# Autonomous Orbit Determination of LLO Satellite Using DRO-LLO Links and Lunar Laser Ranging
## 2 Measurement Model  
### Inter-Satellite Ranging Model  
The inter-satellite ranging model is a bidirectional one-way measurement model, in which spacecraft at both ends of the inter-satellite link send and receive measurement signals from each other. By measuring the difference in signal propagation time and frequency, the distance and clock information between two spacecraft can be obtained. Taking a DRO satellite and an LLO satellite as examples, Figure 3 shows a schematic diagram of the bidirectional one-way ranging model. The bidirectional link measurement models for satellite D (DRO satellite) and satellite L (LLO satellite) can be represented as follows [7; 17; 25]:  
\[ \begin{split} R_{\text{D}}^{\text{L}}(t_{\text{D}})& =\left|r_{\text{D}}(t_{\text{D}})-r_{\text{L}}(\widetilde{t}_{\text{D}}) \right|+c\cdot\left(D_{\text{D}}^{\text{R}}+D_{\text{L}}^{\text{S}}\right)+c \cdot\left(\delta\tau_{\text{D}}(t_{\text{D}})-\delta\tau_{\text{L}}(\widetilde {t}_{\text{D}})\right)+\epsilon_{\text{D}}\\ R_{\text{L}}^{\text{D}}(t_{\text{L}})&=\left|r_{ \text{L}}(t_{\text{L}})-r_{\text{D}}(\widetilde{t}_{\text{L}})\right|+c\cdot \left(D_{L}^{\text{R}}+D_{D}^{\text{S}}\right)+c\cdot\left(\delta\tau_{L}(t_{ \text{L}})-\delta\tau_{\text{D}}(\widetilde{t}_{\text{L}})\right)+\epsilon_{ \text{L}}\end{split}  \tag{1}\]  
where \(R_{\text{D}}^{\text{L}}(t_{\text{D}})\) is the pseudo range received by the DRO satellite at time \(t_{\text{D}}\) and transmitted by the LLO satellite at time \(\widetilde{t}_{\text{D}}\); \(R_{\text{L}}^{\text{D}}(t_{\text{L}})\) is the pseudo range received by the LLO satellite at time \(t_{\text{L}}\) and transmitted by the DRO satellite at time \(\widetilde{t}_{\text{L}}\), and it is assumed that two satellites receive the ranging signal at the same time; \(r_{\text{D}}\) and \(r_{\text{L}}\) are the position vectors of the DRO satellite and the LLO satellite, respectively; \(c\) is the speed of light; \(D_{\text{D}}^{\text{R}}\) and \(D_{\text{L}}^{\text{R}}\) are the device delay for the DRO satellite and the LLO satellite to receive the pseudo range, \(D_{\text{D}}^{\text{S}}\) and \(D_{\text{D}}^{\text{S}}\) are the device delay for the DRO satellite and the LLO satellite to send the pseudo range; \(\delta\tau_{\text{D}}\) and \(\delta\tau_{\text{L}}\) are the clock bias of the DRO satellite and the LLO satellite, respectively; and \(\epsilon_{\text{D}}\) and \(\epsilon_{\text{L}}\) are the measurement thermal noise of the DRO satellite and the LLO satellite receiving the pseudo range, and the measurement thermal noise is modeled as Gaussian white noise. Figure 2: Autonomous orbit determination scheme for the DRO-LLO satellite formation in the Earth–Moon rotating system. By summing and combining the bidirectional one-way measurement model, clock-related terms can be eliminated, achieving the decoupling of distance information and clock information. \[ \begin{array}{ll}z_{\text{com}}&=\frac{R_{\text{D}}^{\text{L}}(t_{\text{D}})+R_ {\text{L}}^{\text{D}}(t_{\text{L}})}{2}\\ &=\frac{1}{2}\cdot\left[\left|\mathbf{r}_{\text{D}}(t_{\text{D}})-\mathbf{r}_{\text{L}} \left(\widetilde{t}_{\text{D}}\right)\right|+\left|\mathbf{r}_{\text{L}}(t_{\text {L}})-\mathbf{r}_{\text{D}}\left(\widetilde{t}_{\text{L}}\right)\right|+c\cdot D_{ \text{com}}+\right.\\ &\left.c\cdot\left(\delta\tau_{\text{D}}(t_{\text{D}})-\delta\tau_{\text{L}} \left(\widetilde{t}_{\text{D}}\right)+\delta\tau_{\text{L}}(t_{\text{L}})- \delta\tau_{\text{D}}\left(\widetilde{t}_{\text{L}}\right)+\varepsilon_{\text {D}}+\varepsilon_{\text{L}}\right)\right]\\ &=\frac{1}{2}\cdot\left[\rho_{\text{D}}^{\text{L}}+\rho_{\text{L}}^{\text{D} }+c\cdot D_{\text{com}}+c\cdot\left(\Delta\delta\tau_{\text{L}}+\Delta\delta \tau_{\text{D}}\right)+\varepsilon_{\text{D}}+\varepsilon_{\text{L}}\right] \end{array}