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A composite finite-time attitude tracking controller is formulated by combining a finite-time disturbance observer (FTDO), the system transformation method and the technique of nonsingular terminal sliding-mode control (NTSMC). Firstly, a FTDO is employed to estimate time-varying disturbances acting upon the rigid spacecraft. Secondly, to tackle the angular velocity constraints, the system transformation method is introduced, which converts the problem of angular velocity constraints into a state-unconstrained one. Subsequently, for the transformed system, a finite-time attitude tracking controller is developed, combining the disturbance estimates derived from the FTDO and the NTSMC principle. Additionally, the unwinding phenomenon associated with quaternions is avoided. The strict proof of finite-time stability on the closed-loop system is provided. 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