## 5 Constellation of Formations
### Dedicated Control  
Every satellite on the formation implements the same control computed for the center of mass of this formation to preserve/achieve the constellation objective. This can be seen as a \"common mode\" control, using absolute orbit navigation of the center of mass. On the other hand, for each satellite on the formation, there is an additional term obtained as the necessary feedback to implement the relative orbit control within the formation with the restriction that the dynamics of the center of mass of the local formation is not perturbed. Following the previous analogy, this can be seen as a \"differential mode\", using relative navigation between the satellites on the same local formation. Consider the two followers and the chief in the previous section's application as a formation; thus, the control acceleration to achieve relative dynamics while preserving the formation's center of mass must be computed. Let \(\underline{a}_{0R}\), \(\underline{a}_{1R}\), and \(\underline{a}_{2R}\) be the relative terms of the control accelerations for the chief (index 0), follower 1 (index 1) and follower 2 (index 2). Because the SAR interferometry requirements are written in terms of the error between the chief and each of the followers rather than the error between the followers, one can begin by stating the desired formation objectives:  
\[-k_{F}\;\delta\underline{a}_{01R}=B(\underline{a}_{1R}-\underline{a}_{0R}) \tag{2}\]  
\[-k_{F}\;\delta\underline{a}_{02R}=B(\underline{a}_{2R}-\underline{a}_{0R}) \tag{3}\]  
where \(k_{F}\) is a proportional gain for the formation control and \(\delta\underline{a}_{01R}\) and \(\delta\underline{a}_{02R}\) are the relative orbital elements of each of the followers with respect to the desired Maneuver orbit (see [31]) written in both cases relative to the same chief:  
\[\delta\underline{a}_{01R}=T_{0}\Big{(}\underline{\xi}_{1}-\partial\underline{ \xi}_{1}-\underline{\xi}_{0}\Big{)} \tag{4}\]  
\[\delta\underline{a}_{02R}=T_{0}\Big{(}\underline{\xi}_{2}-\partial\underline{ \xi}_{2}-\underline{\xi}_{0}\Big{)} \tag{5}\]  
where \(\underline{\xi}_{0},\underline{\xi}_{1},\underline{\xi}_{2}\) are the mean Ustinov parameters (see [14; 15]) of the chief, follower 1 and follower 2 orbits respectively, while \(\partial\underline{\xi}_{1}\) and \(\partial\underline{\xi}_{2}\) are the desired deviation relative to the chief necessary to implement the mission orbit requirement for follower 1 and follower 2, respectively. The matrix \(T_{0}\) is written in terms of the chief parameters and can be found in [30]. Finally, the matrix \(B\) in (2) and (3) is the control input matrix of these relative orbit elements dynamics, which is assumed equal for all the satellites in the formation. These relative orbital elements dynamics are given as (see [31; 32]):  
\[\frac{d\delta\underline{\alpha}_{01R}}{dt}=\underline{f}_{01R}+B(\underline{a }_{1R}-\underline{a}_{0R}) \tag{6}\]  
\[\frac{d\delta\underline{\alpha}_{02R}}{dt}=\underline{f}_{02R}+B(\underline{a }_{2R}-\underline{a}_{0R}) \tag{7}\]  
where \(\underline{f}_{02R}\) and \(\underline{f}_{02R}\) are considered very small disturbances, which can be partially compensated as a feed-forward term by the control law. The common input matrix \(B\) for a formation \(F_{i}\) will be determined by the orbit parameters of its centre of mass orbit, using its orbital elements as follows:  
\[B=\frac{1}{\overline{a}\,\overline{n}} \begin{bmatrix}0&2&0\\ -2&0&0\\ sin\left(\overline{\lambda}\right)&2\cos\left(\overline{\lambda}\right)&0\\ -\cos\left(\overline{\lambda}\right)&2 sin\left(\overline{\lambda}\right)&0\\ 0&0&\cos\left(\overline{\lambda}\right)\\ 0&0&\cos\left(\overline{\lambda}\right)\end{bmatrix}  \tag{8}\]  
where \(\overline{a}\), \(\overline{n}\), and \(\overline{\lambda}\) are the mean orbital elements of the centre of mass of the formation \(F_{i}\), associated respectively to the semi-major _axis_, mean motion and mean argument of latitude.