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Research Papers

# Isoconstrained Parallel Generators of Schoenflies Motion

[+] Author and Article Information
Chung-Ching Lee

National Kaohsiung University of Applied Sciences, 415 Chien Kung Road, Kaohsiung, 80782 Taiwan, R.O.C.cclee@cc.kuas.edu.tw

Jacques M. Hervé

Ecole Centrale Paris, Grande Voie des Vignes, F-92295 Chatenay-Malabry, Francejacques.herve07@orange.fr

J. Mechanisms Robotics 3(2), 021006 (Mar 30, 2011) (10 pages) doi:10.1115/1.4003690 History: Received April 26, 2009; Revised March 04, 2010; Published March 30, 2011; Online March 30, 2011

## Abstract

Based on the Lie-group-algebraic properties of the displacement set, the 4DOF primitive generators of the Schoenflies motion termed $X$-motion for brevity are briefly recalled. An $X$-motion includes 3DOF spatial translation and any 1DOF rotation provided that the axes are parallel to a given direction. The serial concatenation of two generators of 4DOF $X$-motion produces a 5DOF motion called double Schoenflies motion or $X-X$-motion for brevity, which includes 3DOFs of translations and any 2DOFs of rotations if the axes are parallel to two independent vectors. This is established using the composition product of two Lie subgroups of $X$-motion. All possible 5DOF serial chains with distinct general architectures for the generation of $X-X$-motion are comprehensively introduced in the beginning. The parallel setting between a fixed base and a moving platform of two 5DOF $X-X$ limbs, under particular geometric conditions, makes up a 4DOF isoconstrained parallel generator (abbreviated as $IPG-X$) of a Schoenflies motion set. “Isoconstrained” is synonymous with “nonoverconstrianed,” and the corresponding chains are trivial chains of the 6D group of general 6DOF motions and can move in the presence of manufacturing errors. Moreover, related families of $IPG-X$s are also deducted by using the reordering or the commutation of the factor method, which yields more 5D subsets of displacements containing also the $X$-motion of the end effector. In that way, several novel general-type architectures of 4DOF parallel manipulators with potential applications are synthesized systematically in consideration of the actuated pairs near the fixed base.

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Topics: Motion , Generators , Chain

## Figures

Figure 1

General architectures of nine X-motion primitive generators

Figure 2

General architectures for X−1(u)X−2(v) generators of X-X-motion

Figure 3

General architectures of X(u)X−3(v) generators of X-X-motion

Figure 4

A [(HHH)w(HH)u]-∥-[(HHH)w(HH)v]IPG-X with two X-X limbs

Figure 5

General IPG-Xs with X−1(w)X−2(ui) limbs

Figure 6

A [(HHH)u(HH)w]-∥-[(HHH)v(HH)w]IPG-X with two X-X limbs

Figure 7

A [(HH)w(HH)uHw]-∥-[(HH)w(HH)vHw]IPG-X

Figure 8

Two IPG-X having X−2(w)X−2(ui)X−3(w) limbs

Figure 9

A [Hw(HH)u(HH)w]-∥-[Hw(HH)v(HH)w]IPG-X

Figure 10

A [Hw(HHH)uHw]-∥-[Hw(HHH)vHw]IPG-X

Figure 11

Two IPG-Xs having X−3(w)X−1(ui)X−3(w) limbs

Figure 12

IPG-Xs embodying X(u)X−3(w)∩X(v)X−3(w) and their related ones

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