Evolution of Periodic Orbits within the Frame of Formation Satellites
Evolution of Periodic Orbits within the Frame of Formation Satellites
Abouelmagd, Elbaz I.;Doshi, Mitali J.;Pathak, Niraj M.
2020-12-21 00:00:00
Hindawi Advances in Astronomy Volume 2020, Article ID 1348319, 17 pages https://doi.org/10.1155/2020/1348319 Research Article Evolution of Periodic Orbits within the Frame of Formation Satellites 1 2 2 Elbaz I. Abouelmagd , Mitali J. Doshi , and Niraj M. Pathak Celestial Mechanics and Space Dynamics Research Group (CMSDRG), Astronomy Department, National Research Institute of Astronomy and Geophysics (NRIAG), Helwan 11421, Cairo, Egypt Department of Mathematics, Faculty of Technology, Dharmsinh Desai University, Nadiad, Gujarat 3870001, India Correspondence should be addressed to Elbaz I. Abouelmagd; elbaz.abouelmagd@nriag.sci.eg Received 20 July 2020; Revised 20 November 2020; Accepted 30 November 2020; Published 21 December 2020 Academic Editor: Juan L. G. Guirao Copyright © 2020 Elbaz I. Abouelmagd et al. &is 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. In the framework of formation satellites, the periodic orbits of deputy satellite are analyzed when the chief satellite is moving in an elliptical orbit. &is analysis is developed on 1- to 10-loop periodic orbits of the deputy satellite. &ese orbits along with their associated loops are discussed under some specific initial position sets. &e effects of different initial velocities, initial true anomalies, and eccentricities on the initial position and orbital period of periodic orbits of deputy satellite are investigated. theory of averaging is applied to prove the existence of 1. Introduction twelve families of periodic orbits in a 3-dimension for a galactic Hamiltonian dynamical system. Since we are in- &e periodic orbits have substantial and leading role in exploring and understanding the behavior of dynamical terested to evaluate the periodic orbits within frame of systems. At most, they define strange attractors, which lead formation satellites, we will give also an overview about the to chaotic dynamical systems. &e special solution of a literatures and importance of formation satellites in the dynamical system, which repeats and generates itself in time, following paragraphs. is called periodic orbit. From the mathematical point of view, &e formation flying of small multiple satellites as a re- the orbit is a set of points associated by the evolution placement of using single large satellite has shown great in- function of the proposed dynamical system. &ese points are terest for different defense- and science-based space missions. considered as a subset from the phase space, which are Formation flying consists of a set of satellites, which have the covered by the dynamical system trajectory within frame of a same dynamic state and governed by one control law. particular set from the initial conditions. Some recent works, Abundance and precision of the proposed system in terms of formation satellites are more effective tools, which give a job analyzing the periodic orbits, are addressed in [1–4]. &e sufficient condition for the existence of periodic more accuracy than using a conventional large single satellite. orbits is given when the Hamilton system is a function in the Italsoreducesthemaintenanceandlaunchingcosts,extremely action-angle variables; further, these obtained results are expands the surveillance area, and gives more resilience into applied to Hamiltonian of the perturbed Kepler problem in the design of space mission. For example, a sensor of ground [5]. Also, a geometric approach to asymptotically stabilize observation can be loaded on bunch of satellites flying in a with a phase of fixed periodic orbits for global Hamiltonian specified formation for increasing aperture size instead of dynamical system is established in [6]. While in [7], the new constructing a large single satellite with more expense. &ere families of periodic orbits analytically for the Hamilton are chances of aborting the whole mission in the event of satellite failure. Proper management of satellites cluster with system are found, which characterize the local motion in the region around the galaxy center. Furthermore, in [8], the special planning and scheduling reduces the chances of failure. 