## 5 Constellation of Formations
### Allocation of Satellite Masses on Each Formation  
In the studies of a companion satellite for the L-band SAR Argentine MicroWave Observation Satellite: _Satellite Argentino de Observacion COn Microondas (SAOCOM)_ mission [36; 37], the relation of masses between the chief and the follower was around ten times. It is reasonable to fix the same mass for the followers, i.e., \(m_{1}=m_{2}=m_{F}\), and the chief mass is given as \(m_{0}=\beta\ m_{F}\) for \(\beta\ \geq\ 1\). Moreover, it would be convenient to implement on the chief a thruster \(\beta\) times bigger in terms of force and propellant mass, for a given common propulsion technology and specific impulse. Under this mass model, the total mass of the constellation of formation would be \(m_{T}=N_{c}\ (2+\beta)\ m_{F}\), where \(N_{c}\). is the number of formations of three satellites (one chief and two followers). The following particular cases can be identified by inspection of Equations (12)-(14):  
* \(\beta\gg 1\) : In this case, the required chief's control acceleration becomes negligible with respect to the control acceleration of the followers, which tends to be like a classical leader-follower topology on which the control is made by the follower only. * \(\beta=1\) : The required chief's control acceleration authority doubles the required control acceleration authority of each of the followers. * \(\beta=2\) : The required chief's control acceleration authority equals the required control acceleration authority of each of the followers. * \(\beta>2\): The required chief's control acceleration authority is smaller than the required control acceleration authority of each of the followers. In general, if there were \(N_{F_{i}}\) satellites on a formation \(F_{i}\), the mass ratio for equal control acceleration authority for a chief with mass \(m_{0}\ =\ \beta m_{F}\) is given by \(\beta=N_{F_{i}}-1\). Moreover, notice that the case with \(0<\beta<1\) would be feasible, but this case is not practical for a SAR formation, where the chief performs more tasks than the followers and thus requires more satellite mass. Figures 11 and 12 show the control acceleration evolution to implement the same formation reconfiguration as proposed for AT-InSAR, with \(\beta=2\) and \(\beta=10\) respectively. It is a verified fact that for a larger value of \(\beta\), the control authority required on the chief becomes reduced in comparison with the followers'. This also has an impact on the DeltaV of each satellite, as shown in Figure 13. This could be taken for a trade-off on the specific constellation/formation system design under the particular restrictions and mission objectives, which is beyond the scope of this work. Figure 11: Control acceleration results with chief/follower mass ratio \(\beta=2\). Finally, note that the implementation of this distributed control requires knowledge of the satellite masses, which are time-varying; one of the main uncertainties in knowing the satellite mass which is given by the propellant consumption. However, by using very high specific impulse electric propulsion of several thousands of seconds, this becomes negligible. The feasibility of this specific impulse level can be verified with the Field Emission Electric Propulsion (FEEP) technology, which is available now as Commercial Off-The-Shelf (COTS) products for small satellites (see [38; 39]). In order to simplify the implementation of the saturation, the saturation on the control acceleration differences is defined as follows:  
\[\Delta a_{1R}=sat(\underline{a}_{1R}\ -\underline{a}_{0R}) \tag{21}\]  
Figure 12: Control acceleration results with chief/follower mass ratio \(\beta=10\). Figure 13: DeltaV on a formation reconfiguration as a function of the mass ratio \(\beta\).