# Modeling and Control of Satellite Formations: A Survey
## 2 Multiple-Input-Multiple-Output Formation Architecture
Explicit equations of motion that describe the evolution of a system with constraints:  
\[\sigma_{i}=\dot{e}_{i}+c_{i,0}e_{i}+c_{i,-1}\int e_{i}(\tau) \mathrm{d}\tau \tag{28}\]2  
or  
\[\sigma_{i}=\dot{e}_{i}+c_{i,0}e_{i}+c_{i,-1}\int e_{i}(\tau) \mathrm{d}\tau \tag{28}\]3  
where the matrix \(K(q,\dot{q},t)=M^{\frac{1}{2}}(AM^{-\frac{1}{2}})^{\dagger}\), and the upper the subscript <<\(\tau\)>> denotes generalized Moore-Penrose inverse of \(AM^{-\frac{1}{2}}\). (The _Moore-Penrose pseudoinverse_ for any \(n\times m\)-matrix \(H\) is called matrix \(H^{\dagger}=\lim_{\delta\to 0}(H^{\dagger}H\!+\!\delta^{2}\mathbf{I})^{-1}H^{ \dagger}\). If \(H\) is an \(n\times n\)-matrix and \(\det H\
eq 0\), then \(H^{\dagger}=H^{-1}\). If the columns of \(H\) are linearly independent, then \(H^{\dagger}=(H^{\dagger}H)^{-1}H^{\dagger}\). Also it is valid that: \((H^{\dagger})^{\dagger}=(H^{\dagger})^{\dagger}\), \(H^{\dagger}=(H^{\dagger}H)^{\dagger}H^{\dagger}=H^{\dagger}(H^{\dagger}H)^{\dagger}\), \(HH^{\dagger}H\!=\!H,H^{\dagger}HH^{\dagger}\!=\!H^{\dagger}\).) 2. An additional term on the right-hand side of the Equation (37), representing the generalized force of constraints, is explicitly given by the formula  
\[\sigma_{i}=\dot{e}_{i}+c_{i,0}e_{i}+c_{i,-1}\int e_{i}(\tau) \mathrm{d}\tau \tag{28}\]4  
3. Equation (38) after multiplication by \(M^{-1}\) can be rewritten as  
\[\sigma_{i}=\dot{e}_{i}+c_{i,0}e_{i}+c_{i,-1}\int e_{i}(\tau) \mathrm{d}\tau \tag{28}\]5  
or  
\[\sigma_{i}=\dot{e}_{i}+c_{i,0}e_{i}+c_{i,-1}\int e_{i}(\tau) \mathrm{d}\tau \tag{28}\]6  
where vector \(\Delta a=\ddot{q}-a\) represents the deviation (at time \(t\)) of the bounded generalized acceleration \(\ddot{q}\) from the corresponding unbounded acceleration \(a\); the error vector \(e=b-Aa\) represents the degree to which accelerations at time \(t\) corresponding to unbounded motion do not satisfy the constraint equations (36); matrix \(K_{1}=M^{-1}K=M^{-\frac{1}{2}}(AM^{-\frac{1}{2}})^{\dagger}\). (Matrix \(K_{1}\) is called the weighted pseudoinverse Moore--Penrose matrix, [74].) The last form of [74] results in the following fundamental principle of Lagrangian mechanics: _\"The motion of a discrete dynamical system subject to constraints evolves at every moment of time. time in such a way that the deviation of its accelerations from those that it would have at this moment in the absence of restrictions is directly proportional to the degree to which accelerations corresponding to its unbounded motion at time \(t\) do not satisfy restrictions; the proportionality matrix is the weighted generalized inverse Moore-Penrose matrix for the weighted constraint matrix \(A\), and the dissatisfaction measure constraints are provided by the vector \(e\).\"_ The Udwadia-Kalaba equation, in contrast to the Lagrange equation, is equally easily applicable to both holonomic and nonholonomic constraints. In space systems with a large number of satellites, orbit coordination between satellites is required throughout the life of a mission. The main focus has usually been on the limited relative motion between satellites in a constellation, [PERSON] and [PERSON] [83] also considered the additional degree of freedom to manipulate an arbitrary number of orbital elements, which is represented as coordinating the overall orbital transfer and constellation in space. The basic concept uses consensus theory to characterize the properties of a control object, as in a multi-agent system. For this purpose, [PERSON] and [PERSON] [83] assumed that communication in the network satellite system is represented in the form of an undirected graph, and then the dynamics of the control system are realized in an affine-controlled form, as described in the variational Gauss equations. For the general problem of orbital transmission, a controller based on the edge error has been developed and proven to be asymptotically stable. Several strategies for managing the grouping are discussed, namely by changing variables or a two-phase control process based on a dynamic structure.