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In this short paper, we consider the nonlinear control of kinematically redundant robot manipulators. Specifically, we use a Lyapunov technique to design a model-based nonlinear controller that achieves exponential link position and subtask tracking. We note that the control strategy does not require the computation of positional inverse kinematics and does not place any restriction on the self-motion of the manipulator; hence, the extra degrees of freedom are available for subtasks (i.e., maintaining manipulability, avoidance of mechanical limits and obstacle avoidance). Experimental implementations on a redundant robot are also included to illustrate the performance of the proposed control law.