Research Papers

A Unified Approach to Exact and Approximate Motion Synthesis of Spherical Four-Bar Linkages Via Kinematic Mapping

[+] Author and Article Information
Xiangyun Li

School of Mechanical Engineering,
Southwest Jiaotong University,
Chengdu 610031, China

Ping Zhao

School of Mechanical and Automotive Engineering,
Hefei University of Technology,
Hefei 230009, China
e-mail: ping.zhao@hfut.edu.cn

Anurag Purwar, Q. J. Ge

Department of Mechanical Engineering,
Stony Brook University,
Stony Brook, NY 11794-2300

1Corresponding author.

Manuscript received April 28, 2017; final manuscript received October 17, 2017; published online December 20, 2017. Assoc. Editor: Venkat Krovi.

J. Mechanisms Robotics 10(1), 011003 (Dec 20, 2017) (10 pages) Paper No: JMR-17-1127; doi: 10.1115/1.4038305 History: Received April 28, 2017; Revised October 17, 2017

This paper studies the problem of spherical four-bar motion synthesis from the viewpoint of acquiring circular geometric constraints from a set of prescribed spherical poses. The proposed approach extends our planar four-bar linkage synthesis work to spherical case. Using the image space representation of spherical poses, a quadratic equation with ten linear homogeneous coefficients, which corresponds to a constraint manifold in the image space, can be obtained to represent a spherical RR dyad. Therefore, our approach to synthesizing a spherical four-bar linkage decomposes into two steps. First, find a pencil of general manifolds that best fit the task image points in the least-squares sense, which can be solved using singular value decomposition (SVD), and the singular vectors associated with the smallest singular values are used to form the null-space solution of the pencil of general manifolds; second, additional constraint equations on the resulting solution space are imposed to identify the general manifolds that are qualified to become the constraint manifolds, which can represent the spherical circular constraints and thus their corresponding spherical dyads. After the inverse computation that converts the coefficients of the constraint manifolds to the design parameters of spherical RR dyad, spherical four-bar linkages that best navigate through the set of task poses can be constructed by the obtained dyads. The result is a fast and efficient algorithm that extracts the geometric constraints associated with a spherical motion task, and leads naturally to a unified treatment for both exact and approximate spherical motion synthesis.

