# Autonomous Navigation Based on the Earth-Shadow Observation near the Sun-Earth L2 Point
###### Abstract  
This paper is devoted to a novel autonomous navigation method for spacecraft around the Sun-Earth L2 point. In contrast to the previous navigation methods, which rely on ground-based or inter-spacecraft measurements, the proposed method determines the orbit based on Earth-shadow measurements. First, the navigation framework using the Earth-shadow measurement is proposed. Second, the geometric analysis is used to derive the mathematical model of the Earth-shadow measurements. Then, the fifth-degree Cubature Kalman filter (CKF) is designed to estimate the states of the spacecraft. Numerical simulations are implemented to validate the performance of the proposed navigation method. Finally, the simulation results show that the navigation system is observable and that the proposed method could be potentially useful for an autonomous navigation mission near the Sun-Earth L2 point in the future. Sun-Earth L2 point; three-body problem; autonomous navigation; cubature Kalman filter +
Footnote †: journal: Appl. Sci. for future Earth-Moon libration point spacecraft missions, including the CE-4 relay satellite in the L2 region [9]. Nowadays, collinear Lagrange points of the restricted three-body problem are receiving increasing attention because of their special dynamical properties and unique geometrical positions [10; 11; 12]. There are five Lagrange points in the Sun-Earth three-body system. This study focuses on navigation around the Lagrange point L2. Because orbital motion near the Lagrange point L2 is extremely sensitive to the initial value, the small error will be quickly amplified. The spacecraft needs to conduct orbital maneuvers frequently, otherwise, it will deviate from the target area soon. Therefore, the navigation needs to meet high-precision requirements [13; 14; 15]. [PERSON] et al. [16] proposed an autonomous navigation method for deep space spacecraft, designed for measuring the attitude and azimuth of celestial bodies. This method is to determine the spacecraft state using the optical measurement data processed with the least-square-degree Kalman filter algorithm. Focusing on the spacecraft navigation technology, [PERSON] et al. [17] conducted a study about the observability of relative navigation in two-body dynamics by line-of-sight measurement under the influence of J2 perturbation. The results show that the performance of the navigation system can be improved by selecting the orbital parameters properly. [PERSON] et al. [18] investigated the navigation method using space-based angle measurement data. This method calculates the orbital parameters by the Direct Method or Newton method based on the initial value and proves that the spacecraft orbit can be determined by the space-based angle measurement data. [PERSON] et al. [19] considered the relative motion and estimated the parameters of noncooperative targets from the perspective of the nonlinearly constrained optimization problem. The author also proposed a monocular vision method and verified its effectiveness through numerical simulation. [PERSON] [20] took the space-based monitoring system as the research background and analyzed the space-based detectability of noncooperative targets. The author also improved the filtering algorithms and key technologies for initial navigation and combined navigation of space-based noncooperative targets, which are based on the angle measurement information. [PERSON] [21] discussed the autonomous optical navigation technology of the deep space spacecraft, designed the image processing algorithm and information extraction algorithm, and analyzed the observability of the system by the numerical analysis method. The method of determining the relative orbit of two spacecraft by measuring their relative distance is commonly used. Although a great deal of the distance measurement method was put forward earlier, it is difficult to achieve high-precision requirements in a complex space environment.