# Modeling and Control of Satellite Formations: A Survey
## 2 Multiple-Input-Multiple-Output Formation Architecture
]^{\mathrm{T}}\) with time derivatives of the osculating element of the orbit:  
\[\frac{\mathrm{d}\,a}{\mathrm{d}\,t} =\frac{2a^{2}}{h}\Big{(}e\sin f\,u_{r}+\frac{p}{r}u_{\theta}\Big{)}, \tag{4}\] \[\frac{\mathrm{d}\,e}{\mathrm{d}\,t} =\frac{1}{h}\big{(}p\sin f\,u_{r}+\big{(}(p+r)\cos f+re\big{)}u_{ \theta}\big{)},\] (5) \[\frac{\mathrm{d}\,i}{\mathrm{d}\,t} =\frac{r\cos\theta}{h}u_{h},\] (6) \[\frac{\mathrm{d}\,\Omega}{\mathrm{d}\,t} =\frac{r\sin\theta}{h\sin i}u_{h},\] (7) \[\frac{\mathrm{d}\,\omega}{\mathrm{d}\,t} =\frac{1}{h}\big{(}-p\cos f\,u_{r}+(p+r)\sin f\,u_{\theta}\big{)} -\frac{r\sin\theta\,\cos i}{h\sin i}u_{h},\] (8) \[\frac{\mathrm{d}\,M}{\mathrm{d}\,t} =n+\frac{\eta}{h\epsilon}\big{(}(p\cos f-2 re)u_{r}-(p+r)\sin f\,u _{e}\big{)}, \tag{9}\]  
where \(u_{r}\) is directed radially away from the Earth, \(u_{h}\) is directed along the orbital angular momentum vector, \(u_{\theta}\) is orthogonal to the previous directions. Parameter \(f\) is the true anomaly, \(r\) stands for the orbital radius, \(p=a\big{(}1-e^{2}\big{)}\) denotes the focal parameter (_semilatus rectum_), latitude \(\theta=\omega+f\). The average angular velocity \(n\) is determined by the expression  
\[n=\sqrt{\frac{\mu}{a^{3}}}. \tag{10}\]  
These equations in variations were obtained for the Keplerian motion. In matrix form, they take the form \(\dot{e}_{\mathrm{osc}}=\big{[} \begin{array}{cccccc}0&0&0&0&0&n\end{array}  \big{]}^{\mathrm{T}}+\big{[}B(\mathbf{e}_{\mathrm{osc}})\big{]}\,\mathbf{u}\), where \(\mathbf{e}=\big{[} \begin{array}{cccccc}a&e&i&\Omega&M\end{array} \big{]}^{ \mathrm{T}}\) is the vector of the osculating orbit elements, and \(\big{[}B(\mathbf{e}_{\mathrm{osc}})\big{]}\) is the control influence matrix of size \(6\times 3\) corresponding to the right-hand sides of Equations (4)-(9). The vector of the mean orbit \(\mathbf{e}=\big{[} \begin{array}{cccccc}a&e&i&\Omega&M\end{array} \big{]}^{ \mathrm{T}}\) is introduced and the analytical transformation \(\mathbf{e}=\xi(\mathbf{e}_{\mathrm{osc}})\) from osculating elements of the orbit \(\mathbf{e}_{\mathrm{osc}}\) to the mean orbit vector \(\mathbf{e}\). [PERSON] et al. [16] used the first order truncation of [PERSON] analytical solution [29]. Taking into account the influence of the disturbance \(J_{2}\), the Gaussian variational equations for the averaged motion are written in the form  
\[\dot{\mathbf{e}}=\big{[}A(e)\big{]}+\bigg{[}\frac{\partial\xi}{\partial \mathbf{e}_{\mathrm{osc}}}\bigg{]}^{\mathrm{T}}\big{[}B(\mathbf{e}_{\mathrm{ osc}})\big{]}\,\mathbf{u}, \tag{11}\]  
where \((6\times 1)\)-matrix \(\big{[}A(e)\big{]}\) of the controlled plant corresponds to the right-hand sides of (4)-(9). Considering the Brouwer transformation between the oscillating and mean elements of the orbit, one can see that matrix \(\bigg{[}\frac{\partial\xi}{\partial\mathbf{e}_{\mathrm{osc}}}\bigg{]}\) is approximately \(6\times 6\)-identity matrix with off-diagonal elements of the order of \(J_{2}\) or less. Therefore, in [16], the equation for the average velocity of an orbital element is approximated in the form  
\[\dot{\mathbf{e}}\approx\big{[}A(e)\big{]}+\big{[}B(\mathbf{e})\big{]}\,\mathbf{ u}. \tag{12}\]  
The plant model matrix \([A]\) accurately describes the behavior of the elements of the mean orbit. Perturbation \(J_{2}\) has no secular impact on elements \(a,e,i\). The control influence matrix \([B]\), obtained in the Gaussian variational equations, as seen from (4)-(9), makes it possible to calculate the change in the osculating orbital elements due to control vector \(u\). It is assumed that changes in these vibrational elements of the orbit, as follows from (12), are directly reflected in the corresponding changes in the elements of the mean orbit. Working with the elements of the mean orbit has the advantage that short-term period fluctuations are not perceived as tracking errors; only long-term errors are compensated.