The current contribution introduces a nonsingular fixed-time sliding mode control (SMC) scheme for position and velocity tracking of robot manipulators. The approach avoids singularities by introducing a new sliding surface with the special attribute that the exponent employed to achieve fixed time convergence depends on the tracking error and is smaller than one except when the error is exactly zero whereas the exponent becomes one at zero, which makes the derivative at zero to be well defined. A new theoretical result has been introduced in the form of a lemma to prove this innovative property. Furthermore, model uncertainties are handled by means of a time varying gain given by a polynomial of the powers of the norms of the tracking and velocities errors. The fixed-time convergence is proven employing Lyapunov theory and the result holds globally. Simulation outcomes confirm the developed theory and the advantages of the proposed scheme are shown qualitatively by comparing its performance with well-known equivalent control schemes.