# A Sensitivity Study of POD Using Dual-Frequency GPS for CubeSats Data Limitation and Resources
## 3 Orbit Determination under Different Scenarios
### Antenna Attitude  
As shown in the example in Figure 1, it is suspected that the poor data tracking condition is caused by the lack of antenna stabilization. Good antenna stabilization is essential to keep the antenna in a favorable orientation for GNSS tracking, i.e., approximately toward the radial direction. High-accuracy attitude control is expected to be realized also for CubeSats, as found to be feasible in the tests simulating different CubeSat scenarios in [50] and achieved for the CubeSat CanX-2 [20]. Under the condition of appropriate antenna stabilization, the antenna attitude information is often measured by extra sensors like star cameras or IMU on LEO/CubeSats satellites. For CubeSats, however, the limited size and power may limit its capacity to properly measure, store, or transfer the attitude information back to the GPC. As mentioned in Section 1, the data from the IMU onboard the CubeSat STF-1, e.g., was not collected during the GNSS data tracking because of the power allocation, where the IMU used was of a special type tested for a high-accuracy experiment that was carried out at a different time. The attitude information of the satellite (and antenna) is sometimes even ignored in CubeSat simulation studies [21]. Under such circumstances, where no attitude information is available, the antenna attitude could be estimated based on the satellite positions and velocities. Taking the GRACE FO-1 as an example, the antenna was fixed on the satellite and its up direction generally points toward the radial direction from the earth to the LEO satellite. Note that the X-axis in the satellite reference frame (SRF) is here approximately in the anti-flight direction, as GRACE FO-1 is the leading satellite in the test period. Figure 10 depicts the RSW system (green) in the radial, along-track and cross-track directions with the origin in the satellite centre-of-mass, and the antenna reference frame (ARF, red), e.g., for GRACE FO-1, in the north, east, and up (NEU) directions with the origin in the antenna reference point (ARP). To evaluate the differences between the expected and the actual orientations, the angle differences are computed with the help of the east (\(e_{E}\)) and the opposite along-track (\(e_{S}\)) unit vectors, the north (\(e_{N}\)) and the cross-track (\(e_{W}\)) unit vectors, and the up (\(e_{U}\)) and the radial (\(e_{R}\)) unit vectors in the inertial system:  
\[\alpha_{E}=\arccos\bigl{(}-e_{E}^{T}e_{S}\bigr{)},\ \alpha_{N}=\arccos\bigl{(}e_{ N}^{T}e_{W}\bigr{)},\ \alpha_{U}=\arccos\bigl{(}e_{U}^{T}e_{R}\bigr{)}. \tag{11}\]  
Figure 9: 3D RMSE of the reduced-dynamic orbits under different duty-cycles, observation sampling intervals and mean satellite numbers using (**a**) the IGR and (**b**) the IGC products. The cases having 9, 7, and 6 satellites (on average) are given between each pair of the dashed lines. In Equation (11), the unit vectors \(e_{R}\) and \(e_{S}\) are calculated based on the satellite position and velocity vectors, and \(e_{W}\) completes the left-hand system. As the antenna is fixed on the satellite, the antenna NEU unit vectors \(e_{N}\), \(e_{E}\), and \(e_{U}\) are calculated based on the satellite orientations, which are tracked by extra sensors and documented in an attitude file with quaternions. The rotation matrix \(R_{SRF}\) transforming from the SRF to the inertial system can be computed with these quaternions [51]. Based on the definition of the SRF as illustrated in Figure 10 in yellow, the \(e_{N}\), \(e_{E}\), and \(e_{U}\) can be obtained with:  
\[e_{E}=R_{SRF}[1,0,0]^{T},\ e_{N}=R_{SRF}[0,-1,0]^{T},\ e_{U}=R_{SRF}[0,0,-1]^{T} \tag{12}\]  
The angle differences \(\alpha_{E}\), \(\alpha_{N}\), and \(\alpha_{U}\) are given in Figure 11. It can be observed that these angle differences are not large. In case that the antenna attitude file is not given, or the attitude information is not accurately documented, one could approximate the antenna orientation using the RSW directions.