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Research Papers

Mobility of Single-Loop Kinematic Mechanisms Under Differential Displacement

[+] Author and Article Information
Paul Milenkovic

Department of Electrical and Computer Engineering, University of Wisconsin-Madison, Madison, WI 53706phmilenk@wisc.edu

Weisstein, E. W., “Quadratic Surface,” from MathWorld—A Wolfram Web Resource, http://mathworld.Wolfram.com/QuadraticSurface.Html.

This result generalizes to any other set of basis vectors for the null space of A.

Seymour, K., 2007, “JLAPACK Source and Class Files Version 0.8,” http://www.Netlib.org/java/f2j/.

J. Mech. Des 132(4), 041001 (Mar 30, 2010) (9 pages) doi:10.1115/1.4001203 History: Received March 24, 2009; Revised January 31, 2010; Published March 30, 2010; Online March 30, 2010

Linear analysis of motion screws is a means for determining the mobility of a mechanism composed of the parallel combination of serial kinematic chains. Such a mechanism may have one or more degrees of freedom that vanish after a differential displacement from a reference posture. The Lie product, also called the Lie bracket, is known to give the derivative of a motion screw with respect to the displacement along an upstream screw in a serial chain. Serial chains having motion screws that are closed under the Lie product are known to retain their mobility after differential displacement. For a single-loop mechanism, which is composed of a pair of chains that are not closed under the Lie product, mobility is retained when the Lie closures of those chains are within the span of the union of motion screws of the two chains, a new result determined by applying the Baker–Campbell–Hausdorff expansion to the motion screws of the serial chain. When the Lie products have at most one dimension outside the union span, a second-order expression of mobility reduces to a quadratic form, allowing the numerical characterization of constraint singularities under that condition.

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Copyright © 2010 by American Society of Mechanical Engineers
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References

Figures

Grahic Jump Location
Figure 1

Nonsingular and singular postures of a mobile planar four-bar linkage, and a singular posture of an immobile planar four-bar linkage

Grahic Jump Location
Figure 2

Platform motion generated by equal and opposite rotations about the two ends of an R-R dyad, relating the change in the velocity vector of the platform to the Lie product of the two twists

Grahic Jump Location
Figure 3

R-C-C-R linkage in nonsingular (links angled) and singular (links vertical) postures

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