The algorithms for capture of the space garbage using "leier constraint"

In this paper we study the following problem. Let a particle move in a central Newtonian gravitational field near a massive gravitationally oriented body. The body describes a circular orbit and is equipped with a "leier constraint", i.e. a cable with both ends fixed on the body (from the Dutch maritime term "leier"— a rope with both ends fixed). We study the possibility of capturing the particle (i.e. to force the particle to move periodically) by a gripper coasting on the "leier constraint". Any discontinuities in the velocity and acceleration are not allowed. We show that using a sufficiently long cable the capture is possible for almost all values of the system parameters. We also construct a modified algorithm that allows many-fold reduction in the required cable length.