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This paper presents an adaptive proportional-integral-derivative fractional-order nonsingular terminal sliding mode (PID-FONTSM) control for the cable-driven manipulators using time-delay estimation (TDE). The proposed control uses TDE technique to obtain the lumped system dynamics and brings in an effective model-free structure. Then, a PID-FONTSM surface is designed to further enhance the control performance. The proposed PID-FONTSM surface uses the FONTSM error dynamic as its inner loop and utilises a PID-type surface with FONTSM variable as its outer loop. Benefitting from this structure, the proposed PID-FONTSM surface enjoys the advantages from both PID and FONTSM simultaneously. Afterwards, an adaptive algorithm is developed, to timely and precisely regulate the control gain for the robust term. The proposed control method is model-free, highly accurate and essentially continuous thanks to the TDE technique and developed PID-FONTSM surface and adaptive algorithm. The stability of the closed-loop control system under TDE and PID-FONTSM dynamics is proved using Lyapunov stability theory. Finally, comparative simulations were conducted to verify the effectiveness and superiorities of our proposed control method over the existing methods. [ABSTRACT FROM AUTHOR]