## 6 Precise Orbit Determination Methods  
In the initial orbit determination phase, the perturbation factors are usually not taken into consideration [58]. To augment the accuracy of the predicted orbit of the target, it is imperative to iteratively refine the initial orbit based on subsequent observational data and in conjunction with a precise orbit dynamic model. Distinguished from the initial orbit determination, precise orbit determination involves optimizing the estimation of the initial states of space targets and parameters of the orbit dynamic model using a series of observations with associated errors. Essentially, it is a optimal state estimation problem [58]. The methods employed for solving precise orbit determination problems generally revolve around the concept of least squares estimation [79]. Building upon this concept, a series of estimation methods have been developed, including unbiased least squares \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Methods** & **Characteristics** & **Inputs** & **Outputs** & **Main Equations** \\ \hline \multirow{3}{*}{Gauss method} & The most classic method for initial orbit determination & The direction of three observations & \begin{tabular}{c} Geocentric \\ distance of the \\ target \\ \end{tabular} & \(\mathbf{r}_{2}=c_{1}\mathbf{r}_{1}+c_{3}\mathbf{r}_{3}\) \\ \hline \multirow{3}{*}{\begin{tabular}{c} Laplace \\ method \\ \end{tabular} } & Straightforward and practical & The direction of three observations & \begin{tabular}{c} Geocentric \\ distance of the \\ target \\ \end{tabular} & \(\mathbf{r}+\mathbf{R}=\rho\hat{\boldsymbol{p}}\) \\ \hline \multirow{3}{*}{\begin{tabular}{c} Proposed by \\ [PERSON] \\ et al. [64] \\ \end{tabular} } & Proposed method for & relative angle & \begin{tabular}{c} values of several \\ measurements \\ \end{tabular} & \begin{tabular}{c} \([i_{x},i_{y},i_{z}]^{T}=([x_{d}-a_{e}cos(\beta_{0})/2]^{T}_{i}+[y_{d}+a_{e} sin(\beta_{0})]^{T}_{i}+[z_{max}sin(\psi_{0})]\hat{k})\) \\ of the relative & \(/sqrt{\{a_{e}^{2}(1-3 cos^{2}(\beta_{0})/4)+\}}\) \\ orbit elements & \(a_{e}(2y_{d}sin(\beta_{0})-x_{d}cos(\beta_{0}))+x_{d}^{2}+y_{d}^{2}+z_{max} sin^{2}(\psi_{0}))\) \\ \hline \multirow{3}{*}{\begin{tabular}{c} Proposed by \\ [PERSON] \\ et al. [68; 69] \\ \end{tabular} } & Developed novel orbit determination method by & \begin{tabular}{c} three line-of-sight \\ observations for \\ relative motion \\ coating trajectory \\ to the center of \\ mass of an object \\ \end{tabular} & \begin{tabular}{c} \(k_{i}\hat{t}_{los}(i)=\) \\ \(\phi_{rr}(i)\{k_{0}\hat{t}_{los}(0)-\boldsymbol{d}(0)\}+\phi_{rr}(i)\) \\ \(\{\phi_{rr}^{T}(1)[k_{1}\hat{t}_{los}(1)-\) \\ \(\phi_{rr}(1)\{k_{0}\hat{t}_{los}(0)-\boldsymbol{d}(0)\}-\boldsymbol{d}(1)]\}+ \boldsymbol{d}(i)\) \\ \end{tabular} \\ \hline \multirow{3}{*}{\begin{tabular}{c} Proposed by \\ [PERSON] \\ et al. [70] \\ \end{tabular} } & Effectively addressed the nonlinear challenges in orbit determination by using machine & \
\begin{tabular}{c} three sets of \\ bearing angle and \\ the absolute orbit \\ state of the \\ observer \\ \end{tabular} &
\begin{tabular}{c} the initial \\ relative orbit \\ state of the \\ target \\ \end{tabular} & \(L=\dfrac{1}{m}\sum_{i=1}^{n}(y_{i}-\hat{y_{i}})^{2}+\dfrac{\lambda}{2n}\sum_ {\omega}\omega^{2}\) \\ \cline{1-1}  & & & (Loss Function) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of the mentioned initial orbit determination methods in this section. Figure 6: Schematic diagram of angle-only orbit determination based on multiple observation devices. estimation, standard least squares estimation, weighted least squares estimation, and least squares estimation. Typically, the methods grounded in the least squares framework for precise orbit determination begin with the initial orbit determination process. Then, the estimates are incorporated into the precise orbit determination model to update the calculation results iteratively to improve the accuracy.