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Research Papers

Local Kinematic Analysis of Closed-Loop Linkages—Mobility, Singularities, and Shakiness

[+] Author and Article Information
Andreas Müller

Institute of Robotics,
Johannes Kepler University,
Altenberger Strasse 69,
Linz 4040, Austria
e-mail: a.mueller@jku.at

Manuscript received August 8, 2015; final manuscript received February 9, 2016; published online March 18, 2016. Assoc. Editor: Leila Notash.

J. Mechanisms Robotics 8(4), 041013 (Mar 18, 2016) (11 pages) Paper No: JMR-15-1216; doi: 10.1115/1.4032778 History: Received August 08, 2015; Revised February 09, 2016

The mobility of a linkage is determined by the constraints imposed on its members. The geometric constraints define the configuration space (c-space) variety as the geometric entity in which the finite mobility of a linkage is encoded. The aim of a local kinematic analysis of a linkage is to deduce its finite mobility, in a given configuration, from the local c-space geometry. In this paper, a method for the local analysis is presented adopting the concept of the tangent cone to a variety. The latter is an algebraic variety approximating the c-space. It allows for investigating the mobility in regular as well as singular configurations. The instantaneous mobility is determined by the constraints, rather than by the c-space geometry. Shaky and underconstrained linkages are prominent examples that exhibit a permanently higher instantaneous than finite DOF even in regular configurations. Kinematic singularities, on the other hand, are reflected in a change of the instantaneous DOF. A c-space singularity as a kinematic singularity, but a kinematic singularity may be a regular point of the c-space. The presented method allows to identify c-space singularities. It also reveals the ith-order mobility and allows for a classification of linkages as overconstrained and underconstrained. The method is applicable to general multiloop linkages with lower pairs. It is computationally simple and only involves Lie brackets (screw products) of instantaneous joint screws. The paper also summarizes the relevant kinematic phenomena of linkages.

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Figures

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Fig. 1

(a) Planar four-bar linkage in singular configuration q0 and (b) a three-dimensional cut of its c-space

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Fig. 2

(a) Three-loop kinematotropic linkage in a singular configuration, (b) in a 1DOF motion mode, and (c) in a 2DOF motion mode

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Fig. 3

Immobile spherical four-bar linkage

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Fig. 4

Planar five-bar linkage in singular configuration q0

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Fig. 5

A 6R Goldberg linkage in a kinematic singularity, which is a regular point of the c-space V

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Fig. 6

6R linkage in a singular configuration q0

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Fig. 7

(a) Two-loop Assur linkage and (b) its topological graph and FCs Λ4 and Λ6

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Fig. 8

(a) Two-loop linkage in singular configuration q0 and (b) its topological graph with FCs Λ1 and Λ7

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Fig. 10

Topological graph and selected FCs Λ1, Λ4, and Λ9 for the three-loop linkage based on a Peaucellier–Lipkin linkage

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Fig. 11

7R-linkage after Ref. [11] in a singular configuration

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Fig. 12

Decision diagram to identify a kinematic singularity with the information available from the higher-order local analysis. The dotted lines indicate that this decision cannot be made with available information.

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