Research Papers

A Task-Driven Approach to Unified Synthesis of Planar Four-Bar Linkages Using Algebraic Fitting of a Pencil of G-Manifolds

[+] Author and Article Information
Q. J. Ge, Ping Zhao, Shrinath Deshpande

Computational Design Kinematics Laboratory,
Department of Mechanical Engineering,
Stony Brook University,
Stony Brook, NY 11794-2300

Anurag Purwar

Computational Design Kinematics Laboratory,
Department of Mechanical Engineering,
Stony Brook University,
Stony Brook, NY 11794-2300
e-mail: anurag.purwar@stonybrook.edu

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscript received December 17, 2012; final manuscript received December 12, 2016; published online May 16, 2017. Editor: Bahram Ravani.

J. Comput. Inf. Sci. Eng 17(3), 031011 (May 16, 2017) (11 pages) Paper No: JCISE-12-1235; doi: 10.1115/1.4035528 History: Received December 17, 2012; Revised December 12, 2016

This paper studies the problem of planar four-bar motion generation from the viewpoint of extraction of geometric constraints from a given set of planar displacements. Using the image space of planar displacements, we obtain a class of quadrics, called generalized- or G-manifolds, with eight linear and homogeneous coefficients as a unified representation for constraint manifolds of all four types of planar dyads, RR, PR, and PR, and PP. Given a set of image points that represent planar displacements, the problem of synthesizing a planar four-bar linkage is reduced to finding a pencil of G-manifolds that best fit the image points in the least squares sense. This least squares problem is solved using singular value decomposition (SVD). The linear coefficients associated with the smallest singular values are used to define a pencil of quadrics. Additional constraints on the linear coefficients are then imposed to obtain a planar four-bar linkage that best guides the coupler through the given displacements. The result is an efficient and linear algorithm that naturally extracts the geometric constraints of a motion and leads directly to the type and dimensions of a mechanism for motion generation.

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Fig. 2

Geometric constraints of some planar four-bar linkages: (a) RRRR, (b) RRPR, (c) RRRP, and (d) RRPP

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Fig. 3

A right circular hyperboloid of one sheet defined by Z12+(Z2−2Z3)2−4Z32=5

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Fig. 4

A hyperbolic paraboloid defined by Z1Z2−Z3=0

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Fig. 5

Example 6.1: five positions of an aircraft landing gear labeled 1…5 are shown. The moving frame is attached to the top left corner of the housing, while frame XY is the fixed frame.

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Fig. 6

Example 6.1: two resulting constraint manifolds identified from a pencil of G-manifolds that satisfy Eq. (21) are illustrated in this figure by projecting them on hyperplane Z4=1. Intersection of hyperboloid and hyperbolic paraboloid forms constraint manifold of the slider crank mechanism. Five black image points on the intersection curve show projection of five task positions on hyperplane Z4=1.

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Fig. 7

Example 6.2: 11 task positions of ASME Mechanism Design Challenge and its solution as synthesized four-bar mechanism

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Fig. 8

Example 6.2: projection of C-manifolds tabulated in Table 7. Their intersection forms constraint manifold of the four-bar linkage. Eleven black image points on the intersection curve represent projection of task positions on hyperplane Z4=1.

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Fig. 9

Example 6.3: the synthesis starts by finding a 3R triad whose workspace contains the given task positions. The figure shows triad at the first task position.

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Fig. 10

Example 6.3: the figure shows a five-bar linkage, where links 4 and 5 are synthesized relative to the link 2. Figure shows locations of fixed and moving pivots of chosen dyad obtained as a result of relative synthesis.

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Fig. 11

Example 6.3: the synthesized six-bar linkage for the generation of sit-to-stand motion at the first task position. Image at top right corner shows curve traced by shoulder joint of the human skeleton performing sit to stand motion.

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Fig. 12

Example 6.3: the sit-and-stand six-bar linkage at task positions 2, 3 (top) and 4, 5 (bottom)



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