In these notes we give the reader a feel for the mathematical problems involved in describing grasping and fine motion manipulation of objects with multifingered robot hands. Multifingered robot hands can be thought of as several robots (fingers) on a common base (palm) cooperatively manipulating all object. It is clear that positioning an object in space, namely specifying its position and orientation needs 6 degrees of freedom. However, dextrously manipulating objects requires far more degrees of freedom especially in the execution of tasks involving picking up an object, regrasping it and using the object. It is here that the study of multifingered hands is important. The study of multifingered hands has a long history not just in the context of robotics but also in the context of prosthesis. In Chapter 2, we set down a brief discussion of the kinematics of a single rigid body, followed by study of contacts and the kinematics of rolling. Rolling Is an especially important way in which finger tips move over the surface of an object in order both to reposition and regrasp the object. In Section 2.4 we study the kinematics of a multifingered hand in terms of the kinematics of the individual fingers. Finally, we define grasp stability and the manipulability of grasps. The appendix contains a derivation of the contact equations in terms of the metric tensor and connection form of the surfaces in contact at the finger tip and object. In Chapter 3, we develop the dynamics of multifingered hands by aggregating the dynamics of individual fingers with the dynamics of the grasped object and the kinematic equations of contact. In Section 3.3 we describe a few different control techniques to follow a specified trajectory for the body and the grasp forces exerted on it. In Chapter 4, we axiomatize the process of regrasping an object by rolling the finger tips on the surface of the object. We show how the problem of finding geodesics for singular or Carnot-Caratheodory metrics is useful in steering the finger tips from one grasp to another. We conclude with some open problems. The discussion of this paper is a summary of our own work and that of others, notably those at Harvard, in the last few years in this area. Detailed references to these appear in the body of the notes.




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