## 3 Formulation of the Satellite Module Layout Problems
### Constraint Functions
The constraint relative to the angles of inertia is defined as follows: \[g_{AI}(\mathbf{p}_{I}^{c},\mathbf{p}_{\alpha}^{d},\mathbf{p}_{s}^{d})=\sqrt{ \theta_{x^{\prime}y^{\prime}}(\mathbf{p}_{I}^{c},\mathbf{p}_{\alpha}^{d}, \mathbf{p}_{s}^{d})^{2}+\theta_{x^{\prime}z^{\prime}}(\mathbf{p}_{I}^{c}, \mathbf{p}_{\alpha}^{d},\mathbf{p}_{s}^{d})^{2}+\theta_{y^{\prime}z^{\prime}} (\mathbf{p}_{I}^{c},\mathbf{p}_{\alpha}^{d},\mathbf{p}_{s}^{d})^{2}}-\delta_{AI}\] (12) where \(\theta_{x^{\prime}y^{\prime}}\), \(\theta_{x^{\prime}y^{\prime}}\) and \(\theta_{y^{\prime}z^{\prime}}\) are the angles of inertia defined as \[\theta_{x^{\prime}y^{\prime}}(\mathbf{p}_{I}^{c},\mathbf{p}_{\alpha}^{d}, \mathbf{p}_{s}^{d})=\arctan\Biggl{(}-\frac{2I_{x^{\prime}y^{\prime}}(\mathbf{p }_{I}^{c},\mathbf{p}_{\alpha}^{d},\mathbf{p}_{s}^{d})}{(I_{y^{\prime}y^{\prime }}(\mathbf{p}_{I}^{c},\mathbf{p}_{\alpha}^{d},\mathbf{p}_{s}^{d})-I_{x^{\prime} x^{\prime}}(\mathbf{p}_{I}^{c},\mathbf{p}_{\alpha}^{d},\mathbf{p}_{s}^{d})} \Biggr{)}\bigg{/}2\] (13) \[\theta_{x^{\prime}z^{\prime}}(\mathbf{p}_{I}^{c},\mathbf{p}_{\alpha}^{d}, \mathbf{p}_{s}^{d})=\arctan\Biggl{(}-\frac{2I_{x^{\prime}z^{\prime}}(\mathbf{p }_{I}^{c},\mathbf{p}_{\alpha}^{d},\mathbf{p}_{s}^{d})}{(I_{x^{\prime}x^{\prime }}(\mathbf{p}_{I}^{c},\mathbf{p}_{\alpha}^{d},\mathbf{p}_{s}^{d})-I_{z^{\prime }z^{\prime}}(\mathbf{p}_{I}^{c},\mathbf{p}_{\alpha}^{d},\mathbf{p}_{s}^{d})} \Biggr{)}\bigg{/}2\] (14) \[\theta_{y^{\prime}z^{\prime}}(\mathbf{p}_{I}^{c},\mathbf{p}_{\alpha}^{d}, \mathbf{p}_{s}^{d})=\arctan\Biggl{(}-\frac{2I_{y^{\prime}z^{\prime}}(\mathbf{p }_{I}^{c},\mathbf{p}_{\alpha}^{d},\mathbf{p}_{s}^{d})}{(I_{y^{\prime}y^{\prime }}(\mathbf{p}_{I}^{c},\mathbf{p}_{\alpha}^{d},\mathbf{p}_{s}^{d})-I_{z^{\prime }z^{\prime}}(\mathbf{p}_{I}^{c},\mathbf{p}_{\alpha}^{d},\mathbf{p}_{s}^{d})} \Biggr{)}\bigg{/}2\] (15) where the inertias \(I_{x^{\prime}y^{\prime}}\), \(I_{y^{\prime}z^{\prime}}\) and \(I_{y^{\prime}z^{\prime}}\) are expressed as \[I_{x^{\prime}y^{\prime}}(\mathbf{p}_{I}^{c},\mathbf{p}_{\alpha}^{d}, \mathbf{p}_{s}^{d})=\sum_{i=1}^{N} \begin{bmatrix}I_{x^{\prime}i}+m_{i}({p_{ y,i}^{c}}^{2}+{p_{z,i}^{d}}^{2})\\ m_{i}p_{x,i}^{c}p_{y,i}^{c}+\frac{-I_{y^{\prime}i}-m_{i}({p_{x,i}^{c}}^{2}+{p_{z,i} ^{d}}^{2})}{2}\sin(2p_{\alpha,i}^{d})\\ \\ -x_{c}y_{c}m_{tot}\end{bmatrix} \] (16) \[I_{x^{\prime}z^{\prime}}(\mathbf{p}_{I}^{c},\mathbf{p}_{\alpha}^{d}, \mathbf{p}_{s}^{d})=\sum_{i=1}^{N}m_{i}p_{x,i}^{c}p_{z,i}^{d}-x_{c}z_{c}m_{tot}\] (17) \[I_{y^{\prime}z^{\prime}}(\mathbf{p}_{I}^{c},\mathbf{p}_{\alpha}^{d}, \mathbf{p}_{s}^{d})=\sum_{i=1}^{N}m_{i}p_{y,i}^{c}p_{z,i}^{d}-y_{c}z_{c}m_{tot}\] (18)
* Functional constraints: These are also defined in terms of incompatibility between several components. Thermal as well as electromagnetic thresholds are defined such that some components responsible for thermal or electromagnetic fields must be spaced away at given distances. More precisely, a set \(C_{H}\) of pairs of incompatible heat compo nents \(\{i,j\}\in C_{H}\) and a set \(C_{E}\) of pairs of incompatible electromagnetic components \(\{i,j\}\in C_{E}\) are defined. Thus, the thermal functional constraint is defined as  
\[g_{H}(\mathbf{p}_{i}^{c},\mathbf{p}_{a}^{d},\mathbf{p}_{s}^{d})=\sum_{k=1}^{ \text{Card}(C_{H})}d_{H}-d_{k}(\mathbf{p}_{i}^{c},\mathbf{p}_{a}^{d},\mathbf{p}_ {s}^{d}) \tag{19}\]  
where \(\text{Card}(C_{H})\) is the cardinal of the set \(C_{H}\) corresponding to the list of incompatible pairs of thermal radiative components, \(d_{k}\) is the distance between the two components of pair \(k\) if positioned on the same surface and \(d_{H}\) is the minimal distance between the two components of each pair of \(C_{H}\).