Static Stability Analysis of Planar Grasps by Multiple Fingers with Redundant Joints

This paper deals with static stability in planar grasps of an object by multiple fingers. Differently from previous research, we focus on the case that each finger has redundant links and joints. Based on contact constraints between the object and fingers, the relationships among displacements of object’s pose, contact positions, and joint positions are formulated. Using the constraints, the redundant joints are reduced to independent parameters. The relationship between the displacement and reaction torque of each joint is modeled as a linear spring, and potential energy of the grasp is formulated. Not only for frictionless sliding contact but also for pure rolling contact, we derive stable conditions on the contact positions and joint positions. Based on the conditions, partially differentiating the potential energy, a wrench (force and moment) vector and a stiffness matrix applied to the object by each finger are derived. Summing up the wrenches and matrices of all the fingers, we obtain a wrench vector and a stiffness matrix of the grasp, and we evaluate the grasp stability. Because of our analytical formulation, grasp parameters such as local curvatures at contact points, joint stiffnesses, etc., are explicitly included in the derived matrices. Partially differentiating the wrenches and matrices by the grasp parameters, we clarify effects of the parameters on the stability. Moreover, the difference between the frictionless sliding contact and pure rolling contact is derived in the wrench vector and the stiffness matrix. Using numerical examples, we validate our analysis.