The aim of this paper is to design a proposed non-linear fractional order proportional-integral-derivative neural (NFOPIDN) controller by modifying and improving the performance of fractional order PID (FOPID) controller through employing the theory of neural network with cognitive optimization techniques for the differential - drive wheeled mobile robot (WMR) multi-input multi-output (MIMO) system in order to follow a pre-defined trajectory. In this paper a cognitive bacterial foraging optimization algorithm (BFOA) has been utilized to find and tune the parameters of the proposed (NFOPIDN) controller and then find the optimal torque control signals for the differential - drive WMR. The simulation results show that the proposed controller can give excellent performance in terms of compared with other works (minimized tracking error for Ranunculoid-curve trajectory, smoothness of torque control signals obtained without saturation state and no sharp spikes action as well as minimum number of memory units needed for the structure of the proposed NFOPIDN controller).