# A Novel Analytical Approach for Spacecraft Fly-Around Formation Design with a Low-Thrust Maneuver
###### Abstract  
This paper investigates the fly-around formation between the servicer spacecraft and the target spacecraft. Inspired by the spacecraft orbital motion under the Earth's gravity, an intuitive, analytical guidance law for spatial fly-around formation design with the low-thrust maneuver is proposed. Beginning with the relative translational dynamics based on relative position and velocity, the control input of the guidance law is designed to contain two parts. The first part is the feed-forward term, which makes the relative dynamics a second-order integration model. The second part is the artificial gravity term, which has similar expressions to the Earth's gravity, and includes the artificial gravitational coefficient and the vector of the artificial gravity center. The above two parameters can be designed to determine the size, shape, and period of the fly-around trajectory. Specifically, three kinds of fly-around trajectories are discussed in detail. The first two are the spatial ellipses with the target spacecraft locating at the focus and the center of the ellipses, respectively. The third is the spatial circle. The proposed method can be easily extended to the design of planar fly-around formation, which is very systematic and comprehensive, and the fuel consumption of the control input is specifically discussed. Numerical simulations are conducted to demonstrate the efficiency of the proposed method. spacecraft fly-around; guidance law; low-thrust maneuver; artificial gravity +
Footnote †: journal:  
0000-0002-4000-0000-0000-000  
100 m and 50 m, respectively [8]. If the fly-around period is less than the orbital period of the target spacecraft, it is called fast fly-around, and otherwise, slow fly-around. The fast fly-around has great requirements for application in on-orbit missions. If the target spacecraft encounters an emergency, such as sudden failure, space debris impact, or its working state needing to be estimated from the outside in a short time, the fast fly-around can be used to quickly observe the target and obtain the necessary information. Similarly, slow fly-around also has great application requirements. A stable relative configuration of the target spacecraft or detailed observations of its global characteristics can be achieved through slow fly-around. The choice of fast or slow fly-around is mainly based on the mission requirements, but it is also necessary to consider the fuel consumption of the servicer spacecraft. Currently, the most researched is the use of the multi-impulse maneuver to achieve controlled fly-around. [PERSON], [PERSON] et al. [9] reviewed control methods for nanosatellite formation flying, covering both impulsive maneuvers and continuous thrust control strategies, and analyzed the applicability of multi-impulse maneuvers in close-proximity operations. [PERSON], [PERSON] et al. [10] designed a heliocentric orbital formation configuration for the LISA mission and discussed schemes for using multi-impulse maneuvers to maintain high-precision relative orbits between satellites. [PERSON], [PERSON] et al. [11] studied the formation control of solar sail spacecraft near planetary orbits and proposed an event-triggered impulsive cooperative control method suitable for periodic fly-around missions. [PERSON] et al. [12] validated a multi-impulse control algorithm for autonomous formation flying in the PRISMA mission. [PERSON] et al. [13] used relative orbit elements to design the bi-elliptical and bi-teardrop formations, and the fly-around distance and relative view angle were specifically discussed. [PERSON] et al. [14] proposed an impulse control algorithm for establishing parallelogram formation flying, successfully achieving single-loop and multi-loop formation flight configurations. In terms of continuous-thrust-controlled fly-around, [PERSON] et al.