# Autonomous Orbit Determination of LLO Satellite Using DRO-LLO Links and Lunar Laser Ranging
## 4 Autonomous Orbit Determination Simulation
### Analysis of Cramer-Rao Lower Bound  
In order to verify the observability of autonomous orbit determination, the Cramer-Rao Lower Bound (CRLB) analysis is conducted on the DRO-LLO autonomous orbit determination. The Cramer-Rao Lower Bound is a theoretical lower limit for measuring the accuracy of parameter estimation, which is the best estimation accuracy that unbiased estimators can achieve under optimal conditions [20]. So, for the real filtering results:  
\[\mathbf{P}\geq\mathbf{P}^{*} \tag{16}\]  
where \(\mathbf{P}\) is the error covariance matrix corresponding to the unbiased estimation of the parameter to be estimated, and \(\mathbf{P}^{*}\) is the lower limit of the CRLB. For the initial epoch:  
\[\mathbf{P}^{*}(t_{0})=\mathbf{P}^{-1} \tag{17}\]  
The calculation method of the CRLB covariance matrix for other epochs:  
\[\mathbf{P}^{*}(t_{k})^{-1}=\big{(}\mathbf{\Phi}(t_{k},t_{k-1})\mathbf{P}^{*}(t_{k})\mathbf{ \Phi}(t_{k},t_{k-1})^{\text{T}}\big{)}^{-1}+\mathbf{H}(t_{k})^{\text{T}}\mathbf{R}(t_{k}) \mathbf{H}(t_{k}) \tag{18}\]  
The main difference between the EKF and the CRLB is that the EKF uses the estimated orbit of the current epoch, while the CRLB uses the reference orbit and does not contain process noise [20]. In the DRO-LLO satellite formation scenario, an inter-satellite link is constructed between the DRO and the LLO, and a corner reflector is deployed on the lunar surface for the LLO satellite to perform lunar laser ranging. This simulation starts at 0:00 UTC on 1 January 2023 and ends at 0:00 UTC on 20 February 2023. The simulation experiments are divided into two groups. In one group, the laser ranging data are acquired throughout the whole period, while in the other group, the laser measurement is continuously carried out for 20 min every two hours. However, due to the obstruction of the Moon, the laser equipment in these two groups does not continuously carry out ranging to the lunar corner reflectors, but works intermittently. The standard deviation of the initial orbit error is set to 1000 m and 0.1 m/s, and Figure 6 shows the lower limit results of the CRLB for the DRO and LLO satellites. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline DRO process noise/(m\(\cdot\)s\({}^{-2}\)) & \(1\times 10^{-8}\) \\ LLO process noise/(m\(\cdot\)s\({}^{-2}\)) & \(1\times 10^{-7}\) \\ Inter-satellite ranging measurement noise/m & 0.5 [17] \\ Lunar laser ranging measurement noise/m & 0.02 [29] \\ Effective reflection angle of corner reflector/deg & 45 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Simulation setting conditions. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Reference Orbit** & **Simulation Settings** \\ \hline Earth gravity & 70 \(\times\) 70 EGM2008 & 70 \(\times\) 70 EGM2008 \\ Moon gravity & 50 \(\times\) 50 GL0660B & 50 \(\times\) 50 GL0660B \\ Three-body perturbation & Sun, Jupiter, Mars, and & Sun, Jupiter, Mars, and \\ Saturn & Saturn & Saturn \\ Solar pressure coefficient & 1.3 & Estimated \\ Area–mass & 0.02 & 0.02 \\ ratio/(m\({}^{2}\cdot\)kg\({}^{-1}\)) & & \\ Integrator & RungeKutta4 & RungeKutta4 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Force model and integrator settings. The CRLB results show that the following: 1 Within 4 days, the DRO position converges to below 100 m and the LLO position converges to below 10 m. 2 Within 30 days, the DRO position converges to 1 m and the LLO position converges to 0.1 m. 3 The CRLB results of full-time measurement and intermittent measurement differ very little. Based on the CRLB results, estimations of the orbital states of the DRO and the LLO using inter-satellite measurements and lunar laser ranging data are observable and have the potential to achieve a high level of accuracy. In practical engineering, the limitations of laser ranging will not have a significant impact on the orbit determination results. Additionally, the position standard deviation of the DRO and LLO satellites experiences minimal variation after 30 days.