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We present a strategy for grasping of real world objects with two anthropomorphic hands, the three-fingered 9- DOF hydraulic TUM and the very dextrous 20-DOF pneumatic Bielefeld Shadow Hand. Our approach to grasping is based on a reach-pre-grasp-grasp scheme loosely motivated by human grasping. We comparatively describe the two robot setups, the control schemes, and the grasp type determination. We show that the grasp strategy can robustly cope with inaccurate control and object variation. We demonstrate that it can be ported among platforms with minor modifications. Grasping success is evaluated by comparative experiments performing a benchmark test on 21 everyday objects.