## 3 Construction of Mathematical Model
### Dynamical Constraint Equations for Non-Cooperative Targets in Cislunar Space  
The research scenario in this paper focuses on a \"four-body\" problem: the Earth, the Moon, BDS-3 satellites, and non-cooperative targets in cislunar space. For the convenience of study, the problem can be reduced to a \"three-body\" problem (which neglects the BDS-3 satellite as it has a negligible gravitational force on the targets due to its small \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbols** & **Description** \\ \hline \(m_{earth}\) & Mass of the Earth \\ \(m_{mono}\) & Mass of the Moon \\ \(x_{1},y_{1},z_{1}\) & \(x,y,z\)-axis position coordinates of the \(l\)th cislunar targets \\ \(\delta\)_d_ & \(\delta\)_d_ \\ \(\tilde{x}_{1},y_{1},z_{1}\) & \(x,y,z\)-axis velocity components of the \(l\)th cislunar targets \\ \(dd\)_dd_ & \(dd\)_dd_ \\ \(x_{1},y_{1},z_{1}\) & cislunar targets \\ \(\tilde{x}_{1},y_{1},z_{1}\) & position vector of BDS satellites \\ \(\mathbf{r}_{space}\) & position vector of targets \\ \(\delta\tilde{t}_{iBD}\) & clock difference of BDS satellites relative to BDT \\ \(\delta\tilde{t}_{space}\) & clock difference of targets relative to BDT \\ \(\tau_{iBD}\) & hardware delay of BDS satellites \\ \(e_{(lBD\to space)}\) & other errors to be corrected \\ \(n_{i(BD\to space)}\) & observation noise \\ \(L_{i(BD\to space)}\) & interplanetary link transmission losses between the BDS satellites and targets \\ \(L_{i(space\to CEI)}\) & interplanetary link transmission losses between the targets and the ground-based CEI receiving array \\ \(d_{i(BD\to space)}\) & the distances between the GNSS satellite and the targets \\ \(d_{i(space\to CEI)}\) & the distances between the targets and the ground-based \\ \(\lambda_{iBD}\) & the carrier wavelength of the GNSS satellite \\ \(f_{iBD}\) & interplanetary link \\ \(\mathbf{w}_{i(BD\to space)}^{antenna}\) & the antenna phase center offset of Beidou-3 satellites in the Solid Coordinate System(SCS) \\ \(\left(\mathbf{p}_{x},\mathbf{p}_{y},\mathbf{p}_{z}\right)\) & the projection of \(x,y,z\) in the Geocentric Inertial \\ Coordinate System under the SCS \\ \(c_{dry},c_{net}\) & the zenith delay correction models for the dry and wet \\ \(m_{dry}(\varphi)\), \(m_{net}(\varphi)\) & delay components \\ \(\varphi\) & \(\mathbf{\varphi}\) \\ \(\mathbb{C}\) & observed elevation angle of the CEI array to targets \\ \(N_{0}\) & thermal noise power \\ \(C/N_{0}\) & the carrier-to-noise ratio \\ \(T_{\text{transmit}}\) & GNSS interstellar link signal transmit power \\ \(G_{\text{transmit}}(BD\to space)\), \(G_{\text{receive}(space\to CEI)}\) & antenna gain of the GNSS interstellar link transmit and the CEI array receive toward the targets \\ \(T_{antenna}\), \(T_{amplifier}\) & antenna and amplifier noise temperature \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main symbols used in this study. weight): the Earth, the Moon, and the non-cooperative spacecraft in cislunar space, and all three are considered as mass points. The corresponding dynamical constraint diagram of the non-cooperative spacecraft in cislunar space is shown in Figure 5. The line between the Earth and the Moon is used as the \(X\)-axis, and the origin of the coordinates is the center of gravity of both. From the figure, it can be seen that the distances \(\lambda\) and \(1-\lambda\) of the Earth and the Moon from their respective centers of gravity satisfy:  
\[\lambda=\frac{m_{moon}}{m_{earth}+m_{moon}} \tag{1}\]  
\[1-\lambda=\frac{m_{earth}}{m_{earth}+m_{moon}} \tag{2}\]  
where \(m_{earth}\) and \(m_{moon}\) are the masses of the Earth and the Moon, respectively. In addition, from the figure, the distances between the Earth and the Moon and the \(l\)th target satisfy:  
\[r_{es}=\sqrt{\left(x_{l}+\lambda\right)^{2}+{y_{l}}^{2}+{z_{l}}^{2}} \tag{3}\]  
\[r_{ms}=\sqrt{\left(x_{l}+1-\lambda\right)^{2}+{y_{l}}^{2}+{z_{l}}^{2}} \tag{4}\]  
where \(x_{l},y_{l},z_{l}\) are the \(x,y,z\)-axis position coordinates of the \(l\)th cislunar space non-cooperative targets, respectively.