# A Femto-Satellite Localization Method Based on TDOA and AOA Using Two CubeSats
## 3 Results  
When we simulate a deployment, we need to obtain the new Two-Line Element set (TLE) of both the mother-CubeSat and the femto-satellite. After searching for the TLE set that better fits the position and the new velocity for each of them, we found we had a distance error below 6 m and a speed error of less than 0.007 m/s between the theoretical one and the one provided by the new TLE set. The simulations are three days after the deployment, for a time frame of 100 min. We advance the simulation three days since the mission operation proposes to deploy the femto-satellite after the beginning of a strong geomagnetic storm to study the magnetic variations at several points. The duration of the effects of the geomagnetic storm lasts a couple of days based on historical data [11]. We simulated a period of 100 min to guarantee the study of a complete orbit, which is around 94.5 min. We also wanted to calculate the performance for the worst-case scenario. There is a reduction in performance as the satellites separate over time. That is why we also evaluate at the end of three days. We searched for the best localization scenario, assuming an ADCS with perfect accuracy. For this purpose, we simulated the deployment in the positive and negative direction of the three axes of the mother-CubeSat's LVLH frame. The \(x\) axis is in the direction of movement, the \(z\) axis points at the nadir, and the \(y\) axis is orthogonal to them. We selected ten orbit positions to deploy the femto-satellite, with a time step of ten minutes since the apogee. Then, we calculated the average of the root CRB for these ten scenarios. For the root CRB, we used the square root of the trace of the MSE matrix [22]. The root CRB indicates the best possible accuracy of the position estimator. Figure 4 shows the root CRB for the six directions. The worst scenario is when the femto-satellite is launched on the \(x\) axis because it aligns with both CubeSats. The AOA method's performance decreases when there is an alignment between the target and the receivers. We obtained the best results when the deployment was made in the \(y\) axis because the geometry is closer to the optimal for the AOA [25]. The positive direction has lower peaks than the negative one, so we chose that one. Now that we have chosen the deployment speed and direction, we need to find the orbit's point to deploy the femto-satellite. Figure 5 shows different deployment points in the same orbit towards the positive \(y\) axis. Highly elliptical orbits, like Molniya, are more affected by this parameter than this simulated orbit. Still, there are some variations between cases. The lowest average root CRB occurs when the deployment is 10 min after the apogee. After selecting the orbit's point for the deployment, we studied the impact of different attitude control and attitude determination systems on the root CRB. Figure 6 shows the root CRB of the localization system as both \(\sigma_{\text{ADS}}\) and \(\sigma_{\text{ACS}}\) increase. The curves are from three days after the deployment to see the effect on the performance. The root CRB is over 100 m for values of \(\sigma_{\text{ADS}}\) greater than 0.1\({}^{\circ}\). The difference between having an attitude control system, such as reaction wheels or magnetorquers, with an accuracy of 0.01\({}^{\circ}\) or 0.06\({}^{\circ}\) is around 20 m in the root CRB value. This information is useful to decide which devices are worth investing in according to the mission.