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RESEARCH PAPERS

# On Calculating the Degrees of Freedom or Mobility of Overconstrained Linkages: Single-Loop Exceptional Linkages

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

F.I.M.E.E. Universidad de Guanajuato Salamanca, Gto. 38040, México and Departamento de Ingeniería Mecánica,  Instituto Tecnológico de Celaya, Celaya, Gto. 38010, Méxicojrico@salamanca.ugto.mx; mrico@itc.mx

B. Ravani

Department of Mechanical and Aeronautical Engineering,  University of California, Davis, CA 95616bravani@ucdavis.edu

For a review on the basic definitions and results on vector spaces, algebras and matrices, the reader is referred to (15)

The mobility of trivial linkages can be easily determined using Kutzbach-Grübler criterion and does not need the approach of this paper. It is included here only for completeness.

The process indicated here is that of obtaining the closure algebra of a vector space in the Lie algebra, $se(3)$, of the Euclidean group, $SE(3)$, see (13), pp. 483–487.

This paths must be understood in a graph theoretical sense, where the links are regarded as vertex and the kinematic pairs are regarded as edges of a graph. In some cases, see Example 2, the actual position of the links appears to contradict this definition.

The column spaces of these Jacobian matrices are the clockwise and counterclockwise infinitesimal mechanical liaisons of link $j$ with respect to link $i$, denoted in (13) as $Vc(i,j)$ and $Vcc(i,j)$, respectively.

The column space of these closure matrices are the clockwise and counterclockwise closure algebras of link $j$ with respect to link $i$, denoted in (13) as $Ac(i,j)$ and $Acc(i,j)$, respectively.

The column space of this absolute closure matrix is the absolute closure algebra, $Aa(i,j)$ employed in (13).

It is important to note a subtle difference in linear algebra notation, while $[1,2,3]$ indicates the vector space generated by the screws $1,2,3$, $[123]$ indicates the matrix whose columns are given by $1,2,3$.

In fact, any three screws of the sets of screws ${21,32,43,54}$ satisfy the conditions.

In fact, any three screws of the sets of screws ${65,76,87,18}$ satisfy the conditions.

Every cylindrical pair will be represented by a pair of screws, one associated with the rotation and one associated with the traslation. Although other combinations can be used, this seems to be the simplest one. The same approach can be used for other types of kinematic pairs, for example, spherical or planar pairs.

It should be noted that in terms of the original screws of the Jacobian matrix, this Lie product is given by $[21(21,2*2)]$.

It should be noted that in terms of the original screws of the Jacobian matrix, this Lie product is given by $[2*2(212*2)]$.

It should be noted that in terms of the original screws of the Jacobian matrix, this Lie product is given by $[2*2(212*2)]$.

As usual in linear algebra, [ ] represents the vector space generated by the columns of the matrix (see (15)).

Correspondingly, if $[BAa]≠{0⃗}$, Hervè called the constraints dependent.

As indicated in Sec. 4, it is more proper to talk about the mobility characteristics of a pair of links in a kinematic chain. However, by abusing the language, these characteristics are extended to the kinematic chain.

In Table 1, the comparison of ranks of matrices takes the place of the comparison of subalgebras and subspaces employed in 13. The reason for the success of this replacement is that the subalgebras and subspaces involved satisfy the following equations: $Vc(i,j)⩽Ac(i,j)$, $Vcc(i,j)⩽Acc(i,j)$, $Aa(i,j)⩽Ac(i,j)$, $Aa(i,j)⩽Acc(i,j)$, ${0}⩽Aa(i,j)$.

This chain is particularly interesting, if one considers the mobility of link 2 with respect to link 1, the chain is a trivial chain, the same result is obtained by considering the mobility of links 8 with respect to link 1. Moreover, if one considers the mobility of link 6 with respect to link 1, the chain is an exceptional chain. All analyses yield the same number of degrees of freedom.

In this case, the resulting matrix is just the usual Jacobian matrix.

In fact, the relative motion of links 2 or 3, with respect to links 5 or 6 does include a rotation around the $Z$-axis.

Moreover, the vectors $ω⃗$ and $v⃗O$ can be identified as the angular velocity of a rigid body and the velocity of a point $O$, fixed in the rigid body, respectively.

Here, the notation introduced by von Mises is employed. It should be noted that in many differential geometry oriented contributions, a comma after the first operand is included.

J. Mech. Des 129(3), 301-311 (Mar 10, 2006) (11 pages) doi:10.1115/1.2406101 History: Received May 10, 2005; Revised March 10, 2006

## Abstract

This paper reformulates and extends the new, group theoretic, mobility criterion recently developed by the authors, (Rico, J. M., and Ravani, B., 2003, ASME J. Mech. Des., 125, pp. 70–80). In contrast to the Kutzbach-Grübler criterion, the new mobility criterion and the approach presented apply to a large class of single-loop overconstrained linkages. The criterion is reformulated, in terms of the well-known Jacobian matrices, for exceptional linkages and extended to linkages with partitioned mobility and trivial linkages. Several examples are included.

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## Figures

Figure 1

Open-loop kinematic chain

Figure 2

Three basic types of kinematic pairs

Figure 3

Single-loop kinematic chain and the two possible paths between rigid bodies i and j

Figure 4

Figure 5

Figure 6

Kinematic chain with partitioned mobility

Figure 7

Figure 8

Sarrus’s linkage in a singular position

Figure 9

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