This work designed a free-flying space robot (FFSR) that simulates the on-orbit assembly of large space telescopes, drawing inspiration from the flexible movement of water striders on water surfaces. Initially, we developed the system structure of the robot, including the corresponding air-floating ground simulation system. This system enables floating movement of the robot in a gravity-free environment through the utilization of planar air bearings. Subsequently, we established the kinematics and dynamics models for the FFSR. Following that, we propose a novel adaptive boundary layer fuzzy sliding mode control (ABLFSMC) method to achieve trajectory tracking control of the FFSR. The virtual angle and angular velocity are formulated to serve as references for the angle and angular velocity in the body coordinate system. Furthermore, a fuzzy logic system is employed to minimize the chattering effect of the sliding mode control. The global stability of the proposed controller is guaranteed through the Lyapunov stability theory. Finally, we validate the effectiveness of the proposed control method as well as the high trajectory tracking accuracy of the developed FFSR through simulation and experimental results, respectively. Overall, our findings present a crucial experimental platform and development opportunity for the ground-based validation of technologies concerning the on-orbit assembly of large space telescopes.