Research Papers

Kinematic Geometry of Circular Surfaces With a Fixed Radius Based on Euclidean Invariants

[+] Author and Article Information
Lei Cui

King’s College London, University of London, London WC2R 2LS, UKlei.cui@kcl.ac.uk

Delun Wang

 Dalian University of Technology, 2 Linggong Road, Dalian 116024, P.R.C.dlunwang@dlut.edu.cn

Jian S. Dai

King’s College London, University of London, London WC2R 2LS, UKjian.dai@kcl.ac.uk

J. Mech. Des 131(10), 101009 (Sep 16, 2009) (8 pages) doi:10.1115/1.3212679 History: Received October 20, 2007; Revised August 03, 2009; Accepted August 04, 2009; Published September 16, 2009

A circular surface with a fixed radius can be swept out by moving a circle with its center following a curve, which acts as the spine curve. Based on a system of Euclidean invariants, the paper identifies those circular surfaces taking lines of curvature as generating circles and further explores the properties of the principal curvatures and Gaussian curvature of the tangent circular surfaces. The paper then applies the study to mechanism analysis by proving the necessary and sufficient condition for a circular surface to be generated by a serially connected CR, HR, or RR mechanism, where C joint can be visualized as a special H joint with a variable pitch of one degree of freedom. Following the analysis, this paper reveals for the first time the relationship between the invariants of a circular surface and the commonly used D-H parameters of CR, HR, and RR mechanisms.

Copyright © 2009 by American Society of Mechanical Engineers
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Figure 1

A constraint circular surface generated by a series-connected RR mechanism

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Figure 2

A cross section of a circular surface

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Figure 3

The geometric interpretation of the three scalars

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Figure 4

The characteristic of the envelope L passing through point α meeting the circle at singular points P1 and P2

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Figure 5

A canonical surface with a helical spine curve

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Figure 6

A portion of sphere with a radius being larger than r and generated by circles of fixed radius of r

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Figure 7

A tangent circular surface with a helical spine curve

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Figure 8

Vector e1 lies in the normal planes of spine curve

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Figure 9

A serially connected C′R mechanism

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Figure 10

Vector e1 in the coordinate system

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Figure 11

A serially connected HR mechanism

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Figure 12

An HR circular surface




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