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Research Papers

Planar Linkage Synthesis for Mixed Exact and Approximated Motion Realization Via Kinematic Mapping

[+] Author and Article Information
Ping Zhao

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

Xin Ge, Q. J. Ge

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

Bin Zi

School of Mechanical and
Automotive Engineering,
Hefei University of Technology,
Hefei, Anhui 230009, China

Manuscript received September 12, 2015; final manuscript received November 25, 2015; published online May 4, 2016. Assoc. Editor: Andrew P. Murray.

J. Mechanisms Robotics 8(5), 051004 (May 04, 2016) (8 pages) Paper No: JMR-15-1249; doi: 10.1115/1.4032212 History: Received September 12, 2015; Revised November 25, 2015

It has been well established that kinematic mapping theory could be applied to mechanism synthesis, where discrete motion approximation problem could be converted to a surface fitting problem for a group of discrete points in hyperspace. In this paper, we applied kinematic mapping theory to planar discrete motion synthesis of an arbitrary number of approximated poses as well as up to four exact poses. A simultaneous type and dimensional synthesis approach is presented, aiming at the problem of mixed exact and approximate motion realization with three types of planar dyad chains (RR, RP, and PR). A two-step unified strategy is established: first N given approximated poses are utilized to formulate a general quadratic surface fitting problem in hyperspace, then up to four exact poses could be imposed as pose-constraint equations to this surface fitting system such that they could be strictly satisfied. The former step, the surface fitting problem, is converted to a linear system with two quadratic constraint equations, which could be solved by a null-space analysis technique. On the other hand, the given exact poses in the latter step are formulated as linear pose-constraint equations and added back to the system, where both type and dimensions of the resulting optimal dyads could be determined by the solution. These optimal dyads could then be implemented as different types of four-bar linkages or parallel manipulators. The result is a novel algorithm that is simple and efficient, which allows for N-pose motion approximation of planar dyads containing both revolute and prismatic joints, as well as handling of up to four prescribed poses to be realized precisely.

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References

Blaschke, W. , 1911, “ Euklidische kinematik und nichteuklidische Geometrie,” Z. Math. Phys., 60, pp. 61–91.
Hartenberg, R. S. , and Denavit, J. , 1964, Kinematic Synthesis of Linkages, McGraw-Hill, New York.
Suh, C. H. , and Radcliffe, C. W. , 1978, Kinematics and Mechanism Design, Wiley, New York.
Erdman, A. G. , and Sandor, G. N. , 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(6), pp. 661–675. [CrossRef]
Burmester, L. , 1888, Lehrbuch der Kinematik, Verlag Von Arthur Felix, Leipzig, Germany.
Sutherland, G. , 1977, “ Mixed Exact-Approximate Planar Mechanism Position Synthesis,” ASME J. Eng. Ind., 99(2), pp. 434–439. [CrossRef]
Smaili, A. , and Diab, N. , 2005, “ A New Approach for Exact/Approximate Point Synthesis of Planar Mechanisms,” ASME Paper No. DETC2005-84339.
Mirth, J. , 1995, “ Four-Bar Linkage Synthesis Methods for Two Precision Positions Combined With n Quasi-Positions,” ASME Design Engineering Technical Conferences, Boston, MA, Vol. 82, pp. 477–484.
Holte, J. E. , Chase, T. R. , and Erdman, A. G. , 2000, “ Mixed Exact-Approximate Position Synthesis of Planar Mechanisms,” ASME J. Mech. Des., 122(3), pp. 278–286. [CrossRef]
Larochelle, P. , 2015, “ Synthesis of Planar Mechanisms for Pick and Place Tasks With Guiding Positions,” ASME J. Mech. Rob., 7(3), p. 031009. [CrossRef]
Ge, Q. J. , Zhao, P. , Li, X. , and Purwar, A. , 2012, “ A Novel Approach to Algebraic Fitting of Constraint Manifolds for Planar 4R Motion Approximation,” ASME Paper No. DETC2012-71190.
Ge, Q. J. , Zhao, P. , and Purwar, A. , 2013, “ Decomposition of Planar Burmester Problems Using Kinematic Mapping,” Advances in Mechanisms, Robotics and Design Education and Research, Springer, New York, pp. 145–157.
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]
Grunwald, J. , 1911, “ Ein abbildungsprinzip, welches die ebene geometrie und kinematik mit der raumlichen geometrie verknupft,” Sitzungsber. Akad. Wiss. Wien, 120, pp. 677–741.
Bottema, O. , and Roth, B. , 1979, Theoretical Kinematics, North-Holland, Amsterdam, The Netherlands.
McCarthy, J. M. , 1990, Introduction to Theoretical Kinematics, MIT, Cambridge, MA.
Ravani, B. , and Roth, B. , 1983, “ Motion Synthesis Using Kinematic Mappings,” ASME J. Mech., Transm., Autom. Des., 105(3), pp. 460–467. [CrossRef]
Ravani, B. , and Roth, B. , 1984, “ Mappings of Spatial Kinematics,” ASME J. Mech., Transm., Autom, Des., 106(3), pp. 341–347. [CrossRef]
Bodduluri, R. , and McCarthy, J. M. , 1992, “ Finite Position Synthesis Using the Image n 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.
Larochelle, P. , 1996, “ Synthesis of Planar RR Dyads by Constraint Manifold Projection,” ASME Paper No. DETC/MECH-1187.
Larochelle, P. , 2003, “ Approximate Motion Synthesis of Open and Closed Chains Via Parametric Constraint Manifold Fitting: Preliminary Results,” ASME Paper No. DETC2003/DAC-48814.
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]
Hayes, M. J. D. , Luu, T. , and Chang, X.-W. , 2004, “ Kinematic Mapping Application to Approximate Type and Dimension Synthesis of Planar Mechanisms,” Advances in Robotic Kinematics, J. Lenarci and C. Galletti , eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, pp. 41–48.
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, June 19–25, Guanajuato, Mexico.
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]

Figures

Grahic Jump Location
Fig. 1

A planar pose of a moving frame M with respect to the fixed frame F

Grahic Jump Location
Fig. 2

Three types of planar dyad open-chains: RR, PR, and RP

Grahic Jump Location
Fig. 3

Two samples quadratic functions in the form of Eq. (12) plotted in α−β plane with γ set to be 1. These two equations yield up to four intersection points, each denoting one real solution of p in Eq. (11).

Grahic Jump Location
Fig. 4

Eleven-pose example: plot of the 11 poses as listed in Table 1, in which the first, third, eighth, and eleventh poses are critical positions that need to be realized precisely

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Fig. 5

Eleven-pose example: the four-bar linkage is constructed by the two optimal resulting dyads in Table 4. Its coupler motion trajectory is also plotted as comparison to the 11 given poses.

Grahic Jump Location
Fig. 6

Eleven-pose example: plot of the coupler curve and 11 given poses. It can be seen that the first, third, eighth, and eleventh poses are exactly realized.

Grahic Jump Location
Fig. 7

Seven-pose example: seven arbitrary poses in Table 5, in which the third, sixth, and seventh poses require exact realization

Grahic Jump Location
Fig. 8

Seven-pose example: the four-bar linkage is constructed by the first and fourth resulting dyads in Table 7 as well as its coupler motion

Grahic Jump Location
Fig. 9

Seven-pose example: plot of the resulting coupler curve and seven given poses. It can be seen that the third, sixth, and seventh poses are exactly realized while the rest four poses are approximately guided through.

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