Mecanum wheeled robots can exhibit serious slippage problems because of the discontinuous contact between the wheels and the ground which negatively influences the overall navigation quality. Addressing this problem, the aim of this paper is to demonstrate how a learning-based method can be used for the estimation of the drifting error from multiple sensors with distinct measurement types. Here, a recurrent neural network (RNN)-based drift compensation algorithm is proposed for the estimation of the positioning drift. In order to improve the positioning performance in dead reckoning the estimated drift is used within the real-time control loop for proper modification of the motion trajectory. During the training phase, the data acquired from the acceleration sensors attached to the robot chassis and the encoders of the wheels of the robot are used as the main features to train a gated recurrent unit-based RNN. The drift estimator is trained using the computer-generated reference position data, and the response position data which is measured using an optoelectronic motion tracking device. The performance of the proposed learning-based drift estimation and control algorithm is validated through a series of experiments. The responses obtained from the experiments are graphically illustrated and the improvements in the positioning performances are numerically evaluated. The results obtained from the experiments illustrate the effective performance of the proposed algorithm by considerably decreasing the positioning errors.