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The sensor measurement errors have a strong influence on the effective control and the stability of vehicle platoon. Generally, most studies regard the sensor measurement errors as the random series (such as Gaussian, Poisson distribution) whose distributions are known, such that the specific mathematical methods can be employed to remove their impacts. However, there are no mature and effective approaches for the bounded measurement errors in the control system. With respect to this, the nonlinear vehicle dynamics is studied, and consider the bounded measurement errors of onboard sensors and the directed communication topology among vehicles, a sliding mode based control algorithm of vehicle platoon is proposed. This method can effectively address the vehicle platoon control problem with bounded sensor measurement errors. Moreover, by using prescribed performance control, the tracking error can be constrained, such that the string stability of vehicle platoon is guaranteed. Finally, the feasibility and effectiveness of the proposed control method are validated through numerical simulations.