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The backstepping control method provides useful control logic, especially for a cascaded system. Because spacecraft dynamics and kinematics form a cascaded system, the spacecraft slew maneuver problem can be solved using the backstepping control method. However, the simple linear backstepping controller may result in poor design: sluggish motion, trivial nonlinear term cancellation, and excessive control input. To overcome these defects, an effective backstepping control method using a nonlinear tracking function is proposed. The proposed backstepping control method is based on the redesign of the Lyapunov function and careful gain selections. To evaluate the effectiveness of the proposed method, numerical simulations including parameter uncertainties are performed. Simulation results demonstrate that the proposed backstepping controller can achieve the slew maneuver with shorter settling time and smaller peak control torque than existing methods.