Relative Motion Dynamics in the Restricted Three-Body Problem

Abstract

In this paper, a set of equations for relative motion description in lunar orbits is presented. The local-vertical local-horizon frame is selected to describe the relative dynamics of a chaser approaching a target in lunar orbit, allowing the development of relative guidance and navigation systems for rendezvous and docking. The model considers the Earth and Moon gravitational influence on the two spacecraft, which are assumed to have negligible masses. The proposed equations are intended for the study of rendezvous missions with a future cis-lunar space station, whose development is currently investigated by several space agencies as the next step for space exploration.

Closed Loop Guidance During Close Range Rendezvous in a Three Body Problem

“…The aposelene is considered the most feasible area for the docking/berthing. Current literature indicates that as the most likely location, and it has been shown that CR3P equations are sufficiently accurate for the dynamic description of the relative motion [6,21]. In this case, the target has the slowest orbital velocity.…”

“…The close range rendezvous and berthing are assumed to occur during aposelene passage for safety reasons, since the target's velocity is the lowest, this also allows for the simplification of the equations. In this case in fact, the approximation of the primary bodies revolving in circular orbits, (Circular Restricted Three-Body Problem CR3BP assumption) appears appropriate [21]. With this assumption, the number of time-varying parameters in Eq.…”

Fully Safe Rendezvous Strategy in Cis-Lunar Space: Passive and Active Collision Avoidance

“…The NRHO orbit properties, shown in Fig. 1, are given and can be found in [4,7,8], for instance. It has a period of about seven days with 70,000 km as the largest distance from our satellite.…”

“…The key reference frames used in the description of the relative motion are recalled next. They are based on the assumption that the motion of the spacecraft is subjected to the gravitational attraction of both Earth and Moon, and follows the standard definitions in the literature [7,9]. The selected inertial frame is centered at the primaries common center of mass C, with the K̂ unit vector normal to the plane where the primaries revolve, and with the other unit vectors spanning the primaries rotating plane (according to the right-handed rule).…”

“…The equations of relative motion, following the derivation in [7], in the LVLH frame and in nondimensional units are given in Eq. ( 1): where , r , and r em are defined in Fig.…”