Research Papers

Computational Design of Stephenson II Six-Bar Function Generators for 11 Accuracy Points

[+] Author and Article Information
Mark M. Plecnik

Robotics and Automation Laboratory,
Department of Mechanical and
Aerospace Engineering,
University of California, Irvine,
Irvine, CA 92697
e-mail: mplecnik@uci.edu

J. Michael McCarthy

Robotics and Automation Laboratory,
Department of Mechanical and
Aerospace Engineering,
University of California, Irvine,
Irvine, CA 92697
e-mail: jmmccart@uci.edu

Manuscript received March 9, 2015; final manuscript received July 8, 2015; published online August 18, 2015. Assoc. Editor: James Schmiedeler.

J. Mechanisms Robotics 8(1), 011017 (Aug 18, 2015) (9 pages) Paper No: JMR-15-1055; doi: 10.1115/1.4031124 History: Received March 09, 2015; Revised July 08, 2015

This paper presents a design methodology for Stephenson II six-bar function generators that coordinate 11 input and output angles. A complex number formulation of the loop equations yields 70 quadratic equations in 70 unknowns, which is reduced to a system of ten eighth degree polynomial equations of total degree 810=1.07×109. These equations have a monomial structure which yields a multihomogeneous degree of 264,241,152. A sequence of polynomial homotopies was used to solve these equations and obtain 1,521,037 nonsingular solutions. Contained in these solutions are linkage design candidates which are evaluated to identify cognates, and then analyzed to determine their input–output angles in each assembly. The result is a set of feasible linkage designs that reach the required accuracy points in a single assembly. As an example, three Stephenson II function generators are designed, which provide the input–output functions for the hip, knee, and ankle of a humanoid walking gait.

