos ysin ) 2 2 ( ex ey) r = a e E e E β = (7) (8) 3) Calculate three components of RN with short-cycle corrections included r= r + C cos2 f + C sin2f rc rs f = f + C cos2 f + C sin2f (9) λc λs N= C cos2 f + C sin2f Nc Ns 4) Calculate satellite position vectors in orbit coordinate rcos f r = rsin f (1) N 5) Calculate the transformation matrix from orbit coordinate O-PQW to ECI

6 ( Ω) ( i) ( Ω) M = R R R z x z 2 iy ixi y 1 iy 1 + cos i 1 + cos i 2 ii x y i x = 1 ix 1+ cosi 1+ cosi iy ix cosi (11) Where ( ) ( ) i = i + i dot t t x x x oe i = i + i dot t t y y y oe 2 2 ( x y) cosi = 1 i + i (12) Besides that, the transformation matrix from ECI to ECEF is R z ( θ g ), where Where ω e is the angular velocity of the Earth s rotation. ( t t ) θ = θ + ω (13) g g e oe 6) Calculate satellite position vectors in ECEF ( ) R = R M r (14) z θ g 5.Fitting experiments and analysis Set an GEO satellite around 84 E,the models include 1 orders of Earth s gravitational field, Solar and Lunar gravitational perturbation, solar radium pressure, Earth tides and Earth orientation parameters from International Earth Rotation Service (IERS). As GEO satellites need frequency pointing maneuvers, we obtained position vector data in ECEF for ten days. Additionallly, GEO satellites will suffer twice eclipse periods, around the vernal equinox and the autumnal equinox, and each for 46 days. In order to analyze GEO ephemeris fitting during eclipse periods, simulation started from April 6 th 212, where the satellite suffered eclipse for the first seven days, each for about one hour during the eclipse period. When the fitting period includes the data of eclipse periods, fitting accuracy is lost to some degree because the solar radium pressure will intermittently affect the smoothness of the data. As the ephemeris update every hour, we deal with the data by hours to divide the ten days of data into 258 sets, including two, three and four hour periods in each set. In the sets, the time between steps is 3 seconds. With least squares fitting model to calculate ephemeris parameters, do not end the iteration until the residuals mean square (RMS) of adjacent fitting twice is less than.1 [17-18].

7 For the convenience when setting the initial value, we change the ( A M ), to ( Δa Δλ e ),, standing for the deviation between the semi-axis and referenced value a ref,the deviation between sub-satellite point (SSP) longitude and pointing longitude 84 E. λ, where the referenced value of semi-axis is set as m and the pointing longitude is set as e After the iterations, we can calculate the ephemeris parameters from the fitting parameters, as ( ) 1 2 +Δ A= a a ref M = λ +Δ λ + θ e e g (15) We set the initial values of first set of fitting parameters as, and the follow-up set initial values are as the results pf former set. We regard the ephemeris data as the true values of orbit to evaluate the accuracy of fitting ephemeris and fitting user range error (URE). he fitting success rate is 1%, expect for the first set with 5 times iterations, other sets are within four iterations. 5.1.Orbit fitting accuracy Set the position vectors components errors in radial, along-track and normal radial directions are Δ R Δ and Δ N, table 1 shows the RMS of fitting residuals in three sorts of fitting periods. (1) In non-eclipse period (the left plot), the average fitting RMS in N component is the least up to mm accuracy, and it is almost not affected by the length of fitting periods; the accuracies of fitting error in and R components decrease rapidly with the length of fitting periods increasing, and for four hours fitting period, the accuracies are up to.7m and.2m, so the fitting period is better to be set between two to three hours. (2) In eclipse period (the right plot), the relationships between the fitting errors and length of fitting periods keep the same as these in non-eclipse period for three components N, R and. However, the eclipse leads intermittent solar radium pressures and loss of smoothness of ephemeris data, as a result of increasing fitting errors. he fitting errors of position vectors in and R components are up to.5m and.1m for three hour fitting periods. For this case,the length of fitting period should be set within two hours.

