Mathematical Analysis on the Relation between Object Orientation and Contact Forces in the 2D Grasp with Frictions

This paper reports an analysis of 2D grasp where a convex rigid object is grasped by two contact points with frictions. 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 and friction force at each contact point. The formulation of this problem includes 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 takes a local minimal value at the situation that the intersection point of the upper sides of each friction cones at two contact points is located in the opposite direction of the gravity from the center of mass of the grasped object.