Recently, a time-varying quadratic programming (QP) framework that describes the tracking operations of redundant robot manipulators is introduced to handle the kinematic resolutions of many robot control tasks. Based on the generalization of such a time-varying QP framework, two schemes, i.e., the Repetitive Motion Scheme and the Hybrid Torque Scheme, are proposed. However, measurement noises are unavoidable when a redundant robot manipulator is executing a tracking task. To solve this problem, a novel integration-enhanced recurrent neural network (IE-RNN) is proposed in this paper. Associating with the aforementioned two schemes, the tracking task can be accurately completed by IE-RNN. Both theoretical analyses and simulations results prove that the residual errors of IE-RNN can converge to zero under different kinds of measurement noises. Moreover, practical experiments are elaborately made to verify the excellent convergence and strong robustness properties of the proposed IE-RNN.