## 2 Numerical Formulation  
The 3D numerical analysis of multi degrees of freedom electromechanical devices represents a challenging, still open problem, which poses several critical issues. Several commercial codes, typically based on the Finite Elements Method (FEM), provide packages for coupled electro-mechanical analysis, but they seem to be not very effective with multiple degrees of freedom and with sliding contacts. Moving conductors and sliding contacts usually require remeshing of the domains with consequent increase of computational times and potential numerical instabilities. Integral Formulations (e.g., the Method of Moments) seem to work quite well with moving domains since they require the discretization of the active regions only (namely conductors and ferromagnetic materials), so avoiding the problem of coupling meshes with different speeds. Integral formulations implicitly enforce the far field boundary conditions, and are able to produce accurate results by using coarse discretization (when compared with those required by FEM). The numerical investigation of the complete launch system has been performed by the research code EN4 EM. It is based on an integral formulation, and it is under continuous development by the authors for investigating electromechanical systems [27; 28; 29; 30; 31; 32; 33; 34]. EN4 EM applied to the launch package is able to simulate the whole system considering the characteristics of the two devices and taking into account their \"strong interaction\". Figure 1 shows the steps of a conceptual flow chart of the numerical formulation: discretization in elementary volumes, writing and integration of [PERSON]'s law, arrangement in an electric network and writing of the governing electro-mechanical equations. The usual notations for the electromagnetic quantities in Figure 1 are adopted according to the Table 1.  
With reference to the inset in the bottom right of Figure 1, [PERSON]'s law is written in the conductive elementary volumes where a uniform current distribution is assumed (row #1 of the inset). The additive properties of the integrals with respect to the integration domain allows expressing the fields and potentials in the \(k\)-th conductive elementary volume as a summation of contribution due to the currents in the other volumes (row #2 of the inset). Finally integration on the \(k\)-th elementary volume leads to an equation that can be seen as the voltage-current relationship of a branch that is a series connection of a resistor, an inductor coupled with other inductors, and a voltage generator controlled by the currents in other elementary volumes (row #3). The equivalent network of the device is built by connecting the terminals of the branches so obtained. In case of a multi-device system, the code is able to model separately the different devices by their equivalent networks. Then, all these networks are connected together according to the relative positions of the corresponding elementary volumes and to the presence of electrical contacts between them. The described procedure has been applied to model the launcher and the compulsator, obtaining a whole model composed of thousands of branches. Mesh analysis yields to the governing equations written in matrix form:  
\[\mathbf{L}(\mathbf{C}(t))\frac{di}{dt}+\Big{(}\mathbf{R}+\mathbf{K}\Big{(}\mathbf{C}(t),\dot{\mathbf{C} }(t)\Big{)}\Big{)}\mathbf{i}=\mathbf{e}(t) \tag{1}\]  
The values of the elements of the matrices in (1) are function of the system configuration \(\mathbf{C}(t)\) and its derivative \(\dot{\mathbf{C}}(t)\) (i.e., the relative positions and velocities of the elementary volumes used to discretize the devices respectively).