To improve the performance and the robustness of finite control set predictive current control (PCC) under the influence of current measurement error in the actual system, robust exploration based finite control set predictive current control (RE-PCC) is proposed in this paper. Firstly, to analyze directly transient and steady state of system operation, a voltage cost function without weighting factor is designed based on predicted voltage error vector, reference voltage vector and evaluated voltage vector. Secondly, the robust exploration is designed to overcome the influence of measurement error by combing the amplitude and phase attribute between the predicted voltage error vector and the ideal reference voltage vector. Thirdly, based on the robust exploration, the desired sector and the prior vector set is derived for cost function evaluation. It is great benefit to computational burden and switching losses by reducing the seven vectors to three. Finally, the RE-PCC method is experimentally compared with an improved PCC method in terms of robustness, current quality, torque ripple, dynamic characteristics and efficiency. The experimental results verify the high robustness and superior control performance of RE-PCC method in surface-mounted permanent magnet synchronous motor (SPMSM) drives.