# A Femto-Satellite Localization Method Based on TDOA and AOA Using Two CubeSats
## 2 Materials and Methods
\begin{bmatrix}\theta_{m,\text{dep}}\\ \phi_{m,\text{dep}}\end{bmatrix} + \begin{bmatrix}\varepsilon_{\phi}\\ \varepsilon_{\phi}\end{bmatrix} ,\ m=1,2, \tag{2}\]  
where \(\theta_{m,\text{dep}}\) and \(\phi_{m,\text{dep}}\) represent the yaw (azimuth) and pitch (elevation) of the deployment from the CubeSat's LVLH frame, and \(\left[\varepsilon_{\theta},\varepsilon_{\phi}\right]^{T}\) are zero-mean Gaussian noises for the yaw and pitch. This is done to represent the accuracy of the attitude determination system (ADS) and the attitude control system (ACS), with a variance of \(\sigma^{2}_{\text{ADS}}+\sigma^{2}_{\text{ACS}}\). These values move the direction of deployment from the one intended, changing the localization geometry. Figure 3: The Earth-Centered Inertial (ECI) coordinate frame, in black, is used to calculate all the satellites’ positions, while the CubeSat’s Local Vertical/Local Horizontal (LVLH) frame, in red, is used for the deployment simulation. The \(x\) axis points towards the direction of movement, \(z\) to the nadir, and \(y\) is orthogonal to them. With the speed and direction of deployment, we calculate the new velocities for both the femto-satellite and the CubeSat that deploys it. We do this using the law of conservation of momentum. After obtaining the satellites' position and velocity, we search for the TLE set that best describes these two vectors, starting from the original TLE file of the mother-CubeSat. For this purpose, we need to find the new inclination, RAAN, eccentricity, the argument of the perigee, mean anomaly, and mean motion. Since having six nested _for_ loops takes too much time, we decided to select different groups of two to three orbital elements and search for the best fit iteratively. The ballistic coefficient, \(B^{*}\) and the first and second derivatives of the mean motion, \(\dot{n}\) and \(\ddot{n}\), respectively, are set equal to the one provided by the TLE file. This approach does not impact the prediction accuracy for short-term orbit propagations as seen in [24]. The methodology for finding the best localization scenario is as follows. First, we simulate the deployment of the femto-satellite into different directions at 1 m/s. We compare them using the root Cramer-Rao Bound (CRB), which is statistically the lower limit of the error in the position estimation. We select the direction that yields the best results, and then we evaluate if there is any difference in which point of the mother-CubeSat's orbit we deploy the femto-satellite. After selecting these values, we study how both the attitude determination and the attitude control systems' accuracy affect the position estimation's performance. We choose the accuracy of these systems based on the results and on the technology that is currently available. Similarly, we simulate GNSS devices with different accuracies for the CubeSats positions. With the results of the simulations, we choose the accuracy of a GNSS receiver based on what is on the market for space applications. Finally, we evaluate this localization system using the root CRB, the Root Mean Square Error (RMSE), and the estimation bias.