Impact of ECOM Solar Radiation Pressure Models on Multi-GNSS Ultra-Rapid Orbit Determination

: The Global Navigation Satellite System (GNSS) ultra-rapid precise orbits are crucial for global and wide-area real-time high-precision applications. The solar radiation pressure (SRP) model is an important factor in precise orbit determination. The real-time orbit determination is generally less accurate than the post-processed one and may amplify the instability and mismodeling of SRP models. Also, the impact of di ﬀ erent SRP models on multi-GNSS real-time predicted orbits demands investigations. We analyzed the impact of the ECOM 1 and ECOM 2 models on multi-GNSS ultra-rapid orbit determination in terms of ambiguity resolution performance, real-time predicted orbit overlap precision, and satellite laser ranging (SLR) validation. The multi-GNSS observed orbital arc and predicted orbital arcs of 1, 3, 6, and 24 h are compared. The simulated real-time experiment shows that for GLONASS and Galileo ultra-rapid orbits, compared to ECOM 1, ECOM 2 increased the ambiguity ﬁxing rate to 89.3% and 83.1%, respectively, and improves the predicted orbit accuracy by 9.2% and 27.7%, respectively. For GPS ultra-rapid orbits, ECOM 2 obtains a similar ambiguity ﬁxing rate as ECOM 1 but slightly better orbit overlap precision. For BDS GEO ultra-rapid orbits, ECOM 2 obtains better overlap precision and SLR residuals, while for BDS IGSO and MEO ultra-rapid orbits, ECOM 1 obtains better orbit overlap precision and SLR residuals. multi-GNSS ultra-rapid orbit determination. Then we analyze the impact of di ﬀ erent ECOM models on ambiguity resolution and real-time predicted orbit accuracy in terms of overlap discontinuities and satellite laser ranging. Finally, the comparison and selection of proper ECOM models for di ﬀ erent satellites are discussed.


Introduction
The International GNSS (Global Navigation Satellite System) Service (IGS) officially began providing services in 1994. Since 2005, its name has changed from "International GPS Service" to "International GNSS Service". The GLONASS system is integrated into the IGS daily processing operation to provide precise orbit, atmosphere, earth rotation parameters like those of GPS [1]. In 2011, the IGS launched the Multi-GNSS Experiment and Pilot Project (MGEX) with global stations capable of tracking Galileo and BDS signals [2].
To meet the different needs of timeliness, IGS provides the final, rapid, and ultra-rapid orbit products. The real-time precise orbit is a prerequisite for global and wide-area precise positioning of decimeter to centimeter level. For real-time users, IGS has provided the ultra-rapid (IGU) orbits since 2000 [3,4]. Currently, the IGU orbit updates every 6 h, and covers 24 h of observed orbit arc and 24 h predicted orbit arc. The GPS orbital accuracy of the observed part is about 3 cm, and that of prediction (real-time) is about 5 cm. The IGS real-time service (RTS) orbits have a 3D RMS of 5 cm for GPS and 13 cm for GLONASS compared with the IGS final products [5].

Models and Methods
The SRP models of new systems, such as Galileo and BDS, generally adopt the existing models of GPS and GLONASS, which need further optimization. At present, the ECOM SRP model is widely used by IGS analysis centers. This section first describes the ECOM models series and then presents the experiment design and processing strategy of multi-GNSS ultra-rapid precise orbit determination.

ECOM Models
The total acceleration on a satellite due to solar radiation pressure can be written as where a 0 is the selectable a priori model, and u is the satellite's argument of latitude. The ECOM model decomposes the SRP accelerations into three orthogonal directions. The vector e D is the unit vector in the direction from satellite to the sun, e Y points along the satellite solar panels axis, and e B completes the right-hand orthogonal system [19,20].
In the original ECOM, the functions D(u), Y(u) and B(u) are represented as Fourier series truncated after the once-per-revolution (1pr) terms, using the satellite's argument of latitude u as angular argument [19], as shown in The reduced ECOM model [20], i.e., the ECOM 1 model, is often used in practice, Due to deficiencies of ECOM 1 in modeling SRP of GLONASS satellites, CODE developed the ECOM 2 model [21]. In the ECOM 2 model, the truncated Fourier series adopts the angular argument ∆u = u − u s , where u s is the solar argument of latitude in the satellite orbital plane. Compared to the ECOM 1 model, the ECOM 2 model adds the 2pr and 4pr terms in the e D direction shown in D 2i,c cos 2i∆u + D 2i,s sin 2i∆u The main difference of ECOM 2 from the ECOM 1 model is that it accurately models not only the SRP on solar panels but also the SRP changes on the satellite body. This leads to improved orbit accuracy of satellites with elongated bodies, such as GLONASS, especially in the period when the elevation angle of the sun relative to the orbit plane is low. On the other hand, due to the increased estimation parameters, orbit solutions with ECOM 2 might be less stable and more sensitive to other modeling errors such as satellite attitude, which leads to a significant degradation for some GLONASS satellites [27]. The real-time orbit determination is generally less accurate than the post-processed orbit determination, which may amplify the instability and mismodeling of SRP models. Therefore, the impact of different ECOM models on real-time orbits demands further investigations.
