Technical Brief

Special Unitary Matrices in the Analysis and Synthesis of Spherical Linkages

[+] Author and Article Information
Saleh M. Almestiri

Mechanical and Aerospace Engineering,
University of Dayton,
Dayton, OH 45469
e-mail: almestiri@gmail.com

Andrew P. Murray

Mechanical and Aerospace Engineering,
University of Dayton,
Dayton, OH 45469
e-mail: murray@udayton.edu

David H. Myszka

Mechanical and Aerospace Engineering,
University of Dayton,
Dayton, OH 45469
e-mail: dmyszka@udayton.edu

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received April 14, 2018; final manuscript received September 21, 2018; published online November 13, 2018. Assoc. Editor: Hai-Jun Su.

J. Mechanisms Robotics 11(1), 014503 (Nov 13, 2018) (4 pages) Paper No: JMR-18-1102; doi: 10.1115/1.4041633 History: Received April 14, 2018; Revised September 21, 2018

This work seeks to systematically model and solve the equations associated with the kinematics of spherical mechanisms. The group of special unitary matrices, SU(2), is utilized throughout. Elements of SU(2) are employed here to analyze the three-roll wrist and the spherical Watt I linkage. Additionally, the five orientation synthesis of a spherical four-bar mechanism is solved, and solutions are found for the eight orientation synthesis of the Watt I linkage. Using SU(2) readily allows for the use of a homotopy-continuation-based solver, in this case Bertini. The use of Bertini is motivated by its capacity to calculate every solution to a design problem.

Copyright © 2019 by ASME
Topics: Kinematics , Linkages
Your Session has timed out. Please sign back in to continue.


Uicker, J. J. , Ravani, B. , and Sheth, P. N. , 2013, Matrix Methods in the Design Analysis of Mechanisms and Multibody Systems, Cambridge University Press, Cambridge, UK.
Angeles, J. , Hommel, G. , and Kovács, P. , 1993, “ Computational Kinematics,” Solid Mechanics and Its Applications, Vol. 28, Kluwer Academic Publishers, Dordrecht, The Netherlands.
Wampler, W. C. , 2004, “ Displacement Analysis of Spherical Mechanisms Having Three or Fewer Loops,” ASME J. Mech. Des., 126(1), pp. 93–100. [CrossRef]
Dhingra, A. , Almadi, A. , and Kohli, D. , 2000, “ Closed-Form Displacement Analysis of 8, 9 and 10-Link Mechanisms—Part I: 8-Link 1-Dof Mechanisms,” Mech. Mach. Theory, 35(6), pp. 821–850. [CrossRef]
Tsai, L.-W. , and Morgan, A. P. , 1985, “ Solving the Kinematics of the Most General Six- and Five-Degree-of-Freedom Manipulators by Continuation Methods,” ASME J. Mech., Trans., Autom, 107(2), pp. 189–200. [CrossRef]
Wampler, C. W. , 1996, “ Isotropic Coordinates, Circularity, and Bezout Numbers: Planar Kinematics From a New Perspective,” ASME Paper No. DETC96/MECH-1210. https://pdfs.semanticscholar.org/c67f/419901422bc030f4631623f3fb7df6462cd1.pdf
Shuster, M. D. , 1993, “ A Survey of Attitude Representations,” J. Astronaut. Sci., 41(4), pp. 439–517.
Spring, K. W. , 1986, “ Euler Parameters and the Use of Quaternion Algebra in the Manipulation of Finite Rotations: A Review,” Mech. Mach. Theory, 21(5), pp. 365–373. [CrossRef]
Goldstein, H. , Poole, C. , and Safko, J. , 2002, Classical Mechanics, 3rd ed., Addison Wesley, San Francisco, CA.
Rooney, J. , 1977, “ A Survey of Representations of Spatial Rotation About a Fixed Point,” Environ. Plann. B, 4(2), pp. 185–210. [CrossRef]
Altmann, S. L. , 2005, Rotations, Quaternions, and Double Groups, Dover Publications, Mineola, NY.
Roth, B. , and Freudenstein, F. , 1963, “ Synthesis of Path-Generating Mechanisms by Numerical Methods,” ASME J. Eng. Ind., 85(3), pp. 298–304. [CrossRef]
Wampler, C. W. , Morgan, A. P. , and Sommese, A. J. , 1992, “ Complete Solution of the Nine-Point Path Synthesis Problem for Four-Bar Linkages,” ASME J. Mech. Des., 114(1), pp. 153–159. [CrossRef]
Bates, D. J. , Hauenstein, J. D. , Sommese, A. J. , and Wampler, C. W. , 2013, Numerically Solving Polynomial Systems With Bertini, Society of Industrial and Applied Mathematics, Philadelphia, PA.
McCarthy, J. M. , 1990, Introduction to Theoretical Kinematics, MIT Press, Cambridge, UK.
Brunnthaler, K. , Schröcker, H.-P. , and Husty, M. , 2006, “ Synthesis of Spherical Four-Bar Mechanisms Using Spherical Kinematic Mapping,” Advances in Robot Kinematics, Springer, Cham, Switzerland, pp. 377–384.
Plecnik, M. , McCarthy, J. M. , and Wampler, C. W. , 2014, “ Kinematic Synthesis of a Watt I Six-Bar Linkage for Body Guidance,” Advances in Robot Kinematics, Springer, Cham, Switzerland, pp. 317–325.
Ohio Supercomputer Center, 1987, “Ohio Supercomputer Center,” Ohio Supercomputer, Columbus, OH, accessed Oct. 9, 2018, http://osc.edu/ark:/19495/f5s1ph73


Grahic Jump Location
Fig. 1

A spherical Watt I six-bar linkage with the joint angles and physical parameters identified

Grahic Jump Location
Fig. 2

The parameters of the spherical four-bar identified for solving the synthesis problem. All frames are located in the center of the sphere but some are shown on the sphere's surface for clarity.

Grahic Jump Location
Fig. 3

The parameters of the spherical Watt I identified for modeling the synthesis problem. All frames are located in the center of the sphere but are shown on the sphere for clarity.



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In