8 RMS of fit residuals in non-ecliping (m) R-mean -mean N-mean length of fit arcs (hours) RMS of fit residuals in ecliping (m) R-max -max N-max length of fit arcs (hours) Figure 1 RMS of fit residuals vs. the length of fit arcs 5.2 Orbit features analysis of fitting parameters Plot 2 lists time series of some orbit parameters for two hour fitting periods. In the plot (a), the eccentricity and inclination are calculated with related fitting parameters. Set as the orbital cycle period, the inclination varies smoothly and the eccentricity varies as 2 form because there are not short-cycle corrections as 2 form for eccentricity. Plot (b) lists time series of the offsets of semi-axis, mean motion and SSP longitude. As the 2 main cycle parts of semi-axis has been included in the short-cycle parameters, the deviation Δ a varies as 3 form (about eight hours), and the amplitude is up to 5m. According to the Kepler s third law, Δ n and Δ a are linearly correlated. he time series of them reflect the features as the law. Besides, the pointing longitude has drafted.1 to the west during the ten days. Meanwhile, the semi-axis was increasing. We did not take the long-term variable of semi-axis into consideration in the 16 parameters design; we can therefore shorten fitting periods to decrease the accumulation of the semi-axis. hat is the main reason that results for 2 hours are better than those for three and four hours in plot 1. he plot (c) shows time series of the two components of inclination vector. here is a jump every seven days; an effect of eclipse on the fitting results. Additionally, the eclipse should have affected the fitting parameters in the orbital plane and their rates because the number of perturbations parameters is limited in the design; the eclipse effects are shown in direction parameters rates of orbital plane. his indicates that there is space to improve the design of ephemeris parameters.

9 Eccentricity x Inclination ( ) days from h Figure (a) ime series of eccentricity and inclination Δa (m) x 1-1 Δn (rad/s) Δλ ( ) days from h Figure (b) ime series of the offsets of semi-axis, mean motion and SSP longitude

10 1 x 1-1 i x dot (rad/s) 5 i y dot (rad/s) x days from h Figure (c) ime series of the two components of inclination vector Fig.2 ime series of some orbital elements from 2 h fit 5.3 Fitting URE accuracy URE is an important evaluation metric for describing the influences of ephemeris and clock offset errors on positioning. URE includes errors of orbit and clock offset, and fitting errors of broadcast ephemeris and clock offset are also included. We analyze the ephemeris fitting error, formulation as: ( ) 1 2 R N URE =.96Δ +.4Δ +.4Δ (16) he fitting errors in radial direction are the main part. o analyze clearly, we calculate the URE with the middle two hours, each hour before and after the referenced epoch. able 2 shows the RMS of the fitting URE for two, three and four hour fitting periods. he shorter the period is, the higher the fitting URE accuracy is. he RMS of fitting URE for two, three and four hour fitting periods are better than 1cm, 5cm, and.1m respectively. During the eclipse period, the accuracy of fitting URE decreases. Fitting period ab.2 RMS statistics of the fit URE (unit: m) Non-eclipse periods Eclipse periods max min mean max 4h h h During the eclipse season, GEO satellites suffer from the eclipse every day; the eclipse period can be up to 72 minutes [19]. able 3 shows the time series of RMS for a two hour fitting URE. In the first seven days, there is regularly one or two fitting jumps. According to these results, it is better to shorten the fitting periods for increased accuracy during the eclipse periods. RMS of URE (m) days from 212/4/6/12h Figure 3 RMS time series of the fitting URE of 2 h fit

