Object orientation in two dimensional grasp with friction towards minimization of gripping power

This article reports an analysis of two dimensional grasp where a convex rigid object is grasped by two contact points with friction. The purpose is to find the object orientation that minimizes the norm of the contact force vector, each element of which is composed from the normal force and friction force at each contact point. The formulation of this problem requires some equality or inequality conditions. In the analysis, the solution of the equality conditions is parameterized at first. Based on the fact that the norm of the contact force vector becomes monotonic increasing function of this parameter, the minimal parameter values are calculated by means of the piecewise analysis. Using the relation between the friction coefficient and the apex angle of the friction cone effectively, the following result is obtained: the norm of the contact force, i.e, gripping power becomes locally minimal at the object orientation where the intersection point of the upper sides of two friction cones is located in opposite direction of the gravity from the center of mass of the grasped object.