A force-closure test function for an n-finger grasp on a planar object with friction is presented. All n-finger grasps can be represented by an n-dimensional contact space. The critical conditions of the test functions are used to define force-closure curves which are the boundaries of force-closure sets in the contact configuration space. It is shown that the force-closure sets can be decomposed into subsets in which m (m>n) fingers satisfy force closure. It is proven that m=6 is an upper bound on the older of the force-closure subsets. The characteristics of these subsets are discussed, and an algorithm to enumerate them is given. The application of the real function and the contact configuration space formulation to multifinger object manipulation and finger gait planning is demonstrated by an example.
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