# A Femto-Satellite Localization Method Based on TDOA and AOA Using Two CubeSats
## 2 Materials and Methods  
The Two-Line Element set (TLE) consists of two 69-character lines of data provided by NORAD, that describe the orbit of a satellite. It includes orbital elements such as the inclination, the Right Ascension of the Ascending Node (RAAN), the eccentricity, the argument of the perigee, the mean anomaly, and the mean motion, which are necessary to determine a satellite's position and velocity at a given time. For the simulations, we used as an example the TLE data of a 3U CubeSat from Planet Labs called _FLOCK 4P-1_ (Table 1) that is in LEO orbit. This CubeSat has an inclination of 97.4788\({}^{\circ}\), a RAAN of 21.6285\({}^{\circ}\), an eccentricity of 0.0013387, an argument of the perigee of 80.2501\({}^{\circ}\), a mean anomaly of 280.0246\({}^{\circ}\), and a mean motion of 15.2037479 revolutions per day. We assumed that there are two CubeSats in the same orbit, one after the other. This is common for small CubeSat constellations since they are deployed from the same rocket. For this reason, we generated two CubeSats using the same TLE at two different time epochs, with a time-lapse of four seconds (around 30 km of separation). The CubeSat-1, or mother-CubeSat, is the one that deploys the femto-satellite and is behind the second CubeSat. The deployment considers a femto-satellite of 80 g and two CubeSats of 3.2 kg each. The simulation was made for an orbit of 17 November 2020, before the satellites arrive at the South Atlantic Anomaly. We propagated all the orbits using the SGP4 model [23]. We use the Earth Centered Inertial (ECI) coordinate frame for the calculations of the satellite's positions. The femto-satellite's unknown position is represented by the vector \(\textbf{u}=[u_{x},u_{y},u_{z}]^{T}\in\mathbb{R}^{3}\) while the CubeSats are at known positions \(\textbf{s}_{m}=\left[s_{x,m},s_{y,m},s_{z,m}\right]^{T}\in\mathbb{R}^{3}\) \begin{table}
\begin{tabular}{r r}
1 & 44814U 19081L 20321.73053029.0001305 00000-0 63025-4 0 9996 \\
2 & 44814 97.4788 21.6285 0013387 80.2501 280.0246 15.20374749 54001 \\ \end{tabular}
\end{table} 
Table 1: This is the Two-Line Element set (TLE) of the _FLOCK 4P-1_ that we used to simulate two 3U CubeSats. To separate the CubeSats, we used an epoch difference of four seconds, equivalent to around 30 km of distance. \(m=1,2\). The measurement of the CubeSat's position is affected by the accuracy of their GNSS device, as seen in  
\[\mathbf{\hat{s}}_{m}=\mathbf{s}_{m}+\boldsymbol{\varepsilon}_{m,\text{GNSS}},\ m=1,2, \tag{1}\]  
where \(\boldsymbol{\varepsilon}_{m,\text{GNSS}}\) is a zero-mean Gaussian noise to model the accuracy of the GNSS device. The accuracy of each axis is equal to the overall accuracy divided by \(\sqrt{3}\). We use the same measurement model as in [22] with one TDOA and two AOA pairs \((\theta_{m},\phi_{m})\), but the elevation is \(\phi_{m}\in(-\pi/2,\pi/2)\) because we are not limited to the ground. These angles correspond to each CubeSat's Local Vertical/Local Horizontal (LVLH) frame, which is depicted in Figure 3.  
There are five design parameters for this localization system: the deployment direction and speed, the point in the orbit where the deployment takes place, the accuracy of the attitude determination and control systems (ADCS), and the GNSS devices of the CubeSats. For this research, we will assume a speed of deployment of 1 m/s to focus on the rest of the design parameters. We use the LVLH frame of the CubeSat with the Femto-satellite Orbital Deployer (FOD) to conduct the deployment simulations. Figure 3 illustrates the reference frame attached to the CubeSat. Different deployment directions and orbit's point of deployment change the localization geometry. The geometry is directly related to the performance of the localization system. Due to the limitations of the ADCS, the femto-satellite's deployment will not be in the desired direction with exactitude. The model of the deployment direction is  
\[ \begin{bmatrix}\hat{\theta}_{m,\text{dep}}\\ \hat{\phi}_{m,\text{dep}}\end{bmatrix} =