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In this paper, a nonsingular fast terminal adaptive neurosliding mode control for spacecraft formation flying systems is investigated. First, a supertwisting disturbance observer is employed to estimate external disturbances in the system. Second, a fast nonsingular terminal sliding mode control law is proposed to guarantee the tracking errors of the spacecraft formation converge to zero in finite time. Third, for the unknown parts in the spacecraft formation flying dynamics, we proposed an adaptive neurosliding mode control law to compensate them. The proposed sliding mode control laws not only achieve the formation but also alleviate the effect of the chattering. Finally, simulations are used to demonstrate the effectiveness of the proposed control laws.