0
RESEARCH PAPERS

# A More General Mobility Criterion for Parallel Platforms

[+] Author and Article Information
J. M. Rico

Facultad de Ingeniería Mecánica, Eléctrica y Electrónica, Universidad de Guanajuato, Calle Tampico No. 912, 36730 Salamanca, Gto. Méxicojrico@salamanca.ugto.mx

L. D. Aguilera

Departamento de Ingeniería Mecánica, Instituto Tecnológico de Celaya, Celaya, Gto. 38010, México

J. Gallardo

Departamento de Ingeniería Mecánica, Instituto Tecnológico de Celaya, Celaya, Gto. 38010, Méxicogjaime@itc.mx

R. Rodriguez

Departamento de Ingeniería Mecánica, Instituto Tecnológico de Celaya, Celaya, Gto. 38010, Méxicoramunr@itc.mx

H. Orozco

Departamento de Ingeniería Mecánica, Instituto Tecnológico de Celaya, Celaya, Gto. 38010, Méxicohoracio@itc.mx

J. M. Barrera

Centro Regional de Optimización y Desarrollo de Equipo, CRODE-Celaya, Diego Arenas 901, Celaya, Gto. 38020 México

For an in-depth treatment of the basic notions necessary to develop the criterion, the reader is referred to (7-9,16).

It should be noted that the possibility that

$Aam∕f=∩j=1kAjm∕f=∅$
is impossible since $0⃗∊Ajm∕f$ for any subalgebra of the Lie algebra $se(3)$ of the Euclidean group.

The notation $A∕B$ indicates the set of all elements that belong to $A$ and do not belong to $B$.

The same architecture has been also proposed by Carricato and Parenti-Castelli (18-19) and Kong and Gosselin (20-21).

For the mobility analysis, this second pair of rigid bodies will not be counted.

J. Mech. Des 128(1), 207-219 (Jul 15, 2005) (13 pages) doi:10.1115/1.2118687 History: Received January 14, 2005; Revised July 14, 2005; Accepted July 15, 2005

## Abstract

This contribution presents a more general mobility criterion applicable to parallel platforms, unlike previously employed mobility criteria based on the well-known Kutzbach-Grübler criterion that often fails to provide the correct number of degrees of freedom of parallel manipulators, the mobility criterion introduced in this contribution provides the correct number of degrees of freedom for a wider class of parallel manipulators. Furthermore, the analysis provides insight into why the criteria based on the Kutzbach-Grübler criterion often fails. Moreover, the recently developed criterion computes the passive degrees of freedom in parallel platforms and the mobility of some classes of kinematically defficient parallel platforms. Finally, it is important to note that this criterion is based on an analysis of the subalgebras of the Lie algebra, $se$(3), also known as screw algebra, of the Euclidean group, $SE$(3). Moreover, it should be emphasized that, unlike other attempts to develop a criterion, the criterion developed in this contribution does not require any consideration of reciprocal screws. As is common in many areas of kinematics, the criterion presented here can be also obtained by an analysis of the subgroups of the Euclidean group.

<>

## Figures

Figure 1

A parallel manipulator

Figure 2

A kinematically deficient parallel manipulator proposed by Tanev

Figure 3

A parallel manipulator without relative motion between the platforms and without self-motion

Figure 4

A parallel manipulator without relative motion between the platforms and with self-motion

Figure 5

The Cartesian parallel manipulator proposed by Kim and Tsai

Figure 6

Kim and Tsai’s modified cartesian manipulator

Figure 7

The prism parallel manipulator, proposed by Hervé

Figure 8

The Delta parallel manipulator

Figure 9

Parallel manipulator Gough-Stewart

Figure 10

Parallel platform with inactive pairs

## Discussions

Some tools below are only available to our subscribers or users with an online account.

### Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections