Five Position Synthesis of a Planar Four-Bar Linkage

  • Jeffrey GlabeEmail author
  • J. Michael McCarthy
Conference paper
Part of the Mechanisms and Machine Science book series (Mechan. Machine Science, volume 73)


This paper presents a tutorial on the formulation and solution of the five position synthesis equations for a four-bar linkage in a way that is convenient for implementation using polynomial homotopy software. Complex numbers are used to formulate the loop equations for the four-bar linkage with a coupler reference frame in each of five task positions. We use the loop equations for both cranks of the linkage and their conjugates, and the normalization conditions for each of the joint angles in four displaced positions relative to the initial configuration of the linkage, in order to obtain 8 quadratic equations in 8 unknowns. While there are other formulations for these synthesis equations, we show that this formulation is convenient for solution by polynomial homotopy continuation. An example code is provided as well as a sample calculation.


kinematic synthesis four-bar linkage geometric design 


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  1. 1.
    Glabe J., McCarthy J.M., Six-Bar Linkage Design System with a Parallelized Polynomial Homotopy Solver. In: Lenarcic J., Parenti-Castelli V. (eds) Advances in Robot Kinematics 2018. ARK 2018. Springer Proceedings in Advanced Robotics, vol 8. Springer, Cham, 2018.Google Scholar
  2. 2.
    Purwar, A., Deshpande, S., and Ge, Q. J., “MotionGen: Interactive Design and Editing of Planar Four-Bar Motions for Generating Pose and Geometric Constraints,” Journal of Mechanisms and Robotics, April 2017. Scholar
  3. 3.
    Brake, D.A., Hauenstein, J.D., Murray, A.P,. Myszka, D.H., Wampler, C.W., “The Complete Solution of Mixed Burmester Synthesis Problems for Four-Bar Linkages,” ASME. J. Mechanisms Robotics. 2016;8(4):041018-041018-8. Scholar
  4. 4.
    McCarthy J.M., Choe J.. “Diffculty of Kinematic Synthesis of Usable Constrained Planar 6R Robots.” In: Lenarcic J., Stanisic M. (eds) Advances in Robot Kinematics: Motion in Man and Machine. Springer, Dordrecht (2010) Scholar
  5. 5.
    Bates, D. J., Hauenstein, J. D., Sommese, A. J., and Wampler, C. W., Numerically Solving Polynomial Systems With Bertini, (Software, Environments, and Tools), Vol. 25, SIAM, Philadelphia, PA. 2013,Google Scholar
  6. 6.
    Burmester, L., Lehrbuch der Kinematic, Verlag Von Arthur Felix, Leipzig, Germany. 1886Google Scholar
  7. 7.
    Freudenstein, F. “An analytical approach to the design of four-link mechanisms,” ASME Journal of Engineering for Industry Series B, 76(3), 1954.Google Scholar
  8. 8.
    Erdman, A., Sandor, G., and Kota, S., Mechanism Design: Analysis and Synthesis, 4th ed., Vol. 1, Prentice-Hall, Englewood Cliffs, NJ. 2001.Google Scholar
  9. 9.
    McCarthy, J. M., and Soh, G. S., Geometric Design of Linkages, 2nd ed., Springer-Verlag, New York. 2011. Scholar
  10. 10.
    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: 153–159.CrossRefGoogle Scholar
  11. 11.
    Plecnik, M. M. and McCarthy, J. M., “Computational Design of Stephenson II Six-bar Function Generators for 11 Accuracy Points,” ASME Journal of Mechanisms and Robotics, Vol 8(1), February 2016.Google Scholar
  12. 12.
    Plecnik, M. M. and McCarthy, J. M., “Design of Stephenson Linkages that Guide a Point Along a Specified Trajectory,” Mechanism and Machine Theory, Vol 96, Part 1, pp 38-51, February 2016.Google Scholar
  13. 13.
    Plecnik, M. M., Haldane, D. W., Yim, J. K., and Fearing, R. S., “Design Exploration and Kinematic Tuning of a Power Modulating Jumping Monopod,” Journal of Mechanisms and Robotics, 9(1): 011009, 2017.CrossRefGoogle Scholar
  14. 14.
    Haldane, D. W., Pecnik, M. M., Yim, J. K., and Fearing, R. S., “Robotic vertical jumping agility via series-elastic power modulation.” Science Robotics, 1(1), Dec 2016.CrossRefGoogle Scholar
  15. 15.

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of California, IrvineIrvineUSA

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