The still shot below shows a Lego prototype for a desktop paper-manipulating robot.
A roll of 3M removable tape is threaded from a supply reel, around a foot near the ground, to a powered take-up reel. A tension bar helps ensure that the tape keeps the shape of the foot. The entire tape assembly is mounted on a piston. The piston extends through the beams at the top of the device; this keeps the foot pointed roughly toward the ground. The bottom end of the piston is attached to a pair of eccentric cams that are connected with a shaft to stay in sync.
If we start with the foot in the position in the photograph and rotate the cams counterclockwise, the foot descends, hits the floor, lifts the rear wheels and pulls the paper under the robot (as the foot moves along the bottom of the cam toward the front of the robot), then drops the rear wheels on the paper, and lifts between the wheels. This is the basic motion for letting go of paper -- the wheels keep the paper on the ground, so pulling the foot up disconnects the tape. Reversing the cam attaches the paper to the tape. The tape is advanced after every grab operation (or some multiple of them) to ensure that the foot's surface is sticky.
The rear wheels are free to rotate. The front wheel is mounted like a caster; a large gear at the top rotates the wheel in place 90 degrees left or right; a motor mounted on the caster powers the wheel. This allows turning in place (with the wheel rotated 90 degrees left) and moving in a straight line, which are two nice operations for simple navigation.
This animated GIF loop shows an earlier prototype of the robot in action. (The shifting black frame makes up for my lack of a tripod; sorry :v). Observe the motion of the piston. At one end of the animation, the foot is exposed behind the robot, holding the front edge of the paper up so that it can be dragged over other piles of paper on the desk. At the other end of the animation, you can see how the weight on the rear wheels holds the paper in place while the piston lifts free of the paper, releasing it from the sticky tape.