# A Sensitivity Study of POD Using Dual-Frequency GPS for CubeSats Data Limitation and Resources
## 2 Processing Strategy  
In this study, the final LEO orbits (where CubeSats are classified as LEO) are postprocessed following the reduced-dynamic LEO processing scheme of the Bernese GNSS Software V5.2 [34]. After correcting the antenna PCOs, PCVs, and the offsets between the antenna reference point (ARP) and the satellite center-of-mass (CoM), the determined final orbit in this study is referred to the satellite CoM. For the sensitivity analysis purpose, the final reduced-dynamic orbits are produced assuming different observation sampling rates, duty cycling, varying mean number of satellites by changing the elevation mask as will be explained in the next section, arc lengths, using different types of the IGS products, with and without the antenna attitude information. The processing procedure can generally be summarized in the following steps, which is also given in Figure 2 as a flow diagram:  
1. Compute kinematic orbits using single point positioning (SPP) employing the IF combination of the code observations. These kinematic orbits, denoted as vector \(\hat{r}_{K}\), are discrete and have an accuracy of meters. 2. Computation of the code-based reduced-dynamic orbits. The reduced-dynamic orbits are computed with accelerations based on a series of gravitational and nongravitational terms, such as the Earth gravitational terms, the Earth tidal terms, the gravitational attraction from the sun, moon, and other planets, as well as the general relativistic term. Note that mis-modeled effects like the solar radiation pressure and the air drag will be largely absorbed by the estimated dynamic parameters and the stochastic velocity changes or accelerations set up later in the processing [35]. Details of the processing and the dynamic models are given in Table 1. Making use of the kinematic code orbits from the first step, the six Keplerian elements at the initial condition (the semi-major axis of the orbit, the orbital eccentricity, the inclination of the orbital plane, the right ascension of the ascending node, the argument of perigee, and the argument of latitude at the initial condition), and a remaining part of the dynamic models are estimated with a batch least-squares adjustment, which includes at this step nine flight-oriented dynamic parameters. These estimable dynamic parameters contain three constant terms (\(a_{R0}\), \(a_{S0}\), and \(a_{W0}\)) and six periodic terms (\(a_{RC}\), \(a_{SC}\), \(a_{WC}\), \(a_{RS}\), \(a_{SS}\), and \(a_{WS}\)) in the radial (R), along-track (S) and cross-track (W) directions. The total acceleration \(a\) can then be distributed into the term \(a_{0}\), which is assumed known by applying the models given in Table 1, and an additional dynamic term \(a_{dyn}\) that is to be adjusted: \[a=a_{0}+a_{dyn},\] (1) With \[a_{dyn}=a_{R}e_{R}+a_{S}e_{S}+a_{W}e_{W},\] (2) \[a_{R}=a_{R0}+a_{RC}\cos(U)+a_{RS}\sin(U),\] (3) \[a_{S}=a_{S0}+a_{SC}\cos(U)+a_{SS}\sin(U),\] (4)\[a_{W}=a_{W0}+a_{WC}\cos(U)+a_{WS}\sin(U),\] (5) where \(e_{R},e_{S}\), and \(e_{W}\) represent the unit vectors in the radial, along-track, and cross-track directions, respectively. \(U\) denotes the satellite argument of latitude. The code-based kinematic orbits \(\hat{r}_{K}\) obtained from the first step are used as observations to adjust the 15 parameters mentioned above in a least-squares sense. The linearized observation equation at the epoch \(t_{i}\) can be formulated as: \[\mathrm{E}(\hat{r}_{K}-\hat{r}_{0})=\left[A_{rk},\ A_{rd}\right]\left[x_{k},\ x_{d}\right]^{T},\] (6) where \(\hat{r}_{0}\) is the a priori orbit vector obtained based on numerical integration on hand the \(a_{0}\), and the \(\hat{a}_{dyn}\) and the Keplerian elements estimated from the last iteration. The vectors \(x_{k}\) and \(x_{d}\) contain the increments of the six Keplerian elements and the nine dynamic parameters, respectively, and the design matrices \(A_{rk}\) and \(A_{rd}\) contain the partial derivatives of the position vectors with respect to \(x_{k}\) and \(x_{d}\).