Liapunov Stability of Force-Controlled Grasps with a Multi-Fingered Hand

Abstract

Holding an object stably is a building block for dexterous ma nipulation with a multi-fingered hand. In recent years a rather large body of literature related to this topic has developed. These works isolate some desired property of a grasp and use this property as the definition of stable grasp. To varying degrees, these approaches ignore the system dynamics. The purpose of this article is to put grasp stability on a more basic and fundamental foundation by defining grasp stability in terms of the well-established stability theory of differential equations. This approach serves to unify the field and to bring a large body of knowledge to bear on the field. Some relationships be tween the stability concepts used here and the previously used grasp stability concepts are discussed. A hierarchy of three levels of approach to the problem is treated. We consider that the grasp force applied is a basic consideration, in terms of ensuring both that there is sufficient force to prevent dropping the object and that the forces are not too large to cause break age. As a result, we first investigate the use of constant force grasps. It is shown that with the proper combination of finger locations and grasp forces, such grasps can be Liapunov sta ble, and methods are presented that help find such grasps. It is also shown that such grasps cannot be asymptotically stable. To produce asymptotic stability, one must alter the forces ap plied to an object when the object deviates from equilibrium, and a linear feedback force law is given for this. It results in local asymptotic stability guaranteeing convergence to the desired grasp equilibrium from all states within a region of attraction in the state space. Some of the results are similar to results obtained previously, but this time they have a stronger meaning in terms of the dynamic response of the system. In the third level, a nonlinear force law is given that, to within certain limitations, produces global asymptotic grasp stability, so that all initial states are guaranteed to converge to the de sired grasp equilibrium. The method is robust to large classes of inaccuracies in the implementation.