The ECOM models are generally developed for the yaw-steering attitude mode. GPS, GLONASS, and Galileo navigation satellites adopt the yaw-steering attitude mode, while BDS GEO (Geostationary Earth Orbit) satellites adopt the orbit-normal attitude. BDS-2 IGSO (Inclined Geosynchronous Satellite Orbit) and MEO (Medium Earth Orbit) satellites adopt the yaw-steering attitude most of the time, but when the solar elevation angle is less than 4 degrees and the orbital angle approaches 90 degrees, the satellites transition from the yaw-steering to the orbit-normal attitude [28,29]. For orbit-normal attitude satellites, the ECOM vector e D is in the direction from satellite to the sun; the ECOM e Y along the solar panel axis is then perpendicular to the satellite orbital plane, but usually not perpendicular to e D ; and ECOM e B is the cross-product of e D and e Y . Current modified ECOM axis direction for orbit-normal attitude mainly include the modified ECOM 1 model with the virtual Y axis (e Yv ) pointing along the direction of the cross-product of e B and e D [30], and the modified ECOM 1 model with the modified D axis (e D − ) pointing along the direction of the cross-product of e Y and e B [31]. For orbit-normal satellites, the geometric correction, such as the satellite antenna phase center and phase windup, should adopt the actual orbit-normal attitude.

Experiment Methods
The well-distributed global stations from IGS/MGEX and regional stations in China were used in the experiment. The experiment covered DOY 75 to DOY 115 in 2016. The ultra-rapid orbit determination experiment was carried out in simulated real-time mode using the ECOM 1 model [19,20] and ECOM 2 model [21], respectively. The GNSS observation data of all stations were downloaded in real-time and saved as hourly observation files for the experiment period. The ultra-rapid orbit determination adopted the batch processing mode. The accessible hourly observation files were merged into one-day arcs, and the ultra-rapid orbit solution of one-day arc length moved forward with a sliding window of 1 h. The predicted orbit arcs of 1, 3, 6, and 24 h were evaluated by comparison with the corresponding observed orbit arc from the next daily solution, as well as the satellite laser ranging (SLR) observations. Figure 1 shows the distribution of real-time GNSS stations used in this paper.  [31]. For orbit-normal satellites, the geometric correction, such as the satellite antenna phase center and phase windup, should adopt the actual orbit-normal attitude.

Experiment Methods
The well-distributed global stations from IGS/MGEX and regional stations in China were used in the experiment. The experiment covered DOY 75 to DOY 115 in 2016. The ultra-rapid orbit determination experiment was carried out in simulated real-time mode using the ECOM 1 model [19,20] and ECOM 2 model [21], respectively. The GNSS observation data of all stations were downloaded in real-time and saved as hourly observation files for the experiment period. The ultrarapid orbit determination adopted the batch processing mode. The accessible hourly observation files were merged into one-day arcs, and the ultra-rapid orbit solution of one-day arc length moved forward with a sliding window of 1 h. The predicted orbit arcs of 1, 3, 6, and 24 h were evaluated by comparison with the corresponding observed orbit arc from the next daily solution, as well as the satellite laser ranging (SLR) observations. Figure 1 shows the distribution of real-time GNSS stations used in this paper. The Position And Navigation Data Analyst (PANDA) software [32,33] was adapted for this study. The undifferenced ionospheric-free combinations of dual-frequency pseudo-range and carrier phase observations were used to eliminate the first-order ionospheric delays. In the combined multi-GNSS processing, the pseudo-range inter-system biases (ISBs) with respect to GPS were estimated as a daily constant for Galileo and BDS, respectively; the pseudo-range inter-frequency biases (IFBs) with respect to GPS were estimated as a daily constant for each GLONASS satellite. The orbit determination strategy was consistent with the IGS Analysis Center GFZ [34]. Table 1 gives the important processing strategies about the observation model and force model.  