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DISCUSSION

Closure to “Discussion of ‘Kinematics of the Translational 3-URC Mechanism’ ” (2006, ASME J. Mech. Des., 128, pp. 812–813) OPEN ACCESS

[+] Author and Article Information
Raffaele Di Gregorio

Department of Engineering,  University of Ferrara, Via Saragat, 1, 44100 Ferrara, Italyrdigregorio@ing.unife.it

J. Mech. Des 128(4), 814 (Sep 10, 2005) (1 page) doi:10.1115/1.2205876 History: Received May 15, 2005; Revised September 10, 2005
FIGURES IN THIS ARTICLE

Kong and Gosselin (1), with reference to what this author wrote in (2), disagree on the following two points: (i) the inverse position analysis (IPA) of the translational 3-URC has only one solution, and (ii) the translational 3-URC belongs to the class of translational parallel mechanisms (TPMs) with linear input-output equations presented in (3).

Point (i): Figure 1 of (1) shows the ith leg of type URC of a translational 3-URC. With reference to the notations shown in that figure, the inverse position analysis consists of calculating the values of the angle θ1i(i=1,2,3) compatible with an assigned position of the axis of the cylindrical pair (note that the cylindrical pair axis passes through the platform point Bi0 and is parallel to the unit vector, w1i, of the axis of the revolute pair that is embedded in the base, whereas the unit vectors w2i and w3i of the axes of the two intermediate revolute pairs are parallel to each other and perpendicular to w1i). It can be shown through a simple geometric reasoning that the ith translational URC leg can be assembled in four different configurations (assembly modes) once the position of the cylindrical pair axis is assigned. Such configurations can be divided into two groups each of which is composed of two configurations that are symmetric with respect to the plane, the cylindrical pair axis and point Ai belong to (plane σi of Fig. 1), and correspond to only two values of the angle θ1i (the values θ1i(1) and θ1i(2) shown in Fig. 1). Even though the axis of the cylindrical pair keeps itself parallel to w1i during the platform translation, suitable platform translations that bring the ith leg into any configuration out of the four assembly modes, without dismounting and reassembling the leg, exist. Therefore, both the values of θ1i that correspond to the four assembly modes are a solution of the IPA (i.e., Kong and Gosselin are right), and the formulas 16,17, and 17reported in (2) must be changed as follows:Display Formula

w2i=±sinαiw1i×(Bi0Ai)w1i×(Bi0Ai)+cosαiw1i×[w1i×(Bi0Ai)]w1i×(Bi0Ai)
(16)
Display Formula
cosθ1i=viw2i
(17a)
Display Formula
sinθ1i=(w1i×vi)w2i
(17b)
where the angle αi is shown in Fig. 1 (it can be easily computed through the relationships: cosαi=digi and sinαi=[1(digi)2]12 with gi=(Bi0Ai)[(Bi0Ai)w1i]w1i).

Point (ii): Since the IPA of the translational 3-URC has two solutions per leg (i.e., 8 solutions), the translational 3-URC proposed in (2) does not belong to the class of TPMs presented in (3), and Ref. 2 must be correct on this point as Kong and Gosselin observed.

Acknowledgements

This author wishes to thank X. Kong and C. M. Gosselin for having given him the opportunity to discuss and correct his work.

Copyright © 2006 by American Society of Mechanical Engineers
Topics: Kinematics
This article is only available in the PDF format.

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Figures

Grahic Jump Location
Figure 1

Assembly modes of the ith translational URC leg projected onto a plane perpendicular to w1i; each of the projection ((1) or (2)) corresponds to two leg configurations with the same value of θ1i and different values of si

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