Accuracy evaluation of broadcast ephemeris for BDS-2 and BDS-3

6 Abstract: The BDS-3 system was completed in July 2020 and began to provide 7 services to users around the world. The inspection of its operation, especially the 8 detailed evaluation of the orbit, clock error, TGD and other indicators, plays an 9 important role in the subsequent positioning. This study conducts an investigation of 10 the satellite broadcast ephemeris of the BDS-2 and BDS-3. The difference between the 11 satellite orbit position calculated by the broadcast ephemeris and the position calculated 12 by the precise ephemeris is used for analysis. First, the ephemeris form January 2020 13 to February 2020 are investigated. The results show that the broadcast ephemeris 14 accuracy of the BDS-2 MEO satellite is the highest, while the GEO satellite broadcast 15 ephemeris accuracy is the lowest. And their three-dimensional orbit difference is 3m 16 and 7.5m, respectively. Second, the BDS-3 MEO satellite broadcast ephemeris 17 accuracy is higher than the BDS-2, its three-dimensional orbit accuracy is about 0.39m, 18 while its clock error is slightly smaller than the BDS-2. The result of ephemeris 19 calculation is basically equivalent to the clock error of satellite-to-earth observation, 20 which is related to the addition of the clock error of the inter-satellite link in the BDS- 21 3. Finally, the clock error of the BDS-3 MEO satellite with the H clock is basically the 22 same as that of the MEO satellite with the Rb present an assessment of BDS-3 satellites and BDS-2 three kinds of satellites broadcast orbit and clock offset precise as the for the a standard single point positioning (SPP) test used to evaluate the correctness of the results. Satellites positions and clock offsets derived from precise


Introduction 25
The BeiDou Satellite Navigation System (BDS) is a global satellite navigation system 26 developed by China. Its development contains three stages: Demonstration Navigation Satellite System (BDS-1), Regional Navigation Satellite System (BDS-2), and Global And the reference of the broadcast orbit and clock, as well as the precise clock, is the 86 antenna phase center (APC), while the reference of the precise orbit is the center of 87 mass (CoM) of the satellite. Therefore, for the orbit comparison, the differences 88 between CoM and APC need to be carefully corrected. For the clock comparison, the 89 time group delay (TGD) caused by different signal or signal combinations also needs 90 taking into account. 91

BDS MEO and IGSO satellite coordinates (WGS84) calculation method 93
The BDS satellite ephemeris provides 16 ephemeris parameters, including 1 reference 94 moment, 6 Kepler orbit parameters at corresponding reference moments, and 9 orbital 95 perturbation correction parameters. The ephemeris update period is 1h. The meaning of 96 each parameter is as follows: 97 Calculate the mean motion n 0 of the satellite at the reference time t 0 101 (1-1) 102 In the formula: is the earth's gravitational constant in the BDCS coordinate 103 system, μ = 3.986004418 × 10 14 3 2 ⁄ , √ is the square root of the semi-major 104 axis given in the navigation message. 105 Using the difference ∆ between the satellite's average moving speed given in the 106 navigation message and the calculated value, calculated the corrected mean motion: 107 Calculate the satellite mean anomaly at the moment of observation 109 Where, is the ephemeris reference time given in the navigation message; (1-5) 127 Calculate the argument of latitude Φ : 128 (1-6) 129 In the above formula: ω is the angular distance of perigee given in the navigation 130

message. 131
According to the perturbation parameters (1-8) 138 In the above formula: a is the long radius of the satellite orbit, a = (√ ) 3 ，√ 、 139 0 and IDOT are respectively the square root of the semi-major axis and the orbital 140 inclination at the reference moment given by the broadcast ephemeris parameters and 141 the rate of change of orbital inclination. 142 Calculate the coordinates of the satellite in the Cartesian coordinate system of the 143 orbital plane: 144 In the orbital plane Cartesian coordinate system (the coordinate origin is at the 145 center of the earth), the 0 axis is perpendicular to the orbital plane, the 0 axis points 146 to the ascending node, and 0 is perpendicular to the 0 axis in the orbit plane, 147 forming a right-handed system. The satellite's planar Cartesian coordinates are: 148 Calculate the longitude L of the ascending node at the time of observation: 150 (1-10) 151 In the above formula: ̇ and 0 are the rate of change of ascending node right 152 ascension given by the broadcast ephemeris parameters and the ascending node right 153 ascension calculated according to the reference moment; is the earth rotation rate 154 in the CGCS2000 coordinate system = 7.2921150 × 10 −5 ⁄ . 155 Calculate the coordinates of the satellite in the CGCS2000 coordinate system: 156 First rotate the coordinate system as follows: 157 1)Rotate the angle clockwise around the 0 axis to make the 0 axis change 158 from perigee to ascending node; 159 2)Rotate the 0 axis clockwise by the angle to make the 0 axis coincide with 160 the sky axis; 3)Rotate the angle Ω clockwise around the 0 axis so that the 0 axis coincides with 162 the X axis of the celestial coordinate system, thereby obtaining the coordinates of the 163 satellite in the celestial rectangular coordinate system. 164 Because when using BDS positioning, the position of the observation satellite and the 165 observation station should be in a unified coordinate system, and the coordinates in the 166 celestial coordinate system need to be converted to the earth space rectangular 167 coordinate system. The coordinate between the two points only on the X axis. When 168 the direction is different from the Greenwich star, so only one rotation is needed to find 169 the position of the satellite in the instantaneous earth coordinate system. 170 In summary, after knowing the longitude L of the ascending node and the 171 inclination i of the orbital plane, the position coordinates of the satellite in the ground-172 fixed coordinate system can be easily obtained through two rotations. 173 The coordinates of the MEO/IGSO satellite in the CGCS2000 coordinate system are (1-11) 175 In the formula, L is the ascension of the ascending node in the earth-solid system. 176

