Research Papers

Properties of the Bennett Mechanism Derived From the RRRS Closure Ellipse

[+] Author and Article Information
Paul Milenkovic1

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

Morgan V. Brown

Department of Mathematics, University of California, Berkeley, 970 Evans Hall 3840, Berkeley, CA 94720-3840mvbrown@math.berkeley.edu

The negative width in w23=w21 allows Beggs’ alternative mobile form for the Bennett mechanism.


Corresponding author.

J. Mechanisms Robotics 3(2), 021012 (Apr 19, 2011) (8 pages) doi:10.1115/1.4003844 History: Received November 12, 2009; Revised February 04, 2011; Published April 19, 2011; Online April 19, 2011

For many single-loop closed-chain mechanisms, mobility may be characterized by the closure of sets in the theory of Lie groups. The four-revolute (4R) Bennett mechanism remains a persistent exception, requiring the formulation and expression of solutions to the loop closure relations, either directly or indirectly through spatial geometric figures. The simpler loop closure relations of the revolute-revolute-revolute-spherical (RRRS) loop, however, place conditions on the mobility of the 4R mechanism. That loop closure in turn may be interpreted as the congruence of a pair of ellipses. This new result is applied to proving the uniqueness of the Bennett mechanism along with deriving conditions where it is free from singularities. Design parameters are also identified for overconstrained RRRS mechanisms with 1DOF that are neither plane nor line symmetric. Such mechanisms, however, place the S-joint along the revolute axis of an underlying Bennett mechanism.

Copyright © 2011 by American Society of Mechanical Engineers
Topics: Mechanisms , Linkages
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Figure 1

8R superset to the 4R Bennett mechanism

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Figure 2

8R superset to a more general 4R mechanism

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Figure 3

Projections of portions of Fig. 2 into planes normal to the designated line segments

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Figure 4

Karnaugh map: Conditions on link lengths a1 and a2 along with twist angles α1 and α2 where the 4R mechanism from Fig. 2 is immobiled are marked with X

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Figure 5

Bennett mechanism with equal link lengths and 90 deg twist angles exhibiting bifurcation

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Figure 6

Nonsymmetric RRRS linkage with 1DOF




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