F ORMATION flying has received much attention in recent years because of the possible advantages of replacing a single, complex satellite with a cluster of smaller ones. Flying a formation of satellites offers improved flexibility and redundancy and the ability to construct much large virtual sensors than can be flown on a single, monolithic satellite. Several missions have identified formation flying as an enabling technology for increasing the science return and reducing the total mission costs [1,2]. A critical problem in the formation-flying application is the formation design. The purpose of formation design is twofold. First, various flying missions have different requirements for formation design, such as the Aperture Synthesis Radar mission, the Earth Observer-1 mission, the Laser Interferometer Space Antenna mission, etc. The primary purpose of formation design is searching the proper formation array that fulfills the formation mission requirement. The scientific or other need of the mission is the main constraint for the formation design and can be defined as the mission constraint for the formation design. Second, formation-flying satellites operate on orbits with long time spans, in general, from several months to several years, and frequent thruster firings to keeping formation will consume so much propellant and cannot be accepted bymany consideringmissions. And so, it is more important to design nature periodic relative motion orbits and avoid the secular drift. This can be defined as the orbit constraint for formation design. The problem of satellite formation design has been studied by many researchers and many advances have been made. Sabol et al. [3] used the Hill’s equations to design formation and proposed four special formations for various formation-flying missions. Hill’s equations are the constant coefficient differential equations and have simple analytical solutions with an in-track secular term in the solutions. The drift can be avoided by a proper initialization. These simple analytical expressions permit the use of intuitive methods to design formations. However, Hill’s equations can only design valid formations for circular reference orbits and linearized relative motion. Carter et al. [4–6] studied the relative motion of two vehicles in nearby elliptical orbits and presented an analytical solution under assumptions of linearization and without perturbations, and developed an initialization procedure to obtain initial conditions for the formation satellites period relative motion at reference orbit perigee. But, the solutions of Carter’s have a definite integral and are complex in form and cannot be conveniently applied in formation design. Using orbital element differences and Cartesian coordinates, respectively, Lane and Axelrad [7] and Xing et al. [8] derived the simple periodic analytic solutions in eccentric orbits under assumptions of linearization. Theseworks investigated the linearized problem of relative motion of formation flying. Considering nonlinearity and eccentricity, Gurfil [9] and Xing et al. [10,11] independently proposed generalized periodic relative motion conditions (GPRMC) for formation flying on arbitrary Keplerian elliptic orbits. This paper focuses on the formation design for nonlinear relative motion and eccentric reference orbits under GPRMC. Because there are some linearized errors for formation design usingHill’s equations and solutions of Xing’s [8], a new method is proposed to design formation considering nonlinearity and eccentricity which use GPRMC to correct the initial conditions derived from Hill’s equations or Xing’s solutions and get the periodic relative motion orbits for formation flying. This approach is attractive for two reasons. First, the process of formation design uses the GPRMC and removes the secular drift of relative motion orbit designed by Hill’s equations or Xing’s solutions and avoids frequent thruster firings to keep formation. More important, the approach adequately uses the previous research results which were derived from Hill’s equations and could use the intuitive methods to design formations for various formation-flying missions. It is assumed that the formation designed by Hill’s equations fulfilled the requirement of the reality formation-flying mission. Considering nonlinearity and eccentricity, the designed formations were not perfectly consistent with Hill’s. The remainder of this paper developed an optimal impulse formation-keeping maneuver based on the orbit constraint (GPRMC) and the mission constraint (Hill’s solutions). We used the orbit constraint and the mission constraint to keep periodic relativemotion and fulfill the formation-flyingmission requirement, respectively.
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Sep 15, 2023
Gadisa Dinaol 
Open Access