## 5 Constellation of Formations
### Dedicated Control
The columns of matrix \(B\) span the whole vector space \(R^{6}\) every orbit, but locally only can generate a subspace of dimension 3. Therefore (2) and (3) cannot actually be met unless the left-hand sides belong to the column vector space of matrix \(B\), but this can be solved in general by using the pseudo-inverse \(B^{+}\) of the input matrix \(B\) :  
\[\underline{a}_{1R}=\underline{a}_{0R}-k_{F}\cdot B^{+}\cdot\delta\underline{ \alpha}_{01R} \tag{9}\]  
\[\underline{a}_{2R}=\underline{a}_{0R}-k_{F}\cdot B^{+}\cdot\delta\underline{ \alpha}_{02R} \tag{10}\]  
The center of mass constraint for the control accelerations is given as follows, for a chief with mass, and the followers with masses:  
\[\underline{a}_{0R}\cdot m_{0}+\underline{a}_{1R}\cdot m_{1}+\underline{a}_{2 R}\cdot m_{2}=0 \tag{11}\]  
The linear Equations (9)-(11) can be solved for the control vectors \(a_{0R},a_{1R}\), and \(a_{2R}\) as follows:  
\[\underline{a}_{1R}=-k_{F}\cdot B^{+}\bigg{(}\frac{(m_{0}+m_{1})\delta a_{01R}- m_{2}\cdot\delta a_{02R}}{m_{0}+m_{1}+m_{2}}\bigg{)} \tag{12}\]  
\[a_{2R}=-k_{F}\cdot B^{+}\bigg{(}\frac{(m_{0}+m_{2})\cdot\delta a_{02R}-m_{1} \delta a_{01R}}{m_{0}+m_{1}+m_{2}}\bigg{)} \tag{13}\]  
\[a_{0R}=-k_{F}\cdot B^{+}\bigg{(}\frac{m_{1}\cdot(m_{0}+m_{1}-m_{2})\cdot \delta a_{01R}+m_{2}\cdot(m_{0}+m_{2}-m_{1})\cdot\delta a_{02R}}{m_{0}\cdot(m _{0}+m_{1}+m_{2})}\bigg{)} \tag{14}\]  
This method can incorporate disturbance rejection, control saturation, and fuel consumption management as in [31], but the emphasis on the linear combination of the relative orbital elements is maintained. Notice that the control acceleration shown in the previous section's example for each of the followers did not specify the implementation completely, as there were undefined degrees of freedom. For instance, one could define zero relative control acceleration for the chief, as can be found in a non-cooperative leader-follower approach. By exploiting these degrees of freedom more generally, allowing to preserve the center of mass for relative control, and on the other hand, one can compute the control acceleration for the center of mass in order to track the desired constellation objective as a common control acceleration \(\underline{a}_{C}^{F_{i}}\) for a given formation \(F_{i}\). Therefore, the total control accelerations to be implemented on each of the satellites of this formation \(F_{i}\) are as follows:  
\[\underline{a}_{0}^{F_{i}} =\underline{a}_{C}^{F_{i}}+\underline{a}_{0R}^{F_{i}} \tag{15}\] \[\underline{a}_{1}^{F_{i}} =\underline{a}_{C}^{F_{i}}+\underline{a}_{1R}^{F_{i}}\] (16) \[\underline{a}_{2}^{F_{i}} =\underline{a}_{C}^{F_{i}}+\underline{a}_{2R}^{F_{i}} \tag{17}\]  
which is the control law for each formation \(F_{i}\) in the constellation of formations \(C\). For a given objective for the centre of mass of the formation \(F_{i}\) in the constellation, it is defined as a relative orbital element \(\delta\underline{a}_{C}^{F_{i}}\) which determines the control term \(\underline{a}_{C}^{F_{i}}\) as follows:  
\[\underline{a}_{C}^{F_{i}}=-B_{Fi}^{+}\Big{(}k_{C}\cdot\delta\underline{a}_{C} ^{F_{i}}+\underline{f}_{C}^{F_{i}}\Big{)} \tag{18}\]  
where the input matrix corresponds to the formation \(F_{i}\) which is explicitly stated in the notation. The term \(\underline{f}_{C}^{F_{i}}\) may be used for feed-forward compensation of non-conservative dynamics, as aerodynamic drag or solar radiation pressure, as a degree of freedom for the designer. In order to compute the constellation error, it is necessary to compute the desired orbit for the center of mass, which can be performed on-board with a suitable orbit propagator, which should be modified/initialized considering the mission needs. As both control objectives \(O_{C}\) and \(O_{F}\) have different accuracy limits, the proposed separation helps to optimize the application of each of the laws on the specific time periods on which they may be more effective.