The Kinematic Image of 2R Dyads and Exact Synthesis of 5R Linkages

TL;DR

This paper characterizes the kinematic image of 2R dyads as a ruled quadric, enabling a novel synthesis method for 5R linkages by modifying existing algorithms for 6R linkages.

Contribution

It introduces a geometric characterization of 2R dyads and adapts a synthesis algorithm to create 5R linkages with coincident axes.

Findings

Characterization of 2R dyad kinematic image as a ruled quadric

Development of a modified synthesis algorithm for 5R linkages

Explanation of properties using twisted cubics on quadrics

Abstract

We characterise the kinematic image of the constraint variety of a 2R dyad as a regular ruled quadric in a 3-space that contains a "null quadrilateral". Three prescribed poses determine, in general, two such quadrics. This allows us to modify a recent algorithm for the synthesis of 6R linkages in such a way that two consecutive revolute axes coincide, thus producing a 5R linkage. Using the classical geometry of twisted cubics on a quadric, we explain some of the peculiar properties of the the resulting synthesis procedure for 5R linkages.