# Non-Cooperative Target Ranging Based on High-Orbit Single-Star Temporal-Spatial Characteristics
## 1 Introduction  
For situational awareness in space, perceptual measurements of threatening targets, such as space debris, are important safety measures [1; 2; 3]. In particular, distance measurements of non-cooperative threat targets are critical [4; 5]. Visual ranging between two high-orbit satellites requires the host satellite to continuously track and measure the distance to the perceived non-cooperative target satellite [6].  
Traditional visual ranging methods include monocular and binocular visual ranging. Monocular visual ranging typically uses point correspondence calibration to obtain depth information from images [7; 8]. This method deciphers the transformation relationship between corresponding points in different coordinate systems. As point correspondence calibration requires a predetermined camera position and orientation, any parameter change necessitates recalibration, limiting the use of method to fixed camera positions [9; 10; 11]. The applicability of cameras mounted on moving platforms (e.g., a camera mounted on a satellite) is restricted because of changes in camera parameters during motion, which impede accurate ranging.  
For spaceborne monocular cameras, the continuous tracking of non-cooperative targets is necessary to ensure that the target remains at the center of the camera [12]. However, pixel displacement cannot be guaranteed. Moreover, because this process involves imaging with a single camera, the construction of a binocular model is unachievable [13], which restricts the application of ranging algorithms.  
Binocular visual ranging utilizes pixel offsets of the same object on imaging planes to mathematically determine the distance between objects using the camera focal length, pixel offsets, and the actual distance between the two cameras [14; 15; 16]. The principle of binocular ranging is shown in Figure 1, where \(D\) represents the required distance, \(f\) denotes the camera focal length, \(B\) indicates the center distance between the two cameras,\(P\) represents a point on the object, and \(O_{1}\) and \(O_{2}\) are the optical centers of the two cameras. The projections of object point P on the imaging planes of the two cameras are P1 and P2, respectively. By applying the principle of similar triangles (\(\triangle\)PO\({}_{1}\)O\({}_{2}\)\(\sim\)\(\triangle\)PP\({}_{1}\)P\({}_{2}\)), the distance can be calculated using the formula \(D=fB/\left(X_{1}-X_{2}\right)\), where the focal length (\(f\)) and center distance (\(B\)) can be obtained through calibration. Therefore, the distance value can be obtained by determining the value of parallax \(d=X_{1}-X_{2}\)[17].  
In this study, to overcome the limitations of previous methods, we utilized the characteristics of high-orbit satellites in the following ways: (1) spatially--they have low eccentricity and a small inclination angle, which simplifies the motion model of non-cooperative target satellites; (2) temporally--they use the short-term motion of the host satellite to slide and sample pointing vectors continuously to construct a set of equations; and (3), by combining the aforementioned small inclination and short sampling intervals, a further assumption can be made regarding the lack of movement in the Z-axis of the target satellite. Based on these considerations, a monocular visual distance measurement model for non-cooperative targets between high-orbit satellites is proposed.