## 3 Formulation of the Satellite Module Layout Problems
### Objective Function  
The objective function to minimize is the total inertia of the module. Three systems of coordinates are considered:  
* \(Ox^{\prime\prime}y^{\prime\prime}z^{\prime\prime}\): the local coordinate system related to each component. \(O^{\prime\prime}\) corresponds to the center of inertia of the component and the axes are defined with the symmetry planes of the components. * \(O^{\prime}x^{\prime}y^{\prime}z^{\prime}\): the coordinate system related to the system of components. \(O^{\prime}\) stands for the current centroid of the system of components and the axes are defined with the symmetry planes of the module. * \(Oxyz\): the coordinate system related to the module. \(O\) is the geometric center of the container and the axes are defined using its symmetry planes. The three coordinate systems are illustrated in Figure 2.  
Figure 2: Geometrical definition of the multi-container configuration. The \(Oxyz\) system of coordinates is linked to the satellite module while the \(O^{\prime}x^{\prime}y^{\prime}z^{\prime}\) system of coordinates is linked to the system of components. \(O^{\prime}\) stands for the current center of gravity of the components. The \(O^{\prime\prime}x^{\prime\prime}y^{\prime\prime}z^{\prime\prime}\) system of coordinates is linked to one component \(i\), which is located on one plate with the position of its center of inertia \((x_{x,i},x_{y,i})\) and its orientation \(x_{\alpha,i}\). The redder the components, the heavier their corresponding masses. The global inertia \(I_{tot}\) to minimize is calculated in the \(O^{\prime}x^{\prime}y^{\prime}z^{\prime}\) coordinate system, as proposed in [19; 20; 108], and defined as  
\[I_{tot}(\mathbf{p}_{I}^{c},\mathbf{p}_{\alpha}^{d},\mathbf{p}_{s}^{d})=I_{x^{ \prime}x^{\prime}}(\mathbf{p}_{I}^{c},\mathbf{p}_{\alpha}^{d},\mathbf{p}_{s}^{ d})+I_{y^{\prime}y^{\prime}}(\mathbf{p}_{I}^{c},\mathbf{p}_{\alpha}^{d}, \mathbf{p}_{s}^{d})+I_{z^{\prime}z^{\prime}}(\mathbf{p}_{I}^{c},\mathbf{p}_{ \alpha}^{d},\mathbf{p}_{s}^{d}) \tag{2}\]  
With the same formalism as the previous section, the solid inertias specific to each component with relation to their system of coordinates \(O^{\prime\prime}x^{\prime}y^{\prime\prime}z^{\prime\prime}\) are written \(I_{x^{\prime\prime},i},I_{y^{\prime\prime},i}\) and \(I_{z^{\prime\prime},i}\). The inertia of the module along each axis of the \(Oxyz\) coordinate system are written as follows:  
\[I_{xx}(\mathbf{p}_{I}^{c},\mathbf{p}_{\alpha}^{d},\mathbf{p}_{s}^{ d})=I_{x,sat} +\sum_{i=1}^{N}I_{x^{\prime\prime},i}\cos(p_{a,i}^{d})^{2}+I_{y^{ \prime\prime},i}\sin(p_{a,i}^{d})^{2}+m_{i}({p_{y,i}^{c}}^{2}+p_{z,i}^{d}( \mathbf{p}_{s}^{d})^{2}) \tag{3}\] \[I_{yy}(\mathbf{p}_{I}^{c},\mathbf{p}_{\alpha}^{d},\mathbf{p}_{s}^ {d})=I_{y,sat} +\sum_{i=1}^{N}I_{y^{\prime\prime},i}\cos(p_{a,i}^{d})^{2}+I_{x^{ \prime\prime},i}\sin(p_{a,i}^{d})^{2}+m_{i}({p_{x,i}^{c}}^{2}+p_{z,i}^{d}( \mathbf{p}_{s}^{d})^{2})\] (4) \[I_{zz}(\mathbf{p}_{I}^{c},\mathbf{p}_{\alpha}^{d},\mathbf{p}_{s}^ {d})=I_{z,sat} +\sum_{i=1}^{N}I_{z^{\prime\prime},i}+m_{i}({p_{x,i}^{c}}^{2}+{p_ {y,i}^{c}}^{2}) \tag{5}\]  
where \(({p_{x,i}^{c}},{p_{y,i}^{c}}^{2},{p_{z,i}^{d}})\) are the coordinates of component \(i\) in the \(Oxyz\) system of coordinates. The coordinates along the \(z\)-axis, \({p_{z,i}^{d}}\), are discrete and calculated thanks to the value of the \(\mathbf{p}_{s}^{d}\) variables corresponding to the assigned container.