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Research Papers

# A Unified Algorithm for Analysis and Simulation of Planar Four-Bar Motions Defined With R- and P-Joints

[+] Author and Article Information
Xiangyun Li, Xin Ge, Anurag Purwar

Computational Design Kinematics Lab,
State University of New York at Stony Brook,
Stony Brook, NY 11794-2300

Q. J. Ge

Computational Design Kinematics Lab,
State University of New York at Stony Brook,
Stony Brook, NY 11794-2300
e-mail: Qiaode.Ge@stonybrook.edu

1Corresponding author.

Manuscript received September 26, 2014; final manuscript received December 1, 2014; published online December 31, 2014. Assoc. Editor: Carl Nelson.

J. Mechanisms Robotics 7(1), 011014 (Feb 01, 2015) (7 pages) Paper No: JMR-14-1271; doi: 10.1115/1.4029295 History: Received September 26, 2014; Revised December 01, 2014; Online December 31, 2014

## Abstract

This paper presents a single, unified, and efficient algorithm for animating the coupler motions of all four-bar mechanisms formed with revolute (R) and prismatic (P) joints. This is achieved without having to formulate and solve the loop closure equation for each type of four-bar linkages separately. Recently, we developed a unified algorithm for synthesizing various four-bar linkages by mapping planar displacements from Cartesian space to the image space using planar quaternions. Given a set of image points that represent planar displacements, the problem of synthesizing a planar four-bar linkage is reduced to finding a pencil of generalized manifolds (or G-manifolds) that best fit the image points in the least squares sense. In this paper, we show that the same unified formulation for linkage synthesis leads to a unified algorithm for linkage analysis and simulation as well. Both the unified synthesis and analysis algorithms have been implemented on Apple's iOS platform.

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## References

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## Figures

Fig. 1

The screen shot of GUI on iPad

Fig. 2

A planar four-bar mechanism

Fig. 3

Fig. 4

Fig. 5

Fig. 6

The image curve of a RRRR linkage as intersection of two G-manifolds

Fig. 7

Two assembly modes of a RRRR linkage with their corresponding coupler positions are shown in thick and thin lines, respectively

Fig. 8

The image curve of a RRRP linkage as intersection of two G-manifolds

Fig. 9

Two assembly modes of a RRRP linkage with their corresponding coupler positions are shown in thick and thin lines, respectively

Fig. 10

The image curve of a RRPR linkage as intersection of two G-manifolds

Fig. 11

Two assembly modes of a RRPR mechanism with their corresponding coupler positions are shown in thick and thin lines, respectively

Fig. 12

The image curve of a PRPR linkage as intersection of two G-manifolds

Fig. 13

Two assembly modes of a PRPR linkage with their corresponding coupler positions are shown in thick and thin lines, respectively

Fig. 14

The image curve of a PRRP linkage as intersection of two G-manifolds

Fig. 15

A PRRP linkage with their corresponding coupler positions

Fig. 16

The image curve of a RPPR linkage as intersection of two G-manifolds

Fig. 17

A RPPR linkage with its coupler positions

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