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Research Papers

Numerical Synthesis of Six-Bar Linkages for Mechanical Computation

[+] Author and Article Information
Mark M. Plecnik

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

J. Michael McCarthy

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

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received July 31, 2013; final manuscript received April 11, 2014; published online June 17, 2014. Assoc. Editor: Jorge Angeles.

J. Mechanisms Robotics 6(3), 031012 (Jun 17, 2014) (12 pages) Paper No: JMR-13-1145; doi: 10.1115/1.4027443 History: Received July 31, 2013; Revised April 11, 2014

This paper presents a design procedure for six-bar linkages that use eight accuracy points to approximate a specified input–output function. In the kinematic synthesis of linkages, this is known as the synthesis of a function generator to perform mechanical computation. Our formulation uses isotropic coordinates to define the loop equations of the Watt II, Stephenson II, and Stephenson III six-bar linkages. The result is 22 polynomial equations in 22 unknowns that are solved using the polynomial homotopy software Bertini. The bilinear structure of the system yields a polynomial degree of 705,432. Our first run of Bertini generated 92,736 nonsingular solutions, which were used as the basis of a parameter homotopy solution. The algorithm was tested on the design of the Watt II logarithmic function generator patented by Svoboda in 1944. Our algorithm yielded his linkage and 64 others in 129 min of parallel computation on a Mac Pro with 12 × 2.93 GHz processors. Three additional examples are provided as well.

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References

Hartenberg, R. S., and Denavit, J., 1964, Kinematic Synthesis of Linkages, McGraw-Hill, New York.
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]
Plecnik, M., and McCarthy, J. M., 2011, “Five Position Synthesis of a Slider-Crank Function Generator,” ASME Paper No. DETC2011-47581. [CrossRef]
Kim, B. S., and Yoo, H. H., 2012, “Unified Synthesis of a Planar Four-Bar Mechanism for Function Generation Using a Spring-Connected Arbitrarily Sized Block Model,” Mech. Mach. Theory, 49, pp. 141–156. [CrossRef]
Svoboda, A., 1948, Computing Mechanisms and Linkages, McGraw-Hill, New York.
Svoboda, A., 1944, “Mechanism for Use in Computing Apparatus,” U. S. Patent No. 2,340,350.
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]
Freudenstein, F., 1954, “An Analytical Approach to the Design of Four-Link Mechanisms,” Trans. ASME, 76(3), pp. 483–492.
McLarnan, C. W., 1963, “Synthesis of Six-Link Plane Mechanisms by Numerical Analysis,” ASME J. Manuf. Sci. Eng., 85(1), pp. 5–10. [CrossRef]
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]
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]
Wampler, C. W., Sommese, A. J., and Morgan, A. P., 1992, “Complete Solution of the Nine-Point Path Synthesis Problem for Four-Bar Linkages,” ASME J. Mech. Des., 114(1), pp. 153–159. [CrossRef]
Wampler, C. W., 1996, “Isotropic Coordinates, Circularity, and Bezout Numbers: Planar Kinematics From a New Perspective,” ASME Design Engineering Technical Conferences, Irvine, CA, August 18–22, Paper No. DETC/MECH-1210.
Sommese, A. J., and Wampler, C. W., 2005, The Numerical Solution of Systems of Polynomials Arising in Engineering and Science, World Scientific, Singapore.
Erdman, A. G., Sandor, G. N., and Kota, S., 2001, Mechanism Design: Analysis and Synthesis, Prentice Hall, Upper Saddle River, NJ.
Morgan, A. P., and Sommese, A. J., 1989, “Coefficient-Parameter Polynomial Continuation,” Appl. Math. Comput., 29(2), pp. 123–160. [CrossRef]
Morgan, A. P., and Sommese, A. J., 1987, “A Homotopy for Solving General Polynomial Systems That Respects m-Homogeneous Structures,” Appl. Math. Comput., 24(2), pp. 101–113. [CrossRef]
Su, H., McCarthy, J. M., Sosonkina, M., and Watson, L. T., 2006, “Algorithm 857: POLSYS_GLP—A Parallel General Linear Product Homotopy Code for Solving Polynomial Systems of Equations,” ACM Trans. Math. Software, 32(4), pp. 561–579. [CrossRef]
McCarthy, J. M., and Soh, G. S., 2010, Geometric Design of Linkages, 2nd ed., Springer-Verlag, New York.
Verschelde, J., 1999, “Algorithm 795: PHCpack: A General-Purpose Solver for Polynomial Systems by Homotopy Continuation,” ACM Trans. Math. Software, 25(2), pp. 251–276. [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]
Larochelle, P. R., 2000, “Circuit and Branch Rectification of the Spatial 4C Mechanism,” ASME Design Engineering Technical Conferences, Baltimore, MD, September 10–13, ASME Paper No. DETC2000/MECH-14053.
Myszka, D. H., Murray, A. P., and Wampler, C. W., 2012, “Mechanism Branches, Turning Curves, and Critical Points,” ASME Paper No. DETC2012-70277. [CrossRef]

Figures

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Fig. 1

The three types of Watt and Stephenson six-bar linkages that are useful for mechanical computation at fixed pivots. The angle φ at the fixed pivot A is the input value and the angle ψ at the fixed pivot B is the output value of the function.

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Fig. 2

Example of branch sorting and the presence of singularities

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Fig. 3

Criterion used for determining whether a trajectory contains an accuracy point

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Fig. 4

Comparison of (a) Svoboda's logarithm linkage (U.S. Patent 2,340,350, Feb. 1, 1944) and (b) the computed Watt II six-bar linkage

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Fig. 5

Three more Watt II six-bar linkages that fit the eight accuracy points of Svoboda's logarithmic function. Each linkage is displayed in the sixth accuracy position.

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Fig. 6

Design options for the parabolic function for each topology

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Fig. 7

Design option for the range ballistic function

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Fig. 8

The line of the gunsight and the parabolic path of the projectile intersect at a horizontal distance of 15 km

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Fig. 9

Design options for the elevation ballistic function

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