The Position And Navigation Data Analyst (PANDA) software [32,33] was adapted for this study. The undifferenced ionospheric-free combinations of dual-frequency pseudo-range and carrier phase observations were used to eliminate the first-order ionospheric delays. In the combined multi-GNSS processing, the pseudo-range inter-system biases (ISBs) with respect to GPS were estimated as a daily constant for Galileo and BDS, respectively; the pseudo-range inter-frequency biases (IFBs) with respect to GPS were estimated as a daily constant for each GLONASS satellite. The orbit determination strategy was consistent with the IGS Analysis Center GFZ [34]. Table 1 gives the important processing strategies about the observation model and force model. GPS and GLONASS yaw model [43,44]; Galileo nominal yaw-steering [45]; BDS GEO nominal orbit-normal, IGSO and MEO orbit-normal/yaw-steering [28,29] The double-differenced ambiguity resolution method for code division multiple access (CDMA) systems, including GPS, Galileo, and BDS, was consistent with Ge et al. [46]. The double-differenced (DD) wide-lane ambiguities derived from the Hatch-Melbourne-Wübbena (HMW) combination [47][48][49] and the narrow-lane ambiguities derived from the adjustment were sequentially fixed to integers. As the real-time carrier phase ambiguity parameters are generally less accurate than the post-processed ones, stringent criteria were adopted in the experiment. Independent baselines were selected with a length shorter than 3000 km. In the double-differenced ambiguity resolution, only the undifferenced ambiguity arcs of more than 40 min and the double-differenced common-view arcs more than 30 min were considered. The double-differenced ambiguity was resolved only when the fractional cycles of wide-lane and narrow-lane ambiguities were both less than 0.15 cycles [50].
The BDS satellite-induced elevation-dependent pseudo-range variations were corrected in raw observables using the third-order polynomial correction model [51]. For the BDS GEO satellites, there are strong correlations among orbital parameters, SRP parameters, satellite clocks, and ambiguity parameters due to the small variation of geometric observation structure. Considering the poor observation geometry of BDS GEO satellites, the ambiguity parameters are difficult to estimate accurately, and the large orbit error, especially in the along-track direction, is absorbed by the ambiguity parameters in adjustment [52]. Therefore, BDS GEO satellites were excluded from the ambiguity resolution. The frequency division multiple access (FDMA) adopted in GLONASS introduces inter-frequency bias (IFB) at the receiver and different wavelengths, which cannot be eliminated by differences between satellites. Considering the linear relation between phase IFB and the frequency number, Liu et al. [53] proposed a double-differenced ionospheric-free ambiguity resolution method for GLONASS satellites with different frequencies by correcting the carrier phase IFB, which can be applied to baselines with lengths up to several thousand kilometers and with different types of receivers in a global network [54,55]. We adopted this method in the experiment to resolve the double-differenced ambiguities for GLONASS.

Results
In this section, we first assess the ambiguity resolution performance for different SRP models in the experiment. Then we evaluate the precision of real-time predicted orbits based on orbit overlap differences with post-processed observation arcs. Finally, we present the accuracy of microwave-based predicted orbits by satellite laser ranging validation.

Ambiguity Resolution Performance
Ambiguity resolution can significantly improve the GNSS orbit determination. To assess the impact of different SRP models, we first analyze the ambiguity fixing rate of the baselines of various lengths for GPS, GLONASS, Galileo, and BDS, respectively. A higher fixing rate indicates a tighter distribution of double-differenced ambiguities around integers and a better orbit solution. Table 2 gives the average number of GPS independent double-differenced ambiguities and the fixing rates for different baseline lengths. It shows that the relation between GPS ambiguity fixing rate and the baseline length was not obvious, which indicates that GPS orbit precision was high enough for ambiguity resolution at long baselines. Overall, about 95% of the independent ambiguities were fixed, which was consistent with previous studies [46]. The fixing rate is almost the same for ECOM 1 and ECOM 2, which indicates that the two SRP models can achieve similar and high GPS orbit precision.  