## 2 Uvm  
By imaging the target area, an optical sensor can obtain the unit relative position vector of the target according to the target's position relative to a background star catalogue. Other measurement models exist for goniometric systems [21; 22; 23]. In this paper, we assume that a measurement can obtain the right ascension \(\alpha_{t}\) and the declination \(\delta_{t}\) centred on the optical sensor as two independent values, and the measurement error follows the Gaussian distribution [20]. The measured data are defined as follows:  
\[\theta=[\alpha_{t},\delta_{t}]_{t=1,2,3\ldots} \tag{1}\]  
The relationship between the position vector and velocity vector at different time \(t\) is expressed by Lagrangian coefficients \(f_{t}\) and \(g_{t}\). Thus, the relationship between the position of the optical sensor and the target at time \(t\) can be expressed as:  
\[f_{t}\mathbf{r}_{0}+g_{t}\mathbf{v}_{0}=\mathbf{R}_{t}+\rho_{t}\mathbf{L}_{t} \tag{2}\]  
where \(\mathbf{r}_{0}\) and \(\mathbf{v}_{0}\) are the position vector and velocity vector at time \(t_{0}\), respectively; \(\mathbf{R}_{t}\) is the position vector of the optical sensor relative to the centre of the Earth; \(\rho_{t}\) is the distance between the optical sensor and the target; and \(\mathbf{L}_{t}\) is the unit line-of-sight vector of the target relative to the optical sensor, and can be expressed by the observation vector as:  
\[\mathbf{L}_{t}=\begin{bmatrix}\cos\delta_{t}\cos\sigma_{t}\\ \cos\delta_{t}\sin\sigma_{t}\\ \sin\delta_{t}\end{bmatrix} \tag{3}\]  
As defined in the UVM, \(\mathbf{a}_{t}\) and \(\mathbf{b}_{t}\) can be expressed as:  
\[\begin{split}\mathbf{a}_{t}&=\begin{bmatrix}-\sin\alpha_{t }\\ \cos\alpha_{t}\\ 0\end{bmatrix}\\ \mathbf{b}_{t}&=\begin{bmatrix}-\sin\delta_{t}\cos\alpha_{t}\\ -\sin\delta_{t}\sin\alpha_{t}\\ \cos\delta_{t}\end{bmatrix}\end{split} \tag{4}\]  
Thus, the equation for determining the orbit, where the position and velocity vectors are linear, is:  
\[\frac{1}{\rho_{t}}\begin{bmatrix}f_{t}\mathbf{a}_{t}{}^{T}&g_{t}\mathbf{a}_{t}{}^{T} \\ f_{t}\mathbf{b}_{t}{}^{T}&g_{t}\mathbf{b}_{t}{}^{T}\end{bmatrix}\begin{bmatrix}\mathbf{ r}_{0}\\ \mathbf{v}_{0}\end{bmatrix}=\frac{1}{\rho_{t}}\begin{bmatrix}\mathbf{a}_{t}{}^{T}\mathbf{R}_{ t}\\ \mathbf{b}_{t}{}^{T}\mathbf{R}_{t}\end{bmatrix}_{t=1,2,3\ldots} \tag{5}\]  
Assuming that \(f_{t}\) and \(g_{t}\) are independent of \(\mathbf{r}_{0}\) and \(\mathbf{v}_{0}\), the least-squares solutions of \(\mathbf{r}_{0}\) and \(\mathbf{v}_{0}\) are:  
\[\mathbf{X}=\begin{bmatrix}\mathbf{r}_{0}\\ \mathbf{v}_{0}\end{bmatrix}=A^{-1}B \tag{6}\]where \(\mathbf{X}\) is the state vector of the target, and \(A\) and \(B\) are defined as [21]:  
\[\begin{array}{l}A=\sum\limits_{t}\left(\frac{1}{\rho_{t}^{2}}\left[\begin{array} []{cc}f_{t}\mathbf{a}_{t}^{\mathrm{T}}&g_{t}\mathbf{a}_{t}^{\mathrm{T}}\\ f_{t}\mathbf{b}_{t}^{\mathrm{T}}&g_{t}\mathbf{b}_{t}^{\mathrm{T}}\end{array}\right]^{ \mathrm{T}}\left[\begin{array}{cc}f_{t}\mathbf{a}_{t}^{\mathrm{T}}&g_{t}\mathbf{a}_{t }^{\mathrm{T}}\\ f_{t}\mathbf{b}_{t}^{\mathrm{T}}&g_{t}\mathbf{b}_{t}^{\mathrm{T}}\end{array}\right] \right)\\ B=\sum\limits_{t}\left(\frac{1}{\rho^{2}}\left[\begin{array}{cc}f_{t}\mathbf{a}_ {t}^{\mathrm{T}}&g_{t}\mathbf{a}_{t}^{\mathrm{T}}\\ f_{t}\mathbf{b}_{t}^{\mathrm{T}}&g_{t}\mathbf{b}_{t}^{\mathrm{T}}\end{array}\right]^{ \mathrm{T}}\left[\begin{array}{cc}\mathbf{a}_{t}^{\mathrm{T}}\mathbf{R}_{t}\\ \mathbf{b}_{t}^{\mathrm{T}}\mathbf{R}_{t}\end{array}\right]\right)\end{array} \tag{7}\]  
Assuming \(f_{t}=1\), \(g_{t}=t\), and \(\rho_{t}=1\) to be the first iteration, the initial value can be calculated. The values of \(f_{t}\), \(g_{t}\), and \(\rho_{t}\) are updated with the results of the previous iteration, forming an iterative process. It is worth noting that the effect of disturbances can be disregarded when the observed arc is short. The system error in solving \(f_{t}\) and \(g_{t}\) numerically with closed-form orbital mechanics relations is much smaller than when using a time series.