On the Elliptic Path of an End-Effector for an Anthropomorphic Manipulator

Abstract

This paper deals with the problem of an anthropomorphic manipulator with an end-effector traveling along a path whose projection on the rz-plane of a cylindrical coordinate system is an ellipse. The elliptic motion, coupled with an angular movement in the ϕ-direction, provides a smooth transition for motion reversal at both terminal points. To im prove the smoothness of reversal, one may vary the elliptic eccentricity to alter the shape of the path. The analysis reveals that the coordinates along the path may be expressed in terms of (πt/τ + ζ), where ζ( t) is an unknown function of time to be determinedfor an optimal velocity distribution along the path by minimizing the total energy used by the manipu lation system. The computational technique employed in this paper is to express ζ as a finite series, satisfying the bound ary conditions at two terminal points, with the optimal coeffi cients λ_n, f_n, and k_n to be evaluated. For numerical illustra tion, two different values of eccentricity are chosen: one is the smallest value for the problem under investigation, and the other is 0.995 (close to the largest value). The results indicate that the energy optimization is essentially due to λ ₁ and f ₁, which ensures a rapid convergence. The contributions due to k_n are usually quite small, while the contributions due to iteration are relatively insignificant.