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This paper proposes an adaptive controller for a fully free-floating space robot with kinematic and dynamic model uncertainty. In adaptive control design for the space robot, because of high dynamical coupling between an actively operated arm and a passively moving end-point, two inherent difficulties exist, such as non-linear parameterization of the dynamic equation and both kinematic and dynamic parameter uncertainties in the coordinate mapping from Cartesian space to joint space. The proposed method in this study overcomes the above two issues by paying attention to the coupling dynamics. The proposed adaptive controller does not involve any measurement of acceleration; but it is still possible for the system to be linearly parameterized in terms of uncertain parameters and a suitable input torque can be generated in the presence of model uncertainty. A numerical simulation was carried out to confirm the validity of the proposed adaptive control.