# Design of Cislunar Navigation Constellation via Orbits with a Resonant Period
## 2 Fundamental Theory
### Target Area Definition and Gridding  
To compute performance metrics, spatial gridding is implemented, where the performance values at the grid nodes approximate those in their vicinity. Target areas can be defined using either cuboidal spatial grids or spherical spatial grids, as follows:  
* Cuboidal grids are typically applied to global spatial domains (e.g., the entire Earth-Moon system).
* Spherical grids are suited for local regions (e.g., near the Earth or the Moon).  
The cuboidal spatial grid model is defined within the Earth-Moon rotating frame, with coordinate ranges along the three axes specified to delineate the spatial domain. For instance, the X-axis extends from the initial coordinate \(X_{0}\) towards \(X_{f}\). A schematic of the gridded cuboidal spatial domain is illustrated in Figure 1.  
For such a cuboidal spatial grid, uniform discretization can be performed along each axis with specified step sizes. For example, the step size in the X-direction is denoted as \(dX\). The coordinate set \(P\) of all grid points is generated as follows:  
\[P=\left\{(x,y,z)\big{|}x=X_{0}+n_{x}\cdot dX,y=Y_{0}+n_{y}\cdot dY,z=Z_{0}+n_{z }\cdot dZ\right.\big{\}} \tag{12}\]  
where \(n_{x}=0,1,\ldots,\left\lfloor\frac{X_{f}-X_{0}}{dX}\right\rfloor\), \(n_{y}=0,1,\ldots,\left\lfloor\frac{Y_{f}-Y_{0}}{dY}\right\rfloor\), \(n_{z}=0,1,\ldots,\left\lfloor\frac{Z_{f}-Z_{0}}{dZ}\right\rfloor\). Here, \(\left\lfloor\cdot\right\rfloor\) denotes the floor function. The cuboidal grid model enables the description of cuboidal subregions centered at any spatial point within a unified coordinate system. Let the center point coordinates be \(P_{0}=(x_{0},y_{0},z_{0})\); the coordinate set is then updated as follows:  
\[P=\left\{(x,y,z)\big{|}x=X_{0}+n_{x}\cdot dX+x_{0},y=Y_{0}+n_{y}\cdot dY+y_{0},z=Z_{0}+n_{z}\cdot dZ+z_{0}\right.\big{\}} \tag{13}\]  
The spherical spatial grid is defined within a rotating coordinate frame using longitude, latitude, and radial distance. A schematic of the spherical gridding is shown in Figure 2.  
Within this framework, longitude is defined as \(\alpha\in[0,360]^{\circ}\) and latitude as \(\delta\in[-90,90]^{\circ}\). To define the target spherical region, specify the spherical radius \(R\), longitude bounds (start \(\alpha_{0}\), end \(\alpha_{f}\)), and uniform angular step \(d\alpha\). Similarly, latitude bounds (start \(\delta_{0}\); end \(\delta_{f}\)) and step \(d\delta\) must be defined. The spherical grid coordinate set \(P\) is then generated as follows:  
\[P=\{(\alpha,\delta,R)|\alpha=\alpha_{0}+n_{\alpha}\cdot d\alpha,\delta=\delta_ {0}+n_{\delta}\cdot d\delta\} \tag{14}\]  
To facilitate computational analysis, spherical coordinates (longitude and latitude) must be converted to Cartesian coordinates. Assuming the spherical center is located at \(P_{0}=(x_{0},y_{0},z_{0})\), the transformation formulas are as follows:  
\[P=\{(x,y,z)|x=R\cos\delta\cos\alpha+x_{0},y=R\cos\delta\sin\alpha+y_{0},z=R\sin \delta+z_{0}\} \tag{15}\]  
This study focuses on two key regions: the Earth-centered spherical region and the Moon-centered spherical region within the Earth-Moon rotating frame. The Earth's position in this frame is defined as \(\mathbf{r}_{e}=(-\mu,0,0)\), with \(R_{e}\) denoting the radius of the Earth-centered spherical region. Similarly, the Moon's position is \(\mathbf{r}_{m}=(1-\mu,0,0)\), and \(R_{m}\) represents the radius of the Moon-centered spherical region. These definitions enable localized navigation analyses in near-Earth and lunar proximity environments using a unified coordinate framework.