# BDS Orbit Maneuver Detection Based on Epoch-Updated Orbits Estimated by SRIF
## 2 Mathematical Model
### Orbit Maneuver Detection  
First, there is a concept that needs to be redefined, and the schematic diagram of orbit maneuver is shown in Figure 1.  
The \"start time\" and \"end time\" of satellite maneuvers mentioned in GNSS data processing are not the actual time when the satellite propulsion device activates or stops. When the satellite propulsion device is activated, the observation of the satellite will not be changed significantly and instantaneously. Orbit maneuvers change the satellite track with a certain angle with respect to the original orbit. Then, the s  
Figure 1: The schematic diagram orbit maneuver. \(t_{1}\) is the actual start time of orbit maneuver, which means the time when satellite propulsion device is active; \(t_{2}\) is the start time of when the satellite deviates from the original orbit; \(t_{3}\) is actual end time of orbit maneuver, which means the time when satellite propulsion device stops; \(t_{4}\) is the time when the satellite returns to a stable state after the maneuver. the original orbit quickly. The \"start time\" of the orbit maneuver in GNSS data processing is the time when the satellite deviates from the original orbit after orbit maneuvering. In addition, the actual end time of the maneuver is when the satellite propulsion device no longer applies external forces to the satellite. However, external forces will act on the satellite for a short time, which will take some time for the satellite to return to a stable state. The \"end time\" of the orbit maneuver in GNSS data processing is the time when the satellites return to the stable state. We were unable to obtain the active and stop times of the satellite propulsion device. Instead, we can only compare and analyze the orbit of the maneuvering satellite before and after the orbital maneuver to determine the time when the satellite deviated from its original orbit and when it returned to a stable state. Therefore, the \"start time\" and \"end time\" of the satellite orbit maneuver mentioned in this paper refer to the time when the satellite deviates from the original orbit and the time when the satellite returns to a stable state after the maneuver, respectively. The SRIF method can accurately estimate the correct orbit of a maneuvering satellite during both maneuver and non- maneuver periods. High-precision orbital parameters can be obtained by fitting the precise orbit from the day before the maneuver. The correct orbit prior to the maneuver can then be obtained through orbit integration using the high-precision orbital parameters. Similarly, by using the precise orbit from one day after the orbit maneuver, the correct orbit after the completion of the maneuver can also be obtained. The schematic diagram is shown below (Figure 2). The difference between the orbit estimated by the SRIF method and the forward predicted orbit can be used to detect the start time of the orbit maneuver in real time. Similarly, the end time of the orbit maneuver can be detected by using the difference between the orbit estimated using the SRIF method and the orbit predicted by reverse integration, which is the time when the satellite returns to a stable state. \[\Delta\mathbf{X}_{ACR}=\mathbf{R}_{CRS-SAT}(\mathbf{X}_{SRIF}-\mathbf{X}_{PRE}) \tag{7}\]  
where \(\mathbf{X}_{PRE}\) denotes the position vectors of the satellite, which is predicted by using the orbital parameters estimated by batch least squares; \(\mathbf{X}_{SRIF}\) represents the position estimated using square root information filtering, respectively. \(\mathbf{R}_{CRS-SAT}\) is the transformation matrix from the CRS to the satellite coordinate system, which can be calculated using the position and velocity of the satellite in the CRS. \(\Delta\mathbf{X}_{ACR}\) represents the differences between the satellite positions in the along-track, cross-track, and radial directions. This method cannot achieve the real-time detection of the end time of the orbit maneuver, as it relies on the precise orbit obtained one day after the end of the maneuver.