The goal of this paper is to design a spacecraft round trip transfer from a parking orbit to Asteroid 2006\;RH$_{120}$, during its capture time by Earth's gravity, while maximizing the final mass or equivalently minimizing the delta-v. The parking orbit is chosen as a Halo orbit around the Earth-Moon $L_2$ libration point. The round-trip transfer is composed of three portions: a rendezvous transfer departing from the parking orbit to reach 2006\;RH$_{120}$, a lock-in portion with the spacecraft following the asteroid orbit, and finally a return transfer to $L_2$. An indirect method based on the maximum principle is used for our numerical calculations. To partially address the issue of local minima, we restrict the control strategy to reflect an actuation corresponding to up to three constant thrust arcs during each portion of the transfer. The model considered here is the circular restricted four-body problem (CR4BP) with the Sun considered as a perturbation of the Earth-Moon circular restricted three body problem. A shooting method is applied to solve numerically this problem, and the rendezvous point to and departure point from \RH\ are optimized using a time discretization of the trajectory of \RH.