Table 3 gives the number of GLONASS independent double-differenced ambiguities and the fixing rate for different baseline lengths. Considering the relatively lower precision of real-time solution and the short wavelength (about 5.3 cm) of GLONASS ionospheric-free ambiguities, the fixing rate for baselines shorter than 1500 km was analyzed. It shows that the GLONASS fixing rate decreased gradually with the baseline length, which indicates that the geometric errors biased the ambiguity parameters, especially for the long baselines. The longer the baseline is, the better ECOM 2 performs compared to ECOM 1. Overall, the fixing rate is 86.5% and 89.3% for ECOM 1 and ECOM 2, respectively, which implies that ECOM 2 performs better than ECOM 1 for GLONASS ultra-rapid orbits.  Table 4 gives the average number of Galileo independent double-differenced ambiguities and the fixing rate for different baseline lengths. The relation between Galileo fixing rate and the baseline length is obvious. The longer the baseline is, the lower the fixing rate. It indicates that the Galileo orbit accuracy was not high enough, and the geometric errors largely biased the ambiguity parameter. The longer the baseline, the better the ECOM 2 performs compared to ECOM 1. Overall, the fixing rate is 80.0% and 83.1% for ECOM 1 and ECOM 2, respectively, which implies that ECOM 2 performs better than ECOM 1 for Galileo ultra-rapid orbits.  Table 5 gives the average number of BDS independent double-differenced ambiguities and the fixing rate for different baseline lengths. It shows that the BDS fixing rate decreased steeply with the baseline length. It indicates that BDS orbit accuracy was not high enough, and geometric errors hindered the ambiguity resolution, especially for longer baselines. Considering the relatively small number of independent ambiguities, the difference in fixing rate between ECOM 1 and ECOM 2 is not obvious. Overall, the fixing rate is 76.3% and 77.9% for ECOM 1 and ECOM 2, respectively, which is slightly lower than that of Galileo. The ambiguity resolution comparison shows that for GLONASS and Galileo satellites, ECOM 2 achieves higher fixing rates than ECOM 1; and for GPS and BDS IGSO/MEO satellites, ECOM 1 and ECOM 2 achieved similar fixing rates. The different ambiguity resolution performance of ECOM 1 and ECOM 2 is also reflected in the following orbit overlap precision comparison and satellite laser ranging validation.

Orbit Overlap Precision
We first assess the precision of ultra-rapid orbits using the difference of satellite positions in the overlap arcs between two orbit solutions. The current IGS GLONASS final orbits are the combination of mainly float solution orbits from the analysis centers and cannot be used to evaluate the fixed solution orbits accurately. The IGS analysis center CODE enables fixing of all GLONASS ambiguities for baselines shorter than 200 km, but for longer baselines (below 2000 km), only the ambiguities between satellites with the same frequency are fixed [56]. In contrast, the GLONASS ambiguity resolution method in this research can fix the ambiguities with no restrictions on frequency and receiver type for long baselines [53]. For Galileo and BDS, the orbit products of different IGS MGEX analysis centers show some inconsistencies, indicating that it is still difficult to achieve the same accuracy level as GPS and GLONASS. Moreover, for the new GNSS, the lack of enough stations with even global distribution has a negative impact on the IGS MGEX orbit products, especially for BDS. Well-distributed BDS stations in the Asia-Pacific region are still lacking, which can degrade the IGS MGEX orbit products, such as CODE and GFZ products.
To evaluate the orbit precision more objectively, Griffiths and Ray [57] show that the discontinuities in overlap arcs is a better metric. In addition, the overlap arc difference can effectively avoid the biases in inter-AC comparison caused by different strategies and models, such as satellite attitude, antenna phase center, and ambiguity resolution strategies. Thus, the single factor of the solar radiation pressure model on orbits can be analyzed more properly. Therefore, we used this indicator of overlap comparison to evaluate the orbits. Because the overlap difference is calculated by two adjacent orbits, the precision of each orbit is small by about sqrt (2). The predicted orbit arc of the previous daily solution was compared with that of the observed arc of the next daily solution. The predicted orbit arcs of 0 h (00:00 point), 1, 3, 6, and 24 h were analyzed, respectively. The seven-parameter Helmert transformation was used in orbit comparison to remove possible systematic differences between orbits.
