The nonlinear external disturbances and unmodeled dynamics characteristics have crucial impacts on trajectory tracking control accuracy of a four-wheel mobile robot (FWMR) under complicated working conditions. In this work, an adaptive trajectory tracking controller is designed for the FWMR to achieve the prescribed-prediction performance. On the basis of establishing the FWMR’s dynamics equations, an enhanced prescribed performance function (EPPF) is constructed to restrain the tracking errors of the FWMR within a certain range without requiring the exact initial conditions, thus guaranteeing the transient performance of the control system. Then, an optimal-predictive control (OPC) approach is presented to fulfill the asymptotic stability of the tracking errors of the FWMR. Specifically, the radial basis function neural network (RBFNN) incorporating a minimum parameter learning approach that are implanted into the expected controller is designed to attenuate the nonlinear external disturbances and the unmodeled dynamics of the FWMR. Lastly, comparative simulation investigations are carried out to illustrate the superiority of the proposed EPPF-OPC controller, and moreover, the comparative experiments are further performed to validate the practical effectiveness of the EPPF-OPC controller based on a self-established robot operating system (ROS) test platform of the FWMR.