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Research Papers

Triangle Pseudocongruence in Constraint Singularity of Constant-Velocity Couplings

[+] Author and Article Information
Paul Milenkovic

Department of Electrical and Computer Engineering, University of Wisconsin-Madison, Madison, WI 53705

J. Mechanisms Robotics 1(2), 021006 (Jan 07, 2009) (8 pages) doi:10.1115/1.3046142 History: Received February 04, 2008; Revised August 22, 2008; Published January 07, 2009

Congruent triangles establish that a class of intersecting-shaft couplings is constant velocity. These mechanisms employ a pair of linkages in parallel: a spherical joint at the intersection of the shafts and the intersection of straight-line tracks away from the shaft center to transmit rotation. A proof of constant velocity follows from the congruence of an initial pair of triangles with two matching sides and one excluded angle. This side-side-angle (SSA) condition is a pseudocongruence because it allows two different values for the included angle, indicating that such shaft couplings have symmetric and skewed assembly configurations. If the other excluded angle happens to be 90 deg, the SSA condition guarantees congruence because there is a single solution for the included angle. The 90 deg condition, however, occurs at a posture with a constraint singularity, where the shaft coupling is unable to transmit torque. Motion screw analysis establishes the same geometric condition for a coupling based on a revolute-spherical-revolute Clemens linkage. An upper bound on shaft deflection imposed by hyperextension of that linkage, along with a bound on deflection where constraint singularity occurs, identifies couplings where the constraint singularity can occur within the physical limits.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

Congruent triangles that establish equal rotation angle of the input and output shafts of an intersecting-shaft coupling: (a) coupling constrained by the intersection of lines normal to the two shafts; (b) coupling constrained by the intersection of line tracks of general orientation emanating from planes normal to the two shafts

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Figure 2

Triangles matching two sides and excluded angle (SSA)

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Figure 3

Constraint force direction of a R-S-R linkage

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Figure 4

Projection of constraint force vectors into the central symmetry plane for two positions of a R-S-R linkage relative to the center of the coupling

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Figure 5

Relative motion of a pair of rotating shafts

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Figure 6

Articulation of the (R-S-R)∥S coupling under R-S-R linkage hyperextension

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Figure 7

Projection of (R-S-R)∥S coupling in the plane of the hyperextended R-S-R linkage

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Figure 8

Hyperextended (R-S-R)∥S coupling at minimum shaft-deflection angle

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Figure 9

Plane views of the hyperextended (R-S-R)∥S coupling at minimum shaft-deflection angle

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Figure 10

Plane views of the (R-S-R)∥S coupling at the constraint-singularity pose

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