## 4 Experiments and Analysis
### Impact on the Constellation Rotation and Translations  
It is well-known that, when using ISL observations for autonomous orbit determination, there are overall constellation rotation and translation. Existing research has shown that the overall rotation and translation of the constellation can be controlled and improved \begin{table}
\begin{tabular}{c c c} \hline \hline
**Item** & **MEO** & **IGSO** \\ \hline Exp1-Exp2 & 75.12\% & 72.33\% \\ Exp3-Exp4 & 78.57\% & 73.36\% \\ Exp1-Exp3 & 14.91\% & 13.15\% \\ Exp2-Exp4 & 26.70\% & 16.39\% \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Improvement proportion of the 3D orbital accuracy. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Item**} & \multicolumn{3}{c}{**MEO**} & \multicolumn{3}{c}{**IGSO**} \\ \cline{2-7}  & **A** & **C** & **R** & **A** & **C** & **R** \\ \hline Exp1 & 71.83 & 83.07 & 8.94 & 111.01 & 123.12 & 28.40 \\ Exp2 & 16.09 & 20.93 & 9.07 & 24.32 & 28.13 & 28.50 \\ Exp3 & 64.74 & 67.57 & 5.25 & 107.09 & 95.55 & 22.88 \\ Exp4 & 11.96 & 15.80 & 5.20 & 20.73 & 22.43 & 24.06 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Orbital accuracy for two types of satellites (cm). by using ground anchor stations and rotation parameters [43; 44; 45]. Usually, Helmert transformation parameters can be employed to explain any systematic discrepancies between two sets of positions in separate terrestrial reference frames. In other words, by performing a Helmert transformation on two sets of ephemeris products, seven parameters can be estimated, including three translations (TX, TY, TZ), three rotations (RX, RY, RZ), and one scale parameter (Sc). These seven parameters effectively reveal the overall translation and rotation between two ephemeris products. In this section, we focus on analyzing the impact of a priori values on the constellation rotation and translation in combined ISL/SLR orbit determination. Since the impact of adding SLR on the translation and rotation in ISL-only-based orbit determination has already been well-established [44; 45], in this section, we estimated the seven parameters for Experiments 3 and 4, to examine the impact of the a priori orbit on the constellation's translation and rotation during the combined ISL/SLR orbit determination. Taking the DOY 053 as an example, we analyzed the magnitude of these seven parameters, along with an evaluation of the orbital accuracy before and after the Helmert transformation. The two sets of the seven parameters achieved are listed in Table 7. The precision ephemeris and the broadcast ephemeris have been abbreviated to prec and brdc, respectively. Upon comparison, it becomes evident that the magnitudes of the estimated seven parameters are larger when the broadcast ephemeris is used as the a priori orbit, compared to the ones where the precise ephemeris is employed. Moreover, the discrepancies between the two experiments indicate that, for the translation parameters, the difference in the TX direction is most pronounced, at about 20 mm, followed by approximately 10 mm in the TY direction, while the smallest difference occurs in the TZ direction, not exceeding 5 mm. For the rotation parameters, the differences between the two experiments are considerable in all three directions. In detail, there is a nearly ninefold difference in the RX direction, with values of 0.31 mas and 2.74 mas, respectively, and about four times in both the RY and RZ directions. These results may suggest that there are still translation and rotations for the BDS-3 in the combined ISL/SLR orbit determination with the broadcast ephemeris as the a priori orbit. Then, we conducted a detailed analysis of the orbital accuracy before and after the Helmert transformation. Figure 10 illustrates the results for DOY 053. In Figure 10, the left and right columns represent the outcomes using the precise ephemeris and broadcast ephemeris, respectively. The blue bars represent the orbital accuracy before Helmert transformation, while the red bars represent the accuracy after Helmert transformation.