Group Theoretical Setting of Manipulators Kinematics and Control

Abstract

This paper sets some questions on kinematic and dynamic modelling and control of manipulators through vector parametrization of the special orthogonal group SO(3), which presense can not be avoided in the description of the rotation motions. All vectors belonging to the real three dimensional space R3 with definite composition law form a group, isomorphic to a part of SO(3). Remaining part is covered by another chart with the same codomain. Any vector from this group which we shall call vector-parameter has a clear physical sense: it coincides with the rotational axis and its module is equal to the tangent of the half rotational angle. In this aspect all analytic presentations of the direct and inverse kinematic problems of the manipulators systems as of the dynamic equations of motion contains the theory of algebraic or differential equations respectively over a group. The aim of the paper is to show that in the suggested approach, be cause of the simple composition law, the number of the multiplications in the position problem of manipulator systems is reduced with about AO p.c. This makes it more appropriate and convinient for complex spatial anthropomorphic structure description and for on-line manipulator control.