On the elliptic path of an end-effector for an anthropomorphic manipulator

The application of an anthropomorphic manipulator with an end-effector traveling along a path whose projection on the rz-plane of a cylindrical coordinate system as an ellipse is considered. It is shown that the coordinates along the path may be expressed in terms of (pi t + zeta), where zeta(t) is an unknown function of time to be determined for an optimal velocity distribution along the path by minimizing the total energy used by the manipulation system. As a computational technique, zeta is expressed as a finite series, satisfying the boundary conditions at two terminal points, with the optimal coefficients lambda sub n, f sub n, and k sub n to be evaluated. Two different values for numerical illustration are selected: the smallest value for the problem under investigation, and 0.995 (close to the largest value). It is found that the energy optimization is for the most part due to lambda sub 1 and f sub 1, and consequently a rapid convergence is ensured.