A second-order sliding mode fixed-time trajectory tracking controller is designed for the rigid free-floating space robot. The dynamic model of the free-floating space robot system is established by the use of Lagrange method, and a second-order sliding mode fixed-time trajectory tracking controller is designed for the system with the help of the power-addition integral method and the virtual variable strategy. It has been proven through the Lyapunov function method that the controller can ensure the global fixed-time convergence of the system tracking error, and the convergence time is independent of the initial state of the space robot system. The presented method can estimate the upper bound of the minimum convergence time of the system tracking error, which implies the issue of the overestimation of the convergence time existing in traditional fixed-time control methods has been solved, and it can also effectively eliminate the chattering phenomenon of the control torque. The simulation results are in accord with the theoretical analysis, which verified the feasibility and the superiority of the formulated method.