11 6. Summary and conclusions Current GEO satellites broadcast ephemeris keep the same forms as MEO/IGSO, but the broadcast ephemeris parameter fitting algorithm and the calculation of user satellite positions requires rotation of the orbital inclination. Based on GEO orbital features, we designed a new set of 16-parameter GEO broadcast ephemeris and a method for calculation of position based on the second-class non-singular elements. the number of parameters is the same, and the complexities are almost the same as the current ephemeris model,. However, for the designed sets, the improvement is significant with non-singular features with 95% confidence, It is easier to set the initial values and it is not necessary to rotate the inclination for the fitting algorithm and user algorithm with the proposed method. Fitting experiments results show that the new 16-parameters set for a broadcast ephemeris is more suitable to describe the sub-point motion within two hours. In the non-eclipse periods, the accuracyof fitting URE for two hours and three hours is up to.5m and.1m. o decrease the influences of the eclipse, it is better to shorten the fitting periods. In order achieve deeper improvements to increase the accuracy of broadcast ephemeris fitting; we also designed an 18-parameter GEO broadcast ephemeris and position calculation, discussed in another paper. References [1] ZHOU S S, CAO Y L, ZHOU J H, et al. Positioning Accuracy Assessment for the 4GEO/5IGSO/2MEO Constellation of COMPASS[J]. Science China: Physics, Mechanics and Astronomy, 212, 55(12): [2] YANG Yuanxi, LI Jinlong, XU Junyi, et al. Contribution of the Compass Satellite Navigation System to Global PN Users[J]. Chinese Science Bulletin, 211, 56(26): [3] MONENBRUCK O, HAUSCHILD A, SEIGENBERGER P, et al. Initial Assessment of the COMPASS/BeiDou-2 Regional Navigation Satellite System[J]. GPS Solutions, 213, 17(2): [4] DIESPOSI R, DILELLIO J, KELLEY C, et al. he Proposed State Vector Representation of Broadcast Navigation Message for User Equipment Implementation of GPS Satellite Ephemeris Propagation[C]// Proceedings of Institute of Navigation National echnology Meeting 24. San Diego: ION, 24: [5] FU X F, WU M P. Optimal Design of Broadcast Ephemeris Parameters for a Navigation Satellite System [J]. GPS Solutions. 212, 16(4): [6] HAN Xingyuan, XIANG Kaihong, WANG Haihong. Research on Broadcast Ephemeris Parameters Fitting Algorithm Based on the First Class of no Singularity Variables[J]. Spacecraft Engineering, 211, 2(4):

12 [7] HE Feng, WANG Gang, LIU Li, et al. Ephemeris Fitting and Experiments Analysis of GEO Satellite[J]. Acta Geodaetica et Cartographica Sinica, 211, 4(Sup.): [8] CHEN Liucheng, HAN Chunhao, CHEN Jingping. he Research of Satellites Broadcast Ephemeris Parameters Fitting Arithmetic[J]. Science of Surveying and Mapping, 27, 32(3): [9] HUANG Yong, HU Xiaogong, WANG Xiaoya. Precision Analysis of Broadcast Ephemeris for Medium and High Orbit Satellite[J]. Progress in Astronomy, 26, 24(1): [1] RYAN Rengui, JIA Xiaolin, WU Xianbing, et al. Broadcast Ephemeris Parameters Fitting for GEO Satellites Based on Coordinate ransformation[j]. Acta Geodaetica et Cartographica Sinica, 211, 4(Sup.): [11] HUANG Hua, LIU Lin, ZHOU Jianhua, et al. Research on 18 Elements Broadcast Ephemeris Model[J]. Journal of Spacecraft &C echnology, 212, 31(3): [12] LI Hengnian. Geostationary Satellite Orbital Analysis and Collocation Strategies[M]. Beijing: National Defense Industry Press, 21. [13] SOOP E M. Handbook of Geostationary Orbits[M]. Beijing: National Defense Industry Press, (SOOP E M. [14] DU Lan, ZHANG Zhongkai, LIU Li, et al. Design of Fitting Parameters for GEO Broadcast Ephemeris[J]. Chinese Space Science and echnology, 213, 33(3): [15] LIU Lin, HU Songjie, WANG Xin. An Introduction of Astrodynamics[M]. Nanjing: Nanjing University Press, 26. [16] XU G C. GPS: heory, Algorithms, and Applications[M]. Berlin: Springer-Verlag, 27. [17] MONENBRUCK, O, EBERHARD, G. Satellite Orbits: Models, Methods, and Applications[M]. Berlin: Springer-Verlag, 2. [18] APLEY B D, SCHUZ B E, BORN G H. Statistical Orbit Determination [M]. New York: Elsevier Academic Press, 24. [19] MULLINS L D.Calculating Satellite Umbra/Penumbra Entry and Exit Positions and imes[j]. Journal of the Astronautical Sciences, 1991, 39: [2] KUZNESOV E D. he Effect of the Radiation Pressure on the Orbital Evolution of Geosynchronous Objects[J]. Solar System Research, 211, 45(5): [21] GUO Rui, ZHOU Jianhua, HU Xiaogong, et al. A New Strategy of Rapid Orbit Recovery for the Geostationary Satellite[J]. Acta Geodaetica et Cartographica Sinica, 211, 4(Sup.):