Stability Analysis of Error Dynamics in Virtual Target-Based Transfer Trajectories Wthin the Cislunar Environment

In this paper, we analyze the tracking error dynamics between a spacecraft and a virtual target evolving along a desired butterfly orbit in the cislunar space. The spacecraft is controlled by a PD-based law designed to reduce position and velocity errors relative to the virtual target. Although the control input is linear, the resulting closed-loop error dynamics are nonlinear due to the pseudo potential in the restricted three-body problem. We linearize these nonlinear error dynamics around the virtual target trajectory to study their evolution and stability. Stability is assessed through eigenvalue analysis of the linearized system and via Lyapunov candidate functions, which confirms asymptotic stability of the error dynamics. Simulation results demonstrate that the tracking error converges to zero, indicating that the spacecraft successfully transfers to the butterfly orbit under the analyzed control framework. This study provides important insights into the behavior and stability of error dynamics in nonlinear cislunar transfers using virtual target strategies.