Aiming at the high-precision trajectory tracking control in joint space for cable-driven aerial manipulator subjected to lumped disturbances, this article proposes an adaptive robust control strategy based on time delay estimation technique. In the control frame, a time delay estimation technique is introduced to compensate the unmodelled characteristic, external disturbances and dynamic coupling effects. A fractional-order nonsingular terminal sliding mode controller has been designed to improve the convergence speed of the system states and ensure the control precision of trajectory tracking. Meanwhile, an adaptive law is designed to enhance the robustness of the controller. The stability analysis of the closed-loop system has been conducted based on Lyapunov theory. Lastly, the effectiveness of the proposed controller has been verified through a visual simulation and ground tests. The results show that our proposed controller has higher tracking precision, better robustness and stronger ability of disturbance rejection.