Two-point boundary value problem of the relative motion

The two-point boundary value problem of the relative motion is studied, in which the chief spacecraft's motion is known and the motion of the deputy spacecraft should be determined. An accurate numerical solution to this problem is reviewed. Then two types of linearized analytical solutions, one based on linearization of a reference Lambert's problem of the chief and the other based on relative motion's state transition matrix, are derived. These two linearized solutions are shown to be equivalent. Meanwhile, both solutions suffer from singularity, resulting in huge fuel consumption under certain circumstances. The reason of the singularity is analyzed and some analytical expression relations are given when the chief's orbit is a circle. In the end, methods to check and alleviate this singularity are presented. Several examples are also given to demonstrate the findings.