This paper focuses on achieving rapid convergence in finite time for robust trajectory tracking of robotic manipulators, which upper-bound uncertainties are unknown. To address the issues of excessive control torque and control reversal at the initial stage, a novel Robust Adaptive Fast Terminal Sliding Mode Control (RAF-TSMC) method is developed. The proposed method provides faster transient response, higher steady-state tracking accuracy, and can eliminate singularities and mitigate chattering effects. To enhance the physical collaboration in human-robot interactions, a novel Nonlinear Admittance Control method with RAF-TSMC is proposed, which benefits from adaptive compliance characteristics of Nonlinear Admittance Control with RAF-TSMC. This improves suppleness and flexibility of human-robot interaction in uncertain environments. Finally, the proposed controller is validated using a two-link robotic manipulator subjected to uncertainty and external forces. The results demonstrate that the proposed RAF-TSMC controller can effectively minimize tracking errors and convergence time, and the proposed Nonlinear Admittance Control with RAF-TSMC can enhances the flexibility and adaptability of the robot system.