A Practical Quality Index Based on the Octahedral Manipulator

Abstract

The octahedral manipulator is a "3-3" device that is fully in parallel. It has a linear actuator on each of its six legs. The legs connect an equilateral platform triangle to a similar base triangle in a zigzag pattern between vertices. Our proposed quality index takes a maximum value of 1 at a central symmet rical configuration that is shown to correspond to the maxi mum value of the determinant of the 6 x 6 Jacobian matrix of the manipulator. This matrix is none other than that of the nor malized line coordinates of the six leg-lines; for its determi nant to be a maximum, the platform triangle is found to be half of the size of the base triangle, and the perpendicular distance between platform and base is equal to the side of the platform triangle. When the manipulator is actuated so that the octahedron departs from this central configuration, the determinant al ways diminishes, and, as is well known, it becomes zero when a special configuration is reached (the platform then gain ing one or more uncontrollable freedoms). Our quality index λ, 0 ≤ λ ≤ 1, is a constructive measure of acceptable design proportions. The double-spherical joints (there are six) are the source of the critical practical difficulties. Kinematic substitutions can circumvent this problem, but often there is no reasonable alternative to accepting a reduction of maximum quality index through separation by fairly short distances of some or all of the double-ball joints. Too great a separation easily leads too far toward generality; any fully-in-parallel manipulator of more general proportions should be tested for quality against our octahedral form simply because no different form can be superior to a well-designed octahedron from the points of view of structural soundness and ability to apply both forces and controlled displacements to the end-effector platform.