# A Femto-Satellite Localization Method Based on TDOA and AOA Using Two CubeSats
## 3 Results
For the next figures, we display the root-mean-square error and the estimation bias given by Equations (3) and (4)  
\[\text{RMSE}(\mathbf{u})=\sqrt{\frac{1}{L_{0}}\sum_{l_{0}=1}^{L_{0}}\left( \frac{1}{L}\sum_{l=1}^{L}||\hat{\mathbf{q}_{l}}-\mathbf{u}||^{2}\right)} \tag{3}\]  
\[\text{bias}(\mathbf{u})=\sqrt{\frac{1}{L_{0}}\sum_{l_{0}=1}^{L_{0}}\left| \left|\frac{1}{L}\left(\sum_{l=1}^{L}\hat{\mathbf{q}_{l}}\right)-\mathbf{u} \right|\right|^{2}}, \tag{4}\]  
where \(\hat{\mathbf{q}_{l}}\) is the \(l\)th ensemble's position estimate, \(L=5000\) is the number of ensemble runs and \(L_{0}=5000\) is the number of deployment scenarios. In both the RMSE and the estimation bias, we calculate the average of \(L_{0}\) deployments because we are simulating an ADCS without perfect accuracy. For all the simulations, we set \(\sigma_{\text{AOA}}=1^{\circ}\) and \(\sigma_{\text{RD}}=10\) m, where \(\sigma_{\text{AOA}}\) is the standard deviation of the AOA method, and \(\sigma_{\text{RD}}\) is the standard deviation of the TDOA method multiplied by the speed of propagation of the radio waves to transform  
Figure 4: The root CRB indicates the best possible accuracy of the position estimator, and we used it to compare the six deployment directions. This figure shows the root CRB after three days since the deployment of the femto-satellite at 1 m/s. The orbit’s period is around 94.5 min. Each of the lines represents one of the six directions of deployment using the mother-CubeSat’s LVLH frame, and it is the average of 10 different deployment points of the orbit. it into a range difference. We also chose \(\sigma_{\rm ADS}=0.01^{\circ}\) which is equal to 36 arcseconds of accuracy and can be obtained with a low-cost star tracker [26]. For the ACS, we used \(\sigma_{\rm ACS}=0.06^{\circ}\) because Figure 6 shows that below this value, the root CRB decreases by around 20 m.  
Figure 5: Root CRB for deployments towards the positive \(y\) axis, at different points of the orbit. The points are separated by a time-lapse of ten minutes, starting at the apogee. Figure 6: Root CRB for different values of \(\sigma_{\rm ACS}\) and \(\sigma_{\rm ADS}\). These values are related to the accuracy of the deployment system, so we study the effect on the performance of the localization system three days after deployment. The accuracy of the GNSS devices on board the CubeSats also affects the performance of the localization system. To study the consequences, we simulated 5000 deployments towards the positive \(y\) axis. Then, we evaluated the localization system three days after deployment. For each deployment, we generated 5000 zero-mean Gaussian noises with a \(\sigma_{\text{GNSS}}\) ranging from 10 m to 120 m. Figure 7 shows that the RMSE does not change until the accuracy of the GNSS is 50 m.  
For the final simulation, we set \(\sigma_{\text{GNSS}}=10\) m, since there are GNSS devices with that level of accuracy for CubeSats on the market, like the NewSpace Systems (NSS) GPS Receiver from CubeSat Shop. Figure 8 shows that the root CRB is below 30 m for most of the orbit three days after deployment. The purple line depicts the distance between the femto-satellite and the CubeSat that launched it. This distance oscillates and increases over time. This behavior is usual for deployments because, instead of just having a relative speed equal to the deployment speed, what happens is that the orbital parameters change. This change means that the femto-satellite and the CubeSats are now in different orbits. The red line shows the distance between the femto-satellite and the second CubeSat in kilometers. There is no noticeable change in this distance, but this could be related to the scale (kilometers instead of meters like the other curves) and the short time span. These variations may explain the saw-tooth shape of the bias that does not correlate perfectly with the parabolic shape of the angle \(\alpha\) nor with the distance between the femto-satellite and the mother-CubeSat.