The mean RMS of overlap differences in the along-track, cross-track, and radial direction for GPS satellites are shown in Figure 2. ECOM 2 generally obtained higher precision than ECOM 1 for GPS predicted orbits. The GPS predicted orbit precision degraded gradually with the predicted arc length. For the predicted orbit arc of 1 h, ECOM 1 obtained the overlap precision of 3.3 cm, 2.1 cm and 2.6 cm in the along-track, cross-track and radial direction, respectively; ECOM 2 obtained the overlap precision of 3.0, 2.0, and 2.2 cm in the along-track, cross-track, and radial direction, respectively, which was a slight improvement of 9.1%, 4.8%, and 15.4% compared with ECOM 1, respectively. The mean RMS of overlap differences in the along-track, cross-track, and radial direction for GLONASS satellites are shown in Figure 3. ECOM 2 model generally obtained higher 3D precision than the ECOM 1 model for GLONASS predicted orbits. The GLONASS predicted orbit precision degraded gradually with the predicted arc length. For the predicted orbit arc of 1 h, ECOM 1 model obtained the overlap precision of 7.0, 5.0, and 2.7 cm in the along-track, cross-track, and radial direction, respectively; ECOM 2 model obtained the overlap precision of 6.8, 4.3, and 2.5 cm in the along-track, cross-track, and radial direction, respectively, which was a slight improvement of 2.9%, 14.0%, and 7.4% compared with ECOM 1, respectively. The 1-h predicted GLONASS orbits using ECOM 2 obtained the along-track and cross-track precision worse by a factor of about two compared to GPS, and the radial precision close to GPS, which is the main component of the signal-in-space range error. The mean RMS of overlap differences in the along-track, cross-track, and radial direction for GLONASS satellites are shown in Figure 3. ECOM 2 model generally obtained higher 3D precision than the ECOM 1 model for GLONASS predicted orbits. The GLONASS predicted orbit precision degraded gradually with the predicted arc length. For the predicted orbit arc of 1 h, ECOM 1 model obtained the overlap precision of 7.0, 5.0, and 2.7 cm in the along-track, cross-track, and radial direction, respectively; ECOM 2 model obtained the overlap precision of 6.8, 4.3, and 2.5 cm in the along-track, cross-track, and radial direction, respectively, which was a slight improvement of 2.9%, 14.0%, and 7.4% compared with ECOM 1, respectively. The 1-h predicted GLONASS orbits using ECOM 2 obtained the along-track and cross-track precision worse by a factor of about two compared to GPS, and the radial precision close to GPS, which is the main component of the signal-in-space range error.
The mean RMS of overlap differences in the along-track, cross-track, and radial direction for Galileo satellites are shown in Figure 4. The ECOM 2 model generally obtained higher precision than the ECOM 1 model for Galileo predicted orbits. The Galileo predicted orbit precision degraded rapidly with the predicted arc length, especially for the along-track direction, which implies certain mismodeling in observation or force models. For the predicted orbit arc of 1 h, the ECOM 1 model obtained the overlap precision of 11.7, 8.5, and 7.2 cm in the along-track, cross-track, and radial direction, respectively; ECOM 2 model obtained the overlap precision of 8.8, 6.2, and 5.9 cm in the along-track, cross-track, and radial direction, respectively, which was a significant improvement of 24.8%, 27.1%, and 18.1% compared with ECOM 1, respectively. The 1-h predicted Galileo orbits using ECOM 2 were worse by a factor of about three compared to GPS. direction, respectively; ECOM 2 model obtained the overlap precision of 6.8, 4.3, and 2.5 cm in the along-track, cross-track, and radial direction, respectively, which was a slight improvement of 2.9%, 14.0%, and 7.4% compared with ECOM 1, respectively. The 1-h predicted GLONASS orbits using ECOM 2 obtained the along-track and cross-track precision worse by a factor of about two compared to GPS, and the radial precision close to GPS, which is the main component of the signal-in-space range error. The mean RMS of overlap differences in the along-track, cross-track, and radial direction for Galileo satellites are shown in Figure 4. The ECOM 2 model generally obtained higher precision than the ECOM 1 model for Galileo predicted orbits. The Galileo predicted orbit precision degraded rapidly with the predicted arc length, especially for the along-track direction, which implies certain mismodeling in observation or force models. For the predicted orbit arc of 1 h, the ECOM 1 model obtained the overlap precision of 11.7, 8.5, and 7.2 cm in the along-track, cross-track, and radial direction, respectively; ECOM 2 model obtained the overlap precision of 8.8, 6.2, and 5.9 cm in the along-track, cross-track, and radial direction, respectively, which was a significant improvement of 24.8%, 27.1%, and 18.1% compared with ECOM 1, respectively. The 1-h predicted Galileo orbits using ECOM 2 were worse by a factor of about three compared to GPS. The mean RMS of overlap differences for BDS-2 IGSO satellites is shown in Figure 5. The ECOM 1 model generally obtained higher precision than ECOM 2 for BDS-2 IGSO predicted orbits. The BDS-2 IGSO predicted orbit precision degraded rapidly with the predicted arc length, especially for the along-track direction, which also implies certain mismodeling in the observation or force models. For the predicted orbit arc of 1 h, the ECOM 1 model obtained the overlap precision of 28.4 cm and 18.4 cm in the 3D and radial components, respectively; ECOM 2 model obtained the overlap precision of 32.9 cm and 24.7 cm in the 3D and radial components, respectively, which was a significant degradation of 15.8% and 34.2% compared to ECOM 1, respectively. The 1-h predicted BDS-2 IGSO orbits using ECOM 1 were worse by a factor of about seven compared with GPS. The mean RMS of overlap differences for BDS-2 IGSO satellites is shown in Figure 5. The ECOM 1 model generally obtained higher precision than ECOM 2 for BDS-2 IGSO predicted orbits. The BDS-2 IGSO predicted orbit precision degraded rapidly with the predicted arc length, especially for the along-track direction, which also implies certain mismodeling in the observation or force models. For the predicted orbit arc of 1 h, the ECOM 1 model obtained the overlap precision of 28.4 cm and 18.4 cm in the 3D and radial components, respectively; ECOM 2 model obtained the overlap precision of 32.9 cm and 24.7 cm in the 3D and radial components, respectively, which was a significant degradation of 15.8% and 34.2% compared to ECOM 1, respectively. The 1-h predicted BDS-2 IGSO orbits using ECOM 1 were worse by a factor of about seven compared with GPS.
