Studies On Relative Motion Of Spacecraft Formation Flying

Compared with large, single, and expensive one, spacecraft formations offer improved flexibility and redundancy, and reduced cost to space missions. Consequently, much research has been conducted during the past decade. In this thesis, the relative motion of spacecraft close formations in elliptical reference orbits is studied systematically from the perspective of kinematics, where neither perturbations nor active controls are considered. This work sheds light on the inherent characters of relative motion, and has the potential to benefit the application of spacecraft formation flying.First, for close formations, a series of order-of-magnitude analyses about the quantities involved in the reference orbital element approach and orbit element differences are conducted. Consequently, a set of first-order and second-order relative position equations are obtained through Taylor expansion. It is proved that the first-order equations hold the same forms as the periodic solutions of Lawden's equations, and the periodicity condition of the latter is first-order approximate to that of the former, which is the semi-major axis of the follower equals to that of the leader. The first-order relative position equations are expanded as trigonometric series with eccentric anomaly as the angle variable. Except the terms of order one, the trigonometric series'amplitudes are geometric series,and corresponding phases are constant both in the radial and in-track directions. When the trajectory of the in-plane relative motion is similar to an ellipse, a method to seek this ellipse is presented.Second, considering the non-degenerate case in which none of the relative motion in the radial, in-track and cross-track directions vanishes, the relative orbit geometry is studied through algebraic methods. The relative orbit proves to be three-dimensional instead of planar in most cases, and may self-intersect spatially at most once. The conditions for collision, namely, the relative orbit passes through the origin are determined. Furthermore, the conditions for and the number of self-intersections of the relative orbit projected onto the three coordinate planes of the leader local-vertical-local-horizontal frame are obtained, and importantly three types of relative orbit are possible. Most frequently, the relative orbit is on a one-sheet hyperboloid. Otherwise, it rests on an elliptic cone. In the rest of the cases, the relative orbit is on an elliptic cylinder. The criteria for these three types are given respectively.Finally, the two-point boundary value problem of relative motion is studied. For close formations, a set of approximate solutions are derived from linearizing Lagrange's time equation about the classical Lambert problem, and the Newton-Raphson algorithm using the relevant quantities of the leader as the initial values is constructed. In particular, concerning periodic relative orbits, a constraint on leader's true anomalies (implicitly on times) and relative positions is obtained. It is disclosed from the constraint that on the radial/in-track plane of the leader local-vertical-local-horizontal frame, for specified initial and final times, the locus of final positions of the follow with fixed initial position is a straight line, and so is the locus of initial positions with fixed final position. Furthermore, for fixed initial and final positions, the transfer times with either specified initial time or final time can be expressed as the real roots of a cubic equation, for which there are at most three solutions.