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We present a novel robotic grasp controller that allows a sensorized parallel jaw gripper to gently pick up and set down unknown objects once a grasp location has been selected. Our approach is inspired by the control scheme that humans employ for such actions, which is known to centrally depend on tactile sensation rather than vision or proprioception. Our controller processes measurements from the gripper's fingertip pressure arrays and hand-mounted accelerometer in real time to generate robotic tactile signals that are designed to mimic human SA-I, FA-I, and FA-II channels. These signals are combined into tactile event cues that drive the transitions between six discrete states in the grasp controller: Close, Load, Lift and Hold, Replace, Unload, and Open. The controller selects an appropriate initial grasping force, detects when an object is slipping from the grasp, increases the grasp force as needed, and judges when to release an object to set it down. We demonstrate the promise of our approach through implementation on the PR2 robotic platform, including grasp testing on a large number of real-world objects.