LEAST SQUARE APPROXIMATIONS ON FINITE SETS OF LINES AS APPLIED TO THE SYNTHESIS OF ADJUSTABLE ROBOTIC MECHANISMS

MACHINE SCIENCE

Abstract

The problem of determining the special lines of a moving body which in m alternating sets of its given positions deviate least, in a least-square sense, from the coaxial line congruences generated by the moving axes of CC dyads with the common fixed axis of rotation is considered. The sought-for approximation is one which minimizes two sums of the squared angular and linear deviations of such lines from the approximating line congruences associated with the CC dyads to be synthesized. Two theorems describing geometrically the necessary conditions for the best least-square approximation of the given m line-position sets by coaxial circular cones and line congruences are formulated. The loci of the special lines under consideration are studied, and a method for their determination is proposed. The theory and method presented in the paper can be readily applied to the synthesis of adjustable parallel robotic mechanisms based on CCC chains (modules) with the lockable middle joints which serve for adjusting the angle and distance between the moving and fixed axes of rotation to realize multiple kinematic tasks. The synthesis of CCC chain is decomposed into two simpler subproblems: a) synthesis of its spherical indicatrix RRR with the lockable middle joint to determine directions of the moving and fixed axes of rotation and m values of their adjustable twist angle, b) synthesis of CCC chain with the known directions of the rotation axes to determine their locations in the corresponding coordinate systems and m values of the adjustable distance between them.