Copyright © 2016 by ASME
Topics: Linkages , Design , Generators , Knee
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Svoboda, A. , 1948, Computing Mechanisms and Linkages, McGraw-Hill, New York.
Shimojima, H. , and Lian, J. G. , 1987, “Dimensional Synthesis of Dwell Function Generators (On 6-Link Stephenson Mechanisms),” JSME Int. J., 30(260), pp. 324–329. [CrossRef]
Yao, Y. , and Yan, H. , 2003, “A New Method for Torque Balancing of Planar Linkages Using Non-Circular Gears,” Proc. Inst. Mech. Eng., Part C, 217(5), pp. 495–503. [CrossRef]
Freudenstein, F. , 1954, “An Analytical Approach to the Design of Four-Link Mechanisms,” Trans. ASME, 76(3), pp. 483–492.
Dhingra, A. K. , Cheng, J. C. , and Kohli, D. , 1994, “Synthesis of Six-Link, Slider-Crank and Four-Link Mechanisms for Function, Path and Motion Generation Using Homotopy With m-Homogenization,” ASME J. Mech. Des., 116(4), pp. 1122–1131. [CrossRef]
Plecnik, M. , and McCarthy, J. M. , 2015, “Controlling the Movement of a TRR Spatial Chain With Coupled Six-Bar Function Generators for Biomimetic Motion,” ASME Paper No. DETC2015-47876.
Bates, D. J. , Hauenstein, J. D. , Sommese, A. J. , and Wampler, C. W. , 2014, “Bertini: Software for Numerical Algebraic Geometry,” available at: www.bertini.nd.edu
Bates, D. J. , Hauenstein, J. D. , Sommese, A. J. , and Wampler, C. W. , 2013, Numerically Solving Polynomial Systems With Bertini, SIAM Press, Philadelphia, p. 352.
Hartenberg, R. S. , and Denavit, J. , 1964, Kinematic Synthesis of Linkages, McGraw-Hill, New York.
Erdman, A. G. , Sandor, G. N. , and Kota, S. , 2001, Mechanism Design: Analysis and Synthesis, Prentice Hall, Upper Saddle River, NJ.
McCarthy, J. M. , and Soh, G. S. , 2010, Geometric Design of Linkages (Interdisciplinary Applied Mathematics, Vol. 11), 2nd ed., Springer-Verlag, New York.
Svoboda, A. , 1944, “Mechanism for Use in Computing Apparatus,” U.S. Patent No. 2,340,350.
McLarnan, C. W. , 1963, “Synthesis of Six-Link Plane Mechanisms by Numerical Analysis,” J. Eng. Ind., 85(1), pp. 5–10. [CrossRef]
Mohan Rao, A. V. , Erdman, A. G. , Sandor, G. N. , Raghunathan, V. , Nigbor, D. E. , Brown, L. E. , Mahardy, E. F. , and Enderle, E. D. , 1971, “Synthesis of Multi-Loop, Dual-Purpose Planar Mechanisms Utilizing Burmester Theory,” 2nd OSU Applied Mechanism Conference, Stillwater, OK, Oct. 7–8, Paper No. 7.
Liu, A. , Shi, B. , and Yang, T. , 1995, “On the Kinematic Synthesis of Planar Linkages With Multi-Loops,” Ninth World Congress on the Theory of Machines and Mechanisms, Milan, Italy, Aug. 29–Sept. 2, pp. 95–97.
Simionescu, P. A. , and Alexandru, P. , 1995, “Synthesis of Function Generators Using the Method of Increasing the Degree of Freedom of the Mechanism,” Ninth World Congress on the Theory of Machines and Mechanisms, Milan, Italy, Aug. 29–Sept. 2, pp. 139–143.
Akçali, I. D. , 1995, “Modular Approach to Function Generation,” Ninth World Congress on the Theory of Machines and Mechanisms, Milan, Italy, Aug. 29–Sept. 2, pp. 1440–1444.
Kinzel, E. C. , Schmiedeler, J. P. , and Pennock, G. R. , 2007, “Function Generation With Finitely Separated Precision Points Using Geometric Constraint Programming,” ASME J. Mech. Des., 129(11), pp. 1185–1190. [CrossRef]
Hwang, W. M. , and Chen, Y. J. , 2010, “ Defect-Free Synthesis of Stephenson-II Function Generators,” ASME J. Mech. Rob., 2(4), p. 041012. [CrossRef]
Sancibrian, R. , 2011, “Improved GRG Method for the Optimal Synthesis of Linkages in Function Generation Problems,” Mech. Mach. Theory, 46(10), pp. 1350–1375. [CrossRef]
Plecnik, M. , and McCarthy, J. M. , 2014, “Numerical Synthesis of Six-Bar Linkages for Mechanical Computation,” ASME J. Mech. Rob., 6(3), p. 031012. [CrossRef]
Dijksman, E. A. , 1976, Motion Geometry of Mechanisms, Cambridge University Press, Cambridge, UK.
Wampler, C. W. , 1996, “Isotropic Coordinates, Circularity, and Bézout Numbers: Planar Kinematics From a New Perspective,” ASME Paper No. 96-DETC/MECH-1210.
Simionescu, P. A. , and Smith, M. R. , 2001, “ Four-and Six-Bar Function Cognates and Overconstrained Mechanisms,” Mech. Mach. Theory, 36(8), pp. 913–924. [CrossRef]
Roberts, S. , 1875, “ Three-Bar Motion in Plane Space,” Proc. London Math. Soc., s1–7(1), pp. 14–23. [CrossRef]
Chase, T. R. , and Mirth, J. A. , 1993, “Circuits and Branches of Single-Degree-of-Freedom Planar Linkages,” ASME J. Mech. Des., 115(2), pp. 223–230. [CrossRef]
Balli, S. S. , and Chand, S. , 2002, “Defects in Link Mechanisms and Solution Rectification,” Mech. Mach. Theory, 37(9), pp. 851–876. [CrossRef]


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Fig. 1

(a) A Stephenson II linkage in its reference configuration and (b) a Stephenson II linkage displaced from its reference configuration drawn with dotted lines

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Fig. 2

(a) Link vectors of the Stephenson II linkage and (b) link vectors in the jth displaced configuration

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Fig. 3

Three Stephenson II cognate function generators shown in a single overconstrained mechanism. The three cognate mechanisms are comprised of pivots {A,B,C,D,G,H,F}, {A,B,C,D,G′,H′,F′}, and {A,B,C,F′,G″,H″,F}.

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Fig. 4

The coupler curve traced by C as part of the four-bar DGHF is the same for two additional cognate linkages DG′H′F′ and F′G″H″F

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Fig. 5

A humanoid leg is modeled as a planar 3R chain where l1=18.7 and l2=14

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Fig. 6

The specified functions for the hip, knee, and ankle joints: (a) hip joint function ΔyA=fA(t), (b) knee joint function ΔyB=fB(t), and (c) ankle joint function ΔyC=fC(t). These functions are plotted alongside the functions generated by the mechanisms shown in Fig. 7.

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Fig. 7

The hip, knee, and ankle Stephenson II six-bar function generators that generate the desired functions: (a) hip joint ΔyA=fA(t) function generator that reaches 10 accuracy points, (b) knee joint ΔyB=fB(t) function generator that reaches 11 accuracy points, and (c) ankle joint ΔyC=fC(t) function generator that reaches 11 accuracy points

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Fig. 8

Solid model of the hip, knee, and ankle function generators integrated into a humanoid walker. The schematics in Fig. 7 overlay their physical embodiments for each function generator.




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