Mathematical Problems in Engineering

Volume 2018, Article ID 4183941, 14 pages

https://doi.org/10.1155/2018/4183941

## Semianalytical Solutions of Relative Motions with Applications to Periodic Orbits about a Nominal Circular Orbit

^{1}School of Astronomy and Space Science, Nanjing University, Nanjing 210046, China^{2}School of Aeronautics and Astronautics, Sun Yat-Sen University, Guangzhou 510275, China

Correspondence should be addressed to Hanlun Lei; nc.ude.ujn@lhiel

Received 28 March 2018; Revised 6 June 2018; Accepted 24 June 2018; Published 9 July 2018

Academic Editor: Benjamin Ivorra

Copyright © 2018 Qiwei Guo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In the dynamical model of relative motion with circular reference orbit, the equilibrium points are distributed on the circle where the leader spacecraft is located. In this work, analytical solutions of periodic configurations around an arbitrary equilibrium point are constructed by taking Lindstedt-Poincaré (L-P) and polynomial expansion methods. Based on L-P approach, periodic motions are expanded as formal series of in-plane and out-of-plane amplitudes. According to the method of polynomial expansions, a pair of modal coordinates is chosen, and the remaining state variables are expressed as polynomial series about the modal coordinates. In order to check the validity of series solutions constructed, the practical convergence is evaluated. Considering the fact that relative motion model is a special case of restricted three-body problem, the periodic configurations constructed in the model of relative motion are taken as starting solutions to numerically identify the periodic orbits in restricted three-body problem by means of continuation technique with the mass of system as continuation parameter.

#### 1. Introduction

Studying the relative motion among satellites is of significance for configuration design of formation flying, constellation, and distributed spacecraft systems, where the mode of working in cooperation with several probes is required in many space missions.

In the relative motion problem, the commonly adopted model is the Hill-Clohessy-Wiltshire (HCW) equations [1], which are used to describe the relative motion of two point-mass bodies under the gravitational influence of a central body. Later, an important contribution was made by Lawden and Tschauner et al. [2–4], who generalized the HCW equations with an elliptic reference orbit. The results in the case of elliptic reference orbit were studied and extended by many researchers [5–9]. Some researchers studied the relative motions, starting from the solutions of HCW equations and latter moving to consider some perturbations, such as oblateness [10–15], drag [16, 17], and third-body effects [18].

In recent decades, Lindstedt-Poincaré (L-P) method has been applied to the problem of relative motion. High-order solutions have been constructed by several authors [19–21]. In particular, Gómez and Marcote [22] took the L-P method to obtain series solution of periodic configurations up to an arbitrary order about in-plane and out-of-plane amplitudes, and Ren et al. [23] presented the third-order expression for the solutions of elliptic relative problem.

In addition, Shaw et al. [24–26] proposed an invariant manifold approach based on polynomial expansions to perform nonlinear modal analysis. Recently, Qian et al. [27] provided the third-order polynomial relations among the three motion directions for vertical periodic orbits around triangular equilibrium points in the circular restricted three-body problem based on the polynomial expansion method.

To the authors’ knowledge, the periodic motions around an arbitrary equilibrium point in the relative motion model have not been studied. In this work, we aim to investigate analytical expressions for them by taking both the L-P and polynomial expansion approaches. As an application, the periodic objects obtained in the relative motion model are continued to periodic orbits in the circular restricted three-body problem (CRTBP) by means of numerical continuation technique with the mass parameter continuing from zero to the real value.

The remaining part of this paper is structured as follows. In Section 2, the dynamical model of the relative motion is briefly introduced and the equations of motion in the rotating coordinate system centered at the equilibrium point of interest are provided. Sections 3 and 4 discuss how to construct high-order analytical solutions of periodic configurations in the relative motion model by means of the L-P and polynomial expansion techniques, respectively. In Section 5, a continuation methodology is given for identifying periodic orbits in the CRTBP. Finally, the conclusions are drawn in Section 6.

#### 2. Dynamical Model

In the model of relative motion, the leader satellite moves on a circular orbit around the Earth, and the follower satellite moves under the gravitational field generated by the Earth in an elliptic orbit. The masses of the Earth and satellites and are denoted as , , and , respectively. Compared to the Earth mass, the masses of the satellites and are so small such that their gravitational influences upon each other can be ignored. Consequently, the satellites and move in Keplerian orbits around the Earth.

In this work, we focus on the relative motions of the follower with respect to the leader satellite. In order to formulate the equations of motion, we introduce a rotating coordinate system, where the origin is centered at the Earth, the starting axis is directed from the Earth towards the leader satellite, the -axis is parallel to the angular momentum vector of the leader satellite, and the -axis completes the right-handed coordinate system. Let us denote this rotating system as (see Figure 1 for the planar case).