Research Papers

A Novel Approach to Algebraic Fitting of a Pencil of Quadrics for Planar 4R Motion Synthesis

[+] Author and Article Information
Q. J. Ge

e-mail: Qiaode.Ge@stonybrook.edu

Anurag Purwar

e-mail: Anurag.Purwar@stonybrook.edu

Xiangyun Li

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

1Corresponding author.

Contributed by the Computers and Information Division of ASME for publication in the JOURNAL OF Computing and Information Science in Engineering. Manuscript received March 21, 2012; final manuscript received July 28, 2012; published online September 18, 2012. Assoc. Editor: Bahram Ravani.

J. Comput. Inf. Sci. Eng 12(4), 041003 (Sep 18, 2012) (7 pages) doi:10.1115/1.4007447 History: Received March 21, 2012; Revised July 28, 2012

The use of the image space of planar displacements for planar motion approximation is a well studied subject. While the constraint manifolds associated with planar four-bar linkages are algebraic, geometric (or normal) distances have been used as default metric for nonlinear least squares fitting of these algebraic manifolds. This paper presents a new formulation for the manifold fitting problem using algebraic distance and shows that the problem can be solved by fitting a pencil of quadrics with linear coefficients to a set of image points of a given set of displacements. This linear formulation leads to a simple and fast algorithm for kinematic synthesis in the image space.

Copyright © 2012 by ASME
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Grahic Jump Location
Fig. 1

A planar displacement of a moving frame M with respect to the fixed frame F

Grahic Jump Location
Fig. 2

A planar 4R linkage with a fixed frame F and a moving frame M. The ground pivots are located at (–2.2, 0.1) and (1.15, 0.38) relative to F; the moving pivots are located at (1.24, 0.1) and (4.59, 1.34) relative to M; and the length of the input link and output links are r1 = 1.2377 and r2 = 4.6712, respectively.

Grahic Jump Location
Fig. 3

Two “eigen-quadrics” defined by two singular vectors in Table 4: v1 defines an ellipsoid while v2 defines a hyperboloid of two sheets. The highlighted image curve shows the intersection of two quadrics.

Grahic Jump Location
Fig. 4

Constraint manifolds of the two dyads identified from a pencil of quadrics. In the hyperplane Z4 = 1, the two constraint manifolds are given by (X + 22)2 + (Y − 01)2 = 1.53 and (X + 115)2 + (Y - 038)2 = 21.82.

Grahic Jump Location
Fig. 5

In-between quadrics obtained from vector p for three values of θ = 80.21 deg, 70.73 deg, 60.00 deg. Also, shown is an “eigen-quadric” (hyperboloid of two sheets) defined by v2 (θ = 90 deg).

Grahic Jump Location
Fig. 6

In-between quadrics obtained from vector p for three values of θ = −48.60 deg, −42.84 deg, −36.87 deg. Also, shown is an “eigen-quadric” (ellipsoid) defined by v1 (θ = 0).

Grahic Jump Location
Fig. 7

“Eigen-quadrics” (ellipsoid and hyperboloid of two sheets) and the constraint manifolds (hyperboloid of one sheet) of a planar 4R mechanism for approximate motion synthesis



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