New fly-around formations for an elliptical reference orbit

Highlights

• A series of analytic natural fly-around formations are proposed.

• The reference orbit is elliptical orbit, which can also be applied to circular orbit.

• A two-stage constant thrust control method solved through the linearized sensitive matrix is presented.

• Several maintaining strategies are presented to guarantees periodic revisit specified relative position.

Abstract

Fly-around technology, as applied to maintain a chasing spacecraft flying periodically around the chief spacecraft, has received widespread attention in recent years. The conventional impulsive or continuous-thrust control approaches for maintaining non-natural formations put a high demand on the engine and limit the applicability of engineering. In this study, which is based on the relative orbital elements, we propose a series of analytic natural fly-around formations for the two-body case, which can revisit specific points or be maintained within a certain range. In addition to maintaining the fly-around formation when considering the J2 perturbation, we present a two-stage constant thrust control method solved with the linearized sensitive matrix. Furthermore, based on this method, we present several maintaining strategies that guarantee a periodic revisit to a specified relative position. We conduct numerical simulations to demonstrate the efficacy of these proposed methods.

Introduction

In recent years, on-orbit servicing (OOS) missions [[1], [2], [3], [4]], including fuel filling, proximity operation [[5], [6], [7], [8]], and autonomous rendezvous, have aroused the interest of many scholars. In addition, space debris has also attracted wide concern because the increasing debris poses a fatal threat to the life of on-orbit satellites or spacecraft [[9], [10], [11]]. In order to accomplish OOS missions or remove space debris, the position and status of the target spacecraft (or space debris) should be continuously inspected. Furthermore, to guarantee the safety and efficiency of the mission-carrying spacecraft, the distance between both spacecraft should be kept within a certain range. These requirements can be realized by fly-around control technology, which usually refers to a chaser spacecraft that flies around a target spacecraft periodically with a bounded orbit.

Fly-around technology is mainly based on the relative motion theory, and large amounts of research have been conducted [12]. The CW equations [13] and the TH equations [14] are the most well-known relative motion models used for the circular reference orbit and the elliptical reference orbit, respectively. More precise results were obtained by analyzing higher order terms [[15], [16], [17]]. In addition, if the relative motion equations are established in the curvilinear coordinate system [18,19], the accuracy of the equations can be improved. Furthermore, Koenig et al. [20] present new state transition matrices for the relative motion that considers the J₂ perturbation. To analyze the relative motion in detail, orbital element methods are also adopted. The concept of an eccentricity/inclination vector was defined by D'Amico and Montenbruck [21]. This concept is applied to orbit control and proximity analyses. Gim and Alfriend [22] proposed a precise analytic solution and analyzed the orbital element differences with a geometric approach that considered the reference orbit eccentricity and the differential perturbations. By using the Cartesian configuration space in conjunction with classical orbital elements, Gurfil and Kholshevnikov [23] established a methodology for obtaining the solution for general spacecraft relative motion. Based on a linearized approximation, Healy and Henshaw [24] defined the geometric relative orbital elements (ROEs). Based on the ROEs, Han and Yin [25] proposed a new relative motion model that considered both elliptical and near-circular cases.

Fly-around formation refers to the periodic relative trajectory between the chaser spacecraft and target spacecraft. Based on whether external control is applied to the chaser spacecraft, fly-around formations are divided into two categories: natural fly-around and forced fly-around. The natural fly-around period is equal to the chief spacecraft's orbiting period, which means that the relative motion trajectory can naturally form a periodic closed trajectory without external force control. Based on the Hill equations, Sabol et al. [26] proposed four basic natural formation flying designs: in-plane formation, in-track formation, circular, and projected circular formations. In addition, Lin [27] obtained the necessary conditions for a closed trajectory by using the Hill equations. Moreover, by using ROEs, Lovell and Tragesser [28] analyzed the natural fly-around motions, which can be described by an instantaneous elliptical path with a drifting center. Furthermore, He [29] also obtained three typical coplanar fly-around formations through an analysis of long-term relative motion in the orbital plane. For an elliptical reference orbit, Chang [30] deduced the necessary conditions for the fly-around formation and proposed the impulse-control strategy for maintenance. Furthermore, Jiang et al. [31] presented the concepts of fly-around curvature and fly-around torsion, as well as proposed continuous-thrust strategy for keeping the formation. Although these natural formations can save fuel, they make it difficult to meet the requirements of conventional engineering, such as revisiting specific points or boundary fly-around motion. Moreover, to maintain the formation, they need well defined and constrained requirements for the engine.

The forced fly-around formations are also widely studied. Based on a linearization method, an adaptive law to fly around the chief spacecraft was provided by Kojima [32]. Qi and Jia [33] and Wang et al. [34] presented a continuous-low-thrust control law to form a fast fly-around space circle. Gehler et al. [35] presented continuous-thrust control for the gravity tractor spacecraft to maintain the distance to the reference spacecraft. Zhu et al. [36] also solved the problem of optimal finite-thrust control for spacecraft fly-around motion. However, these methods require a variable continuous-thrust engine to realize the designed trajectory. To overcome the high demand on the engine, several related researches have been conducted [[37], [38], [39], [40]]. Bennett et al. [41] presented a way point guidance solution for fast circumnavigation. Masutani et al. [42] proposed a bi-elliptic fly-around formation maintained by an impulse control scheme. Furthermore, based on the work of the ROEs, Zhang et al. [43] also presented a fast bi-elliptic fly-around formation controlled by effective impulsive control strategies for a circular reference orbit. When compared with natural fly-around formations, the forced fly-around formations can accomplish various goals of space missions, despite consuming more fuel. Similarly, these formations have tight requirements for the engines, which limit the engineering applications.

By combining the advantages of the two types of fly-around formations, we can determine our main objective of this study, which is to design a method for the natural fly-around formation. Furthermore, in order to reduce the tight requirement on the engine, this paper also improves the approach for formation maintenance for the J₂ case. First, a series of new fly-around formations for an elliptical reference orbit is designed analytically and described by the ROEs. Second, based on two-stage constant thrust equations and the sensitive matrix, different control strategies are proposed for maintaining a fly-around formation. Finally, the effectiveness of the proposed method is verified with numerical simulations. When compared with the previous fly-around formations, the natural fly-around formations proposed in this paper can revisit specific points or be maintained within a certain range. These could meet the goals of various space missions. Moreover, the presented formation maintenance strategies for the J₂ case are based on a two-stage constant thrust, which greatly reduces engine requirements. Thus, it can provide a practical choice for engineering applications.

The outline of this paper is as follows. The new fly-around formations based on the relative orbital elements are introduced in Sec. II. The mathematical model of the two-stage constant thrust control method and closed-loop control strategies for maintaining formation are presented in Sec. III. Finally, Sec. IV summarizes the paper.