This paper introduces an adaptive generalized super-twisting sliding mode control algorithm (AGSTC) designed for n-degree-of-freedom (DOF) robot manipulators. Compared to the traditional super-twisting algorithm (STA), the standout feature of the proposed method is its substitution of the discontinuous term in the conventional STA with a fractional power term. This alteration significantly enhance the overall performance of the traditional STA method. When the upper and lower bounds of the disturbance are unknown, the corresponding parameters are adjusted by adaptive control to enhance the robustness of the system. In order to verify the superiority of the control algorithm, the experiment is carried out with 6-DOF industrial manipulator under different working conditions, and the experimental results show that the tracking accuracy of AGSTC is much higher than that of STA.