## 5 Constellation of Formations  
The formation presented in the previous section has a ground-track repeatability described by the repeatability of any of the ground tracks of the satellites in the formation. These ground tracks limit the observation coverage for a given instrument's field of view. A constellation of satellite formations is proposed to increase the repeatability and coverage of the desired DSS while keeping the formation advantages. Figure 10 shows an example of two formations flying in a constellation. Notice that each formation preserves the single-pass  
Figure 8: Control acceleration for the follower 1—chief formation, where red is the Radial, green is the Transverse, and blue is the Normal component.  
Figure 9: Safe condition evaluation for follower 1—chief formation. 5. Constellation of Formations.  
SAR interferometry objective. On the other hand, the resulting constellation can be designed by using the center of mass of each formation as the constellation-equivalent satellite.  
Let the constellation \(C=\left\{F_{1},\ F_{2},\ \ldots,\ F_{Nc}\right\}\) be a set of \(N_{c}\) formations \(F_{i}\), each of them composed by \(N_{F_{i}}\) satellites, i.e., \(F_{1}=\left\{s_{1,1},\ s_{1,2},\ldots,s_{1N_{F_{1}}}\right\}\), \(F_{2}=\left\{s_{2,1},s_{2,2},\ldots,s_{2\times N_{F_{2}}}\right\}\), etc. As each satellite \(s_{i,ij}\) belongs to the formation \(F_{i}\) and to the constellation C, there are at least two objectives for the orbit controller:  
\(O_{F}\)--To keep the relative orbit of the satellite \(s_{i,ij}\) within the formation \(F_{i}\).  
\(O_{C}\)--To keep the formation \(F_{i}\) in the constellation C.  
The formation objective \(O_{F}\) has been treated in the previous section by using the relative orbital elements to describe the feedback error. However, there is flexibility in the implementation of the control law that designers can now use, as for any given acceleration control determined by the follower relative orbit feedback law, designers can implement it on the follower alone, on the chief satellite alone, or on both. In this case, the freedom is used to implement the formation orbit control laws in such a way that _it does not modify the dynamics of the formation's center of mass_.  
The constellation objective \(O_{C}\) is stated considering the absolute orbit elements as generated by the constellation design, which is typically free of non-conservative forces, i.e., without drag and solar pressure effects, and with a certain reduced order model of the gravitational effects. On the other hand, to build the constellation control error, we need to estimate the center of mass of each formation \(F_{i}\). This requires the knowledge of the centre of mass and the mass of each of the satellites but has the benefit to enable a smoothing of the navigated orbit, as the center of mass is a weighted average. In particular, if all the satellites have the same mass and independent identical navigation error distributions, the navigated position and velocity of the center of mass will have their standard deviation reduced by a factor of \(\sqrt{N_{Fi}}\). On the other hand, if there is a satellite dominant in mass, the centre of mass navigation error is dominated by the navigation error of this satellite. In this way, the constellation objective has been reduced to the problem of the control of the center of mass of the systems or particles determined by each formation. Once the control acceleration for the center of mass is computed, this is translated into the specific forces  
Figure 10: A constellation of two formations with three satellites.  
to be implemented on each satellite, considering their masses. Additional features of this proposal are developed in the following sections.