2 Advances in Astronomy Using formation satellite in space-based missions has in Section 1 as a part of literature review. Model description many advantages, but at the expense of increased complexity and derived governing equations of motion are covered in Section 2. Analysis of the given sets of initial positions for and different challenges like high-precision relative navi- gation [9], distributed communication [10], stable formation deputy satellite, which generate periodic orbits, is investi- design [11], trajectory optimization and control [12], and gated in Section 3. While in Section 4, we compare the effect attitude synchronization [13]. In formation satellite, tra- of variation in eccentricity of chief satellite’s orbit on pe- jectory optimization and control problem are two important riodic orbits of deputy satellite with number of loops. Fi- tasks to achieve a successful launching of satellites set in the nally, conclusion is drawn from the analysis becomes the space. &ese tasks comprise maintaining the small satellites part of Section 5. in a stable formation within frame of enough accuracy against different perturbations of orbit and maneuvering of 2. Model Description formation for guiding and performing control command for reconfiguring from perturbed satellite formation to one Consider two spacecrafts orbiting a common primary and its stable formation. mass is m. Mainly, the motion of these two spacecrafts is &e precise model of relative motion in order to analyze governed by the Kepler model or the dynamical system of two satellites formation flying is a basic need which covers ac- bodies [23–26]. One of the spacecraft is termed as a chief curate linear and nonlinear satellites models of relative satellite, and the second is referred as a deputy satellite, where motion taking into account J perturbations. Different their masses are m 2 and m , respectively. &en, the equations 0 1 relative dynamic models are proposed in the literature using of relative motion of deputy satellite with respect to chief different assumptions with many methodologies. It is nec- satellite under the setup of the Keplerian two-body problem essary to make a comparative study to choose appropriate areobtainedasfollows:weconsiderachief-fixed,localvertical models for specified missions with perturbation that should local horizontal (LVLH) rotating frame, also known as be considered for definite applications. EulerHill frame. Here, the origin is located at the position of A considerable work is accomplished into satellite for- chief satellite, as shown in Figure 1. mation flying for libration point mission with different From two-body problem, the motion of the chief and models that characterize the relative motion satellites be- deputy satellites around the primary (Earth or any planet) in tween two or more in low Earth orbit (LEO). &e major inertial frame of reference are given by fundamental of this work is carried out by Hill in [14]. While the relative motion within frame of Clohessy–Wiltshire €r � −μ , 0 0 equations is written in terms of a Cartesian or curvilinear (1) coordinates tracing a circular reference orbit around the r � −μ , Earth and models by using orbit elements differences to 1 1 characterize relative orbits [15]. &e extended version of the Hill equations was given in [16] that involves the influent of where μ � m + m and μ � m + m , but m , m ≪ m; then, 0 0 1 1 0 1 the zonal harmonic parameter J using a force gradient we can approximate μ ≈ μ � μ. &ereby, the solutions of 0 1 method to time-varying form. It was verified and applied to equation (1) are controlled by linear quadratic regulator design and evaluated for the station-keeping task in [17]. a 1 − e 0 0 In [18], the force gradient modelling approach for sat- r � , 1 + e cos f 0 0 ellite formation flying around the libration point L using (2) periodic halo motion as a reference is investigated. &e a 1 − e optimal maneuver problem can be characterized as a state r � , 1 + e cos f transition problem based on Hill’s system and maximum 1 1 principle of Pontryagin. &e optimal solution can be ob- where a (a ), e (e ), and f (f ) are the semimajor axis, 0 1 0 1 0 1 tained by solving the state transition equations and per- eccentricity, and true anomaly of chief (deputy) satellite’s forming the simulation study [19]. In a formation satellite, a orbit, respectively. magnetic field approach helps a large number of closely Now, we assume that ρ is the position vector of deputy located satellites in tracking each other in six degrees of satellite relative to chief satellite; hence, ρ � r − r , and the 1 0 freedom without disturbing their positions and orientation relative motion of deputy satellite is relative to each other (see [20, 21] for details). r + ρ r &e relative motion control is an important task required 0 0 ρ � −μ + μ , (3) 3 3 in the formation of the flying missions. Different control r + ρ methods withoutfuel consumption are ofa specific interest. A T T number of these methods based on atmosphere drag effects, where r � [r ,0,0] , ρ � [x, y, z] , f � θ − ω, and ω is 0 0 0 0 electrostatic magnetic field, and the Lorentz force have been the argument of periapsis. But ω is a constant. &ereby, we proposed, but exchanging mass between satellites is a novel can define the angular velocity vector Ω � [0,0,θ ] . technique for formation flying relative motion control [22]. &e general relation between the velocity and acceleration &is paper is organized into four sections. &e impor- in the inertial frame and the rotating by the angular velocityΩ tance and applications of formation satellites are discussed is controlled by Advances in Astronomy 3 Deputy Chief Figure 1: Configuration of the LVLH frame. I R h dx ρ _ � ρ _ + Ω ∧ ρ, x _ �