## 2 Autonomous Orbit Determination Model  
### Observation Model for Autonomous Orbit Determination  
Based on the concurrent spatial time division duplexing technology of the phased array antenna, bidirectional one-way distance measurement between the BDS-3 satellites was completed within 3.0 s [30]. At different times within these 3.0 s, the observation equations of the two satellites \(SAT_{A}\) and \(SAT_{B}\) can be expressed as.  
\[\rho_{AB}=|R_{B}(t_{1})-R_{A}(t_{1}-\Delta t_{1})|-\delta t_{A}+\delta t_{B}+c \cdot\tau_{A}^{send}+c\cdot\tau_{B}^{rec}+\Delta\rho_{AB}+\varepsilon_{AB} \tag{1}\]  
\[\rho_{BA}=|R_{A}(t_{2})-R_{B}(t_{2}-\Delta t_{2})|+\delta t_{A}-\delta t_{B}+c \cdot\tau_{B}^{send}+c\cdot\tau_{A}^{rec}+\Delta\rho_{BA}+\varepsilon_{BA} \tag{2}\]  
where \(R_{A}\) and \(R_{B}\) are the positions of the two satellites \(SAT_{A}\) and \(SAT_{B}\), respectively; \(c\) is the speed of light; \(\delta t_{A}\) and \(\delta t_{B}\) are the clock offsets of the two satellites \(SAT_{A}\) and \(SAT_{B}\), respectively; \(\Delta t_{1}\) and \(\Delta t_{2}\) are the propagation times of \(\rho_{AB}\) and \(\rho_{BA}\), respectively; \(\tau^{send}\) and \(\tau^{rec}\) are the emission delay and reception delay of the satellite, respectively; \(\Delta\rho_{AB}\) and \(\Delta\rho_{BA}\) are the error terms that can be modeled, including the relativistic effect and so on; and \(\varepsilon_{AB}\) and \(\varepsilon_{BA}\) are the noises of the respective observed values.  
In addition, it is necessary to further reduce the inter-satellite bidirectional observations at different times to the same time to participate in the satellite orbit determination. First, utilizing the predicted orbit and the predicted satellite clock offset parameters, the observed values measured at \(t_{1}\) and \(t_{2}\) are reduced to the nearest full 3 s at time \(t_{0}\). Considering that the bidirectional observed values are completed in a very short time, the prediction errors for the orbit and the clock offset can be ignored.  
\[\begin{array}{l}\rho_{AB}(t_{0})=\rho_{AB}+|R_{B}(t_{0})-R_{A}(t_{0})|-|R_{B}(t _{1})-R_{A}(t_{1}-\Delta t_{1})|+\\ c\cdot(\delta t_{B}(t_{0})-\delta t_{A}(t_{0}))-c\cdot(\delta t_{A}(t_{1})- \delta t_{B}(t_{1}-\Delta t_{1}))\\ \\ \rho_{BA}(t_{0})=\rho_{BA}+|R_{A}(t_{0})-R_{B}(t_{0})|-|R_{A}(t_{2})-R_{B}(t_{2} -\Delta t_{2})|+\\ c\cdot(\delta t_{A}(t_{0})-\delta t_{B}(t_{0}))-c\cdot(\delta t_{A}(t_{2})- \delta t_{B}(t_{2}-\Delta t_{2}))\\ \end{array} \tag{3}\]  
Then, adding the bidirectional pseudorange at time \(t_{0}\), the observation equation that only contains the satellite orbit parameters can be obtained by sorting  
\[\frac{\rho_{AB}(t_{0})+\rho_{BA}(t_{0})}{2}=|R_{B}(t_{0})-R_{A}(t_{0})|+\frac{ c\cdot\tau_{A}^{+}}{2}+\frac{c\cdot\tau_{B}^{+}}{2}+\varepsilon^{+} \tag{5}\]  
The above equation is the orbit determination observation equation in autonomous orbit determination, where \(\tau^{+}\) is the sum of the emission delay and reception delay of the satellite, which is referred to as the time delay sum parameter in autonomous orbit determination.  
In this contribution, we adopt an extended Kalman filter (EKF) to estimate the satellite's orbit parameters to ensure the real-time performance, in each of which we simultaneously estimate all satellite orbit parameters in a distribute processing mode.