This paper addresses the problem of robust trajectory tracking for uncertain robot manipulators within a priori fixed-time. A new sliding surface is first proposed and a robust control is developed for ensuring global approximate fixed-time convergence. Global approximate fixed-time convergence of tracking errors is proven that the position tracking errors globally converge to an arbitrary small set centered on zero within a uniformly bounded time and then go to zero exponentially. It is also proved that there exists a uniformly bounded a priori convergence time and such a bound is independent of the initial states. Advantages of the proposed approach include approximate fixed-time stability featuring faster transient and higher steady-state tracking precision and singularity-free. Numerical simulations and experimental results validate the effectiveness and improved performance of the proposed approach.