# Non-Cooperative Target Ranging Based on High-Orbit Single-Star Temporal-Spatial Characteristics
## 2 Establishment of the Distance Measurement Model  
To obtain monocular visual distance measurements between a host satellite and non-cooperative targets in high Earth orbit, a model for measuring the distance between satellite S and target satellite T needs to be established. Satellite S is assumed to be equipped with a star sensor camera capable of real-time tracking and pointing toward the target satellite T. The star sensor camera's star-sensitive function provides the rotation matrix M from the celestial coordinate system (which aligns the Z-axis with the Earth's rotation axis) to the star sensor coordinate system (the star sensor camera can obtain the rotation matrix \(\mathbf{M}\) through star map matching) [18; 19]. The rotation matrix \(\mathbf{M}_{\mathrm{J2000}}\) from the star sensor coordinate system to the celestial coordinate system is calculated through matrix transposition, as shown in Equation (1). \[\mathbf{M}_{\mathrm{J2000}}=\mathbf{M}^{\mathrm{T}} \tag{1}\]  
In this paper, the superscript T is used to denote the transpose operation, and the subscript T or t is used to represent the target satellite. Calculating the pointing vector \(\mathbf{v_{T}}\) of the onboard camera's optical axis to the target satellite T based on the rotation matrix \(\mathbf{M}_{\mathrm{J2000}}\) from the star sensor coordinate system to the celestial coordinate system can be achieved using Equation (2). \[\mathbf{v_{T}}=\mathbf{M}_{\mathrm{J2000}}\mathbf{U} \tag{2}\]  
where \(\mathbf{U}\) represents the vector \(\left[0,\ 0,\ 1\right]^{T}\). As shown in Figure 2, [0, 0, 1] represents the unit vector in the z-direction of the camera within the star sensor coordinate system. This vector  
Figure 1: Schematic of principle of binocular ranging. is transformed into the pointing vector of the camera axis to the target within the celestial coordinate system through the MJ2000 matrix. During the ranging process, both satellites are assumed to maintain their respective orbits. The positional changes that occur during the motion of the two satellites are illustrated in Figure 3, in which OXYZ represents the celestial coordinate system. This assumes the movement positions of satellite \(S\) in its orbit as \(S\)1\(\rightarrow\)_S_2\(\rightarrow\)_S_3, and simultaneously, the movement positions of target satellite \(T\) in its orbit as \(T\)1\(\rightarrow\)_T_2\(\rightarrow\)_T_3. Satellite \(S\) moves from position \(S\)1 to position \(S\)2, with the coordinates corresponding to these positions denoted as \(S\)1 (\(x_{\text{s1}},y_{\text{s1}},z_{\text{s1}}\)) and \(S\)2 (\(x_{\text{s2}},y_{\text{s2}},z_{\text{s2}}\)). When satellite \(S\) is at positions \(S\)1 and \(S\)2, the assumed coordinates for the corresponding positions of target satellite \(T\) are \(T\)1 (\(x_{\text{t1}},y_{\text{t1}},z_{\text{t1}}\)) and \(T\)2 (\(x_{\text{t2}},y_{\text{t2}},z_{\text{t2}}\)). At position \(S\)1, satellite \(S\) points to the position of target satellite \(T\) at \(T\)1; similarly, at position \(S\)2, satellite S points to the position of target satellite \(T\) at \(T\)2. The corresponding pointing vectors are \(\mathbf{v_{1}}(v_{\text{v1}},v_{\text{s1}},v_{\text{s1}})\) and, \(\mathbf{v_{2}}(v_{\text{v2}},v_{\text{s2}},v_{\text{s2}})\), which can be calculated using Equations (3) and (4) derived from Equations (1) and (2), respectively:  
\[\mathbf{v_{1}^{T}} =\mathbf{M}\mathbf{I}_{\text{J2000}}\mathbf{U} \tag{3}\] \[\mathbf{v_{2}^{T}} =\mathbf{M}\mathbf{I}_{\text{J2000}}\mathbf{U} \tag{4}\]  
where \(\mathbf{M}\mathbf{I}_{\text{J2000}}\) and \(\mathbf{M}\mathbf{I}_{\text{J2000}}\) are the rotation matrices from the star sensor coordinate system to the celestial coordinate system at positions \(S\)1 and \(S\)2, respectively. Assuming the unknown coefficients \(k_{1}\) and \(k_{2}\), we construct the vector equation \(\mathbf{v_{1}}\) for satellite \(S\) pointing to target satellite \(T\) at position \(T\)1 when \(S\) is at position \(S\)1 and perform a similar construction for vector equation \(\mathbf{v_{2}}\).