Algebraic test of relative form closure grasps with its implementation

This paper presents an approach to algebraic detection of relative form closure for a multi-fingered grasp G and gives an attention to the classification of relative form closures. Whether or not a grasp G is of relative form closure has been verified by the geometric criterion 0？rintC[G]（Xiong Y L, Xiong X R. Algebraic structure and geometric interpretation of rigid complex fixture system. IEEE Trans Autom Sci Eng, 2007, 4： 252–264; Xiong Y L. Theory of point contact restraint and qualitative analysis of robot grasping. Sci China Ser A-Math, 1994, 37： 629–640）. Our aim is at transferring the geometric criterion into an algebraic formulation by a theorem that a grasp G is relative form closure, if, and only if the origin 0 can be represented as a positive convex combination of G. So we develop a constructive procedure called ri-simplex algorithm to find a positive convex combination of G when it has relative form closure. Form closure is considered as a special case of relative form closure, when the grasp G can affinely span the whole space, while partly relative form closure is referred to as form closed relative to a subset G1 of G.