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Research Papers: Design of Mechanisms and Robotic Systems

An Application of Yaglom's Geometric Algebra to Kinematic Synthesis of Linkages for Prescribed Planar Motion of Oriented Lines

[+] Author and Article Information
E. Pennestrì

Department of Enterprise Engineering,
University of Rome Tor Vergata,
via del Politecnico,
Roma 100133, Italy
e-mail: pennestri@mec.uniroma2.it

P. P. Valentini

Department of Enterprise Engineering,
University of Rome Tor Vergata,
via del Politecnico,
Roma 100133, Italy
e-mail: valentini@ing.uniroma2.it

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received April 27, 2017; final manuscript received December 1, 2017; published online January 30, 2018. Assoc. Editor: Oscar Altuzarra.

J. Mech. Des 140(3), 032302 (Jan 30, 2018) (8 pages) Paper No: MD-17-1298; doi: 10.1115/1.4038924 History: Received April 27, 2017; Revised December 01, 2017

Planar motion coordination of an unoriented line passing through a point or tangent to a conic is a well-known problem in kinematics. In Yaglom's algebraic geometry, oriented lines in a plane are represented with dual numbers. In the present paper, such algebraic geometry is applied in the kinematic synthesis of an inverted slider–crank for prescribed three and four finitely separated positions of the coupler. No previous application of Yaglom's algebraic geometry in the area of linkage kinematic synthesis is recorded. To describe the planar finite displacement of an oriented line about a given rotation pole, new dual operators are initially obtained. Then, the loci of moving oriented lines whose three and four homologous planar positions are tangent to a circle are deduced. The paper proposes the application of findings to the mentioned kinematic synthesis of the inverted slider–crank. Numerical examples show the reliability of the proposed approach. Finally, it is also demonstrated that, for a general planar motion, there is not any line whose five finitely separated positions share the same concurrency point. For the case of planar infinitesimal displacements, the same property was established in a paper authored by Soni et al. (1978, “Higher Order, Planar Tangent-Line Envelope Curvature Theory,” ASME J. Mech. Des., 101(4), pp. 563–568.)

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References

Beyer, R. , 1963, The Kinematic Synthesis of Mechanisms, Chapman & Hall, London.
Sandor, G. N. , and Erdman, A. G. , 1984, Advanced Mechanism Design: Analysis and Synthesis, Vol. II, Prentice Hall, Englewood Cliffs, NJ.
Bottema, O. , 1970, “On Some Loci of Lines in Plane Kinematics,” J. Mech., 5(4), pp. 541–548. [CrossRef]
Kaufman, R. , 1973, “Singular Solutions in Burmester Theory,” ASME J. Eng. Ind., 95(2), pp. 572–576. [CrossRef]
Soni, A. H. , Siddhanty, M. N. , and Ting, K. L. , 1979, “Higher Order, Planar Tangent-Line Envelope Curvature Theory,” ASME J. Mech. Des., 101(4), pp. 563–568. [CrossRef]
Gibson, D. , and Kramer, S. N. , 1984, “Kinematic Design and Analysis of the Rack-and-Gear Mechanism for Function Generation,” Mech. Mach. Theory, 19(3), pp. 369–375. [CrossRef]
Yaglom, I. M. , 1968, Complex Numbers in Geometry, Academic Press, New York.
Claudio, M. , and Kramer, S. N. , 1986, “Synthesis and Analysis of Rack and Gear Mechanism for Four Point Path Generation With Prescribed Input Timing,” ASME J. Mech., Transm. Autom. Des., 108(1), pp. 10–14. [CrossRef]
Ananthasuresh, G. K. , and Kramer, S. N. , 1992, “Kinematic Synthesis and Analysis of the Rack and Pinion Multi-Purpose Mechanism,” ASME J. Mech. Des., 114(3), pp. 428–432. [CrossRef]
Siddhanty, M. N. , 1979, “Unified Higher Order Path and Envelope Curvature Theories in Kinematics,” Ph.D. thesis, Oklahoma State University, Stillwater, OK. https://shareok.org/bitstream/handle/11244/24510/Thesis-1979D-S568u.pdf?sequence=1&isAllowed=y
Yaglom, I. M. , 1979, A Simple Non-Euclidean Geometry and Its Physical Basis, Springer-Verlag, New York.
Coxeter, H. S. M. , 1968, “The Problem of Apollonius,” Am. Math. Mon., 75(1), pp. 5–15. [CrossRef]
Di Benedetto, A. , and Pennestrì, E. , 1985, “On the Calculation of Burmester's Point Pairs for Five Finite Positions of the Moving Plane,” Ninth OSU Applied Mechanism Conference, Kansas City, MO, Oct. 28–30, pp. III-1–III-5. https://www.researchgate.net/publication/305656470_On_the_Calculation_of_Burmster%27s_Point_Pairs_for_Five_Finite_Positions_of_a_Moving_Plane
Di Benedetto, A. , and Pennestrì, E. , 1993, Introduction to the Kinematics of Mechanisms, Vol. I, Casa Editrice Ambrosiana, Milano, Italy (in Italian).
McCarthy, J. M. , and Soh, G. S. , 2011, Geometric Design of Linkages, Springer Verlag, New York. [CrossRef]

Figures

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Fig. 1

Circles tangent to three not oriented lines. The oriented circle C is associated with the oriented lines z1, z2, and z3.

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Fig. 2

Oriented line and dual number representation of general and special cases

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Fig. 3

Angle between oriented lines

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Fig. 4

Distance between oriented lines (Nomenclature)

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Fig. 6

Oriented circle with center in the origin O and radius r

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Fig. 5

Four oriented lines tangent to a circle (Nomenclature)

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Fig. 7

Centric inverted slider–crank or swinging-block mechanism

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Fig. 8

Eccentric inverted slider–crank mechanism

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Fig. 10

Solution of Problem 2

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Fig. 11

Solution of Problem 3

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Fig. 9

Solution of Problem 1

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