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We present a novel efficient method addressing the inverse differential kinematics problem for redundant manipulators in the presence of different hard bounds (joint range, velocity, and acceleration limits) on the joint space motion. The proposed SNS (Saturation in the Null Space) iterative algorithm proceeds by successively discarding the use of joints that would exceed their motion bounds when using the minimum norm solution and reintroducing them at a saturated level by means of a projection in a suitable null space. The method is first defined at the velocity level and then moved to the acceleration level, so as to avoid joint velocity discontinuities due to the switching of saturated joints. Moreover, the algorithm includes an optimal task scaling in case the desired task trajectory is unfeasible under the given joint bounds. We also propose the integration of obstacle avoidance in the Cartesian space by properly modifying on line the joint bounds. Simulation and experimental results reported for the 7-dof lightweight KUKA LWR IV robot illustrate the properties and computational efficiency of the method.