## 5 Initial Orbit Determination Methods  
In the initial stages of orbit determination, challenges arise due to the limited field of view of monitoring camera and the high relative velocity between space targets and monitoring platforms. A single monitoring spacecraft often struggles to maintain continuous tracking of the target motion. Consequently, the estimation of a space target orbit is usually based on limited observational data, a technique referred to as short-arc orbit determination [53]. Representing the orbit of a space target uniquely requires six orbital \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Developers** & **Characteristics** & **Inputs** & **Outputs** & **Main Equations** \\ \hline \multirow{3}{*}{[PERSON] et al. [45]} & \multirow{3}{*}{\begin{tabular}{c} Regard the gravity as a constraint on \\ the motion of targets and utilized \\ the orbital altitudes as a constraint to \\ construct the feasible domain \\ \end{tabular} } & \multirow{3}{*}{\begin{tabular}{c} Arc of observations \\ and the \\ corresponding optical \\ attributable \\ \end{tabular} } & \multirow{3}{*}{\begin{tabular}{c} Admissible region \\ of the orbit of \\ target \\ \end{tabular} } & \multirow{3}{*}{\(\mathcal{C}=\mathcal{C}_{1}\cap\mathcal{C}_{2}\)} \\  & & & \\ \cline{1-1} \cline{4-5}  & & & \\ \cline{1-1} \cline{4-5}  & & & \\ \cline{1-1} \cline{4-5}  & & & \\ \hline \multirow{3}{*}{\begin{tabular}{c} [PERSON] \\ et al. [46] \\ \end{tabular} } & \multirow{3}{*}{\begin{tabular}{c} Simplified the association process \\ and further limited the size of \\ admissible region in the range \\ range-rate plane \\ \end{tabular} } & \multirow{3}{*}{\
\begin{tabular}{c} Arc of observations \\ and the \\ corresponding optical \\ attribute \\ \end{tabular} } & \multirow{3}{*}{
\begin{tabular}{c} Admissible region \\ of the orbit of \\ target \\ \end{tabular} } & \multirow{3}{*}{\(\mathcal{C}=\sum\limits_{i=1}^{A}\mathcal{C}_{i}\)} \\ \cline{1-1} \cline{4-5}  & & & \\ \cline{1-1} \cline{4-5}  & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the orbit association methods mentioned in this section. elements. However, a single optical observation can only provide information about two angles of the target and their corresponding times. Hence, achieving short-arc orbit determination during the initial phase typically requires obtaining six sets of observations, including angles and times from three separate instances as depicted in Figure 5.  
The most classic algorithms for initial orbit determination are the Gauss method and the Laplace method shown in Table 3[54; 55]. Among them, the Gauss method initiates by formulating a system of equations for target positioning, involving 18 unknowns. This process leverages the geometric relationship between the observation platform and the target, along with the spatial target position vector expression based on Lagrange coefficients and the conservation theorem of angular momentum. Finally, the Gauss method utilizes multiple observations, incorporating target angle information and the position of the observation platform, to solve these equations [56]. However, the accuracy of the Gauss method diminishes when the curvature of the observed arc is low since the curvature of the observed trajectory is used as a divisor during the computation, and it may encounter singularity and instability problems during its usage [50]. Considering the impact of observational errors, results obtained by the Gauss method may face challenges in subsequent differential corrections for least squares fitting, potentially resulting in divergence in the orbit determination process [57]. On the other hand, the Laplace method solves the geocentric distance of the target iteratively by utilizing the azimuth of the observed vector, the azimuthal velocity, and the angular acceleration derived differentially from the observed data.