along-track direction, which also implies certain mismodeling in the observation or force models. For the predicted orbit arc of 1 h, the ECOM 1 model obtained the overlap precision of 28.4 cm and 18.4 cm in the 3D and radial components, respectively; ECOM 2 model obtained the overlap precision of 32.9 cm and 24.7 cm in the 3D and radial components, respectively, which was a significant degradation of 15.8% and 34.2% compared to ECOM 1, respectively. The 1-h predicted BDS-2 IGSO orbits using ECOM 1 were worse by a factor of about seven compared with GPS.  The mean RMS of overlap differences for BDS-2 MEO satellites are shown in Figure 6. The ECOM 1 model generally obtained higher precision than the ECOM 2 model for BDS-2 MEO predicted orbits. The BDS-2 MEO predicted orbit precision degraded rapidly with the predicted arc length, especially for the along-track direction, which also implies certain mismodeling in observation or force models. For the predicted orbit arc of 1 h, the ECOM 1 model obtained the overlap precision of 15.6 cm and 5.7 cm in the 3D and radial components, respectively; ECOM 2 model obtained the overlap precision of 16.1 cm and 6.6 cm in the 3D and radial components, respectively, which was a degradation of 3.2% and 15.8% compared with ECOM 1, respectively. The 1-h predicted BDS-2 MEO orbits using ECOM 1 were similar to Galileo, and worse by a factor of about three compared with GPS in the radial direction.
Remote Sens. 2020, 12, x FOR PEER REVIEW 11 of 17 The mean RMS of overlap differences for BDS-2 MEO satellites are shown in Figure 6. The ECOM 1 model generally obtained higher precision than the ECOM 2 model for BDS-2 MEO predicted orbits. The BDS-2 MEO predicted orbit precision degraded rapidly with the predicted arc length, especially for the along-track direction, which also implies certain mismodeling in observation or force models. For the predicted orbit arc of 1 h, the ECOM 1 model obtained the overlap precision of 15.6 cm and 5.7 cm in the 3D and radial components, respectively; ECOM 2 model obtained the overlap precision of 16.1 cm and 6.6 cm in the 3D and radial components, respectively, which was a degradation of 3.2% and 15.8% compared with ECOM 1, respectively. The 1-h predicted BDS-2 MEO orbits using ECOM 1 were similar to Galileo, and worse by a factor of about three compared with GPS in the radial direction. For the BDS GEO satellites in the experiment, the ECOM 2 model generally obtained better overlap precision than the ECOM 1 model in the 3D and radial components. For the predicted orbit arc of 24 h, the ECOM 1 and ECOM 2 models obtained the radial overlap precision of 282.4 cm and 220.6 cm, respectively, and ECOM 2 showed an improvement of 21.9% compared with ECOM 1. For BDS GEO satellites, the SRP model should accurately describe the SRP force while reducing the correlation between parameters, which needs further SLR validation. For the BDS GEO satellites in the experiment, the ECOM 2 model generally obtained better overlap precision than the ECOM 1 model in the 3D and radial components. For the predicted orbit arc of 24 h, the ECOM 1 and ECOM 2 models obtained the radial overlap precision of 282.4 cm and 220.6 cm, respectively, and ECOM 2 showed an improvement of 21.9% compared with ECOM 1. For BDS GEO satellites, the SRP model should accurately describe the SRP force while reducing the correlation between parameters, which needs further SLR validation.
