# Design of Cislunar Navigation Constellation via Orbits with a Resonant Period
## 2 Fundamental Theory
### Navigation Performance Modeling  
Current metrics for evaluating constellation performance include coverage, visibility, and DOP, with DOP being a critical indicator for assessing the positioning accuracy of navigation constellations. DOP reflects the error characteristics in pseudorange-based positioning solutions. Numerous studies have utilized DOP for the design and optimization of constellations; the evaluation methodologies are extensively investigated in Refs. [28; 29; 30]. This study accordingly analyzes the DOP performance of resonant constellations. To ensure position resolution, constellation satellites must establish communication links with receivers to transmit pseudorange signals [31]. By definition, DOP-based solutions require the simultaneous geometric visibility of at least four satellites to the receiver. This work assumes geometric visibility (lines of sight unobstructed by large celestial bodies) as the link criterion, disregarding practical communication constraints. DOP values are computed only when a receiver achieves four-satellite visibility, with the calculation formula following standard definitions. In the Earth-Moon rotating frame, the unit vector \(\bar{\mathbf{r}}\) from the user receiver to the \(i\)-th constellation satellite is expressed as  
\[\bar{\mathbf{r}}=\left[ \begin{array}{cc}\frac{x_{i}-x}{R_{i}}&\frac{y_{i}-y}{R_{ i}}&\frac{z_{i}-z}{R_{i}}\end{array} \right] \tag{6}\]  
where \(R_{i}\) represents the distance between the receiver and the \(i\)-th constellation satellite. \[R_{i}=\sqrt{\left(x_{i}-x\right)^{2}+\left(y_{i}-y\right)^{2}+\left(z_{i}-z \right)^{2}} \tag{7}\]  
Assuming \(n(n\geq 4)\) satellites in the constellation are engaged in the positioning solution at a specific epoch, the pseudorange measurement residual matrix \(\mathbf{H}\) is computed. \[\mathbf{H}=\left[ \begin{array}{cccc}\frac{x_{1}-x}{R_{1}}&\frac{y_{1}-y}{R_{1}}& \frac{z_{1}-z}{R_{1}}&1\\ \frac{x_{1}-x}{R_{2}}&\frac{y_{2}-y}{R_{2}}&\frac{z_{2}-z}{R_{2}}&1\\ \ldots&\ldots&\ldots&1\\ \frac{x_{n}-x}{R_{n}}&\frac{y_{n}-y}{R_{n}}&\frac{z_{n}-z}{R_{n}}&1\end{array} \right] \tag{8}\] \begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Type Num** & **Position** & **Type** & **Code** & \
\begin{tabular}{l} **Minimum** \\ **Period/\(t^{*}\)** \\ \end{tabular} &
\begin{tabular}{l} **Maximum** \\ **Period/\(t^{*}\)** \\ \end{tabular} \\ \hline
1 & Moon & DRO & DRO & 2.7261 & 5.9373 \\
2 & & Lyapunov & L1L & 2.6930 & 7.4280 \\
3 & L1 & North Halo & L1 NH & 1.8037 & 2.7875 \\
4 & & South Halo & L1 SH & 1.8037 & 2.7875 \\
5 & & Vertical & L1V & 2.7695 & 5.6891 \\
6 & & Lyapunov & L2L & 3.3735 & 6.1671 \\
7 & & North Halo & L2 NH & 1.3739 & 3.4155 \\
8 & & South Halo & L2 SH & 1.3739 & 3.4155 \\
9 & & Vertical & L2V & 3.5177 & 5.7857 \\
10 & & Lyapunov & L3L & 6.2184 & 6.2272 \\
11 & & North Halo & L3 NH & 6.2356 & 6.2391 \\
12 & & South Halo & L3 SH & 6.2356 & 6.2391 \\
13 & & Vertical & L3V & 6.2499 & 6.2502 \\
14 & & Planar & L4P & 6.2832 & 6.2869 \\
15 & & Vertical & L4V & 6.5391 & 6.5827 \\
16 & & Planar & L5P & 6.2832 & 6.2869 \\
17 & & Vertical & L5V & 6.5391 & 6.5827 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Specification and coding of orbit types in the Operational Orbit Library. The first three columns of the matrix represent the components of the unit vector from the receiver to each constellation satellite, while the last element in each row corresponds to the partial derivative of the pseudorange with respect to the user clock bias. The covariance matrix is computed via the least squares method:  
\[\mathbf{Q}=\left(\mathbf{H}^{T}\mathbf{H}\right)^{-1}=\left[ \begin{array}{ cccc}\sigma_{x}^{2}&\sigma_{xy}&\sigma_{xz}&\sigma_{xt}\\ \sigma_{xy}&\sigma_{y}^{2}&\sigma_{yz}&\sigma_{yt}\\ \sigma_{xz}&\sigma_{yz}&\sigma_{z}^{2}&\sigma_{zt}\\ \sigma_{xt}&\sigma_{yt}&\sigma_{zt}&\sigma_{t}^{2}\\ \end{array} \right] \tag{9}\]  
In navigation constellations, the Position Dilution of Precision (PDOP) and GDOP are key metrics, denoted as \(D_{p}\) and \(D_{g}\) respectively.