## 2 Mathematical Modeling
### Optimization Problem  
From the assumptions made in Sections 2.1-2.4, the position dilution of precision _PDOP_ has the functional dependence described in (32).  
\[\begin{split} PDOP=f(\mathbf{x}_{\rm E},\mathbf{x}_{5,1}, \mathbf{x}_{5,2},\ldots,\mathbf{x}_{5,k},\ldots,\mathbf{x}_{5,n})\\ =f(\alpha_{\rm GR}(t),\Lambda,\lambda,\Omega_{1},\Omega_{2},\ldots, \Omega_{k},\ldots,\Omega_{n},\
u_{1},\
u_{2},\ldots,\
u_{k},\ldots,\
u_{n},r) \end{split} \tag{32}\]  
In (32), convenient values of \(\lambda=0\) and \(\alpha_{\rm GR}=0\) can be assumed because from a relative perspective, there exists an equivalent shift in the values of the right ascension of the ascending nodes \(\Omega_{1},\Omega_{2},\ldots,\Omega_{k},\ldots,\Omega_{n}\). This shift is the same for every satellite and is not relevant for the purposes of this study because it is focused on the relative geometry among satellites (i.e., differences in \(\Omega\)). So, without loss of generalization, the functional dependence reported in (33) can be considered.  
\[PDOP=f(\Lambda,\Omega_{1},\Omega_{2},\ldots,\Omega_{k},\ldots,\Omega_{n},\
u_ {1},\
u_{2},\ldots,\
u_{k},\ldots,\
u_{n},r) \tag{33}\]  
An optimization problem needs variables, domains, and constraints.  
As previously investigated in [26], this kind of problem can be defined considering the right ascension of the ascending node \(\Omega_{1},\Omega_{2},\ldots,\Omega_{k},\ldots,\Omega_{n}\) and the true anomaly \(\
u_{1},\
u_{2},\ldots,\
u_{k^{\prime}},\ldots,\
u_{n}\) as a set of variables, while \(\Lambda\) and \(r\) are considered assigned parameters of a single instance of the problem. By definition, the domains of the right ascension of the ascending node and the true anomaly are \(\Omega\in[0,2\pi]\) and \(\
u\in[0,2\pi]\), so the whole set is \(\mathcal{D}=[0,2\pi]^{2n}\).  
Two main physical constraints are considered in this study. The first one is related to the fact that both the emitter and the satellites cannot be inside the surface of the Earth, so \(h\geq 0\) and \(r\geq r_{\oplus}\) (equivalently, \(h_{\rm S}\geq 0\)). The second condition is on the LOS. As derived in [26], indicating with \(\Omega_{\rm K,0}\) and \(\
u_{\rm K,0}\) the angular position for the \(k\)-th satellite corresponding to the Nadir direction with respect to the emitter position, there exists a maximum angle \(\delta_{\rm k}\) for the k-th satellite such that the condition on the satellite angles \(\Omega_{\rm k}\) and \(\
u_{\rm k}\) is expressed as in (34).  
\[(\Omega_{\rm k}-\Omega_{\rm K,0})^{2}+(\
u_{\rm k}-\
u_{\rm K,0})^{2}\leq \delta_{\rm k}^{2} \tag{34}\]  
Thus, (34) is an absolute condition on the k-th satellite. However, considering also that the absolute angle \(\delta_{\rm k}\) can be expressed as  
\[\delta_{\rm k}=\text{acos}\bigg{(}\frac{r_{\oplus}}{r_{\oplus}+h_{\rm S_{k}}} \bigg{)} \tag{35}\]  
and the assumption that all the satellites have the same orbit radius, the absolute condition (34) has the same maximum angle for all the satellites \(\delta_{1}=\delta_{1}=\cdots=\delta_{\rm k}=\cdots=\delta_{\rm n}=\delta\).  
As derived also in [26], (36) shows the problem of spatial optimization with the standard deviation on the position \(\sigma_{\rm p}\) as an objective function and \(\Omega_{\rm K,0}=\
u_{\rm K,0}=0\) for every \(k\)-th satellite (i.e., the \(k\)-th LOS condition is written with respect to the Nadir alignment between the \(k\)-th satellite and the emitter).  
\[\left\{\begin{aligned} &\min_{(\Omega_{1},\Omega_{2},\ldots, \Omega_{k},\ldots\Omega_{n},\Omega_{n+1},\
u_{12},\ldots,\
u_{k},\ldots,\
u_{n })\in\mathcal{D}=[0,2\pi]^{2n}}\big{(}\sigma_{\mathrm{P}})\\ & s.t.\quad\Omega_{1}^{2}+\
u_{1}^{2}\leq\delta^{2}\\ & s.t.\quad\Omega_{2}^{2}+\
u_{2}^{2}\leq\delta^{2}\\ &\quad\quad\quad\quad\vdots\\ & s.t.\quad\Omega_{\mathrm{k}}^{2}+\
u_{\mathrm{k}}^{2}\leq\delta^ {2}\\ &\quad\quad\quad\quad\vdots\\ & s.t.\quad\Omega_{\mathrm{n}}^{2}+\
u_{\mathrm{n}}^{2}\leq\delta^ {2}\\ \end{aligned}\right. \tag{36}\]