Takeya Shima

New State Transition Matrix for Formation Flying in J2-Perturbed Elliptic Orbits

A state transition matrix for formation flying in eccentric orbits is considered in this paper. The relative motion of the formation flying obeys the Tschauner–Hempel equations, which are extended for the presence of a J2 perturbation. Furthermore, the state transitionmatrix of the Tschauner–Hempel equations is extended for J2, and it is demonstrated that the extended state transition matrix becomes the solution of the extended Tschauner–Hempel equations. Because the extended state transition matrix only uses osculating orbital elements in the initial state and does not use mean orbital elements, an analytical treatment of the state transition matrix is possible. When the state transition matrix uses only the first-order term of J2, the errors sometimes become large compared with the relative distancemultiplied by J2 in a time range that is longer thanone orbit period.Amethod to suppress errors numerically is also proposed.

Effect of J2 Perturbations on Relative Spacecraft Position in Near-Circular Orbits

F ORMATION flying, in which several spacecraft maintain their relative positionswith each other, has attractedmuch attention in recent years, and various missions for Earth and astronomical observations using spacecraft flying in formation have been proposed [1,2]. A spacecraft that moves in a reference orbit is called the chief spacecraft, and a spacecraft flying in formation with the chief spacecraft is called the deputy spacecraft. The relative equations of motion for a circular reference orbit are given by the well-known Hill–Clohessy–Wiltshire (HCW) equations [3]. The HCWequations are a set of linearized equations that describe themotion of the deputy relative to the chief under the assumption that the Earth is perfectly spherical. Because the effects of the Earth’s oblateness (as described by the J2 term in the Earth’s gravitational potential) are ignored in the derivation of the HCWequations, they cannot be used to accurately predict relative motion over a long time period. Many studies have incorporated the effects of the J2 perturbation. Schweighart and Sedwick extended the HCWequations to include the effect of J2 and derived a set of constant-coefficient linearized equations of motion [4,5]. Ross [6] and Roberts and Roberts [7] derived a set of timevarying coefficient linearized equations of motion using the gradient of acceleration due to J2. Even when the chief orbit is nominally circular, a small eccentricity usually occurs because of J2. In general, the relative equations of motion for a chief with an elliptic orbit are known as the Tschauner–Hempel equations [8]. The relative motion in an elliptic orbit becomes much more complicated because of J2. Gim and Alfriend derived a state transition matrix for the relative motion under such conditions with the effects of J2 [9]. Hamel and Lafontaine also derived a state transition matrix for the relative motion. They approximated the time variation in the difference between the osculating orbit elements of the chief and deputy by using the difference between the mean orbit elements and obtained a simple form of the matrix [10]. In addition, Humi and Carter focused on cases in which the chief follows an equatorial orbit and a polar orbit and derived relative equations of motion under J2 [11]. An averaged solution of the relative motion when the chief is in an elliptical orbit with the J2 perturbation was also obtained [12]. Moreover, a method to modify the initial conditions of a formation through numerical integration of the equations ofmotion by using the Gaussian least-square method was proposed [13]. In this study, we use the solutions to the HCWequations to focus on two formations in which the relative distance between the chief and deputy is constant: an along-track formation and a circular formation. We analyze the relative distance variation due to J2. This distance variation is particularly crucial when formation flying is used to create a telescopewith a long focal distance. In relation to the distance variation caused by J2, Sabatini et al. used a genetic algorithm and examined the conditions required to minimize the deviation from a circular formation; they found that the chief orbit had two special inclinations at which the deviation becomes extremely small [14]. Moreover, the condition for the agreement of in-plane and outof-plane fundamental frequencies of the relative motion under the effects of J2 is reduced to the same inclinations as that of the chief orbit [15]. The main objective of this study is to analyze the relative distance variation within one orbit period for a mission in which the relative distance is precisely maintained. For this purpose, we take a novel approach to derive a state transition matrix without using the mean orbit elements of the chief, assuming that the eccentricity of the chief is small, i.e., of the same order of magnitude as J2. From this state transition matrix, we obtain the relative distance variation occurring in one orbit period in an analytical form. Furthermore, we also derive the relation between the initial values of the deputy and the relative distancevariation. The new formulation of the deputy initial values in terms of the osculating orbit elements is derived to reduce the distance variation. Finally, numerical simulations are carried out to validate the analytical results.