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Bottema, O. , and Roth, B. , 1979, Theoretical Kinematics, North-Holland, Amsterdam, The Netherlands.
McCarthy, J. M. , 1990, Introduction to Theoretical Kinematics, MIT Press, Cambridge, MA.
Suh, C. H. , and Radcliffe, C. W. , 1978, Kinematics and Mechanism Design, Wiley, New York.
Sandor, G. N. , and Erdman, A. G. , 1997, Mechanism Design: Analysis and Synthesis, 3rd ed., Prentice Hall, Englewood Cliffs, NJ.
Angeles, J. , and Bai, S. , 2005, “ Some Special Cases of the Burmester Problem for Four and Five Poses,” ASME Paper No. DETC2005-84871.
Husty, M. L. , Pfurner, M. , Schrocker, H.-P. , and Brunnthaler, K. , 2007, “ Algebraic Methods in Mechanism Analysis and Synthesis,” Robotica, 25, pp. 661–675. [CrossRef]
McCarthy, J. M. , and Soh, G. S. , 2010, Geometric Design of Linkages, 2nd ed., Springer, New York.
Zi, B. , Ding, H. , Wu, X. , and Kecskemthy, A. , 2014, “ Error Modelling and Sensitivity Analysis of a Hybrid-Driven Based Cable Parallel Manipulator,” Precis. Eng., 38(1), pp. 197–211. [CrossRef]
Ge, Q. J. , Zhao, P. , Purwar, A. , and Li, X. , 2012, “ A Novel Approach to Algebraic Fitting of a Pencil of Quadrics for Planar 4R Motion Synthesis,” ASME J. Comput. Inf. Sci. Eng., 12(4), p. 041003. [CrossRef]
Zhao, P. , Ge, X. , Zi, B. , and Ge, Q. J. , 2016, “ Planar Linkage Synthesis for Mixed Exact and Approximated Motion Realization Via Kinematic Mapping,” ASME J. Mech. Rob., 8(5), p. 051004. [CrossRef]
Zhao, P. , Li, X. , Zhu, L. , Zi, B. , and Ge, Q. J. , 2016, “ A Novel Motion Synthesis Approach With Expandable Solution Space for Planar Linkages Based on Kinematic-Mapping,” Mech. Mach. Theory, 105, pp. 164–175. [CrossRef]
Blaschke, W. , 1911, “ Euklidische kinematik und nichteuklidische geometrie,” Z. Math. Phys., 60, pp. 61–91.
Grünwald, J. , 1911, “ Ein abbildungsprinzip, welches die ebene geometrie und kinematik mit der raumlichen geometrie verknüpft,” Sitzungsber. Akad. Wiss. Wiss. Wien, 120, pp. 677–741.
Ravani, B. , and Roth, B. , 1983, “ Motion Synthesis Using Kinematic Mappings,” ASME J. Mech. Transm. Autom. Des., 105(3), pp. 460–467. [CrossRef]
Bodduluri, R. M. C. , and McCarthy, J. M. , 1992, “ Finite Position Synthesis Using the Image Curve of a Spherical Four-Bar Motion,” ASME J. Mech. Des., 114(1), pp. 55–60. [CrossRef]
Bodduluri, R. , 1990, “ Design and Planned Movement of Multi-Degree of Freedom Spatial Mechanisms,” Ph.D. thesis, University of California, Irvine, CA. http://rep443.infoeach.com/view-NDQzfDE3NDcyMDg=.html
Ge, Q. J. , and Larochelle, P. , 1999, “ Algebraic Motion Approximation With NURBS Motions and Its Application to Spherical Mechanism Synthesis,” ASME J. Mech. Des., 121(4), pp. 529–532. [CrossRef]
Wu, J. , Purwar, A. , and Ge, Q. J. , 2010, “ Interactive Dimensional Synthesis and Motion Design of Planar 6R Single-Loop Closed Chains Via Constraint Manifold Modification,” ASME J. Mech. Rob., 2(3), p. 031012. [CrossRef]
Purwar, A. , Anantwar, S. , and Zhao, P. , 2012, “ An Interactive Approach to Designing Planar Parallel Manipulators Using Image Space Representation,” ASME Paper No. DETC2012-70880.
Hayes, M. J. D. , and Rusu, S. R. , 2011, “ Quadric Surface Fitting Applications to Approximate Dimensional Synthesis,” 13th World Congress in Mechanism and Machine Science, Guanajuato, Mexico, June 19--23, Paper No. A7-561. http://faculty.mae.carleton.ca/John_Hayes/Papers/A7-561.pdf
Purwar, A. , Jin, Z. , and Ge, Q. J. , 2008, “ Computer Aided Synthesis of Piecewise Rational Motions for Spherical 2R and 3R Robot Arms,” ASME J. Mech. Des., 130(11), p. 112301. [CrossRef]
Brunnthaler, K. , Schrëocker, H.-P. , and Husty, M. , 2006, “ Synthesis of Spherical Four-Bar Mechanisms Using Spherical Kinematic Mapping,” Advances in Robot Kinematics, Springer, Dordrecht, The Netherlands, pp. 377–385. [CrossRef]
Ruth, D. A. , and McCarthy, J. M. , 1999, “ The Design of Spherical 4R Linkages for Four Specified Orientations,” Mech. Mach. Theory, 34(5), pp. 677–692. [CrossRef]
Léger, J. , and Angeles, J. , 2015, “ A Solution to the Approximate Spherical Burmester,” Multibody Mechatronic Systems, Springer, Cham, Switzerland, pp. 521–529.
Li, X. , Zhao, P. , Ge, Q. J. , and Purwar, A. , 2013, “ A Task Driven Approach to Simultaneous Type Synthesis and Dimensional Optimization of Planar Parallel Manipulator Using Algebraic Fitting of a Family of Quadrics,” ASME Paper No. DETC2013-13197.
Burmester, L. , 1888, Lehrbuch der Kinematik, Verlag Von Arthur Felix, Leipzig, Germany.
Chiang, C. H. , 1988, Kinematics of Spherical Mechanisms, Cambridge University Press, Cambridge, UK.
Li, X. , Ge, X. , Purwar, A. , and Ge, Q. J. , 2015, “ A Unified Algorithm for Analysis and Simulation of Planar Four-Bar Motions Defined With R- and P-Joints,” ASME J. Mech. Rob., 7(1), p. 011014. [CrossRef]
Larochelle, P. , Dees, S. , and Ketchel, J. , 2001, “ Fan Structure Having a Spherical Four-Bar Mechanism,” Florida Institute of Technology, Melbourne, FL, U.S. Patent No. 6,213,715 B1. http://www.google.co.in/patents/US6213715


Grahic Jump Location
Fig. 1

The spherical RR dyad with A as the fixed axis, B as the moving axis, and α is the angular dimension of the chain

Grahic Jump Location
Fig. 2

The projected C-manifold of a spherical circular constraint with its dimensions given as follows: its center lies on the axis A = (0.4243, 0.5657, −0.7071) and its radius that represented by the sphere central angle is α = 64.9 deg

Grahic Jump Location
Fig. 3

Coupling of two C-manifolds in the image space, where each intersection curve represents one circuit/assembly mode of the resulting spherical four-bar linkage, and the task positions are converted to a set of image points

Grahic Jump Location
Fig. 4

Example 1: 12 given poses sampled from a four-bar motion

Grahic Jump Location
Fig. 5

Example 1: Two resulting C-manifolds are shown with the 12 black image points which lie on the intersection curve in the figure and represent 12 given positions in Table 1

Grahic Jump Location
Fig. 6

Example 2: The first figure shows p1 and p4, which are close to the two resulting C-manifolds in the first example. The second one shows p2 and p3. Similarly, the 12 black image points lying on the intersection curve in the figure represent 12 approximate poses in Table 5.

Grahic Jump Location
Fig. 7

Example 2: The resulting spherical four-bar mechanism constructed by p1 and p4

Grahic Jump Location
Fig. 8

Example 3: Infinity fan driven by a spherical four-bar mechanism




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