# Distributed Localization of Non-Cooperative Targets in Non-Coplanar Rendezvous Processes
## 1 Introduction  
With the continuous exploration and utilization of the space environment by humans, a growing number of non-cooperative targets have emerged, including space debris, defunct satellites, and other orbital fragments. If these targets are not effectively tracked and managed, they pose a serious risk to the operational safety and sustainability of spacecraft in orbit. Therefore, research on the localization and rendezvous of these non-cooperative targets is of great significance. Most existing studies have been conducted under the assumption that a target and trackers are within the same orbital plane, and there has been little research on targets in non-coplanar orbits. However, in practice, most of these non-cooperative targets are in non-coplanar orbits. Achieving coplanar rendezvous requires significant orbital maneuvers to transfer the tracking spacecraft to the target's orbital plane. This process is complex, time-consuming, and inefficient for conducting multiple rendezvous missions concurrently. Therefore, the ability to directly intercept and track non-cooperative targets in non-coplanar orbits could greatly enhance the efficiency of space operations. This highlights the significant research value of developing methods for the tracking and localization of non-cooperative targets in non-coplanar orbits. The foundation for achieving precise rendezvous with non-cooperative targets lies in accurately localizing these targets throughout the rendezvous process. Typically, non-cooperative targets are tracked using non-collaborative sensors, such as optical systems or ranging devices [1]; however, for long-distance targets in non-coplanar orbits, the effectiveness of ranging devices is limited due to their restricted operational range. Given the need for lightweight, power-efficient solutions, microsatellites equipped with optical cameras present an optimal choice for this task. Current research on single-LOS positioning systems has demonstrated that such systems are poorly observable due to lack of distance information. In order to solve the observability problem, scholars have conducted extensive studies; the main methods are as follows: 1. the orbital maneuver method, 2. the camera bias method, 3[PERSON] the non-linear dynamics method, and 4. the multi-satellite multi-sensor method[PERSON] [PERSON] and [PERSON] [2] proposed the idea of improving observability by orbital maneuvering, which was further studied by [PERSON] [3], who proposed using potential relative distance information from orbital maneuvers to improve observability. However, the limitation of this method is that as the relative distance becomes greater, the observability becomes worse, rendering it unsuitable for long-range non-coplanar targets. [PERSON] [4] also proposed the camera bias method to indirectly introduce distance information, thereby enhancing LOS system observability. On the basis of this method, [PERSON] [5] analyzed the observability conditions for an angle-only relative navigation system applicable to close-range orbital rendezvous based on T-H equations. This method is also limited due to the need for a close range. Additionally, researchers such as [PERSON] [6], [PERSON] [7], [PERSON] [8], and [PERSON] [9] have studied non-linear dynamics methods to improve single-LOS system observability[PERSON] However, this class of methods is computationally intensive and the initial orbit determination consumes a lot of time, making it difficult to meet the requirements of rapidity. Multi-satellite collaboration can effectively solve the problem of poor observability in LOS positioning. The studies on this method can be divided into two main categories: The first includes studies on the relative navigation problem of close-range coplanar targets based on the C-W/T-H equations: Based on the dual-LOS-measurement relative navigation method, [PERSON] et al. [10] proposed an autonomous rendezvous approach guidance law with triangular constraints, which achieved close-range observation under the constraint of a no-fly zone.