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Research Papers

A Unified Approach to Exact and Approximate Motion Synthesis of Spherical Four-Bar Linkages Via Kinematic Mapping

[+] Author and Article Information
Xiangyun Li

School of Mechanical Engineering,
Southwest Jiaotong University,
Chengdu 610031, China

Ping Zhao

School of Mechanical and Automotive Engineering,
Hefei University of Technology,
Hefei 230009, China
e-mail: ping.zhao@hfut.edu.cn

Anurag Purwar, Q. J. Ge

Department of Mechanical Engineering,
Stony Brook University,
Stony Brook, NY 11794-2300

1Corresponding author.

Manuscript received April 28, 2017; final manuscript received October 17, 2017; published online December 20, 2017. Assoc. Editor: Venkat Krovi.

J. Mechanisms Robotics 10(1), 011003 (Dec 20, 2017) (10 pages) Paper No: JMR-17-1127; doi: 10.1115/1.4038305 History: Received April 28, 2017; Revised October 17, 2017

This paper studies the problem of spherical four-bar motion synthesis from the viewpoint of acquiring circular geometric constraints from a set of prescribed spherical poses. The proposed approach extends our planar four-bar linkage synthesis work to spherical case. Using the image space representation of spherical poses, a quadratic equation with ten linear homogeneous coefficients, which corresponds to a constraint manifold in the image space, can be obtained to represent a spherical RR dyad. Therefore, our approach to synthesizing a spherical four-bar linkage decomposes into two steps. First, find a pencil of general manifolds that best fit the task image points in the least-squares sense, which can be solved using singular value decomposition (SVD), and the singular vectors associated with the smallest singular values are used to form the null-space solution of the pencil of general manifolds; second, additional constraint equations on the resulting solution space are imposed to identify the general manifolds that are qualified to become the constraint manifolds, which can represent the spherical circular constraints and thus their corresponding spherical dyads. After the inverse computation that converts the coefficients of the constraint manifolds to the design parameters of spherical RR dyad, spherical four-bar linkages that best navigate through the set of task poses can be constructed by the obtained dyads. The result is a fast and efficient algorithm that extracts the geometric constraints associated with a spherical motion task, and leads naturally to a unified treatment for both exact and approximate spherical motion synthesis.

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Figures

Grahic Jump Location
Fig. 1

The spherical RR dyad with A as the fixed axis, B as the moving axis, and α is the angular dimension of the chain

Grahic Jump Location
Fig. 4

Example 1: 12 given poses sampled from a four-bar motion

Grahic Jump Location
Fig. 2

The projected C-manifold of a spherical circular constraint with its dimensions given as follows: its center lies on the axis A = (0.4243, 0.5657, −0.7071) and its radius that represented by the sphere central angle is α = 64.9 deg

Grahic Jump Location
Fig. 3

Coupling of two C-manifolds in the image space, where each intersection curve represents one circuit/assembly mode of the resulting spherical four-bar linkage, and the task positions are converted to a set of image points

Grahic Jump Location
Fig. 5

Example 1: Two resulting C-manifolds are shown with the 12 black image points which lie on the intersection curve in the figure and represent 12 given positions in Table 1

Grahic Jump Location
Fig. 7

Example 2: The resulting spherical four-bar mechanism constructed by p1 and p4

Grahic Jump Location
Fig. 8

Example 3: Infinity fan driven by a spherical four-bar mechanism

Grahic Jump Location
Fig. 6

Example 2: The first figure shows p1 and p4, which are close to the two resulting C-manifolds in the first example. The second one shows p2 and p3. Similarly, the 12 black image points lying on the intersection curve in the figure represent 12 approximate poses in Table 5.

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