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 *i*th-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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# Local Kinematic Analysis of Closed-Loop Linkages—Mobility, Singularities, and Shakiness

Andreas Müller

Andreas Müller

Institute of Robotics,

Johannes Kepler University,

Altenberger Strasse 69,

Linz 4040, Austria

e-mail: a.mueller@jku.at

Johannes Kepler University,

Altenberger Strasse 69,

Linz 4040, Austria

e-mail: a.mueller@jku.at

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Andreas Müller

Johannes Kepler University,

Altenberger Strasse 69,

Linz 4040, Austria

e-mail: a.mueller@jku.at

*J. Mechanisms Robotics*. Aug 2016, 8(4): 041013 (11 pages)

**Published Online:**March 18, 2016

Article history

Received:

August 8, 2015

Revised:

February 9, 2016

Citation

Müller, A. (March 18, 2016). "Local Kinematic Analysis of Closed-Loop Linkages—Mobility, Singularities, and Shakiness." ASME. *J. Mechanisms Robotics*. August 2016; 8(4): 041013. https://doi.org/10.1115/1.4032778

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