We consider the problem of reorienting a rigid object with arbitrary known shape on a table using a two-finger pinch gripper. Reorienting problem is challenging because of its non-smoothness and high dimensionality. In this work, we focus on solving reorienting using pivoting, in which we allow the grasped object to rotate between fingers. Pivoting decouples the gripper rotation from the object motion, making it possible to reorient an object under strict robot workspace constraints. We provide detailed mechanical analysis to the 3D pivoting motion on a table, which leads to simple geometric conditions for its stability. To solve reorienting problems, we introduce two motion primitives: pivot-on-support and roll-on-support, and provide an efficient hierarchical motion planning algorithm with the two motion primitives to solve for the gripper motions that reorient an object between arbitrary poses. To handle the uncertainties in modeling and perception, we make conservative plans that work in the worst case, and propose a robust control strategy for executing the motion plan. Finally we discuss the mechanical requirements on the robot and provide a "two-phase" gripper design to implement both pivoting grasp and firm grasp. We demonstrate the effectiveness of our method in simulations and multiple experiments. Our algorithm can solve more reorienting problems with fewer making and breaking contacts, when compared to traditional pick-and-place based methods.
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