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The work presented here investigates task-space trajectory tracking control for free-floating space robots with uncertain kinematics and dynamics. To deal with these two kinds of uncertainties, we propose a prediction error based adaptive Jacobian controller, which includes a modified task-space computed torque controller and two modified least-squares estimators. By defining a new variable that is termed as the estimate of the spacecraft angular acceleration, the proposed controller can work without requiring measurement of the spacecraft angular acceleration. The kinematic and dynamic parameter adaptations are driven by prediction errors. Using input-output stability analysis, we obtain explicit stability results of the closed-loop system and show the asymptotic convergence of the end-effector motion tracking errors of the free-floating manipulator. Furthermore, it is shown that the performance of the proposed controller has a tight relationship with the estimated generalized Jacobians of free-floating manipulators. Simulation results are presented to show the performance of the proposed controller, and in addition some implementation issues of the proposed control algorithm are discussed.