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Research Papers: Mechanisms and Robotics

# An Algorithm for Analytically Calculating the Positions of the Secondary Instant Centers of Indeterminate Linkages

[+] Author and Article Information
Raffaele Di Gregorio

Department of Engineering, University of Ferrara, Via Saragat 1, 44100 Ferrara, Italyrdigregorio@ing.unife.it

For any mechanical system with $m$ rigid bodies, if all the relative motions of the links with respect to a given link, for instance, the frame, let us say Link 1 (i.e., $(m−1)$ relative motions), are known, the generic $ij$ relative motion between Link $i$ and Link $j$ can be determined by composing the two known relative motions $i1$ and $1j$. Since this fact brings to write ${[m(m−1)∕2]−(m−1)}$ linear six-dimensional-vector equations involving the $[m(m−1)∕2]$ twists that identify all the relative motions, if any set of $(m−1)$ relative motions, instead of the ones relative to the frame, is assigned, the remaining ${[m(m−1)∕2]−(m−1)}$ relative motions can be uniquely determined by solving the linear system of vector equations.

It is worth noting that, in a mechanism, the indices $i$ and $j$ of a primary instant center must correspond to two rows of the table whose two intersections with any column do not simultaneously identify two other primary instant centers. In fact, if this condition occurred, that instant center would be overdetermined since it should also lie on the line identified by the other two instant centers. On the contrary, in a structure, this condition can occur.

J. Mech. Des 130(4), 042303 (Feb 28, 2008) (9 pages) doi:10.1115/1.2839008 History: Received September 05, 2006; Revised July 04, 2007; Published February 28, 2008

## Abstract

In a planar mechanism, the position of the instant centers reveals important pieces of information about its static and kinematic behaviors. Such pieces of information are useful for designing the mechanism. Unfortunately, when the mechanism architecture becomes complex, common methods to locate the instant centers, which are based on the direct application of the Aronold–Kennedy theorem, fail. Indeterminate linkages are single-degree-of-freedom (single-dof) planar linkages where the secondary instant centers cannot be found by direct application of the Aronold–Kennedy theorem. This paper presents an analytical method to locate all the instant centers of any single-dof planar mechanism, which, in particular, succeeds in determining the instant centers of indeterminate linkages. In order to illustrate the proposed method, it will be applied to locate the secondary instant centers of the double butterfly linkage and of the single flier eight-bar linkage.

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## Figures

Figure 1

Flowchart of the algorithm illustrated in Sec. 3

Figure 2

Table that can be used to determine the secondary instant centers of a mechanism with m links and q slipping contacts

Figure 3

A linkage with six bars and seven revolute pairs: (a) table; (b) sketch of the linkage

Figure 4

Mechanism obtained from the linkage of Fig. 3 by substituting a rolling contact and a prismatic pair for two revolute pairs

Figure 5

Figure 6

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