# Modeling and Control of Satellite Formations: A Survey
## 2 Multiple-Input-Multiple-Output Formation Architecture
In [27], it is shown that even in the presence of differential interference \(J_{2}\), the proposed approach to control the movement of the formation is very efficient, which requires fuel consumption (expressed in terms of the change in velocity during the orbit) \(\Delta V\in[5,15]\) mm/s/orbit, depending on the scenario. Simulations showed that [PERSON]'s equations [24] are needed to determine the desired state for periodic relative motion, but the Hill equations are sufficient for posing the linear programming control problem. This result is important because the use of the time-independent Hill equations [22] significantly reduces the computational costs required to solve a linear programming problem. Two nonlinear feedback control laws for recovering the desired \(J_{2}\)-invariant relative orbit are presented in [16]. According to [28], there are two necessary conditions for two adjacent orbits to be invariant to the perturbation of the gravitational harmonic \(J_{2}\) in the sense that both orbits will demonstrate the same secular angular drift velocities. In particular, these conditions ensure that the velocities of the ascending node \(\dot{\Omega}\) and the mean values of the latitude \(\dot{\theta}_{M}=\dot{M}+\dot{\omega}\), where \(M\) is the mean anomaly and \(\omega\) is the argument of perigee, which are equal to each other. Since it is convenient to describe the relative orbit of the follower (\"daughter\") satellite relative to the lead (\"parent\") satellite in terms of the mean differences of the orbital elements and since the conditions for the relative invariance of \(J_{2}\) relative to the orbit are expressed in terms of the mean orbit elements, the first control law returns errors in terms of mean orbit elements. Using the designations of the orbital elements with the index \(1\) for the lead satellite, the differences in the momenta of the \(J_{2}\)-invariant relative orbit elements of the slave satellite are written in [16]:  
\[\delta\eta =-\frac{\eta_{1}}{4}\tan i_{1}\delta_{1} \tag{1}\] \[\delta a =2 Da_{1}\delta\eta,\] (2) \[D =\frac{J_{2}}{4a_{1}^{2}\eta_{1}^{5}}(4+3\eta_{1})(1+5\cos^{2}i_{ 1}), \tag{3}\]  
where \(a\) is the semi-major axis of the orbit, \(i\) is the inclination, the eccentricity index \(\eta\) is defined as \(\eta=\sqrt{1-e^{2}}\), \(e\) is the eccentricity. After choosing a certain middle element of the difference \(\delta a\), \(\delta e\) or \(\delta i\), the remaining two differences of the momentum elements are given by two constraints in the Equations (1) and (2). Since only the pulse elements \(a\), \(e\) and \(i\) affect the secular drift caused by \(J_{2}\), the mean angles \(M\), \(\omega\) and \(\Omega\) can be chosen arbitrarily. To find the corresponding vectors of inertial position and velocity, the elements of the mean orbit are first translated into the corresponding oscillating elements of the orbit using the theory of the artificial Brouwer satellite [29]. Since the relative orbit is described in terms of the relative differences in the mean orbit elements when establishing \(J_{2}\)-invariant relative orbits, [PERSON] et al. [16] constructed the feedback law in terms of the mean orbit elements instead of the more traditional feedback approach on position and velocity vector errors. Not all orbital position errors are the same. The ascending node error should be controlled at a different position in orbit than the pitch error. Gaussian variational equations of motion (see [30]) provide a convenient set of equations relating the influence of the control vector of accelerations \(\mathbf{u}=[ \begin{array}{cccc}u_{r}&u_{\theta}&u_{h}\end{array}