Technical Brief

The Coupler Surface of the RSRS Mechanism

[+] Author and Article Information
Nicolas Rojas

Department of Mechanical Engineering and
Materials Science,
Yale University,
9 Hillhouse Avenue,
New Haven, CT 06511
e-mail: nicolas.rojas@yale.edu

Aaron M. Dollar

Department of Mechanical Engineering and
Materials Science,
Yale University,
15 Prospect Street,
New Haven, CT 06511
e-mail: aaron.dollar@yale.edu

1Corresponding author.

Manuscript received February 16, 2015; final manuscript received May 27, 2015; published online August 18, 2015. Assoc. Editor: Raffaele Di Gregorio.

J. Mechanisms Robotics 8(1), 014505 (Aug 18, 2015) (5 pages) Paper No: JMR-15-1036; doi: 10.1115/1.4030776 History: Received February 16, 2015

Two degree-of-freedom (2-DOF) closed spatial linkages can be useful in the design of robotic devices for spatial rigid-body guidance or manipulation. One of the simplest linkages of this type, without any passive DOF on its links, is the revolute-spherical-revolute-spherical (RSRS) four-bar spatial linkage. Although the RSRS topology has been used in some robotics applications, the kinematics study of this basic linkage has unexpectedly received little attention in the literature over the years. Counteracting this historical tendency, this work presents the derivation of the general implicit equation of the surface generated by a point on the coupler link of the general RSRS spatial mechanism. Since the derived surface equation expresses the Cartesian coordinates of the coupler point as a function only of known geometric parameters of the linkage, the equation can be useful, for instance, in the process of synthesizing new devices. The steps for generating the coupler surface, which is computed from a distance-based parametrization of the mechanism and is algebraic of order twelve, are detailed and a web link where the interested reader can download the full equation for further study is provided. It is also shown how the celebrated sextic curve of the planar four-bar linkage is obtained from this RSRS dodecic.

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Grahic Jump Location
Fig. 1

General RSRS spatial mechanism and its associated notation

Grahic Jump Location
Fig. 2

Examples of the RSRS dodecic: (a) s1,2 = 4,s1,3 = 65,s1,4 = 17,s2,3 = 65,s2,4 = 21,s3,5 = 30,s3,6 = 19,s4,5 = 66,s4,6 = 41,s4,7 = 42,s5,6 = 9,s5,7 = 36,s6,7 = 49, (b) s1,2 = 4,s1,3 = 73,s1,4 = 21,s2,3 = 65,s2,4 = 33,s3,5 = 53,s3,6 = 35,s4,5 = 69,s4,6 = 37,s4,7 = 33,s5,6 = 14,s5,7 = 62,s6,7 = 66, and (c) s1,2 = 4,s1,3 = 50,s1,4 = 25,s2,3 = 50,s2,4 = 29,s3,5 = 21,s3,6 = 34,s4,5 = 36,s4,6 = 21, s4,7 = 22,s5,6 = 9,s5,7 = 22,s6,7 = 21



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