Relative Lambert Problem of Near Coplanar Long-Range Relative Motion

Abstract

The relative Lambert problem based on a near-coplanar long-range relative motion in circular reference orbit is investigated. The relative Lambert problem consists of determining the relative velocity given relative positions at two known times. As for the short distance relative Lambert problem, an analytic solution is obtained by using matrix inversion based on the Clohessy-Wiltshire equation. Then a second-order relative motion Lambert solver is derived from the second-order Clohessy-Wiltshire equation. However, the admissible relative distance between spacecrafts is still far smaller than the semi-major axis of the chief spacecraft. This paper aims to enlarge the admissible relative distance. To solve the corresponding relative Lambert problem, a time-explicit analytic description of a near-coplanar long-range relative motion in circular reference orbit is introduced to provide a two-pulse relative transfer strategy. It is worth pointing out that the angle between the inertial position vector of the chief and the inertial position vector of the deputy can be any value, which means the admissible relative distance can be very large. The choice of the transfer time is also important. As a result of the existence of singular points, the transfer time should be far away from these singular points. In addition, the transfer time should ensure that the initial impulse meets the corresponding condition of the analytical solution to the long range relative motion. Four numerical results show that the proposed method to solve the relative Lambert problem is indeed effective and feasible.