## 3 Formulation of the Satellite Module Layout Problems
### Mathematical Formulation  
Thus, multi-container SMLPs can be mathematically defined as follows:  
\[\begin{split}\min_{\mathbf{p}_{i}^{c},\mathbf{p}_{a}^{d}, \mathbf{p}_{s}^{d}}& I_{tot}(\mathbf{p}_{i}^{c},\mathbf{p}_{a}^{d },\mathbf{p}_{s}^{d})\\ \text{w.r.t.}&\mathbf{p}_{i}^{c}\in F_{\mathbf{p}_{i}^{c}} \subseteq\mathbb{R}^{n\mathbf{p}_{i}^{c}},\mathbf{p}_{a}^{d}\in F_{\mathbf{p}_ {a}^{d}}\subseteq\mathbb{N}^{n\mathbf{p}_{a}^{d}},\mathbf{p}_{s}^{d}\in F_{ \mathbf{p}_{s}^{d}}\subseteq\mathbb{N}^{n\mathbf{p}_{s}^{d}}\\ \text{s.t.}&\mathbf{h}_{overlap}^{C}(\mathbf{p}_{i}^ {c},\mathbf{p}_{a}^{d},\mathbf{p}_{s}^{d})=0\\ &\mathbf{h}_{overlap}^{E}(\mathbf{p}_{i}^{c},\mathbf{p}_{a}^{d}, \mathbf{p}_{s}^{d})=0\\ & g_{CG}(\mathbf{p}_{i}^{c},\mathbf{p}_{a}^{d},\mathbf{p}_{s}^{d}) \leq 0\\ & g_{AI}(\mathbf{p}_{i}^{c},\mathbf{p}_{a}^{d},\mathbf{p}_{s}^{d}) \leq 0\\ &\mathbf{g}_{H}(\mathbf{p}_{i}^{c},\mathbf{p}_{a}^{d},\mathbf{p}_ {s}^{d})\leq 0\\ &\mathbf{g}_{E}(\mathbf{p}_{i}^{c},\mathbf{p}_{a}^{d},\mathbf{p}_ {s}^{d})\leq 0\\ \end{split} \tag{21}\]  
Remarks on the Inertia Equations  
In all the papers dealing with the SMLP [19; 108; 112; 117; 48; 108], the inertias are calculated along with the axes of the \(O^{\prime}x^{\prime}y^{\prime}z^{\prime}\) system corresponding to the coordinate system linked to the current centroid of the whole module. They are obtained by calculating the inertia in the \(Oxyz\) system and then by applying [PERSON] theorem, which leads to Equations (6)-(8). However, as the objective function is to minimize the sum of the three aforementioned equations, it will also lead to a maximization of the three positive second terms:  
\[\begin{array}{l}(y_{x}^{2}+z_{c}^{2})m_{tot},\\ (x_{c}^{2}+z_{c}^{2})m_{tot},\\ (x_{c}^{2}+y_{c}^{2})m_{tot}.\end{array}\]  
Thus, the formulation of the inertia also contributes to maximize the coordinates of the global center of gravity of the module. As the balancing constraint (Equation (10)) bounds these coordinates to a tolerance zone, the optimization process will lead to a positioning of the center of gravity to the boundaries of the balancing constraint. However, the goal of the optimization process would rather be to minimize the inertia along with positioning the center of gravity as close as possible to a position of reference, which often corresponds to the geometrical center of the module or to the center of gravity of the empty module if different.  
Moreover, the first terms of the inertia equations are calculated in the _Oxyz_ system of coordinates, which is linked to the bottom of the satellite. Minimizing these quantities includes minimizing the terms \(m_{i}p_{x,i}^{c}\) and \(m_{i}p_{y,i}^{c}\) but also \(m_{i}p_{z,i}^{d}\). Then, the optimization process based on these inertia equations might overload the bottom surfaces \(S_{3}\) and \(S_{4}\), which leads to an unbalanced satellite module.  
Consequently, in order to avoid the two aforementioned issues, the inertia equations should be written in the system of coordinates linked to the position of reference where the center of gravity has to be positioned, as developed in Appendix B. However, in the present study, for the sake of numerical comparison with existing works, the equations employed in the literature, i.e., Equations (6)-(8), are used in the following experiments.