| In this thesis we develop a mathematical theory for optimizing the kinematic performance of robotic mechanisms, and as a main application obtain a collection of analytical tools for robot design. Judging from the sheer variety of kinematic chains found in nature, and given the wide range of robotic assembly tasks, it is unlikely that a single universal kinematic chain for robotics can be found. Nevertheless, two clearly desirable properties of general robotic mechanisms are that their workspace be large, and that they be able to generate motions and apply forces in arbitrary directions as easily as possible. This latter quality of a mechanism is generally what is meant by its dexterity. Clearly dexterity and workspace volume are intrinsic to a mechanism, so that any mathematical formulation of these properties should necessarily be independent of the particular coordinate representation of the kinematics.;Our focus in this thesis will be on the kinematic aspects of dexterity. By regarding the forward kinematics of a mechanism as defining a mapping between Riemannian manifolds, the coordinate-free language of differential geometry provides a natural setting for addressing the kinematic dexterity and workspace volume of a mechanism. An important consequence of this approach is that the geometric and topological structures of both the joint and work spaces are respected. One novel aspect of this thesis in this context is an engineering application of the theory of harmonic maps. Specifically, we show that the functional associated with harmonic mapping theory provides a natural measure of the kinematic dexterity of a mechanism; we call this measure the kinematic distortion. Extremizing this measure of dexterity then determines a unique design for the basic classes of mechanisms. We explore the relationship between kinematic dexterity and workspace volume, and compare this coordinate-free measure of kinematic dexterity with some other coordinate-free dexterity measures. Our results indicate that in order to fairly compare the kinematic dexterity between two mechanisms, the kinematic distortion of each mechanism should be normalized by the corresponding workspace volume. We also suggest some ways in which the dynamics of a mechanism might be included into a mathematical formulation of dexterity. |
