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Research Papers: Mechanisms and Robotics

# New Considerations on the Theory of Type Synthesis of Fully Parallel Platforms

[+] Author and Article Information
José M. Rico

Facultad de Ingeniería Mecánica, Eléctrica y Electrónica, Departamento de Ingeniería Mecánica, Universidad de Guanajuato, Salamanca, 36730 Guanajuato, Mexicojrico@salamanca.ugto.mx

J. Jesús Cervantes-Sánchez

Facultad de Ingeniería Mecánica, Eléctrica y Electrónica, Departamento de Ingeniería Mecánica, Universidad de Guanajuato, Salamanca, 36730 Guanajuato, Mexicojecer@salamanca.ugto.mx

Facultad de Ingeniería Mecánica, Eléctrica y Electrónica, Departamento de Ingeniería Mecánica, Universidad de Guanajuato, Salamanca, 36730 Guanajuato, Mexicoaltadeo@itesi.edu.mx

Gerardo Israel Pérez-Soto

ITESM, Campus Laguna Paseo del Tecnológico No. 751, Colonia Ampliación La Rosita, Torreón, 27250 Coahuila, Mexicog.perez@itesm.mx

Juan Rocha-Chavarría

Universidad Politécnica de San Luis Potosí, Iturbide No. 140, Centro Histórico, San Luis Potosí, 78000 San Luis Potosí, Mexicojuan.rocha@upslp.edu.mx

All these contributions, as well as this present contribution, do not deal with the dimensional synthesis of parallel platforms, where McCarthy and co-workers (13-14) produced quite remarkable results.

In the rest of the paper, only the relative displacements of the moving platform with respect to the fixed platform are of interest; therefore, the subscript $m/f$ will be removed.

For an in-depth treatment of the basic notions necessary to develop this analysis, the reader is referred to Ref. 43.

Otherwise, this case would be equal to case No. $2$.

Of course, it is also possible that these subsets of the Euclidean group that still represent possible displacements between the moving and fixed platforms of a parallel platform may be produced by the composition of three or more subgroups whose composition is not a subgroup. However, this possibility will not be considered here.

In order to show the parallel platform in a more detailed manner, this triangular frame is not shown completely.

It is easy to note the presence of passive prismatic pairs in the parallel platform shown in Fig. 1. These passive prismatic pairs are prone to self-locking. However, at this stage, only the type synthesis is considered. The remotion of the passive prismatic pairs can be dealt with at the mechanical design stage. One possible solution is the substitution of these passive prismatic pairs by quadrilaterals connected to parallel revolutes.

J. Mech. Des 130(11), 112302 (Sep 23, 2008) (9 pages) doi:10.1115/1.2976447 History: Received March 20, 2007; Revised June 09, 2008; Published September 23, 2008

## Abstract

This contribution presents new considerations on the theory of type synthesis of fully parallel platforms. These considerations prove that the theory of type synthesis of fully parallel platforms can be dealt with by analyzing two types of fully parallel platforms, where the displacements of the moving platform generate a subgroup of the Euclidean group, $SE(3)$, including in this type, 6DOF parallel platforms and lower mobility platforms or, more precisely, parallel platforms where the displacements of the moving platforms generate only a subset of the Euclidean group. The theory is based on an analysis of the subsets and subgroups of the Euclidean group, $SE(3)$, and their intersections. The contribution shows that the different types of parallel platforms are determined by the intersections of the subgroups or subsets, of the Euclidean group, generated by the serial connecting chains or limbs of the parallel platform. From an analysis of the intersections of subgroups and subsets of the Euclidean group, this contribution presents three possibilities for the type synthesis of fully parallel platforms where the displacements of the moving platform generate a subgroup of the Euclidean group, and two possibilities for the type synthesis of fully parallel platforms where the displacements of the moving platforms generate only a subset of the Euclidean group. An example is provided for each one of these possibilities. Thus, once these possible types of synthesis are elucidated, the type synthesis of fully parallel platforms is just reduced to the synthesis of the serial connecting chains or limbs that generate the required subgroups or subsets of the Euclidean group.

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Copyright © 2008 by American Society of Mechanical Engineers
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## Figures

Figure 1

Classification and inclusion relationships between the subgroups of the Euclidean group

Figure 2

A parallel manipulator

Figure 3

Intersection of subgroups equal to a common subgroup HR when k=4

Figure 4

Intersection of subgroups Hj equal to a subgroup HR when k=4

Figure 5

The subgroup HR contained in the intersection of subsets Lj equal to a subset La when k=4

Figure 6

Spatial translation parallel platform where the serial connector chains generate spatial translation subgroup themselves

Figure 7

Spatial translation parallel platform where the serial connector chains generate subgroups whose intersection generates spatial translation subgroup

Figure 8

Spatial translation parallel platform where the serial connector chains generate subsets whose intersection contains the spatial translation subgroup

Figure 9

Intersection of subsets, Lj, that is equal to the required subset, LR, when k=4

Figure 10

Intersection of subsets, Lj, that is equal to the required subset, LR, when k=4

Figure 11

Parallel platform where the required displacement set forms only a subset of the Euclidean group SE(3), and all the serial connector chains have the same required displacement set

Figure 12

Parallel platform where the required displacement set forms only a subset of the Euclidean group SE(3), and not all the serial connector chains have the same required displacement set

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