Under-actuated nonlinear dynamic systems trajectory tracking, such as space robots and manipulators with structural flexibility, has recently been investigated for hierarchical sliding mode control since these systems require complex computations. However, the instability phenomena possibly occur especially for long-term operations. In this paper, a new design approach of an adaptive fuzzy hierarchical terminal sliding-mode controller (AFHTSMC) is proposed. The sliding surfaces of the

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... construct the hierarchical structure of the proposed method in which the top layer includes all of the subsystems' sliding surfaces. Moreover, a terminal-sliding mode has been implemented in each layer to ensure the error convergence to zero in finite time besides chattering reduction. In addition, online fuzzy models are employed to approximate the two nonlinear dynamic system's functions. Finally, a simulation example of an inverted pendulum is proposed to confirm the effectiveness and robustness of the proposed controller.