This paper investigates the problem of vehicular platoon control with asymmetric actuator saturation. First, an improved Gaussian error function (IGEF)-based approximation strategy is presented to replace saturation nonlinearity in a smooth way. Then, an adaptive fixed-time control algorithm via sliding mode control is proposed with which the individual vehicle stability and string stability can be guaranteed in a given time, and the settling time can be prescribed in advance as needed. Besides, a set of adaptation laws are introduced to deal with uncertain parameters and unknown disturbances. Finally, the effectiveness of the proposed control scheme is illustrated by numerical simulations.