A curvature-based bound on the number of frictionless fingers required to immobilize three-dimensional objects

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This paper presents a curvature-based bound on the number of frictionless fingers or fixtures required to immobilize 3D objects. A recently developed second-order mobility theory has shown that in addition to first-order geometrical effects, second-order or curvature effects play an important role in the kinematics of contact. We show that when second-order effects are included, four convex fingers or fixtures with sufficiently flat curvature can immobilize any generic smooth or polyhedral 3D object. The derivation of the improved bound proceeds by first constructing a suitable equilibrium grasp of the given object, called a pre-immobilizing grasp. Depending on the object's geometry, a pre-immobilizing grasp may have two, three, or four contacts. The conversion of a pre-immobilizing grasp to a four-finger immobilizing grasp depends on the object's curvature at the contacts. This curvature can be convex, concave, or saddle-like. Since a pre-immobilizing grasp can have k = 2, 3, 4 contacts, there are 31 cases to consider. We construct immobilizing grasps for all of these cases, and present simulation results showing the immobilization of selected object types.