Title: Computing the developable forms of planar and spherical four-bar linkages

Authors: Andrew P. Murray; J. Michael McCarthy

Addresses: Design of Innovative Machines Laboratory, University of Dayton, USA ' Robotics and Automation Laboratory, University of California, Irvine, USA

Abstract: A developable mechanism is a linkage that has its axes located on a developable ruled surface and its links shaped to fit this surface in a stowed configuration. Our goal is to find configurations for both planar and spherical four-bar linkages in which points on the four joint axes lie on a circle. This configuration can form the stowed position of the linkage with its links shaped to form a cylinder in the planar case and a cone for the spherical case called the conformed position. The result is a cylindrical or conical developable mechanism respectively. We provide two derivations for both the planar and spherical cases. One uses the formulas for the radii of the open and crossed form of these mechanisms and the other computes the configuration directly. We demonstrate these results with examples.

Keywords: developable; planar; spherical; linkage; mechanism; four-bar; 4R; cyclic quadrilateral; conformed position.

DOI: 10.1504/IJMRS.2021.115153

International Journal of Mechanisms and Robotic Systems, 2021 Vol.5 No.1/2, pp.22 - 38

Received: 03 Aug 2020
Accepted: 21 Nov 2020

Published online: 08 May 2021 *

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