Necessary and sufficient conditions of the rendezvous between a single spacecraft and three non-coplanar Walker constellation satellites

The probability of the rendezvous between a single spacecraft and three non-coplanar constellation satellites is studied, and the necessary and sufficient conditions of the rendezvous without orbital maneuver are deduced. The rendezvous orbit design can be transformed into the patching of two spacecraft orbits, either of which can achieve the rendezvous with two satellites. Firstly, due to the precious quality of spherical geometry, the unique existence of the rendezvous orbit for two constellation satellites is proved. Then, according to the difference between equispaced and non-equispaced orbital planes of three satellites, the necessary and sufficient conditions are given respectively, and the calculating method of the spacecraft orbit is proposed. At last, the constraint conditions between two different rendezvous orbits is derived, while the relative position of two groups of objects are under specific distribution. The results can be applied to the rendezvous between a single spacecraft and multiple constellation satellites without orbital maneuver.