# Modeling and Control of Satellite Formations: A Survey
## 2 Multiple-Input-Multiple-Output Formation Architecture
In hybrid optimization, the genetic algorithm performs a global search to create two-impulse trajectories at each transition time, and vector analysis of basis vectors serves for local optimization with two-impulse trajectories as initial approximations. The final optimal trajectory has multiple pulses and also lower fuel consumption than two-pulse trajectories. The mission planner must investigate the trade-off between fuel saved by multiple impulses and the increased mission complexity due to additional impulses. In many missions, the trajectory obtained by the genetic algorithm will have an advantage for the transition due to its simplicity and almost optimal fuel consumption. In a series of papers, [PERSON] and [PERSON] [74,75], [PERSON] [76], [PERSON] and [PERSON] [77], [PERSON] [78,79,80,81], [PERSON] and [PERSON] [82] the approach is proposed based on the Gauss principle to obtaining models of dynamical systems with constraints. In particular, [PERSON] and [PERSON] [74] noted: \"The principles of analytical mechanics laid down by [PERSON], [PERSON] (1787), and [PERSON] (1829) are all-encompassing, and therefore it naturally follows that there can be no new fundamental principles of the theory of motion and equilibrium of discrete [finite-dimensional] dynamical systems. Despite this, additional perspectives can be helpful in understanding the laws of Nature from new perspectives, in particular, if they can help in solving problems of particular importance and in providing a deeper understanding of how Nature works. Although the general problem of limited motion was formulated at least as early as the time of [PERSON], the definition of explicit equations of motion for discrete dynamical systems with constraints, even within the limited view of [PERSON] mechanics, was a serious obstacle. The Lagrange multiplier method is based on problem-oriented approaches to determining the multipliers; they are often very difficult to find and, therefore, to obtain explicit equations of motion (both analytically and numerically) for systems that have a large number of degrees of freedom and many non-integrable constraints\". In [74], a finite-dimensional dynamical system whose configuration is described by \(n\) generalized coordinates \(q=[q_{1},q_{2},\ldots,q_{n}]^{\mathrm{T}}\) is considered. Its motion can be described, using the Newtonian or Lagrangian formalism, by equations  
\[M(q,t)\bar{q}=Q(q,\dot{q},t), \tag{34}\]  
where the \(n\times n\) matrix \(M\) is symmetric and positive definite. Generalized accelerations \(a\) of the system without restrictions, therefore, are given by the expression  
\[\ddot{q}M^{-1}Q\equiv a(q,\dot{q},t). \tag{35}\]  
Now, let the system be subject to \(m\) consistent constraints (not necessarily linearly independent)  
\[\sigma_{i}=\dot{e}_{i}+c_{i,0}e_{i}+c_{i,-1}\int e_{i}(\tau) \mathrm{d}\tau \tag{28}\]0where \(A\) is \(m\times\) matrix, commonly called the \"constraint matrix\", \(b\) is the \(m\) -dimensional vector. Differentiation of the usual constraint equations used in Lagrangian mechanics, which often have a Pfaffian form, leads to equations of the form (36). These constraint equations therefore include, among others, conventional holonomic, nonholonomic, scleronomic, rheonomic types of restrictions and their combinations. Thus, Equation (35) describe an unbounded system, and the Equation (36) describe the connections superimposed on this system, which covers all Lagrangian mechanics. Constraints (36) are more general than those in the usual framework of the Lagrangian mechanics. The presence of constraints (36) sets additional \"generalized constraint forces\" on the system so that the explicit equations of motion for the constrained system take the form  
\[\sigma_{i}=\dot{e}_{i}+c_{i,0}e_{i}+c_{i,-1}\int e_{i}(\tau) \mathrm{d}\tau \tag{28}\]1  
where the additional term \(Q_{c}\) on the right-hand side arises due to the constraint imposed according to (36). The main result for a constrained system is formulated in [74] in three equivalent forms. 1.