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We address the problems of computing form-closure regions as well as computing optimal fingertip locations yielding n-finger form-closure grasps of three-dimensional (3-D) objects. Given grip points of k fingers not in form-closure, we propose a sufficient and necessary condition for the n-k fingers to achieve an n-finger form-closure grasp. Based on the condition, it is demonstrated that the problem of computing form-closure regions can be formulated as an existence problem of a solution for a set of inequalities. A sufficient condition is also proposed to simplify the computation of form-closure regions. As to searching for optimal fingertip locations, we transform the problem to a nonlinear programming problem subject to a set of constraints arising from that sufficient and necessary condition. Furthermore, a performance index is defined by measuring the distance between the center of mass of the grasped object and …