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Research Papers

Simplified Kinematics for a Parallel Manipulator Generator of the Schönflies Motion

[+] Author and Article Information
Jaime Gallardo-Alvarado

Department of Mechanical Engineering,
Instituto Tecnologico de Celaya, TecNM,
Celaya 38010 GTO, Mexico
e-mail: jaime.gallardo@itcelaya.edu.mx

Mohammad H. Abedinnasab

Department of Biomedical Engineering,
Rowan University,
Glassboro, NJ 08028
e-mail: abedin@rowan.edu

Daniel Lichtblau

Wolfram Research,
100 Trade Center Drive,
Champaign, IL 61820
e-mail: danl@wolfram.com

1Corresponding author.

Manuscript received March 4, 2016; final manuscript received September 19, 2016; published online October 25, 2016. Assoc. Editor: Yuefa Fang.

J. Mechanisms Robotics 8(6), 061020 (Oct 25, 2016) (10 pages) Paper No: JMR-16-1055; doi: 10.1115/1.4034884 History: Received March 04, 2016; Revised September 19, 2016

This work is devoted to simplify the inverse–forward kinematics of a parallel manipulator generator of the 3T1R motion. The closure equations of the displacement analysis are formulated based on the coordinates of two points embedded in the moving platform. Afterward, five quadratic equations are solved by means of a novel method based on Gröbner bases endowed with first-order perturbation and local stability of parameters. Meanwhile, the input–output equations of velocity and acceleration are systematically obtained by resorting to reciprocal-screw theory. In that concern, the inclusion of pseudokinematic pairs connecting the limbs to the fixed platform and a passive kinematic chain to the robot manipulator allows to avoid the handling of rank-deficient Jacobian matrices. The workspace of the robot is determined by using a discretized method associated to its inverse–forward displacement analysis, whereas the singularity analysis is approached based on the input–output equation of velocity. Numerical examples are provided with the purpose to show the application of the method.

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Figures

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Fig. 1

The layout of the parallel manipulator under study

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Fig. 2

Two direct singularities, the lines $i(i=1,2,3,4) are (i) coplanar and (ii) parallel

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Fig. 3

Example of inverse singularity (left) and escapement from the singularity (right)

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Fig. 7

Time history of the kinematics of the moving platform

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Fig. 5

The two poses of example 1

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Fig. 6

Workspace of the robot

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Fig. 4

Example 1: Geometry of the moving platform

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