Analysis of fourth order manipulator kinematics using conic sections

A technique for analyzing robots with fourth-order inverse kinematic solutions is introduced. The solution of the inverse kinematics problem is restated as a pencil of conics. In this representation, the order of the solution reduces when one of the conics in the pencil becomes degenerate and noncentral. Properties of conics are exploited to define and develop special manipulator geometries. The method is illustrated by a detailed examination of all revolute (R) robots which have a three-axis wrist (i.e. 3R regional manipulators). All criteria for reduction of the inverse kinematics of this type of robot are determined, resulting in some new robot designs. These designs constitute all regional robots with kinematics that require the solution of only quadratic equations