Designing Rotary Linkages for Polar Motions

Abstract

Polar linkages have two degrees-of-freedom (DOF) where one input joint angle controls the length of a radial segment while another controls its angle. Considering a theoretical planar robot model, this mapping between joint angles to output motions can be shown to be energetically advantageous over the ubiquitous two-revolute linkage. Since a polar linkage’s typical construction involves a moving prismatic joint, it is cumbersome to implement alongside rotary electromagnetic actuators offsetting any advantage. In this paper, we present a procedure for designing polar linkages using only revolute joints. The procedure starts with a pre-existing single DOF straight line linkage and then finds the dimensions of a three-link attachment to produce the second DOF. In the end, the straight line linkage actuates the polar length and the attachment actuates the polar angle. The design process is framed under optimization with an objective that is both polynomial and invariant to the number of discretization points. This enables the techniques of numerical continuation to efficiently find complete sets of minima. We demonstrate our procedure with an example in which multiple minima are found including the global minimum. This computed design solution is then fabricated in order to validate the designed kinematics.