The multi-GNSS ultra-rapid orbits in the experiment obtained comparable precision with the current predicted orbit products [5][6][7][8][9]. The orbit overlap comparison shows that for GPS, GLONASS, Galileo, and BDS GEO satellites, the ECOM 2 model generally achieved better orbital prediction than the ECOM-1 model, while for BDS IGSO and MEO satellites, the ECOM 1 model generally achieved better orbital prediction than the ECOM 2 model.

Satellite Laser Ranging Validation
Satellite laser ranging (SLR) observables are usually used as external validation to evaluate GNSS satellite orbit quality. The SLR residuals, i.e., the differences between the SLR observables and the range calculated from microwave-based satellite positions, including the observed and predicted orbit arc, mainly show the GNSS orbit accuracy in the radial direction. In the experiment, the GLONASS, Galileo, and BDS GEO C01, IGSO C08, and C10, MEO C11 satellites equipped with laser retroreflector arrays were observed by the SLR stations from the International Laser Ranging Service (ILRS) [58]. The SLR station coordinates were fixed to the a priori reference frame, and the station displacements were corrected consistently with the microwave-based solutions. The tropospheric delays, relativistic effects, and the Laser Retro-Reflector Arrays offsets with respect to the satellites' center of mass were corrected in the SLR observables. Outliers exceeding 0.6 m were excluded for GLONASS, Galileo, and BDS MEO satellites, and those exceeding 3.0 m and 10.0 m were excluded for BDS IGSO and GEO satellites, respectively. This outlier check excluded approximately 2.5%, 3.7%, 9.3%, and 1.2% of the data points for GLONASS, Galileo, BDS MEO, and BDS IGSO satellites, respectively, and no data points were excluded for BDS GEO satellites. After the removal of outliers, the number of normal points for observed orbit arcs of 24 h (predicted 0 h) and various predicted orbit arcs is shown in Table 6. The RMS of the SLR residuals for the observed and predicted orbits are given in Figure 7. The predicted orbit precision of GLONASS, Galileo, and BDS MEO satellites generally degraded gradually with the predicted arc length. The RMS of the SLR residuals was slightly larger than the RMS of the orbit overlap differences in the radial direction in the previous section. Considering SLR as the external validation from a different technique, the SLR residuals generally agreed with the orbit overlap precision.
For the GLONASS predicted orbit arc of 1 h, the SLR residuals RMS was 6.7 cm for the ECOM 1 model and 6.4 cm for the ECOM 2 model, which corresponds to an improvement of 5.2% of ECOM 2 compared to ECOM 1. For the GLONASS predicted orbit arc of 24 h, the SLR residuals RMS was 9.8 cm for ECOM 1 and 8.9 cm for ECOM 2, which corresponds to an improvement of 9.2%. For the Galileo predicted orbit arc of 1 h, the SLR residuals RMS was 11.1 cm for the ECOM 1 model and 7.0 cm for ECOM 2, which corresponds to an improvement of 36.9% of ECOM 2 compared to ECOM 1. For the Galileo predicted orbit arc of 24 h, the SLR residuals RMS was 15.9 cm for ECOM 1 and 11.5 cm for ECOM 2, which corresponds to an improvement of 27.7%. For the BDS MEO predicted orbit arc of 24 h, the SLR residuals RMS was 12.8 cm for ECOM 1 and 13.2 cm for ECOM 2, which corresponds to a degradation of 3.1% of ECOM 2 compared to ECOM 1. The RMS of the SLR residuals for the observed and predicted orbits are given in Figure 7. The predicted orbit precision of GLONASS, Galileo, and BDS MEO satellites generally degraded gradually with the predicted arc length. The RMS of the SLR residuals was slightly larger than the RMS of the orbit overlap differences in the radial direction in the previous section. Considering SLR as the external validation from a different technique, the SLR residuals generally agreed with the orbit overlap precision. For the GLONASS predicted orbit arc of 1 h, the SLR residuals RMS was 6.7 cm for the ECOM 1 model and 6.4 cm for the ECOM 2 model, which corresponds to an improvement of 5.2% of ECOM 2 compared to ECOM 1. For the GLONASS predicted orbit arc of 24 h, the SLR residuals RMS was 9.8 cm for ECOM 1 and 8.9 cm for ECOM 2, which corresponds to an improvement of 9.2%. For the For the BDS IGSO satellites, the ECOM 1 model obtained the SLR residuals RMS of 6.8 cm and 74.2 cm for the observed and predicted orbit arc of 24 h, respectively, whereas ECOM 2 model obtained the SLR residuals RMS of 7.7 cm and 99.7 cm, which was a degradation of 13.2% and 34.4% compared to ECOM 1. For the BDS GEO satellites, the respective values for the ECOM 1 model are 27.5 cm and 508.5 cm, and for the ECOM 2 model, they are 13.2 cm and 166.4 cm, which was an improvement of 52.0% and 67.3% compared to ECOM 1.
