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In this paper, the kinematics and dynamics of free-floating space robot system with dual-arms are analyzed. Considering the Uncertainties of the system, the state function of the space robot system with dual-arms is established. Using backstepping method, a new adaptive robust tracking control method using GCMAC(Gauss basis function Cerebellar Model Articulation Controller) is proposed. The Lyapunov functions with state tracking error and GCMAC weight error is constructed, and the neural networks weight turn laws is designed, which, together with the robust adaptive controller, can guarantee the output tracking errorpsilas converging to zero while the states of the nonlinear system and the approximation errors of neural networks are bounded. The simulation verified the proposed controller.