BeiDou GEO satellite coordinates (WGS84) calculation method 177
Due to the small inclination of the GEO orbit, the use of GPS ephemeris parameters 178 to fit the GEO satellite orbit may not converge due to the singularity of the matrix. Ascension of the ascending node in the inertial frame is 197 (1-12) 198 3)Solving the Greenwich side angle GAST of the instantaneous epoch will bring a lot 199 of computation to the receiver and bring inconvenience to the design of the receiver. (1-14) 215 In the above formula, L is the right ascension of the ascending node in the inertial 216 coordinate system 217 The coordinates of the GEO satellite in the CGCS2000 coordinate system are 218 (1-15) 219 In the above formula, 220

BDS Satellite Clock Offsets 251
The clock difference between the broadcast ephemeris and precise ephemeris is a 252 systematic deviation. Considering that the indicator SISRE refers to the ranging error 253 from the user receiver to the phase center of the satellite antenna when evaluating the 254 accuracy of the broadcast ephemeris, there should be no systematic deviation. Therefore, 255 we must eliminate the systematic error of satellite clock error. 256 The hardware delay, for example the group delay, is part of the systemic error. The 257 group delay (time group delay, TGD) parameter of the BDS broadcast ephemeris is one 258 of the important components for dual-frequency users to achieve positioning. The BDS 259 broadcast clock offset is based on the B3 frequency. The precision clock difference 260 between BDS-2 and BDS-3 is based on B1/B3 frequency. Therefore, in order to unify 261 the time base, processing needs to be performed when evaluating the satellite broadcast (1-18) 264 In the above formula, 1 and 3 are the frequencies of B1 and B3 frequency 265 points, respectively; 3 is the satellite clock difference calculated from the broadcast 266 ephemeris parameter; 1 is the group delay parameter of the BDS broadcast 267 ephemeris (Jiao et al. 2020). 268 In this paper, a method is also used to calculate the mean value of the difference 269 between the broadcast ephemeris and the precise ephemeris in one day for a single Where , and denote the orbit errors in radial, along-track and cross-track 280 direction, while represents clock errors. 1 and 2 are weight factors for the 281 global SISRE related to a specific constellation. If we neglect clock errors , we obtain 282 the orbit-only SISRE formulation, that is, 283 − = √ 1 2 2 + 2 2 ( 2 + 2 ) (1-20) 284 The detailed value of the weight factors for BDS-2 and BDS-3 satellites have been 285 computed and presented by Montenbruck et al. (2015). 286 The process of broadcasting ephemeris fitting and evaluation (Fig. 1) 287

Fig. 1 Broadcast ephemeris accuracy assessment flowchart 289
Results and discussion 290  Fig. 2, Fig. 3, Fig. 4 and Fig. 5 show the time series of the orbit error and clock error of 294 the three types of BDS satellite broadcasting in 60 days. Fig. 2, Fig. 3 and Fig. 4 are the 295 root mean square (RMS) orbit errors of a single GEO satellite (C04), a single IGSO satellite (C09) and a single MEO satellite (C11) in the R, A and C directions . Fig. 5  297 shows the root mean square of the three-dimensional orbit error, clock error and SISRE 298 time series of a single GEO satellite (C04), a single IGSO satellite (C09) and a single 299 MEO satellite (C11). It can be seen that the accuracy of BDS-2's MEO satellite 300 broadcast ephemeris is better than that of IGSO satellites and GEO satellites, and its 301 three-dimensional orbit error is basically around 2.6m. 302 For BDS-2 GEO satellites, it can be seen from Fig. 2

Comparison of MEO with H clock and MEO with Rb clock in BDS-3 342
It can be seen from Fig. 6, Fig. 7, and Fig. 8 that the MEO satellite with Rb clock 343 and the MEO satellite with H clock in BDS-3 have similar three-dimensional orbit 344 errors, and the former is slightly better than the latter. It can be seen from Figure 8 that 345 the clock error of the MEO satellite with the H clock is not much different from that of 346 the MEO satellite with the Rb clock, and the former is slightly better than the latter, 347 which may be related to the higher stability of the H clock. The MEO satellite clock 348 difference with the Rb clock is about 0.47m, and the MEO satellite clock difference 349 with the H clock is about 0.45m. 350 From Fig. 9, Fig. 10 and Fig. 11, it can be seen that the root mean square error (RMS) 364 of the three-dimensional orbit of the BDS-3 satellite is better than that of the BDS-2 365 satellite, which is about 0.39m. The root mean square (RMS) difference of the clock 366 errors between BDS-3 and BDS-2 satellites is very small. The root mean square (RMS) 367 of SISRE of the BDS-3 satellite is generally better than that of the BDS-2 satellite, 368 which is about 0.39m. 369