The satellite laser ranging comparison shows that for GLONASS, Galileo, and BDS GEO satellites, the ECOM 2 model generally achieved better predicted orbits than the ECOM 1 model while for BDS IGSO and MEO satellites the ECOM 1 model generally achieved better predicted orbits than the ECOM 2 model. The satellite laser ranging results generally agreed well with the orbit overlap precision comparison, which confirms the findings with the external validation.

Discussion
In the experiment, for GLONASS and Galileo ultra-rapid orbit determination, the ECOM 2 model performed better than the ECOM 1 model in ambiguity resolution, orbit overlap precision, and satellite laser ranging validation, which demonstrates the improved orbits of satellites with markedly elongated bodies [21]. For GLONASS, the ECOM 2 model achieved the fixing rate of 89.3% in ultra-rapid orbit determination, and for the GLONASS predicted orbit arc of 1 h, the overlap precision reached 6.8, 4.3, and 2.5 cm in the along-track, cross-track, and radial direction, respectively, and the SLR residuals RMS reached 6.4 cm. For Galileo, the ECOM 2 model achieved the fixing rate of 83.1% in ultra-rapid orbit determination, and for the Galileo predicted orbit arc of 1 h, the overlap precision reached 8.8, 6.2, and 5.9 cm in the along-track, cross-track, and radial direction, respectively, and the SLR residuals RMS reached 7.0 cm.
For GPS satellites, ECOM 2 obtained a similar fixing rate as ECOM 1 and a slightly better overlap precision in ultra-rapid orbit determination, which may suggest that for GPS satellites with a slightly elongated body, both orbits are accurate enough for ambiguity resolution [27]. For the GPS predicted orbit arc of 1 h, ECOM 2 reached the overlap precision of 3.0, 2.0, and 2.2 cm in the along-track, cross-track, and radial direction, respectively. For BDS IGSO and MEO ultra-rapid orbits, ECOM 1 obtained a similar fixing rate as ECOM 2, and improved orbit overlap precision and SLR residuals, which suggests that ECOM 1 better fits BDS IGSO and MEO satellites with bodies of a nearly cube shape. BDS MEO ultra-rapid orbits obtained similar overlap precision and SLR residuals as Galileo, while BDS IGSO ultra-rapid orbits were worse. For BDS GEO ultra-rapid orbits, ECOM 2 obtained better overlap precision and SLR residuals than ECOM 1, which may be due to the elongated satellite bodies including the communication antenna. ECOM 2 achieved BDS GEO orbit accuracy of 13.2 cm and 166.4 cm for the observed and predicted orbit arc of 24 h, respectively. These ultra-rapid orbits can be applied in global and wide-area precise positioning of the decimeter to centimeter level [6][7][8][9].

Conclusions
Multi-GNSS ultra-rapid orbits are crucial for real-time high-precision GNSS applications, and the solar radiation pressure model is one of the main factors influencing predicted orbit accuracy. We focused on analyzing the impact of ECOM solar radiation pressure models on multi-GNSS ultra-rapid orbit determination. The different performances of the ECOM 1 and ECOM 2 models were evaluated in terms of ambiguity resolution, orbit overlap precision, and satellite laser ranging validation in the experiment.
For GLONASS and Galileo ultra-rapid orbits, the ECOM 2 model performed better than the ECOM 1 model in ambiguity resolution, orbit overlap precision, and satellite laser ranging validation. ECOM 2 increased the ambiguity fixing rate to 89.3% and 83.1%, respectively, and improved the predicted orbit accuracy by 9.2% and 27.7%, respectively. For GPS ultra-rapid orbits, ECOM 2 obtained a similar ambiguity fixing rate as ECOM 1 and a slightly better orbit overlap precision. For BDS GEO ultra-rapid orbits, ECOM 2 obtained better overlap precision and SLR residuals, while for BDS IGSO and MEO ultra-rapid orbits, ECOM 1 obtained better orbit overlap precision and SLR residuals.
We focused on analyzing the impact of the ECOM 1 and ECOM 2 solar radiation pressure models on multi-GNSS ultra-rapid orbit determination. The multi-GNSS ultra-rapid orbits obtained high accuracy comparable to that of the current predicted products. The performances of other high-precision solar radiation pressure models on multi-GNSS real-time orbits should be investigated in further studies.